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! ! ! ! This!accepted!author!manuscript!is!copyrighted.!Changes!resulting!from!the! publishing!process!–!such!as!editing,!corrections,!structural!formatting,!and! other!quality!control!mechanisms!–!are!not!reflected!in!this!version!of!the! text.!For!any!quotation,!please!refer!to!the!definitive!version!published!in:! «Rosenzweig+Jahrbuch+/+Rosenzweig+Yearbook»,+5+(2010),+pp.+141A159! Luca Bertolino Die Rolle des Chores in Franz Rosenzweigs Stern der Erlösung Im Aufsatz von 1925 „Das neue Denken" – nicht zufällig mit dem Untertitel „Einige nachträgliche Bemerkungen zum ‚Stern der Erlösung'" – gibt Franz Rosenzweig dem Leser seines vier Jahre zuvor publizierten Hauptwerks verschiedene Leitlinien an die Hand. So lädt er den hilflosen general reader philosophischer Bücher dazu ein, sich nicht bei jedem Satz oder Absatz aufzuhalten, gleich jener „ancien régime-Strategie, die keine Festung unerobert im Rücken lassen zu dürfen meint", sondern sich des Stern der Erlösung gleichsam „napoleonisch" zu bemächtigen: „in kühnem Vorstoss auf die feindliche Hauptmacht, nach deren Besiegung die kleinen Grenzfestungen schon von selber fallen werden." (GS III, 141f.) Solchen hermeneutischen Hinweisen folgend, sehe ich hier den napoleonischen Sieg über den Stern der Erlösung als bereits erfolgt an und kehre auf die kleine Festung des Chores im Buch zurück. Denn von dieser Warte aus erschliesst sich Rosenzweigs Ästhetik und der Blick fällt – allgemeiner betrachtet – auf jenes Gebiet, von dem sein ganzes Werk wie eine Landkarte ausgebreitet werden kann. Im Stern der Erlösung wird nicht oft vom „Chor" gesprochen. Das Wort selbst erscheint elfmal, davon abgeleitete Formen insgesamt achtmal (einmal „Chorform", sechsmal „Choral", einmal „choralmässig"). Diese Okkurrenzen lassen sich m. E. drei weiter gefassten Begriffen zuordnen: dem Chor in der (antiken) Tragödie, dem Chor der Erlösung, dem Chor in der Kirchenmusik. Mit Bezug auf Tragödie und Kirchenmusik ist leicht zu verstehen, warum Rosenzweigs Ästhetik von der Chor-Festung aus in den Blick kommt, auch wenn dieser unweigerlich eingeschränkt bleibt. Rosenzweigs Theorie über die Kunst und das Schöne spielt in allen drei Büchern des ersten und zweiten und im zweiten Buch des dritten Teiles seines Systems der Philosophie eine Rolle: „Während der erste Band [Teil] nur die geläufigen ästhetischen Grundbegriffe abhandelt und der zweite zwar sowohl in der ganzen Einordnung wie in der letzten Zuspitzung die Ästhetik aus der hier besonders fest, weil unbewusst gebliebenen Bindung in die idealistische Tradition befreite, aber zwischenhinein doch den üblichen Inhalt einer Ästhetik entwickelte, lässt der dritte Band [Teil] sie gipfeln in einer angewandten Ästhetik und verbrennt in dieser Rechtfertigung der Kunst durch das Kunstgewerbe alle Schiffe, die aus diesem Neuland in das klassische Ursprungsland der Wissenschaft vom ,zweckfreien Wohlgefallen' zurücktragen könnten." (GS III, 157; vgl. 140f. [„Das neue Denken"])1 Hier beschränke ich mich darauf, zum einen Rosenzweigs Gegnerschaft zur idealistischen Ästhetik zu betonen und zum anderen hervorzuheben, dass er sich der Tatsache bewusst war, eine in gewisser Weise „klassische" Kunsttheorie zu entwickeln. Doch vermeidet er eine idealistisch „reine" Auffassung der Kunst, da es deren Bestimmung sei, „angewandt" zu werden.2 1 Mit dem Begriff vom „zweckfreien Wohlgefallen" bezieht sich Rosenzweig vermutlich auf I. Kant, Kritik der Urteilskraft (1790), Akademie-Ausgabe, Bd. 5, 204f., 209f., 354. 2 Ein umfassender Überblick über die Ästhetik Rosenzweigs findet sich u. a. in folgenden Beiträgen: S. Mosès, „L'esthétique de Franz Rosenzweig", in: O. Mongin, J. Rolland, A. Derczanski (Hg.), Franz Rosenzweig (Les Cahiers de La nuit surveillée, Nr. 1), Paris 1982, 119-135; A. Mayer, „Die Bedeutung der Kunst in Franz Rosenzweigs Werk", in: W. Licharz, M. Keller (Hg.), Franz Rosenzweig und Hans Ehrenberg. Bericht einer Beziehung, Frankfurt a. M. 1986, 35-54; A. Mayer, „Rosenzweigs Stellung zur Kunst", und G. Baffo, „Die ästhetische Dimension im Denken Rosenzweigs", in: W. Schmied-Kowarzik (Hg.), Der Philosoph Franz Rosenzweig (1886-1929). Internationaler Kongress Kassel 1986, Freiburg/München 1988, Bd. 2, 951-964 bzw. 965-978; F. P. Ciglia, „Pigmalione in ginocchio", in: Ders., Fra Atene e Gerusalemme. Il «nuovo pensiero» di 3 Rosenzweigs Verwendung des Bildes von gan 'eden, d. h. vom bereits verlorenen Gottesgartens (1. Mose 2, 8.15)3, ist emblematisch dafür, dass er dem Idealismus vorwirft, die Kunst vergöttert zu haben,4 um Abhilfe für die Armut seiner eigenen Logik zu schaffen und um den Kontakt zum Reichtum des lebendigen Daseins nicht zu verlieren. Denn nach Rosenzweig ist es gerade die idealistische Philosophie, die den „Sündenfall" (das ist bezeichnenderweise das einzige Mal, wo dieses Wort im Stern der Erlösung erscheint) beging, als „sie der eigenen Weisheit mehr vertraute als der sichtbar sie umfangenden Schöpfermacht Gottes [...]. An Stelle des geschaffenen Gottesgartens der Sprache, in dem sie ohne das Misstrauen und die Hintergedanken der Logik gelebt hatte, und den sie durch eigene Schuld verlassen musste, suchte sie nach einem Menschengarten, einem Menschenparadies. Es musste ein Garten sein, den der Mensch selbst gepflanzt hätte und der doch nicht sein bewusstes Werk wäre; denn wäre er das, so hätte er keinen Ersatz für den verlorenen Garten bieten können, den Gott selber gepflanzt hatte. Wie jener verlorene Garten musste es einer sein, der den Menschen umgab, er selber wusste nicht woher; er musste ihn wohl gepflanzt haben, aber er durfte es selber nicht wissen; er musste sein Werk sein, aber sein bewusstloses, alle Zeichen zweckvoller Arbeit tragen und doch zwecklos entstanden sein, gewirktes Werk und doch pflanzenhaft gewachsen. So kam es, dass der Idealismus in dem Augenblick, wo er die Sprache verwarf, die Kunst vergötterte." (GS II, 162f.) Seinem „Grunddogma" (GS II, 165) der Identität von Sein und Denken getreu stellt der Idealismus die sozusagen totalitäre Forderung, das All zu erzeugen. Hierbei soll die mathematische Logik mit ihren Gründen, ihrer Rechenschaft und ihrer Errechenbarkeit als „Organon des Denkens" (GS II, 23) im Menschenparadies die Sprache des Gottesgartens ersetzen. Doch besteht sie aus stummen Zeichen und ist daher unfähig, dem Leben vollen Ausdruck zu verleihen. Deshalb vergöttert der Idealismus die Kunst, indem er in ihren Werken, die zwar sein eigenes Erzeugnis sind, ihm aber als Stück Natur erscheinen, einen Schutz vor der „Methode des ‚panlogistisch' reinen Erzeugers" (GS II, 163) entdeckt. Anstatt den gan 'eden der Sprache zu bebauen und bewahren, als ursprüngliche Natur, die zuvörderst von Gott gepflanzt und dann dem Menschen übertragen wurde, wird der Mensch vom Idealismus dazu verurteilt, im hortus conclusus der Kunst zu leben. So tritt Kultur an die Stelle der Natur, wo das Kunstwerk als bewusstloses Erzeugnis des Geistes gilt und in Unbewusstheit seines Werdens als Offenbarung der Wirklichkeit bzw. als sichtbare Erscheinung eines Absoluten angenommen wird. Einerseits beklagt Rosenzweig also den Exilzustand des Menschen und verweist auf seine Absurdität, andererseits scheint er ihn zu rehabilitieren, indem er von ihm als von einer „immerwährende[n] Vorwelt" (GS II, 1) ausgeht. Insbesondere zeigt er, dass gerade in jenem heidnischen Menschengarten die ästhetischen Grundbegriffe (äussere Form, innere Form und Gehalt: Vgl. GS II, 41, 65f., 87f.) aufzufinden sind: Auf diesen baut er in der „allzeiterneuerte[n] Welt" (GS II, 101) der „Reihe Schöpfung-Offenbarung-Erlösung" (GS II, 210) in einer ganz systematischen, stammbaumartigen Art und Weise (vgl. GS II, 167) seine „Theorie der Kunst" (GS II, 161, 209, 270) auf. Es handelt sich demnach um eine „klassische" Ästhetik, um eine Ästhetik, die das Phänomen der Kunst in seinen wesentlichen Elementen untersucht: Urheber (sei er Genie, Dichter, Künstler oder Mensch: Vgl. GS II, Franz Rosenzweig, Genova/Milano 2009, 106-144. 3 „ER, Gott, pflanzte einen Garten in Eden, morgenseits, / und setzte darein den Menschen, den er gebildet hatte." „ER, Gott, nahm den Menschen und setzte ihn in den Garten von Eden, / dass er ihn baue und hüte" – so nach der von Martin Buber gemeinsam mit Rosenzweig unternommenen Verdeutschung: Das Buch Im Anfang (Die Schrift. Die fünf Bücher der Weisung. Erstes Buch), Berlin o. J. [1926], 12f. 4 Vgl. hierzu F. P. Ciglia, „Arte, profezia della rivelazione", in: Ders., Scrutando la «Stella». Cinque studi su Rosenzweig, Padova 1999, 97-122, bes. 101-107, und M. Bienenstock, „Rosenzweig und die Vergötterung der Kunst", in: ND I, 418-430. 4 165f., 215, 272), Werk (sei es episch, lyrisch oder dramatisch: Vgl. GS II, 216f., 272f.), Publikum bzw. Betrachter (vgl. GS II, 270-272). Ausserdem zeigt Rosenzweig eine regelrechte Systematik der unterschiedlichen Künste auf: Bildkunst, Tonkunst, Dichtung (vgl. GS II, 217-221, 273-276). Rosenzweigs Ästhetik soll jedoch, wie übrigens auch seine Logik und Ethik, eine vollkommene Erneuerung des Denkens herbeiführen (vgl. GS III, 140 [„Das neue Denken"]). Die Kunstlehre, die im zweiten Teil des Stern der Erlösung lediglich als „Episode" – sofern „notwendig" – erscheint (vgl. GS II, 213, 221), erweist sich letztlich als wesentlich mehr denn je episodisch und „klassisch". Die Kunst als tiefreichendes Bindeglied zum wirklichen alltäglichen Leben kann nämlich den Menschen in den verlorenen Gottesgarten zurückführen. Von hier stammt Rosenzweigs Soziologie der Kunst (vgl. GS II, 393-397, 399-403, 412-415), die deren Bedürfnis, religiös werden zu müssen, aufzeigen will. Denn in der liturgischen Anwendung reinigt sich die Kunst von ihrer idealistischen Reinheit und stellt schliesslich die Nähe des Menschen zur „ewige[n] Überwelt" (GS II, 293) her. 1. Der Chor in der Tragödie Der Chor tritt auf der Bühne des Stern der Erlösung erstmals mit Bezug auf die Rolle auf, die er in der Tragödie einnimmt: Er spielt eine positive Rolle in der antiken, überhaupt keine Rolle dagegen in der modernen Tragödie. Betrachtet man Rosenzweigs Systematik innerhalb der unterschiedlichen Kunstgattungen, so hat man es hier mit der Dichtung zu tun, die sich im Verhältnis zur Bildkunst und zur Tonkunst als „die eigentlich lebendige Kunst" und folglich auch als „die unentbehrlichste Kunst" (GS II, 273) durchsetzt. Die Dichtung ist wesentlich mit dem vorstellenden Denken verbunden, in dem nach Rosenzweig auch der Ursprung für Raum und Zeit liegt, in denen die bildende Kunst resp. die Musik entstehen. Das Dramatische des poetischen Kunstwerks schlechthin rührt daher, dass es eine breit/räumlich epische und zugleich eine unmittelbar/zeitlich lyrische Dimension enthält (vgl. GS II, 272f.). Das poetische Kunstwerk und somit auch die Tragödie bilden sich durch die Idee, die als Form verstanden Körper/Gestalt und Stimme/Melos verleiht (vgl. GS II, 275f.). Rosenzweig lenkt seine Aufmerksamkeit auf jene Ideen, die die moderne Tragödie von der antiken unterscheiden. Vor allem zeigt er, weshalb „man sie mit Recht als Charaktertragödie jener als der Handlungstragödie entgegengestellt hat" (GS II, 234). Während sich in der antiken Tragödie die Handlungen unterscheiden, die alle darauf zielen, das stets mit sich identische Selbst des tragischen Helden zu akzentuieren, begegnen uns in der modernen Tragödie verschiedenartige Charaktere, die je ganz persönliche, individuelle Weltanschauungen verkörpern. Auf diesen grundsätzlichen Unterschied zwischen den beiden Tragödien-Formen lassen sich auch die anderen von Rosenzweig genannten Unterscheidungen zurückführen. Protagonist der antiken Tragödie ist der Held, der sich stets der eigenen, in sich abgeschlossenen Identität bewusst ist; Protagonist der modernen Tragödie ist der Philosoph, d. h. ein Mensch, dessen Bewusstsein nach Klarheit verlangt, vor allem dann, wenn er mit sich allein ist – absoluter Mensch angesichts des Absoluten, so wie in den „Höhepunkte[n] der modernen Tragödie überhaupt: Hamlet, Wallenstein, Faust" (GS II, 235). Der Held der antiken Tragödie spricht nicht, er schweigt (man denke z. B. an die Tragödien Aischylos') und „erhebt sich aus den Gefilden der Persönlichkeit [...] in die eisige Einsamkeit des Selbst" (GS II, 84); dagegen geht die individuelle Persönlichkeit der Protagonisten der modernen Tragödie aus dem dramatischen Zwiegespräch bzw. – genauer gesagt – aus der Debatte mit den anderen Gestalten hervor. Der tragische Held der Antike drückt sich – wenn überhaupt – in lyrischen Monologen aus (wie z. B. bei Sophokles und Euripides) und tritt dabei in seiner ganzen monolithischen Bewusstheit hervor; der Monolog der modernen Tragödie ist dagegen als eine Art Denkpause der Protagonisten gestaltet, als „Selbstbetrachtung, Einordnung der eigenen Existenz in die Welt, Entschlussklärung, Niederschlagung von Zweifeln" (GS II, 5 234). Da der Held in der antiken Tragödie vollkommen auf sich selbst bezogen ist, fehlen in ihr auch Überredungsund Liebesszenen, in denen die Personen der modernen Tragödie einander gegenüberstehen (vgl. GS II, 85). Auch im Chor, der in der modernen Tragödie nicht erscheint, während er in der antiken eine für die dramatische Handlung ausschlaggebende Funktion hat, sieht Rosenzweig einen wichtigen Unterschied voller Implikationen zwischen den beiden Tragödien-Modellen. In der antiken Tragödie hat der Chor die Aufgabe, Blick und Aufmerksamkeit des Betrachters auf den tragischen Helden zu lenken, indem er diesen der Debatte entzieht, um ihm alsdann das Wort zum Monolog zu erteilen, d. h. zu „einer jener lyrischen Monologe, zu denen das Dasein des Chors immer wieder den Anlass gibt [...]. Die ungeheure Wichtigkeit dieser lyrisch-musikalischen Partien in der Ökonomie des dramatischen Ganzen beruht eben darauf, dass die Attiker im eigentlich Dramatischen, im Dialog, nicht die Form fanden, das Heroisch-Tragische zum Ausdruck zu bringen." (GS II, 84) „Nichts andres ist ja der Chor in der antiken Tragödie als dies Heranwallen der Aussenwelt an den Helden, dies Angesprochenwerden der marmorstummen Gestalt. Auf der Bühne dargestellt musste das werden; es genügte nicht, es dem Empfinden des Zuschauers zu überlassen; denn an sich wäre es allerdings sehr natürlich, wenn dem stummen Helden gegenüber der Beschauer sich verstummen, dem blinden gegenüber auch er sich erblinden fühlte. Aber das soll eben nicht sein; der Held soll sichtbare Gestalt sein, in der Welt stehen, auch wenn er es selbst weder weiss noch wahrhaben will; und eben dies Gefühl, dass es so ist, zwingt dem Beschauer der Chor auf, der den Helden ansieht, anhört, anredet." (GS II, 230f.) In der modernen Tragödie dagegen „ist er [der Mensch] nun unmittelbar sichtbar und hörbar geworden. Er kann ja jetzt sein Gesehenund Gehörtwerden erzwingen; er ist nicht mehr starres Marmorbild wie der tragische Held des Altertums; nein, er spricht. Deswegen fällt in der neueren Tragödie der Chor als überflüssig weg. Es braucht dem Zuschauer nicht mehr vor Augen geführt zu werden, dass der Held trotz seiner Blindheit sichtbar, trotz seiner Taubstummheit ansprechbar ist; es braucht ihm nicht vor Augen geführt zu werden, er sieht es selbst." (GS II, 233) Welche Implikationen liegen nun in der Präsenz des Chores in der antiken Tragödie, und warum fehlt er in der modernen? Im Vordergrund stehen ganz offensichtlich die Verschiedenartigkeit der Protagonisten in ihrem Verhältnis zur Welt sowie die Unterschiedlichkeit im Blick des Zuschauers. Im Hintergrund stehen, so will es mir scheinen, unterschiedliche ästhetische Theorien und hermeneutische Perspektiven. Oben wurde bereits erwähnt, dass der Held der Protagonist der antiken Tragödie ist: Er ist schweigsam, allein und ewig. Er ist in der „Sphäre der reinen erhabenen Stummheit, des Selbst" (GS II, 83); er isoliert sich von der Welt und den Göttern, unbesorgt um das Verständnis seines eigenen Schicksals (wie Ödipus in Kolonos und ganz anders als Hiob: Vgl. GS II, 85); er „verlangt nach der Einsamkeit des Untergangs, weil es keine grössere Einsamkeit gibt als diese. Deshalb stirbt der Held eigentlich auch nicht" (GS II, 86). Obwohl der tragische Held sich verschliesst, ist er für die Welt zugänglich: Es ist der Chor, der diesen Zugang gewährt und dem Betrachter erschliesst. Ganz anders der Protagonist der modernen Tragödie: Er befindet sich in ständigem Dialog mit den anderen Gestalten, er ist in die Angelegenheiten der Welt verstrickt und ist schrecklich sterblich. Er ist „ganz in das Hinund Widerströmen der Welt hineingeworfen, ganz lebendig und voll unverhehlter Scheu vor dem offenen Grab. Diesen Helden [...] sieht der Zuschauer zu voller Lebendigkeit erwachen im Dialog" (GS II, 233). Diesen Unterschieden zwischen den Protagonisten entspricht laut Rosenzweig ein Unterschied in der – durch die Tragödie hervorgerufenen – Gemütsverfassung des Publikums. Der Chor der antiken Tragödie ruft im Zuschauer ein Gefühl für sich selbst hervor, das dem 6 des Helden ähnlich ist und folglich bei ihm Furcht und Mitleid auslöst: „Und wenn er mir, in diesem Augenblick, / Wie die Antike starr entgegenkömmt, / Thut er mir leid, und ich muss ihn bedauern!"5 In der modernen Tragödie hingegen kommt es zu einer Art „Einfühlungsprozess" des Betrachters in den Protagonisten: Bleiben in der antiken Tragödie Held und Betrachter voneinander getrennt, zwingt die moderne Tragödie den Menschen im Parkett auf die Bühne, gleichsam mitten unter die verschiedenen Gestalten und ihre Gefühle; „nicht zu Furcht und Mitleid steigert ihn [den Betrachter] das Geschehen auf der Bühne, sondern zu Widerspruch und Mitgerissenheit." (ebd.) Diese unterschiedlichen Reaktionen des Publikums erscheinen um so wichtiger, als es – gemäss Rosenzweigs Theorie der Kunst – das Kunstwerk und dessen Urheber ihrer Vorläufigkeit entreissen. Denn das Kunstwerk wird unheimlich in dem Moment, in dem der Künstler es freigibt, der Künstler ist Autor nur im Augenblick, in dem er das Werk erschafft (vgl. GS II, 270-272). Man könnte also den Schluss ziehen, dass sowohl die antiken Tragödien und Tragödiendichter als auch die modernen Dramen und Dramatiker mit dem Betrachter leben: Die ersteren wecken in ihm Gefühle für den Helden, der gleichwohl in seinem aphasischen und unbeweglichen Selbst weit entfernt auf dem Proszenium verharrt; die letzteren ziehen den Zuschauer auf die Bretter hinauf, ohne aber je in seine reale Welt einzugreifen. Folgt man den Überlegungen Rosenzweigs, so kann abschliessend der unterschiedliche Einfluss der antiken und modernen Tragödien auf das Publikum zwei verschiedenen ästhetischen Theorien zugeschrieben werden: einmal der philosophischen Kunstlehre des Altertums, die „in der lebendigen Schöne das Werk Gottes erschaut, bei Platon, Plotin, Augustin und weniger bewusst noch bei manchen andern", zum anderen der idealistischen Ästhetik, die „von vornherein nicht das lebendige Schöne überhaupt, sondern die ,schöne Kunst' auf den Schild" (GS II, 163)6 erhoben hat. Rosenzweig verbirgt keinesfalls seine Vorliebe für die ästhetische Konzeption der antiken Tragödie: Sie erscheint ihm kongenialer, eben weil – so argumentiert er in den Fussstapfen von Aristoteles – sie im Betrachter „Furcht und Mitleid" (GS II, 88, 233) erwecken kann und sich nicht mit einer einfach „dianoetisch[en]" (GS II, 84) Debatte begnügt.7 So heisst es bedeutungsvoll im Stern der Erlösung: „[S]chon Aristoteles voll ahnenden Tiefsinns formulierte[:] ,Furcht und Mitleid'. Im Beschauer werden sie wach und richten sich sofort in sein eigenes Innere, machen ihn zum Selbst. Würden sie im Helden selber wach, so hörte er auf, stummes Selbst zu sein; ,Phobos' und ,Eleos' würden sich als ,Ehrfurcht und Liebe' enthüllen, die Seele Sprache gewinnen und das neugeschenkte Wort von Seele zu Seele ziehen. Nichts von solchem Zueinanderkommen hier. [...] Jeder bleibt für sich, jeder bleibt Selbst. Es entsteht keine Gemeinschaft. Und dennoch entsteht ein gemeinsamer Gehalt. Die Selbste kommen nicht zueinander, und dennoch klingt in allen der gleiche Ton, das Gefühl des eigenen Selbst." (GS II, 88)8 Die anscheinend uneinholbare Distanz zwischen Held und Betrachter in der antiken Tragödie, das sie trennende Schweigen mag man der anscheinend nahen Auseinandersetzung von Zuschauer und Protagonisten in der modernen Tragödie bevorzugen. Denn laut Rosenzweig findet dieser dianoetische Vergleich im engen Raum einer vom einsamen Leser nur erdachten 5 H. von Kleist, Prinz Friedrich von Homburg. Ein Schauspiel (1821), Brandenburger Ausgabe, Bd. I/8, 73, V. 786-788, von Rosenzweig partiell zitiert („Wie die Antike starr entgegenkömmt": GS II, 233). 6 Als Beispiel für „schöne Kunst"-Theorien vgl. I. Kant, Kritik der Urteilskraft (wie Anm. 1), 304ff. und G. W. F. Hegel, Enzyklopädie der philosophischen Wissenschaften im Grundrisse (1830), §§ 562f., TheorieWerkausgabe, Bd. 10, 370-372. 7 Was die Begriffe „Furcht und Mitleid" (gr. phobos und eleos) und „dianoetisch" (von gr. dianoia) betrifft, vgl. Aristoteles, Poetica, 1449 b 24-28, 1452 b 28 1454 a 15 bzw. 1449 b 36 1450 b 20. 8 Zum Ausdruck „Ehrfurcht und Liebe" siehe unten. 7 Bühne statt, d. h. – so das desideratum der idealistischen Ästhetik – in einer idealen Welt, die von der real existierenden eben nur eine Idee hat: „Das Heim der Poesie [...], in dem sie ihre Gefangenschaft absitzt, ist der Bücherschrank. Der Raum zwischen den zwei Deckeln eines Buchs – das ist die einzige Stätte, wo die Poesie wahrhaft ,reine' Kunst ist; dort ist sie in ihrer reinen Gedankenwelt, jedes Werk in seiner eignen. [...] Gar wenn sie als dramatische Dichtung etwa aufs Theater kommt, ist es um ihre ästhetische ,Idealität' geschehen; das rechte Drama ist das Buchdrama; dass es theatralisch sei, gilt im Munde des Ästheten für ein Verbrechen" (GS II, 412f.). 2. Der Chor der Erlösung Die zweite Stelle, an der der Chor im Stern der Erlösung thematisiert wird, steht im Zusammenhang mit der Erlösung. Sie ist das zeitliche Ereignis, in dem der von Gott geliebte und von ihm zur Liebe befähigte Mensch die eigene Liebestat nun auf seinen Nächsten (im Sinne von demjenigen, der ihm rea', plesios, proximus, zu-nächst das jeweils Nächste ist: Vgl. GS II, 243, 262) bzw. auf die Welt als lebendiges Reich richtet und damit indirekt auch auf Gott. So stehen Mensch und Welt hier in Wechselwirkung zueinander und miteinander: Der Mensch erlöst die Welt und wird von ihr erlöst. Doch nur Gott ist Erlöser im wahrsten Sinne. Indem er die menschliche Liebe in der Offenbarung ermöglicht und die lebendige Welt in der Schöpfung geschaffen hat, wird er zur Grundlage ihrer wechselseitigen Erlösung: Gott ist Erlöser der Welt durch den Menschen, Erlöser des Menschen in der Welt, Erlöser seiner selbst in der Erlösung beider (vgl. GS II, 229-265). In der Erlösung herrscht „die Sprache des Chors" (GS III, 151 [„Das neue Denken"]): „[D]ie Erlösung der Seele an den Dingen, der Dinge durch die Seele geschieht im gleichatmenden Zwiegesang der beiden, im Satz, der aus den Stimmen der beiden Worte zusammenklingt." (GS II, 255) Das ist die „Chorform" (GS II, 258) der Erlösung: Mensch und Welt – die „Mitsprecher des Chors" (GS III, 151 [„Das neue Denken"]) – vereinigen sich in einem Lobund Dankgesang an Gott, weil er die Welt machtvoll erschaffen und sich dem Menschen liebevoll offenbart hat. „Er ist gut", „denn er ist gut" (GS II, 258)9: Dies ist der „strophisch sich steigernde[...] Gesang. Und Urgesang, der stets Gesang von mehreren ist; der Einzelne singt nicht [...]. [U]rsprünglich ist der Gesang vielstimmig gleichen Tons und Atems, und über allem Inhalt des Gesanges steht die Form dieser Gemeinsamkeit. Ja der Inhalt ist selbst weiter gar nichts als die Begründung für diese Form. Man singt nicht gemeinsam um eines bestimmten Inhalts willen, sondern man sucht sich einen gemeinsamen Inhalt, damit man gemeinsam singen kann." (ebd.) Im Lobund Dankgesang der Erlösung wie in den Psalmen als Gesängen der Gemeinde (vgl. GS II, 278f.) erheben sich also die Stimmen der erlösten Seele und der erlösten Welt. Zunächst ermahnen sie einander, um sich dann zum mächtigen Unisono des Wir zu vereinen. Das Wir – es sind „Wir alle", „Wir alle, die wir hier beisammen sind" (GS II, 263) und im Chor singen. Die beiden Nominative „Mensch" und „Welt" mit ihren jeweils unterschiedlichen Akkusativen vereinigen sich im Dativ „Gott", dem ihr Gesang gewidmet ist (vgl. GS II, 259f.). So gehen die einzelnen Stimmen von Mensch und Welt, die Singularitäten des „Ich" und des „Du" auf dem Wege des Duals, mit dem sie sich gegenseitig anspornen, in die Allheit über (wohlgemerkt nicht in die Pluralität): „Im Wir also hebt die Schlussstrophe des Gesangs der Erlösung an; im Kohortativ hatte er mit dem Aufruf der Einzelnen, die aus dem Chor hervortraten, und den Responsen des Chors darauf begonnen; 9 Vgl. Psalmen 106,1, 107,1, 118,1.29 und 136,1. 8 im Dual ging es in einem zweistimmigen Fugato, an dem sich immer neue Instrumente beteiligten, fort; im Wir endlich sammelt sich alles zum choralmässig gleichen Takt des vielstimmigen Schlussgesangs." (GS II, 264) Es leuchtet ein, dass dieses „sprechende", „grammatische[...] Denken" (GS III, 151 [„Das neue Denken"]), diese „Pronominalphilosophie"10 Rosenzweigs, in den gan 'eden gehört, d. i. in den ursprünglich geschaffenen Gottesgarten der Sprache und nicht in seinen künstlichen Ersatz, den Menschengarten der Kunst. Verglichen mit dem Chor der Erlösung ist der Chor der Tragödie unfähig, den Gipfel des Wir zu erklimmen oder auch nur den Weg des Duals einzuschlagen. Er stellt höchstens eine stimmliche Pluralität dar, die in der dritten Person Singular entstanden ist (vgl. GS II, 264), d. h. im tragischen Selbst des antiken Helden. Denn der Chor der antiken Tragödie und der tragische Held bleiben in die eigenen Akkusative eingeschlossene Nominative und wissen nichts von einem Dativ; sie ermahnen sich nicht wechselseitig, sie kennen nichts anderes als die eigene Singularität. Obwohl die Kunst – gemeint ist nun die poetische Kunst der Tragödie – der lebendigen Sprache des verlorenen Gartens so fern ist, betont Rosenzweig doch mehrfach ihre Bedeutung: Sie ist nämlich die „Sprache [...] des Unaussprechlichen, die Sprache solang es noch keine Sprache gibt, Sprache der Vor-welt" (GS II, 164; vgl. 87, 89, 139, 212). Wie die Sprache ist auch die Kunst positiv im Wort begründet; im Unterschied zu jener präsentiert sie sich jedoch negativ nur als Wort, und niemals als wirkliche Sprache der Grammatik. Dennoch ist sie nach Rosenzweig ein Wort, das nicht ungesprochen bleiben darf (vgl. GS II, 213). Die Kunst ist nämlich nicht nur Vor-sprache, sondern sie besitzt auch die Eigenschaft, keine „Umkehrung" (GS II, 26, passim) ihrer aus der Vorwelt stammenden Grundbegriffe zu verlangen, so dass das Kunstwerk in unmittelbarer Kontinuität zur Vorwelt entsteht: Über den drei Grundpfeilern der äusseren Form, der inneren Form und des Gehalts wölben sich die Bögen, die, indem sich je zwei verbinden und ineinander überführen, das Kunstwerk aufbauen (vgl. GS II, 165). Eine solche Kontinuität erleichtert nicht zuletzt das Erkennen (nicht jedoch schon das Erleben) des Wunders der Offenbarung, soweit die Kunst eine notwendige Voraussetzung, sozusagen eine Voraus-sage dieses Wunders und der ihm eigenen Sprache bildet: „[W]enn es keine Künstler gäbe, dann wäre die Menschheit ein Krüppel; denn es fehlte ihr dann die Sprache vor der Offenbarung, durch deren Dasein allein die Offenbarung ja die Möglichkeit hat, als historische Offenbarung in die Zeit einmal einzutreten und dort sich zu erweisen als etwas, was schon von uran ist." (GS II, 212) So glaube ich, dass es möglich ist, eine Verbindung zwischen dem Chor der Erlösung und dem der antiken Tragödie anzunehmen: Diesen kann man als „Vorstufe" bezeichnen, ja als „Voraus-schauender" jenes Wunders, zu dessen Ehren der Chor der Erlösung seinen Gesang auf Gott anstimmt. Wie oben ausgeführt, hat der Chor in der antiken Tragödie die Funktion, die Aufmerksamkeit des Betrachters auf den Helden zurückzulenken, um diesem das Wort zum Monolog zu erteilen. Im Zentrum der Handlung des Chores steht das tragische Selbst: Es umfasst das Selbst des Helden in seiner stummen Isolation, aber auch das Selbst des Zuschauers, der von Furcht und Mitleid für den Helden zum Selbst gemacht wird. Schliesslich geht es um das Selbst der Choreuten, d. h. um die Pluralität singulärer Stimmen, die als Gegenüber der dritten Person des Helden entstanden ist. Dieses Selbst, diese Singularität, die der Chor der Tragödie kennt und uns erkennen lehrt, kann ein tiefes Erlebnis (nicht nur eine Erkenntnis) des Wunders der Offenbarung Gottes an den Menschen nicht ignorieren. Nur 10 Hiermit beziehe ich mich auf D. Di Cesare, „Per una filosofia dei pronomi. Rosenzweig e la grammatica", in: Teoria 28 (2008), Heft 1, 137-147. 9 wem dieses In-sich-verschlossen-Sein bewusst ist, der kann das Ausmass der Sprengkraft göttlicher Liebe vollkommen erkennen: Mit ihrem Imperativ, mit ihrem Gebot „Liebe mich" (GS II, 197) ermöglicht sie in der Erlösung, dass sich der Mensch der Welt öffnet. Phobos und eleos, Furcht und Mitleid werden so zu „Ehrfurcht und Liebe"; die Worte, mit denen Aristoteles die Gefühle umschreibt, die die kathartische Wirkung der Tragödie hervorbringen, klingen ähnlich wie diejenigen, die im jüdischen Gebet an Sukkot beim Eintritt in die Sukka gebräuchlich sind (vgl. GS II, 456): „Das Wort des griechischen Theoretikers der Tragödie und das Wort, das die Offenbarung, als sie griechisch reden lernte, sich wählte, ist ein und dasselbe: Phobos. Die Ehrfurcht reisst die in der Kunstund Scheinwelt der Tragödie Getrennten, den Helden und den Zuschauer in eins; das leblose Bild wird nun selber erfüllt von dem Leben, das es bisher bloss im Betrachter erweckt, und also lebendig; es kann nun seinen Mund auftun und reden." (GS II, 188) Schreitet man auf diesem Weg der Interpretation voran, so lässt sich paradoxerweise auch eine Verbindung zwischen dem Chor der Erlösung und demjenigen herstellen, dessen Präsenz im modernen Drama fehlt. Denn der Dramatiker verzichtet auf den Chor, weil dieser nicht mehr nötig ist, um die Hauptfiguren hervortreten zu lassen, die sich mit ihrer eigenen philosophischen Weltanschauung dem Absoluten gegenüberstellen. Nach Rosenzweig stehen sie wohl dem Absoluten gegenüber, aber leben noch nicht in ihm: „[I]m Grunde versuchen sie alle [die Tragiker], [...] die Weltanschauungszur Lebenstragödie zu steigern. Das kaum gewusste Ziel dabei ist dies: an Stelle der unübersehbaren Vielheit der Charaktere den einen absoluten Charakter zu setzen, einen modernen Helden, der ebenso ein einer und immergleicher ist wie der antike. Dieser Konvergenzpunkt, in dem sich die Linien aller tragischen Charaktere schneiden würden, dieser absolute Mensch, der dem Absoluten nicht nur wissend gegenübersteht, sondern der es erlebt hat und der aus diesem Erlebnis heraus nun in ihm lebt, [...] ist kein andrer als der Heilige." (GS II, 235) In diesem Fall scheint der philosophische Held der modernen Tragödie zu einem „Zeichen", nahezu zu einem „Pro-pheten" des Wunders zu werden, das sich im Chor der Erlösung ereignet. Denn er kündigt den Menschen an, der zusammen mit der Welt singt, indem er sie erlöst und durch sie Erlösung findet, da sie beide der Erlösung Gottes (verstanden als genitivus subjectivus und genitivus objectivus) teilhaftig sind. Dieser erlösende und erlöste Mensch lebt – eben im Unisono des Wir – absolut im Absoluten, als der „zum Höchsten entschlossene Mensch, im Gegensatz zu dem in der einen immergleichen Finsternis des Selbst verschlossenen Helden" (GS II, 236). An die Stelle, die das tragische Selbst in der Vorwelt einnahm und an die die moderne Tragödie den philosophischen Helden zu setzen versucht hatte, tritt in der allzeiterneuerten Welt schliesslich der Heilige, der Knecht Gottes. 3. Der Chor in der Kirchenmusik Das Finale seines Auftretens hat die kleine Festung des Chores auf der Landkarte des Stern der Erlösung im Kontext von Rosenzweigs Soziologie der Tonkunst11 als Chor in der Kirchenmusik. Die Tonkunst wird nach Rosenzweig durch Vereinigung von Rhythmus und Harmonie in der Melodie gekennzeichnet: Die Harmonie beseelt und lässt den einzelnen Takt ertönen, der im Rhythmus als stummes Glied des Ganzen erscheint; die Melodie, die Linie 11 Vgl. hierzu Th. Eicker, Einsäen der Ewigkeit ins Lebendige. Impulse der Ästhetik Franz Rosenzweigs für eine Theologie gottesdienstlicher Musik, Paderborn/München/Wien/Zürich 2004; K.-J. Sachs, „Musik – erfahren und erörtert durch Franz Rosenzweig (1886-1929)", in: M. Beiche, A. Riethmüller (Hg.), Musik – Zu Begriff und Konzepten. Berliner Symposion zum Andenken an Hans Heinrich Eggebrecht, München 2006, 87-113; K.-J. Sachs, „Musik im Denken Franz Rosenzweigs", in: ND I, 446-455. 10 des Melos, ist in dem Masse das Lebendige an der Musik, in dem sie sich über den charakteristischen Rhythmus und die stimmungsvolle Harmonie des Musikstücks erhebt (vgl. GS II, 219-221, 274). So „ist die Musik überwiegend ,lyrisch', denn sie stellt ihre Werke in den Fluss der Zeit, und die Zeit ist die Form, die jeweils immer nur einen einzelnen Augenblick ins Bewusstsein treten lässt" (GS II, 217). Vom Chor der Erlösung ist schon gesagt worden, dass sich alles im Wir „zum choralmässig gleichen Takt" des vielstimmigen Lobund Dankgesangs sammelt: „Alle Stimmen sind hier selbständig geworden, jede singt die Worte nach der eigenen Weise ihrer Seele, doch alle Weisen fügen sich dem gleichen Rhythmus und binden sich zur einen Harmonie." (GS II, 264) Indem Rosenzweig die Einheit von Rhythmus und Harmonie im Wir des Chores der Erlösung behauptet, führt seine Theorie der Tonkunst ohne weiteres dazu, in jenem die Verwirklichung der Melodie der Musik zu sehen. Mir scheint aber, dass diese Behauptung noch etwas Wichtigeres enthält: Dass sich die Melodie der Musik nur im WirGesang der Erlösung voll verwirklichen kann. Auf den Seiten im Stern der Erlösung, die abschliessend der Kunst und damit der Musik gewidmet sind, scheint diese Interpretation sehr naheliegend zu sein – dies zeigt ein längerer bedeutungsvoller Abschnitt: „[W]o würde die Einzelseele auf den Ton gestimmt, der sie mit andern zusammenstimmen liesse in harmonischer Stimmung? Solche Stimmung, unbewusst und doch die Seele geleitend auf den Weg der höchsten Bewusstheit, des schweigenden Einverständnisses mit andern, kommt der Seele nur von einer einzigen Gewalt her: von der Kunst. Und nicht von der Kunst, wie sie sich selbst samt ihrem Schöpfer und ihrem Geniesser am liebsten absondern würde von aller Welt in ein letztes Abseits, sondern allein von einer Kunst, die aus jenem Sonderreich den Rückweg ins Leben gefunden, wirklich gefunden hat, den schon jene in ihr Sonderreich gebannte Kunst allenthalben als ihre Erlösung von sich selbst gesucht hatte. Erst die Künste, die man mit einem als Herabwertung gemeinten, in Wahrheit sie adelnden Namen als angewandte bezeichnet, erst sie führen den Menschen, ohne auch nur einen Funken von ihrer Herrlichkeit einzubüssen, ganz wieder ins Leben zurück, aus dem er, solange er sich dem ,reinen' Kunstgenuss hingab, sich entfernt hatte. Ja, sie sind es allein, die ihn ganz heilen können von jener Krankheit der Weltentfremdung, die den Kunstfreund in den trügerischen Wahn höchster Gesundheit einwiegte grade dann, wenn er sich der Krankheit widerstandslos öffnete. Die Kunst entgiftet so sich selber; sie reinigt sich und den Menschen von ihrer eigenen Reinheit, sie wird aus einer anspruchsvollen Geliebten seine gute Frau, die ihn durch die tausend kleinen Dienste des Alltags und die Pflege des Hauses kräftig macht für den Markt und die grossen Stunden des öffentlichen Lebens, und dabei selber in ihrer Würde als Herrin des Hauses erst zur Vollreife ihrer Schönheit aufblüht." (GS II, 393) Die gleiche Kunst, die in ihrer Reinheit den Menschen dazu verdammt, als Ersatz für den verlorenen in einem menschlichen, allzumenschlichen Garten zu leben, kann also – als angewandte Kunst – den Menschen nach gan 'eden, in den göttlichen Garten des Lebens zurückführen. Die Kunst kann den Menschen wirklich dazu erheben, nicht nur als Gegenüber des Absoluten, sondern in diesem zu leben. Aber nochmals: nicht die reine Kunst, die vom Idealismus irrtümlich mit der sichtbaren Erscheinung eines Absoluten verwechselt worden ist, sondern die angewandte Kunst, die als bildende Kunst den Raum schafft, als tönende Kunst die Zeit angibt und als darstellende Kunst die Gebärde lehrt.12 Denn in diesem Raum, in dieser Zeit und in dieser Gebärde kann der Mensch die Nähe Gottes erfassen. Folglich müssen die Kunstwerke dem magischen Kreis des Idealismus, seinem reinen Raum, seiner reinen Zeit 12 Rosenzweigs Unterschied zwischen reiner und angewandter Kunst findet ein Echo in seiner Unterscheidung zwischen reinem und angewandtem Denken, wo nur das letztere denkt und das System der Denkbestimmungen ermöglicht, während das rein auf das Sein gerichteten Denken wohl als einheitlicher Ursprung gedacht und vorausgesetzt, aber nicht erwiesen werden kann (vgl. GS II, 46f.). 11 und seiner reinen Gedankenwelt entrissen und in den wirklichen Raum, die wirkliche Zeit und die wirkliche Welt eingefügt werden. Was die Musik anbetrifft, sollte man nach Rosenzweig vor allem darauf verzichten, sie als „ideale Zeit" zu sehen, d. h. als „Flucht aus den Aufregungen oder je nachdem auch der lähmenden Langeweile [d]es wirklichen Lebens" (GS II, 400), also sozusagen als Realität anderer Art. So verstanden würde sie eine Gefahr für den Menschen darstellen: Sie entzieht ihn der realen Zeit, sie lässt ihn sich selbst vergessen und Hunderte von imaginären Leben leben, sie verleitet ihn dazu, mit offenen Augen zu träumen, soweit dass – wie Rosenzweig bissig bemerkt – „der Hund, der sich höllenheiss betrübt, weil seine Dame Flügel spielte, [...] echter, ja wenns erlaubt ist ,menschlicher' [lebt], als der ,Musikalische'." (GS II, 401) Um die Musik von ihrem Reinseinwollen zu reinigen, muss man sie in die wirkliche Zeit der religiösen Liturgie eingliedern: „Indem die Musik sich diesen Festen und überhaupt dem Kirchenjahr einfügt, steigt das einzelne Musikwerk heraus aus dem künstlichen Rahmen seiner idealen Zeit und wird ganz lebendig, weil es gepfropft wird auf den säftereichen Stamm der wirklichen Zeit." (ebd.) Darin besteht die Bedeutung des Chores in der Kirchenmusik: „Wer einen Choral mitsingt, wer Messe, Weihnachtsoratorium oder Passion hört, der weiss ganz genau, in welcher Zeit er ist; er vergisst sich nicht und will sich nicht vergessen; er will sich nicht aus der Zeit flüchten, sondern im Gegenteil: er will seine Seele mit beiden Beinen in die Zeit, in die allerwirklichste Zeit, in die eine Zeit des einen Welttags, dessen alle einzelnen Welttage nur Teile sind, hineinstellen. Dahin soll ihm die Musik das Geleit geben." (GS II, 401f.) Rosenzweig denkt also an den Choral, insbesondere an den Choral Bachs,13 aber auch an den „Gregorianischen" der katholischen Kirche.14 Der Choral dient ihm als Modell, sogar als „eigentliche Grundlage der kirchlichen Anwendung der Musik" (GS II, 402). Denn in der Harmonisierung des Chorals verschmelzen die einzelnen Stimmen zur Einheit einer nicht kontrapunktischen Polyphonie; die verschiedenen melodischen Linien erklingen trotz ihrer Unabhängigkeit voneinander im Unisono und werden Ausdruck der Zusammengehörigkeit aller Versammelten: „Im Choral ist die Sprache, die sonst aus jedes Einzelnen Mund ihr eigenes und besonderes Wort zu reden hat, zum Schweigen gebracht. Nicht zu jenem Schweigen, das einfach stumm dem verlesenen Wort zuhört, sondern zum Schweigen seiner Eigenheit in der Einmütigkeit des Chors." (ebd.) Ausserdem stimmt der Choral nicht nur die einzelnen Stimmen des Chores harmonisch aufeinander ab, indem er sie auf die Gemeinschaft des Wir einstimmt, sondern er erleichtert auch die gemeinsame Andacht derjenigen, die diese Musik mithören, aber selbst nicht mitsingen. Es handelt sich in diesem spezifischen Fall angewandter Musik um das Publikum der musikalischen Liturgie, deren Zuhörer – ein jeder für sich – zum Gleichklang der Gefühle bewegt werden: Jeder schweigt und jeder spricht „die allen gemeinsamen Worte zur Musik." (GS II, 403) Mit diesem crescendo, mit diesem „gleichrhythmischen Chor des gleichen Worts" (GS II, 13 Über Rosenzweigs „Lieblingsidee [...] der Gewinnung Bachscher Vokalmusik für den jüdischen Gottesdienst" siehe seinen Brief an Hermann Geiger vom 31. Dezember 1926 (GS I/2, 1119), erstmals erschienen unter dem Titel „Bach in die Synagogen! Zur Reform der Reform", in: Die Gemeinschaft. Hefte für die religiöse Erstarkung des Judentums 13/14 (18. August 1928), 20. Vgl. auch Rosenzweigs Brief an Eugen Mayer vom 26. September 1928, GS I/2, 1198f. 14 Über den „Gregorianischen Choral" als von der katholischen Kirche weitergepflegte Tradition schreibt Rosenzweig in seinen Musikbesprechungen „Der Konzertsaal auf der Schallplatte" (1928/1929), GS III, 448. 12 413) kommt die Parabel des Chores im Stern der Erlösung zum Abschluss. Eine Parabel, deren aufsteigende Bahn über die Anwesenheit des Chores in der antiken Tragödie und seine Abwesenheit im modernen Drama den Extrempunkt im Chor der Erlösung erreicht, um auf der absteigenden Bahn eine Rolle im Chor der Kirchenmusik zu übernehmen. Eine Parabel, die gestattet, das Hauptwerk Rosenzweigs im Lichte seiner Kunsttheorie neu zu lesen: vom Ausgangspunkt der poetischen Gattung der Tragödie zum Endpunkt der Tonkunst und ihrer Bedeutung in der liturgischen Praxis. Eine Parabel, die zeigt, wie die Kunst – idealistisch in ihrer Reinheit verstanden – den Menschen dazu verdammt, in seinem hortus conclusus in Verbannung zu bleiben, fern von Gott und dem wirklichen Leben. Diese Kunst kann jedoch, wenn sie korrekt angewandt wird, den Menschen in den verlorenen gan 'eden zurückversetzen. Also eine Parabel, die die Interpretation des Stern der Erlösung in dem Sinne verstärkt, dass das „neue" Denken Rosenzweigs seine definitive Form im Bruch mit der „alten" idealistischen Philosophie erlangt, da diese unfähig ist, die Gestaltung des wirklichen Lebens zu verstehen. Eine Parabel aber, die jetzt auch ermöglicht, Rosenzweig zwischen diesem Idealismus und der religiösen Erfahrung treffender einzuordnen. Schliesslich eine Parabel, die – wie Rosenzweigs gesamtes Buch – das Tor ins Leben öffnet: „Besser Schreiben als Lesen, / besser Dichten als Schreiben, / besser Leben als Dichten!" hat der Verfasser des Stern der Erlösung am 8. Oktober 1906 in sein Tagebuch geschrieben (GS I/1, 58). Und so kann man mit den Worten Goethes An Lina enden, auf die Rosenzweig zum Teil selbst verweist (vgl. GS II, 275): „Lass die Saiten rasch erklingen / Und dann sieh in's Buch hinein; / Nur nicht lesen! immer singen! / Und ein jedes Blatt ist dein."15 Deutsch: Lieselotte Mangels, Luca Bertolino 15 J. W. von Goethe, An Lina (1800), Weimarer Ausgabe, Bd. I/1, 104.

UNIVERSIDADE FEDERAL DA PARAÍBA CENTRO DE CIÊNCIAS SOCIAIS APLICADAS PROGRAMA DE PÓS-GRADUAÇÃO EM CIÊNCIAS CONTÁBEIS CURSO DE DOUTORADO EM CIÊNCIAS CONTÁBEIS EMANOEL TRUTA DO BOMFIM FATORES CONTINGENCIAIS, ESTRATÉGIA COMPETITIVA E DESEMPENHO FINANCEIRO: UM ESTUDO EM EMPRESAS DE PAÍSES MEMBROS DO BRICS E DO G7 JOÃO PESSOA – PB 2020 EMANOEL TRUTA DO BOMFIM FATORES CONTINGENCIAIS, ESTRATÉGIA COMPETITIVA E DESEMPENHO FINANCEIRO: UM ESTUDO EM EMPRESAS DE PAÍSES MEMBROS DO BRICS E DO G7 Tese de doutorado apresentada ao Programa de Pós-Graduação em Ciências Contábeis da Universidade Federal da Paraíba – UFPB, como requisito parcial para a obtenção do título de Doutor em Ciências Contábeis. Orientador: Prof. Dr. Aldo Leonardo Cunha Callado. Área de Concentração: Informação Contábil. Linha de Pesquisa: Informação Contábil para Usuários Internos. JOÃO PESSOA – PB 2020 B695f Bomfim, Emanoel Truta do. FATORES CONTINGENCIAIS, ESTRATÉGIA COMPETITIVA E DESEMPENHO FINANCEIRO: UM ESTUDO EM EMPRESAS DE PAÍSES MEMBROS DO BRICS E DO G7 / Emanoel Truta do Bomfim. - João Pessoa, 2020. 134 f. Tese (Doutorado) UFPB/CCSA. 1. Estratégia competitiva. 2. Fatores contingenciais. 3. Desempenho financeiro. I. Título UFPB/BC Catalogação na publicação Seção de Catalogação e Classificação A Deus, pela graça de perseverar e nunca desistir. À minha mãe, Ivanize (in memoriam), e ao meu pai, Alfeu, que não mediram esforços para a realização deste sonho. À minha esposa, Dayane, e à minha filha, Maria Clara, pelo amor e carinho. Aos professores, colegas e amigos, pelos ensinamentos e pela amizade. AGRADECIMENTOS A Deus, por sua graça e por sempre me acompanhar em tudo o que faço em minha vida, abençoando-me e proporcionando-me coragem e disposição para concretizar sonhos e vitórias, bem como à sua Mãe, a Virgem Maria, que me cobriu com seu manto de amor. Minha eterna gratidão! À minha mãe, Ivanize Truta (in memoriam), que não mediu esforços para que eu, com a graça de Deus, tivesse alcançado esta conquista. Minha eterna professora. Te amo! À minha esposa, Dayane, e à minha filha, Maria Clara, por todo amor e incentivo durante este período. Muito obrigado! Amo vocês. Ao meu pai, Alfeu Bomfim, que sempre me apoiou com amor e carinho, e por entender a minha ausência. Obrigado por sempre estar do meu lado! Aos meus irmãos, Kleber, Afrânio, Adimael, Ana Maria e Aristócles, e aos meus sobrinhos Nicolás e Hugo, por sempre estarem do meu lado e entenderem a minha ausência. Às minhas primas Marlene e Brenda, pelo apoio e pela gentileza sempre dispensada. Ao Prof. Dr. Aldo Callado, pela paciência, por toda a orientação e pela atenção que me dispensou durante todo o curso, e cujas contribuições ajudaram-me não só a realizar um sonho, mas a melhorar minhas habilidades como profissional e pesquisador. Muito obrigado! A todos os professores do Programa de Pós-Graduação em Ciências Contábeis da UFPB, pelos ensinamentos compartilhados: Dr. André Callado, Dr. Edilson Paulo, Dr. Paulo Amilton, Dra. Márcia Reis, Dr. Paulo Aguiar, Dr. Paulo Roberto, Dra. Simone Bastos, Dr. Orleans Martins e Dr. Wenner Lucena. Aos amigos e colegas da turma 2016: Augusto, Ariane, Fábia, Gilberto, Paulo, Rone, Kléber e Evaldo. Foram muitos momentos ao lado de todos vocês, os quais irei lembrar com muito carinho. Aprendi muito com cada um de vocês. Aos amigos Gilberto Magalhães e Kléber, com os quais pude contar em todos os momentos durante todo o curso. Aos(às) colegas Ingrid, Vinícius Martins, Luciana, Kallyse, Yara, Lineker, Carlos André, Davi, Luís Manuel e os demais, com quem aprendi muito durante todo o curso. À secretária do Programa de Pós-Graduação em Ciências Contábeis da UFPB, na pessoa de Wilma, pelo carinho, atenção e gentileza sempre dispensada. In memoriam, à minha tia Josefa, ao seu esposo, José Mateus, e ao o meu amigo Sr. Xavier, pelo apoio no início desta jornada, que Deus os recompensem com o céu. À minha madrinha, Josefa (Zezita), que desde a minha tenra idade sempre me apoiou. Muito obrigado! Aos meus amigos Severino, Claudivam, Hugo Macedo, Aderaldo Barbosa, Mamadou, e aos colegas da UEPB, com quem pude contar em muitos momentos. A todos que, de uma forma ou de outra, também contribuíram para a realização deste trabalho. A todos, meus sinceros agradecimentos! "Doravante todas as gerações me chamarão bem-aventurada, porque o Senhor fez em mim maravilhas". (São Lucas 1, 34-55) RESUMO O objetivo desta tese foi analisar a relação da estratégia competitiva (liderança em custos ou diferenciação) com os fatores contingenciais (incerteza do ambiente, nível de competição, tamanho e idade da organização) sobre o desempenho financeiro de empresas, considerando o ambiente em que estão localizadas (países membros do BRICS e do G7). Para alcançar este objetivo, foi necessário avaliar a relação da estratégia competitiva com o desempenho financeiro das empresas, considerando o ambiente em que atuam; verificar os fatores contingenciais que afetam o desempenho financeiro, de acordo com ambiente organizacional; identificar a estratégia competitiva que torna o desempenho financeiro das empresas sustentável; e investigar a relação entre a estratégia competitiva e os fatores contingenciais, de acordo com o ambiente organizacional. Para tanto, desenvolveu-se uma pesquisa de cunho empírico-analítico, com o intuito de avaliar a relação entre as variáveis investigadas. Compuseram a amostra da pesquisa 775 empresas (5.425 observações), sendo 172 localizadas na China, 33 na Índia, 48 na Alemanha, 25 no Canadá, 307 nos Estados Unidos da América, 169 no Japão e 21 no Reino Unido. Coletaram-se os dados no banco de dados da Thomson Reuters Eikon® e, quando necessário, no sítio e nos demonstrativos financeiros das empresas na Internet. Para estimar as relações entre as variáveis da pesquisa, aplicaram-se modelos de dados em painel dinâmico. Os resultados apontaram que, apenas para o ambiente do BRICS, a estratégia de liderança em custos afeta positivamente o desempenho financeiro das empresas. Essa evidência aponta que a estratégia de liderança em custos aumenta o resultado financeiro das organizações. As evidências encontradas sugerem que, para as empresas sediadas nos países do BRICS (China e Índia), a estratégia de liderança em custos parece reduzir os efeitos da incerteza e da idade da organização sobre o retorno dos ativos, bem como que a estratégia de diferenciação modera a influência da idade da organização sobre o desempenho. Já com relação ao ambiente de países membros do G7, os resultados apontaram que a estratégia de liderança em custos parece atenuar a influência da incerteza e que a estratégia de diferenciação modera os efeitos do tamanho sobre o desempenho financeiro das companhias. Por fim, concluiu-se que as estratégias competitivas podem moderar os efeitos dos fatores contingenciais sobre o desempenho financeiro das empresas, dependendo do ambiente em que a organização atua, conforme previsto na tese de pesquisa. Além disso, estes resultados demonstraram a aplicabilidade da teoria da contingência, indicando que as organizações devem se adequar ao contexto em que atuam, bem como que se pode utilizar a estratégia competitiva como um meio para se moderar as influências dos fatores contingenciais sobre o desempenho das organizações. Palavras-chave: Estratégia competitiva. Fatores contingenciais. Desempenho financeiro. ABSTRACT This thesis aimed to analyze the relationship between the competitive strategy (cost leadership or differentiation) with the contingency factors (environmental uncertainty, level of competition, size and age of organization) on the financial performance of firms, considering the environment in which they are located (countries of BRICS or G7). To achieve this aim, it was necessary to evaluate the relationship among competitive strategy and the financial performance of firms, considering the environment in which they operate; verify the contingency factors that affect financial performance, according to organizational environment; identify the competitive strategy that makes sustainable the financial performance of firms; and investigate the relationship between competitive strategies and contingency factors, according to the organizational environment. For this purpose, an empirical-analytical research was developed in order to evaluate the relationship between the investigated variables. The research samples was composed by 775 companies (5,425 observations), 172 located in China, 33 in India, 48 in Germany, 25 in Canada, 307 in the United States of America, 169 in Japan and 21 in the United Kingdom. The data were collected from the Thomson Reuters Eikon® database and, when necessary, on the website and in the financial statements of firms on Internet. To estimate the relationship between the research variables, dynamic panel data models were used. The results showed that, only for the BRICS environment, the cost leadership strategy positively affects the financial performance of firms. This evidence points out that the cost leadership strategy increases the financial results of organizations. The results found suggests that, for companies located in BRICS countries (China and India), the cost leadership strategy seems to decrease the effects of uncertainty and age of the organization on the return of assets, and differentiation strategy moderates the influence of organization's age on financial performance. Already the environment of the G7 member countries, the evidence also showed that cost leadership strategy can reduce the influence of uncertainty and that the differentiation strategy moderates the effects of size on the financial performance of firms. Lastly, it was concluded that competitive strategies can moderate the effects of contingency factors on the financial performance of companies, depending on the environment in which the organization operates, as provided for in the research thesis. Moreover, these results demonstrated the applicability of the contingency theory, indicating that firms must adapt to the context in which they operate, as well as that competitive strategy can be used as a means to moderate the influences of contingency factors on organizational performance. Keywords: Competitive strategy. Contingency factors. Financial performance. LISTA DE QUADROS Quadro 1 Principais fatores contingenciais de Otley (2016) ................................................ 34 Quadro 2 Características da estratégia prospectora de Miles e Snow (1978) ....................... 40 Quadro 3 Características da estratégia defensora de Miles e Snow (1978) .......................... 40 Quadro 4 Características da estratégia analítica de Miles e Snow (1978) ............................ 41 Quadro 5 Requisitos das estratégias competitivas de Porter (1980) ..................................... 46 Quadro 6 Padrões estratégicos identificados por Robinsin Jr. e Pearce II (1988) ................ 50 Quadro 7 Estratégias do relógio de Bowman e Falkner (1997) ............................................ 54 Quadro 8 Características das estratégias de Hax e Wilde (1999) ......................................... 56 Quadro 9 Estratégias competitivas de Mintzberg e Quinn (2001) ........................................ 57 Quadro 10 Resumo das tipologias de estratégias competitivas ............................................ 62 Quadro 11 Estudos associando fatores contingenciais, estratégia e desempenho financeiro 70 Quadro 12 Relação esperada entre as variáveis explicativas e a dependente ....................... 97 Quadro 13 Resumo dos procedimentos metodológicos ........................................................ 98 Quadro 14 Resultados das hipóteses de pesquisa ............................................................... 116 LISTA DE FIGURAS Figura 1 Modelo SARFIT ..................................................................................................... 30 Figura 2 Estratégias institucionais competitivas como um movimento organizacional ....... 60 Figura 3 Modelo conceitual da pesquisa ............................................................................... 74 LISTA DE TABELAS Tabela 1 Amostra da pesquisa por ambiente ......................................................................... 82 Tabela 2 Classificação das empresas por indústria e país ..................................................... 83 Tabela 3 Análise fatorial exploratória e confirmatória dos constructos: empresas de países membros do BRICS ................................................................................................................ 100 Tabela 4 Análise fatorial exploratória e confirmatória dos constructos: empresas de países membros do G7 ...................................................................................................................... 102 Tabela 5 Estatísticas descritivas: empresas de países membros do BRICS (2012-2018) ... 103 Tabela 6 Estatísticas descritivas: empresas de países membros do G7 (2012-2018) .......... 104 Tabela 7 Correlação entre as variáveis: empresas de países membros do BRICS .............. 105 Tabela 8 Análise de correlação das variáveis: empresas de países membros do G7 .......... 106 Tabela 9 Resultados da estimação do modelo para empresas de países membros do BRICS (2012-2018) ............................................................................................................................ 109 Tabela 10 Análise da sustentabilidade do desempenho das empresas localizadas em países membros do BRICS (2016-2018) ........................................................................................... 111 Tabela 11 Resultados da estimação do modelo para empresas de países membros do G7 (2012-2018) ............................................................................................................................ 113 Tabela 12 Análise da sustentabilidade do desempenho das empresas de países membros do G7 (2016-2018) ...................................................................................................................... 115 LISTA DE ABREVIATURAS E REDUÇÕES AGFI Goodness of Fit Index Adjusted for Degrees of Freedom AIL Ativo Imobilizado Líquido AR1 Autocorrelação de Primeira Ordem AR2 Autocorrelação de Segunda Ordem AT Ativo Total BDTD Base Digital de Teses e Dissertações BRICS Brasil, Rússia, Índia e China Capes Coordenação de Aperfeiçoamento de Pessoal de Nível Superior CAPEX Capital Expenditures CBS Custo dos Bens e Serviços CE Crescimento Econômico CFI Comparative Fit Index CFROI® Cash Flow Return on Investment CULT Aspecto Cultural do País DF Desempenho Financeiro DIF Diferenciação DInd Dummy Setor Industrial DMC Dominant Market Competition (Competição em Mercado Dominante) Dpaís Dummy país DVGA Despesas com Vendas, Gerais e Administrativas EC Estratégia Competitiva END Endividamento EVA® Economic Value Added FCE Fatores Contingenciais Externos FCI Fatores Contingenciais Internos FMI Fundo Monetário Internacional (International Monetary Fund) G7 Grupo dos Sete Países mais Industrializados do Mundo GFI Goodness of Fit Index GICS® Global Industry Classification Standards GMM Generalized Method of Moments (Método dos Momentos Generalizados) GP Empresas de Grande Porte HI Índice de Herfindahl ID Idade da Organização IC-S Institutional Competition – Suport (Competição Institucional por Apoio) IC-G Institutional Competition – Governance (Competição Institucional por Governança) INC Incerteza KMO Kaiser-Meyer-Olklin LC Liderança em Custos LNE Logaritmo do Número de Empregados NC Nível de Competitividade NE Número de Empregados NFI Normed Fit Index NMC Niche Market Competition (Competição em Mercado de Nicho) P&D Pesquisa e Desenvolvimento PEQ Empresas de Pequeno Porte PIB Produto Interno Bruto RL Receita Líquida RMR Root Mean Square Residual ROA Return On Assets (Retorno sobre os Ativos) ROE Return On Equity (Retorno sobre o Patrimônio Líquido) ROI Return On Investiment (Retorno sobre os Investimentos) SARFIT Structural Adjustment to Regain Fit TAM Tamanho Organizacional VA Valor Adicionado SUMÁRIO 1 INTRODUÇÃO ................................................................................................................... 16 1.1 CONTEXTUALIZAÇÃO DO TEMA ................................................................................ 16 1.2 PROBLEMA DE PESQUISA ............................................................................................. 18 1.3 OBJETIVOS ....................................................................................................................... 20 1.3.1 Objetivo geral ................................................................................................................. 20 1.3.2 Objetivos específicos ...................................................................................................... 20 1.4 JUSTIFICATIVA E CONTRIBUIÇÕES ........................................................................... 20 1.4.1 Originalidade ................................................................................................................. 21 1.4.2 Contribuições e implicações práticas ........................................................................... 23 1.5 TESE ................................................................................................................................... 23 1.6 DELIMITAÇÃO DO TEMA .............................................................................................. 24 1.7 ESTRUTURA DA TESE .................................................................................................... 25 2 REFERENCIAL TEÓRICO .............................................................................................. 26 2.1 TEORIA DA CONTINGÊNCIA ........................................................................................ 26 2.1.1 Fatores contingenciais ................................................................................................... 33 2.1.1.1 Fatores contingenciais externos .................................................................................... 34 2.1.1.2 Fatores contingenciais internos .................................................................................... 36 2.2 ESTRATÉGIAS COMPETITIVAS ................................................................................... 38 2.2.1 Tipologia estratégica de Miles et al. (1978) ................................................................. 38 2.2.2 Estratégias competitivas propostas por Porter (1980) ............................................... 43 2.2.3 Missão estratégica de Gupta e Govidarajan (1984) .................................................... 48 2.2.4 Estratégias competitivas de Robinsin Jr. e Pearce II (1988) ..................................... 49 2.2.5 Estratégias competitivas de Treacy e Wiserma (1993) ............................................... 51 2.2.6 Estratégias competitivas de Bowman e Faulkner (1997) ........................................... 53 2.2.7 Estratégias competitivas de Hax e Wilde II (1999) ..................................................... 55 2.2.8 Estratégias competitivas de Mintzberg e Quinn (2001) ............................................. 57 2.2.9 Estratégias competitivas de Udayasankar e Das (2004) ............................................. 58 2.2.10 Resumo das principais estratégias competitivas ....................................................... 61 2.3 MENSURAÇÃO DE DESEMPENHO ............................................................................... 63 2.3.1 Conceitos e objetivos ..................................................................................................... 64 2.3.2 Medidas para mensuração de desempenho ................................................................. 66 2.4 EVIDÊNCIAS EMPÍRICAS .............................................................................................. 69 2.5 MODELO CONCEITUAL DA PESQUISA ...................................................................... 74 3 PROCEDIMENTOS METODOLÓGICOS ..................................................................... 76 3.1 CARACTERIZAÇÃO DA PESQUISA ............................................................................. 76 3.2 HIPÓTESES DE PESQUISA ............................................................................................. 76 3.2.1 Incerteza do ambiente e desempenho financeiro ........................................................ 76 3.2.2 Nível de competição e desempenho financeiro ............................................................ 77 3.2.3 Tamanho e desempenho financeiro ............................................................................. 78 3.2.4 Idade da organização e desempenho financeiro ......................................................... 79 3.2.5 Papel moderador da estratégia competitiva ................................................................ 79 3.3 POPULAÇÃO E AMOSTRA ............................................................................................. 82 3.4 COLETA E TRATAMENTO DOS DADOS ...................................................................... 84 3.5 VARIÁVEIS DA PESQUISA ............................................................................................ 84 3.5.1 Variável dependente: desempenho financeiro ............................................................ 84 3.5.2 Variáveis explicativas .................................................................................................... 85 3.5.3 Variáveis de controle ..................................................................................................... 90 3.6 MÉTODOS DE ANÁLISE ................................................................................................. 91 3.6.1 Análise fatorial exploratória e confirmatória ............................................................. 91 3.6.2 Análise de dados em painel ........................................................................................... 93 3.6.3 Modelos empíricos ......................................................................................................... 93 3.7 RESUMO DOS PROCEDIMENTOS METODOLÓGICOS ............................................. 98 4 APRESENTAÇÃO E ANÁLISE DOS RESULTADOS .................................................. 99 4.1 ANÁLISE FATORIAL EXPLORATÓRIA E CONFIRMATÓRIA DAS ESTRATÉGIAS COMPETITIVAS ..................................................................................................................... 99 4.2 ESTATÍSTICAS DESCRITIVAS .................................................................................... 103 4.3 ANÁLISE ECONOMÉTRICA ......................................................................................... 107 4.3.1 Análise das estimações de empresas de países membros do BRICS ....................... 107 4.3.2 Análise das estimações de empresas de países membros do G7 .............................. 112 4.3.3 Síntese dos resultados das hipóteses de pesquisa ...................................................... 116 5 CONSIDERAÇÕES FINAIS ............................................................................................ 118 5.1 CONCLUSÕES ................................................................................................................ 118 5.2 LIMITAÇÕES E RECOMENDAÇÕES DA PESQUISA ................................................ 120 REFERÊNCIAS ................................................................................................................... 121 16 1 INTRODUÇÃO Neste capítulo apresentam-se a contextualização do tema, o problema de pesquisa, os objetivos geral e específicos, a justificativa que fundamenta o presente estudo, a tese proposta, a delimitação da pesquisa e a estrutura da tese. 1.1 CONTEXTUALIZAÇÃO DO TEMA As empresas estão sendo cada vez mais desafiadas a buscarem soluções para se manterem competitivas em um mercado complexo e marcado por constantes mudanças, em que as relações passaram a ocorrer globalmente, de acirrada concorrência (ACQUAH, 2013; BLAHOVA; PALKA; HAGHIRIAN, 2017) e cercado por incertezas que afetam o desempenho organizacional (CHENHALL, 2003; OTLEY, 2016). Para permanecerem competitivas em um mercado tão dinâmico, as corporações precisam se adaptar às contingências (ou fatores contingenciais) inerentes ao ambiente e às suas atividades, para assim, manterem ou alcançarem o nível de desempenho desejado (MILES et al., 1978; DONALDSON, 1987, 1999). A adaptação aos fatores contingenciais intrínsecos ao ambiente corporativo se faz necessária para que cada corporação possa enfrentar as incertezas de mercado e tornar-se competitiva frente aos concorrentes, por exemplo, uma vez que as mudanças no ambiente ocorrem de forma contínua e ininterrupta. Contudo, de acordo com a teoria da contingência, não há apenas uma estrutura organizacional que seja aplicável e efetiva para todas as organizações. A escolha por uma determinada estrutura ou sistema corporativo dependerá de determinados fatores que podem estar relacionados ao ambiente (fatores contingenciais externos) ou à organização (fatores contingenciais internos) e que moldam a estrutura corporativa a qualquer momento, interferindo na forma como uma entidade desenvolve e estrutura seus processos, impactando o desempenho organizacional (DONALDSON, 1999; MCKINLEY; MONE, 2003; OTLEY, 2016; HAMANN, 2017). Os fatores contingenciais podem ser definidos como qualquer variável do ambiente em que a organização está inserida que modera o efeito de uma característica organizacional no desempenho. Neste sentido, para ser lucrativa, uma empresa precisa ajustar a sua estrutura aos fatores contingentes presentes no ambiente de atuação, sendo que cada aspecto da estrutura organizacional pode ser contingente a um ou mais fatores, dentre os quais estão: estratégia, 17 tamanho, hostilidade, incerteza do ambiente (DONALDSON, 1999; CHENHALL, 2003; OTLEY, 2016). Dentre os fatores contingenciais, entende-se que a estratégia corporativa pode ser considerada o meio pelo qual a organização pode ajustar a sua estrutura às contingências, por ser um padrão ou fluxo de decisões tomadas, capaz de ser utilizada para alcançar a combinação mais favorável entre a organização e o ambiente externo (HOFER, 1975; KIM; LIM, 1988; CHENHALL, 2003; OTLHEY, 2016; ANWAR; HASNU, 2017). Com base na estratégia, a organização pode estabelecer um planejamento unificado, abrangente e integrado de metas e objetivos básicos de curto e longo prazo, a serem realizados por meio da adoção de medidas necessárias para a alocação de recursos indispensáveis para a realização dos objetivos corporativos, de forma a responderem rapidamente às mudanças ocorridas no ambiente corporativo (PORTER, 1996; MCGEE, 2015). A estratégia corporativa é vista como uma variável mediadora entre a organização e o ambiente (CHILD, 1972; MILES et al., 1978; GUPTA; GOVIDARAJAN, 1984; HOQUE, 2004), capaz de permitir que as empresas adotem ações e planos para enfrentar as incertezas futuras, criando alternativas que melhor se ajustem às contingências relativas ao ambiente (PORTER, 1996; MCGEE, 2015). A estratégia ainda pode ser considerada como sendo uma variável diferente das demais, pois se trata de um elemento do contexto interno, a partir do qual os gestores podem influenciar a natureza do ambiente externo, como o comportamento dos concorrentes, bem como os demais fatores internos ao ambiente organizacional (tecnologia, estrutura organizacional, cultura organizacional, controle e outros), com o objetivo de melhorar o desempenho organizacional (CHENHALL, 2003; OTLEY, 2016; MAKADOK; BURTON; BARNEY, 2018). Para tanto, a organização precisa questionar, verificar e redefinir a maneira com que interage com o ambiente, por meio de um padrão estratégico que lhe possibilite manter um efetivo alinhamento com o mercado (MILES et al., 1978), buscando utilizar suas principais competências e capacidades para estabelecer e manter produtos e serviços que produzirão uma vantagem competitiva sustentável em relação aos seus concorrentes (PORTER, 1996; COLLINS; ROMÁN; CHAN, 2011). Na busca por alcançar uma vantagem competitiva sustentável, a estratégia a ser adotada pela organização precisa gerar ganhos no longo prazo, tornando-a competitiva frente aos concorrentes e possibilitando alcançar um desempenho superior à média do setor, tornando-a menos susceptível às mudanças ocorridas no ambiente (PORTER, 2004; BANKER; MASHUWALA; TRIPATHY, 2014). 18 Entretanto, segundo a teoria da contingência, os efeitos das atividades corporativas sobre o desempenho organizacional estão condicionados à estratégia competitiva (HUOR et al., 2014), que, por sua vez, para ser efetiva precisa estar alinhada as variáveis do ambiente e as características organizacionais (HOFER, 1975; DONALDSON, 1987; ZOTT; AMIT, 2008). Nota-se, desta forma, que as organizações precisam adequar à estratégia de acordo com as diversas mudanças ocorridas no ambiente, moldando-a para adaptá-la à complexidade de cada setor, buscando utilizar as experiências e o conhecimento adquirido para enfrentar as incertezas de cada cenário. Portanto, observa-se que as contingências podem afetar a estratégia competitiva e o desempenho organizacional, levando uma entidade a adotar a estratégia que lhe permitirá escolher as atividades que lhes diferencie dos rivais (PORTER, 1996), descrevendo como conduzirá suas operações e negócios para se adaptar ao ambiente, buscando, por meio de planos, alcançar o desempenho ou a lucratividade almejada de forma sustentável (MINTZBERG, 1978; MILLER, 1982; COLE, 2001; BANKER; MASHUWALA; TRIPATHY, 2014; OTLEY, 2016). Diante disto, considera-se relevante analisar a estratégia competitiva, que pode moderar as influências dos fatores contingenciais sobre o desempenho organizacional, considerando-se, para tanto, o ambiente em que as empresas operam, como países do grupo dos mais ricos do mundo (G7) ou membros do BRICS (Brasil, Rússia, Índia, China e África do Sul). 1.2 PROBLEMA DE PESQUISA Segundo a teoria da contingência, para uma organização alcançar um desempenho superior, ela precisa se ajustar aos fatores presentes no ambiente, ou seja, aos fatores contingenciais internos e externos (MILLER, 1992; DONALDSON, 2015). Isto exige que a entidade adeque estruturas e processos com aspectos do contexto externo (MILLER, 1992), sendo esta uma condição necessária para obter maior desempenho (MARTINS, 2019). Para isso, se faz necessário que a estrutura organizacional, por exemplo, esteja alinhada com a estratégia (DONALDSON, 1999), uma vez que a estratégia é quem determina a estrutura (CHANDLER, 1962; JUNQUEIRA et al., 2016). Segundo Gupta e Govindarajan (1984), Chenhall (2003) e Anwar e Hasnu (2017), a estratégia é considerada um fator contingencial um pouco diferente de outras variáveis contingenciais. Não é considerado um elemento de contexto externo, mas um meio pelo qual os gerentes podem influenciar a natureza do ambiente externo, as tecnologias da organização, 19 os arranjos estruturais, entre outros fatores (CHENHALL, 2003). Ou seja, é um fator contingencial que pode ser utilizado para a organização se ajustar aos demais fatores contingenciais e obter desempenho diferenciado. Para uma organização alcançar um desempenho financeiro superior, é necessário que obtenha e sustente uma vantagem competitiva no longo prazo (BANKER; MASHUWALA; TRIPATHY, 2014; HUANG et al., 2015). De acordo com Porter (2004), há apenas três estratégias competitivas capazes de gerar vantagem competitiva no longo prazo, permitindo que uma empresa crie uma posição defensável e capaz de enfrentar as forças de mercado, são elas: liderança em custos, diferenciação e enfoque. Estas três estratégias competitivas são fundamentadas em duas fontes de vantagem competitiva: custo ou diferenciação. Neste sentido, percebe-se que, para uma organização obter um desempenho superior, ela precisa ajustar sua estrutura aos fatores contingenciais presentes no ambiente, sendo que a estratégia competitiva é o meio pelo qual a entidade pode minimizar a influência desses fatores sobre o desempenho, gerando e mantendo um elevado desempenho financeiro no longo prazo. Nos últimos anos, estudos vêm sendo desenvolvidos sobre o papel moderador da estratégia competitiva, avaliando sua influência sobre os impactos dos fatores contingenciais no desempenho organizacional (HUO et al., 2014; SANTOS, 2015; ACQUAAH; AGYAPONG, 2016; TENJIÄLÄ; LAAMANEM, 2018; CHEN et al., 2018). Contudo, a maioria desses estudos, limita-se a analisar a mediação da estratégia competitiva sobre fatores contingenciais internos, como: integração da cadeia de suprimentos (HUO et al., 2014), clima organizacional (SHIN, 2014), práticas de gestão operacional (JAYARAM; TAN; LAOSIRIHONGTHONG, 2014), capabilidades (SANTOS, 2015), sistema de controle gerencial (ACQUAAH; AGYAPONG, 2016), nível de produção (MOHSENZADEH; AHMADIAN, 2016) e sistema de remuneração (TENJIÄLÄ; LAAMANEM, 2018). Ou seja, pesquisas sobre moderação da estratégia competitiva acerca dos fatores contingenciais externos ainda são escassas. Segundo Huo et al. (2014) e Parnell, Long e Lester (2015), é preciso que pesquisas abordem diferentes ambientes, como países em desenvolvimento, uma vez que a maioria das evidências reportadas se limita aos ambientes de países desenvolvidos (JAYARAM; TAN; LAOSIRIHONGTHONG, 2014). Diante do exposto, e tendo o entendimento de que tantos os fatores contingenciais internos como os externos afetam o desempenho organizacional, bem como que a estratégia competitiva pode ser considerada um meio pelo qual se pode reduzir a influência desses fatores sobre o desempenho financeiro, dependendo do ambiente, país em que a empresa atua, 20 apresenta-se o seguinte problema de pesquisa: qual a relação da estratégia competitiva com os fatores contingenciais sobre o desempenho financeiro das empresas? 1.3 OBJETIVOS 1.3.1 Objetivo geral O objetivo geral desta pesquisa consiste em analisar a relação da estratégia competitiva (liderança em custos ou diferenciação) com os fatores contingenciais (incerteza do ambiente, nível de competição, tamanho e idade da organização) sobre o desempenho financeiro de empresas, considerando o ambiente em que estão localizadas (países membros do BRICS e do G7). 1.3.2 Objetivos específicos Para alcançar o objetivo geral, foram traçados os seguintes objetivos específicos: • avaliar a relação da estratégia competitiva com o desempenho financeiro das empresas, considerando o ambiente em que atuam; • verificar os fatores contingenciais que afetam o desempenho financeiro, de acordo com ambiente organizacional; • identificar a estratégia competitiva que torna o desempenho financeiro das empresas sustentável; e • investigar a relação entre as estratégias competitivas e os fatores contingenciais, de acordo com o ambiente organizacional. 1.4 JUSTIFICATIVA E CONTRIBUIÇÕES A relação entre a estratégia e o desempenho organizacional, bem como entre as escolhas estratégicas e o desempenho, ainda é foco de debate para a gestão estratégica, para contabilidade gerencial e para outras disciplinas e áreas de conhecimentos que buscam identificar os direcionadores do desempenho corporativo e respostas para explicar por que algumas empresas fracassam enquanto outras são bem-sucedidas (MAKADOK; BURTON; BARNEY, 2018). 21 Neste sentido, a importância deste estudo, centra-se no entendimento de que a estratégia competitiva é o meio pelo qual uma organização pode reduzir as influências dos fatores contingenciais sobre o seu desempenho financeiro, variando de acordo com o ambiente em que opera (país membro do BRICS e do G7). Em síntese, para que a organização obtenha melhores desempenhos, a estratégia competitiva adotada precisa reduzir os efeitos das contingências presentes no ambiente sobre suas atividades. 1.4.1 Originalidade Para justificar a originalidade desta pesquisa, realizou-se uma revisão sistemática da literatura nas bases de dados do portal de periódicos da Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) e na Base Digital de Teses e Dissertações (BDTD). A revisão da literatura abrangeu o período de 2014 a 2018, incluindo as seguintes bases de dados: BDTD, EBSCO, Emerald, JSTOR, Sage, SciELO, ScienceDirect, Spring, Taylor & Francis e Wiley. Realizou-se a busca por estudos (artigos, teses e dissertações) com base em seis palavras-chave, nas línguas inglesa e portuguesa: strategy (estratégia), competitive strategy (estratégia competitiva), contingency (contingência), theory of contingency (teoria da contingência), performance (desempenho) e financial performance (desempenho financeiro). Para refinar a busca, utilizaram-se os seguintes filtros: texto completo, publicado em revistas acadêmicas na área de administração, ciências contábeis e gestão, contendo um dos vocábulos no título e nas palavras-chave do resumo. Desconsideraram-se os trabalhos das demais áreas de conhecimento, por não fazerem parte das áreas supracitadas. Inicialmente, considerando-se apenas as palavras-chave strategy (estratégia) e competitive strategy (estratégia competitiva) no título dos trabalhos, a pesquisa nas bases de dados retornou 19.509 estudos com os termos citados. Após a inserção dos demais vocábulos como palavras-chave, selecionaram-se 2.534 estudos, sendo que, com a leitura dos resumos, restaram 212 trabalhos para leitura do texto completo, dos quais apenas 72 foram considerados relacionados com o tema da pesquisa. A partir da leitura dos textos identificados, pôde-se perceber que as pesquisas existentes abordam diversas relações entre a estratégia competitiva e o desempenho financeiro, sendo que tratam os fatores contingenciais de forma isolada, a exemplo de integração de cadeia de suprimentos x estratégia competitiva x desempenho (HUO et al., 2014); estratégia competitiva x práticas operacionais x desempenho (JAYARAM; TAN; LAOSIRIHONGTHONG, 2014); 22 sistemas de trabalho x estratégia competitiva x desempenho (SHIN, 2014); estrutura de governança x estratégia competitiva x tamanho (MERCHANT, 2014); estratégia competitiva x capacidades x incerteza (PARNELL; LONG; LESTER, 2015); estratégia competitiva x incerteza ambiental x estrutura organizacional (VARGAS; TREZ, 2017); sistema de controle gerencial x desempenho x estratégia competitiva (ORO, 2015; ACQUAAH; AGYAPONG, 2016); estratégia competitiva x inovação x desempenho (BAYAKTAR et al., 2017); e estratégia competitiva x sistemas de remuneração x desempenho (TENHIÄLÄ; LAAMANEN, 2018). Notou-se ainda que poucos são os estudos que atribuem à estratégia competitiva o papel de variável contingencial moderadora entre os demais fatores contingenciais e o desempenho organizacional (HUO et al., 2014; JAYARAM; TAN; LAOSIRIHONGTHONG, 2014; SHIN, 2014; SANTOS, 2015; ACQUAAH; AGYAPONG, 2016; HERNÁNDEZ-PERLINES; MORENO-GARCÍA; YAÑEZ-ARAQUE, 2016; MOHSENZADEH; AHMADIAN, 2016; TENJIÄLÄ; LAAMANEM, 2018; CHEN et al., 2018). Sendo esta uma lacuna a ser explorada, justificando a originalidade e relevância deste estudo, uma vez que, pela revisão da literatura, verificou-se que não há pesquisas abordando o papel da estratégia competitiva na mitigação (redução) das influências de fatores contingenciais sobre o desempenho financeiro, nem mesmo comparando tal aspecto por ambiente de atuação (países desenvolvidos e em desenvolvimento). Tais resultados demonstram a importância da presente pesquisa, evidenciando a necessidade de analisar se a estratégia competitiva pode reduzir os efeitos dos fatores contingenciais sobre o desempenho financeiro organizacional em diferentes ambientes. Devido às características de cada ambiente, os fatores contingenciais que afetam o desempenho das empresas podem diferir entre regiões e países, cabendo às entidades identificarem a estratégia competitiva capaz de reduzir os efeitos dessas variáveis sobre o desempenho financeiro, tornando-as mais lucrativas (CHANHALL, 2003; PORTER, 2004; HUO et al., 2014; OTLEY, 2016; ANWAR; HASNU, 2017). De acordo com Parnell, Long e Lester (2015), evidências sobre as relações entre estratégia competitiva, fatores contingenciais (incerteza) e desempenho organizacional em diferentes países ainda são incipientes, sendo necessário o desenvolvimento de pesquisas que abranjam economias emergentes e desenvolvidas fornecendo achados adicionais. Neste sentido, a presente tese contribuirá com evidências associadas ao aspecto moderador que a estratégia competitiva pode causar nos efeitos dos fatores contingenciais (internos e externos) sobre o desempenho financeiro de empresas localizadas em diferentes ambientes. 23 1.4.2 Contribuições e implicações práticas As práticas e as pesquisas em contabilidade gerencial têm passado por mudanças nas últimas décadas, incorporando novas perspectivas no sentido de torná-las mais estratégicas (ITTNER; LARCKER, 2001; TRIPATHY, 2006), gerando mudanças nos papéis dos contadores gerenciais, permitindo-lhes maior participação e envolvimento no processo de decisão, planejamento e sistemas de mensuração de desempenho, bem como na formulação e implementação da estratégia organizacional (INSTITUTE OF MANAGEMENT ACCOUNTANTS, 2019). Neste sentido, esta pesquisa, ao analisar a relação da estratégia competitiva com os fatores contingenciais sobre o desempenho financeiro, visa contribuir com a literatura em contabilidade gerencial, demonstrando para contadores, administradores, gestores e controllers como a escolha e implementação da estratégia podem moderar os efeitos dos fatores contingenciais no desempenho organizacional. Ademais, tais informações podem ser úteis na formulação e escolhas de métricas para avaliação de desempenho, possibilitando alinhar os indicadores de desempenho com a estratégia organizacional, com base no ambiente onde as organizações operam. As evidências também podem ser relevantes para que gestores e diretores avaliem, de acordo com o ambiente (BRICS e G7), qual estratégia competitiva pode gerar vantagens competitivas, identificando os direcionadores capazes de sustentar tais fontes de vantagens no longo prazo, bem como o desempenho organizacional. Desta forma, cabe aos controllers e aos contadores gerenciais, por exemplo, um papel mais ativo na formulação e implementação da estratégia organizacional para terem maior entendimento do ambiente e das atividades corporativas, tanto para identificar os fatores contingenciais que podem afetar a estratégia competitiva e o desempenho organizacional, como para apoiar as decisões de diretores e gestores com informações necessárias para que se façam as melhores escolhas, com base nos objetivos organizacionais. 1.5 TESE O presente estudo propõe como tese que a estratégia competitiva pode reduzir os efeitos dos fatores contingenciais (incerteza do ambiente, nível de competição, tamanho e idade da organização) sobre o desempenho financeiro das empresas, variando de acordo com o ambiente onde as empresas operam (países membros do BRICS e do G7). 24 1.6 DELIMITAÇÃO DO TEMA Diversas abordagens teóricas foram desenvolvidas sobre estratégia, como as escolas do design, de planejamento, de posicionamento, ambiental, entre outras, cada uma apresentando um ponto de vista acerca da formulação, implementação e execução da estratégia organizacional. Mintzberg, Ahlstrand e Lampel (2010) classificaram as correntes sobre estratégia em 10 escolas, a saber: escola do design, escola de planejamento, escola de posicionamento, escola empreendedora, escola cognitiva, escola de aprendizado, escola de poder, escola cultural, escola ambiental e escola de configuração. Não foi objetivo desta pesquisa analisar a relação da estratégia com os fatores contingenciais sobre o desempenho financeiro de empresas de acordo com as perspectivas de cada uma das escolas de pensamento estratégico individualmente. Mas este estudo limitou-se a analisar a relação da estratégia competitiva com os fatores contingenciais sobre o desempenho financeiro na ótica da escola do posicionamento, considerando especificamente as estratégias genéricas propostas por Porter (1980, 2004), liderança em custos e diferenciação, não levando em consideração a estratégia baseada no foco adotado pela empresa, uma vez que o modelo utilizado para identificar as estratégias competitivas não contempla informações acerca da estratégia baseada no foco, dificultando sua utilização. Além disso, para observar se uma empresa utiliza uma estratégia baseada no foco, fazse necessário, por exemplo, a obtenção de informações mais detalhadas acerca de atividades voltadas para clientes ou segmentos específicos. Contudo, devido ao estudo ter sido desenvolvido a partir de dados divulgados pelas companhias, nem sempre tais informações são publicadas, não permitindo identificar se a organização utiliza ou não uma estratégia baseada no foco. Ressalta-se, também, que diversos fatores contingenciais podem afetar a estratégia competitiva de uma organização, visto que a estratégia organizacional pode ser ajustada para se adaptar às condições do ambiente em que a corporação atua (CHENHALL, 2003; OTLEY, 2016). Entretanto, para fins deste estudo, consideraram-se apenas os seguintes fatores contingenciais: incerteza do ambiente, nível de competição, tamanho (ou porte corporativo) e idade da organização. Com relação ao desempenho financeiro, destaca-se que não se pretendeu avaliar o desempenho das empresas com base na eficiência ou eficácia da estratégia genérica adotada, mas tão somente mensurou-se o resultado financeiro (LEBAS, 1995) e, posteriormente, 25 verificou-se qual(is) fator(es) contingencial(ais) afeta(m) tal desempenho, considerando-se o ambiente em que a organização atua. 1.7 ESTRUTURA DA TESE A presente tese encontra-se estruturada em cinco capítulos. No primeiro capítulo, encontra-se a introdução. O segundo capítulo apresenta o referencial teórico, que aborda a teoria da contingência, as estratégias competitivas, a mensuração de desempenho, as evidências empíricas e o modelo conceitual da pesquisa. No terceiro capítulo, apresentam-se os procedimentos metodológicos adotados para o alcance dos objetivos estabelecidos. O quarto capítulo apresenta as análises dos resultados da pesquisa. E, por fim, no quinto capítulo, apresentam-se as considerações finais da pesquisa, destacando-se as conclusões da tese, as limitações e recomendações para futuras pesquisas. 26 2 REFERENCIAL TEÓRICO Neste capítulo, apresenta-se o referencial teórico que serviu de base para o desenvolvimento da presente pesquisa, sendo abordadas: as origens, premissas e características da teoria da contingência, bem como os fatores contingenciais; as principais estratégias competitivas; a avaliação e mensuração de desempenho, destacando-se os conceitos, objetivos e as medidas utilizadas na avaliação de desempenho; e, por fim, as evidências empíricas. 2.1 TEORIA DA CONTINGÊNCIA A origem da teoria da contingência se deu a partir dos estudos sobre estrutura organizacional desenvolvidos no início da década de 1960, quando apareceram as primeiras críticas à abordagem clássica da administração, surgindo como uma alternativa a esta corrente teórica (DONALDSON, 1999, 2015; ESPEJO, 2008; HAMANN, 2017). O modelo clássico da administração defende que uma estrutura organizacional seja ideal para todas as organizações de todos os tipos (DONALDSON, 1999). Opostos a essa visão, os estudos da escola de relações humanas preconizavam que o sucesso das organizações dependia diretamente das relações desenvolvidas e criadas com os empregados; e os da abordagem sistêmica postulavam que uma entidade está inserida em um sistema mais amplo, sendo moldada pelo ambiente em que atua, ou seja, são sistemas abertos que sofrem influência de diversos fatores (DONALDSON, 1999, 2015; ESPEJO, 2008; HAMANN, 2017). A partir desse embate, a teoria da contingência surgiu defendendo que não há uma estrutura organizacional que seja ótima para todas as organizações, tendo como pressuposto básico que a estrutura deve se ajustar ao contexto para que a organização tenha um bom desempenho (DRAZIN; VAN DE VEN, 1985; DONALDSON, 1999; HAMANN, 2017). Com base neste pressuposto, três elementos compõem o núcleo central da teoria da contingência: (a) deve haver uma associação entre o fator contingencial e o sistema organizacional; (b) uma mudança na contingência acarreta uma mudança na estrutura corporativa; e (c) um ajuste entre a variável contingencial e o sistema organizacional melhora o desempenho da corporação (DONALDSON, 1999; HAMANN, 2017). Nota-se que, de acordo com a teoria da contingência, uma organização só conseguirá obter um melhor desempenho quando adequar sua estrutura, sistemas e outros fatores com os fatores contingenciais presentes no seu ambiente de atuação. 27 O desenvolvimento da teoria da contingência se deu a partir dos trabalhos desenvolvidos por Burns e Stalker (1961), Chandler Jr. (1962), Woodward (1965), Lawrence e Lorch (1967), Perrow (1967), Thompson (1967), Greiner (1972) e Khandwalla (1972). Dentre os estudos mais recentes que abordam aspectos associados à teoria da contingência, destacam-se os desenvolvidos por Child (1972), Donaldson (1987), Hill, Hitt e Hoskisson (1992), Jennings e Seaman (1994), Sousa e Voss (2008), Verwall, Commandeur e Verbeke (2009), Qiu e Donaldson (2010) e O'Brien e Sasson (2017). Burns e Stalker (1961) foram os primeiros autores a estudarem a relação entre contingência e estrutura organizacional. Ao investigarem o impacto da tecnologia sobre a estrutura organizacional de firmas inglesas e escocesas, observaram que o ambiente (estável e dinâmico) influenciava a estrutura organizacional (mecanicista e orgânica), sendo que, quando uma entidade atua em um ambiente estável, a estrutura mecanicista é mais adequada; porém, se a corporação enfrenta um alto índice de mudança tecnológica e de mercado, uma estrutura orgânica passa a ser desejável. Chandler Jr. (1962) examinou como as escolhas estratégicas de uma entidade afetam a estrutura organizacional. Os resultados demonstraram que a estratégia determina a estrutura, indicando que alterações na estrutura corporativa são provocadas por mudanças na estratégia que estão associadas com mudanças no ambiente onde a corporação atua. Por sua vez, Woodward (1965) analisou empresas inglesas e identificou que as variáveis estruturais estavam relacionadas diretamente com a natureza tecnológica das firmas. Segundo a autora, uma adequação entre a estrutura organizacional e a tecnologia adotada gera um melhor desempenho para as entidades. Os primeiros autores a utilizarem o termo teoria da contingência foram Lawrence e Lorch (1967). Em um estudo realizado em diferentes indústrias, eles encontraram evidências de que: o nível de mudança ambiental influencia a diferenciação e a integração da organização; diferentes mercados e ambientes tecnológicos demandam diferentes estruturas corporativas; as empresas de sucesso em cada setor estudado utilizavam distintas formas de integração, consistentes com o ambiente de atuação das mesmas; e as firmas que obtinham melhores resultados apresentavam estruturas ajustadas (adequadas) ao ambiente. Perrow (1967) identificou que a tecnologia era contingente à estrutura corporativa, estando relacionada com todos os processos organizacionais. Além disso, segundo o autor, quanto mais explícito fosse o conhecimento utilizado nas atividades e quanto menos exceções fossem encontradas, mais o processo de tomada de decisão seria centralizado. 28 Thompson (1967) agrupou as organizações em sistemas fechados e abertos, realizando observações à medida que havia uma mudança de ambiente. Após esse procedimento, verificou que as entidades buscam proteger tecnologias de produção em um sistema fechado, tornandoas eficientes, procurando protegê-las dos concorrentes. O autor ainda ressalta que o ambiente influencia diretamente a estrutura organizacional, levando à especialização de diferentes partes da estrutura para atender às demandas do ambiente onde a corporação opera. Greiner (1972) observou que a idade e o porte afetam a estrutura organizacional, evidenciando que, ao mesmo tempo em que as companhias crescem, desenvolvem-se ao longo de cinco fases ou ciclo de vida organizacional. Esse resultado indica que a estrutura organizacional é influenciada pelo tamanho e pelo ciclo de vida de cada empresa, levando-as a efetuar ajustes em suas estruturas ao perpassarem o ciclo de vida organizacional. Por sua vez, Khandwalla (1972) constatou que o desempenho corporativo não estava associado a uma característica particular, mas estaria correlacionado a diversos fatores contingenciais que afetam conjuntamente a estrutura organizacional. Com base nos estudos apresentados, pode-se afirmar que a teoria da contingência se desenvolveu dentro de um contexto em que a influência da incerteza ambiental sobre a entidade gera a busca pela adaptação de sua estrutura organizacional à(s) contingência(s) existente(s), possuindo como premissa básica que uma mudança na variável contingencial leva a uma mudança na estrutura organizacional. Contudo, o pressuposto básico da teoria da contingência, de que uma alteração em uma contingência provoca uma mudança na estrutura organizacional, foi criticado devido à falta de relação direta entre algumas contingências e a estrutura corporativa (DONALDSON, 1987). Além disso, a relação direta ou o determinismo da abordagem contingencial (uma mudança na contingência provoca uma mudança na estrutura) é passível de críticas, pois é um tipo de construção teórica simplista, oriundo de associações entre as variáveis contingenciais e o sistema organizacional (CHILD, 1972). Child (1972) foi o primeiro a rejeitar a hipótese do determinismo contingencial. Para o autor, havia outros fatores que influenciavam a mudança na estrutura organizacional, dentre os quais a escolha estratégica. Segundo Child (1972), a mudança na estrutura organizacional sofreria influência política de atores organizacionais, os quais, a partir de escolhas estratégicas, buscariam ajustar a estrutura de acordo com suas escolhas. Tal pressuposto ficou conhecido como o postulado das escolhas estratégicas ou do funcionalismo estrutural. Para o modelo da teoria da contingência postulado pelo funcionalismo estrutural, diferentemente da visão do determinismo contingencial, a influência das contingências sobre a 29 estrutura é um conjunto de processos mais demorado, podendo ser descrito da seguinte forma: uma mudança nas variáveis contingenciais leva ao desequilíbrio organizacional, produzindo um declínio no desempenho que gera pressão por mudanças, provocando uma adaptação estrutural na organização restaurando a eficácia ou desempenho (DONALDSON, 1987; MCKINLEY; MONE, 2003). Percebe-se que, pela abordagem do funcionalismo estrutural, os efeitos provocados pelas contingências na corporação não ocorrem de forma instantânea, mas podem ocorrer por etapas ou processos, gerando desajustes na estrutura corporativa, que, por sua vez, impactam o desempenho, gerando mudanças e adequações para que se reestabeleça o equilíbrio organizacional e o desempenho desejável. A partir do pressuposto do funcionalismo estrutural e da integração de diversos estudos sobre contingências, uma vez que não havia uma teoria geral da contingência, Donaldson (1987) apresentou uma nova abordagem para a teoria da contingência conhecida como nova teoria da contingência (MCKINLEY; MONE, 2003; JUNQUEIRA et al., 2016). Para a nova teoria da contingência, um sistema organizacional (estrutura, sistema de planejamento ou controle etc.) deve estar adequado ao seu contexto para ser eficiente, sendo que uma organização que está ajustada produz um desempenho superior à que se encontra em desajuste com as contingências (HAMMAN, 2017). O modelo proposto por Donaldson (1987) engloba as três premissas comuns existentes nas diferentes abordagens contingenciais: (a) há uma relação entre um fator contingencial e a estrutura organizacional; (b) uma alteração em uma variável contingencial acarreta mudanças no sistema organizacional; e (c) um ajuste (fit) entre o(s) fator(es) contingencial(ais) e a estrutura organizacional afeta(m) positivamente o desempenho corporativo (DONALDSON, 1987, 1999; JUNQUEIRA, 2010; HAMMAN, 2017). A partir das referidas premissas, Donaldson (1987, 1999) apresentou um modelo geral, baseado em uma análise conjunta das variáveis contingenciais e de seus efeitos na estrutura e no desempenho organizacional (JUNQUEIRA et al., 2016; HAMMAN, 2017), conforme pode ser visualizado na Figura 1, a seguir. Este modelo ficou conhecido como teoria da adaptação estrutural para readquirir adaptação (Structural Adjustment to Regain Fit (SARFIT)). 30 Figura 1 Modelo SARFIT Fonte: Junqueira et al. (2016). A teoria da adaptação estrutural para readquirir adaptação (SARFIT), conforme apresentado na Figura 1, pressupõe que uma entidade inicialmente apresenta uma estrutura organizacional ajustada (fit) aos fatores contingenciais, refletindo positivamente no desempenho corporativo. Todavia, ao ocorrer alterações nas contingências, sem mudanças na estrutura, a corporação entra em desajuste (misfit) com o nível de variáveis contingenciais existentes. Com isso, há uma redução no desempenho, que, gerando insatisfação para os proprietários, leva a firma a uma mudança para se adaptar, com a finalidade de ajustar uma nova estrutura ou sistema organizacional aos novos níveis contingenciais, recuperando o nível de desempenho desejado (DONALDSON, 1987, 1999; JUNQUEIRA et al., 2016). Para validar o SARFIT, Donaldson (1987) testou dados de pesquisas de estratégia e estrutura realizadas em cinco países (Alemanha, Estados Unidos da América (EUA), França, Japão e Reino Unido). Os resultados indicaram que as empresas mudam da adequação para inadequação devido ao aumento no nível de contingência; e a inadequação leva a uma mudança organizacional na busca de adequação estrutural, ou seja, as empresas realizam uma mudança adaptativa para ajustar sua estrutura à contingência ou contingências existentes, confirmando cada passo do SARFIT (DONALDSON, 1987, 1999). Neste sentido, com base no modelo SARFIT, Donaldson (1987, 1999) observou que alterações nos fatores (variáveis) contingenciais podem provocar mudanças na estrutura corporativa, levando as organizações a adaptarem suas estruturas às contingências existentes no ambiente, buscando manter o nível de desempenho desejado, ou manter o nível de eficiência anterior as mudanças provocadas pelos fatores contingenciais. Em estudos mais recentes, baseados na teoria da contingência, novas evidências foram encontradas. Hill, Hitt e Hoskisson (1992), ao examinarem se fatores organizacionais 31 influenciam a relação entre a estratégia de diversificação e o desempenho econômico, encontraram evidências de que a melhor adequação entre estratégia, estrutura e sistemas de controle produz melhores desempenhos econômicos. Além disso, as empresas que obtiveram ganhos de escopo apresentavam melhores desempenhos se os arranjos organizacionais possibilitassem a cooperação entre as unidades corporativas, enquanto as firmas que buscavam obter ganhos com os controles internos eficientes demonstravam melhor desempenho se estimulassem a competição entre as unidades. Tal resultado demonstra que o desempenho não é apenas contingente à estratégia de diversificação, mas do grau de ajuste entre a estratégia e os arranjos organizacionais internos. Ou seja, para que otimizem o desempenho, as entidades precisam ajustar sua estrutura à estratégia organizacional, de forma que os fatores internos cooperem com a realização das prioridades estratégicas. Jennings e Seaman (1994) examinaram os níveis de adaptação organizacional de acordo com as mudanças ocorridas no ambiente e suas relações com a estratégia, estrutura e desempenho organizacional. As evidências sugerem que entidades com níveis específicos de adaptação tendem a ter arranjos específicos de estrutura e estratégia que geram certos resultados no desempenho, enquanto as empresas que apresentaram uma melhor combinação entre estratégia e estrutura apresentaram os melhores desempenhos. Observa-se que o alinhamento da estratégia e da estrutura corporativa ao ambiente e as mudanças ocorridas no ambiente podem gerar resultados positivos no desempenho, porém cabe às entidades identificar o melhor ajustamento entre estratégia, estrutura e ambiente para obterem os melhores resultados possíveis. Sousa e Voss (2008) apresentaram um exame crítico acerca da aplicação universal de tais práticas pelas organizações e seus efeitos sobre o desempenho. Segundo os autores, a aplicabilidade de novas práticas de gestão e seus benefícios para as entidades estão condicionados a fatores contingentes existentes no contexto em que as empresas operam, sendo necessário investigar seu impacto sobre o desempenho operacional e que variáveis podem explicar o sucesso na implantação dessas ferramentas de gestão operacional. Percebe-se que, de acordo com a teoria da contingência, a adoção de novas práticas de gestão organizacional que se mostraram eficazes em determinado contexto talvez não seja efetiva em outro ambiente, devendo-se realizar ajustes nas ferramentas de gestão com base nas contingências presentes no ambiente organizacional, para que sejam eficazes. Verwall, Commandeur e Verbeke (2009), ao analisarem a criação de valor em estratégias de terceirização, encontraram evidências de que, nas decisões de terceirização, o 32 valor dos recursos e os custos de transação são considerados simultaneamente e que o processo de terceirização é dependente do alinhamento entre as características dos recursos e das transações. Tal resultado demonstra a aplicação dos pressupostos da teoria da contingência em processos de terceirização, em que, mais especificamente, as características dos recursos moldam as transações realizadas entre as empresas. Qiu e Donaldson (2010), em uma visão mais ampla, propõem um novo modelo para explicar o ajuste entre estratégia e estrutura para empresas multinacionais, a partir da teoria da contingência: o modelo contingencial tridimensional ou cúbico (the cubic contingency model). Segundo Qiu e Donaldson (2010), o primeiro modelo contingencial sobre a relação entre estratégia e estrutura em companhias multinacionais foi desenvolvido por Stopford e Wells Jr. (1972). Para Stopford e Wells Jr. (1972), o ajuste entre estratégia e estrutura dar-se-ia de três modos: (a) pela adaptação entre a estrutura de divisões internacionais com o baixo nível de diversificação geográfica e de produtos estrangeiros; (b) pelo ajuste entre a estrutura de divisão de produtos com a alta diversificação de produtos estrangeiros e com baixa diversificação geográfica; e (c) pelo ajustamento entre a estrutura de divisões geográficas com a baixa diversificação de produtos estrangeiros e com a alta diversificação de áreas. Nota-se que, de acordo como Stopford e Wells Jr. (1972), o ajuste entre estratégia e estrutura em nível de empresas multinacionais seria alcançado pela adequação entre as estruturas de divisão internacional com o nível de diversificação de produtos e com o grau de diversificação geográfica pretendido pela organização. De acordo com Qiu e Donaldson (2010), tal matriz de diversificação apresenta limitações, pois não considera o nível de integração entre as filiais estrangeiras e o controle geral realizada pela matriz, nem a responsabilidade de cada subsidiária. Desta forma, os autores propuseram o modelo contingencial tridimensional, que integra os conceitos de integração global, responsabilidade local e diversificação de produtos estrangeiros. Segundo Qiu e Donaldson (2010), o modelo contingencial tridimensional fornece um conjunto amplo de escolhas estratégicas, possibilitando que as empresas multinacionais possam realizar ajustes entre a estratégia e a estrutura com base na integração entre as matrizes locais e as filiais estrangeiras, no nível de responsabilidade delegada aos gestores das filiais internacionais e com base no grau de diversificação de produtos em nível global. Assim, observa-se que o modelo proposto por Qiu e Donaldson (2010) abrange uma gama mais ampla de opções de ajustes entre a estratégia e a estrutura de empresas multinacionais, cabendo-lhes adequá-las de acordo com as contingências existentes, buscando obter os melhores resultados possíveis. 33 Em outra perspectiva, O'Brien e Sasson (2017) propõem uma teoria contingencial para pequenos empreendedores que buscam captar recursos em bancos. Tal teoria é baseada no alinhamento entre as características fundamentais para a captação de recursos (especificidade de ativos, incerteza e frequência das transações) com a natureza da relação entre a organização e a instituição bancária (direta, isto é, puramente comercial; e parceria, relação mais estreita). Com base em estudo realizado com pequenos empreendedores, os autores encontraram evidências de que o alinhamento entre a natureza da relação entre a empresa e o banco e as características da negociação dependem da especificidade dos ativos, da incerteza e da frequência das operações, variando de acordo com as práticas de governança. Desta forma, observa-se que, até em transações para captação de recursos, as organizações precisam adequar os mecanismos de gestão de acordo com as características e natureza das operações, buscando reduzir o nível de incerteza associada a cada transação, com base na natureza e características da transação. Em síntese, constata-se que os fundamentos da teoria da contingência podem ser aplicados não somente para avaliar o ajuste entre a estrutura e a estratégia organizacional, mas poderão ser utilizados para identificar a adequação de práticas de gestão, sistemas de informação, terceirização de serviços, diversificação de produtos, estrutura da cadeia de suprimentos, políticas de captação de recursos, entre outros, bem como a implicação dessas práticas e políticas sobre o desempenho organizacional, permitindo a realização de ajustes quando necessários. 2.1.1 Fatores contingenciais Para manter o nível de desempenho desejado, uma entidade precisa ajustar sua estrutura à(s) contingência(s) existente(s), identificando as variáveis contingenciais que afetam a sua estrutura e, por conseguinte, o desempenho corporativo (eficiência, eficácia, lucratividade e outros). Os fatores contingenciais podem ser definidos como quaisquer variáveis relacionadas ao ambiente ou ao contexto empresarial (fatores externos), ou interno à entidade (fatores internos), que moldam a estrutura organizacional a qualquer momento, ou seja, influenciam na maneira como uma corporação desenvolve e estrutura suas atividades impactando o desempenho organizacional (CHENHALL, 2003; JUNQUEIRA et al., 2016; OTLEY, 2016). 34 As variáveis ou fatores contingenciais podem ser elencados em fatores contingenciais externos e fatores contingenciais internos, conforme apresentado no Quadro 1 (MCKINLEY; MONE, 2003; CHENHALL, 2003; GANESCU, 2012; BURKERT et al., 2016; OTLEY, 2016). Quadro 1 Principais fatores contingenciais de Otley (2016) Descrição Itens Fatores Contingenciais Externos Tecnologia Competição ou hostilidade de mercado Incerteza ambiental Cultura nacional Fatores Contingenciais Internos Tamanho Estrutura Estratégia Sistema de compensação Sistema de informação Ciclo de vida dos produtos ou organizacional Variáveis psicológicas Participação de empregados em sistemas de controle Posição no mercado Mudanças nos sistemas Fonte: adaptado de Otley (2016). A seguir, apresentam-se os conceitos e as características de alguns dos principais fatores contingenciais externos (tecnologia, competição ou hostilidade de mercado, incerteza ambiental e cultura nacional) e internos (tamanho, estrutura, estratégia, sistema de compensação, sistema de informação e outros). 2.1.1.1 Fatores contingenciais externos Os principais fatores contingenciais externos englobam a tecnologia, o nível de competitividade ou hostilidade do mercado, a incerteza ambiental e a cultura nacional (MCKINLEY; MONE, 2003; CHENHALL, 2003; JUNQUEIRA, 2010; BURKERT et al., 2016; JUNQUEIRA et al., 2016; OTLEY, 2016). Nota-se que os principais fatores contingenciais externos, propostos por Otley (2016), englobam características do ambiente que interferem nos arranjos corporativos, demandando da organização a implementação de planos e medidas capazes de mitigar seus efeitos sobre a estrutura organizacional. A tecnologia representa como os processos operacionais são elaborados (conduzidos) pela organização (a forma como se transforma os inputs em outputs) e inclui bens tangíveis (máquinas, ferramentas, hardwares etc.), materiais, processos, software e conhecimento (CHENHALL, 2003). Existem três tipos genéricos de tecnologia que são identificados na 35 literatura organizacional: complexidade, incerteza das tarefas e interdependência (WOODWARD, 1965; CHENHALL, 2003; DONALDSON, 1999). Evidências indicam que o tipo de tecnologia utilizado pode influenciar a estrutura organizacional, o sistema de controle gerencial, o sistema de informação corporativo e contábil, entre outros (XIAO; DYSON, POWELL, 1996; CHENHALL, 2003). Por sua vez, a competição ou hostilidade de mercado representa um aspecto do ambiente externo, sendo caracterizada pela intensa competição entre as empresas no que se refere à qualidade, variedade, promoção e distribuição, entre outros aspectos na produção e comercialização de produtos (KHANDWALLA, 1972; OTLEY, 2016). Diferentes níveis de competitividade (hostilidade) afetam o preço, o marketing e a distribuição de bens e serviços no mercado, influenciando a estrutura e o desempenho organizacional (KHANDALLA, 1972; CHENHALL, 2003). A incerteza pode ser agrupada, de acordo com o ambiente, em incerteza ambiental interna e externa. Na incerteza ambiental interna, consideram-se todas as forças que atuam na própria entidade, como: objetivos e metas da empresa, natureza dos produtos e serviços da organização, processos e redes de comunicação e histórico educacional dos empregados. A incerteza ambiental externa refere-se à taxa de mudança ou variabilidade do contexto (ambiente) externo à organização, compreendendo principalmente clientes, concorrentes, regulamentações governamentais e sindicatos (TUNG, 1979; CHENHALL, 2003; HABIB; HOSSAIN; JIANG, 2011). A literatura tem demonstrado que a incerteza afeta a estrutura, os sistemas organizacionais (comunicação, informação, controle, gerencial etc.), moldando as características das entidades (TUNG, 1979; CHENHALL, 2003; OTLEY, 2016). Assim, verifica-se que a incerteza é uma variável contingencial que interfere no modo com que as organizações moldam sua estrutura, sistemas e processos, sendo necessários ajustes para mitigar possíveis influências sobre as atividades corporativas e seu desempenho. Por sua vez, a cultura nacional é composta por tradições padronizadas e interrelacionadas, sendo transmitidas ao longo do tempo e do espaço por dispositivos fundamentados nas capacidades simbolizantes linguísticas e não linguísticas desenvolvidas exclusivamente pelo homem (CHENHALL, 2003). De acordo com Chenhall (2003), a cultura refere-se a características inerentes a uma nação, tais como: conhecimento, crença, arte, moral, leis, costumes e hábitos adquiridos pelos membros de uma sociedade. Há indícios de que a cultura nacional afeta as práticas gerenciais, o processo orçamentário entre empresas localizadas em diversos países, a participação no orçamento e na 36 avaliação de desempenho praticado na matriz e em suas subsidiárias, o sistema de controle gerencial e o sistema de remuneração (HOFSTEDE, 1983; HARRISON, 1993; SNODGRASS; GRANT, 1986; CHENHALL, 2003; FILATOTCHEV; ALLCOCK, 2010; OTLEY, 2016). A seguir, apresentam-se os principais fatores contingenciais internos, ou seja, os fatores contingenciais que estão ligados ao ambiente interno das organizações, destacando-se os conceitos e as principais características de cada fator. 2.1.1.2 Fatores contingenciais internos Os fatores contingenciais internos incluem o tamanho ou porte corporativo, estrutura, estratégia, sistemas de remuneração ou compensação, sistemas de informação, estágios do ciclo de vida do produto ou da firma, entre outros (HOFER, 1975; MCKINLEY; MONE, 2003; CHENHALL, 2003; BURKERT et al., 2016; JUNQUEIRA et al., 2016; OTLEY, 2016). O crescimento do tamanho corporativo representa uma possibilidade para as entidades melhorarem sua eficiência, aproveitarem as oportunidades de mercado e de especialização de tarefas e divisão de tarefas. Empresas de grande porte têm mais poder para controlar suas operações e podem empregar métodos de produção em larga escala, reduzindo a incerteza das tarefas (CHENHALL, 2003). Em relação à estrutura organizacional, a literatura apresenta diversas definições, distinguindo-a em estrutura de resultados e mecanismos estruturais. De um modo geral, a estrutura organizacional pode ser definida como o modelo pelo qual uma entidade estabelece as funções dos gerentes nas subunidades (diferenciação) e como estas subunidades atuarão de acordo com os objetivos corporativos (integração) (LAWRENCE; LORSCH, 1967; CHENHALL, 2003). O tipo de estrutura pode influenciar a eficiência do trabalho, a motivação dos indivíduos, o fluxo de informações e o sistema de controle, e podem moldar o futuro da entidade (CHENHALL, 2003). O sistema de remuneração compreende um conjunto de compensações, incentivos e demais ordenados oferecidos pela firma aos seus executivos, composto por proventos financeiros e não financeiros, por meio de benefícios fixos e variáveis, com o objetivo de motivá-los a atingir os objetivos corporativos (BALKIN; GOMEZ-MEJIA, 1987). Entende-se que o sistema de remuneração pode ser utilizado como um mecanismo de incentivos para alinhar as ações dos gestores e empregados aos objetivos organizacionais, devendo as entidades identificarem a política de remuneração que gere os melhores resultados possíveis. 37 Estudos indicam que o sistema de compensação exerce um papel importante no recrutamento, motivação e retenção de empregados e executivos, podendo ser utilizado como um meio estratégico para a obtenção de gerentes habilidosos e de capital humano necessário para o desenvolvimento organizacional, sendo influenciado por variáveis contingenciais, como, por exemplo: cultura nacional, incerteza e dinâmica organizacional (BOYD; SALAMIN, 2001; COMBS; SKILL, 2003; FILATOTCHEV; ALLCOCK, 2010). No que se refere aos estágios de ciclo de vida organizacional ou dos produtos, a literatura aponta que as entidades ou os seus produtos possuem estágios de desenvolvimento, que vão desde a existência ao desaparecimento (falência), sem, contudo, evoluir de fase para fase, mas podendo sair do estágio de nascimento para o de maturidade, por exemplo. Além disso, não seguem uma ordem cronológica, ou seja, uma firma no estágio de nascimento pode mudar para o estágio de declínio sem passar pelos demais estágios (MILLER; FRIESEN, 1984; DICKINSON, 2011; COSTA et al., 2017). Já o sistema de informação contempla um conjunto de informações acerca das atividades corporativas que permitem às entidades atingirem seus objetivos. Por sua vez, um sistema de informação gerencial é um sistema que permite que as informações internas e externas sejam reportadas no nível operacional, tático e estratégico, contemplando dimensões como qualidade, flexibilidade e inovação, bem com finanças, dentro de um modelo apropriado de controle (BRIGNALL, 1997; OTLEY, 1980). A definição de estratégia pode ser vista em diversas disciplinas, sendo conceituada como a determinação de metas e objetivos básicos de longo prazo de uma organização a serem realizados pela adoção de ações e alocação de recursos necessários para concretizá-los (MINTZBERG, 1978). Para Mintzberg (1978), a estratégia pode ser conceituada por cinco ângulos: como um plano de ação para o futuro; como um padrão de comportamento da organização ao longo do tempo; como uma manobra utilizada para ludibriar um concorrente; como uma posição, a forma como a entidade se posiciona no mercado; e como uma perspectiva, o modo como uma empresa olha para si (exerce suas atividades) e para o ambiente (visão de mercado). A seguir, apresentam-se as principais estratégias competitivas, discutindo-se as características de cada tipologia, por ser considerado um fator contingente mediador entre as contingências externas e as demais variáveis contingenciais internas com o desempenho organizacional. 38 2.2 ESTRATÉGIAS COMPETITIVAS A estratégia competitiva, também denominada de estratégia genérica competitiva ou estratégia principal, entre outras denominações, envolve o estabelecimento de um posicionamento estratégico geral que pode ser adotado por qualquer organização, independentemente do ramo de negócio (HERBERT; DERESKY, 1987; CHAVES; BENEDETE; POLO, 2009). A estratégia competitiva também pode ser definida como escolhas fundamentais, pretendidas ou realizadas, concebida entre padrões alternativos de decisões e ações, cuja aplicação tem efeito substancial no funcionamento e desempenho de uma organização (RUGMAN; VERBEKE, 1994). Assim, entende-se que a estratégia competitiva ou genérica é um padrão de comportamento adotado por uma organização para se posicionar no setor, com base na combinação de ações que visam alinhar atividades e processos organizacionais para alcançar o desempenho desejado. Uma estratégia competitiva pode ser definida por inúmeros caminhos ou visões, apresentando pontos comuns que a caracterizam como um conjunto de diretrizes que buscam determinar uma decisão no futuro, ou o meio que uma organização escolheu para atingir seus objetivos, metas e resultados (MINTZBERG, 1978; RUGMAN; VERBEKE, 1994; CHAVES; BENEDETE; POLO, 2009; CAPALONGA; DIEHL; ZANINI, 2014). A literatura apresenta algumas tipologias de estratégias competitivas, dentre as quais: Miles et al. (1978); Porter (1980); Gupta e Govidarajan (1984); Robinsin Jr. e Pearce II (1988); Treacy e Wiserma (1993); Bowman e Faulkner (1997) e Udayasankar e Das (2004). 2.2.1 Tipologia estratégica de Miles et al. (1978) O modelo de estratégias competitivas proposto por Miles et al. (1978) foi concebido com base no pressuposto de que as organizações precisam se adaptar (mudança adaptativa) constantemente às incertezas e às mudanças ocorridas no ambiente (processo dinâmico) para manter um alinhamento com o ambiente enquanto gerencia as interdependências internas (ciclo adaptativo). Para os autores, o problema de adaptação organizacional pode ser visto verificando-se três problemas relacionados à estrutura corporativa: problema do empreendedor (escolha de domínio do produto e de mercado), problema de engenharia/tecnologia (escolha dos sistemas 39 técnicos) e problema de administração (estrutura e processo organizacional). O problema do empreendedor, principalmente em novas firmas ou em firmas de rápido crescimento, enfatiza que cabe ao gestor a definição dos bens e serviços a serem explorados pela entidade, bem como o mercado-alvo ou o segmento de mercado (MILES et al., 1978; CHAVES; BENEDETE; POLO, 2009; ANDRADE et al., 2013; PLETSCH et al., 2015). Por sua vez, o problema de engenharia envolve a criação de um sistema gerencial para solucionar o problema do empreendedor. Este sistema requer que a administração selecione uma tecnologia apropriada para produzir e distribuir os produtos ou serviços escolhidos para padronizar novos meios de informação, comunicação e controle para garantir a operação adequada do modelo tecnológico. A partir da solução desses problemas ocorre a implementação do sistema administrativo (MILES et al., 1978). O problema de administração consiste em reduzir a incerteza dentro do sistema organizacional, ou seja, aperfeiçoar as atividades que foram solucionadas com sucesso pela organização durante as fases de empreendedorismo e de engenharia. A solução do problema administrativo vai além de aperfeiçoar o sistema para a redução da incerteza, mas deve permitir a formulação e implementação de processos que façam a organização evoluir ou inovar de forma contínua (MILES et al., 1978). A partir dessa análise, Miles et al. (1978) apontaram que o ciclo de adaptação leva as organizações a empregarem estratégias para solucionar os problemas de empreendedorismo, engenharia e administração, sendo que cada empresa tem seu tipo de estratégia para relacionarse com o mercado escolhido e uma configuração de tecnologia, de estrutura e de processo que seja alinhada com a estratégia de mercado. Neste sentido, Miles e Snow (1978) identificaram quatro tipos de estratégias (prospectora, defensiva, analítica e reativa), que se diferenciam conforme algumas dimensões dos problemas do empreendedor são solucionadas. Uma estratégia prospectora é adotada por empresas que atuam em ambientes dinâmicos, sendo caracterizada pela busca e exploração de oportunidades de produtos e mercado, em que a entidade almeja o pioneirismo e a inovação, mesmo que tenha que sacrificar sua lucratividade. Além disso, uma entidade prospectora deve utilizar tecnologias flexíveis em suas atividades, cabendo à administração facilitar e coordenar inúmeras e diversas operações (MILES et al., 1978; ANDRADE et al., 2013; PLETSCH et al., 2015). O Quadro 2 apresenta as principais características de uma estratégia prospectora. 40 Quadro 2 Características da estratégia prospectora de Miles e Snow (1978) PROBLEMAS Empreendedorismo Engenharia Administração Problema: como identificar e explorar novos produtos e oportunidades de mercado? Soluções: (a) ampliar e desenvolver o domínio continuamente; (b) monitorar, de forma ampla, as condições e eventos ambientais; (c) criar mudanças na indústria; (d) crescer por meio do desenvolvimento de produtos e mercados. Custos e benefícios: inovações de produto e de mercado protegem a entidade de um ambiente em mudanças, mas a organização corre o risco de baixa lucratividade etc. Problema: como evitar compromissos de longo prazo para ou com um único processo tecnológico? Soluções: (a) adotar tecnologias múltiplas e flexíveis e protótipos; (b) baixo grau de rotinas e mecanização, tecnologia incorporada pelas pessoas. Custos e benefícios: a flexibilidade tecnológica permite uma resposta rápida a um domínio em mudança, mas a organização não pode desenvolver a máxima eficiência em seu sistema de produção e distribuição, devido a múltiplas tecnologias. Problema: como facilitar e coordenar inúmeras e diversas operações? Soluções: (a) dar às equipes de marketing e P&D o papel central na coalizão dominante; (b) planejar com foco nos problemas; (c) tender a uma estrutura pouco focada na divisão do trabalho e com baixo grau de formalismo. Custos e benefícios: o sistema administrativo é necessário para manter a flexibilidade e eficácia, mas pode gerar mau uso dos recursos. Fonte: adaptado de Miles et al. (1978). Em outra direção, por meio de uma estratégia defensiva, busca-se manter a estabilidade do ambiente por meio da produção, comercialização de bens ou pela prestação de serviços a um segmento de mercado, sendo o seu sucesso decorrente da eficiência no atendimento desse mercado. A manutenção da estabilidade do ambiente é considerada a solução para o problema do empreendedor. A partir dessa estratégia, almeja-se o domínio estável de produtos, focando em um segmento de mercado, praticando preços competitivos ou produtos com qualidade, por meio de eficiência tecnológica e forte controle organizacional, conforme características descritas no Quadro 3 (MILES et al., 1978; KALD; NILSSON; RAPP, 2000; ANDRADE et al., 2013; PLETSCH et al., 2015). Quadro 3 Características da estratégia defensora de Miles e Snow (1978) PROBLEMAS Empreendedorismo Engenharia Administração Problema: como dominar uma porção total de mercado e criar um conjunto estável de produtos e clientes? Soluções: (a) reduzir e estabilizar o domínio; (b) manter o domínio agressivamente; (c) tender a ignorar inovações fora do domínio. Custos e benefícios: é difícil para os concorrentes deslocarem a organização de seus pequenos nichos de mercado, mas uma grande mudança no mercado poderia ameaçar a sobrevivência. Problema: como produzir e distribuir bens e serviços de forma eficiente? Soluções: (a) adotar a tecnologia eficiente, sob a ótica de custo e com foco central único; (b) tender à integração vertical; (c) aprimorar de modo contínuo a tecnologia para manter a eficiência. Custos e benefícios: a eficiência tecnológica é central para o desempenho organizacional, mas pesados investimentos nessa área exigem que os problemas tecnológicos permaneçam conhecidos e previsíveis por longos períodos de tempo. Problema: como manter estrito controle da organização, de modo a garantir eficiência? Soluções: (a) dar às equipes de finanças e produção o papel central na coalizão dominante; (b) planejar com foco em custo; (c) tender a uma estrutura baseada na divisão de trabalho com alto grau de formalismo etc. Custos e benefícios: o sistema administrativo é necessário para manter a estabilidade e a eficiência, mas não é adequado para localizar e responder a novos produtos ou oportunidades de mercado. Fonte: adaptado de Miles et al. (1978). 41 A estratégia analítica caracteriza-se pela combinação de aspectos da estratégia defensora com a prospectora, buscando reduzir o risco e aumentar a oportunidade de lucro e mantendo um equilíbrio entre ambas. Ao adotar essa estratégia, uma corporação busca localizar e explorar novos produtos e oportunidades de mercado, mantendo simultaneamente uma base de clientes e produtos tradicionais, diferenciando processos e estruturas para operar em ambientes estáveis e dinâmicos, conforme características apresentadas no Quadro 4 (MILES et al., 1978; PLETSCH et al., 2015). Quadro 4 Características da estratégia analítica de Miles e Snow (1978) PROBLEMAS Empreendedorismo Engenharia Administração Problema: como identificar e explorar novas oportunidades de produtos e de mercado, mantendo simultaneamente uma base sólida de produtos e clientes tradicionais? Soluções: (a) instituir mecanismos de monitoramento focados em marketing e P&D em pequena extensão; (b) crescer por meio a penetração de mercado etc. Custos e benefícios: baixo investimento em P&D combinado com imitação de produtos de sucesso minimiza o risco, mas o domínio do mercado deve ser balanceado em momentos de estabilidade e flexibilidade. Problema: como ser eficiente em partes estáveis do mercado (domínio) e flexível em partes variáveis? Soluções: (a) adotar tecnologias ao mesmo tempo estáveis e flexíveis; (b) operar com grau moderado de racionalidade técnica etc. Custos e benefícios: o duplo núcleo tecnológico está apto para ser utilizado em um mercado híbrido (estáveldinâmico), mas a tecnologia pode não ser completamente efetiva ou eficiente. Problema: como diferenciar a estrutura e os processos organizacionais para adaptá-los às áreas de operações estáveis e dinâmicas? Soluções: (a) dar às equipes de marketing e engenharia e, num segundo plano, à de produção, o papel central na coalizão dominante; (b) planejar com foco amplo; (c) adotar estrutura matricial etc. Custos e benefícios: o sistema administrativo é capaz de equilibrar estabilidade e flexibilidade, mas, se esse equilíbrio for perdido, pode ser difícil restaurá-lo. Fonte: adaptado de Miles et al. (1978). Já a estratégia reativa, ou o fracasso estratégico, é aquela em que a entidade apenas reage ao mercado, como se não tivesse estratégia, procurando apenas novas oportunidades de mercado ou novos produtos quando se sente ameaçada pelos concorrentes, não apresentando alinhamento entre estratégia, tecnologia, estrutura e processos de gestão organizacional (MILES et al., 1978; KALD; NILSSON; RAPP, 2000; CHAVES; BENEDETE; POLO, 2009; ANDRADE et al., 2013; PLETSCH et al., 2015). Miles et al. (1978) indicam três motivos para que uma entidade venha utilizar uma estratégia reativa, a saber: (a) os executivos principais não têm claramente articulada a estratégia organizacional; (b) não há uma adequação entre estrutura e processos organizacionais com a estratégia escolhida; e (c) tendência em manter a relação atual existente entre estratégia e estrutura, mesmo que haja fortes mudanças no ambiente. Uma empresa que adota uma estratégia reativa apresenta as seguintes características: padrão de adaptação ao ambiente que é inconsistente e instável; exibe um estado de 42 instabilidade quase permanente; necessita de mecanismos de respostas às mudanças no ambiente; o ciclo de adaptação responde inadequadamente às mudanças e incertezas do mercado; apresenta um desempenho insatisfatório; e reluta em reagir às mudanças ambientais (MILES et al., 1978). Destarte, uma estratégia reativa pode ser considerada como um comportamento organizacional caracterizado pela ausência de padrão estratégico previamente definido, levando a entidade a reagir às mudanças ocorridas no mercado apenas quando considerar ameaçada sua sobrevivência. Percebe-se, com base nas estratégicas genéricas de Miles et al. (1978), que, para se adaptar às incertezas do ambiente, uma corporação deve identificar a estratégia que melhor se adapta ao mercado em que atua (prospectora, defensora e analítica), buscando solucionar os problemas de empreendedorismo, engenharia e administração, ou apenas reagir aos fatores que ameaçam sua continuidade no mercado (estratégia reativa). As estratégias propostas por Miles et al. (1978) receberam uma nova abordagem, a partir do trabalho desenvolvido por Miles e Snow (1986). Segundo Miles e Snow (1986), as empresas estavam passando por mudanças, levando-as a repensarem suas estratégias competitivas, dando origem a um novo formato corporativo que combinava estratégia, estrutura e processos de gestão, denominado de rede dinâmica. A rede dinâmica caracteriza-se pela atuação de empresas em rede, por meio de uma combinação única de estratégia, estrutura e processos gerenciais, sendo decorrente e resultado das novas mudanças no ambiente competitivo (MILES; SNOW, 1986). Em que pese os novos arranjos organizacionais, para os autores, as estratégias genéricas propostas por Miles et al. (1978) continuam a ser válidas. Contudo, Miles e Snow (1986) afirmam que o sucesso das estratégias genéricas seria resultado do grau de ajuste às condições do ambiente e da conformidade da estrutura e dos processos à estratégia competitiva. Portanto, percebe-se que as estratégias genéricas propostas por Miles et al. (1978) continuam a ser efetivas, desde que estejam em conformidade com as mudanças ocorridas no ambiente e a estrutura e os processos corporativos estejam alinhados com a estratégia adotada pela organização. A seguir, apresentam-se as estratégias competitivas propostas por Porter (1980), destacando-se as premissas, requisitos e principais características das estratégias de liderança em custos, diferenciação e foco (ou enfoque). 43 2.2.2 Estratégias competitivas propostas por Porter (1980) O modelo de estratégias competitivas ou prioridades estratégicas propostas por Porter (1980) baseia-se nas premissas da escola de posicionamento estratégico, a saber: estratégias são posições genéricas, sendo identificáveis no setor; o mercado é econômico e competitivo; a formulação da estratégia parte de posições ou escolhas; e o mercado influencia as escolhas estratégicas, que, por conseguinte, afeta a estrutura organizacional (MINTZBERG; AHLSTRAND; LAMPEL, 2010). A construção de uma estratégia competitiva, na visão de Porter (1980, 2004), parte do desenvolvimento de uma concepção abrangente de como competir, dos objetivos a serem estabelecidos e das políticas que deveriam ser implementadas para alcançar tais objetivos. Para tanto, Porter (1980) identifica cinco forças que atuam no ambiente de uma organização que influenciam a concorrência, a saber: ameaça de novos participantes, poder de barganha dos fornecedores da empresa, poder de barganha dos clientes, ameaça de produtos substitutos e intensidade de rivalidade entre empresas concorrentes (PORTER, 1980, 2004; KALD; NILSSON; RAPP, 2000; CHAVES; BENEDETE; POLO, 2009; MINTZBERG; AHLSTRAND; LAMPEL, 2010). A partir da análise dessas cinco forças, as entidades deveriam desenvolver (ou adotar) uma estratégia competitiva que lhe permitiria implementar ações, tanto defensivas como ofensivas, buscando tomar uma posição frente às cinco forças que permeiam a competição no mercado (PORTER, 2004; ALLEN; HELMS, 2006; CHAVES; BENEDETE; POLO, 2009; MINTZBERG; AHLSTRAND; LAMPEL, 2010; JUNQUEIRA et al., 2016). Porter (1980, 2004) afirma que haveria apenas três estratégias genéricas capazes de gerar vantagens competitivas para uma organização em longo prazo, tornando-as competitivas frente aos seus concorrentes e possibilitando alcançar um desempenho superior à média do setor, quais sejam: liderança em custos, diferenciação e foco (ou enfoque). A estratégia pautada na liderança no custo total fundamenta-se na criação de uma posição de baixo custo em relação aos concorrentes de um determinado setor (liderança de custo total na indústria), por meio de políticas e ações capazes de atingir tal objetivo (PORTER, 2004; MINTZBERG; AHLSTRAND; LAMPEL, 2010; BANKER; MASHRUWALA; TRIPATHY, 2014). A liderança em custos é alcançada baseada no baixo custo de produção de bens e serviços por meio de instalações eficientes, grandes investimentos em maquinário, reduções de custos com base na aprendizagem e na experiência organizacional, controle rígido de custos e 44 despesas, redução na formação de contas adicionais de clientes e cortes de custos com pesquisa e desenvolvimento (P&D), equipe de vendas, propaganda e publicidade, entre outros gastos (PORTER, 2004; ALLEN; HELMMS, 2006). Para que a estratégia de liderança em custos tenha êxito, a organização precisa realizar investimentos em diversas áreas para aperfeiçoar seu processo produtivo, obter ganhos de escala, produção em massa, distribuição em massa, eficiência tecnológica, desenho de produtos (MINTZBERG; AHLSTRAND; LAMPEL, 2010; BANKER; MASHRUWALA; TRIPATHY, 2014), padronização de produtos (KALD; NILSSON; RAPP, 2000), redução de custos de insumos, eficiência da capacidade de produção e na utilização de recursos, acesso a matériasprimas e alta participação no mercado (PORTER, 2004; JUNQUEIRA et al., 2016). A adoção de um posicionamento de baixo custo leva a entidade a se proteger contra as cinco forças competitivas presentes no mercado, produzindo rentabilidade acima da média do setor. A liderança em custos propicia defesa contra a competição dos concorrentes, pois custos mais baixos possibilitam-na obter retornos superiores aos seus competidores, que utilizaram suas margens de lucro para se manterem competitivas. A liderança de custo possibilita, também, que a companhia se posicione frente aos seus principais fornecedores, tornando-a capaz de enfrentar o crescimento de custo de matérias-primas (PORTER, 2004). A posição de baixo custo, de forma geral, também permite a criação de barreiras de entrada para novos competidores, devido às economias de escala e vantagens obtidas com as reduções de custos. Além disso, a liderança em custos proporciona um posicionamento favorável no que se refere aos produtos substitutos das outras empresas que competem no mercado (PORTER, 2004). Em síntese, a implementação de estratégia de liderança em custos pode proteger a organização contra as cinco forças existentes no mercado, gerando vantagens competitivas que se traduzem em lucros anormais para a entidade, ou seja, acima das demais empresas do setor. Por outro lado, a estratégia de diferenciação caracteriza-se pela oferta de bens e serviços por uma entidade que apresentam características distintas dos oferecidos pelos seus concorrentes em determinada indústria, sendo reconhecida pelos atributos que os produtos ou serviços apresentam no mercado (PORTER, 2004; BANKER; MASHRUWALA; TRIPATHY, 2014; JNQUEIRA et al., 2016). A adoção da estratégia de diferenciação requer investimentos em P&D (busca constante pela inovação), desenho do produto (qualidade e atributos desejados pelos clientes) e pelo marketing (construção da imagem dos bens e serviços e da marca) (MILLER, 1978; BALSAM; FERNANDO; TRIPATHY, 2011). 45 Entende-se, desta forma, que a estratégia de diferenciação é aquela que distinguirá a organização de seus concorrentes, tornando-a conhecida pela qualidade de seus produtos e serviços, sendo sua imagem e reputação construída a partir do reconhecimento de seus clientes, que estão dispostos a pagar um sobrepreço pelos bens e serviços que lhes são ofertados. Para que a estratégia de diferenciação seja alcançada, a organização precisa se distinguir em várias dimensões, tais como: projeto ou imagem da marca, tecnologia, características dos produtos, serviços sob encomenda, rede de fornecedores, canais de distribuição, rede de vendedores, qualidade e durabilidade dos produtos, entre outras (PORTER, 2004; ALLEN; HELMMS, 2006). A estratégia de diferenciação, quando implementada de forma efetiva, proporciona vantagens competitivas para as empresas, gerando lucros acima da média do setor e criando mecanismos para enfrentar as cincos forças competitivas que permeiam o ambiente e o mercado (PORTER, 2004). Ao adotar a diferenciação como estratégia, uma companhia consegue se proteger contra a competição, uma vez que a diferenciação dos produtos fabricados e ofertados proporciona maior fidelidade dos clientes e há uma menor variação na política de preços. A política de diferenciação estratégica impõe barreiras de entrada aos concorrentes devido à possível lealdade do consumidor e à necessidade daqueles em superar as qualidades e os atributos dos produtos da firma (PORTER, 2004). Um posicionamento estratégico baseado na diferenciação produz altas margens de lucros, possibilitando à firma enfrentar os fornecedores e reduzir o poder dos compradores, visto não ser refém de variações nos preços. E a fidelização dos consumidores, produzida pela diferenciação, defende a organização contra a entrada de produtos substitutos no mercado (PORTER, 2004). A terceira estratégia defendida por Porter (1980, 2004) reside no foco (ou enfoque) em um nicho de mercado, grupo comprador ou um segmento da linha de produtos. A estratégia de enfoque fundamenta-se no pressuposto de que uma companhia é capaz de atingir sua estratégia atuando em um nicho específico de um setor ou segmento, no lugar de concorrer de uma forma mais abrangente, ou seja, em todo o setor (PORTER, 2004). Mediante essa estratégia, uma companhia direciona e estrutura suas atividades para atender um determinado tipo de cliente ou focalizar linhas de produtos e serviços ou segmento industrial, buscando adaptar sua vantagem estratégica (foco na diferenciação ou no baixo custo) com base nas características do mercado para desenvolver suas competências e aprimorar a aprendizagem organizacional (MINTZBERG; AHLSTRAND; LAMPEL, 2010). 46 Na visão de Porter (2004), a organização que implantar de forma eficiente e eficaz a estratégia de enfoque pode obter vantagens competitivas e alcançar retornos superiores em comparação aos demais competidores do setor, adaptando sua estratégia de acordo com as peculiaridades do mercado, identificando setores de mercado onde o foco no baixo custo ou na diferenciação seja mais efetivo. Além disso, a estratégia de enfoque também possibilita a empresa a enfrentar as forças competitivas, proporcionando escolher nichos de mercado onde seus produtos sejam menos suscetíveis de ser substituídos e onde seus competidores apresentem maior vulnerabilidade. Percebe-se que a adoção do enfoque estratégico permite que as entidades selecionem mercados onde possam utilizar o foco estratégico que seja mais eficaz, podendo definir seu escopo de atuação e voltando suas atividades para atender seu alvo estratégico melhor que os seus competidores, que atuam de forma mais ampla no setor. No Quadro 5, apresentam-se os principais requisitos para implantação das estratégias genéricas propostas por Porter (1980). Quadro 5 Requisitos das estratégias competitivas de Porter (1980) Estratégia Recursos/habilidades em geral requeridos Requisitos organizacionais comuns Liderança em custos Investimento em capital sustentado e acesso ao capital; boa capacidade de engenharia de processo; supervisão intensa de mão de obra; sistema de distribuição com baixo custo etc. Controle rígido de custos; relatórios de controle frequentes e detalhados; incentivos baseados em metas estritamente quantitativas etc. Diferenciação Grande habilidade de marketing; engenharia do produto; criatividade; grande capacidade em pesquisa básica; forte cooperação de canais etc. Forte coordenação entre P&D, desenvolvimento do produto e marketing; avaliações baseadas em medidas não financeiras etc. Enfoque/foco Combinação das políticas acima dirigidas para a meta estratégica particular. Combinação das políticas acima dirigidas para a meta estratégica particular. Fonte: Porter (2004). Conforme se denota no Quadro 5, a adoção de uma estratégia genérica por uma empresa requer que esta possua atributos e recursos capazes de facilitar a implantação da estratégia desejada. Ainda segundo Porter (2004), a implementação da estratégia também pode requerer da empresa perfis diferentes de liderança ou gestão, adaptando-se a diferentes ambientes e culturas organizacionais. De acordo com Porter (2004), para que uma organização obtenha vantagens competitivas frente aos seus competidores, precisa escolher uma das três estratégias genéricas, pois, caso decida adotar simultaneamente as referidas estratégias, ao que denomina estratégia do meio-termo, pode sofrer perdas de rentabilidade, não conseguindo sustentar e criar vantagens competitivas no longo prazo. 47 Em que pese tal ressalva de Porter (1980, 2004), estudos encontraram evidências de que combinar as estratégias de diferenciação com eficiência de custos não afeta a lucratividade das empresas, mas melhora o desempenho (MILLER, 1992; GOPALAKRISHNA; SUBRAMANIAN, 2001; RICHARDSON; DENNIS, 2003; SPANOS; ZARALIS; LIOUKAS, 2004). Talvez tal fato seja explicado pelas características do ambiente em que as empresas competem, onde, para se manterem competitivas, precisam apresentar produtos e serviços eficientes, mas que possuam certo grau de qualidade, ou seja, apresentar um padrão mínimo de qualidade exigido pelo mercado, de forma que, para isso, seja necessária a aplicação de aspectos relacionados a duas ou mais estratégias competitivas. Além disso, segundo Miller (1992), organizações que adotam exclusivamente um tipo de estratégia podem cair na "armadilha da estratégia genérica", tornando-se vulneráveis frente aos seus competidores, que exploram as mudanças ocorridas no mercado para implementar estratégias que aliam eficiência e qualidade ao mesmo tempo, ganhando participação no mercado e, consequentemente, melhorando seus resultados. Porter (1985), por meio do livro Competitive advantange, reforçou os aspectos conceituais de seu modelo de estratégias competitivas indicando que, para uma organização se manter à frente de seus concorrentes, deveria ser capaz de criar valor para os seus clientes, havendo apenas duas maneiras de se obter vantagem competitiva: a liderança em custos e a diferenciação. Com base no conceito de vantagem competitiva, Porter (1985) ajustou sua tipologia de estratégias competitivas defendendo que os dois tipos de vantagem competitiva combinados com o enfoque de atuação conduziriam a três estratégias competitivas anteriormente definidas por ele: liderança em custos, diferenciação e foco. Além disso, a estratégia baseada no foco admitiria dois escopos de atuação, com enfoque na diferenciação ou no baixo custo (CHAVES; BENEDETE; POLO, 2009). Segundo Porter (1985), a adoção de uma estratégia genérica, exclusivamente, não garantirá um desempenho superior à média do setor, sendo alcançado apenas se as vantagens competitivas obtidas mediante a estratégia utilizada fossem sustentáveis em longo prazo. Percebe-se que apenas a adoção de uma estratégia competitiva não implica desempenho superior ao dos concorrentes, mas é necessário que a estratégia utilizada produza vantagens competitivas de forma contínua, permitindo-lhe estar sempre à frente de seus concorrentes, com lucratividade crescente e sustentável em longo prazo. 48 O modelo desenvolvido por Porter (1980, 1985) indica que uma organização só apresentará desempenho superior aos seus concorrentes caso possa criar valor para os seus clientes, gerando vantagens competitivas por meio de estratégias baseadas na liderança em custos, na diferenciação ou no foco, do contrário, estará ameaçada por seus concorrentes. Apresenta-se, abaixo, a tipologia de estratégias competitivas proposta por Gupta e Govidarajan (1984), evidenciando a classificação das estratégias competitivas, premissas e características. 2.2.3 Missão estratégica de Gupta e Govidarajan (1984) Gupta e Govidarajan (1984), a partir da análise da relação entre mercado e o ciclo de vida do produto/organização, construíram um modelo de estratégias competitivas com base no ciclo de vida, utilizando como base o conceito de missão estratégica ou estratégia de portfólio. Uma estratégia de portfólio ou missão estratégica pode ser considerada uma política estratégica que envolve características de unidades organizacionais que perpassa pela escolha entre o aumento na participação de mercado e na maximização de lucro e fluxo de caixa no curto prazo (GUPTA; GOVIDARAJAN, 1984; KALD; NILSSON; RAPP, 2000; CAPALONGA; DIEHL; ZANINI, 2014). Seguindo os conceitos de missão estratégica, Gupta e Govidarajan (1984) identificaram quatro tipos de estratégias genéricas, a saber: construção, colheita, manutenção e renúncia. A estratégia de construção é adotada por empresas que buscam aumentar sua participação ou domínio de mercado e a posição competitiva, mesmo que para isso tenham que sacrificar lucros e fluxos de caixa no curto prazo. Por outro lado, uma estratégia baseada na colheita tem como objetivo maximizar o lucro e o fluxo de caixa do negócio no curto prazo, embora que para isso a entidade tenha que abrir mão de uma maior participação de mercado (KALD; NILSSON; RAPP, 2000; ANDRADE et al., 2013). Uma empresa que adota uma missão estratégica fundamentada na manutenção almeja combinar uma política estratégica de construção e colheita ao mesmo tempo, ou seja, busca por uma maior maximização de lucro e de fluxos de caixa no curto prazo por meio de uma maior participação no mercado. Por sua vez, a estratégia genérica de renúncia significa que a entidade não deseja continuar suas operações, ou seja, implica o fim da atividade, não sendo considerada uma estratégia propriamente dita (GUPTA; GOVIDARAJAN, 1984; KALD; NILSSON; RAPP, 2000; ANDRADE et al., 2013). 49 As estratégias propostas por Gupta e Govidarajan (1984) estão relacionadas com o ciclo de vida dos produtos ou do mercado, cabendo à entidade selecionar a missão estratégica que proporcione um equilíbrio entre as metas ou objetivos de crescimento de participação de mercado e da maximização de lucro almejado pela organização no curto prazo. Em seguida, demonstram-se as estratégias competitivas desenvolvidas por Robinsin Jr. e Pearce II (1988), enfatizando as premissas, as características e o foco de cada estratégia competitiva, segundo os citados autores. 2.2.4 Estratégias competitivas de Robinsin Jr. e Pearce II (1988) Diferentemente das estratégias competitivas apresentadas anteriormente, Robinsin Jr. e Pearce II (1988) desenvolveram um modelo de estratégias com base nos conceitos de padrões de formação de estratégia desenvolvidos por Mintzberg (1978), analisando simultaneamente a relação entre a estratégia pretendida (padrão de comportamento estratégico priorizado pela administração) e o planejamento de processos (meio pelo qual as atividades organizacionais serão articuladas para facilitar a implantação da estratégia) sobre o desempenho corporativo. As estratégias propostas por Robinsin Jr. e Pearce II (1988) partem da premissa de que os executivos não buscam apenas a efetiva realização dos processos planejados (planejamento de processos), mas que estes estejam de acordo com a estratégia almejada (pretendida) pela empresa, buscando utilizá-los como ferramenta para construção e implementação de padrões estratégicos efetivos. Para desenvolverem seu modelo de padrões estratégicos, Robinsin Jr. e Pearce II (1988) realizaram um estudo com 97 empresas de 60 indústrias diferentes. A análise de clusters foi utilizada para agrupar as entidades de acordo a estratégia utilizada. Com base nesse estudo, os autores identificaram cinco padrões de comportamento estratégicos, a saber: eficiência e serviço; serviços de altos preços; inovação e desenvolvimento de produtos; construção da marca; e empresas sem estratégia definida. Os métodos e as características de cada padrão estratégico são apresentados no Quadro 6. 50 Quadro 6 Padrões estratégicos identificados por Robinsin Jr. e Pearce II (1988) Padrão estratégico Características Eficiência e serviço Foco: eficiência de produtos e serviços. Métodos: treinamento de pessoal, rigoroso controle de qualidade, baixo custo unitário, busca por inovação nos processos produtivos etc. Sem estratégia definida O padrão estratégico é incerto, reflete indecisão e a empresa não sabe o que quer. Serviço para mercados de alto preço Foco: desenvolvimento de produtos e serviços de alto valor, com amplo atendimento aos clientes. Métodos: atendimento ao cliente, construir uma reputação na indústria e atuação em mercados de alto valor. Inovação e desenvolvimento de produtos Foco: maior ênfase na inovação e desenvolvimento de produtos, buscando criar reputação por meio de serviços de alto valor aos seus clientes. Métodos: desenvolvimento de novos produtos, desenvolvimento e melhoramento dos produtos existentes, ênfase em produtos especializados e processo orientado por P&D. Construção da marca Foco: balanceamento entre a eficiência estratégica e a construção da marca por meio de canais de influência. Métodos competitivos: construir uma marca de reputação, influenciar nos canais de distribuição, desenvolvimento de novos produtos e inovação das técnicas de marketing. Fonte: adaptado de Robinsin Jr. e Pearce II (1988). De acordo com o Quadro 6, pode-se dizer que as estratégias propostas por Robinsin Jr. e Pearce II envolvem aspectos relacionados a: eficiência de produtos e serviços, com baixo custo (eficiência e serviço); desenvolvimento de produtos e serviços com alto valor aquisitivo, visando atender às necessidades do mercado (serviços para mercados de alto preço); pioneirismo na criação e desenvolvimento de produtos (inovação e desenvolvimento de produtos); criação de uma marca renomeada, a partir de produtos e serviços com elevada eficiência e qualidade (construção da marca); e incerteza em relação aos objetivos corporativos (sem estratégia definida). Percebe-se, também, que os padrões estratégicos (eficiência e serviço e serviços para mercados de altos preços) identificados por Robinsin Jr. e Pearce II (1988) apresentam semelhanças com as estratégias competitivas propostas por Miles et al. (1978) (defensora e prospectora) e Porter (1980) (custo e diferenciação), diferenciando-se por evidenciar uma integração entre a estratégia perseguida pela administração e a natureza do planejamento de processos. Pode-se dizer que as estratégias identificadas por Robinsin Jr. e Pearce II (1988) são construídas a partir do alinhamento entre a estratégia pretendida e o planejamento de processos, sendo este utilizado como meio para a realização efetiva da estratégia organizacional. Apresentam-se, logo após, as estratégias competitivas propostas por Treacy e Wiserma (1993), destacando-se os pressupostos, as características e a classificação adotada pelos autores, cujo foco é a geração de valor pelas empresas. 51 2.2.5 Estratégias competitivas de Treacy e Wiserma (1993) Treacy e Wiserma (1993) propõem uma tipologia de estratégia baseada no valor esperado (percebido) pelos clientes ou no valor distribuído aos clientes, agrupadas em disciplinas de valor que são necessárias para que uma empresa realize e mantenha uma liderança de mercado. O conceito de valor não é novo, mas a maneira como os clientes percebem o valor hoje é diferente, levando as empresas a desenvolverem meios para atender a todas as expectativas dos clientes, como comodidade nas compras, atendimento e serviço pós-vendas, entre outros (TREACY; WISERMA, 1993). Com base na premissa de que as organizações que conseguem agregar valor aos seus bens e serviços alcançam sucesso, Treacy e Wiserma (1993) defendem que as empresas podem distribuir valor aos seus clientes por meio de três estratégias competitivas: excelência operacional; intimidade com clientes e liderança dos produtos. A estratégia de excelência operacional é similar ao posicionamento de liderança no custo total proposto por Porter (1980), mas não se limita somente à eficiência em custos, pois descreve um modelo estratégico para produção e distribuição eficiente de produtos e serviços (TREACY; WISERMA, 1993; WEBER; POLO, 2010). A partir da excelência operacional, busca-se combinar qualidade, preço e facilidade de compra em que nenhum outro concorrente do mercado possa igualar, por meio de ofertas com os menores custos possíveis (TREACY; WISERMA, 1993). Além disso, apresenta como características: produção eficiente; produtos desenhados para eficiência em custos; processos com operações padronizadas, simplificadas, planejadas e centralizadas; sistema de gestão e controle focados em operações integradas, confiáveis e de alta velocidade; política de desperdício zero e recompensa para a eficiência; e distribuição eficiente de bens e serviços (WEBER; POLO, 2010). Pode-se dizer que, por meio da excelência operacional, a organização deve buscar atingir o máximo de eficiência em suas operações para alcançar o menor custo para seus produtos e serviços, sem que estes percam um nível desejável de qualidade, levando a obterem um posicionamento de liderança no mercado por meio do valor observado pelos clientes. Já pela estratégia de intimidade com os clientes, busca-se uma contínua adaptação e melhoramento dos produtos e serviços para atender às expectativas dos clientes. Nessa estratégia, a firma almeja conquistar a lealdade do cliente em longo prazo, buscando construir 52 uma relação que lhe gere valor não apenas por meio de uma única transação, mas por toda a vida do cliente (TREACY; WISERMA, 1993). A estratégia de intimidade com o cliente apresenta as seguintes características: foco nas necessidades do cliente; visão de longo prazo; soluções específicas para os clientes; descentralização de decisões; avaliação de resultados com base em clientes selecionados; relacionamentos de longo prazo; e quadro de pessoal talentoso, flexível e multifuncional (TREACY; WISERMA, 1993; WEBER; POLO, 2010). Nota-se que, na estratégia baseada na intimidade com o cliente, a construção de uma relação em longo prazo com o cliente é foco principal, levando a empresa a investir em produtos diferenciados para um determinado segmento do mercado, semelhante à estratégia de enfoque defendida por Porter (1980), mas almejando a criação de valor por meio da fidelização do cliente. Por sua vez, a estratégia de liderança de produtos fundamenta-se na busca contínua pelo melhor produto, por meio de uma produção contínua de um fluxo de produtos e serviços de ponta, ou de última geração. Baseadas nessa estratégia, as corporações se caracterizam por serem criativas, inovadoras e pioneiras na busca de soluções para os problemas que seu produto ou serviço acaba de resolver (TREACY; WISERMA, 1993). Empresas que utilizam a estratégia de liderança no produto apresentam estruturas mais flexíveis, especializadas; sistema de gestão baseados em resultados; recompensam resultados positivos com o desenvolvimento de novos produtos, sem punir a criatividade; focam em P&D e valorizam a imaginação dos indivíduos, orientada para o futuro. Além disso, apresentam semelhanças com a estratégia de diferenciação defendida por Porter (1980) (WEBER; POLO, 2010). Percebe-se que, na estratégia de liderança de produtos, há uma busca constante pela inovação e o pioneirismo, tendo como base, principalmente, a criatividade e a P&D, para alcançar a liderança de mercado por meio de bens e serviços de última geração que visam atender às necessidades dos clientes. Treacy e Wiserma (1993) propõem que, para se tornar líder de um mercado ou de um segmento, uma empresa precisar criar valor para os seus clientes, diferenciando-se de seus concorrentes por meio de uma estratégia baseada na excelência operacional, na intimidade com clientes ou na liderança de produtos. A seguir, apresenta-se a tipologia de estratégia competitiva desenvolvida por Bowman e Faulkner (1997), conhecida como estratégia do relógio, evidenciando suas premissas, características e classificação adotada. 53 2.2.6 Estratégias competitivas de Bowman e Faulkner (1997) Bowman e Fallkner (1997) apresentam uma tipologia de estratégia competitiva denominada de estratégia do relógio (strategy clock). A estratégia do relógio, também conhecida como matriz de cliente, explora as opções de posicionamento estratégico, identificando como um produto ou serviço deve ser posicionado para garantir a posição que seja mais competitiva no setor (BOWMAN; FALKNER, 1997; CARLISLE; FALKNER, 2005). A estratégia do relógio foi construída a partir de uma análise desenvolvida sobre as estratégias competitivas de Porter (1980). De acordo com Falkner e Bowman (1992), as estratégias apresentadas por Porter (1980) – liderança no custo e diferenciação – apresentam algumas limitações, tornando-as frágeis do ponto de vista prático e teórico. Segundo Falkner e Bowman (1992), a estratégia de liderança de custo é quase imperceptível para o cliente; baixos custos não garantem rentabilidade acima da média do setor; e a estratégia de liderança em custos requer o conhecimento dos custos dos concorrentes, sendo que os gestores procuram focar em fatores internos em vez de fatores externos à organização. Para os autores, a estratégia de diferenciação também apresenta limitações, pois não fica claro, por exemplo, se uma organização que busca a diferenciação de seus produtos irá obter vantagens competitivas com base na cobrança de um prêmio sobre o preço ou pelo aumento na participação de mercado. Além disso, não há fundamentos teóricos e práticos que impeçam uma corporação a utilizar aspectos da estratégia de baixo custo para aumentar a participação de mercado, mesmo adotando uma política de diferenciação para seus produtos e serviços (FALKNER; BOWMAN, 1992). Identifica-se que, para Falkner e Bowman (1992), as estratégias desenvolvidas por Porter (1980, 1985) necessitam de fundamentos teóricos e práticos mais consistentes, gerando dúvidas de como a estratégia de liderança em custos ou de diferenciação poderia garantir vantagem competitiva para as empresas. De acordo com Falkner e Bowman (1992), para que uma organização adote uma estratégia sustentável, é necessário que seus produtos e serviços apresentem as seguintes características: (a) melhores do que os dos concorrentes, mas com preços similares ou mais altos; (b) tão bons quanto os da concorrência, mas com preços menores; e (c) melhores do que os dos competidores e com preços menores, ou, caso possível, melhor do que os oferecidos na situação (a) ou (b). Assim, segundo Falkner e Bowman (1992), esses três posicionamentos estratégicos refletem a situação que é mais perceptível para o cliente e influenciarão a decisão de compra 54 deste. Além disso, as estratégias serão influenciadas pelos critérios externos de sucesso, que variam de ambiente para ambiente, sendo os mais comuns: preço, confiabilidade, nome da marca, reputação da empresa ou qualidade percebida pelo cliente (FALKNER; BOWMAN, 1992). Com base nessas premissas, Bowman e Falkner (1997) propuseram as seguintes estratégias, ou posicionamentos estratégicos: (a) diferenciação; (b) diferenciação com foco; (c) aumento de preço, mas valor padronizado; (d) aumento de preço, mas baixo valor; (e) baixo valor com preço padronizado; (f) preço baixo/baixo valor adicionado; (g) preço baixo; e (h) híbrida. As características de cada posicionamento estratégico ou estratégia do relógio são apresentadas no Quadro 7. Quadro 7 Estratégias do relógio de Bowman e Falkner (1997) Estratégias Características Diferenciação Oferecer para os clientes o nível mais elevado de valor agregado percebido; a marca e a qualidade desempenham um papel fundamental para a estratégia de diferenciação, pois, mediante a qualidade do produto e do reconhecimento da marca aliada à lealdade do cliente, pode-se obter preços relativamente altos e valor agregado. Diferenciação com foco Posiciona o produto nos mais altos níveis de preço, em que os clientes compram o produto devido ao alto valor percebido; adotada por marcas de luxo, que buscam atingir prêmios sobre o preço por segmentação, mediante promoção e distribuição; se realizada com sucesso, pode gerar altas margens de lucro, mas apenas os melhores produtos e marcas podem sustentar esta estratégia em longo prazo. Aumento de preço, mas com valor padronizado Estratégia de alto risco; há aumento de preços, mas sem aumento de valor percebido pelos clientes; pode ser viável em curto prazo, mas, em longo prazo, os clientes podem encontrar outros produtos pelo mesmo preço e com maior valor agregado; pouco competitiva. Aumento de preço, mas com baixo valor Eficiente em mercados onde há monopólio, com apenas uma empresa fornecendo o produto; a empresa define o preço que desejar; não há alternativa para o cliente; as empresas que atuam em monopólios têm os preços regulamentados. Baixo valor, mas com preço padronizado Estratégia considerada como receita para o insucesso; caso a empresa não atue em um monopólio, haverá perda de participação de mercado. Preço baixo/baixo valor adicionado Não é uma posição muito competitiva; o produto não é diferenciado e o cliente percebe muito pouco valor no produto, mesmo com preço baixo; considerada uma estratégia de barganha; a solução para ser competitivo é utilizar os preços mais baixos. Preço baixo Estratégia utilizada por empresas líderes em baixo custo no mercado; a minimização de custos está associada com economias de escalas; baixas margens de lucros, mas com alto volume de produção; alta competição entre as empresas, envolvendo guerra de preços. Híbrida Posição híbrida que envolve elementos de baixo custo (em relação à concorrência) e de diferenciação de produtos; busca convencer os clientes de que há um bom valor agregado por meio da combinação de um preço razoável e de uma diferenciação aceitável do produto; pode ser um posicionamento eficaz, desde que o valor agregado percebido seja oferecido de forma consistente. Fonte: Bowman e Falkner (1997) e Carlisle e Falkner (2005). Percebe-se que as estratégias propostas por Bowman e Falkner (1997) apresentam um aprimoramento do modelo de Porter (1980), buscando relacionar o posicionamento estratégico com o valor percebido pelos clientes, seja por meio de uma estratégia de baixo preço ou diferenciação de produtos, ou ambas. Além disso, a reputação organizacional é considerada 55 uma fonte de diferenciação capaz de gerar vantagens competitivas, sendo os posicionamentos estratégicos meios para atingir tal objetivo. Em seguida, demonstra-se a tipologia de estratégias competitivas desenvolvidas por Hax e Wilde II (1999), conhecida como modelo delta, destacando-se as principais premissas, as características e a classificação utilizada pelos autores. 2.2.7 Estratégias competitivas de Hax e Wilde II (1999) O modelo de estratégias competitivas propostas por Hax e Wilde II (1999) parte de uma análise crítica das estratégias defendidas por Porter (1980). Segundo Hax e Wilde II (1999), a estrutura apresentada por Porter (1980) não engloba todas as possibilidades pelas quais as organizações podem competir no ambiente atual, marcado por constantes mudanças, desafios complexos e elevados níveis de incerteza. A partir de um estudo envolvendo mais de 100 empresas norte-americanas, Hax e Wilde II (1999) propuseram o modelo delta, que apresenta algumas diferenças em relação às demais tipologias, tais como: define como os posicionamentos estratégicos devem refletir principalmente novas fontes de lucratividade; alinhamento com as atividades; e necessidade de implantar processos de adaptação capazes de responder continuamente a um ambiente de incertezas. O modelo delta é representado por um triângulo que reflete as diversas possibilidades de competir atualmente no mercado, propondo três opções de posicionamento estratégico: (a) melhor produto; (b) soluções para o cliente; e (c) lock-in de sistema, ou sistema fechado (HAX; WILDE II, 1999). A estratégia de melhor produto relaciona-se com as tipologias clássicas de competitividade a partir de baixo custo ou diferenciação, sendo impulsionada pela concentração em um produto ou serviço. Uma organização pode obter menores custos mediante economias de escalas, simplificação de produtos e processos e maior participação de mercado, permitindolhe explorar a experiência e o aprendizado obtido. Já com base na diferenciação, almeja-se aperfeiçoar os atributos do produto, buscando agregar valor para os clientes. A diferenciação é alcançada por meio de tecnologia, imagem da marca, características adicionais ou serviços especiais (específicos) (HAX; WILDE II, 1999). Busca-se, com base na estratégia de melhor produto, uma maior intimidade com o cliente, criando laços de fidelidade com a superioridade de produtos e serviços, rápida introdução de novos produtos no mercado e o estabelecimento de uma marca dominante (HAX; WILDE II, 1999). 56 A estratégia de soluções para o cliente, por sua vez, fundamenta-se em uma oferta mais abrangente de bens e serviços que satisfaça todas ou a maioria das necessidades dos clientes. Baseada nesse posicionamento, a corporação foca no cliente, oferecendo uma ampla gama de produtos e serviços personalizados, conforme as preferências de cada cliente. Além disso, com base nessa estratégia, a empresa busca um maior relacionamento com o cliente para prever suas necessidades, aprimorar o aprendizado e personalizar os produtos e serviços, exigindo a construção de parcerias e alianças que podem incluir fornecedores, clientes e até mesmo concorrentes (HAX; WILDE II, 1999). No posicionamento de sistema fechado ou lock-in de sistema, o foco não é no produto ou no cliente, mas a entidade busca ampliar o escopo de atuação, levando em consideração todos os participantes importantes do sistema que contribuem para a criação de valor econômico (HAX; WILDE II, 1999). De acordo com Hax e Wilde II (1999), com base na estratégia de sistema fechado, os laços entre fornecedor e cliente se tornam fundamental, levando a organização a estimular, atrair e manter fornecedores e os demais participantes normais do setor para aprimorar os produtos e serviços ofertados no mercado. Além disso, o ponto máximo dessa estratégia é tornar a corporação a dona do produto ou serviço padrão de mercado. Para Hax e Wilde II (1999), as estratégias propostas no modelo delta não são excludentes, e a organização pode adotar uma estratégia híbrida, com formas diferentes de competir, variando o alcance, a escala e os laços de relacionamento, ou seja, de acordo com as características do ambiente, do produto ou do cliente. No Quadro 8, pode-se visualizar as possibilidades e as características do modelo delta. Quadro 8 Características das estratégias de Hax e Wilde (1999) Melhor produto Soluções para o cliente Lock-in do sistema Alcance Descaracterizado: Baixo custo; Cheio de qualidades; Diferenciado. Ampla gama de produtos: Pacotes; Desenvolvimento conjunto; Terceirização. Estimular os complementares: Variedade e número; Arquitetura aberta. Escala Produto: Participação no mercado. Cliente: Participação do cliente. Sistema: Participação do complementar. Laços Conexão com o produto: O primeiro no mercado; Projeto dominante. Conexão com os clientes: - "Captação" do cliente; Aprendizado; Personalização. Conexão com o sistema: Afastamento do concorrente; Padrões próprios. Fonte: Hax e Wilde II (1999). Entende-se que o modelo delta também amplia as estratégias desenvolvidas por Porter (1980), englobando dimensões como cooperação e relacionamentos complementares e com 57 importantes participantes de mercado, havendo a possibilidade de combinar os três posicionamentos estratégicos (melhor produto, soluções para o cliente e sistema fechado) com diferentes alcances, escalas e laços de relacionamento. A seguir, apresentam-se as estratégias competitivas propostas por Mintzberg e Quinn (2001), evidenciando-se os principais pressupostos, as características e a classificação adotada pelos autores. 2.2.8 Estratégias competitivas de Mintzberg e Quinn (2001) Mintzberg e Quinn (2001) propõem um modelo de estratégias competitivas abrangentes, baseado nos estudos de Porter (1980) e Ansof (1965). Ainda segundo Mintzberg e Quinn (2001), as estratégias competitivas devem seguir uma sequência lógica, englobando todos os processos da cadeia de valor, iniciando desde a criação da empresa, com a escolha do setor de atuação, identificando as demandas e restrições de mercado, levando a empresa a definir se atuará em um mercado amplo ou segmentado, estruturando suas operações para adotar o posicionamento estratégico que lhe torne competitiva frente aos seus concorrentes (WEBER; POLO, 2010). De acordo com Mintzberg e Quinn (2001), para que uma organização mantenha uma posição competitiva no setor e possa se diferenciar de seus concorrentes, precisa se destacar em aspectos como qualidade, preço, design, suporte e imagem, ofertando produtos e serviços que sejam diferenciados de alguma forma. A partir desses elementos, apresentam seis estratégias de diferenciação, a saber: diferenciação de preço; diferenciação de imagem; diferenciação de suporte; diferenciação de qualidade; diferenciação de design; e não diferenciação. As características de cada estratégia são apresentadas no Quadro 9. Quadro 9 Estratégias competitivas de Mintzberg e Quinn (2001) Estratégia Características Diferenciação de preço Similar à estratégia de liderança no custo total de Porter (1980); baixo custo dos produtos; eficiência operacional; alto volume de vendas; baixa margem de lucratividade. Diferenciação de imagem A empresa busca diferenciar seus produtos ou serviços por meio de ações de marketing, criando uma imagem bonita; relaciona-se com a apresentação do produto no mercado; buscase vender o produto pela sua imagem. Diferenciação de suporte Diferencia-se o produto mediante suporte especializado, assistência técnica, condições especiais de venda, entrega rápida, entre outros atributos. Diferenciação de qualidade Oferta de produtos com qualidade superior em relação aos dos concorrentes, com desempenho mais confiável, mais durável e com maior valor percebido pelos clientes. Diferenciação de design Consiste em oferecer produtos com design diferenciado, fora do padrão e com diversas funções. Não diferenciação A empresa não apresenta estratégia definida; busca copiar os concorrentes; não apresenta habilidade nem vontade de obter alguma diferenciação. Fonte: Mintzberg e Quinn (2001). 58 Com base no Quadro 9, percebe-se que a tipologia proposta por Mintzberg e Quinn (2001) defende que uma organização, para ser competitiva, precisa se diferenciar de seus concorrentes de alguma forma (preço, imagem, suporte, qualidade e design), ou, caso não apresente uma estratégia definida, não se diferenciar, restando-lhe a opção de imitar ou copiar as demais entidades. Além dessas estratégias, Mintzberg et al. (2007) indicam outras que estendem ou aprimoram o contexto do produtos oferecidos, tais como: estratégia de penetração (oferecer o mesmo produto no mesmo mercado, mas de forma intensa, por meio de forte propaganda e publicidade); estratégia de pacote (oferta de dois ou mais produtos conjuntos); estratégia de desenvolvimento de mercado (oferecer os produtos em novos mercados, setores etc.); e estratégia de diversificação (oferta de vários produtos e serviços em diferentes mercados). Portando, para Mintzberg e Quinn (2001), o núcleo da estratégia competitiva é a diferenciação, ou seja, a organização deve se diferenciar de seus concorrentes de alguma forma, seja pela qualidade dos seus produtos, imagem, preço, design, suporte, ou até mesmo não se diferenciar, sendo obtida por meio da estruturação de suas atividades ao longo da cadeia de valor, ampliando a oferta de produtos e serviços mediante políticas de penetração, estratégia de pacote, de desenvolvimento e diversificação de mercado. Na sequência, apresenta-se a tipologia de estratégias competitivas desenvolvidas por Udayasankar e Das (2004), cuja premissa básica está fundamentada na relação entre a estrutura do ambiente institucional e na posição competitiva da empresa no mercado. 2.2.9 Estratégias competitivas de Udayasankar e Das (2004) Udaysankar e Das (2004) propõem uma nova tipologia de estratégias genéricas baseada na relação entre a estrutura do ambiente institucional e a posição competitiva, denominada de estratégias institucionais competitivas. O ambiente institucional, segundo Udaysankar e Das (2004), afeta tanto os meios como os fins de uma organização, sendo o campo onde ela atua e desenvolve suas atividades. Além disso, as operações corporativas são influenciadas por dois componentes do ambiente institucional: apoio institucional e influência nos padrões de governança. O apoio institucional é caracterizado pelas instituições que afetam a capacidade de uma organização manter suas atividades e competir. As funções das instituições são variadas e nem todas as entidades podem se beneficiar dos mesmos tipos de apoios institucionais, dentre os quais: financeiro, de mercado, regulatório e a recursos (UDAYSANKAR; DAS, 2004). Ou seja, 59 a obtenção de um apoio institucional, seja financeiro, com bancos públicos, por exemplo, em alguns casos, é necessário para que uma entidade se mantenha competitiva. Igualmente, as instituições também têm diferentes expectativas quanto aos padrões de governança que as empresas devem seguir, particularmente se as empresas desejam alcançar apoio perante as instituições. As organizações podem reconsiderar e expandir seus padrões de governanças com base em três tipos de influências: de mercado, de partes interessadas da comunidade e de propriedade (UDAYSANKAR; DAS, 2004). Constata-se que, na visão dos autores, o apoio institucional e a expectativa sobre os padrões de governança são componentes institucionais importantes para o desenvolvimento das operações empresariais, podendo interferir na relação das companhias com as demais instituições do ambiente. Para Udaysankar e Das (2004), o apoio institucional e a influência sobre a governança corporativa são os dois eixos do ambiente institucional competitivo. Segundo os autores, com base nesses dois componentes, pode-se classificar o nível de apoio institucional e a qualidade dos padrões de governança em quatro quadrantes distintos. A partir destes quadrantes, pode-se classificar as empresas com base no apoio institucional e na qualidade dos padrões de governança, bem como avaliar a posição da entidade em relação à média dos competidores do ambiente, ou seja, o seu poder em comparação aos demais competidores. A análise do ambiente institucional competitivo, segundo Udaysankar e Das (2004), pode ser aplicável a diferentes contextos, sendo utilizado para avaliar a posição de uma empresa no nível de país, setor ou grupo, se os eixos refletirem a faixa prática de suporte institucional ou os padrões de governança aplicáveis ao contexto. De acordo com Udaysankar e Das (2004), tanto a influência competitiva (pressão que as empresas sofrem para melhorar seu desempenho em relação aos concorrentes, por meio do aumento do poder comparativo) como as influências institucionais (pressões para aumentar os níveis de apoio, ou padrões de governança, por outros motivos que não sejam apenas a vantagem competitiva) atuam sobre a estrutura do ambiente institucional competitivo, levando as empresas a se moverem entre os quadrantes da estrutura para atingir níveis mais altos de apoio institucional e de governança corporativa na busca de maior poder comparativo, conforme é apresentado na Figura 2. 60 Figura 2 Estratégias institucionais competitivas como um movimento organizacional Fonte: Udaysankar e Das (2004). Neste sentido, segundo Udaysankar e Das (2004), as empresas podem se mover na estrutura do ambiente institucional competitivo por meio de quatro estratégias, a saber: competição em mercado dominante (dominant market competition (DMC)); competição em mercado de nicho (niche market competition (NMC)); competição institucional por apoio (institutional competition – suport (IC-S)); e competição institucional por governança (institutional competition – governance (IC-G)). A competição em mercado dominante é uma estratégia seguida por companhias que desejam assumir a liderança competitiva no mercado, geralmente por meio de táticas agressivas de mercado. Essas empresas não pretendem competir com base no aumento de poder seu comparativo, pois possuem maior poder comparativo do que os seus concorrentes (UDAYSANKAR; DAS, 2004). Na estratégia de competição em mercado de nicho, busca-se competir com base no desempenho, por um nicho de mercado, nas linhas das estratégias competitivas predominantes, quando a organização percebe seu poder comparativo inferior ao de seus concorrentes (UDAYSANKAR; DAS, 2004). Com base na estratégia de competição em mercado de nicho, a entidade buscará concorrer em um segmento de mercado onde seu poder comparativo seja equivalente ou superior ao dos potenciais concorrentes, buscando atingir o desempenho esperado. Já a estratégia de competição institucional por apoio é aquela em que a empresa tem por finalidade aumentar seu poder comparativo aumentando o nível de apoio disponível para si própria de forma distinta e individual. As entidades que utilizam essa estratégia são diferenciadas por atividades com altos níveis de interação com instituições, como lobby, 61 negociações com instituições, ativismo político e formação de alianças (UDAYSANKAR; DAS, 2004). No tocante à estratégia de competição institucional por governança, caracteriza-se por ser uma estratégia na qual a empresa visa especificamente aumentar seu poder comparativo e reduzir o poder comparativo de seus concorrentes, aumentando a qualidade de seus padrões de governança corporativa, de forma distinta e individualmente (UDAYSANKAR; DAS, 2004). Verifica-se que, a partir da estratégia de competição institucional por governança, o objetivo da empresa é aumentar seu poder comparativo em relação aos seus concorrentes, e, para isso, aprimora os mecanismos de governança corporativa para se diferenciar dos demais participantes do mercado. Assim, percebe-se que cada estratégia institucional competitiva descreve a forma como uma organização pode atuar em um ambiente institucional competitivo, buscando aumentar o apoio institucional, aperfeiçoar os padrões de qualidade de governança corporativa, bem como aumentar o poder comparativo em relação aos seus concorrentes. A escolha da estratégia específica perpassa as percepções da empresa em relação ao seu poder comparativo e aos de seus concorrentes, bem como de suas metas em relação ao tempo e à concorrência. Além disso, as organizações que pretendem maximizar o desempenho no curto prazo tendem a optar por estratégias de competição no mercado, ou seja, por estratégias de mercado dominante ou de nicho (UDAYSANKAR; DAS, 2004). Portanto, pode-se dizer que as estratégias institucionais competitivas indicam meios pelos quais as organizações podem atuar em um ambiente institucional competitivo (competição em mercado dominante, competição em mercado de nicho, competição institucional por apoio e competição institucional por governança), buscando obter apoio institucional e aperfeiçoar os padrões de governança corporativa para aumentar o poder comparativo e obter vantagens competitivas frente aos seus concorrentes. 2.2.10 Resumo das principais estratégias competitivas Após discorrer sobre as principais tipologias de estratégias genéricas, apresenta-se, no Quadro 10, um resumo com a principal premissa de cada modelo de posicionamento estratégico. Analisando-se as estratégias genéricas expostas, verifica-se que cada tipologia parte de um pressuposto para definir como uma organização precisa definir sua estratégia de atuação no setor. 62 Quadro 10 Resumo das tipologias de estratégias competitivas Autores Estratégias Premissa Miles et al. (1978) Prospectora. Defensora. Analítica. Reativa. Necessidade de se adaptar ao ambiente alinhando estratégia, tecnologia, estrutura e processos (ciclo de adaptação). Porter (1980) Liderança de custo. Diferenciação. Foco. Vantagem competitiva. Gupta e Govidarajan (1984) Construção. Colheita. Manutenção. Renúncia. Relação entre o mercado e o ciclo de vida do produto. Robinsin Jr. e Pearce (1988) Eficiência e serviço. Sem estratégia definitiva. Serviços para mercado de alto preço. Inovação e desenvolvimento de produtos. Construção da marca. Relação entre estratégia pretendida e planejamento de processos. Treacy e Wiserma (1993) Excelência operacional. Intimidade com clientes. Liderança de produtos. Valor esperado ou percebido pelos clientes. Bowman e Falkner (1997) Diferenciação. Diferenciação com foco. Aumento de preço, mas com valor padronizado. Aumento de preço, mas com baixo valor. Baixo valor, mas com preço padronizado. Preço baixo/baixo valor adicionado. Preço baixo. Híbrida. Percepção dos clientes em relação aos produtos e serviços (valor e preço). Hax e Wilde II (1999) Melhor produto. Soluções para o cliente. Sistema fechado. Fontes de lucratividade, alinhamento com as atividades e adaptação às mudanças no ambiente. Mintzberg e Quinn (2001) Diferenciação de preço. Diferenciação de imagem. Diferenciação de suporte. Diferenciação de qualidade. Diferenciação de design. Não diferenciação. Ofertar de produtos e serviços diferenciados (qualidade, preço, design, suporte, imagem ou sem diferenciação). Udaysankar e Das (2004) Competição em mercado dominante. Competição em mercado de nicho. Competição institucional por apoio. Competição institucional por governança. Estrutura do ambiente institucional. Fonte: elaboração própria, com base no referencial teórico. De um modo geral pode-se dizer que os modelos propostos por Miles et al. (1978), Gupta e Govidarajan (1984), Robinsin Jr. e Pearce II (1988), Hax e Wilde II (1999) e Udaysançar e Das (2004) partem de premissas que estão mais ligadas à análise das mudanças ocorridas no ambiente para definir as estratégias organizacionais. Por sua vez, as estratégias propostas por Porter (1980), Treacy e Wiserma (1993) e Mintzberg e Quinn (2001), além de utilizarem pressupostos relacionados ao ambiente organizacional, defendem que as empresas devem se posicionar no setor, oferecendo produtos de baixo custo ou de alta qualidade, cujo valor seja percebido pelos clientes, levando-as a obterem vantagens competitivas frente aos seus concorrentes. 63 Ademais, observa-se que, em que pese as críticas recebidas, a tipologia proposta por Porter (1980) continua a ser referência, sendo utilizada como base para o desenvolvimento de outras tipologias de estratégias genéricas (TREACY; WISERMA, 1993; BOWMAN; FALKNER, 1997; HAX; WILDE II, 1999; MINTZBERG; QUINN, 2001), bem como é utilizada por ser o paradigma de diversos estudos empíricos da área de gestão estratégica e contabilidade gerencial (KIM; LIM, 1988; ALLENS; HELMS, 2006; BAACK; BOGGS, 2008; BALSAM; FERNANDO; TRIPATHY, 2011; BANKER; MASHUWALA; TRIPATHY, 2014; BRENES; MONTOYA; CIRAVEGNA, 2014; JUNQUEIRA et a., 2016; FERNANDO; SCHNEIBLE JR; TRIPATHY, 2016). Desta forma, as estratégias genéricas propostas por Porter (1980) – liderança no custo e diferenciação – serão utilizadas para identificar o posicionamento estratégico das empresas analisadas neste estudo. A partir da identificação do posicionamento estratégico, pode-se analisar sua relação com os fatores contingenciais e com o desempenho, avaliando-se a influência da estratégia competitiva sobre o desempenho financeiro das empresas. Para tanto, a seguir, apresentam-se as principais medidas de avaliação de desempenho organizacional, destacando-se objetivos, tipologias e sistemas de avaliação e mensuração de desempenho. Após apresentar as principais tipologias de estratégias competitivas, em seguida, discorre-se sobre a mensuração de desempenho, destacando-se os conceitos, objetivos, classificação e tipos de indicadores de desempenho. 2.3 MENSURAÇÃO DE DESEMPENHO A competição por clientes, insumos e capital transformou a mensuração de desempenho, tornando-a essencial para a sobrevivência e sucesso organizacional (RICHARD; DEVINNEY; YIP, 2009), uma vez que não se pode administrar o que não se mensura ou avalia (KONSTA; PLOMARITOU, 2012). A seguir, apresentam-se os conceitos, objetivos, medidas (ou indicadores) para mensuração de desempenho, evidenciando que, em um ambiente que está mudando constantemente (BITITCI et al., 2012; OTLEY, 2016), mensurar e administrar o desempenho organizacional é de suma importância para que as organizações busquem o melhoramento contínuo. 64 2.3.1 Conceitos e objetivos Para Lebas (1995), a mensuração é uma atividade complexa, frustrante, difícil, desafiante, importante e, muitas vezes, mal utilizada, sendo definida como a capacidade, ou o ponto de partida, para avaliação e implementação de ações futuras, a fim de alcançar objetivos e metas. O processo de mensuração de desempenho é considerado um dos processos mais críticos no meio corporativo (ITTNER; LARCKER, 1998), apresentando um papel-chave no desenvolvimento e avaliação de estratégias (VENKATRAMA; RAMANUJA, 1986), na avaliação da realização dos objetivos organizacionais, na compensação dos gestores (ITTNER; LARCKER, 1998; TANGEN, 2003) e no planejamento e controle empresarial (CHENHALL; LANGFIELD-SMITH, 2007). Já a mensuração de desempenho pode ser definida como o processo de quantificar a eficiência e a eficácia de uma ação, atividade etc., sendo o indicador a métrica utilizada para medir a eficiência e eficácia dessa ação, entre outros (NEELY; GREGORY; PLATTS, 1995; RICHARD; DEVINNEY; YIP, 2009). É um processo mais amplo, envolvendo a escolha do indicador ou métrica para mensurar a eficiência e eficácia das ações tomadas, buscando avaliar os resultados alcançados para serem tomadas medidas corretivas ou de aprimoramento dos processos para aperfeiçoar os seus resultados. O desempenho organizacional difere da eficácia organizacional, enquanto aquele engloba resultados financeiros, desempenho dos produtos no mercado e retorno dos acionistas, esta é mais ampla e captura o desempenho organizacional associado aos resultados internos (eficiência operacional) e externos (avaliação econômica) e a responsabilidade social corporativa (RICHARD; DIVINNEY; YIP, 2009). Destarte, entende-se que a avaliação de desempenho é um processo contínuo e complexo que se inicia com a escolha de medidas e indicadores, internos ou externos, financeiros ou não financeiros, utilizados para mensurar, gerir, avaliar e implementar estratégias na busca de atingir as metas e os objetivos organizacionais. A mensuração de desempenho pode ser realizada por diversos motivos, variando de acordo com objetivo a ser alcançado. Para Lebas (1995), há pelo menos cinco objetivos (razões) para se mensurar o desempenho: (a) para saber como se estava antes (desempenho passado); (b) para identificar a situação atual (desempenho atual); (c) para apoiar objetivos e metas futuras, no desenvolvimento de planos e ações (suportar metas futuras); (d) indicar, por meio de planejamento e orçamentos, as atividades necessárias para se alcançar as metas e objetivos 65 traçados (melhoramento contínuo); e (e) ser utilizado como feedback para avaliar se as metas e objetivos foram alcançados (controle e feedback). Para cada um desses objetivos, métricas de desempenho podem ser criadas por diferentes usuários e para propósitos distintos. Os gestores, por exemplo, podem formular indicadores de desempenho para avaliar o próprio desempenho e a aprendizagem. Já os supervisores podem elaborar medidas para monitorar e controlar ações delegadas para outros, buscando seu melhoramento (LEBAS, 1995). Outros objetivos para mensuração de desempenho organizacional podem ser encontrados na literatura (KAPLAN; NORTON, 1992; NEELY; GREGORY; PLATTS, 1995; ITTNER; LARCKER, 1998; TANGEN, 2003, 2004; CHENHALL; LANGFIELD-SMITH, 2007; KONSTA; PLOMARITOU, 2012), tais como: • reduzir custos; • flexibilizar as operações para adaptá-las às mudanças; • agilizar as operações para atender os clientes; • melhorar a qualidade das operações para reduzir desperdícios; • tornar as operações mais confiáveis (seguras) para atender aos pedidos planejados; • verificar se a estratégia traçada está sendo alcançada; • alinhar as atividades à estratégia organizacional; • corrigir rumos; • realizar mudanças no ambiente de trabalho; • aumentar a competitividade; • mudar os papéis da organização; • aumentar a eficiência organizacional; • base de mensuração e incentivos para executivos e empregados; • obter avaliações sobre ações, atividades, projetos; • melhorar a tomada de decisão; • aprimorar o planejamento estratégico e orçamentário; • melhoramento contínuo, entre outros. Percebe-se que, por meio da mensuração de desempenho, pode-se alcançar inúmeros objetivos. Contudo, deve-se escolher um indicador de performance que reflita, em termos qualitativos e quantitativos, o resultado alcançado para que se possa adotar medidas para 66 correção de rumos, aprimoramento de ações ou encerramento de atividades cujo desempenho esteja abaixo do esperado. Assim, diferentes medidas podem ser criadas e utilizadas no processo de mensuração e avaliação de desempenho, variando de acordo com os objetivos desejados. Contudo, a escolha de medidas de desempenho não é uma tarefa simples, pois há diferentes tipos de métricas. Apresentam-se, a seguir, as principais características, vantagens e desvantagens de cada tipo de medida de desempenho. 2.3.2 Medidas para mensuração de desempenho Uma métrica é uma medida verificável, expressa em termos quantitativos ou qualitativos, sendo definida em relação a uma base de referência. As métricas possuem algumas características, tais como: devem ser verificáveis; são medidas que capturam características ou resultados em uma forma numérica ou nominal; o resultado de uma métrica deve ser comparado com uma base de referência padrão ou desenvolvida; devem ser compreendidas para que sejam efetivas; devem ser baseadas em valor; devem estar vinculadas com a operação que se deseja mensurar, entre outras (MELNYK; STEWART; SWINK, 2004). As métricas possuem três funções básicas, a saber: (a) controle: são utilizadas como medidas de controle, permitindo que se avalie e controle o desempenho de uma atividade, negócio, entre outras; (b) comunicação: comunicam o desempenho para todos os gestores, trabalhadores e demais partes interessadas; e (c) melhoria: apontam as diferenças entre o resultado esperado e o alcançado, fornecendo informações para ajustes nas operações etc. (MELNYK; STEWART; SWINK, 2004). São medidas que podem ser verificáveis em termos quantitativos e qualitativos, sendo utilizadas para mensurar, controlar, comunicar e avaliar o resultado de uma atividade, projeto, departamento ou entidade. As medidas utilizadas para mensuração e avaliação de desempenho podem ser agrupadas em diversas classes (MELNYK; STEWART; SWINK, 2004), sendo a mais comum a que classifica os indicadores em financeiros e não financeiros. Cada tipo de medida reflete um aspecto da mensuração de desempenho: os indicadores financeiros indicam, em termos financeiros, os resultados dos recursos aplicados, ou o valor criado por um negócio (medidas de valor); e os indicadores não financeiros geram informações não monetárias sobre as atividades operacionais, satisfação de clientes, qualidade, entre outras. Os indicadores ou índices financeiros surgiram no final do século XIX, a partir da relação entre os recursos investidos por uma entidade e os fluxos financeiros registrados nos 67 demonstrativos contábeis, sendo utilizados para avaliação do sucesso e da lucratividade de uma entidade (LIESZ; MARANVILLE, 2008). Inúmeras medidas financeiras podem ser obtidas a partir da análise das demonstrações contábeis, tais como: lucro, lucro por ação, fluxo de caixa, fluxo de caixa descontado, retorno sobre os ativos (return on assets (ROA)), retorno sobre o patrimônio líquido (return on equity (ROE)), giro do ativo, liquidez corrente, endividamento, retorno sobre os investimentos (return on investiment (ROI)), lucro operacional, giro dos estoques, prazo médio de recebimento, capital circulante líquido (capital de giro líquido), entre outros (TANGEN, 2003; CHENHALL; LANGFIELD-SMITH, 2007; LIESZ; MARANVILLE, 2008). Os indicadores financeiros podem ser agrupados em índices de liquidez de: curto prazo (indicam a capacidade de pagamento); eficiência (grau de eficiência no emprego dos ativos operacionais); lucratividade (indicam a rentabilidade das operações); alavancagem financeira (nível de endividamento corporativo para aumentar a lucratividade); mercado (desempenho das ações da empresa no mercado); e medidas de valor (riqueza criada por um negócio). As medidas financeiras são consideradas importantes no processo de avaliação de desempenho, pois indicam a eficiência dos recursos administrados pelos gestores, representam os resultados das ações passadas e evidenciam como as decisões passadas influenciam nos resultados atuais. Entretanto, com o passar do tempo, os índices financeiros ou contábeis foram tornandose insuficientes, sendo objeto de diversas críticas, devido às seguintes limitações: foco apenas em informações de custos e controle; refletem informações decorrentes de decisões passadas; podem gerar informações distorcidas, devido à utilização de critérios subjetivos na avaliação de custos; não englobam informações sobre satisfação de clientes, produtividade e qualidade; são inconsistentes com as necessidades de cada área da organização devido à padronização; não levam ao melhoramento contínuo; não estão diretamente alinhados com a estratégia corporativa; não levam em consideração o custo de capital na avaliação desempenho, entre outras (KAPLAN; NORTON, 1992; GHALAYINI; NOBLE, 1996; ITTNER; LACKER, 1998; TANGEN; 2004). Devido aos indicadores financeiros tradicionais não levarem em consideração o custo de capital na mensuração de desempenho, foram criados indicadores financeiros de valor, também conhecidos como medidas de valor, sendo métricas utilizadas para avaliar a geração de riqueza para os acionistas e a viabilidade econômica de um investimento. As medidas de valor são indicadores financeiros que indicam o quanto de riqueza foi gerada por uma entidade, sendo obtidas a partir do lucro residual, isto é, do lucro remanescente 68 após a dedução da remuneração exigida pelos acionistas (GHALAYINI; NOBLE, 1996; ITTNER; LACKER, 1998). Dentre as medidas financeiras de valor, destacam-se o Economic Value Added (EVA®) e o Cash Flow Return on Investment (CFROI®). O EVA® pode ser definido como o a diferença entre o retorno de um investimento e o custo para financiar este investimento; caso o retorno seja superior ao custo de capital, pode-se dizer que o empreendimento gerou riqueza ou valor, sendo calculado pela dedução do custo de capital do lucro antes dos impostos sobre o lucro (SHARMA; KUMAR, 2010). Nota-se que o EVA® é uma medida residual que indica o quanto de valor uma empresa gerou utilizando suas atividades, indicando se as ações desenvolvidas pelos gestores aumentaram ou não a riqueza dos acionistas. O CFROI®, por sua vez, é uma medida que busca identificar a lucratividade gerada pelos ativos de uma companhia, representando a taxa interna de retorno dos fluxos de caixa de um empreendimento ajustados pela inflação do investimento (MADDEN, 1998). Observa-se que o CFROI® é uma versão modificada da taxa interna de retorno, com a diferença de que considera o efeito da inflação nos recursos aplicados, indicando a rentabilidade real de um investimento. As medidas de valor EVA® e CFROI® também apresentam algumas limitações. Para a obtenção do EVA®, por exemplo, devem-se realizar diversos ajustes nas variáveis utilizadas para que se obtenha o mais fiel possível a realidade econômica de uma empresa. E o cálculo do CFROI® é muito centrado nos ativos imobilizados. Tais limitações podem gerar distorções na correta avaliação do desempenho organizacional. Além disso, há evidências de que as medidas de valor são de difícil entendimento por parte dos empregados, bem como que os indicadores financeiros tradicionais e os não financeiros explicarem melhor o retorno das ações e o desempenho futuro, respectivamente (ITTNER; LACKER, 1998). Entretanto, do mesmo modo que os indicadores financeiros tradicionais, as medidas de valor também apresentam limitações, como o não alinhamento com a estratégia organizacional, bem como a necessidade de diversos ajustes nas métricas utilizadas para o cálculo da riqueza gerada por uma organização. Com base nas limitações apresentadas pelos indicadores financeiros, foram propostos indicadores não financeiros. Os indicadores não financeiros englobam informações sobre satisfação de clientes, processos internos, melhoramento de atividades, qualidade de produtos e serviços, eficiências dos empregados, entre outras, servindo como direcionadores de desempenho financeiro futuro, sendo desenvolvidos tendo como modelo a estratégia 69 organizacional (KAPLAN; NORTON, 1992; ITNER; LACKER, 1998; BANKER; POTTER; SRINIVASAN, 2000; AHMAD; ZABRI, 2016). De acordo com Ghalayiani e Noble (1996), são características dos indicadores não financeiros: medidas relacionadas à estratégia organizacional; utilização diária; medidas simples, que são compreendidas e utilizadas por todos os empregados; promovem a melhoria das atividades; auxiliam no melhoramento contínuo; e podem mudar de acordo com as necessidades e com o ambiente organizacional. Nota-se que as medidas não financeiras, por não serem oriundas das demonstrações contábeis, possuem maior flexibilidade, tempestividade, sendo de simples compreensão por parte de todos os empregados e diretores, bem como estão relacionadas diretamente à estratégia corporativa. As medidas não financeiras também podem ser vistas como medidas de desempenho futuro ou preditivas, pois podem oferecer indícios sobre o resultado financeiro futuro, por meio de indicadores de satisfação de clientes, evolução na participação de mercado, redução de defeitos, aumento de produtividade, entre outros. Observa-se que, para serem efetivas, as medidas não financeiras devem ser utilizadas de acordo com o contexto em que a organização opera, buscando mitigar os efeitos das contingências presentes no ambiente organizacional. Em resumo, considera-se que cada tipo de indicador é importante no processo de mensuração e avaliação de desempenho organizacional, devendo a organização adotar as medidas que lhes sejam úteis em cada processo, de forma que possam atingir os objetivos e metas traçadas pela corporação, com base na estratégia organizacional. Posteriormente, apresentam-se as evidências empíricas relacionadas aos temas da pesquisa, levantadas com base na revisão sistemática da literatura, ressaltando-se as principais evidências reportadas pelos estudos. 2.4 EVIDÊNCIAS EMPÍRICAS A partir da revisão sistemática da literatura, conforme critérios definidos na justificativa, apresentam-se, no Quadro 11, os estudos relacionados ao tema desta pesquisa, destacando-se os autores, o título, os métodos utilizados, os períodos investigados e os resultados encontrados por cada estudo. 70 Quadro 11 Estudos associando fatores contingenciais, estratégia e desempenho financeiro (continua) Autores Título Método-período Resultados Banker, Mashruwala e Tripathy (2014) Does a differentiation strategy lead to more sustainable financial performance than a cost leadership strategy? Dados de 12.849 empresas que negociam ações na NYSE, AMEX e NASDAQ entre 1989-2003; análise fatorial e análise de regressão. Tanto a estratégia de liderança e custos e de diferenciação impactam positivamente o desempenho atual; a estratégia de diferenciação permite que o desempenho seja sustentável no futuro do que a estratégia baseada no custo, contudo a estratégia de diferenciação está associada a um maior risco sistemático e instabilidade no desempenho. Huo et al. (2014) The impact of supply chain integration on firm performance: the moderating role of competitive strategy. Survey com 604 indústrias chinesas; regressão linear hierárquica. As estratégias competitivas influenciam as práticas de integração da cadeia de suprimentos; não apresentaram efeito moderador significativo na relação entre a integração da cadeia de suprimentos e o desempenho operacional. Jayaram, Tan e Laosirihongthong (2014) The contingency role of business strategy on the relationship between operations practices and performance. Survey com 329 empresas tailandesas; análise de clusters; regressão múltipla. Relação entre práticas de gestão operacional e o desempenho; a interação entre as práticas de produção enxuta e a gestão da cadeia de suprimentos afetam significativamente o desempenho das empresas com estratégia baseada no foco. Shin (2014) Unions and the adoption of high-performance work systems in Korea: moderating roles of firms' competitive strategies. Survey com 301 empresas coreanas realizado em 2009; análise de regressão. Os resultados sugerem que a estratégia de diferenciação é um fator-chave na resolução de conflitos entre empregadores e sindicatos em relação à adoção de sistemas de trabalho de alto desempenho. Parnell, Long e Lester (2015) Competitive strategy, capabilities and uncertainty in small and medium sized enterprises (SMEs) in China and the United States. Survey com gestores da China (166) e dos EUA (176); análise de variância (ANOVA). Os resultados dão suporte à integridade das estratégias genéricas de Miles e Snow; empresas que apresentam estratégias prospectora, analítica e defensora apresentaram melhor desempenho em relação às que apresentam uma estratégia reativa. Santos (2015) As relações entre ambidestria organizacional, capabilidades e seus impactos no desempenho organizacional, moderado pela estratégia. Questionário aplicado a 119 empresas brasileiras dos setores industrial, comercial e de serviços; modelo de equação estrutural. Os dados apontam que a variável latente capabilidade exerce efeito mediador entre ambidestria organizacional e desempenho organizacional, sendo que o efeito varia de total a parcial, dependendo do tipo de orientação estratégica adotada pelas empresas, segundo o modelo de Miles e Snow (1978). Acquaah e Agyapong (2015) The relationship between competitive strategy and firm performance in micro and small businesses in Ghana: the moderating role of managerial and marketing capabilities. Survey aplicado a 581 micros e pequenas empresas de Gana; análise de regressão hierárquica múltipla. A estratégia de diferenciação influencia o desempenho das empresas; capacidades gerenciais e de marketing moderam a relação entre a estratégia competitiva (liderança em custos e diferenciação) e o desempenho das micros e pequenas organizações, sendo que a capacidade gerencial fortalece a relação entre a estratégia de liderança em custos e o desempenho e a capacidade de marketing aumenta a influência da diferenciação sobre o desempenho organizacional. 71 Quadro 11 Estudos associando fatores contingenciais, estratégia e desempenho financeiro (continuação) Autores Título Método-período Resultados Acquaah e Agyapong (2016) Dynamic tensions from management control systems and performance in a sub-Saharan African economy: mediating effects of competitive strategy. Survey aplicado a 106 firmas de Gana; análise fatorial confirmatória. A tensão dinâmica do sistema de controle gerencial influencia a estratégia competitiva, mas também o desempenho por meio da estratégia competitiva. Agyapong, Ellis e Domeher (2016) Competitive strategy and performance of family businesses: moderating effect of managerial and innovative capabilities. Questionário aplicado a 265 micros e pequenas empresas familiares de Gana; análise de correlação e análise fatorial confirmatória. A busca por uma posição baseada na diferenciação ou no custo deve ser construída com fortes capacidades gerenciais internas; empresas familiares inovadoras devem adotar uma estratégia competitiva baseada na diferenciação. Oyewobi et al. (2016) Relationship between competitive strategy and construction organisation performance: the moderating role of organisational characteristics. Survey aplicado a 72 grandes organizações da África do Sul; regressão múltipla hierárquica. As características organizacionais moderam parcialmente a relação entre a estratégia competitiva e o desempenho organizacional. Fernando, Schneible Jr. e Tripathy (2016) Firm strategy and market reaction to earnings. Dados de 2.101 empresas extraídos do Compustat de 2001 a 2009; análise fatorial confirmatória e análise multivariada. Em empresas que adotam uma estratégia de liderança em custos, o anúncio dos lucros é mais comumente interpretado e resultam em uma maior crença na mudança do preço das ações; a divulgação de lucros pelas empresas que buscam uma estratégia de diferenciação são interpretados de forma heterogênea, resultando uma menor mudança no preço das ações. Mohsenzadeh e Ahmadian (2016) The mediating role of competitive strategies in the effect of firm competencies and export performance. Survey aplicado a gestores de 200 companhias do Irã; análise fatorial confirmatória e modelo de equação estrutural. As estratégias competitivas mediam a relação entre a capacidade de produção e o desempenho das exportações, mas não interferem na relação entre o efeito da competência em marketing com o resultado das exportações. HernándezPerlines, MorenoGarcía e YañezAraque (2016) The mediating role of competitive strategy in international entrepreneurial orientation. Questionário aplicado a 174 empresas da Espanha; modelo de equação estrutural. A influência da orientação empreendedora internacional sobre o desempenho internacional melhora com a adoção de uma estratégia competitiva, sendo a inovação uma condição necessária para que a estratégia competitiva exerça um efeito mediador. Anwar e Hasnu (2016) Business strategy and firm performance: a multiindustry analysis. Dados financeiros de 320 empresas paquistanesas de 2008 a 2013; estatística descritiva e análise de variância. Os resultados indicam que há diferença no desempenho das organizações de acordo com a estratégia competitiva, variando conforme o setor e o tamanho das empresas. 72 Quadro 11 Estudos associando fatores contingenciais, estratégia e desempenho financeiro (conclusão) Autores Título Método-período Resultados Bayraktar et al. (2017) Competitive strategies, innovation, and firm performance: an empirical study in a developing economy environment. Survey/entrevista com gestores de 140 empresas da Turquia; modelo de equação estrutural. A estratégia competitiva (diferenciação e liderança em custos) influencia a inovação e, por conseguinte, aumenta o desempenho das empresas; a inovação apresenta-se como uma variável significativa na relação entre estratégia competitiva e desempenho organizacional. Anwar e Hasnu (2017) Strategic patterns and firm performance: comparing consistent, flexible and reactor strategies. Dados financeiros de 307 empresas paquistanesas; modelos multivariados e de regressão. A maioria das empresas do Paquistão possui uma posição estratégica consistente, seguidas por flexíveis e reativas; as diferenças médias no desempenho indicam que as estratégias consistentes e flexíveis superam aquelas que apresentam uma posição reativa; há uma variação significativa no desempenho de acordo com tipo estratégico devido à variação no tamanho das empresas e do setor. Chen et al. (2018) How business strategy in non-financial firms moderates the curvilinear effects of corporate social responsibility and irresponsibility on corporate financial performance. Dados de 1.461 companhias entre 2003 e 2009; análise de regressão. A estratégia de diferenciação e de liderança em custos moderam positivamente a relação entre a responsabilidade social corporativa e o desempenho financeiro, mas afetam negativamente a relação entre a irresponsabilidade social corporativa e o desempenho financeiro. Laari, Töyli e Ojala (2018) The effect of a competitive strategy and green supply chain management on the financial and environmental performance of logistics service providers. Survey com 266 prestadores de serviços logísticos e dados financeiros; análise fatorial confirmatória. As empresas líderes com excelência operacional e marcas fortes são mais avançadas em termos de gestão de cadeia de suprimentos verdes do que as que não se destacam em qualquer prioridade estratégica, estando as práticas de gestão de cadeia de suprimentos verdes positivamente relacionadas com o desempenho ambiental, mas não com o desempenho financeiro. Tenhiälä e Laamanen (2018) Right on the money? The contingent effects of strategic orientation and pay system design on firm performance. Survey e dados financeiros de 86 empresas da Finlândia. Empresas com estratégia de prospecção apresentam um desempenho melhor quando utilizam sistemas de remuneração com ênfase em alto nível de dispersão salarial horizontal; já as empresas com estratégia defensiva utilizam sistemas de remuneração que têm alto nível de dispersão salarial vertical e baixa remuneração básica. Fainshmidt et al. (2019) When do dynamic capabilities lead to competitive advantage? The importance of strategic fit. Survey com 249 firmas de Israel; análise qualitativa comparativa fuzzy-set (fsQCA). A relação entre as capacidades dinâmicas e a vantagem competitiva depende da adequação estratégica entre fatores organizacionais e ambientais. Fonte: elaboração própria, com base na revisão sistemática da literatura. Com base nos estudos evidenciados no Quadro 11, percebeu-se que a maioria das pesquisas desenvolvidas nos últimos anos foi feita em ambientes considerados em desenvolvimento, sendo realizados em Gana e na China, por exemplo. Dentre os países mais 73 avançados economicamente, destacam-se os EUA, onde foram realizadas três pesquisas envolvendo desempenho e estratégia nos últimos cinco anos. Dentre as 20 pesquisas empíricas analisadas, 15 utilizaram dados primários para avaliar a relação entre fatores contingenciais, estratégia competitiva e desempenho organizacional. A maioria dos estudos examinou os dados por meio de análise de regressão, modelos de equações estruturais e análise fatorial confirmatória, evidenciando a utilização de vários métodos estatísticos para avaliar as relações entre as variáveis. De acordo com as informações expostas no Quadro 11, verificou-se que as pesquisas existentes abordam diversos fatores contingenciais, relacionando-os com a estratégia competitiva e o desempenho financeiro. Dentre os principais fatores contingenciais descritos por Otley (2016), o tamanho organizacional (MERCHANT, 2014; ANWAR E HASNU, 2016; ANWAR E HASNU, 2017), a incerteza do ambiente (PARNELL; LONG; LESTER, 2015; VARGAS; TREZ, 2017), a estrutura organizacional (HUO et al., 2014; JAYARAM; TAN; LAOSIRIHONGTHONG, 2014; VARGAS; TREZ, 2017; OYEWOBI et al., 2016; BAYAKTAR et al., 2017; LAARI, TÖYLI; OJALA, 2018), o sistema de controle gerencial (ORO, 2015; ACQUAAH; AGYAPONG, 2016) e o sistema de remuneração (TENHIÄLÄ; LAAMANEN, 2018) foram investigados. No que se refere aos estudos que utilizaram a estratégia competitiva como um fator que modera a influência dos fatores contingenciais sobre o desempenho organizacional, observouse tal relação com práticas de integração de cadeias de suprimentos (HUO et al., 2014), práticas operacionais e gestão da cadeia de suprimentos (JAYARAM; TAN; LAOSIRIHONGTHONG, 2014), adoção de sistemas de trabalho de alto desempenho (SHIN, 2014), sistema de controle gerencial (ORO, 2015; ACQUAAH; AGYAPONG, 2016), orientação empreendedora (HERNÁNDEZ-PERLINES; MORENO-GARCÍA; YAÑEZ-ARAQUE, 2016), capabilidades e capacidades organizacionais (SANTOS, 2015; MOHSENZADEH; AHMADIAN, 2016) e com a responsabilidade social corporativa (CHEN et al., 2018). Apenas o estudo desenvolvido por Parnell, Long e Lester (2015) buscou avaliar a relação entre fatores contingenciais (incerteza do ambiente), estratégia competitiva e desempenho organizacional, avaliando empresas dos EUA e da China, ou seja, comparando organizações que atuam em um ambiente desenvolvido com as que operam em um mercado em desenvolvimento. Segundo estes autores, devido às mudanças ocorridas na China, as companhias evitam a incerteza e buscam informações mais claras antes de implementarem estratégias, algo que não ocorre nos EUA, onde o ambiente é mais estável e as mudanças ocorrem de forma mais gradual. 74 Em resumo, pode-se dizer que a maioria dos estudos realizados não buscou verificar se o papel moderador da estratégia competitiva varia de acordo com o ambiente em que as empresas operam (países desenvolvidos ou em desenvolvimento), conforme predito pela teoria da contingência. Além disso, a maior parte das evidências encontradas se refere à influência da estratégia competitiva sobre fatores contingenciais relacionados ao ambiente interno das organizações, tais como: estrutura organizacional, sistema de controle gerencial, práticas operacionais, entre outros. 2.5 MODELO CONCEITUAL DA PESQUISA Com base no referencial teórico e na revisão da literatura realizada, elaborou-se o modelo conceitual da pesquisa (Figura 3), ou modelo teórico, em que se postula que a estratégia competitiva (diferenciação ou liderança em custos) reduz as influências dos fatores contingenciais sobre o desempenho financeiro organizacional, variando de acordo com o ambiente (país membro do BRICS e do G7). Figura 3 Modelo conceitual da pesquisa Fonte: elaboração própria, com base na revisão sistemática da literatura. De um modo geral, a Figura 3 pode ser interpretada da seguinte forma: em um ambiente corporativo, há diversos fatores que influenciam as atividades de uma organização, alguns estão presentes no ambiente externo, sendo denominados de fatores contingenciais externos (FCE), e outros são internos à organização, conhecidos como fatores contingenciais internos (FCI). 75 Para que a organização possa obter o melhor ajuste entre sua estrutura, por exemplo, e os fatores presentes no ambiente externo e interno, ela precisa escolher uma estratégia competitiva que lhe permita se adaptar e atenuar a influência dessas contingências sobre suas atividades, de forma a aperfeiçoar o seu desempenho financeiro. Neste sentido, considera-se que a estratégia competitiva atua como uma variável contingente mediadora, sendo utilizada como um meio para adequar a organização com o seu ambiente (fatores contingenciais externos) e para ajustar as políticas e procedimentos, de acordo com os fatores contingenciais internos inerentes às suas atividades, aumentando o desempenho financeiro das empresas (MILES et al., 1978; HAMBRICK, 1983; GUPTA; GOVIDARAJAN, 1984; HOQUE, 2004; CHENHALL, 2003; OTLEY, 2016; MAKADOK; BURTON; BARNEY, 2018). Já os fatores contingenciais representam as variáveis internas e externas que influenciam as características organizacionais em um determinado momento, impactando o desempenho corporativo (MCKINLEY; MONE, 2003; CHENHALL, 2003; JUNQUEIRA et al., 2016). Nesta pesquisa, os fatores contingenciais utilizados foram: incerteza do ambiente, nível de competição, tamanho e idade da organização. Consideram-se como o ambiente de atuação das empresas os países reputados como membros dos BRICS (Brasil, Rússia, Índia, China e África do Sul) e do G7 (Alemanha, Canadá, EUA, França, Itália, Japão e Reino Unido), conforme classificação adotada pelo International Monetary Fund (Fundo Monetário Internacional (FMI, 2019)). E, por fim, o desempenho financeiro pode ser visto como o resultado econômico obtido pelas entidades no final de cada período, oriundo de suas operações, sendo expresso em termos monetários. Para tanto, considerou-se o retorno dos ativos de cada empresa no período analisado. 76 3 PROCEDIMENTOS METODOLÓGICOS Neste capítulo apresentam-se os procedimentos metodológicos adotados para atingir os objetivos propostos na pesquisa. O capítulo se encontra dividido nos seguintes itens: caracterização da pesquisa; hipóteses da pesquisa; população e amostra; coleta de dados; variáveis da pesquisa; e métodos de análise. 3.1 CARACTERIZAÇÃO DA PESQUISA O presente estudo foi desenvolvido por meio de uma pesquisa empírico-analítica, que se fundamenta na utilização de técnicas estatísticas para coleta, tratamento e análise dos dados (análise fatorial exploratória e confirmatória e dados em painel dinâmico), buscando analisar a relação entre as variáveis investigadas (MARTINS; THEÓPHILO, 2016). Assim, com base no objetivo proposto, a partir de técnicas estatísticas, buscou-se analisar a relação da estratégia competitiva com os fatores contingenciais (internos e externos) sobre o desempenho financeiro de empresas, considerando o ambiente organizacional (países membros do BRICS e do G7). 3.2 HIPÓTESES DE PESQUISA 3.2.1 Incerteza do ambiente e desempenho financeiro A incerteza do ambiente é definida como a taxa de mudança ou variabilidade no ambiente externo da organização que afeta suas principais atividades, tornando imprevisíveis as ações dos clientes, fornecedores, competidores, governos e sindicatos (CHILD, 1972; CHENHALL, 2003; GOSH; OLSEN, 2009). Dentre os fatores contingenciais ou aspectos do ambiente externo, a incerteza é considerada um dos principais, visto que elevados níveis de incerteza requerem, por exemplo, a adoção de sistemas flexíveis e adaptáveis para gerenciar eventos inesperados quando ocorrem (CHENHALL, 2003; OTLEY, 2016). Devido à incerteza decorrer de fatores externos à organização, as mudanças nesses fatores afetam as principais atividades de uma corporação e, consequentemente, sua produtividade e receitas (GOSH; OLSEN, 2009). Cheng e Kesner (1997) encontraram evidências de que a incerteza do ambiente impacta o desempenho organizacional. Segundo os autores, a alta incerteza ambiental acarreta 77 variabilidade nas receitas e nos lucros e, por conseguinte, afeta a magnitude e a variabilidade do desempenho organizacional. Outros estudos têm demonstrado que a incerteza do ambiente também aumenta a participação no processo orçamentário (GOVIDARAJAN, 1984; KREN, 1992), influencia o desenho do sistema de controle gerencial (CHENHALL, 2003; OTLEY, 2016) e impacta a integração da cadeia de suprimentos e o desempenho operacional (WONG, BONN-ITT; WONG, 2011). Ainda de acordo com Parnell et al. (2015), a incerteza do ambiente influencia a formulação de estratégia, que, por sua vez, influencia o desempenho organizacional. Para os autores, um alto nível de incerteza é provavelmente associado com baixo nível de desempenho. Considerando-se que o nível de incerteza do ambiente pode afetar as atividades de uma organização, provocando alterações na estrutura organizacional, por exemplo, e, consequentemente, no desempenho, apresenta-se a seguinte hipótese de pesquisa: • H1: a maior incerteza do ambiente afeta negativamente o desempenho financeiro das organizações. 3.2.2 Nível de competição e desempenho financeiro Outro aspecto do ambiente organizacional é o nível de competição entre as empresas. O nível de competição pode ser considerado, essencialmente, como uma luta no setor, seja de bens e serviços, ou de fatores de mercado (KHANDWALLA, 1972). Em seu estudo, Khandwalla (1972) encontrou evidências de que o aumento da competitividade aumenta o controle gerencial, ou seja, que quanto maior for a competição, mais as empresas necessitam controlar os custos, avaliar a produção, marketing, finanças e outros. Nickell (1996) ao analisar a competição e o desempenho corporativo de 670 companhias do Reino Unido, verificou que a competição por mercado de produtos aumenta a produtividade e é um determinante da lucratividade das empresas. Os resultados de Kahyarara (2013) também indicam que o nível de competição aumenta a produtividade e influencia os lucros das empresas. Outros estudos sugerem que o nível de competitividade impacta os incentivos gerenciais, as operações corporativas e o risco dos negócios (KARUNA, 2007; BEINER; SCHIMID; WANZENRIED, 2011), afeta positivamente o custo das dívidas bancárias (VALTA, 2012), reduz o valor das empresas (BEINER; SCHIMID; WANZENRIED, 2011) e o desempenho (LIU; QU; HAMAN, 2018). 78 Desta forma, considerando-se que o nível de competitividade pode afetar as atividades organizacionais, gerando impactos na produtividade e na lucratividade das empresas, levantase a seguinte hipótese de pesquisa: • H2: o nível de competição do ambiente afeta negativamente o desempenho financeiro das companhias. 3.2.3 Tamanho e desempenho financeiro O tamanho de uma empresa é considerado um dos principais fatores contingenciais (OTLEY, 2016). Em comparação com entidades menores, grandes organizações apresentam maiores lucros (HALL; WEISS, 1967) e maior participação de mercado (AMATO; WILDER, 1985), tendem a ter mais poder para controlar seu ambiente operacional (CHENHALL, 2003), obtêm maiores vantagens com economias de escala (SERRASQUEIRO; NUNES, 2008), e outros. Estudos têm apresentado evidências entre tamanho e desempenho financeiro. Hall e Weiss (1967) encontraram evidências que apontam que o tamanho apresenta um efeito positivo sobre os lucros, indicando que as barreiras geradas pelas empresas, devido ao seu porte, aumentam a lucratividade. Por sua vez, Amato e Wilder (1985) indicam que não há uma relação entre tamanho e lucratividade, mas que a participação e a concentração de mercado parecem influenciar mais o desempenho das companhias. Serrasqueiro e Nunes (2008), em um estudo considerando pequenas empresas portuguesas, apontam que o desempenho é positivamente relacionado com o tamanho das firmas, sendo que as economias de escala, a diversificação de atividades e a maior habilidade das grandes empresas com mudanças de mercado contribuem para a relação positiva entre tamanho e lucratividade. Já Lee (2009), ao analisar 7.000 empresas norte-americanas no período de 1987-2006, demonstrou que as taxas de lucro eram positivamente relacionadas com o tamanho das firmas, sendo tal relação não linear e influenciada por características específicas das empresas e do setor industrial. Os resultados da pesquisa de Dogãn (2013), considerando dados de 200 companhias listadas na Turquia, indicam uma relação positiva entre o tamanho e a lucratividade das empresas. Vu et al. (2019) também encontraram uma relação positiva entre o porte das firmas 79 e o desempenho. Sendo assim, com base nas evidências empíricas, sabendo-se que o tamanho pode influenciar positivamente o desempenho das organizações, apresenta-se a seguinte hipótese de pesquisa: • H3: o tamanho (porte) afeta positivamente o desempenho financeiro das empresas. 3.2.4 Idade da organização e desempenho financeiro A idade da organização é vista como um fator contingencial que afeta a estrutura organizacional. Segundo Greiner (1972), as organizações buscam efetuar ajustes em suas estruturas ao longo dos anos para se adequarem às mudanças ocorridas no ambiente. Anthony e Ramesh (1992), ao analisarem a influência do ciclo de vida, mensurado com base na idade das companhias, sobre a relação entre desempenho e valor das ações, apontam que há uma redução no crescimento das vendas e investimentos de capital na transição da fase de crescimento para a de estagnação. Já Yazdanfar e Öhman (2014) encontraram evidências que sugerem que pequenas empresas mais jovens tendem a apresentar melhor desempenho em termos de crescimento e lucratividade do que as empresas mais maduras. Além disso, as maiores empresas apresentam melhor desempenho do que as pequenas. Warusawitharana (2018) em um estudo com companhias do Reino Unido, reporta que a lucratividade aumenta com o tempo para organizações mais jovens, permanecendo elevada, reduzindo lentamente quando as companhias atingem a maturidade. Diante disso, propõe-se a seguinte hipótese: • H4: a idade da organização pode afetar positivamente o desempenho financeiro. 3.2.5 Papel moderador da estratégia competitiva De acordo com a teoria da contingência, a estratégia competitiva deve ser utilizada para adequar a estrutura organizacional aos fatores contingenciais presentes no ambiente corporativo. Contudo, não basta que haja somente adequação entre a estrutura e as características do ambiente, mas que a corporação possa utilizar uma estratégia competitiva para melhor se defender das forças competitivas ou influenciá-las (PORTER, 2004), ajustando a sua estrutura corporativa, uma vez que a estratégia é quem determina a estrutura, e sua 80 adequação é relevante para o desempenho organizacional (CHANDLER, 1962; JUNQUEIRA et al., 2016). A estratégia organizacional é vista como um meio utilizado pelas organizações para interagir com o seu ambiente, buscando reduzir os impactos dos diversos fatores contingenciais sobre suas operações e alcançar o desempenho esperado (CHANDLER, 1962; CHILD, 1972; MILES et al., 1978; PORTER, 1980; GUPTA; GOVINDARAJAN, 1984; ANWAR; HASNU, 2017). Segundo Porter (2004), para enfrentar as forças competitivas presentes no mercado, as organizações precisam adotar uma estratégia que seja capaz de gerar vantagens competitivas no longo prazo, tornando-as lucrativas. Para o autor, apenas três estratégias competitivas podem criar uma posição defensável em longo prazo para as companhias, garantindo-lhes vantagens competitivas frente às ameaças de mercado, são elas: liderança em custos, diferenciação e enfoque. Estas três estratégias competitivas são fundamentadas em duas fontes de vantagem competitiva: custo ou diferenciação. Nesse caso, considera-se que a estratégia competitiva adotada pela organização pode atuar para reduzir a influência de múltiplos fatores contingentes presentes no ambiente organizacional (OTLEY, 2016) e que desempenha um papel importante na interação da entidade com o ambiente (MILES et al., 1978; GUPTA; GOVINDARAJAN, 1984; ANWAR; HASNU, 2017), contribuindo para a geração de vantagens competitivas que colaboram para o desempenho financeiro (PORTER, 2004). Estudos sobre o papel moderador da estratégia competitiva ainda são escassos. Huo et al. (2014) apresentaram evidências de que a estratégia competitiva influencia significativamente a efetividade das práticas de integração de cadeia de suprimentos, incluindo integração de produtos (interna e de processos). Segundo os autores, a integração interna afeta significativamente o desempenho financeiro das empresas líderes em custos, enquanto a integração de processos contribui mais para o desempenho financeiro das empresas com estratégia competitiva baseada na diferenciação. Jayaram, Tan e Laosirihongthong (2014) ao analisarem o papel contingencial da estratégia sobre a relação entre as práticas operacionais e o desempenho, apontaram que a estratégia baseada em foco influencia na relação entre as práticas de gestão operacional e o desempenho. Já Shin (2014), apresentou evidências de que a estratégia de diferenciação pode ser considerada efetiva na resolução de conflitos entre empregadores e sindicatos quando da adoção de sistemas de trabalho de alto desempenho. Santos (2015) mostrou que a orientação estratégica pode influenciar a relação entre as capabilidades, a ambidestria e o desempenho organizacional, 81 sendo que o efeito pode variar de parcial a total, dependendo da estratégia, conforme modelo sugerido por Miles e Snow (1978). Hernández-Perlines e Mancebo-Lozano (2016) reportaram que a estratégia competitiva modera a relação entre a internacionalização das empresas e o desempenho internacional, sendo influenciada pelo ambiente. Mohsenzadeh e Ahmadian (2016) também acharam indícios de que as estratégias competitivas mediam a relação entre a capacidade de produção e o desempenho das exportações, mas não encontraram sinais de que interferem na relação entre o efeito da competência em marketing com o resultado das exportações. Acquaah e Agyapong (2016) identificaram que a tensão dinâmica do sistema de controle gerencial influencia a estratégia competitiva adotada pelas empresas, sendo que a estratégia competitiva afeta a relação entre a tensão dinâmica do sistema de controle gerencial e o desempenho organizacional. Tenjiälä e Laamanem (2018) apontaram que empresas com um sistema de remuneração baseada em alto nível de desempenho e com estratégia prospectora apresentam melhor desempenho corporativo. Já as empresas com estratégia defensiva apresentam melhores resultados quando utilizam sistemas de remuneração que têm alto nível de dispersão salarial vertical e baixa remuneração básica. Em seu estudo, Chen et al. (2018), encontraram evidências de que a estratégia de diferenciação e de liderança em custos moderam positivamente a relação entre a responsabilidade social corporativa e o desempenho financeiro, mas afetaram negativamente a relação entre a falta de responsabilidade social corporativa e o desempenho financeiro. Os resultados desses estudos indicam que a estratégia competitiva pode atuar como um fator contingencial mediador, moderando (reduzindo) os efeitos da estrutura da cadeia de suprimentos, do sistema de controle gerencial, do sistema de remuneração, da responsabilidade social corporativa, entre outros sobre o desempenho organizacional. Com base nessas evidências, entende-se que a estratégia competitiva também pode moderar as influências de outros fatores contingenciais, relacionados ao ambiente externo (nível de competitividade, incerteza do ambiente e cultura nacional) e interno (tamanho e ciclo de vida), sobre o desempenho financeiro das empresas. Com isso, apresenta-se a seguinte hipótese de pesquisa: • H5: a estratégia competitiva pode reduzir (moderar) a influência dos fatores contingenciais sobre o desempenho financeiro, variando de acordo com o ambiente organizacional. 82 3.3 POPULAÇÃO E AMOSTRA A população desta pesquisa foi composta por empresas de capital aberto listadas em bolsas de valores de países membros do BRICS (Brasil, Rússia, Índia, China e África do Sul) e do G7 (grupo dos países mais industrializados do mundo). De acordo com o FMI (2019), o G7 é formado pelas economias mais avançadas e desenvolvidas do mundo, sendo composto por Alemanha, Canadá, EUA, França, Itália, Japão e Reino Unido. Já o BRICS é um grupo de países composto por Brasil, Rússia, Índia, China e África do Sul. Os BRICS são países que se destacaram por apresentarem elevado potencial e rápido crescimento econômico, tornando-se líderes no desenvolvimento de novos bens e serviços (FEDATO; PIRES; TREZ, 2017). Desta forma, a escolha das empresas pertencentes aos países membros do G7 e do BRICS possibilita analisar a relação da estratégia competitiva com os fatores contingenciais (internos e externos) sobre o desempenho financeiro das organizações, considerando-se a característica do ambiente em que atuam. Por sua vez, a amostra foi constituída pelas companhias não financeiras que possuíssem dados para análise considerando-se o período de 2012 a 2018. Foram excluídas também da amostra, empresas de utilidade pública, visto que essas organizações atuam em ambiente regulado, podendo influenciar no desempenho e na estratégia competitiva (BANKER; MASHRUWALA; TRIPATHY, 2014). A amostra final da pesquisa foi constituída por 775 empresas (5.425 observações), sendo 172 da China, 33 da Índia, 48 da Alemanha, 25 do Canadá, 307 dos EUA, 169 do Japão e 21 do Reino Unido. As empresas do Brasil, da África do Sul, da Rússia, da França e da Itália foram excluídas da amostra, visto que não apresentaram informações sobre gastos com P&D e número de empregados suficientes para estimar a estratégia competitiva. A distribuição das empresas por ambiente (desenvolvido ou em desenvolvimento) e por país encontra-se apresentada na Tabela 1. Tabela 1 Amostra da pesquisa por ambiente Ambiente País Empresas % Observações % BRICS China 172 22,19 1.204 22,19 Índia 33 4,26 231 4,26 G7 Alemanha 48 6,19 336 6,19 Canadá 25 3,23 175 3,23 EUA 307 39,61 2.149 39,61 Japão 169 21,81 1.183 21,81 Reino Unido 21 2,71 147 2,71 Total 775 100,00 5.425 100,00 Fonte: elaboração própria, a partir de dados da Thomson Reuters Eikon®. 83 De acordo com a Tabela 1, é possível notar que a maioria das empresas é de países membros do G7, concentrando mais de 70% das observações da pesquisa. Dentre eles, destacam-se os EUA, que apresentam a maioria das firmas da amostra (40,10%). Em seguida, encontra-se a China, com cerca de 22% das companhias da amostra. Em seguida, vêm o Japão (21,81%), a Alemanha (6,19%), a Índia (4,26%), o Canadá (3,23%) e o Reino Unido (2,71%). Isso demonstra a heterogeneidade da amostra da pesquisa. Apresenta-se, na Tabela 2, a classificação das empresas por indústria, agrupadas com base na classificação da Global Industry Classification Standards (GICS®). Tal procedimento visou padronizar a classificação das empresas por indústrias. Tabela 2 Classificação das empresas por indústria e país Indústria China Índia Alemanha Canadá EUA Japão Reino Unido Total Alimentos - - - 12 13 25 Automóveis e componentes 19 10 8 9 8 54 Bens de capital 4 9 71 62 2 148 Bens de consumo e aparelhos - - - 5 - 05 Equipamentos e serviços de saúde - 4 41 6 51 Farmácia e biotecnologia 12 7 25 28 7 79 Hardware e equip. tecnológicos - - - 25 - 25 Materiais 153 7 4 18 18 45 3 248 Produtos pessoais e domésticos - - - 7 7 14 Semicondutores e equipamentos - 6 - - - 06 Serviços profissionais e comerciais - - - 4 - 04 Software e serviços - 10 7 90 9 116 Total 172 33 48 25 307 169 21 775 Fonte: elaboração própria, a partir de dados da Thomson Reuters Eikon®. Com base na Tabela 2, observa-se que a indústria que mais apresentou empresas na amostra foi a de materiais (248), agrupando firmas de todos os países, seguida pela de bens de capital (148) e de software e serviços (116). A indústria que menos apresentou firmas foi a de bens de consumo e aparelhos, com apenas 5 empresas. Ainda de acordo com a Tabela 2, pode-se dizer que a amostra da pesquisa, considerandose a classificação das indústrias, está mais diversificada nos países do G7, enquanto que da China e Índia, duas indústrias (automóveis e componentes e materiais) concentram a maioria das empresas. Por fim, ressalta-se que a análise dos resultados foi realizada separando-se as empresas com base no ambiente (BRICS e G7), sendo a amostra final composta por 775 companhias de 7 países, classificadas em 10 indústrias, no período de 2012 a 2018, totalizando 5.425 observações. 84 3.4 COLETA E TRATAMENTO DOS DADOS Os dados para a realização deste estudo foram coletados no banco de dados da Thonsom Reuters® e, quando necessário, nas demonstrações contábeis e/ou em outros relatórios disponibilizados pelas empresas, como o relatório da administração ou de sustentabilidade, e, quando necessário, no sítio das companhias na Internet. Já as informações sobre o crescimento econômico de cada país foram obtidas no sítio do FMI (2019). Para a obtenção das variáveis financeiras da pesquisa, consideraram-se as demonstrações consolidadas das companhias, quando disponíveis na base de dados da Thonsom Reuters®, ou os dados não consolidados quando aqueles não estavam disponíveis. Além disso, todos os dados contábeis-financeiros foram padronizados para o dólar norte-americano, uma vez que tais informações eram divulgadas nas moedas de cada país. Esse procedimento permitiu a comparabilidade dos dados contábeis, bem como reduziu os efeitos da variação na taxa de câmbio no desempenho financeiro das organizações (PAREDES; WHEATLEY, 2017). 3.5 VARIÁVEIS DA PESQUISA As variáveis utilizadas nesta pesquisa foram determinadas com base na literatura observada em estudos anteriores (CHENHALL, 2003; BALSAM; FERNANDO; TRIPATHY, 2011; BANKER; MASHUWALA; TRIPATHY, 2014; OTLEY, 2016). A seguir, apresentamse as variáveis dependente, explicativas e de controle. 3.5.1 Variável dependente: desempenho financeiro A variável dependente deste estudo foi o desempenho financeiro de cada companhia para o período analisado, sendo obtido com base no retorno dos ativos (ROA). O retorno dos ativos é a medida financeira mais utilizada para avaliar o desempenho financeiro de uma organização e é um dos indicadores de desempenho mais utilizados por analistas (BARBER; LYON, 1996; BANKER; MASHUWALA; TRIPATHY, 2014; LUOMA, 2015), justificando sua escolha como proxy para desempenho financeiro. O ROA foi calculado conforme Equação 1: ROAit = LAJTit ATMit (1) Em que, 85 ROAit é o retorno dos ativos da empresa i no período t; LAJTit é o lucro antes dos juros e tributos sobre o lucro da entidade i no período t; e ATMit é o ativo total médio da organização i no período t, calculado pela soma do ativo total inicial com o ativo total no final do período dividido por dois. 3.5.2 Variáveis explicativas As variáveis explicativas utilizadas nesta pesquisa foram: a estratégia competitiva, o nível de competitividade, a incerteza do ambiente, a cultura nacional, o ciclo de vida organizacional e o tamanho. a) Estratégia competitiva Para identificar a estratégia competitiva de cada organização, adaptaram-se os constructos desenvolvidos por Tripathy (2006), fundamentados nas estratégicas genéricas de Porter (1980, 2004). Segundo Balsam, Fernando e Tripathy (2011), com base na literatura e nas estratégias competitivas de Porter (1980), Tripathy (2006) propôs seis índices para mensurar e identificar a estratégia de liderança no custo e de diferenciação a partir dos dados contábeis, ou seja, das informações contábeis divulgadas pelas empresas. Desta forma, a estratégia baseada no foco não será considerada neste estudo, uma vez que os referidos constructos não contemplam variáveis para sua identificação, bem como por se necessária a obtenção de informações sobre as operações, como gastos com publicidade e propaganda para cada linha de produtos e serviços por grupos de clientes, que não são divulgadas pelas companhias. As variáveis utilizadas por Tripathy (2006) envolvem informações sobre gastos com P&D, número de empregados, investimento em ativo imobilizado, despesas com vendas, gerais e administrativas, custos dos bens e serviços vendidos e ativo imobilizado liquido. Contudo, optou-se por utilizar o índice valor adicional em relação à receita líquida (VA/RL) em vez da relação entre custos e receitas, conforme sugerido por Hambrick (1983), visto que tal índice pode indicar o valor diferenciado gerado pela empresa ao cliente, ou seja, é um elemento capaz de captar a estratégia de diferenciação. Os índices foram obtidos conforme descrito a seguir: ● DVGA/RL: relação entre despesas de vendas, gerais e administrativas (DVGA) e receitas líquidas (RL). Captura os investimentos da empresa nas atividades para diferenciar seus produtos e serviços dos ofertados por seus concorrentes; 86 ● P&D/RL: relação entre despesas com pesquisa e desenvolvimento e receitas líquidas (RL). Alto investimento em P&D pode indicar que a empresa possui uma estratégia de diferenciação; ● VA/RL: relação entre o valor adicionado bruto (diferença entre as receitas líquidas e os custos dos bens e serviços vendidos) e as receitas líquidas. Esta variável pode captar tanto a estratégia de diferenciação, visto que empresas com essa estratégia tendem a gerar um valor único ou diferenciado para os clientes, ou capturar uma estratégia baseada no foco, visto que representam o valor agregado na cadeia produtiva. Altos valores para a relação VA/RL indicam uma estratégia de diferenciação (HAMBRICK, 1983); ● CAPEX/RL: relação entre as despesas de capital em imobilizado (capital expenditures (CAPEX)) e as receitas líquidas (RL). A relação entre CAPEX/RL indica uma utilização mais eficiente dos ativos; ● AIL/RL: relação entre o ativo imobilizado líquido (AIL) e as receitas líquidas (RL). A relação entre AIL/RL também indica uma utilização mais eficiente dos ativos; ● AT/NE: relação entre os ativos totais (AT) e o número de empregados (NE). É utilizada para medir a eficiência dos empregados. Baixos valores nas variáveis CAPEX/RL, AIL/RL e AT/NE são consistentes com a estratégia de liderança em custos, visto que empresas com essa estratégia primam pela eficiência operacional para reduzir seus custos (HAMBRICK, 1983; BALSAM; FERNANDO; TRIPATHY, 2011; BANKER; MASHUWALA; TRIPATHY, 2014). Já os maiores valores nas variáveis DVGA/RL, P&D/RL e VA/RL são consistentes com uma estratégia de diferenciação, uma vez que organizações que buscam diferenciar seus produtos no mercado investem ativamente em P&D, marketing e publicidade, por exemplo, para obter lucros superiores aos dos seus concorrentes (PORTER, 1980; HAMBRICK, 1983; BALSAM; FERNANDO; TRIPATHY, 2011; BANKER; MASHUWALA; TRIPATHY, 2014). A partir das variáveis CAPEX/RL, AIL/RL e AT/RL, foram obtidos os constructos para identificar as empresas que utilizam uma estratégia competitiva baseada na liderança no custo total, enquanto os resultados das medidas DVGA/RL, P&D/RL e VA/RL compuseram o outro constructo que indicará as firmas que apresentam uma estratégia de diferenciação dos produtos e serviços. Para identificar a orientação estratégica de cada companhia (liderança em custos e diferenciação), calculou-se a média dos cinco anos para os períodos 2011-2015, 2012-2016 e 87 2013-2017 dos dados analisados, conforme sugerido por Tripathy (2006), Balsam, Fernando e Tripathy (2011) e Banker, Mashuwala e Tripathy (2014). Posteriormente, a pontuação fatorial de cada variável foi utilizada para calcular, com base na observação anual de cada empresa, o fator padronizado para cada constructo, e a pontuação padronizada de cada fator foi utilizada como as medidas para estratégia, ou seja, como proxy para a estratégia de liderança em custos e de diferenciação. b) Nível de competição O nível de competição ou hostilidade do setor representa um aspecto do ambiente externo, sendo caracterizado pela intensa competitividade entre as empresas no que se refere à qualidade, variedade, promoção e distribuição de bens e serviços (KHANDWALLA, 1972; OTLEY, 2016). Diferentes níveis de competição (hostilidade) afetam o preço, a qualidade, a promoção e comercialização de produtos e serviços no mercado, podendo influenciar a estrutura e o desempenho organizacional (KHANDALLA, 1972; CHENHALL, 2003). Neste sentido, espera-se que, quanto maior for a competição, maior será a variação na lucratividade das empresas, porém se presume que a utilização das estratégias competitivas (baixo custo e diferenciação) possa mitigar os efeitos da alta competitividade no mercado. Para calcular o nível de competição no setor, utilizou-se como proxy o índice de Herfindahl, que indica o grau de concorrência entre as companhias de um determinado setor, podendo ser calculado da seguinte forma: HI = ∑ (pi) 2K i=1 (2) Em que, HI é o nível de competitividade, ou índice de Herfindahl; pi é a parcela decimal de mercado de cada uma das firmas; e K é o número de empresas que atuam no mercado (setor). O HI considera todas as firmas do mercado. Desse modo, a entrada de novas empresas ou saída de companhias já estabelecidas no mercado afeta o resultado do índice. O valor do resultado do índice varia entre 1 e 1/K, sendo que um HI =1 denota a presença de monopólio, um valor abaixo de 0,10 indica alta baixa concentração no mercado; entre 0,10 e 0,18 sugere um ambiente com competição moderada e um índice acima de 0,18 aponta um mercado com 88 baixa competitividade. A presença de um grande número de firmas no mercado (K elevado) fará com que o índice HI tenda a zero, ou seja, apresentará uma situação de alta competitividade (DANTAS et al., 2012; VALTA, 2012; FERREIRA; CIRINO, 2013). Para identificar o nível de competição de cada empresa no setor industrial, utilizou-se a participação de cada companhia na receita total do setor industrial, conforme calculado pelo índice HI. Dessa forma, a partir do nível de competição entre as empresas, pode-se observar a influência desta variável contingente sobre o desempenho financeiro das empresas, bem como se a estratégia competitiva atenua seus efeitos. c) Incerteza do ambiente O ambiente organizacional pode ser classificado em interno e externo. O ambiente interno refere-se a todos os fatores que atuam dentro da organização, como os objetivos e metas da empresa, natureza dos produtos e serviços, processos de comunicação, conhecimento dos empregados, entre outros. Já o ambiente externo compreende todas as variáveis externas à entidade, como clientes, fornecedores, concorrentes, governos e sindicatos (TUNG, 1979). Para a mensuração da incerteza do ambiente, considerou-se apenas a incerteza do ambiente externo, sendo definida como a mudança ou a variabilidade ocorrida nos fatores externos à organização (TUNG, 1979; GOSH; OLSEN, 2009). A incerteza do ambiente pode ser mensurada por características do mercado e tecnológicas. Contudo, Gosh e Olsen (2009) e Habib, Hossain e Jiang (2011) sugerem que as vendas de bens e serviços, por representarem uma característica do mercado das empresas, podem ser consideradas uma medida que melhor reflete a incerteza do ambiente, pois refletiria as mudanças provocadas pelos clientes, concorrentes, entre outras variáveis presentes no mercado externo. Destarte, a incerteza do ambiente externo foi mensurada pelo coeficiente de variação das vendas para cada empresa, sendo calculada da seguinte forma: CV(Zi) = √∑ (zi−??) 2 ?? n k=1 ?? (3) Em que, Zi é a incerteza de mercado para a organização i no período k, representado pelo coeficiente de variação das vendas; e z é a média das vendas para o período. 89 Para reduzir os efeitos do setor sobre a incerteza de cada empresa, Gosh e Olsen (2009) e Habib, Hossain e Jiang (2011) sugerem ajustar a medida de incerteza ponderando-a pela incerteza do setor em que a companhia atua. Ou seja, pelo coeficiente de variação de vendas de cada setor ao qual a entidade pertence. Espera-se que a incerteza do ambiente tenha um efeito negativo sobre o desempenho financeiro, uma vez que o nível de incerteza afeta operações corporativas e, posteriormente, podem reduzir o desempenho financeiro. d) Idade da organização A idade de cada organização foi mensurada em anos, sendo calculada pela diferença entre o ano mais recente e o ano de fundação da organização (ANTHONY; RAMESH, 1992; BANKER; MASHUWALA; TRIPATHY, 2014). e) Tamanho O tamanho refere-se ao porte de uma organização, podendo ser mensurado pelo número de empregados, total do ativo, volume de vendas, entre outras variáveis (BANKER; MASHUWALA; TRIPATHY, 2014; DONALDSON, 2015). À medida que uma empresa aumenta de tamanho, precisa realizar ajustes na sua estrutura corporativa para reduzir a incerteza das atividades e continuar operando de forma eficiente, como: contratar novos empregados, criar novos departamentos, aumentar decisões administrativas, maior controle gerencial, e outros (DONALDSON, 2015). O tamanho organizacional foi calculado a partir da Equação 4: Tit = LNEit (4) Em que, Tit é o tamanho da organização i no período t; e LNEit é o logaritmo natural do número de empregados da entidade i no período t. Devido a problemas de multicolinearidade, como medida alternativa, classificaram-se as companhias a partir de cada quartil de Tit em pequenas, médias e grandes. O quartil 1 representou as firmas consideradas de pequeno porte, os quartis 2 e 3 as de porte médio e o quartil 4 as empresas de grande porte. A partir dessa classificação, foram criadas duas variáveis 90 dummies, nomeadas de PEQit e GPit, que representam as 25% menores e maiores entidades, respectivamente, em cada período. Neste sentido, espera-se que a variável GPit apresentem uma relação positiva com o desempenho organizacional, uma vez que firmas de maior porte podem deter maior participação no setor, melhores controles operacionais, gerando ganhos de eficiência que aumentam o desempenho organizacional. 3.5.3 Variáveis de controle Nos modelos econométricos foram incluídas como variáveis de controle o nível de endividamento e o setor econômico de cada empresa. A inclusão dessas variáveis visa controlar os efeitos de mudanças no nível de endividamento e de características do setor industrial sobre o desempenho financeiro das empresas. O endividamento é um índice que indica de que maneira uma organização financia suas atividades, revelando o quanto de capital de terceiros está sendo utilizado para alavancar as operações de uma empresa (MACHADO; MEDEIROS; EID JÚNIOR, 2010). O nível de endividamento foi obtido por meio da seguinte equação: Endit = POit ATit (5) Em que, Endit é o endividamento da entidade i no período t; POit é o passivo oneroso da firma i no período t; e ATit é o ativo total da organização i no período t. Segundo Myers (1984), o grau de endividamento afeta o custo de capital e influencia o desempenho organizacional. Portando, espera-se uma relação negativa entre o nível de endividamento e o desempenho financeiro, pois, quanto maior for o grau de endividamento das empresas, maiores serão as despesas com juros e, por conseguinte, menor será a lucratividade. A variável setor industrial foi utilizada com o objetivo de controlar os efeitos do ramo de atividade cada organização (BANKER; MASHUWALA; TRIPATHY, 2014; FERNANDO; SCHNEIBLE JR.; TRIPATHY, 2016). O setor foi mensurado a partir de variáveis dummies (DInd), que assumem valor igual a 1 ou 0, de acordo com o setor industrial de atuação de cada empresa. 91 Além disso, por se tratar de uma pesquisa que envolve diversos ambientes (países), decidiu-se adicionar ao modelo econométrico duas variáveis para captar as características dos países, são elas: país e crescimento econômico. A variável país buscou controlar as características de cada ambiente (país) sobre o desempenho das empresas, sendo calculada por meio de uma variável dummy (Dpaís), que assume valor igual a 1 ou 0, conforme o ambiente (país) em que a empresa está localizada. Já a variável crescimento econômico visou captar a influência da economia de cada país, uma vez que cada nação possui diferenças no nível de crescimento econômico. O crescimento econômico de cada país foi mensurado pela taxa de crescimento anual do produto interno bruto (PIB), em dólares americanos, calculada em relação ao ano anterior. 3.6 MÉTODOS DE ANÁLISE 3.6.1 Análise fatorial exploratória e confirmatória A análise fatorial exploratória foi utilizada com o objetivo de capturar os padrões comuns entre os itens que compõem os constructos das estratégias competitivas (liderança em custos e diferenciação). Com base na análise fatorial exploratória, observou-se a adequação dos itens para cada constructo (HAIR JR. et al., 2009). Para tanto, foram verificados os seguintes critérios: (a) tipos de variáveis: a maioria foi contínua; (b) tamanho da amostra: deve ser de, no mínimo, 5 observações para cada variável, sendo utilizadas 3 para o constructo de liderança em custos e 3 para diferenciação, totalizando 6 variáveis, e sendo necessárias, no mínimo, 30 observações, tendo amostra deste estudo considerado mais de 5.000 observações; e (c) padrão de correlação: o padrão de correlação entre as variáveis deve apresentar cargas fatoriais maiores que 0,30 na análise fatorial confirmatória (HAIR JR. et al., 2009). A adequação da amostra foi obtida por meio do teste de Kaiser-Meyer-Olklin (KMO). A estatística KMO varia entre 0 e 1, sendo que, quanto mais próximo de 1, melhor é adequação dos dados. O valor aceitável para o teste de KMO é de 0,50. Já a significância foi verificada a partir do teste de esfericidade de Bartlett. Esse teste mensura a significância estatística de que a matriz de correlações apresenta correlações significantes entre as variáveis. O valor da significância do teste (p-valor) deve ser inferior a 0,05 (p < 0,05) (HAIR JR. et al., 2009). Para extrair os fatores, empregou-se o método dos componentes principais. Esse método considerou a variância total dos dados. Já o critério utilizado para avaliação dos fatores foi a análise de comunalidades. A comunalidade indica o total de variância que uma variável 92 compartilha com as demais variáveis incluídas na análise. Deve-se avaliar a comunalidade para verificar se as variáveis atendem aos níveis aceitáveis de explicação, sendo admitidos variáveis com valores maiores ou iguais a 0,50 (HAIR JR. et al., 2009). Por sua vez, a análise fatorial confirmatória foi empregada para testar se as variáveis utilizadas para identificar as estratégias competitivas (liderança em custos e diferenciação) representam adequadamente esses constructos. O objetivo da análise fatorial confirmatória é medir o grau de ajustamento entre os dados observados com o previsto na teoria. Para avaliar a validade dos constructos, deve-se analisar a validade convergente (variância comum compartilhada entre as variáveis do construto) e a validade discriminante (indica em que grau um construto é diferente das demais) (HAIR JR. et al., 2009). A validade convergente pode ser obtida: (a) por meio das cargas padronizadas, as quais devem apresentar valores entre 0,50 ou mais, sendo valores iguais ou maiores que 0,70 o ideal; (b) pelo percentual médio da variância extraída (average variance extracted), cujos valores aceitáveis devem ser superiores a 0,50; (c) pelo alfa de Cronbach, em que um valor entre 0,6 e 0,7 é considerado aceitável e um valor acima de 0,7 ou mais sugere um bom nível de confiabilidade (FIELD, 2009; HAIR JR. et al., 2009). Para avaliar a validade convergente, observou-se o percentual médio da variância extraída. Dois testes podem ser utilizados para mensurar a validade discriminante da análise fatorial confirmatória: a análise das cargas cruzadas e o critério Fornell-Lacker. Na análise das cargas cruzadas, observa-se se a carga do indicador relativa ao construto latente associado está mais alta do que a carga por ele apresentada nos demais constructos do modelo. Por sua vez, no critério de Fornell-Lacker, verifica-se se um construto latente compartilha mais variância com seus indicadores do que com qualquer outra medida latente do modelo. Nesse critério, o valor da variância média extraída de cada constructo latente deve ser maior que a correlação quadrática mais alta de qualquer outro constructo latente (HAIR; RINGLE; SARSTEDT, 2011). O critério de Fornell-Lacker foi utilizado para analisar a validade discriminante dos constructos. Portanto, a partir da análise fatorial confirmatória, verifica-se se as variáveis utilizadas representam as estratégias de diferenciação e liderança em custos, conforme previsto pela literatura, ou seja, que empresas que competem com baixo custo buscam maior eficiência de suas atividades, enquanto firmas que buscam diferenciar seus produtos no mercado realizam altos investimentos em P&D, por exemplo. 93 3.6.2 Análise de dados em painel Para analisar a relação da estratégia competitiva com os fatores contingenciais (internos e externos) sobre o desempenho financeiro de empresas localizadas em países desenvolvidos e em desenvolvimento, utilizou-se a metodologia de dados em painel na forma dinâmica. A aplicação de dados em painel permite a análise das mesmas empresas em vários períodos, o que possibilita uma análise mais acurada do relacionamento entre as variáveis, podendo o painel ser balanceado ou desbalanceado (GUJARATI; PORTER, 2011). Segundo os mesmos autores, a técnica de dados em painel apresenta as seguintes vantagens: considera a heterogeneidade das variáveis, oferece mais informações acerca das observações, é mais adequada para examinar a dinâmica das variáveis, entre outras. O modelo de dados em painel na forma dinâmica é aplicável quando há a inclusão da variável dependente defasada entre as variáveis explicativas (BALTAGI, 2005). A estimação do modelo econométrico a partir de outras técnicas estatísticas, como o método dos mínimos quadrados ordinários, poderia gerar coeficientes enviesados e inconsistentes, uma vez que a variável dependente defasada é correlacionada com os erros da regressão e devido à existência de endogeneidade das variáveis explicativas (BALTAGI, 2005). Para resolver tais problemas, Arrelano e Bond (1991) propuseram o método dos momentos generalizados em primeira diferença (generalized method of moments (GMM)). O método consiste em eliminar os efeitos fixos por meio da primeira diferença entre as equações do modelo e na utilização de valores da variável defasada em um ou mais períodos como variáveis instrumentais válidas na equação, ou, em último caso, em dois ou mais períodos, caso as primeiras variáveis instrumentais não sejam válidas (ARRELANO; BOND, 1991). A consistência e a robustez do modelo dependem da premissa de ausência de correlação serial nos erros da regressão e da validade dos instrumentos adicionais (BALTAGI, 2005; LABRA; TORRECILLAS, 2018). Para que os estimadores da regressão sejam consistentes, deve-se rejeitar a hipótese de ausência de autocorrelação de primeira ordem, mas aceitar o de segunda ordem. Em seguida, aplicou-se o teste de Hansen para sobreidentificação e para aferir se os instrumentos utilizados são válidos. 3.6.3 Modelos empíricos Para analisar a estratégia competitiva (liderança em custos e diferenciação) que pode reduzir a influência dos fatores contingenciais (incerteza do ambiente, nível de competitividade, 94 cultura nacional, ciclo de vida e tamanho organizacional) sobre o desempenho financeiro de empresas, considerando-se o ambiente em que estão localizadas (países desenvolvidos e em desenvolvimento), adaptou-se o modelo proposto por Banker, Mashuwala e Tripathy (2014). Com base nos objetivos da pesquisa, a relação entre as variáveis da pesquisa foi estimada, a partir do modelo empírico de Banker, Mashuwala e Tripathy (2014), acrescentado das variáveis referentes aos fatores contingenciais, crescimento econômico, setor industrial e país, conforme Equação 6, a seguir: DFit+1 = α0 + α1DFit + α2LCit + α3DIFit + α4NCjt + α5INCit + α6IDit + α7PEQit + α8GPit + α9ENDit + α10CEjt + α11DInd + α12DPaís + it (6) Em que, DFit é o desempenho financeiro da companhia i no período t, calculado com base nos retornos dos ativos (ROA); DFit-1 é o desempenho financeiro da companhia i no período t-1, calculado com base nos retornos dos ativos (ROA); LCit o escore fatorial para a estratégia liderança em custos para a empresa i no período t; mensurada com base no constructo obtido a partir da média de t-1 a t-5 das variáveis RL/CAPEX, RL/AIL e NE/AT; DIFit é o escore fatorial para a estratégia de diferenciação da empresa i no período t; mensurada com base no constructo obtido, a partir da média de t-1 a t-5 das variáveis DVGA/RL, P&D/RL e VA/RL; NCit é o nível de competividade no setor j no período t, calculado com base no índice de Herfindahl; INCjt é o grau de incerteza da companhia i no período t, mensurado com base no coeficiente de variação das receitas; IDit é a idade da organização i no período t, mensurada com base na idade de cada empresa; PEQit é uma variável dummy que representa valor igual a 1, se a empresa i no período t for considerada de pequeno porte, ou 0, caso contrário, mensurada a partir da classificação em quartis do logaritmo natural do número de empregados; 95 GPit é uma variável dummy que representa valor igual a 1, se a empresa i no período t for considerada de grande porte, ou 0, caso contrário, mensurada a partir da classificação em quartis do logaritmo natural do número de empregados; ENDit é o grau de endividamento da empresa i no período t, mensurado pela relação entre o passivo oneroso e o ativo total; CEjt é o crescimento econômico do ambiente (país) j no período t, mensurado pela variação no PIB; DInd é uma variável dummy que assume valor igual a 1 ou 0, de acordo com a indústria organizacional; DPaís é uma variável dummy, que assume valor igual a 1, se o ambiente (país) for considerado desenvolvido, ou 0, caso contrário; e εit são os erros da regressão. A partir das estimações da Equação 6 pode-se avaliar as relações entre as variáveis explicativas e o desempenho financeiro das companhias, possibilitando identificar a estratégia competitiva e os fatores contingencias que afetaram a performance das firmas no período investigado. Já para avaliar a sustentabilidade do desempenho financeiro das empresas foram incluídas na Equação 6 as variáveis referentes as interações entre o desempenho financeiro anterior e as estratégias competitivas (liderança em custos e diferenciação), sendo reescrita, conforme Equação 7: DFit = α0 + α1DFit−1 + α2DFit−1 × LCit + α3DFit−1 × DIFit + α4NCjt + α5INCit + α6Idit + α7PEQit + α8GPit + α9ENDit + α10DInd + α11CEjt + α12DPaís + it (7) Em que: DFit é o desempenho financeiro da companhia i no período t, calculado com base nos retornos dos ativos (ROA); DFit-1 é o desempenho financeiro da companhia i no período t-1, calculado com base nos retornos dos ativos (ROA); LCit o escore fatorial para a estratégia liderança em custos para a empresa i no período t; mensurada com base no constructo obtido a partir da média de t-1 a t-5 das variáveis RL/CAPEX, RL/AIL e NE/AT; 96 DIFit é o escore fatorial para a estratégia de diferenciação da empresa i no período t; mensurada com base no constructo obtido, a partir da média de t-1 a t-5 das variáveis DVGA/RL, P&D/RL e VA/RL; NCit é o nível de competividade no setor j no período t, calculado com base no índice de Herfindahl; INCjt é o grau de incerteza da companhia i no período t, mensurado com base no coeficiente de variação das receitas; IDit é a idade da organização i no período t, mensurada com base na idade de cada empresa; PEQit é uma variável dummy que representa valor igual a 1, se a empresa i no período t for considerada de pequeno porte, ou 0, caso contrário, mensurada a partir da classificação em quartis do logaritmo natural do número de empregados; GPit é uma variável dummy que representa valor igual a 1, se a empresa i no período t for considerada de grande porte, ou 0, caso contrário, mensurada a partir da classificação em quartis do logaritmo natural do número de empregados; ENDit é o grau de endividamento da empresa i no período t, mensurado pela relação entre o passivo oneroso e o ativo total; DInd é uma variável dummy que assume valor igual a 1 ou 0, de acordo com a indústria organizacional; CEjt é o crescimento econômico do ambiente (país) j no período t, mensurado pela variação no PIB; DPaís é uma variável dummy, que assume valor igual a 1, se o ambiente (país) for considerado desenvolvido, ou 0, caso contrário; e εit são os erros da regressão. A Equação 7 supõe que o desempenho financeiro sustentável de uma empresa depende da estratégia competitiva adotada, sendo influenciada pelos fatores contingenciais internos e externos a organização, pelo setor econômico, pelo nível de endividamento e pelas características do ambiente (país) em que atuam. Com base na Equação 7 pode-se avaliar, por meio do coeficiente e do sinal, os fatores contingenciais que afetam o desempenho financeiro, considerando o ambiente de atuação (desenvolvido ou em desenvolvimento), o setor e o nível de endividamento das empresas. Para avaliar a estratégia competitiva (EC) que pode reduzir as influências dos fatores contingenciais externos (FCE) e internos (FCI) sobre o desempenho financeiro das empresas, 97 conforme tese proposta por este estudo, foram adicionadas na Equação 7 variáveis de interação entre os fatores contingenciais externos (NCjt e INCit) e os fatores contingenciais internos (IDit, PEQit e GPit) com as estratégias competitivas (liderança no custo e diferenciação), conforme descrito na Equação 8: DFit = β0 + α1DFit−1 + ∑ γk 3 k=1 FCEit × ECit + ∑ δm 2 m=1 FCIit × ECit ∑ φn 4 n=1 VCit+ it (8) Em que, DFit+1 é o desempenho financeiro da companhia i no período t+1, calculado com base nos retornos dos ativos (ROA); DFit é o desempenho financeiro da companhia i no período t, calculado com base nos retornos dos ativos (ROA); FCEit são os fatores contingenciais externos do ambiente da organização i no período t, mensurados pelas variáveis NCjt e INCit; FCIit são os fatores contingenciais internos do ambiente da organização i no período t, mensurados pelas variáveis IDit, PEQit e GPit; ECit indica a estratégia competitiva da empresa i no período t baseada na liderança em custos (LCit), ou na diferenciação (DFIit); VCit são as variáveis de controle do modelo (grau de endividamento, setor industrial, crescimento econômico e país em que a organização atua); e εit são os erros da regressão. A relação da estratégia competitiva com os fatores contingenciais foi obtida com base no sinal dos coeficientes angulares de cada interação entre essas variáveis. Quadro 12 Relação esperada entre as variáveis explicativas e a dependente Variável Sigla Sinal esperado Referências Liderança em custos LCit + Child (1972); Miles et al. (1978); Porter (1980, 2004); Gupta e Govindarajan (1984); Banker, Mashuwala e Tripathy (2014); Anwar e Hasnu (2016, 2017). Diferenciação DIFit + Nível de competição NCjt Khandwalla (1972); Nickell (1996); Karuna (2007); Beiner, Schimid e Wanzenried (2011); Kahyarara (2013); Liu, Qu e Haman, (2018). Nível de incerteza INCit Govidarajan (1984); Kren (1992); Cheng e Kesner (1997); Gosh; e Olsen (2009); Wong, Bonn-Itt; Wong (2011); Parnell et al. (2015). Idade da organização IDit + Anderson e Zeithaml (1984); Anthony e Ramesh (1992); Yazdanfar e Öhman (2014); Warusawitharana (2016); Costa et al. (2017). Tamanho PEQit e GPit + Hall e Weiss (1967); Amato e Wilder (1985); Serrasqueiro e Nunes (2008); Lee (2009); Dogãn (2013); Vu et al. (2019). Fonte: elaboração própria, com base na revisão sistemática da literatura. 98 Com base na hipótese H5, espera-se que a interações entre as estratégias competitivas e os fatores contingencias sejam positivas e significativas, indicando que as estratégias moderam (reduzem) os efeitos dos fatores contingenciais sobre o desempenho financeiro. No Quadro 12, acima, apresenta-se a relação esperada entre as variáveis explicativas com a variável dependente da pesquisa. 3.7 RESUMO DOS PROCEDIMENTOS METODOLÓGICOS Apresenta-se, no Quadro 13, o resumo dos procedimentos metodológicos desenvolvidos, bem como os objetivos geral e específicos, hipóteses e análise dos dados. Quadro 13 Resumo dos procedimentos metodológicos Problema de pesquisa: qual a relação da estratégia competitiva com os fatores contingenciais sobre o desempenho financeiro das empresas? Objetivos Hipóteses Análise dos dados Geral Específicos Analisar a relação da estratégia competitiva (liderança em custos ou diferenciação) com os fatores contingenciais (incerteza do ambiente, nível de competição, idade da organização e tamanho organizacional) sobre o desempenho financeiro de empresas, considerando o ambiente em que estão localizadas (países membros do BRICS e do G7). Avaliar a relação da estratégia competitiva com o desempenho financeiro das empresas, considerando o ambiente em que atuam. Não se aplica. Análise de dados em painel. Verificar os fatores contingenciais que afetam o desempenho financeiro, de acordo com ambiente organizacional. H1: a maior incerteza do ambiente afeta negativamente o desempenho financeiro das organizações. H2: o nível de competição do ambiente afeta negativamente o desempenho financeiro das companhias. H3: o tamanho (porte) afeta positivamente o desempenho financeiro das empresas. H4: a idade da organização pode afetar positivamente o desempenho financeiro. Identificar a estratégia competitiva que torna o desempenho financeiro das empresas sustentável. Não se aplica. Investigar a relação entre a estratégia competitiva e os fatores contingenciais, de acordo com o ambiente organizacional. H5: A estratégia competitiva pode reduzir (moderar) a influência dos fatores contingenciais sobre o desempenho financeiro, variando de acordo com o ambiente organizacional. Fonte: elaboração própria. Portanto, com base nos procedimentos metodológicos, espera-se encontrar evidências acerca da relação estratégia competitiva (liderança em custos ou diferenciação) com os fatores contingenciais (incerteza do ambiente, nível de competição, tamanho organizacional e idade da organização) sobre o desempenho financeiro de empresas, considerando o ambiente em que estão localizadas (países membros do BRICS e do G7). 99 4 APRESENTAÇÃO E ANÁLISE DOS RESULTADOS 4.1 ANÁLISE FATORIAL EXPLORATÓRIA E CONFIRMATÓRIA DAS ESTRATÉGIAS COMPETITIVAS Para capturar a estratégia competitiva de longo prazo de cada empresa, foi computada a média das variáveis DVGA/RL, P&D/RL, VA/RL, CAPEX/RL, AIL/RL e AT/NE considerando o período de 2011-2017. Depois, foi realizada uma análise fatorial exploratória para capturar o padrão comum para as seis variáveis descritas. Os resultados estão apresentados no Painel A da Tabela 4. Inicialmente, analisando-se as comunalidades das variáveis, conforme consta no Painel A da Tabela 3, apenas as variáveis P&D/RL (0,34) e AT/NE (0,188) apresentaram comunalidades inferiores a 0,50, sendo necessário excluir tais variáveis, conforme Hair et al. (2009). Contudo, decidiu-se deixar essas variáveis, uma vez que, apesar de apresentarem uma comunalidade baixa, tais itens contribuem com uma carga fatorial superior a 0,30 para os fatores, bem como por serem apontadas por outros estudos como indicadores da estratégia de diferenciação e de liderança em custos, respectivamente (HAMBRICK, 1983; BALSAM; FERNANDO; TRIPATHY, 2011; BANKER; MASHUWALA; TRIPATHY, 2014; ASDEMIR et al., 2017). Além disso, os resultados da análise fatorial confirmatória (Painel B da Tabela 3) indicaram que tais variáveis também representam os constructos. Com base nas cargas fatoriais expostas no Painel A da Tabela 3, as variáveis foram agrupadas em duas estratégias competitivas, a saber: liderança em custos e diferenciação. As variáveis DVGA/RL, P&D/RL e VA/RL foram agrupadas no fator diferenciação, pois apresentaram as maiores cargas fatoriais nesse constructo. Por exemplo, a variável DVGA/RL contribui com maior carga fatorial para o fator diferenciação (carga fatorial = 0,83), mas contribui negativamente para o constructo liderança em custos (carga fatorial = -0,43). Já as variáveis CAPEX/RL, AIL/AIL e AT/NE, com base nas cargas fatoriais, foram agregadas na estratégia baseada na liderança em custos. Tais resultados estão de acordo com o previsto na literatura (HAMBRICK, 1983; BALSAM; FERNANDO; TRIPATHY, 2011; BANKER; MASHRUWALA; TRIPATHY, 2014), indicando que essas variáveis podem explicar a estratégia competitiva adotada pelas companhias. 100 Tabela 3 Análise fatorial exploratória e confirmatória dos constructos: empresas de países membros do BRICS Painel A – Análise fatorial exploratória (2011-2017) Variáveis Liderança em custos (cargas fatoriais) Diferenciação (cargas fatoriais) Comunalidades DVGA/RL -0,43 0,81 0,85 P&D/RL -0,12 0,57 0,34 VA/RL -0,30 0,88 0,86 CAPEX/RL 0,77 0,52 0,86 AIL/RL 0,83 0,37 0,83 AT/NE 0,43 -0,07 0,19 Variância explicada 1,77 2,16 Alfa de Cronbach 0,65 0,75 KMO 0,521 Teste de Bartlett 1.513,455*** Painel B – Analise fatorial confirmatória (2011-2017) Variáveis Liderança em custos (cargas fatoriais) Diferenciação (cargas fatoriais) Coeficiente de confiabilidade Fornell e Larcker (1981) Variância média extraída (VME) DVGA/RL 0,85 0,86 0,70 P&D/RL 0,36 VA/RL 0,99 CAPEX/RL 0,91 0,77 0,59 AIL/RL 0,84 AT/NE 0,17 Estatísticas de ajuste do modelo Chi2 118,138*** Root mean square residual (RMR) 0,067 Goodness of fit index (GFI) 0,943 Goodness of fit index adjusted for degrees of freedom (AGFI) 0,868 Normed fit index (NFI) 0,922 Comparative fit index (CFI) 0,928 Notas: Significância estatística: *** p < 0,01. Todas as variáveis foram padronizadas (média = 0; d.p. = 1). DVGA/RL é a relação entre as despesas gerais, de vendas e administrativas com a receita líquida; P&D/RL é a relação entre os gastos com P&D e a receita líquida; VA/RL é a relação entre o valor adicionado bruto e a receita líquida; CAPEX/RL é a relação entre as despesas de capital e a receita líquida; AIL/RL é a relação entre o ativo imobilizado líquido e a receita líquida; AT/NE é a relação entre o ativo total e o número de empregados. Fonte: elaboração própria, a partir de dados da Thomson Reuters Eikon® e dos sítios, relatórios e demonstrativos financeiros das empresas. A consistência interna dos fatores foi avaliada por meio do alfa de Cronbach. Um alfa igual ou maior que 0,70 é desejável (HAIR JR. et al., 2009). Os resultados do teste demonstraram que a consistência interna para os dois fatores, liderança em custos (alfa > 0,60) e diferenciação (alfa > 0,70), segundo Hair Jr. et al. (2009), podem ser considerados aceitáveis. Além disso, devido a cada fator possuir apenas três variáveis, optou-se por avaliar a confiabilidade composta por cada constructo, conforme sugerido por Hair Jr. et al. (2009). A partir do coeficiente de confiabilidade de Fornell e Lacker (1981), pode-se dizer que os constructos apresentam consistência interna, com índices superiores a 0,70, sugerindo que as 101 medidas representam adequadamente o mesmo constructo latente (FORNELL; LACKER, 1981; HAIR JR. et al., 2009). A adequação da análise fatorial foi analisada a partir do teste KMO e de Bartlett. Os resultados dos testes indicaram que a análise fatorial é adequada e pode ser utilizada (KMO = 0,521), indicando que a correlação entre as variáveis examinadas é estatisticamente significante (Teste de Bartlett = 1.513,455; p < 0,01). Posteriormente, foi realizada a análise fatorial confirmatória para validar as medidas de estratégia competitiva. Os resultados estão apresentados no Painel B da Tabela 3. As estatísticas de ajuste do modelo indicam que o modelo se mostrou ajustado. O teste de ajuste geral do modelo (GFI) foi de 0,943, indicando um bom ajuste geral do modelo. As demais estatísticas de ajustes do modelo também apontaram para a sua adequação. Com relação à validade convergente da análise fatorial confirmatória, observando-se o resultado da variância média extraída (VME), pode-se afirmar que a variância comum compartilhada entre as variáveis dos dois constructos é convergente e confiável. Já o coeficiente de confiabilidade conjunta de Fornell e Lacker (1981) aponta que o modelo apresenta validade convergente, isto é, confiabilidade. Por fim, com base nos resultados da análise fatorial confirmatória, sugere-se que os constructos das estratégias são confiáveis e aceitáveis para a amostra das empresas dos países membros do BRICS (China e Índia). Na Tabela 4, apresentam-se os resultados das análises fatoriais exploratórias e confirmatórias para as empresas sediadas nos países membros do G7. Os dados apresentados no Painel A da Tabela 4 indicam que apenas a variável AT/NE apresentou comunalidade inferior a 0,50. Neste caso, também se decidiu continuar com o item, haja vista que ele contribui com uma carga fatorial de quase 0,60, sendo considerando um item que contribui para o fator de liderança em custos. A partir das cargas fatoriais exibidas no Painel A da Tabela 4, observa-se que, assim como para as empresas dos países membros do BRICS, as variáveis DVGA/RL, P&D/RL e VA/RL foram agrupadas na estratégia de diferenciação e os itens CAPEX/RL, AIL/RL e AT/NE na estratégia de liderança em custos. Tais resultados confirmam o reportado na literatura (HAMBRICK, 1983; ITTNER; LARCKER; RAJAN, 1997; BALSAM; FERNANDO; TRIPATHY, 2011; BANKER; MASHRUWALA; TRIPATHY, 2014; ASDEMIR et al., 2017), demonstrando que, a partir desses itens, pode-se identificar as estratégias competitivas das empresas, independente do ambiente em que atuam. No tocante à confiabilidade dos constructos, o alfa de Cronbach revelou um nível de confiabilidade aceitável (alfa > 0,60), sendo confirmado pelo índice de confiabilidade 102 composta, exposto no Painel B da Tabela 4, em que as duas medidas alcançaram um índice superior a 0,80. Tabela 4 Análise fatorial exploratória e confirmatória dos constructos: empresas de países membros do G7 Painel A – Análise fatorial exploratória (2011-2017) Variáveis Liderança em custos (cargas fatoriais) Diferenciação (cargas fatoriais) Comunalidades DVGA/RL 0,37 0,76 0,72 P&D/RL 0,54 0,61 0,66 VA/RL 0,53 0,67 0,73 CAPEX/RL 0,67 -0,63 0,84 AIL/RL 0,62 -0,71 0,85 AT/NE 0,59 -0,20 0,39 Variância explicada 1,88 2,32 Alfa de Cronbach 0,76 0,67 KMO 0,576 Teste de Bartlett 4.391,544*** Painel B – Analise fatorial confirmatória (2011-2017) Variáveis Liderança em custos (cargas fatoriais) Diferenciação (cargas fatoriais) Coeficiente de confiabilidade Fornell e Larcker (1981) Variância média extraída (VME) DVGA/RL 0,81 0,87 0,69 P&D/RL 0,65 VA/RL 0,77 CAPEX/RL 0,85 0,85 0,69 AIL/RL 1,01 AT/NE 0,32 Estatísticas de ajuste do modelo Chi2 284,612*** Root mean square residual (RMR) 0,034 Goodness of fit index (GFI) 0,950 Goodness of fit index adjusted for degrees of freedom (AGFI) 0,882 Normed fit index (NFI) 0,917 Comparative fit index (CFI) 0,920 Notas: Significância estatística: *** p < 0,01. Todas as variáveis foram padronizadas. DVGA/RL é a relação entre as despesas gerais, de vendas e administrativas com a receita líquida; P&D/RL é a relação entre os gastos com P&D e a receita líquida; VA/RL é a relação entre o valor adicionado bruto e a receita líquida; CAPEX/RL é a relação entre as despesas de capital e a receita líquida; AIL/RL é a relação entre o ativo imobilizado líquido e a receita líquida; AT/NE é a relação entre o ativo total e o número de empregados. Fonte: elaboração própria, a partir de dados da Thomson Reuters Eikon® e dos sítios, relatórios e demonstrativos financeiros das empresas. Com base nos testes de KMO e de Bartlett, pode-se apresentar que a análise fatorial se mostrou também adequada para as companhias sediadas em países desenvolvidos (KMO = 0,576), sugerindo que as correlações entre os itens dos constructos são significantes (teste de Bartlett = 4.391,544; p < 0,01). Ademais, os resultados da análise fatorial confirmatória apontam a validade dos constructos, uma vez que as estatísticas de ajuste indicam para a adequação do modelo, bem 103 como validade convergente, conforme resultado da variância média extraída (VME > 0,50), e confiabilidade composta maior que 0,70. 4.2 ESTATÍSTICAS DESCRITIVAS As estatísticas descritivas das variáveis dependentes e das variáveis independentes, para as empresas de países membros do BRICS (China e Índia), encontram-se evidenciadas na Tabela 5, destacando-se, entre outros, a média de cada variável. Tabela 5 Estatísticas descritivas: empresas de países membros do BRICS (2012-2018) Variáveis Observações Média Desvio padrão Mínimo Máximo ROAit 1.435 0,05 0,07 -0,50 0,47 ROAit-1 1.434 0,05 0,07 -0,50 0,47 LCit 1.435 0,00 1,00 -25,26 1,44 DIFit 1.435 0,00 1,00 -4,28 9,51 NCjt 1.435 0,01 0,06 0,00 0,86 INCit 1.435 0,06 0,07 0,00 0,71 IDit 1.435 28,08 18,92 1,00 156,00 PEQit 1.435 0,25 0,43 0,00 1,00 GPit 1.435 0,25 0,43 0,00 1,00 ENDit 1.435 0,26 0,17 0,00 0,72 CEjt 1.435 0,07 0,00 0,05 0,08 Notas: ROAit é o retorno dos ativos da empresa i no período t; LCit o escore fatorial para a estratégia liderança em custos para a empresa i no período t; DIFit é o escore fatorial para a estratégia de diferenciação da empresa i no período t; NCit é o nível de competividade no setor j no período t, calculado com base no índice de Herfindahl; INCjt é o grau de incerteza da companhia i no período t, mensurado com base no coeficiente de variação das receitas; IDit é a idade da organização i no período t, mensurada com base na idade de cada empresa; PEQit é uma variável dummy que assume valor igual a 1, se a empresa i no período t for considerada de pequeno porte, ou 0, caso contrário, mensurada a partir da classificação dos quartis do logaritmo natural do número de empregados; GPit é uma variável dummy que assume valor igual a 1, se a empresa i no período t for considerada de grande porte, ou 0, caso contrário, mensurada a partir da classificação dos quartis do logaritmo natural do número de empregados; ENDit é o grau de endividamento da empresa i no período t, mensurado pela relação entre o passivo oneroso e o ativo total; CEjt é o crescimento econômico do ambiente (país) j no período t, mensurado pela variação no PIB. Fonte: elaboração própria, a partir de dados da Thomson Reuters Eikon® e dos sítios, relatórios e demonstrativos financeiros das empresas. De acordo com as informações exibidas na Tabela 5, observou-se que, em média, as empresas da China e da Índia apresentam um ROAit de 0,05, isto é, uma lucratividade de 5%; com um endividamento médio de 26% (ENDit = 0,26) e com uma idade média de 28 anos. Além disso, possuem um nível de incerteza em relação ao ambiente externo de 0,06, demonstrando pouca variação nas receitas das empresas. Esse resultado está de acordo com o reportado por Parnell, Long e Lester (2015), que encontram para o ambiente chinês, por exemplo, um nível baixo de incerteza. No que se refere ao ambiente das empresas, as estatísticas descritivas indicaram que, em média, a China e a Índia apresentaram um crescimento econômico médio de 7% (CEjt = 0,07) 104 Esse resultado pode sugerir que as firmas que atuam nesses ambientes possuem oportunidades de crescimento. Na Tabela 6, a seguir, apresentam-se os resultados das estatísticas descritivas para as organizações dos países membros do G7. Os resultados exibidos na Tabela 6 apontam que as companhias sediadas na Alemanha, Canadá, EUA, Japão e Reino Unido possuem, em média, uma lucratividade menor do que as empresas que atuam nos países em desenvolvimento (ROAit = 0,04), com um nível de endividamento médio de 15% e idade média de 55 anos. Isso indica que, apesar de apresentarem uma baixa lucratividade e endividamento em relação às entidades do BRICS, essas firmas possuem mais tempo de atuação no mercado. Observou-se, também, que, no período, a taxa de incerteza decorrente da variação nas receitas foi cerca de 4% (INCit = 0,04), demonstrando que essas companhias atuam em ambiente com baixa incerteza. Tabela 6 Estatísticas descritivas: empresas de países membros do G7 (2012-2018) Variáveis Observações Média Desvio padrão Mínimo Máximo ROAit 3.990 0,04 0,14 -1,52 1,29 ROAit-1 3.989 0,03 0,14 -1,52 1,29 LCit 3.990 0,00 1,00 -20,30 1,39 DIFit 3.990 0,00 1,00 -3,66 13,81 NCjt 3.990 0,01 0,06 0,00 0,88 INCit 3.990 0,04 0,05 0,00 0,52 IDit 3.990 55,86 48,42 1,00 653,00 PEQit 3.990 0,25 0,43 0,00 1,00 GPit 3.990 0,25 0,43 0,00 1,00 ENDit 3.990 0,15 0,15 0,00 0,71 CEjt 3.990 0,02 0,02 0,00 0,07 Notas: ROAit é o retorno dos ativos da empresa i no período t; LCit o escore fatorial para a estratégia liderança em custos para a empresa i no período t; DIFit é o escore fatorial para a estratégia de diferenciação da empresa i no período t; NCit é o nível de competividade no setor j no período t, calculado com base no índice de Herfindahl; INCjt é o grau de incerteza da companhia i no período t, mensurado com base no coeficiente de variação das receitas; IDit é a idade da organização i no período t, mensurada com base na idade de cada empresa; PEQit é uma variável dummy que assume valor igual a 1, se a empresa i no período t for considerada de pequeno porte, ou 0, caso contrário, mensurada a partir da classificação dos quartis do logaritmo natural do número de empregados; GPit é uma variável dummy que assume valor igual a 1, se a empresa i no período t for considerada de grande porte, ou 0, caso contrário, mensurada a partir da classificação dos quartis do logaritmo natural do número de empregados; ENDit é o grau de endividamento da empresa i no período t, mensurado pela relação entre o passivo oneroso e o ativo total; CEjt é o crescimento econômico do ambiente (país) j no período t, mensurado pela variação no PIB. Fonte: elaboração própria, a partir de dados da Thomson Reuters Eikon® e dos sítios, relatórios e demonstrativos financeiros das empresas. Com relação aos aspectos do ambiente (país), observa-se que o crescimento econômico dos países do G7, no período analisado, é considerado baixo (CEjt = 0,02). Ou seja, com base nesse resultado, pode-se dizer que o ambiente possibilitou poucas oportunidades de crescimento para as organizações, devido ao baixo crescimento econômico dos países. Após a análise das estatísticas descritivas das variáveis, nas Tabelas 7 e 8 apresenta-se a análise de correlação, sendo que na Tabela 7 estão exibidos os resultados para as firmas dos 105 países membros do BRICS e os da Tabela 8 para as companhias dos G7. Os resultados evidenciados na Tabela 7, para as empresas sediados na China e na Índia, indicam uma correlação positiva e estatisticamente significativa entre a estratégias competitivas liderança nos custos (LCit) e diferenciação (DIFit) e o desempenho financeiro atual (ROAit) e anterior (ROAit-1), demonstrando que a estratégia competitiva adotada associa-se positivamente com o desempenho financeiro das organizações. Tabela 7 Correlação entre as variáveis: empresas de países membros do BRICS (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (1) ROAit 1,00 (2) ROAit-1 0,53** 1,00 (3) LCit 0,16** 0,06* 1,00 (4) DIFit 0,17** 0,21** -0,10** 1,00 (5) NCjt 0,07** 0,08** 0,01 0,12** 1,00 (6) INCic -0,06* -0,11** -0,14** -0,01 -0,07* 1,00 (7) IDit -0,00 0,01 0,13** 0,21** 0,17** 0,05 1,00 (8) PEQit 0,06* 0,04 -0,03 0,15** -0,08** 0,06* 0,02 1,00 (9) GPit 0,01 0,04 -0,02 -0,08** 0,15** -0,04 -0,05 -0,33** 1,00 (10) ENDjt -0,34** -0,29** -0,17** -0,30** -0,01 0,01 -0,10** -0,33** 0,31** 1,00 (11) CEit 0,001 0,06* -0,02 -0,06* -0,00 -0,07* -0,06* 0,06* -0,02 0,57 1,00 Notas: Significância * p < 0,05 e ** p < 0,01. ROAit é o retorno dos ativos da empresa i no período t; LCit o escore fatorial para a estratégia liderança em custos para a empresa i no período t; DIFit é o escore fatorial para a estratégia de diferenciação da empresa i no período t; NCit é o nível de competividade no setor j no período t, calculado com base no índice de Herfindahl; INCjt é o grau de incerteza da companhia i no período t, mensurado com base no coeficiente de variação das receitas; ; Idit é a idade da organização i no período t, mensurada com base na idade de cada empresa; PEQit é uma variável dummy que assume valor igual a 1, se a empresa i no período t for considerada de pequeno porte, ou 0, caso contrário, mensurada a partir da classificação dos quartis do logaritmo natural do número de empregados; GPit é uma variável dummy que assume valor igual a 1, se a empresa i no período t for considerada de grande porte, ou 0, caso contrário, mensurada a partir da classificação dos quartis do logaritmo natural do número de empregados; ENDit é o grau de endividamento da empresa i no período t, mensurado pela relação entre o passivo oneroso e o ativo total; CEjt é o crescimento econômico do ambiente (país) j no período t, mensurado pela variação no PIB. Fonte: elaboração própria, a partir de dados da Thomson Reuters Eikon® e dos sítios, relatórios e demonstrativos financeiros das empresas. Com relação às demais variáveis, encontrou-se uma correlação positiva entre o nível de competição e o retorno dos ativos, mas uma associação negativa entre a incerteza com o desempenho financeiro, sendo estatisticamente significantes (p < 0,05 e p < 0,01). Isso pode indicar que, devido ao baixo nível de competição no ambiente, as organizações conseguem manter suas estruturas adequadas e obter resultados positivos. Verificou-se, também, uma correlação positiva e significativa, ao nível de 5%, entre a variável que designa as companhias de pequeno porte e os retornos dos ativos, podendo indicar que essas firmas obtiveram maior desempenho financeiro no período analisado. Além disso, encontrou-se uma correlação negativa entre o nível de endividamento e o desempenho 106 financeiro, podendo indicar que, quanto maior o nível de endividamento das corporações, menor é a lucratividade. Em resumo, a análise de correlação aponta que, para as empresas de pequeno porte sediadas na China e na Índia, por exemplo, as estratégias competitivas e o nível de competição apresentam uma associação positiva com o desempenho financeiro (ROAit e ROAit-1), porém o nível de incerteza e de endividamento se relacionam negativamente com o desempenho corporativo. A análise de correlação entre as variáveis, para as companhias sediadas nos países membros do G7, encontra-se exposta na Tabela 8. Os resultados exibidos sugerem que a estratégia de diferenciação (DIFit) correlaciona-se negativamente com o retorno dos ativos, sendo estatisticamente significante ao nível de 1% (p < 0,01). Por outro lado, a estratégia de liderança em custos não apresentou correlação significativa. Tabela 8 Análise de correlação das variáveis: empresas de países membros do G7 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (1) ROAit 1,00 (2) ROAit-1 0,65** 1,00 (3) LCit 0,01 -0,01 1,00 (4) DIFit -0,34** -0,28** 0,08** 1,00 (5) NCjt 0,07** 0,05** -0,02 -0,02 1,00 (6) INCic -0,18** -0,17** -0,04** 0,14** -0,05** 1,00 (7) IDit 0,17** 0,15** 0,01 -0,27** 0,13** -0,16** 1,00 (8) PEQit -0,37** -0,32** 0,12** 0,30** -0,10** 0,17** -0,23** 1,00 (9) GPit 0,18** 0,16** -0,05** -0,16** 0,27** -0,13** 0,28** -0,33** 1,00 (10) ENDit 0,05** 0,07** -0,14** -0,21** 0,09** -0,02 0,18** -0,24** 0,31** 1,00 (11) CEjt -0,02 -0,04* 0,04* 0,09** 0,04* 0,13** -0,09** 0,04** -0,03 -0,02 1,00 Notas: Significância * p < 0,05 e ** p < 0,01. ROAit é o retorno dos ativos da empresa i no período t; LCit o escore fatorial para a estratégia liderança em custos para a empresa i no período t; DIFit é o escore fatorial para a estratégia de diferenciação da empresa i no período t; NCit é o nível de competividade no setor j no período t, calculado com base no índice de Herfindahl; INCjt é o grau de incerteza da companhia i no período t, mensurado com base no coeficiente de variação das receitas; Idit é a idade da organização i no período t, mensurada com base na idade de cada empresa; PEQit é uma variável dummy que assume valor igual a 1, se a empresa i no período t for considerada de pequeno porte, ou 0, caso contrário, mensurada a partir da classificação dos quartis do logaritmo natural do número de empregados; GPit é uma variável dummy que assume valor igual a 1, se a empresa i no período t for considerada de grande porte, ou 0, caso contrário, mensurada a partir da classificação dos quartis do logaritmo natural do número de empregados; ENDit é o grau de endividamento da empresa i no período t, mensurado pela relação entre o passivo oneroso e o ativo total; CEjt é o crescimento econômico do ambiente (país) j no período t, mensurado pela variação no PIB. Fonte: elaboração própria, a partir de dados da Thomson Reuters Eikon® e dos sítios, relatórios e demonstrativos financeiros das empresas. Verificou-se, também, conforme consta na Tabela 8, correlações positivas entre as variáveis nível de competição, idade da organização, grande porte e nível de endividamento com o ROAit, e uma associação negativa entre incerteza, pequeno porte e desempenho financeiro, sendo todas estatisticamente significantes ao nível de 1% (p < 0,01). 107 De um modo geral, essas correlações podem sugerir que, apesar de as estratégias competitivas não contribuírem para o desempenho financeiro e da incerteza do ambiente, as empresas de países membros do G7 de grande porte e com mais tempo de mercado conseguiram se manter competitivas e com baixo nível de endividamento. 4.3 ANÁLISE ECONOMÉTRICA 4.3.1 Análise das estimações de empresas de países membros do BRICS Inicialmente, destaca-se que apenas foram considerados para análise os modelos e as variáveis que apresentaram significância estatística ao nível de 1% e 5%. Todos os modelos exibidos na Tabela 9 foram estimados pelo GMM-sistema utilizando-se o procedimento twostep, com erros padrões robustos para corrigir o problema de heterocedasticidade, bem como que o painel se encontra balanceado. Destaca-se que os modelos foram estimados com o número de instrumentos inferior ao número de grupos, uma vez que o número de instrumentos não pode ser superior ao número de grupos, conforme sugerido pela literatura (ROADMAN, 2009; LABRA; TORRECILLAS, 2018). De acordo com os resultados do teste de Wald, pode-se dizer que as estimações expostas na Tabela 9 apresentam significância estatística (p < 0,01). Com relação aos pressupostos do modelo de painel dinâmico, o teste para autocorrelação de segundo grau (AR2) indica a ausência de autocorrelação para todas as estimações (modelos 1, 2, 3, 4 e 5), demonstrando que os estimadores dos modelos são consistentes. Além disso, mediante a estatística variance inflation fator (VIF), observou-se que as variáveis presentes nos modelos não apresentaram problema de multicolinearidade. Os resultados expostos na Tabela 9 sugerem que o desempenho anterior (ROAit-1) foi persistente durante o período analisado, variando de 0,606 a 0,670, mostrando-se estatisticamente significante, ao nível de 1%, para as todas as estimações. Com relação às estratégias competitivas, considerando-se o modelo 1, observa-se que apenas a estratégia de liderança em custos (LCit) apresentou uma relação positiva e estatisticamente significante, ao nível de 1%, com o desempenho financeiro (ROAit), indicando que essa estratégia contribuiu para o desempenho das empresas. Este resultado está consistente com a literatura (PORTER, 1980; HAMBRICK, 1983; BANKER; MASHRUWALA; TRIPATHY, 2014; JUNQUEIRA et al., 2016; CHO; LEE, 2018), que prevê que uma estratégia baseada na liderança em custos possibilita às companhias alcançarem desempenho superior ao de seus concorrentes. 108 Já para a estratégia baseada na diferenciação (modelo 1), as evidências apontam uma relação negativa com o desempenho financeiro, mas sem significância estatística. Tal resultado pode ser explicado pelas características dos ambientes em que as empresas operam. Tanto a Índia como a China passaram por um processo de transição na economia, permitindo maior liberação econômica (GOPALAKRISHNA; SUBRAMANIAN, 2001; PARNELL; LONG; LESTER, 2015; FEDATO; PIRES; TREZ, 2017). Além disso, segundo Parnell, Long e Lester (2015), no contexto chinês, por exemplo, as empresas tendem a utilizar uma estratégia baseada na liderança em custos ou no foco, devido à baixa incerteza do ambiente. No que se refere às demais variáveis explicativas, os resultados do modelo 1 sugerem que a incerteza afeta positivamente o desempenho das empresas, sendo significativa ao nível 1%. Conforme mencionado anteriormente, tal evidência pode ser explicada devido ao baixo nível de competitividade e de incerteza presente no mercado chinês e indiano, conforme citado na literatura (GOPALAKRISHNA; SUBRAMANIAN, 2001; PARNELL; LONG; LESTER, 2015). Com base nesses resultados, rejeita-se as hipóteses H1 para este contexto, uma vez que se esperava uma relação negativa entre a incerteza e o desempenho financeiro das entidades. De acordo com o modelo 1, não se encontraram as relações esperadas entre o nível de competição e as variáveis relativas ao tamanho organizacional com o desempenho financeiro, rejeitando-se, portanto, as hipóteses H2 e H3, respectivamente. Ainda com base no modelo 1, verificou-se uma associação negativa e significativa ao nível de 5% entre a idade da organização e o desempenho financeiro, podendo sugerir que, no período em análise, a evolução das empresas ao longo do tempo não contribui para aumentar o retorno dos ativos. A partir desse resultado, rejeita-se a hipótese H4. A partir do modelo 2 até o modelo 5 pôde-se observar as interações entre as estratégias competitivas (LCit e DIFit) e os fatores contingenciais externos (NCjt, e INCit) e os fatores contingenciais internos (PEQit, GPit e IDit). Com base nos resultados apresentados no modelo 3, conforme Tabela 10, observou-se que o coeficiente de interação entre o nível de incerteza e a estratégia de liderança em custos mostrou-se positivo e estatisticamente significante, ao nível de 5%. Tal resultado pode sugerir que uma estratégia de liderança em custos auxilia as empresas a enfrentarem a incerteza do ambiente, gerando resultados financeiros positivos, conforme previsto pela literatura (PORTER, 1980; HAMBRICK, 1983; ALLEN; HELMMS, 2006; HUO et al., 2014). Esse resultado corrobora o previsto na hipótese H6, indicando que a estratégia competitiva liderança em custos reduz os efeitos da incerteza sobre o desempenho das empresas. 109 Tabela 9 Resultados da estimação do modelo para empresas de países membros do BRICS (2012-2018) Variáveis Modelo 1 Modelo 2 Modelo 3 Modelo 4 Modelo 5 Coeficiente Coeficiente Coeficiente Coeficiente Coeficiente ROAit-1 + 0,653** 0,620** 0,670** 0,650** 0,606** LCit + 0,015** 0,015** 0,001 0,013** 0,004 DIFit + -0,004 -0,004 -0,006 -0,001 -0,009 NCjt 0,050 -0,019 0,036 0,031 0,058 INCit 0,229** 0,202** 0,197** 0,147** 0,194** PEQit + 0,010 0,010 -0,008 0,002 0,009 GPit + -0,021 -0,008 -0,004 -0,012 -0,013 IDit + -0,001* -0,001* -0,001 -0,001* -0,001* ENDit - -0,023 -0,048 -0,064 -0,037 -0,037 CEjt -0,209 -0,361 -0,021 -0,280 -0,319 NCit x DIFit 0,036 NCit x LCit 0,008 INCit x DIFit 0,011 INCit x LCit 0,048* PEQit x DIFit 0,002 PEQit x LCit -0,007 GPit x DIFit 0,002 GPit x LCit -0,008 IDit x DIFit 0,0002** IDit x LCit 0,0004** DInd Sim Sim Sim Sim Sim DPaís Sim Sim Sim Sim Sim Constante 0,100* 0,121* 0,097 0,090* 0,085* Observações 1.434 1.434 1.434 1.434 1.434 Instrumentos 103 123 123 143 123 Grupos 205 205 205 205 205 Teste de Wald 240,44** 316,91** 330,74** 358,74** 269,81** AR1 (p-valor) 0,000** 0,000** 0,000** 0,000** 0,000** AR2 (p-valor) 0,328 0,364 0,292 0,344 0,275 Teste de Hansen (p-valor) 0,321 0,264 0,293 0,659 0,373 Notas: Significância: * p < 0,05 e ** p < 0,01. ROAit é o retorno dos ativos da empresa i no período t; LCit o escore fatorial para a estratégia liderança em custos para a empresa i no período t; DIFit é o escore fatorial para a estratégia de diferenciação da empresa i no período t; NCjt é o nível de competividade no setor j no período t, calculado com base no índice de Herfindahl; INCjt é o grau de incerteza da companhia i no período t, mensurado com base no coeficiente de variação das receitas; IDit é o tempo de mercado organização i no período t, mensurada com base na idade de cada empresa; PEQit é uma variável dummy que assume valor igual a 1, se a empresa i no período t for considerada de pequeno porte, ou 0, caso contrário, mensurada a partir da classificação dos quartis do logaritmo natural do número de empregados; GPit é uma variável dummy que assume valor igual a 1, se a empresa i no período t for considerada de grande porte, ou 0, caso contrário, mensurada a partir da classificação dos quartis do logaritmo natural do número de empregados; ENDit é o grau de endividamento da empresa i no período t, mensurado pela relação entre o passivo oneroso e o ativo total; CEjt é o crescimento econômico do ambiente (país) j no período t, mensurado pela variação no PIB. Fonte: elaboração própria, a partir de dados da Thomson Reuters Eikon® e dos sítios, relatórios e demonstrativos financeiros das empresas. Além disso, essa evidência do papel moderador da estratégia de liderança em custos em relação ao nível de incerteza pode ser explicada devido às características do ambiente chinês e indiano, onde o baixo nível de incerteza possibilita a padronização e formalização de tarefas e rotinas organizacionais, permitindo que a estratégia de liderança em custos seja efetiva, conforme sugerido por Huo et al. (2014) e Parnel, Long e Lester (2015). 110 Já com base no modelo 5, encontraram-se relações positivas e significativas ao nível de 1% nas interações entre a idade da organização e as estratégias competitivas. Essas evidências sugerem que tanto a estratégia de liderança em custos como a estratégia de diferenciação podem moderar os efeitos do tempo sobre o desempenho das empresas, conforme previsto na hipótese H5. Esse resultado também pode demonstrar que, com o passar dos anos, as empresas que utilizam uma estratégia baseada na liderança em custos buscam empregar o conhecimento adquirido ao longo do tempo para otimizar suas atividades, tornando-se eficientes e melhorando o desempenho organizacional. Já para as firmas que buscam empregar uma estratégia de diferenciação, tal evidência pode indicar que, ao longo do tempo, por exemplo, essas companhias buscaram empregar mais recursos para melhorar os produtos, tornando-os diferenciados e mais conhecidos no mercado (reputação). Com base nos resultados estimados para as empresas localizadas na China e na Índia, pode-se dizer que há indícios de que as estratégias competitivas podem atuar para reduzir os efeitos dos fatores contingenciais sobre o desempenho financeiro das companhias, conforme previsto na hipótese H5. Além disso, os resultados reportados estão em consonância com os achados de Huo et al. (2014) e Chen et al. (2018), apontando que as estratégias competitivas podem moderar os efeitos de fatores contingenciais sobre o desempenho financeiro. A seguir, na Tabela 10, apresenta-se a análise da sustentabilidade do desempenho financeiro das empresas sediadas em países membros do BRICS (China e Índia), considerandose as estratégias competitivas. Todos os modelos foram estimados pelo método two-step, com erros padrões robustos para corrigir a heterocedasticidade. Com base no teste de Wald, pode-se dizer que os modelos apresentados na Tabela 11 são estatisticamente significantes. No que se refere às premissas do modelo de painel dinâmico, os testes para autocorrelação de segunda ordem (AR2) apontam a presença de autocorrelação, sugerindo a validade dos modelos. Já o teste de Hansen também indica a validade dos instrumentos utilizados para todas as estimações. Além disso, a partir da aplicação do teste VIF, não se encontraram problemas de multicolinearidade entre as variáveis. De acordo com os resultados exibidos na Tabela 10, verificou-se que o desempenho das empresas não se mostrou persistente no período analisado. Além disso, as análises das interações entre o desempenho financeiro e as estratégias competitivas, conforme modelos 6, 7 e 8, apontam que as estratégias de liderança em custos e diferenciação não contribuíram para a sustentabilidade do retorno dos ativos das empresas, no período de 2016 a 2018. 111 Tabela 10 Análise da sustentabilidade do desempenho das empresas localizadas em países membros do BRICS (2016-2018) Variáveis Sinal esperado Modelo 6 (ROAit+1) Modelo 7 (ROAit+2) Modelo 8 (ROAit+3) ROAit + -0,219* 0,112 -0,024 ROAit x LCit + -0,018 0,009 -0,030 ROAit x DIFit + 0,025 0,029 0,018 NCjt 0,177* 0,039 0,006 INCit 0,109 -0,133* 0,193** PEQit -0,013 -0,038* -0,014 GPit -0,321* 0,001 0,005 IDit -0,001 -0,001 -0,001 ENDit 0,006 -0,237** -0,236** CEjt -0,119 0,656 0,426 DInd Sim Sim Sim DPaís Sim Sim Sim Constante 0,12** 0,132** 0,155** Observações 1.432 1.432 1.432 Instrumentos 105 139 105 Grupos 205 205 205 Teste de Wald 76,61** 13,30** 76,91** AR1 (p-valor) 0,000** 0,000** 0,004** AR2 (p-valor) 0,870 0,514 0,450 Teste de Hansen (p-valor) 0,758 0,228 0,209 Notas: Significância: * p < 0,05 e ** p < 0,01; A variável CULTjt foi omitida devido a problemas de multicolinearidade. ROAit é o retorno dos ativos da empresa i no período t; LCit o escore fatorial para a estratégia liderança em custos para a empresa i no período t; DIFit é o escore fatorial para a estratégia de diferenciação da empresa i no período t; NCit é o nível de competividade no setor j no período t, calculado com base no índice de Herfindahl; INCjt é o grau de incerteza da companhia i no período t, mensurado com base no coeficiente de variação das receitas; IDit é a idade da organização i no período t, mensurada com base na idade de cada empresa; PEQit é uma variável dummy que assume valor igual a 1, se a empresa i no período t for considerada de pequeno porte, ou 0, caso contrário, mensurada a partir da classificação dos quartis do logaritmo natural do número de empregados; GPit é uma variável dummy que assume valor igual a 1, se a empresa i no período t for considerada de grande porte, ou 0, caso contrário, mensurada a partir da classificação dos quartis do logaritmo natural do número de empregados; ENDit é o grau de endividamento da empresa i no período t, mensurado pela relação entre o passivo oneroso e o ativo total; CEjt é o crescimento econômico do ambiente (país) j no período t, mensurado pela variação no PIB. Fonte: elaboração própria, a partir de dados da Thomson Reuters Eikon® e dos sítios, relatórios e demonstrativos financeiros das empresas. Com base na literatura (PORTER, 1980; HAMBRICK, 1983; BANKER; MASHRUWALA; TRIPATHY, 2014; JUNQUEIRA et al., 2016), esperava-se encontrar uma relação positiva e significativa na interação entre o desempenho financeiro e as estratégias competitivas. Entretanto, os resultados observados demonstraram que nem a estratégia de liderança em custos nem a estratégia de diferenciação contribuíram para a sustentabilidade da performance das firmas que atuam na China e na Índia. Isso pode indicar, por exemplo, que características desses ambientes, como baixo nível de competição e incerteza (GOPALAKRISHNA; SUBRAMANIAN, 2001; PARNELL; LONG; LESTER, 2015), afetam a aplicação dessas estratégias, tornando-as menos efetivas. 112 4.3.2 Análise das estimações de empresas de países membros do G7 Apresentam-se na Tabela 11 os resultados das estimações do modelo de painel dinâmico para as companhias sediadas em países membros do G7 (Alemanha, Canadá, EUA, Japão e Reino Unido). Todos os modelos foram estimados com base no método GMM-sistema, pelo método two-step, com erros padrões robustos para correção da heterocedasticidade e o painel encontra-se balanceado. Apenas foram consideradas para análise os modelos e as variáveis que apresentaram significância estatística ao nível de 1% e 5%. Os resultados apresentados na Tabela 11 indicam que todos os modelos (9, 10, 11, 12 e, 13) foram estimados com o número de instrumentos inferior ao número de grupos, conforme sugerido pela literatura (ROADMAN, 2009), bem como que, pelo teste de Wald, todos os modelos apresentam significância estatística. Além disso, o teste para autocorrelação de segundo grau (AR2) e o teste de Hansen indicam a consistência dos estimadores e a validade dos instrumentos, respectivamente, para todos os modelos. Ressalta-se, ainda, que, com base na estatística VIF, observou-se a ausência de multicolinearidade entre as variáveis presentes nos modelos. De acordo com o exposto na Tabela 11, verificou-se uma relação positiva entre o desempenho anterior e o desempenho atual das empresas, sendo tal relação estatisticamente significante ao nível de 1% para todos os modelos, indicando que as companhias apresentaram resultados persistentes ao longo do período analisado. No que se refere às relações entre as estratégias competitivas e o desempenho financeiro, conforme modelo 9, os resultados apontam que as estratégias de liderança em custos e diferenciação não contribuíram para a performance das companhias, no período analisado. Ainda de acordo com o modelo 9, não se observaram as associações esperadas entre as variáveis NCit, INCit, PEQit, GPit e IDit com o desempenho das empresas, rejeitando-se as hipóteses H1, H2, H3 e H4, respectivamente. A partir do modelo 10 até o modelo 13 pôde-se verificar os resultados para as interações entre os fatores contingenciais (NCit, INCit, PEQit, GPit e IDit) e as estratégias competitivas (LCit e DIFit). De acordo com o modelo 11, encontrou-se uma relação positiva e significativa, ao nível de 1%, entre a estratégia de liderança em custos e o nível de incerteza, sugerindo que essa estratégia pode reduzir os efeitos da incerteza sobre o desempenho das firmas, contribuindo para o aumento do desempenho financeiro. Tal resultado confirma o predito na hipótese H6, indicando que a estratégia competitiva pode moderar os efeitos dos fatores contingenciais sobre o desempenho corporativo. 113 Tabela 11 Resultados da estimação do modelo para empresas de países membros do G7 (2012-2018) Variáveis Modelo 9 Modelo 10 Modelo 11 Modelo 12 Modelo 13 Coeficiente Coeficiente Coeficiente Coeficiente Coeficiente ROAit-1 + 0,670** 0,627** 0,678** 0,680** 0,671** LCit + 0,0001 -0,005 -0,004 -0,001 0,029* DIFit + -0,065 -0,014 -0,007 -0,013 -0,021* NCjt 0,036 0,081 -0,006 -0,107 0,097 INCit 0,128 0,365 0,111 0,150 0,047 PEQit + -0,009 -0,093 -0,0002 0,007 -0,027 GPit + 0,021 -0,029 0,013 0,026 -0,014 IDit + 0,0001 -0,0001 0,0001 0,0001 0,0001 ENDit - -0,023 -0,141 -0,004 -0,019 0,019 CEjt 0,325** 0,134 0,333** 0,308** 0,227* NCit x DIFit 0,031 NCit x LCit 0,045 INCit x DIFit 0,011 INCit x LCit 0,087** PEQit x DIFit -0,013 PEQit x LCit 0,022 GPit x DIFit 0,042* GPit x LCit 0,006 IDit x DIFit -0,0003 IDit x LCit 0,0002 DInd Sim Sim Sim Sim Sim DPaís Sim Sim Sim Sim Sim Constante 0,100* 0,121* 0,097 0,090* 0,085* Observações 3.989 3.989 3.989 3.989 3.989 Instrumentos 96 88 112 128 136 Grupos 570 570 570 570 570 Teste de Wald 309,38** 238,14** 359,69** 323,35** 420,23** AR1 (p-valor) 0,000** 0,000** 0,000** 0,000** 0,000** AR2 (p-valor) 0,058 0,082 0,074 0,070 0,065 Teste de Hansen (p-valor) 0,100 0,382 0,171 0,178 0,131 Notas: Significância: * p < 0,05 e ** p < 0,01; A variável CULTjt foi omitida devido a problemas de multicolinearidade. ROAit é o retorno dos ativos da empresa i no período t; LCit o escore fatorial para a estratégia liderança em custos para a empresa i no período t; DIFit é o escore fatorial para a estratégia de diferenciação da empresa i no período t; NCit é o nível de competividade no setor j no período t, calculado com base no índice de Herfindahl; INCjt é o grau de incerteza da companhia i no período t, mensurado com base no coeficiente de variação das receitas; IDit é a idade da organização i no período t, mensurada com base na idade de cada empresa; PEQit é uma variável dummy que assume valor igual a 1, se a empresa i no período t for considerada de pequeno porte, ou 0, caso contrário, mensurada a partir da classificação dos quartis do logaritmo natural do número de empregados; GPit é uma variável dummy que assume valor igual a 1, se a empresa i no período t for considerada de grande porte, ou 0, caso contrário, mensurada a partir da classificação dos quartis do logaritmo natural do número de empregados; ENDit é o grau de endividamento da empresa i no período t, mensurado pela relação entre o passivo oneroso e o ativo total; CEjt é o crescimento econômico do ambiente (país) j no período t, mensurado pela variação no PIB. Fonte: elaboração própria, a partir de dados da Thomson Reuters Eikon® e dos sítios, relatórios e demonstrativos financeiros das empresas. Tal evidência pode indicar que, para enfrentar a incerteza do ambiente, as organizações que utilizam uma estratégia baseada na liderança em custos buscam, por exemplo, realizar investimentos para otimizar a estrutura organizacional, aumentando a eficiência da capacidade de produção e na utilização de recursos (PORTER, 1980; HAMBRICK, 1983; BANKER; 114 MASHRUWALA; TRIPATHY, 2014; HUO et al., 2014), buscando manter sua estrutura ajustada e sustentar o desempenho em longo prazo. No que se refere às interações entre as variáveis que representam o tamanho organizacional (PEQit e GPit) e as estratégias competitivas (LCit e DIFit), conforme modelo 12, os resultados apontam um coeficiente de interação positivo e significativo, ao nível de 5%, entre a variável GPit e a estratégia de diferenciação, sugerindo que essa estratégia pode reduzir os efeitos do porte corporativo sobre o desempenho financeiro das companhias. Essa evidência corrobora o previsto na hipótese H6. Esse resultado pode indicar que empresas de grande porte, que utilizam uma estratégia de diferenciação por possuírem mais recursos e maior controle do ambiente operacional (CHENHALL, 2003), conseguem investir mais em P&D, por exemplo, tornando seus produtos conhecidos pelos atributos e qualidades distintas, mantendo-se sempre à frente de seus competidores, em termos de inovação e desempenho diferenciados dos outros (PORTER, 1980; HAMBRICK, 1983; HUO et al., 2014; BANKER; MASHRUWALA; TRIPATHY, 2014; JUNQUEIRA et al., 2016; CHO; LEE, 2018). As evidências reportadas reforçam as evidenciadas nos estudos de Santos (2015), Hernández-Perlines e Mancebo-Lozano (2016), Mohsenzadeh e Ahmadian (2016) e Chen et al. (2018), apontando que as estratégias competitivas podem moderar os efeitos dos fatores contingenciais externos e internos sobre o desempenho financeiro. Portanto, a partir dos resultados expostos, aponta-se que, para o ambiente das empresas localizadas em países membros do G7, no período analisado, parece que a estratégia de liderança em custos contribuiu para reduzir os efeitos da incerteza e que a estratégia baseada na diferenciação modera a influência do porte corporativo sobre o desempenho das empresas, gerando resultados financeiros positivos. A seguir, na Tabela 12, apresenta-se a análise da sustentabilidade do desempenho financeiro das empresas sediadas em países desenvolvidos, considerando-se a influência das estratégias competitivas. Os modelos exibidos na Tabela 12 foram estimados com robustez para os erros padrões devido à presença de heterocedasticidade, e para estimar o GMM utilizou-se o método two-step. Por meio do teste de Wald, pode-se dizer que todos os modelos estimados apresentam significância estatística. Ademais, as estatísticas para ausência de autocorrelação de segundo grau (AR2) e o teste de Hansen apontam a validade e a consistência dos estimadores GMM, conforme sugerido pela literatura (RODMAN, 2009; LABRA; TORRECILLAS, 2018). 115 Tabela 12 Análise da sustentabilidade do desempenho das empresas de países membros do G7 (2016-2018) Sinal esperado Modelo 14 (ROAit+1) Modelo 15 (ROAit+2) Modelo 16 (ROAit+3) ROAit + -0,012 0,040 -0,087 ROAit x LCit + 0,111 0,011 -0,041 ROAit x DIFit + -0,025 -0,006 0,045 NCjt 0,073 -0,020 -0,031 INCit -0,205 -0,316 0,076 PEQit -0,051 -0,052 -0,027 GPit 0,011 0,019 0,033 IDit -0,001 -0,0002 -0,0001 ENDit -0,230** -0,101 -0,105 CEjt -0,228 -0,128 0,122 DInd Sim Sim Sim DPaís Sim Sim Sim Constante 0,102** 0,084** 0,058* Observações 3.985 3.986 3.987 Instrumentos 92 92 112 Grupos 570 570 570 Teste de Wald 37,43* 57,68** 58,76** AR1 (p-valor) 0,000** 0,000** 0,000** AR2 (p-valor) 0,470 0,312 0,308 Teste de Hansen (p-valor) 0,467 0,291 0,432 Notas: Significância: * p < 0,05 e ** p < 0,01. ROAit é o retorno dos ativos da empresa i no período t; LCit o escore fatorial para a estratégia liderança em custos para a empresa i no período t; DIFit é o escore fatorial para a estratégia de diferenciação da empresa i no período t; NCit é o nível de competividade no setor j no período t, calculado com base no índice de Herfindahl; INCjt é o grau de incerteza da companhia i no período t, mensurado com base no coeficiente de variação das receitas; ; PEQit é uma variável dummy que assume valor igual a 1, se a empresa i no período t for considerada de pequeno porte, ou 0, caso contrário, mensurada a partir da classificação dos quartis do logaritmo natural do número de empregados; GPit é uma variável dummy que assume valor igual a 1, se a empresa i no período t for considerada de grande porte, ou 0, caso contrário, mensurada a partir da classificação dos quartis do logaritmo natural do número de empregados; IDit é a idade da organização i no período t, mensurada com base na idade de cada empresa; ENDit é o grau de endividamento da empresa i no período t, mensurado pela relação entre o passivo oneroso e o ativo total; CEjt é o crescimento econômico do ambiente (país) j no período t, mensurado pela variação no PIB. Fonte: elaboração própria, a partir de dados da Thomson Reuters Eikon® e dos sítios, relatórios e demonstrativos financeiros das empresas. Com base nas evidências reportadas na Tabela 12, verifica-se que o desempenho financeiro das empresas não se mostrou persistente no período analisado, sugerindo que as firmas não conseguiram manter o nível de retorno de ativos ao longo dos anos. Aliado a isso, os resultados também apontaram que as estratégias competitivas não contribuíram para a sustentabilidade da performance das companhias. A partir dessas evidências, pode-se dizer que, no período analisado, as empresas localizadas em países membros do G7 não apresentaram desempenho persistente e que as estratégias de liderança em custos e diferenciação também não contribuíram para o aumento e a sustentabilidade do desempenho financeiro das empresas sediadas nesses ambientes. Esses resultados não corroboram os achados de Banker, Mashruwala e Tripathy (2014), os quais, por exemplo, estudando empresas dos EUA, encontraram evidências de que as estratégias de liderança em custos e diferenciação contribuíram para a sustentabilidade do 116 desempenho das firmas. Nesse sentido, esperava-se que, para os países membros do G7, tais evidências fossem convergentes. Contudo, parece que, no período analisado, outras características ainda não consideradas nos estudos anteriores reportados na revisão sistemática da literatura contribuíram para que essas estratégias competitivas não fossem efetivas para as empresas dos demais países do G7. 4.3.3 Síntese dos resultados das hipóteses de pesquisa No Quadro 14, apresenta-se um resumo dos resultados das hipóteses de pesquisa, considerando-se o ambiente em que a organização está sediada, isto é, se o país é membro do BRICS (China e Índia) ou do G7 (Alemanha, Canada, EUA, Japão e Reino Unido). Quadro 14 Resultados das hipóteses de pesquisa Hipóteses Resultados Países membros do BRICS Países membros do G7 H1: a incerteza do ambiente afeta negativamente o desempenho financeiro das organizações. Não confirmada (conforme modelo 1) Não confirmada (conforme modelo 9) H2: o nível de competição do ambiente afeta negativamente o desempenho financeiro das companhias. Não confirmada (conforme modelo 1) Não confirmada (conforme modelo 9) H3: o tamanho (porte) afeta positivamente o desempenho financeiro das empresas. Não confirmada (conforme modelo 1) Não confirmada (conforme modelo 9) H4: a idade da organização pode afetar positivamente o desempenho financeiro. Não confirmada (conforme modelo 1) Não confirmada (conforme modelo 9) H5: A estratégia competitiva pode reduzir (moderar) a influência dos fatores contingenciais sobre o desempenho financeiro, variando de acordo com o ambiente organizacional. Confirmada para a estratégia de liderança em custos (reduz os efeitos da incerteza e da idade da organização), conforme modelos 3 e 5; e para a estratégia de diferenciação (modera os efeitos da idade da organização), conforme modelo 5. Confirmada para a estratégia de liderança em custos (reduz os efeitos da incerteza), conforme modelo 11; e para a estratégia de diferenciação (modera a influência do porte corporativo), conforme modelo 12. Fonte: elaboração própria. Com base nos resultados das estimações de dados em painel, conforme Quadro 14, a hipótese H1 não foi confirmada, uma vez que não se encontrou relação negativa entre o nível de incerteza e o desempenho financeiro das empresas sediadas em países do BRICS ou do G7. Com relação à hipótese H2, também não se verificou uma relação negativa entre o nível de competição e o desempenho financeiro das entidades. No que se refere às hipóteses H3 e H4, para ambos os ambientes (BRICS e G7), também não foram confirmadas. Ou seja, não se encontrou uma relação positiva entre o porte 117 corporativo e a idade da organização com o desempenho financeiro, conforme previsto na literatura. Por fim, para a hipótese H5, as evidências indicam que a estratégia competitiva parece moderar os efeitos dos fatores contingenciais sobre o desempenho financeiro das entidades. No ambiente dos países do BRICS (China e Índia), a estratégia de liderança em custos reduz os efeitos da incerteza e da idade da organização sobre os resultados financeiros das empresas, e a estratégia de diferenciação modera a influência da idade da companhia. Por outro lado, para os países membros do G7 (Alemanha, Canada, EUA, Japão e Reino Unido), a estratégia de liderança em custos parece moderar os efeitos da incerteza e a estratégia de diferenciação atenua os efeitos do porte corporativo (tamanho) sobre o desempenho financeiro, no período analisado. Esses resultados sugerem que as estratégias competitivas (liderança de custos e diferenciação) podem moderar (reduzir) os efeitos dos fatores contingenciais sobre o desempenho financeiro das empresas, dependendo do ambiente (país membro do BRICS ou do G7) em que a organização atua, conforme previsto na tese de pesquisa. 118 5 CONSIDERAÇÕES FINAIS 5.1 CONCLUSÕES O ambiente organizacional é caracterizado por diversos fatores contingenciais que moldam a estrutura corporativa. Dentre esses fatores, a estratégia competitiva é vista como uma variável mediadora que pode ser utilizada pelas entidades para adequar suas atividades e reduzir as influências dos demais fatores contingenciais sobre o desempenho financeiro. Neste sentido, a presente tese de doutorado teve por objetivo analisar a relação da estratégia competitiva (liderança em custos ou diferenciação) com os fatores contingenciais (incerteza do ambiente, nível de competição, tamanho e idade da organização) sobre o desempenho financeiro de empresas, considerando-se o ambiente em que estão localizadas (países membros do BRICS e do G7). O primeiro objetivo específico buscou avaliar a relação da estratégia competitiva com o desempenho financeiro das empresas, considerando-se o ambiente em que as companhias atuam. As evidências indicaram que, no ambiente dos países do BRICS, a estratégia de liderança em custos afeta positivamente o desempenho financeiro das empresas. Esse resultado indica que a estratégia de liderança em custos parece aumentar o desempenho financeiro das organizações. Por sua vez, o segundo objetivo consistiu em verificar os fatores contingenciais que afetam o desempenho financeiro, de acordo com ambiente organizacional. De acordo com os resultados da análise de dados em painel, não foram encontradas as relações esperadas entre os fatores contingenciais e o desempenho financeiro. O terceiro objetivo buscou identificar a estratégia competitiva que torna o desempenho financeiro das empresas sustentável. Com base nos modelos estimados, não se encontraram relações positivas e significativas na interação entre o desempenho financeiro e as estratégias competitivas para ambos os ambientes, sugerindo que, no período investigado, as estratégias de liderança em custos e diferenciação não contribuíram para a sustentabilidade do desempenho das companhias. E o quarto objetivo consistiu em investigar a relação entre as estratégias competitivas e os fatores contingenciais, de acordo com o ambiente organizacional. As evidências encontradas, por meio do sinal do coeficiente de interação entre a estratégia competitiva e os fatores contingenciais, apontam que, para as empresas sediadas em países membros do BRICS (China e Índia), a estratégia de liderança em custos parece reduzir os efeitos da incerteza e da idade da 119 organização sobre os resultados financeiros das empresas. Além disso, verificou-se que a estratégia de diferenciação pode moderar a influência da idade da organização sobre o desempenho financeiro. Com relação aos países membros do G7 (Alemanha, Canada, EUA, Japão e Reino Unido), os resultados sugerem que, no período analisado, a estratégia de liderança em custos modera os efeitos da incerteza sobre o retorno dos ativos, e que a estratégia de diferenciação parece atenuar a influência do porte corporativo sobre o desempenho financeiro. A partir das evidências encontradas, pode-se dizer que há indícios de que a tese de pesquisa proposta pode ser aceita, uma vez que, no ambiente dos países do BRICS, a estratégia de liderança em custos parece reduzir os efeitos da incerteza e da idade da organização e a estratégia de diferenciação modera a influência da idade da organização sobre o desempenho financeiro das companhias, bem como que, nos países membros do G7, a estratégia de liderança em custos parece moderar os efeitos da incerteza e a estratégia de diferenciação do tamanho sobre o desempenho financeiro, no período analisado. Esses resultados também demonstram a aplicabilidade da teoria da contingência, confirmando a premissa de que as organizações devem se adequar ao contexto em que atuam, bem como que a estratégia competitiva pode ser utilizada como um meio para moderar as influências dos fatores contingenciais sobre o desempenho das organizações. Como implicações práticas e empíricas, os resultados sugerem que os gestores, controllers e demais envolvidos na gestão organizacional devem buscar escolher a estratégia competitiva que se adeque ao ambiente organizacional e que possa moderar as influências dos fatores contingenciais, buscando otimizar o desempenho financeiro das companhias, tornandoo sustentável ao longo do tempo. Salienta-se que os resultados apresentados devem ser considerados com diligência e que se restringem à amostra de empresas investigadas no período abordado. Ademais, não se pode afirmar, de forma categórica, que uma determinada estratégia competitiva específica é mais eficiente em moderar as influências dos fatores contingenciais sobre o desempenho financeiro das empresas, conforme o ambiente em que atuam (países desenvolvidos ou em desenvolvimento), sendo que outras variáveis, não contempladas por esta pesquisa, podem atuar na escolha da estratégia competitiva a ser utilizada por uma organização. Desta forma, devem-se analisar as evidências apresentadas no contexto do presente estudo, sendo as evidências válidas apenas para as entidades analisadas e para o período avaliado. 120 5.2 LIMITAÇÕES E RECOMENDAÇÕES DA PESQUISA Algumas limitações potenciais desta pesquisa devem ser levadas em consideração quando da interpretação dos resultados, tais como: número de períodos analisados; o reduzido número de empresas estudadas de países em desenvolvimento; diferenças no período de encerramento demonstrações contábeis; e a não inclusão de outros fatores contingenciais, como variáveis psicológicas, mudanças nos sistemas de controle, cultura nacional, entre outras que podem afetar o desempenho das companhias. O pequeno número de períodos analisados se deu devido à utilização de dados em painel balanceado, visto que, para tal tipo de painel, faz-se necessário que todas as entidades apresentem dados disponíveis. Contudo, o número de períodos utilizados foi suficiente para estimar a estratégia competitiva das companhias. Já o reduzido número de companhias sediadas em países em desenvolvimento deve-se ao fato de que muitas das variáveis necessárias para a pesquisa não eram divulgadas pelas organizações, reduzindo consideravelmente o número de firmas estudadas. Outro fator limitante refere-se a diferenças no período de encerramento das demonstrações contábeis. Tal fato foi observado para algumas entidades de algumas indústrias da China, Japão e EUA. Essas diferenças referem-se ao fato que algumas empresas preferem encerrar suas demonstrações com base no ciclo operacional ao ano civil. Contudo, esse fato não altera os resultados da pesquisa, uma vez que a maioria das empresas da referida indústria também apuravam seus resultados com base no ciclo operacional. Por fim, a não inclusão de outros fatores contingenciais, como variáveis psicológicas, mudanças nos sistemas de controle, entre outras que podem afetar o desempenho das companhias, deve-se ao fato de que as empresas não divulgam informações sobre essas variáveis. No caso da cultura nacional, poder-se-ia utilizar os dados do modelo de Hofstede (1983), contudo só havia informações disponíveis para apenas um período, dificultando sua análise ao longo dos anos. 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Perspective Finding Our Way through Phenotypes Andrew R. Deans1*, Suzanna E. Lewis2, Eva Huala3,4, Salvatore S. Anzaldo5, Michael Ashburner6, James P. Balhoff7, David C. Blackburn8, Judith A. Blake9, J. Gordon Burleigh10, Bruno Chanet11, Laurel D. Cooper12, Mélanie Courtot13, Sándor Csösz14, Hong Cui15, Wasila Dahdul16, Sandip Das17, T. Alexander Dececchi16, Agnes Dettai11, Rui Diogo18, Robert E. Druzinsky19, Michel Dumontier20, Nico M. Franz5, Frank Friedrich21, George V. Gkoutos22, Melissa Haendel23, Luke J. Harmon24, Terry F. Hayamizu25, Yongqun He26, Heather M. Hines1, Nizar Ibrahim27, Laura M. Jackson16, Pankaj Jaiswal12, Christina James-Zorn28, Sebastian Köhler29, Guillaume Lecointre11, Hilmar Lapp7, Carolyn J. Lawrence30, Nicolas Le Novère31, John G. Lundberg32, James Macklin33, Austin R. Mast34, Peter E. Midford35, István Mikó1, Christopher J. Mungall2, Anika Oellrich36, David OsumiSutherland36, Helen Parkinson36, Martın J. Ramırez37, Stefan Richter38, Peter N. Robinson39, Alan Ruttenberg40, Katja S. Schulz41, Erik Segerdell42, Katja C. Seltmann43, Michael J. Sharkey44, Aaron D. Smith45, Barry Smith46, Chelsea D. Specht47, R. Burke Squires48, Robert W. Thacker49, Anne Thessen50, Jose Fernandez-Triana51, Mauno Vihinen52, Peter D. Vize53, Lars Vogt54, Christine E. Wall55, Ramona L. Walls56, Monte Westerfeld57, Robert A. Wharton58, Christian S. Wirkner38, James B. Woolley58, Matthew J. Yoder59, Aaron M. Zorn28, Paula M. Mabee16 1 Department of Entomology, Pennsylvania State University, University Park, Pennsylvania, United States of America, 2 Genome Division, Lawrence Berkeley National Lab, Berkeley, California, United States of America, 3 Department of Plant Biology, Carnegie Institution for Science, Stanford, California, United States of America, 4 Phoenix Bioinformatics, Palo Alto, California, United States of America, 5 School of Life Sciences, Arizona State University, Tempe, Arizona, United States of America, 6 Department of Genetics, University of Cambridge, Cambridge, United Kingdom, 7 National Evolutionary Synthesis Center, Durham, North Carolina, United States of America, 8 Department of Vertebrate Zoology and Anthropology, California Academy of Sciences, San Francisco, California, United States of America, 9 The Jackson Laboratory, Bar Harbor, Maine, United States of America, 10 Department of Biology, University of Florida, Gainesville, Florida, United States of America, 11 Muséum national d'Histoire naturelle, Département Systématique et Evolution, Paris, France, 12 Department of Botany and Plant Pathology, Oregon State University, Corvallis, Oregon, United States of America, 13 Molecular Biology and Biochemistry Department, Simon Fraser University, Burnaby, British Columbia, Canada, 14 MTA-ELTE-MTM, Ecology Research Group, Pázmány Péter sétány 1C, Budapest, Hungary, 15 School of Information Resources and Library Science, University of Arizona, Tucson, Arizona, United States of America, 16 Department of Biology, University of South Dakota, Vermillion, South Dakota, United States of America, 17 Department of Botany, University of Delhi, Delhi, India, 18 Department of Anatomy, Howard University College of Medicine, Washington D.C., United States of America, 19 Department of Oral Biology, College of Dentistry, University of Illinois, Chicago, Illinois, United States of America, 20 Stanford Center for Biomedical Informatics Research, Stanford, California, United States of America, 21 Biocenter Grindel and Zoological Museum, Hamburg University, Hamburg, Germany, 22 Department of Computer Science, Aberystwyth University, Aberystwyth, Ceredigion, United Kingdom, 23 Department of Medical Informatics & Epidemiology, Oregon Health & Science University, Portland, Oregon, United States of America, 24 Department of Biological Sciences, University of Idaho, Moscow, Idaho, United States of America, 25 Mouse Genome Informatics, The Jackson Laboratory, Bar Harbor, Maine, United States of America, 26 Unit for Laboratory Animal Medicine, Department of Microbiology and Immunology, Center for Computational Medicine and Bioinformatics, and Comprehensive Cancer Center, University of Michigan Medical School, Ann Arbor, Michigan, United States of America, 27 Department of Organismal Biology and Anatomy, University of Chicago, Chicago, Illinois, United States of America, 28 Cincinnati Children's Hospital, Division of Developmental Biology, Cincinnati, Ohio, United States of America, 29 Institute for Medical Genetics and Human Genetics, Charité-Universitätsmedizin Berlin, Berlin, Germany, 30 Department of Genetics, Development and Cell Biology and Department of Agronomy, Iowa State University, Ames, Iowa, United States of America, 31 Signalling ISP, Babraham Institute, Babraham, Cambridgeshire, UK, 32 Department of Ichthyology, The Academy of Natural Sciences, Philadelphia, Pennsylvania, United States of America, 33 Eastern Cereal and Oilseed Research Centre, Ottawa, Ontario, Canada, 34 Department of Biological Science, Florida State University, Tallahassee, Florida, United States of America, 35 Richmond, Virginia, United States of America, 36 European Molecular Biology Laboratory European Bioinformatics Institute, Wellcome Trust Genome Campus, Hinxton, United Kingdom, 37 Division of Arachnology, Museo Argentino de Ciencias Naturales CONICET, Buenos Aires, Argentina, 38 Allgemeine & Spezielle Zoologie, Institut für Biowissenschaften, Universität Rostock, Universitätsplatz 2, Rostock, Germany, 39 Institut für Medizinische Genetik und Humangenetik Charité – Universitätsmedizin Berlin, Berlin, Germany, 40 School of Dental Medicine, University at Buffalo, Buffalo, New York, United States of America, 41 Smithsonian Institution, National Museum of Natural History, Washington, D.C., United States of America, 42 Knight Cancer Institute, Oregon Health & Science University, Portland, Oregon, United States of America, 43 Division of Invertebrate Zoology, American Museum of Natural History, New York, New York, United States of America, 44 Department of Entomology, University of Kentucky, Lexington, Kentucky, United States of America, 45 Department of Biological Sciences, Northern Arizona University, Flagstaff, Arizona, United States of America, 46 Department of Philosophy, University at Buffalo, Buffalo, New York, United States of America, 47 Department of Plant and Microbial Biology, Integrative Biology, and the University and Jepson Herbaria, University of California, Berkeley, California, United States of America, 48 Bioinformatics and Computational Biosciences Branch, Office of Cyber Infrastructure and Computational Biology, National Institute of Allergy and Infectious Diseases, National Institutes of Health, Bethesda, Maryland, United States of America, 49 Department of Biology, University of Alabama at Birmingham, Birmingham, Alabama, United States of America, 50 The Data Detektiv, 1412 Stearns Hill Road, Waltham, Massachusetts, United States of America, 51 Canadian National Collection of Insects, Ottawa, Ontario, Canada, 52 Department of Experimental Medical Science, Lund University, Lund, Sweden, 53 Department of Biological Sciences, University of Calgary, Calgary, Alberta, Canada, 54 Universität Bonn, Institut für Evolutionsbiologie und Ökologie, Bonn, Germany, 55 Department of Evolutionary Anthropology, Duke University, Durham, North Carolina, United States of America, 56 iPlant Collaborative University of Arizona, Thomas J. Keating Bioresearch Building, Tucson, Arizona, United States of America, 57 Institute of Neuroscience, University of Oregon, Eugene, Oregon, United States of America, 58 Department of Entomology, Texas A & M University, College, Station, Texas, United States of America, 59 Illinois Natural History Survey, University of Illinois, Champaign, Illinois, United States of America PLOS Biology | www.plosbiology.org 1 January 2015 | Volume 13 | Issue 1 | e1002033 Abstract: Despite a large and multifaceted effort to understand the vast landscape of phenotypic data, their current form inhibits productive data analysis. The lack of a community-wide, consensusbased, humanand machine-interpretable language for describing phenotypes and their genomic and environmental contexts is perhaps the most pressing scientific bottleneck to integration across many key fields in biology, including genomics, systems biology, development, medicine, evolution, ecology, and systematics. Here we survey the current phenomics landscape, including data resources and handling, and the progress that has been made to accurately capture relevant data descriptions for phenotypes. We present an example of the kind of integration across domains that computable phenotypes would enable, and we call upon the broader biology community, publishers, and relevant funding agencies to support efforts to surmount today's data barriers and facilitate analytical reproducibility. Introduction Phenotypes, i.e., observable traits above the molecular level, such as anatomy and behavior, underlie, and indeed drive, much of the research in the life sciences. For example, they remain the primary data we use to define most species and to understand their phylogenetic history. Phenotype data are also used to recognize, define, and diagnose pathological conditions in plants, animals, and other organisms. As such, these data represent much of what we know of life and are, in fact, necessary for building a comprehensive tree of life [1]. Our observations of organismal phenotypes also inspire science aimed at understanding their development, functions, evolution, and interactions with the environment. Research in these realms, for example, has uncovered phenotypes that could be used to create antimicrobial materials [2] and efficient microrobots [3], yield novel approaches for drug delivery [4], treat the adverse effects of aging [5], and improve crop traits [6], among many other applications. Disease phenotypes, likewise, provoke us to research their genomic and environmental origins, often through manipulations of model organisms and/or by exploring the wild populations and ancestors, especially in the case of plants. The gamut of research on phenotype is very broad, but given the lack of computability across phenotype data (Fig. 1, bottom panel), there exists minimal cross-domain interaction. By not investing in the infrastructure needed to share phenotype data, we are missing opportunities for extraordinary discoveries. Annotation strategies for genomes, in contrast to phenomes, are well advanced, with common methodologies, tools, syntaxes, and standards for articulating a precise description of nearly every type of genomic element [7–12]. Genomic data are also aggregated into large datasets, e.g., NCBI [7], EBI [8], DDBJ [9], and others [10–13]. Researchers lack these similarly well-established, linked, and consolidated resources for describing phenotypes and the contexts in which they arise, despite previous calls for more investment in this area [14–17]. Phenotype data (Table 1), although abundant and accumulating rapidly-e.g., species descriptions, image databases, analyses of induced variation, physiological measurements, whole genome knockout studies, highthroughput assays, electronic health records-are extremely heterogeneous, largely decentralized, and exist predominantly as free text. Thus, phenotype data are difficult to locate and impractical to interpret. In some areas of research, such as crop genetics and patient care, a great majority of the phenotype data underlying published research is not publicly available [18]. There also exists a divide between quantitative data and qualitative phenotype data, requiring reference measures or populations and statistical cutoffs to support interoperability (for example, ''large head'' versus a head circumference measurement). Finally, phenotypes change over time-be it evolutionary time, disease-course time, or developmental time- and the timing and ordering of phenotypic presentation is specific in any given context yet is rarely communicated. In short, while phenotype data are as complex, diverse, and nuanced as genomic data, they have not seen data standardization and analyses applied with the same broad strokes as we have seen for genomics. Nevertheless, a small quantity of phenotype data, for a handful of species, is indeed formalized, such that it can be reliably searched, compared, and analyzed computationally (see below). However, with many disparate approaches to formalizing phenotypes, including different annotation strategies, the use of unrelated vocabularies, and the use of incomparable models and formats-these data are not fully unified or interoperable between taxa. Given the latent potential of phenotype data and the emerging approaches to representing and computing across phenotypes, we members of the Phenotype Research Coordination Network (Phenotype RCN) [19], feel that the time is ripe for system-wide investment in the development of the needed tools and standards. As described in Box 1, many projects, sometimes working together but often independently, have begun building the foundation. There is now an opportunity for the large cross-domain phenomics research community to take advantage of new technologies for analyzing and managing the vast and diverse landscape of phenotype data, if attention and resources are applied to build in a consistent fashion on the current foundation. Building a Phenomics Discovery Environment How do we develop an environment in which researchers can readily make discoveries concerning the intimate connections among phenotypes, environment, and genetics? Three requirements must Citation: Deans AR, Lewis SE, Huala E, Anzaldo SS, Ashburner M, et al. (2015) Finding Our Way through Phenotypes. PLoS Biol 13(1): e1002033. doi:10.1371/journal.pbio.1002033 Published January 6, 2015 Copyright:  2015 Deans et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Funding: This effort was funded by the US National Science Foundation, grant number DEB-0956049. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. * Email: [email protected] The Perspective section provides experts with a forum to comment on topical or controversial issues of broad interest. PLOS Biology | www.plosbiology.org 2 January 2015 | Volume 13 | Issue 1 | e1002033 Fig. 1. How to discover branching phenotypes? (Bottom panel) Phenotype data exhibiting various forms of branchiness are not easily discerned from diverse natural language descriptions. (A) Bee hairs are different from most other insect hairs in that they are plumose, which facilitates pollen collection. (B) A mutant of Drosophila melanogaster exhibits forked bristles, due to a variation in mical. (C) In zebrafish larvae (Danio rerio), angiogenesis begins with vessels branching. (D) Plant trichomes take on many forms, including trifurcation. (Top) Phenotypes involving some type of ''branched'' are easily recovered when they are represented with ontologies. In a semantic graph, free text descriptions are converted into phenotype statements involving an anatomy term from animal or plant ontologies [56,118] and a quality term from a quality ontology [106], connected by a logical expression (''inheres_in some''). Anatomy (purple) and quality (green) terms (ontology IDs beneath) relate phenotype statements from different species by virtue of the logic inherent in the ontologies, e.g., plumose, bifurcated, branched, and tripartite are all subtypes PLOS Biology | www.plosbiology.org 3 January 2015 | Volume 13 | Issue 1 | e1002033 be met for this vision to become a reality across large-scale data. First, phenotype descriptions must be rendered in a computable format, which usually involves the use of appropriate ontology terms (via Uniform Resource Identifiers [URIs]) to represent the phenotypic descriptions found in narrative text or data sources. Each bit of text is thereby imbued with properties and relationships to other terms (Fig. 1, top panel). Second, these semantically represented phenotype data, which integrate the phenotypes (Fig. 1, top panel) across species and also with their genetic and environmental contexts, must be stored in a way that is broadly accessible on the Internet in a nonproprietary format, e.g., in a Resource Description Framework (RDF). The third requirement is to grow a set of algorithms that enable users to analyze the data. That is, these algorithms combine the logical connections inherent in the ontologies with statistical analyses to, for example, identify similar phenotypes and their correlations with specific genetic or environmental factors. Examples of systems that have the potential to transform their fields come from several domains. For instance, by computing from natural species phenotypes to the phenotypes resulting from gene disruption in model organisms, the Phenoscape project [20] demonstrated that genes underlying evolutionarily novel phenotypes can be proposed for experimental testing [21–23]. Uniting these previously unlinked data from evolutionary and biomedical domains provided a way to virtually automate the formulation of evolutionary developmental (evo-devo) hypotheses. The reinvention of descriptive taxonomy as a 21st century information science, likewise, requires computable phenotypic data and resources [24], including those for taxonomy [25] and for evolutionary biology [26–28]. This process is an active research focus of the Hymenopteran Anatomy Ontology project [29], which is developing computational methods to allow descriptions of species' phenotypes to be made in explicit and searchable forms [30,31]. Other successes have come from linking human disease phenotypes to annotated genetic data from model organisms, thus yielding insights into the genes involved in human disease [32,33]. Similarly, the Gramene project [34] developed the plant Trait Ontology (TO) to annotate the Quantitative Trait Locus (QTL) [35] for several crop plants, including rice, maize, and wheat. Remarkably, and despite their significantly different aims, much of the phenotypic data that have been amassed through these projects can be made comparable- an outcome that until recently would have been impossible-because each of these groups shared common ontologies (i.e., semantics) and data annotation strategies. The systems they used are thus logically interoperable, and the bodies of phenotypic data emerging from their work can be compared and aggregated without further intervention. For these limited and domain-specific successes to be brought to bear more generally, approaches to ontology development and data annotation must be scaled up. Several hurdles must be overcome. First, only a small fraction of the phenotypic diversity of life is currently represented in phenotype ontologies. Ontology development is time-consuming, requires expert knowledge and community buy-in, and is ideally paired with data-driven research that iteratively checks the soundness of the ontology as it simultaneously seeks discovery. New approaches are needed to expedite ontology development. Second, current methods of phenotypic data annotation are largely manual, thus requiring substantial resources for personnel to translate data from the published literature into a computable format. Semiautomated approaches for extracting phenotypes and other data from text [36–38] must be further developed. Though timeconsuming, the transformation of legacy data in relation to these resources should be a one-time investment. It is only possible, however, if current and future projects co-develop and adopt common standards, and actively contribute to their ongoing development and maintenance, of ''branched.'' Image credits: bumble bee with pollen by Thomas Bresson, seta with pollen by István Mikó, Arabidopsis plants with hair-like structures (trichomes) by Annkatrin Rose, Drosophila photo by John Tann, Drosophila bristles redrawn from [119], scanning electron micrograph of Arabidopsis trichome by István Mikó, zebrafish embryos by MichianaSTEM, zebrafish blood vessels from [120]. Figure assembled by Anya Broverman-Wray. doi:10.1371/journal.pbio.1002033.g001 Table 1. Finding phenotypes. Phenotype data source Characteristics Example/Reference published literature from biological and biomedical domains highly dispersed corpus, mainly digitized, but still in natural language; contains abundant phenotypes publisher websites, reviews that summarize important reference phenotype datasets [79,80] supplementary data spreadsheets, text files publisher repositories, open repositories (e.g., Dryad [81]) trait databases and large corpora relational databases containing free text phenotype descriptions phenotype repositories specific to a particular field of study [82], Biodiversity Heritage Library [83], Encyclopedia of Life [62], Plant Trait Database [84], morphology databases [85–87] images annotated with keywords (free text); dispersed across many databases and repositories; phenotype or genotype data contained in these images are not computationally accessible [78]. biodiversity image stores [85–89], patient MRI images, X-rays, brightfield micrographs, image-bases of plant phenotypes [90] natural history collections .3,000,000,000 biological specimens worldwide, some with free text descriptions and associated images iDigBio [91] auto-generated data quantitative data from satellite tracking devices, environmental sensors, and high-throughput phenotyping processes National Ecological Observatory Network (NEON) [92], high throughput [26–28], tracking sensors [93] The rich legacy of research in the life sciences includes a wealth of phenotype data contained in many sources, for millions of extinct and extant species. Some important sources of phenotypes date from more than 250 years ago [74–77]. With very few exceptions, phenotype data are not computationally accessible [78]. doi:10.1371/journal.pbio.1002033.t001 PLOS Biology | www.plosbiology.org 4 January 2015 | Volume 13 | Issue 1 | e1002033 and if researchers avoid practices that may create errors [39] by writing their descriptions in ambiguous or locally idiosyncratic ways. Thus we must involve authors, editors, publishers, and funding agencies in the entire scholarly communication process in establishing the needed resources needed for data interoperability. Predicting an individual organism's phenotypic characteristics based on the combination of its genetic heritage, development, and environmental context is a challenge for research at the intersection of the physical and life sciences [40] and is a driving force behind a major cyberinfrastructure investment by the United States National Science Foundation (NSF) [41]. With focused attention on the requirements for a phenomics-based system, we can expedite this goal. Integrating species phenotypes with data across all levels of the biological hierarchy is possible if strategies for data management are co-developed and coordinated. Achieving Data Integration Researchers who attempt to explore biological data using a multidisciplinary approach are aware that it is nearly impossible to integrate comparable data from multiple species and multiple publications. We manually assemble an example (Fig. 2) of how large-scale availability of logically structured phenotype descriptions could inform and relate disparate fields of research and help address this significant problem. Past efforts, however, have largely involved manual integration of limited datasets. In the future, the study of phenotypic causality will be increasingly reliant on large and rapidly growing data stores that can only be effectively searched with automated or semi-automated methods. At this juncture, discoveries in many areas of biology rely on integrating genomic data with phenotypic data, and such integration is at an impasse because of the lack of computable and accessible phenotypic data within and across species [42]. Linking Phenotypes to Genomic and Genetic Variation Data Given that genomic data are now relatively inexpensive to collect (approximately US$5,000 per individual genome and rapidly approaching US$100 [43]), a growing number of independent projects are explicitly linking genetic variants to related phenotypes at costs upwards of US$1 million per species genome. For example, the NCBI databases [7,44] capture data concerning human variants related to disease using semantic terms [45–47]. Large-scale integration of such variants, including computable descriptions of disease phenotypes in humans, model and non-model organisms, are collected and semantically integrated to help support disease diagnosis and mechanism discovery by the Monarch Initiative [33]. The National Institutes of Health (NIH) Undiagnosed Disease Program [48] captures individual patient phenotype profiles using the Human Phenotype Ontology (HPO) and submits these phenotype data to the database of Genotypes and Phenotypes (dbGaP) [49] and to PhenomeCentral [50] to aid patient matching based on semantic comparisons. Multiple projects and institutions have collaborated to develop an approach for the capture of standardized human pathogen and vector sequencing metadata designed to support epidemiologic and genotype–phenotype association studies [51]. The NIH Knockout Mouse Phenotyping Program (KOMP2) [52] and the International Mouse Phenotype Consortium (IMPC) [53] provide both their quantitative and qualitative phenotype assay data for the mouse using the Mammalian Phenotype Ontology (MP) [54]. Both HP and MP classes (i.e., descriptive terms) are linked to upper-level classes in the UBERON anatomy ontology [55,56]. Thus, the phenotypes and associated variations from these autonomous projects can be compared automatically, as evident in cross-species resources such as PhenomeNET [57] and others [58,59]. Similarly, the Gramene project [34] developed the plant Trait Ontology (TO) to annotate the Quantitative Trait Locus (QTL) [35] for several crop plants, including rice, maize, and wheat. As noted above, however, the paths between genotype and phenotype are not one-to-one. Any successful strategy must also account for environmental contributions, and, as with phenotypes and genotypes, a wellstructured, consistent means of describing environmental differences is essential. Linking Phenotypes to Environment An organism's phenotypes result from the interplay of environment with genetics and developmental processes. The meaning of ''environment'' differs according to biological context. For biodiversity, environment refers to the specific conditions and geographical location in which any given organism is found. For model organisms, environment comprises the experimental perturbations relative to what is ''normal'' for an organism of that time, for example, changes in exposure to a drug or in the concentration of salt in the water that serves as an organism's home. For epidemiological studies, environment may refer to features in the physical proximity, such as to a nuclear plant, or relate to prior personal behavior, such as a history of smoking. Although the phenoBox 1. Methodologies to Make Phenotypes Computable The prospects of computable phenotype data have slowly improved over the past several years, with several domain-specific initiatives yielding results [21,30,32,94,95] and a larger framework of data integration resources [96–100]. These pioneering projects have achieved several goals: (i) more standardized measurements of complex phenotypes (e.g., PhenX [101]); (ii) an integrative phenotype semantic representation (in Web Ontology Language [OWL] [102]) and its use [103–105] to capture the genetic and environmental context of an observed phenotype [106]; (iii) an ontology of classes defining the anatomical, behavioral, and biological function terms and the relevant phenotypic qualities needed to describe phenotypes effectively in detail; and (iv) algorithms, such as OWLSim [107,108], combining the logical connections inherent in the ontologies with statistical analyses to identify phenotypes that are correlated with specific genetic makeups. These tools have been used effectively in both the model organism biomedical and biodiversity domains, for example to discover new genes involved in gene networks underlying human disease [95,109–111], to prospect for candidate genes associated with crop improvement using Genome-Wide Association Studies (GWAS) experiments [112,113], to propose candidate genes for evolutionary novelties [21], to integrate and organize diverse functional data [114], to understand the characteristics used to diagnose species [30,31] and, when combined with systems biology data such as protein–protein interactions or pathway resources, to augment the analysis used in a clinical setting for diagnostics [95,115–117]. The use of computable phenotypes is expected to be a powerful approach to discovery of the genetic contribution to phenotypes, and it applies across all categories of genetic elements. PLOS Biology | www.plosbiology.org 5 January 2015 | Volume 13 | Issue 1 | e1002033 Fig. 2. Phenotypes shared across biology. Phenotype data are relevant to many different domains, but they are currently isolated in data ''silos.'' Research from a broad array of seemingly disconnected domains, as outlined here, can be dramatically accelerated with a computable data store. (A) Domains: Diverse fields such as evolutionary biology, human disease and medicine, and climate change relate to phenotypes. (B) Phenotypes: insects, vertebrates, plants, and even forests all have features that are branched in some way, but they are described using different terms. For a computer to discover this, the phenotypes must be annotated with unique identifiers from ontologies that are logically linked. Under ''shape'' in the PATO quality ontology [106], ''branchiness'' is an encompassing parent term with subtypes ''branched'' and ''increased branchiness.'' From left to right, top layer, insects, vertebrates and plants have species that demonstrate phenotypes for which the genetic basis is not known. Often their companion model species, however, have experimental genetic work that is relevant to proposing candidate genes and gene networks. Insects (1): An evolutionary novelty in bees (top layer) is the presence of branched setae used for pollen collection. Nothing is known about the genetic basis of this feature. One clue to the origin of this evolutionary feature comes from studies of Drosophila (bottom layer), where Mical overexpression in unbranched wild-type bristles generates a branched morphology [119]. Mical directly links semaphorins and their plexin receptors to the precise control of actin filament dynamics [119]. Vertebrates (2): In humans, aberrant angiogenesis, including excessive blood vessel branching (top layer), is one of the six central hallmarks of cancer [121]. Candidate genes have been identified using data from model organisms. In zebrafish (middle layer), studies of the control of sprouting in blood vessel development show that signaling via semaphorins [122] and their plexin receptors is required for proper abundance and distribution [123]; disruption of plxnd1 results in increased branching [120,124,125]. In mouse (bottom layer), branching of salivary glands is dependent on semaphorin signaling [126], as is the branching of various other epithelial organs [127]. Plants (3): The uppermost canopy of trees of the rainforest (top layer) undergo a marked increase in branching associated with climate change [128]. Nothing is known about the genetic basis of this feature. The branching of plant trichomes (bottom layer), tiny outgrowths with a variety of functions including seed dispersal, has been studied in the model Arabidopsis thaliana. Branching occurs in association with many MYB-domain genes [129], transcription factors that are found in both plants and animals [130]. (C) Environment: Diverse input from the environment influences organismal phenotype. (D) Genes: At the genetic level, previously unknown associations with various types of ''branchiness'' between insects and vertebrates are here made to possibly a common core or network of genes (the semaphorin-plexin signaling network). No association between genes associated with plant branching (Myb transcription factors) and animal branching is obvious from the literature. Image credit: Anya Broverman-Wray. doi:10.1371/journal.pbio.1002033.g002 PLOS Biology | www.plosbiology.org 6 January 2015 | Volume 13 | Issue 1 | e1002033 type data collected in these different types of environments may at first glance seem mutually irrelevant, there is, in fact, often a need to combine them. Exposure to an environmental toxin, for example, could similarly affect the phenotype of local flora and fauna populations and of human patients, and it could be related to phenotypic outcomes identified via experiments involving perturbation of the environments of model organisms. Neither environment nor phenotype is a static entity; both change over developmental and evolutionary time [15,16]. Very few efforts have attempted to relate phenotypic data captured in these varied contexts, in part due to the vastly different mechanisms by which the environmental variables and measures are described. Building blocks to capture these pieces include the Environment Ontology (EnvO) [60] and the Exposure Science Ontology (ExO) [61], which provide controlled, structured vocabularies designed to enable representation of the relationships between organisms and biological samples to their environment. EnvO has been used by projects as disparate as the Encyclopedia of Life [62] and the International Census of Marine Microbes [63]. It is also one of the ontologies incorporated into the Experimental Factor Ontology (EFO) [64] used for systematic description of experimental variables available in European Bioinformatics Institute (EBI) databases [8] and for National Human Genome Research Institute's catalog of published GWAS [65]. Ontologies and associated tools provide a powerful, rational means for discovering connections between data from multiple projects. This potential can only be realized by reusing and combining classes from core primary ontologies. This is the strategy used by numerous successful cases, such as the EFO's incorporation of EnvO and other ontologies, and has dual benefits. It allows projects to tailor their ontology to suit their own particular needs, while retaining the powerful capability to semantically integrate their data with data from multiple other projects. This approach brings convergence, avoids duplication of effort and enables joint analysis of combined data. Remarkable advances are being made in measuring environmental data, ranging from fine-scale measurements across the surface of a leaf to variation across a planted field to high-resolution environmental layers at a global scale (e.g., [66,67]). As environmental data rapidly accumulate as a result of these new technologies, now is an opportune moment to ensure the usability and longevity of these data by adopting systematic standards. Towards this end, recent workshops funded by NSF [68] and National Institute of Environmental Health Sciences (NIEHS) [69] brought together diverse sets of experts to aid in developing vocabularies and standards for describing environment. Recommendations Recommendation 1 We urge all biologists, data managers, and clinicians to actively support the development, evaluation, refinement, and adoption of methodologies, tools, syntaxes, and standards for capturing and computing over phenotypic data and to collaborate in bringing about a coordinated approach. And we urge university lecturers to introduce their students to these tools and concepts and integrate them into the standard basic curriculum in all relevant fields. The resultant increase in interoperability will enhance broad access to large stores of phenotypic data required or already existing across many areas of biology. It will accelerate discoveries across biological domains and increase significantly the return on the huge past and present investment made to generate the data. Although there are daunting challenges to this critical and enormous undertaking, its success will increase efficiency, greatly reduce the loss of data and duplication of effort, and facilitate reuse of phenotype data [70]. Recommendation 2 We urge publishers to require contribution of structured phenotype data in semantic-enabled ways as the technology is developed, to enable us to compute beyond the impasse of the free-text narrative. Moreover, funding agencies should request appropriate metadata for phenotypic descriptions, and they should require that all phenotypic screening made with their funds result in open and interoperable data. Recommendation 3 With the community, conceptual, and methodological framework falling into place, the next steps require a new set of resources for phenotypes, including tools for the conversion of important legacy phenotype datasets to the newly established computable formats, putting into place mechanisms to scale up acquisition of new phenotypes, methods that ensure appropriate mark-up and deposition of phenotypic data upon publication [71], organization of the data into accessible online resources, new tools to visualize and analyze the data, and the development of a comprehensive cross-species and crossdomain phenotypic resource. These needs are urgent and reach across the research spectrum, from understanding biodiversity loss and decline, to interpreting genomes of the new ''nonmodel'' systems that are coming online, to elevating the health of the expanding human population. The use of computable phenotypes is expected to be a powerful approach to discovery of the genetic contribution to phenotypes [72,73], and it applies across all categories of genetic elements. Science revolves around gathering facts and making theories, a repeating cycle of improvement and increasing knowledge. In the history of science, the iterative accumulation of well-integrated facts- starting with the creation of a common system of units-has over and over again determined accelerated growth in scientific understanding. As our base of phenotypic knowledge grows ever larger, it will only become ever more difficult to navigate and comprehend, without the coordinated improvements in infrastructure and culture that will expedite scientific discovery. Acknowledgments We thank Anya Broverman-Wray for her expert preparation of Figs. 1 and 2 and the photographers who availed their images for Fig. 1. References 1. Burleigh JG, Alphonse K, Alverson AJ, Bik HM, Blank C, et al. 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International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 4, April – 2019, Pages: 1-8 www.ijeais.org 1 Artin's Characters Table of the Group (Q2n×D3) When n= , and are Primes Number Lecturer Naba Hasoon Jaber University of Krufa, College of Education for Girls, Department of Mathematics, Iraq Email:[email protected] Abstract: The main purpose of this paper is to find Artin's characters table of the group (Q2n×D3)when n= ,and are primes number, which is denoted by Ar(Q2n×D3) where Q2m is denoted to Quaternion group and D3 is the Dihedral group of order 6 . Keywords: Artin, characters,group, Q2n , D3 ,prime. 1. INTRODUCTION Let G be a finite group, two elements of G are said to be Г-conjugate if the cyclic subgroups they generate are conjugate in G and this defines an equivalence relation on G and its classes are called Г-classes [3]. Let H be a subgroup of G and let be a class function on H, the induced class function on G, is given by: ′(g) = ∑ , g where is defined by: (x) = { [2]. Let H be a subgroup of G and be a character of H, then is a character of G, and it is called the induced character on G[7]. In 1976 ,David.G[3] studied "Artin Exponent of arbitrary characters of cyclic subgroup ", Journal of Algebra,61,p 58-76. In 1996, Knwabusz .K[9] studied "Some Definitions of Artin's Exponent of finite Group", USA, National foundation Math,GR. In this work we find Artin's characters table of the qroup (Q2n×D3) when n= ,and are primes number. 2. PRELIMINARIES 2.1The Generalized Quaternion Group Q2m [7 ] For each positive integer m,the generalized Quaternion Group Q2m of order 4m with two generators x and y satisfies Q2n={x h yk ,0 ,k=0,1} which has the following properties {x2n=y4=I, yxny-1=x-n}. 2.2The Dihedral Group D3 [9 ] The Dihedrael Group D3 is generate by a rotation r of order 3 and reflection s of order 2 then 6 elements of D3 can be written as: {1,r,r2,s,sr,sr2} . 2.3The Rational valued characters table: Definition(2.3.1) [5 ] A rational valued character θ of G is a character whose values are in Z,which is θ(g) Z for all g G. Theorem (2.3.2)[9 ] Every rational valued character of G be written as a linear combination of Artin's characters with coefficient rational numbers. Corollary (2.3.3)[5 ] The rational valued characters ∑ Form a basis for ,where are the irreducible characters of G and their numbers are equal to the number of conjugacy classes of a cyclic subgroup of G. Proposition(2.3.4)[9 ] The number of all rational valued characters of finite G is equal to the number of all distinct Γ-classis. Definition (2.3.5)[5 ] The information about rational valued characters of a finite group G is displayed in a table called a rational valued characters table of G.We denote it by ≡⃰(G)which is matrix whose columns are Γ-classes and rows are the valuses of all rational valued characters where is the number of Γ-classes. International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 4, April – 2019, Pages: 1-8 www.ijeais.org 2 The rational character table of Q2m where m is an odd number( 2.3.6) [7 ] Table(1) Where 0≤r≤ɭ ,ɭ is the number of Γ-classes C2m ,θj such that 1≤j≤ɭ+1 are the rational valued characters of group Q2m and if we denote Cij the elements of ≡⃰(Cm) and hij the elements of H as defined by: Hij={ The rational character table of Q2n when n= ,and are primes number (2.3.7)[7] Table(2) The rational character table of D3(2.3.8)[6] ≡⃰(D3) Table(3) 3. ARTIN'S CHARACTER TABLES: Theorem(3.1):[5] Let H be a cyclic subgroup of G and h1,h2,...,hm are chosen representatives for the m-conjugate classes of H contained in CL(g) in G,then: [y] Γ-classes of c2m X2r+1 X2r 1 1 1 1 ≡⃰(Cm) 1 1 1 ≡⃰(Cm) 0 0 -1 1 1 1 H 1 1 1 ≡⃰(Cm) 0 0 0 -2 -2 -2 2 2 2 [y] [x] [xn] [x2] [1] Γ-classes 1 1 1 1 1 0 -1 n-1 -1 n-1 1 1 1 1 1 0 1 1-n -1 n-1 0 -2 -2 2 2 [s] [r] [I] classes -Γ 3 2 1 2 3 6 0 -1 2 1 1 1 1 1 1 International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 4, April – 2019, Pages: 1-8 www.ijeais.org 3 (g)={ ∑ Propostion(3.2)[5 ] The number of all distinct Artin's characters on a group G is equal to the number of Γ-classes on G.Furthermore , Artin's characters are constant on each Γ-classes. Theorem(3.3) [10 ] The Artin's characters table of the Quaternion group Q2n when m is odd number is given as follows: Table(4) Where 0≤r≤m-1 , ɭ is the number of Γ-classes of C2m and are the Artin characters of the Quaternion group Q2m,for all 1≤j≤l+1. Artin characters table of Q2n when n= ,and are primes number (4.4)[ 8] The general form of Artin's characters of Q2n when n= ,and are primes number Table(5) The Artin characters of D3 (4.5)[9 ] Table(6) Γ-classes of C2m Γ-classes X2r X2r+1 [y] | 1 2 2 1 2 2n | 4n 2n 2n 4n 2n 2 2Ar(C2n) 0 0 0 m 0 0 n 0 0 1 Γ-classes [1] [x2] [xn] [x] [y] l l 1 2 1 2 2n l l 4n 2n 4n 2n 2 4n 0 0 0 0 4 4 0 0 0 2n 0 2n 0 0 2 2 2 2 0 n 0 n 0 1 Γ-classes [I] [r] [s] l l 1 2 3 ( )l 6 3 2 6 0 0 2 2 0 3 0 1 International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 4, April – 2019, Pages: 1-8 www.ijeais.org 4 4. THE MAIN RESULTE Propostion(4.1) If n = ,and are primes number, then The Artin's character table of the group (Q2n D3) is given as: The general form of the Artin characters of the group(Q2n D3)when n= ,and are primes number Table(7) which is (5 5) square matrix . Proof: Let g 2nxD3); g=(q,d),q Q2n,d D3 Case(I): If H is a cyclic subgroup of (Q2n {I}),then 1H= (x,I) 2H= (y,I) And the principle character of H, Artin's characters of Q2n,1≤j≤l+1, then by using theorem (4.1) { ∑ } 1H= (i) If g=(1,I) (j,1)(1,I)= . (g)= = = =6. (1) since H (ii) If g=(x n ,I),g then (j,1)(g)= = = =6. ( ) since H (iii) If g=(x 2 ,I) or g=(x,I) and g then (j,1)(g)= | | = (1+1)= .2= =6. ( since H and (g)=ø( )=1 and since g=(q,I),q Q2n,q x n (iv) if g H then Γ-classes [1,I][x2,I][xn,I][x,I][y,I] [1,r][x2,r][xn,r][x,r][y,r] [1,s][x2,s][xn,s][x,s][y,s] 1 2 1 2 2n 1 2 1 2 2n 1 2 1 2 2n 24n 24n 12n 12 24n 12n 24n 12n 12 24p 12n 24n 12n 12 6Ar(Q2n) 0 0 2Ar(Q2n) 2Ar(Q2n) 0 3Ar(Q2n) 0 Ar(Q2n) International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 4, April – 2019, Pages: 1-8 www.ijeais.org 5 (j,1)(g)=0 since H CL(g)= 2If H=(y,I)={(1,I),(y,I)(y2,I)(y3,I)} then (i) If g=(1,I) then (l+1,1)(g)= = =6.n=6. l+1(1) since H CL(1,I)={(1,I)} (ii) If g=(x n ,I)=(y 2 ,I) and g then (l+1,1)(g)= = =6.n=6. l+1(x n) since H CL(g)={g}, (g)=1 (iii) If g (x n ,I) and g H ,i.e.{g=(y,I) or g=(y 3 ,I)} then (l+1,1)(g)= -1))= =3.2=6. l+1(y)since H CL(g)={g,g -1} and (g)= (g-1)=1 Otherwise (l+1,I)(g)=0 since H CL(g)= Case(II): If H is a cyclic subgroup of (Q2nx{r}) then: 1H=(x,r) 2H=(y,r) 1-H=(x,r) and the principle character of H, then by using theorem (4.1) { ∑ } (i) If g=(1,I),(1,r) then (j,2)(g)= = = = =2. j(1) since H CL(g)={(1,I),(1,r),(1,r2)} (ii) g=(1,I),(xn,I),(xn,r),(1,r); g if g=(1,I),(1,r) then (j,2)(g)= (g)= since H CL(g)={g} and (g)=1 = = =2 j(1) (iii) if g= (xn,I),(xn,r) then (j,2)(g)= (g)= = = =2 j( ) (iv) if g (xn,I),(xn,r) and g then (j,2)(g)= (g)+ (g-1))= since H CL(g)={g,g-1} and (g)= (g-1)=1 = = Since g=(q,r),q Q2n,q x n (v) then (j,2)(g)=0 = j(q) since H CL(g)= 2if H=(y,r)={(1,I),(y,I),(y2,I),(y3,I),(1,r),(y,r),(y2,r),(y3,r)} (i) if g=(1,I),(1,r) then (l+1,2)(g)= = =2n=2 l+1(g) (ii) if g=(y2,I)=(xn,I),(y2,r) and g then International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 4, April – 2019, Pages: 1-8 www.ijeais.org 6 (l+1,2)(g)= = =2n=2 l+1(g) since H CL(g)={g} and (g)=1 (iii) if g (x n ,I) and g i.e. g={(y,I),(y,r)} or g={(y 3 ,I),(y 3 ,r)} then (l+1,2)(g)= (g)+ (g-1))= = 2 l+1(g) since H CL(g)={g,g-1} and (g)= (g-1)=1 otherwise (l+1,2)(g)=0 since H CL(g)= case(III): if H is a cyclic subgroup of (Q2nx{s}) then 1H=(x,s), 2H=(y,s) and the principle character of H, then by using theorem (4.1) { ∑ } 1H=(x,s) (i) If g=(1,I) then (j,3)(g)= | | (g)= = since H CL(g)={(1,I)} If g={(1,s)} then (j,3)(g)= | | (g)= = since H CL(g)={(1,s)} (ii) If g=(1,I),(xn,I),(xn,s),(1,s); g then If g=(1,I) then (j,3)(g)= (g)= since H CL(g)={g} and (g)=1 = = =3 j(1) If g={(1,s)} then (j,3)(g)= | | (g)= = since H CL(g)={g} and (g)=1 (iii)If g= (xn,I) then (j,3)(g)= (g)= = = =3 j( ) If g= (xn,s) then (j,3)(g)= (g)= = = = j( ) (iv)If g (xn,I),(xn,s) and g If g (xp,I) then (j,3)(g)= (g)+ (g-1)) = since H CL(g)={g,g-1} and (g)= (g-1)=1 = = j Since g=(q,I),q Q2n , q x n If g (xn,s) then (j,3)(g)= (g)+ (g-1)) = since H CL(g)={g,g-1} and (g)= (g-1)=1 = = j Since g=(q,s),q Q2n , q x n then (j,3)(g)=0 = j(q) since H CL(g)= International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 4, April – 2019, Pages: 1-8 www.ijeais.org 7 2-if H=(y,s)={(1,I),(y,I),(y2,I),(y3,I),(1,s),(y,s),(y2,s),(y3,s)} then (i)If g=(1,I) then (l+1,3)(g)= = =3.n=3 l+1(g) If g=(1,s) then (l+1,3)(g)= = =n= l+1(g) (ii)If g=(y2,I)=(xn,I) and g then (l+1,3)(g)= = = 3.n=3 l+1(g) since H CL(g)={g} and (g)=1 If g=(y2,s) and g then (l+1,3)(g)= = =n= l+1(g) since H CL(g)={g} and (g)=1 (iii)If g (xn,I) and g i.e. g={(y,I),(y,s)} or g={(y3,I),(y3,s)} then (l+1,3)(g)= (g)+ (g-1))= = 3 l+1(g) since H CL(g)={g,g -1 } and (g)= (g -1 )=1 (iv)If g=(y2,s), g H then (l+1,3)(g)= (g)= = = l+1(g) (v)If g=(y,s) then (l+1,3)(g)= (g)+ (g-1))= = = since H CL(g)={g,g-1} and (g)= (g-1)=1 otherwise (l+1,3)(g)=0 sinceH CL(g)= Example (4.2): To find Artine's character table of the group (Q66xD3) . Ar(Q66xD3)= Ar(Q2.3.11xD3)= Table(8 ) Γ-classes [1,I] [x2,I] [x33,I] [x,I] [y,I] [1,r] [x2,r] [x33,r] [x,r] [y,r] [1,s] [x2,s] [x33,s] [x,s] [y,s] | | 1 2 1 2 2n 2 2 2 2 2n 3 3 3 3 6n | 792 396 792 396 12 396 396 396 396 12 264 264 264 264 4 (1,1) 792 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (2,1) 264 264 0 0 0 0 0 0 0 0 0 0 0 0 0 (3,1) 396 0 132 0 0 0 0 0 0 0 0 0 0 0 0 (4,1) 24 0 0 24 0 0 0 0 0 0 0 0 0 0 0 (5,1) 8 8 0 8 8 0 0 0 0 0 0 0 0 0 0 (1,2) 12 0 4 12 0 4 0 0 0 0 0 0 0 0 0 (2,2) 396 0 0 0 0 0 396 0 0 0 0 0 0 0 0 (3,2) 132 132 0 0 0 0 132 132 0 0 0 0 0 0 0 (4,2) 198 0 66 0 0 0 198 0 66 0 0 0 0 0 0 (5,2) 12 0 0 12 0 0 12 0 0 12 0 0 0 0 0 (1,3) 4 4 0 4 4 0 4 4 0 4 4 0 0 0 0 (2,3) 6 0 2 6 0 2 6 0 2 6 0 2 0 0 0 (3,3) 198 0 0 0 0 0 198 0 0 0 0 0 6 0 0 (4,3) 66 66 0 0 0 0 66 66 0 0 0 0 2 2 0 (5,3) 99 0 33 0 0 0 99 0 33 0 0 0 3 0 1 REFERENCES: [1] M.J.Hall "The Theory of Group ",Macmillan,Neyork,1959. International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 4, April – 2019, Pages: 1-8 www.ijeais.org 8 [2] I. M. Isaacs, "On Character Theory Of Finite Groups", Academic Press, Newyork,1976. [3] David.G "Artin Exponent of arbitrary characters of cyclic subgroup ", Journal of Algebra, 1976,p 58-76. [4] M.S.Kirdar,"The factor Group of the Z-valued class function Modulo the group of the Generalized characters"University of Birmingham1980 [5] C.Curits and I.Reiner,"Methods of Representation Theory with Application to finite Groups and order",John Wily and sons,NewYork,1981. [6] H.H.Abass,"On the factor Group of class function over the Group of Generalized characters of Dn",M.S.C.thesis,Technology University,1994. [7] N.R.Mahamood"The cyclic Decomposition of the factor group of ",M.SC. thesis University of Technology,1995. [8] A.S.Abid,"Artin's characters table of Dihedral group for odd number",M.S.C.thesis,University of Kufa,2006. [9] K . Knwabusz, "some Definitions of Artin's Exponent of finite Group", USA, National foundation Math,GR,1996. [10] A.H.Abdul-Mun'em,"On Artin Cokernel of the Quaternion Group Q2m when m is odd number ",2008.

Volume 03, Issue 03 (July-August 2020), PP 01-14 www.ijmsdr.org ISSN: 2581-902X 1 International Journal of Medical Science and Dental Research Improving the Access of the Indonesian Community to Qualified Health Services Santriani Hadi1, Hasta Handayani Idrus2 1 Department of Parasitology Faculty of Medicine,, University Muslim Indonesia, Makassar, Indonesia 2Department of MicrobiologyFaculty of Medicine, University Muslim Indonesia, Makassar, Indonesia Abstract Health development is faced with a variety of important issues including health status disparities; double burden of disease; quality, equity and affordability of health services; community protection in the field of medicine and food; and clean and healthy life behavior. Methods: The method used in this short communication is descriptive-comparative where we review Safety Culture in Indonesian Health Services in five aspects, namely Health Services for the Poor, nutritional problems that are never complete, Extraordinary Events of Communicable Diseases, Poor health in Disaster areas, and finally the number of health workers still lacking and comparing the problem of Human Resources Health problems in Indonesia according to WHO (2011) and the Indonesian Ministry of Health (2009). Results: The results obtained in this brief communication are that we get new information in the form of problems encountered in Safety Culture in Indonesian Health Services, examples of cases that occur, policies taken by the government in handling them and the results obtained after the implementation of the policy. All of these are reviewed in five aspects. Conclusion: The conclusion we can take in this brief communication is that health problems that occur in Indonesia have not been resolved even though the government has implemented policies related to these problems but has not been resolved to date Keywords: qualified health services, safety culture, Indonesian community I. Introduction Health development is faced with a variety of important issues including health status disparities; double burden of disease; quality, equity and affordability of health services; community protection in the field of medicine and food; and clean and healthy life behavior. Some other important issues that need to be addressed immediately are increasing the access of the poor to health services, handling malnutrition problems, tackling infectious disease outbreaks, health services in disaster areas, and fulfilling the number and distribution of health workers [1]. Health development is an effort to fulfill one of the basic rights of the people, namely the right to obtain health services. Health development must be seen as an investment to improve the quality of human resources and support economic development, and have an important role in poverty reduction efforts [2]. Steps that have been taken are increasing access to health, especially for the poor through free health services; increased prevention and control of infectious diseases including polio and bird flu; improving the quality, affordability and equity of basic health services; increasing the quality and quantity of health workers; quality assurance, safety and efficacy of drugs and food; health management in disaster areas; and increasing health promotion and community empowerment [3]. Volume 03, Issue 03 (July-August 2020), PP 01-14 www.ijmsdr.org ISSN: 2581-902X 2 As a follow up, health development is directed at increasing the equity and affordability of health services; improve the quality of health services; improve hygiene and healthy behavior; increase efforts to prevent and eradicate diseases; improve the state of community nutrition; and improve handling of health problems in disaster areas [4]. The main problems of health development at present include the high disparity in health status between socio-economic, interregional levels, and between urban and rural areas. In general, the health status of populations with high socioeconomic levels, in western Indonesia, and in urban areas, tends to be better [5]. Conversely, the health status of the population with low socioeconomic status, in eastern Indonesia and in rural areas is still lagging behind [6]. Other important problems faced are the double burden of the disease, namely not yet overcoming infectious diseases suffered by the community such as pulmonary tuberculosis, acute respiratory infections, malaria, and diarrhea, and the reemergence of polio and bird flu [7]. However, at the same time there was an increase in noncommunicable diseases such as heart and blood vessel diseases, as well as diabetes mellitus and cancer. On the other hand, the quality, equity, and affordability of health services is also still low. Quality of service is a constraint because medical personnel are very limited and inadequate equipment [8]. In terms of numbers, the ratio of health workers to the number of people that must be served is still low. The affordability of services is closely related to the number and equity of health facilities. In 2002, for every 100,000 residents there were only 3.5 Puskesmas. Even then, some residents, especially those living in remote areas, do not use the Puskesmas because of limited transportation facilities and geographical constraints [9]. In addition, the original Indonesian medicine has not been fully developed, even though its potential is huge. Community behavior also often does not support clean and healthy living [10]. This can be seen from the widespread smoking habit, the low level of exclusive breastfeeding, the high prevalence of undernutrition and over nutrition in children under five, as well as the tendency of increasing numbers of people with HIV / AIDS, sufferers of narcotics, psychotropic, addictive substances, and accidental death [11]. II. Methods In this descriptive-comparative study, we review Safety Culture in Indonesian Health Services in five aspects, namely Health Services for the Poor, nutritional problems that are never complete, Extraordinary Events of Communicable Diseases, Poor health in disaster areas, and finally The number of health workers is still lacking (Table 1). We compare the issue of Human Resources Health problems in Indonesia according to WHO (2011) and the Indonesian Ministry of Health (2009) (Table 2). III. Results The results obtained in this brief communication are that we get new information in the form of problems encountered in Safety Culture in Indonesian Health Services, examples of cases that occur, policies taken by the government in handling them and the results obtained after the implementation of the policy. We can see in at the table 1. and in table 2. we can see a comparison of Issues of problems of Health Human Resources in Indonesia according to WHO (2011) and Indonesian Ministry of Health (2009). Volume 03, Issue 03 (July-August 2020), PP 01-14 www.ijmsdr.org ISSN: 2581-902X 3 Table 1. Shows an overview of Safety Culture in Indonesian Health Services reviewed in five aspects Problems encountered Examples of Problems Government Policy Steps The Results Achieved Health Services for the Poor Society Jamkesmas is a social assistance program for health services for the poor and disadvantaged. This program is organized securely so that there will be cross-subsidies in order to create comprehensive health services for the poor [3]. 1. The implementation of the 1945 Constitution article 34 paragraph 1 and 2, since 1998 has been carried out several efforts to maintain the health of the poor population. The aim of this effort is to maintain and improve the quality and access to health services for the poor, especially health services in puskesmas and hospitals [12] . 2. Free health services for the poor have been sought by the Government since the economic crisis in 1997 [4]. 3. The implementation of Law No. 40 of 2004 concerning the National Social Security System, efforts to improve access of the poor to health services are further enhanced through efforts to maintain the health of the poor with a health insurance / insurance system [6]. 4. The health service program for the poor is carried out with the principles of (a) comprehensive services in accordance with health standards; and (b) basic health services in the Puskesmas and outpatient and inpatient referrals in hospitals [5]. 1. The National Social Security System improves health services for the poor through efforts to maintain the health of the poor with a health insurance / insurance system [7]. 2. The poor are included in health insurance with a premium paid by the Government [8]. 3. The implementation of the health service program for the poor is carried out with the principle, among others, comprehensive services in accordance with health standards and basic health services in Puskesmas and outpatient and inpatient referrals in class III hospitals [2]. 4. The poor can get free health services both at the puskesmas and in the hospital [7]. 5. The poor can get access to the Puskesmas and operational costs and motorbikes have been provided for officers, four-wheeled mobile Puskesmas, and Puskesmas around motorized boats [13]. nutrition problem that is never complete The problem of malnutrition and measles in Papua is a clear example of cases of malnutrition that occurred in Indonesia. The Ministry of Health sent 39 health workers in response to the incidence of malnutrition in this area. In addition, assistance was also provided by the Indonesian national 1. Organizing a coordination meeting to reactivate the Food and Nutrition Team [14]. 2. Implementation of the Food and Nutrition Alert System (SKPG), as well as case investigations to all potential areas [15]. 3. Interventions conducted in dealing with malnutrition are directed at preventing death and disability through early discovery of cases 1. The results achieved include the enactment of an extraordinary event of malnutrition in the province of West Nusa Tenggara and the exemption of fees for sufferers of malnutrition who are hospitalized [15]. 2. One tonne of complementary breastfeeding food has been sent to sufferers through the regency / city government in West Nusa Tenggara Province and 1.5 tonnes of complementary breastfeeding food for East Volume 03, Issue 03 (July-August 2020), PP 01-14 www.ijmsdr.org ISSN: 2581-902X 4 army by sending health workers [14]. of malnutrition and providing professional management at the community, health center, and hospital level [15]. Nusa Tenggara Province, as well as assigning special staff to carry out investigations [15]. 3. Additional food assistance was provided to 26,200 toddlers, 8,400 pregnant women and 38,000 school children in West Nusa Tenggara Province and 23,200 toddlers, 6,700 pregnant women, and 43,000 school children in East Nusa Tenggara Province [16]. Extraordinary Occurrence of Communicable Diseases The extraordinary incident of polio in Indonesia was last reported in 20052006 for type 1 polio virus originating from the Middle East. This extraordinary incident occurred in 10 provinces and 47 regencies / cities throughout Indonesia, with a total of reported cases of 305. On the other hand, the availability of polio vaccine in Indonesia is nothing to worry about [17] . 1. increase coverage of routine immunizations for infants to the village level that is given free of charge. 2. Carry out additional immunizations through the National Immunization Week, National Immunization Sub-Week for 5 provinces and carry out the Immunization Month for School Children [18]. 3. Perform routine surveillance of Acute Flaccid Paralysis or sudden paralysis in children under 15 years routinely [19]. 1. The results of the implementation of routine polio immunization activities nationally in the last three years the coverage reached more than 90 percent but the activity was still not evenly distributed in all villages. This happens because the village Universal Child Immunization in the last three years has not reached 80 percent [9]. 2. With the outbreak of Polio in the Provinces of West Java, Banten, Central Java, Lampung and DKI Jakarta, Outbreak Response Immunization has been carried out in these provinces to prevent transmission of the polio virus [10]. 3. Well implemented Outbreak Response Immunization activities that are strived to prevent transmission of wild polio virus around patients. 4. The implementation of limited mass immunization or mopping up to break the wider chain of transmission of wild polio virus [4]. Poor health in the Disaster area The impact of disasters on public health varies, among other things depending on the type and magnitude of the disaster that occurred. Injury cases that require medical treatment, for example, are relatively more common in earthquake disasters 1. For the management of natural disasters, policy measures have been taken to address emergency health with the aim of providing immediate health services to the living population. 2. Also carried out rehabilitation of various health facilities and infrastructure so that health 1. Built and operated 7 Satellite Health Posts in several dwellings, while 23 Satellite Health Posts are still in the completion stage [22]. 2. For the management of the Satellite Health Post, 880 health workers have been recruited, consisting of 110 doctors, 165 midwives, 110 public health scholars, 55 nutritionists, 55 environmental health workers, 330 Volume 03, Issue 03 (July-August 2020), PP 01-14 www.ijmsdr.org ISSN: 2581-902X 5 compared to injury cases due to floods and tidal waves. Conversely, floods that occur in a relatively long time can cause damage to the sanitation system and clean water, as well as lead to the potential for extraordinary events such as diseases transmitted through water-borne diseases such as diarrhea and leptospirosis [20]. services can function as before. All this was done in a coordinated way, especially with volunteers from within and outside the country [21]. 3. A more comprehensive health and rehabilitation plan for the disaster area has been prepared, as stated in the blueprint for the Rehabilitation and Reconstruction Plan for the disaster area [21]. nurses, and 55 pharmacist assistants [23]. 3. The implementation of health services by the mobile medical team using the facilities of 4-wheeled vehicles and wheels 2. In an effort to meet the needs of specialist doctors and skilled nurses, initially carried out through special assignments [24]. 4. Specialist doctors and nurses are further developed into permanent staff placements currently being recruited [24]. 5. The recovery of the function of health services is carried out by rehabilitating health service facilities that were slightly damaged by provincial and district / city health offices in collaboration with local community social institutions [25]. 6. Conducting counseling training for health workers on a regular basis followed by early detection of community psychiatric disorders and treatment in health care facilities. The number of health workers is still lacking The number of general practitioners in Indonesia has exceeded the quota, but the uneven distribution makes 728 health centers in Indonesia do not have general practitioners, 728 health centers are only filled with health workers such as midwives and nurses. While supporting health workers other than doctors are also needed for promotive and preventive efforts such as environmental health workers, pharmacists, nutritionists, public health and medical laboratory experts [26]. 1. In the context of increasing the quantity and quality of health human resources, the policy step taken is the recruitment of medical personnel, especially for puskesmas and hospitals in remote areas [27]. 2. The preparation of the plan for the placement of health workers shall be allocated from the regular appointment of doctors for nonpermanent employees and prospective civil servants [28]. 1. As a result, 1,040 doctors, 139 dentists, and 3,937 midwives were appointed to Non-Permanent Employees in 2004. There are 466 doctors, 77 dentists and 1,651 midwives who are placed in disadvantaged areas. To improve equity, of these 276 doctors, 31 dentists, and all midwives were placed in remote and very remote areas [29]. 2. Selection of Prospective Civil Servants of the Ministry of Health in 2004 was conducted on 28,929 applicants to fill the formation of 2,384 people. From the area it was reported that 409,746 applicants had been selected to be placed in disadvantaged areas as many as 6,235 people [29]. 3. In addition, the Presidential Decree No. revision Volume 03, Issue 03 (July-August 2020), PP 01-14 www.ijmsdr.org ISSN: 2581-902X 6 concept has also been prepared. 37 of 1991 and Presidential Decree No. 77 of 2000 concerning Strategic Health Workers for Non-permanent Employees and the concept of the Presidential Regulation on Strategic Health Workers as Nonpermanent Employees [29]. 4. There was also a discussion on the draft of the Presidential Regulation on the Appointment of Strategic Health Workers for Non-Permanent Employees and the Policy for Providing Incentives for health workers in disadvantaged areas with provincial health offices, several district health offices, several regional public hospitals, and cross-sectoral officials [29]. table 2. Comparison of Issues of problems of Health Human Resources in Indonesia according to WHO (2011) and Indonesian Ministry of Health (2009). Issues of problems with Health Human Resources in Indonesia according to WHO (2011) Issues of problems with Health Human Resources in Indonesia according to the Indonesian Ministry of Health (2009) Issues of problems with Health Human Resources in Indonesia according to WHO (2011) Issues of problems with Health Human Resources in Indonesia according to the Indonesian Ministry of Health (2009) The development of health workers has not been able to meet the needs of health workers for health services / development. Health workers continue to improve in number, quality and distribution, but are still unable to meet the needs of health services in all regions, especially in disadvantaged, remote, border and island areas. The quality of health workers does not yet have the competitiveness in meeting the demand for health workers from abroad [30]. Development and empowerment of Human Resources for health has not been able to meet the needs of Human Resources for health development. Because the amount is not enough and the distribution of health workers that has not been resolved [31]. Regulations to support efforts to develop health workers are still limited. Much of the training needed by health workers is paid and quite expensive so that many health workers cannot upgrade their knowledge due to cost constraints [30]. Health human resource policy and program planning is still weak and has not been supported by an adequate health human resource information system [31]. Planning for health personnel needs still needs to be improved and not yet supported by an adequate health workforce information system. The overall plan for the need for health There is still a lack of harmony between the needs and procurement of various types of health human resources. The quality of health human resource education and health Volume 03, Issue 03 (July-August 2020), PP 01-14 www.ijmsdr.org ISSN: 2581-902X 7 workers has not been prepared as expected, so it has not been fully used as a reference in the procurement / education of health workers, utilization of health workers, as well as guidance and quality control of health workers [32] . training in general is still inadequate [30]. There is still a lack of harmony between the needs and the procurement / education of various types of health workers. The study of the type of health workers needed has not been carried out properly. The quality of the education and training of health workers in general is still inadequate. There are still many educational institutions which are not accredited and meet the standards. This will have an impact on the competence and quality of graduates of health workers. Problems in the education of health workers in general are systemic, among others there is a mismatch in the competence of education graduates with health services needed by the community, weak cooperation between actors in health development and health workforce education, more dominant education of health workers oriented to hospitals compared to Primary Health Care [32] . In the utilization of Health Human Resources, the distribution of quality Health Human Resources is still lacking. Career development, reward and sanction systems have not been as appropriate. Regulations to support Human Resources for health are still limited [33]. In the utilization of health workers, even distribution and utilization of quality health workers is still lacking, especially in disadvantaged, remote, border, island and less desirable areas. This is due to the socio-economic, cultural and regional government disparities, including geographical conditions between regions, reducing the interest of health workers to be placed in the area. In addition, the development and implementation of career development patterns, rewards and sanctions systems have not been implemented as expected. Continuing professional development (Continue Professional Development / CPD), and Training Need Assessment (TNA) still needs to be developed [34]. Development and supervision of health Human Resources and health Human Resources support is still lacking. Mainly in remote areas because many doctors do not want to be placed in remote areas and they are only concentrated in urban areas [2]. The guidance and supervision of the quality of health workers cannot be carried out as expected. Registration and certification of health workers is still limited to doctors and dentists. Dissemination and application of laws and regulations in the field of health personnel development have not been carried out adequately. Resources supporting the development and empowerment of health workers are still limited. Health personnel information systems have not been able to provide accurate, reliable and timely data. Support of financial resources and other resources has not been sufficient [34]. The problem of Human Resources for Health at the Global level that has not been resolved even though many programs are in line [5]. Volume 03, Issue 03 (July-August 2020), PP 01-14 www.ijmsdr.org ISSN: 2581-902X 8 IV. Discussion A. Problems Encountered Basic problems in the last ten months, there have been at least five important issues in the health sector that need immediate treatment, namely guaranteeing poor people's access to health services, handling malnutrition problems, handling infectious disease outbreaks, health services in disaster areas, and meeting the number and distribution of health workers [5]. 1. Health Services for the Poor Society Nationally the status of public health has improved. However, the disparity in health status between the able population and the poor population is still quite large. Various data shows that the health status of the poor is lower when compared to the rich population. This can be seen, among others, from the high infant mortality rate and infant mortality rate in the poor population group. According to the 2002-2003 Indonesian Demographic and Health Survey, the infant mortality rate in the poorest group was 61 compared to 17 per 1,000 live births in the richest group. Likewise, the under-five mortality rate in the poorest population (77 per 1,000 live births) is much higher than the under-five mortality rate in the richest population (22 per 1,000 live births). Infectious diseases that are the leading cause of death in infants and toddlers, diarrhea, neonatal tetanus and birth complications, are also more common in poor populations [7], [13]. 2. Nutrition problem that is never complete Health problems that have caused considerable public attention lately are problems of malnutrition and poor nutrition. Although since 1989 there has been a relatively sharp decrease in the prevalence of undernutrition, starting in 1999 the decline in the prevalence of undernutrition and malnutrition in children under five is relatively slow and tends to remain unchanged. Currently there are 10 provinces with a prevalence of malnutrition above 30, and some even above 40 percent, namely in the provinces of Gorontalo, West Nusa Tenggara, East Nusa Tenggara, and Papua. Lack of energy and protein at a severe or more popular level called malnutrition, can cause major health problems and can even cause death in children [15], [35]. 3. Extraordinary Occurrence of Communicable Diseases Another health problem that is of concern to the community is the outbreak of various infectious diseases. Most infectious diseases suffered by the community are infectious diseases such as pulmonary tuberculosis which currently ranks third in the world, acute respiratory infections, malaria, and diarrhea. In addition, Indonesia also faces emerging diseases (newly developing diseases) such as HIV / AIDS and Severe Acute Respiratory Syndrome and re-emerging diseases (diseases that previously began to decline, but increased again) such as dengue hemorrhagic fever and Infection lung [36]. One infectious disease that has recently emerged is the emergence of polio cases in several regions such as West Java, Banten, Central Java, Lampung and DKI Jakarta. Polio is a very dangerous infectious disease caused by a virus that attacks the nervous system and can cause permanent paralysis or death. One in 200 cases of viral infection will cause paralysis, 5-10 percent of patients die from paralysis of the respiratory muscles. There is no cure for polio. This disease can only be prevented by immunization. The vaccine for immunization is safe and by the Indonesian Ulema Council declared halal [15]. 4. Poor health in the Disaster area The natural disasters of the earthquake and tsunami that occurred in Aceh, Nias, Alor and Nabire have had a major impact on the health sector. Many victims died, disappeared and were injured. Many health facilities and infrastructure were destroyed and did not function optimally, such as hospitals, puskesmas, supporting puskesmas, health offices, health laboratory centers, pharmaceutical warehouses, vaccine warehouses, health polytechnics, and port health offices . The tsunami disaster in Aceh caused damage to 9 hospitals, 43 health centers, 59 sub-health centers, 700 village polyclinics, and 55 mobile health centers, and other facilities such as hospitals, laboratories and health offices. The number of health workers who died or disappeared was 683 people [17], [37]. Volume 03, Issue 03 (July-August 2020), PP 01-14 www.ijmsdr.org ISSN: 2581-902X 9 5. The number of health workers is still lacking The condition of health workers in 2004 was not much different from that because the education system was still unable to produce sufficient numbers of health workers, and the recruitment system and incentive patterns for health workers were less than optimal[38]. In addition, the number and distribution of community health workers is still inadequate so that many puskesmas do not have doctors and community health workers. This limitation is exacerbated by the uneven distribution of health workers. For example, more than two-thirds of specialist doctors are in Java and Bali. Doctor ratio disparity general per 100,000 population among regions is also still high and ranges from 2.3 in Lampung to 28.0 in Yogyakarta [3], [30]. B. Examples of Problems 1. Health Services for the Poor Society The 1945 Constitution Article 28H and Law Number 36 of 2009 concerning Health, stipulates that everyone has the right to receive health services. Therefore every individual, family and community has the right to obtain protection for their health, and the state is responsible for regulating the fulfillment of the right to a healthy life for its population, including for the poor and disadvantaged. The establishment of the National Social Security System is realized through Law Number 40 of 2004 concerning the National Social Security System which has a health insurance program, work accident insurance, old age insurance, pension insurance and death insurance. This social security is the government's effort in dealing with the monetary crisis. As is known, the crisis began in 1997 until now, due to multidimensional factors including the transfer of the subsidy program for the poor in the form of fuel subsidies for the health sector for the poor to the Health Safety Net program for the Poor [33], [39]. 2. Nutrition problem that is never complete One of the most terrible problems that occur in Indonesia is the problem of malnutrition. The problem of malnutrition in general can be divided into problems of over nutrition as well as nutritional deficiencies. There are 8.4 million children in Indonesia who are malnourished and experience what doctors call the term stunting. Stunting is a condition where a child has a very small body size compared to children their age [4]. The problem of malnutrition and measles in Papua is an extraordinary event. The Ministry of Health has deployed 39 health workers in response to an outbreak of severe malnutrition and measles. In addition, assistance was also provided by the Indonesian National Army by sending health units. The local government also formed a team that was immediately sent to the field to conduct prevention and treatment and supplementary feeding. Habits of people who do not care about health are the cause of the outbreak [6]. 3. Extraordinary Occurrence of Communicable Diseases From the beginning until now, outbreaks of disease have been attacking the people of Indonesia, and not infrequently cause extraordinary events. The Ministry of Health implements Extraordinary Events to classify an outbreak of a disease. Indonesia has struggled to face a variety of Extraordinary Events. Here are some examples of the extraordinary events of infectious diseases that have hit Indonesia is the outbreak of polio in Indonesia was last reported in 2005-2006 for type 1 polio virus originating from the Middle East. This extraordinary incident occurred in 10 provinces and 47 regencies / cities throughout Indonesia, with a total of reported cases of 305. On the other hand, the availability of polio vaccine in Indonesia is nothing to worry about [18], [19]. 4. Poor health in the Disaster area One of the impacts of the disaster on the declining quality of life of the population can be seen from various public health problems that occur. Disasters that are followed by displacement have the potential to cause health problems that are actually preceded by problems in other fields / sectors. Earthquakes, floods, landslides and volcanic eruptions, in the short term can have an impact on the death toll, victims of severe injuries that require intensive care, increased risk of infectious diseases, damage to health facilities and water supply systems. The emergence of health problems, among others, starts from a lack of clean water which Volume 03, Issue 03 (July-August 2020), PP 01-14 www.ijmsdr.org ISSN: 2581-902X 10 results in poor personal hygiene, poor environmental sanitation which is the beginning of the proliferation of several types of infectious diseases [24], [25]. 5. The number of health workers is still lacking World Health Organization said, Indonesia is included in the group of countries with the most serious shortages of health workers. WHO identifies Indonesia, Bangladesh, Bhutan and India as countries with less than 23 health workers including doctors, midwives and nurses, per 10,000 population. The ratio of 23 health workers per 10,000 population is considered the minimum limit to reach the 80 percent coverage of the most essential health interventions. A study on human resources carried out in February 2012 found that a country that experienced a crisis of health workers could not increase the number of health workers to a minimum. Even financial assistance is not enough to achieve the desired progress in this field [28], [29]. C. Government Policy Steps and The Results Achieved 1. Health Services for the Poor Society Recognizing the importance of sustainable handling of the health problems of the poor as an effort to fulfill the mandate of the 1945 Constitution article 34 paragraph 1 and 2, since 1998 a number of efforts have been taken to maintain the health of the poor. The aim of this effort is to maintain and improve the quality and access to health services for the poor, especially health services in puskesmas and hospitals [4]. Free health services for the poor have been sought by the Government since the economic crisis in 1997. The results of monitoring the implementation of health services for the poor indicate that there are several obstacles, including inefficient use of funds. Therefore, in 2005, in line with Law No. 40 of 2004 concerning the National Social Security System, efforts to increase access of the poor to health services are further enhanced through efforts to maintain the health of the poor with a health insurance / insurance system[40]. With this system, poor people are included in health insurance with premiums paid by the Government [41]. 2. Nutrition problem that is never complete Learning from the experiences of nutrition management in Indonesia and experiences in various countries, overcoming the problem of malnutrition is carried out with a holistic approach involving all parties including families, communities, governments and economic actors. Interventions to overcome the problem of malnutrition consist of short-term (emergency), medium-term, and long-term stages. The efforts that have been made are mass weighing to find cases early, holding a coordination meeting to reactivate the Food and Nutrition Team, implementing the Food and Nutrition Alert System, and investigating cases to all potential areas. Interventions conducted in dealing with malnutrition are directed at preventing death and disability through early discovery of malnutrition cases and providing professional management at the community, health center, and hospital level [2], [15]. 3. Extraordinary Occurrence of Communicable Diseases To achieve eradication of polio, various efforts have been carried out, namely (1) increasing coverage of routine immunization in infants to the village level that is given free of charge; (2) conduct additional immunizations through the National Immunization Week, SubNational Immunization Week for 5 provinces and carry out the Immunization Month for School Children; and (3) routine surveillance of Acute Flaccid Paralysis with sudden paralysis in children under 15 years old [4], [17]. 4. Poor health in the Disaster area The earthquake and tsunami disaster in the Province of Nangroe Aceh Darussalam and Nias, North Sumatra in addition to causing casualties, missing and injured, has also destroyed thousands of houses, infrastructure, and various public service facilities including health care facilities . For the prevention of natural disasters in Nangroe Aceh Darussalam Province, policy measures have been taken to address emergency health in order to provide immediate health services to the living population [24]. In addition, rehabilitation of various health facilities and infrastructure was carried out so that health services could function as before. All this was done in a coordinated way, especially with volunteers from Volume 03, Issue 03 (July-August 2020), PP 01-14 www.ijmsdr.org ISSN: 2581-902X 11 within and outside the country. In addition, a more comprehensive rehabilitation and reconstruction plan for Aceh and Nias in the health sector, such as the blueprint or blueprint for the Rehabilitation and Reconstruction Plan for Aceh and Nias, North Sumatra [41]. 5. The number of health workers is still lacking In order to improve the quantity and quality of health human resources, the policy steps taken are the recruitment of medical personnel, especially for puskesmas and hospitals in remote areas. The preparation of the planned placement of health workers is allocated from the regular appointment of non-permanent employees doctors and prospective civil servants [29]. In addition, the draft revision of Presidential Decree No. has also been prepared. 37 of 1991 and Presidential Decree No. 77 of 2000 concerning Strategic Health Workers for Non-permanent Employees and the concept of the Presidential Regulation on Strategic Health Workers as non-permanent Employees. Do also a discussion on the draft of the Presidential Decree on the Appointment of non-permanent Employees Strategic Health Workers and the Policy on Providing Incentives for health workers in disadvantaged areas with provincial health offices, several district health offices, several regional public hospitals, and cross-sectoral officials [30]. D. The next step must be taken By considering the problems encountered, the policy steps taken, and the results that have been achieved as mentioned above, the necessary follow-up plans can be described as follows. 1. Health Services for the Poor Society In order to improve the quality of health services, efforts will be made to appoint and place health workers, such as doctors and nursing staff, especially in remote areas, increasing the proportion of puskesmas that have doctors; an increase in the proportion of district / city hospitals that are have basic specialist doctors, and improve the quality of education and training of health workers [2]. Planning for the need for health workers needs to be increased to meet the needs of health workers, especially for health services in puskesmas and their networks, as well as district / city hospitals, especially in remote and disaster areas. This step needs to be followed by improving the skills and professionalism of health workers through education and training of health workers, fostering health workers including career development of health workers; and the preparation of competency standards and regulations of the health profession [42], [43]. 2. Nutrition problem that is never complete In order to improve the nutritional status of the community, especially in pregnant women, infants, and children under five, nutrition education and community empowerment need to be carried out for the achievement of nutritionally aware families. Reduction of protein energy, iron nutrient anemia, disorders caused by iodine deficiency, lack of vitamin A, and other micronutrient deficiencies need to be improved, in line with over nutrition control, and nutritional surveillance [14]. 3. Extraordinary Occurrence of Communicable Diseases In order to improve clean and healthy living behaviors activities will be carried out (1) developing health promotion media and communication, information and education technology; (2) developing communitybased health efforts, (such as integrated service posts, village maternity halls, and school health businesses) and young people; and (3) improvement of public health education [17]. Efforts to improve clean and healthy living behavior need to be supported by improving the quality of the environment carried out through the provision of clean water and basic sanitation facilities, especially for the poor; maintenance and supervision of environmental quality; controlling the impact of environmental pollution risks; and developing healthy areas [17], [18]. Volume 03, Issue 03 (July-August 2020), PP 01-14 www.ijmsdr.org ISSN: 2581-902X 12 4. Poor health in the Disaster area In the context of mitigating the effects of disasters occurring in various regions, efforts that will be continued include rehabilitation and reconstruction of damaged health service facilities, fulfillment of health personnel, prevention and eradication of diseases, provision of medicines and health equipment, nutrition improvement, and efforts to restore the function of health services in the affected areas. Furthermore, in the context of mitigating the effects of the earthquake and tsunami disaster in Nangroe Aceh Darussalam and Nias, North Sumatra, to improve the efficiency and effectiveness of health services, in the rehabilitation and reconstruction phase, cross-sectoral and cross-program cooperation will be further enhanced, especially with the Rehabilitation Agency and Reconstruction of Aceh and Nias, North Sumatra, including the availability of funding sources [21], [22]. 5. The number of health workers is still lacking Increasing the equitable distribution and affordability of public health services is carried out through, among others, the provision of free health services for the poor in the puskesmas and its networks, as well as in hospitals. Through this effort it is expected that the level of disparity in health status among the population rich and poor are decreasing [26]. To anticipate various technical obstacles in the field faced by the poor in getting proper services, for example administrative and procedural barriers, socialization and advocacy to implementing institutions will be further improved, in addition to strengthening monitoring and safe guarding. In addition, an increase in facilities and infrastructure at the puskesmas and its networks was also carried out; construction and repair of hospitals, especially in the affected areas and selectively lagging; procurement of medicines, procurement of equipment and health supplies; and providing operational and maintenance costs [27], [28]. V. Conclusion The conclusion we can take in this brief communication is that health problems that occur in Indonesia have not been resolved even though the government has implemented policies related to these problems but has not been resolved to date. therefore the role required by various parties in solving this problem is primarily the role of the wider community in the application of government programs so that problems in the health sector can be immediately resolved mainly in five aspects namely health status disparities; double burden of disease; quality, equity and affordability of health services; community protection in the field of medicine and food; and clean and healthy life behavior. Ethical considerations Ethical issues (Including plagiarism, informed consent, misconduct, data fabrication and/or fal-sification, double publication and/or submission, redundancy, etc.) have been completely observed by the authors. Acknowledgments Thank you to the Ministry of Research and Technology / National Research and Innovation Agency of the Republic of Indonesia for the opportunity that has been given to make publications at Coventry University UK, and thank my supervisor Prof. Benny Tjahjono who has guided me in writing this manuscript. Conflict of interest The authors declare that there is no conflict of interests. References [1] K. Rachmawati, T. Schultz, and L. Cusack, "Translation , adaptation and psychometric testing of a tool for measuring nurses ' attitudes towards research in Indonesian primary health care," Wiley Nurs. Open, pp. 96–107, 2017. Volume 03, Issue 03 (July-August 2020), PP 01-14 www.ijmsdr.org ISSN: 2581-902X 13 [2] J. Mulyanto, D. S. Kringos, and A. E. Kunst, "The evolution of income-related inequalities in healthcare utilisation in Indonesia , 1993 – 2014," PLoS One, pp. 1–15, 2019. [3] M. A. 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Open Theology 2016; 2: 53–78 Olga Louchakova-Schwartz* Theophanis the Monk and Monoimus the Arab in a Phenomenological-Cognitive Perspective DOI 10.1515/opth-2016-0005 Received August 31, 2015; accepted October 26, 2015 Abstract: Two brief Late Antique religious texts, respectively by the monk Theophanis and by Monoimus the Arab, present an interesting problem of whether they embody the authors' experience, or whether they are merely literary constructs. Rather than approaching this issue through the lens of theory, the article shows how phenomenological analysis and studies of living subjectivity can be engaged with the text in order to clarify the contents of introspective experience and the genesis of its religious connotations. The analysis uncovers a previously unnoticed form of embodied introspective religious experience which is structured as a ladder with a distinct internal structure with the high degree of synchronic and diachronic stability. This approach also helps one identify the specific introspective techniques in the canonical and non-canonical literature of early Christian tradition, as related to the concepts of "theosis" and "kenosys", as well as to suggest some neurological correspondents of religious cognition. Keywords: cognitive historiography, human neuroscience, introspection, subjectivity, ladder imagery, Monoimus the Arab, phenomenology, Philokalia, Prayer of the Heart, religious experience, Theophanis the Monk In this paper, we will explore two brief Late Antique religious texts, respectively by the monk Theophanis and by Monoimus the Arab. These texts present an interesting problem of whether they embody the authors' experience, or whether they are merely literary constructs. Rather than theorizing around this issue, I shall show how phenomenological analysis and studies of living subjectivity can be engaged with the text in order to clarify the contents of introspective experience and the genesis of its religious connotations1. This approach also helps one identify the specific introspective techniques in the canonical and non-canonical literature of early Christian tradition, as well as to suggest some neurological correspondents of religious cognition. To illustrate the problem, here I cite a quotation from Theophanis's poem in the Philokalia, a compilation of Patristic sources written in Greek during the period from the 4th to 15th century. Philokalia was put together by St. Nikodemos of the Holy Mountain (1749-1809) and St. Makarios of Corinth (17311805)2 as a reference manual for Orthodox monastics. The poem entitled "The Ladder of Divine Graces which experience has made known to those inspired by God" begins as follows: 1 For the methods of phenomenological analysis, see Schmicking, "A Toolbox". 2 For more on Philokalia, see McGuckin, "Making"; Ware, "St. Nicodemos". © 2016 Olga Louchakova-Schwartz, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. *Corresponding author: Olga Louchakova-Schwartz, University of California at Davis; Patriarch Athenagoras Orthodox Institute, Graduate Theological Union, Berkeley; Sofia University, Palo Alto, CA, E-mail: [email protected] Cognitive Science of Religion Open Access Unauthenticated Download Date | 2/29/16 5:15 PM 54 O. Louchakova-Schwartz (1) "The first step is that of purest prayer, (2) From this there comes warmth of heart, (3) And then a strange, a holy energy, (4) Then tears wrung from the heart, God-given. (5) Then peace from thoughts of every kind. (6) From this arises purging of the intellect, (7) And next the vision of heavenly mysteries. (8) Unheard of light is born from this ineffably, (9) And thence, beyond all telling, the heart's illumination. (10) Last comes – a step that has no limit (11) Though compassed in a single line – (12) Perfection that is endless. (13) The Ladder's lowest step (14) Prescribes pure prayer alone. (15) But prayer has many forms: (16) My discourse would be long (17) Were I now to speak of them. (18) And, friend, know that always (19) Experience teaches one, not words.3" The poem is valued in the Orthodox tradition for its apologetics of experience4, as is manifest directly in (19), and in (46) "experience alone can teach these things, not talk", and also in (33) "your heart can inwardly experience it". While practitioners of traditional Orthodox introspective prayer assert that the steps of the Ladder are an actual experience5, non-contemplatives view it as a symbol. In the constitution of the text, experience may be masked by historical ideas, amalgamated with the religious symbolism of the time, metaphorized, creatively changed by the author, altered in cultural stimulus diffusion, or simply invented, as a literary exercise6. Since I wish to find out whether the author's references to experience of the Ladder indicate an actual experience, and what kind of experience that would be, I will bracket out the philosophical questions of the reality of experience as such or the questions of relationships between experience and language, and focus on the contents and structure 3 Theophanis, "The Ladder". 4 See the introductory note to the poem by the editors of Philokalia G.E.H. Palmer, P. Sherrard and K. Ware, in Theophanis, "The Ladder", 66. For apologetics of experience in Eastern Orthodox mystical theology, see Lossky, The Mystical Theology, Lourié, "Theophaneia School"; Louth, "Influence", 59: "...[W]e participate in this [Patristic] tradition not just by learning..., but by praying, by living out the theology we discern and proclaim. The Philokalia... initiates us into a participation in the divine life, the divine energies, by... a process of purification, illumination, and perfection". 5 For and example of ascertainments of the poem being a literary description of experience, see Cutsinger, "Yoga". 6 For articulation of mystical experience, see Sells, Mystical Languages; Louchakova-Schwartz, "Approach", 1056. For problems with the generalizability of introspective data, see Nisbett and Wilson, "Telling more". For an opposite point of view in classical phenomenology, see Mohanty "Noema"; Mohanty, "Phenomenology". For the ontological status of experience, see Searle, Intentionality. For the reality of experience, see Molyneux, "The Logic". For epistemological issues in religious experience, see the review by Webb, "Religious Experience". For introspective self-knowledge disputed, see Brueckner and Ebbs, Debating; for self-knowledge vindicated, see Elshof, Introspection. For a problem of experience in cogntitve historiography, see Taves, Religious Experience; for more on difficulties in the study of religious experience, including in the texts, see Flannery et al., "Introduction". Unauthenticated Download Date | 2/29/16 5:15 PM Theophanis and Monoimus the Arab 55 of experience as it is lived by all of us, that is, in the natural attitude7. Such bracketing is routinely done in human science and phenomenological psychology in the study of human experience; neuroscience does the same in the research of cognition. Since we already have an indication that live subjects experience a similar introspective Ladder8, our task here will be to understand the reason for their resonance with the experience allegedly lived by the early antique author while being incorporated into the textual evidence. Clarifying a presence of introspective experience in historical evidence seems to me important for the following reasons: first, introspection can be of different kinds9. Even though introspection is "the privileged point of departure" for the formation of a religious sense10, no definitive connection has yet been established between the particular kind of introspective experience and a particular religious idea11. With diachronic changes in the sociology of experience, experience is reinterpreted, and some forms of it become obscure12. Therefore, the study of experience in the texts of antiquity may help to rescue such obscure or even commonly unknown forms of experiential consciousness13. Second, the naturalization of religion has explained rituals, beliefs, magic and morality in terms of social cognition and by application of evolutionary biology14. Experience per se remains outside of the scope of these studies. Even though Slingerland points to the importance of the study of experience for cognitive historiography15, the theories of social cognition or evolutionary biology provide no means for the study of experience. At the same time, both data-driven and theory-laden approaches in 7 For more on the natural attitude, see Luft, Subjectivity; in science, see Gurwitsch, Phenomenology. For the study of human experience in phenomenological psychology and human science, see Frie, Understanding; Giorgi, Psychology. For the natural attitude in religious experiencing, see Louchakova-Schwartz, "The Seal"; for an example of an early mystical text in which the problem of experience is treated specifically as a philosophical problem, and therefore, the author departs from the natural attitude, see Louchakova-Schwartz, "Phenomenological Approach". For more on the "innate" character of religiosity, see Ales Bello, The Divine; Barrett, Born Believers. 8 Flood, The Truth. 9 For the introspective origins of religious experience, see Alston, Perceiving; Chittick, The Self-Disclosure of God; Dadosky, The Structure; Ḥāʾirī Yazdī, Principles; Crowe, Theology, 124-143; Louchakova-Schwartz, "Intuition of Life", 2011; Louchakova-Schwartz, "Direct Intuition"; Louchakova-Schwartz, "Di(a)Logos"; Louchakova-Schwartz, "The Seal". For a history of introspection, see Lyons, Disappearance. For introspection as an exegetical category for a number of different processes, see Stern-Gillet, "Consciousness", 145. For scientific or philosophical research of introspection, see Alderson-Day and Fernyhough, "Relations"; Allo, "Many Faces"; Zedelius, Broadway, and Schooler, "Motivating Meta-Awareness". For discussions on introspection, and its various types, see Brueckner and Ebbs, Debating; Crisp, Oxford Handbook; Schwitzgebel, "Introspection?"; Paul, "How we know"; Feest, "Introspection"; Zahavi, "Varieties". For spatiality of introspection, see Mehta, "Beyond Transparency", 2. For introspection in antiquity, see Diuk et al., "Quantitative Philology"; Stramara, Gregory; Stramara, Introspection. 10 Ales Bello, "Husserlian Approach", 65. 11 Crowe, Theology, 124-143. 12 For an example of a reinterpretation of the forms of experience, see Lukoff, "Spiritual Emergency". For the forms of experience "invisible" in the zeitgeist, see Greenwell, "Energies". 13 Cf. the analysis of the uncommon levels of the cognition evidently participating in the formation of religious experience according to Indian and Tibetan texts, in Bronkhorst, "Levels of Cognition". 14 For the biological origin of religious thought, see Atran, Gods; Boyer, Cognitive Aspects; Boyer, Religion; Bloom, Babies; Cohen, Mundry, and Kirschner, "Religion"; Heintz, "Cognitive History"; Parker, "Commentary"; Whitehouse, Modes; Whitehouse and McCauley, Mind; Whitehouse and Laidlaw, Religion; Xygalatas, Burning Saints. For social cognition as the substratum of religion, see McCorkle and Xygalatas, Mental Culture. For the naturalization of religion, see Sperber, Coubray, and Schmitt, "Entretien". 15 For the importance of the study of experience in cognitive historiography, see Slingerland, "Towards a Second Wave"; for an example of early studies of introspective religious experience in the natural attitude, in which experience is taken for granted, without philosophical reflection or historical analysis, see Wilber, Engler and Brown, Transformations. Unauthenticated Download Date | 2/29/16 5:15 PM 56 O. Louchakova-Schwartz cognitive science use a heuristic decomposition of normative cognition for their hypotheses building16. Whether or not the same can be performed on introspective or religious experience, that is, whether the latter has a structure which can be decomposed, has not been researched and remains unclear17. To find out whether there is a structure of experience in Theophanis' poem, I shall implement a sequential research design with steps hermeneutically building on one another18. I shall first examine whether the overarching metaphor of the Ladder19 may possibly refer to the phenomenological structure of experience20. This should furnish a direction of analysis, as reflected in the section entitled Research Design. In the first stage of analysis, I shall treat the text as a compound of ideas, subject to cultural stimulus diffusion (see below the section entitled Poetic Metaphor and the Structure of Experience)21, in order to extract the phenomenological structure of experience out of its historical matrix and separate it from the number symbolism clearly present in the poem (see the section entitled Ideas in Theophanis' Poem). Then, I shall search for a similar experience in the oral (non-textual) tradition of introspective prayer associated with Philokalia, and examine whether the experience in question has diachronic and synchronic stability (see the section entitled An Anthropological-Phenomenological Bridge to Experience: "Purest Prayer"). To overcome the constraint of self-referencing I will use secondperson reporting22. In order to fully understand the ways this experience is incorporated in the tradition, I shall examine a fragment from another early writer, Monoimus the Arab, whose doctrine was recorded, among other heresies, by Hippolytus of Rome23. Finally, I shall show how this kind of introspection generates the ideas of the Gnostic kenosis (emptying oneself) and the Christian theosis (union with God). In the Conclusion, I shall discuss the relevance of the findings to cognitive science. 16 For data-driven approaches, see the recent literature on neural networks analysis: Pedone et al., "Efficacy"; Silva et al., "Real-Time Ultrawide Field Image Evaluation". For an example of a theory-laden approach, see Keil, "Committee Report", 2: "The predictions should directly relate to the theories and previous findings". For the heuristic decomposition of experience in the creation of a cognitive experiment, see Kosslyn, "Imagery"; Kosslyn et al., "Imagery Used"; Koshevnikov et al., "The Enhancement"; Krans, de Bree, and Moulds, "Cognitions", Kreplin and Fairclough, "Effects"; Louchakova-Schwartz, "Cognitive Phenomenology". For heuristics in the study of cognition, see Gonzalez and Lebier, "Cognitive Architectures". For normative cognition, see Jensen, "Normative Cognition". For secularization of cognition, see Swatos and Olson, Secularization Debate, 37. For the creation of the term "religious experience" in the nineteenth century, see Sharf, "Rhetoric". The exclusion of religious experience from dominant discourse must be contrasted with earlier centrality of religious experience, e.g. in "Is God what the mind first knows?" in St. Thomas Aquinas, Faith, Sent. 1, d.3, q1, a 2; De veritate 10.12; Contra gentiles 1.10-11; Summa theol. 1.2.1, 1.88.3, a. 17 For the variability of introspective experience, see Nisbett and Wilson, "Telling more"; Lutz et al., "Guiding". Petitmengin et al. in "Gap" suggests overcoming variability by using second person research approaches, such as interviewing a large number of subjects. This approach was utilized in the present paper. For a critique of isomorphism in designing a scientific experiment, see Gallagher, "Phenomenology and Experimental Design". For a review of classic phenomenological writings on the structures of religious experience, see Dadosky, The Structure. 18 For mixed methods design in social sciences, see Creswell and Plano, Designing. The hermeneutics of sequential mixed methods design are applied here to historical research in order to integrate the data from a variety of data streams in critical historical analysis and phenomenological research of living participants. 19 As an example, the folk-psychological Islamic understanding of the Mi'rāj journey as a real event should be compared with Hämeen-Anttila's "Descent" for an analysis of the same myth before any correlation to experience can be claimed. 20 For an analysis of the somatic aspects of mystical perception, see Louchakova and Warner, "Via Kundalini"; Louchakova, "Ontopoiesis and Union"; Louchakova-Schwartz, "The Seal"; Louchakova-Schwartz, "Phenomenological Approach". For the loss of the embodied component in translation, listen to Louchakova, O. "Mediating intimacy: Essential Ibn 'Arabi for education and psychotherapy". 21 Kroeber, "Stimulus". 22 For rigor in studying human experience, see Colaizzi, Reflection; Frie, Understanding; Giorgi, Descriptive Phenomenological Method. 23 St. Hippolytus, Refutatio. Unauthenticated Download Date | 2/29/16 5:15 PM Theophanis and Monoimus the Arab 57 Research Design How do we extract the structure of experience from under the layer of ideas in the poem? In following Table 1 below, I indicate some aspects of the problem. Table 1. Analytic problems in the extraction of experience from textual evidence 24252627 Problem Example Solution (1) The experience is masked by ideas or literary conventions. The poem is viewed as a purely symbolic Ladder which has no correspondence to any experience. Traditional critical historical and philological analysis of ideas in the text helps separate cultural construction from the core structure of possible experience. In the case of the Ladder, the core structure will be not the number of steps, but the fact of stratification (see below the section entitled Ideas in Theophanis' Poem). (2) The experience does not exist. The poem is a mere exercise in a literary genre. Phenomenological correlation of the text with reports of live subjects in first (autoethnographic)25 and second person (interviewing) approaches.26 Positive only in the presence of a generalizable phenomenological structure.27 (See the section titled An AnthropologicalPhenomenological Bridge to Experience: "Purest Prayer"). (3) The experience is unrecognizable.28 There is no experience which has the neat stratification of the ten steps (number symbolism). Same as above. (4) Evidence of experience in the text is skimpy or confusing. For example, the fragment from Monoimus by itself is insufficient for claims to the experiential character of evidence. Comparative analysis with other texts (see below the section entitled Attentional Strategies of Theophanis the Monk and Monoimus the Arab). After the solutions in Table 1 are implemented, the next step of analysis must focus on the references to the "heart" (lines 4, 33, 57 of the poem) that bridge the ideas in the poem with embodiment28; this was addressed via anthropological data (see below the section entitled An Anthropological-Phenomenological Bridge to Experience: "Purest Prayer"). This included the analysis of the structure of practice in the oral tradition, as well as finding practitioners, screening them to select those whose experience resonated with the poem, interviewing the latter and performing the phenomenological analysis of their experience29. Finally, the structures were verified in a reproduction of the experience in a guided introspective practice (a so-called quasi-experiment). In the Figure 1 I summarize the stages of design and the tasks pertaining to each stage. 24 For the reproducibility of phenomenological structures of experience, see the method of imaginal variations in Giorgi, Descriptive Phenomenological Method. 25 For references to first-person experience in research, see Desbordes and Negi, "New Era"; Díaz, "Narrative Method" 26 For situated and generalizable structures, see Giorgi, Descriptive Phenomenological Method. 27 In addition, according to the lament of St. Hesychios already in the fifth century C.E., the experience of purity was already rare; nowadays, it may appear to be extinguished. 28 For the experiential aspects of the concept of the "heart", see Louchakova, "Spiritual Heart"; Morris, Reflective Heart. 29 Xygalatas ("On the Way") mentions that historians attempting to understand experience might benefit from experimental work in three ways: first, by employing "existing, experimental evidence from living subjects to make inferences about past people"; second, by using historical data to design experiments that test cognitive historiographical hypotheses; and finally, through "natural experiments" in which one applies systematic quantification and statistical analysis to historical material. Unauthenticated Download Date | 2/29/16 5:15 PM 58 O. Louchakova-Schwartz !Figure 1. Stages of analysis of Theophanis' poem Participants The interviews for the analysis of phenomenological data were collected between 1994 and 2011 from selfdirected practitioners of the Prayer of the Heart (N = 273) and from those who received guidance from a spiritual director (N = 175). The phenomenological structure discovered through the interviews was verified in guided practice of the prayer in the focus groups (N = 500)30. About 75% of all participants were female, the rest were male. Participants were solicited from various monastic communities and churches, as well as California centers such as the Esalen Institute31 and the Mercy Center, and among the students of John F. Kennedy University, the California Institute of Integral Studies, the Institute of Transpersonal Psychology, the Starr King School for the Ministry, and the Patriarch Athenagoras Orthodox Institute in Berkeley. The data were compared with reports of regular introspection in non-praying subjects. Note on Phenomenological Method Phenomenological data were considered together with the data of historical analysis and textual research32. The study used a formalized genetic analytic phenomenological approach developed in psychology and phenomenology of religion to overcome the interpretive errors in the research of experience33. Poetic Metaphor and the Structure of Experience The poem and its overarching metaphor of the Ladder must be understood in the context of experienceoriented Orthodox mystical theology and the corresponding practice of the Prayer of the Heart (PH)34, both 30 For a similar approach, see Price and Barrell, "Inner Experience", 16-19. For an adaptation of the method of the Prayer of the Heart for New Age audiences, see Louchakova, "Essence". For more on the growth of popularity of the Philokalia, see Ware, "St. Nicodemos". 31 A New Age version of the PH in non-traditional communities is described in Louchakova, "Essence". 32 Smart, "Foreword", xi. 33 For examples of strict analytic approaches to experience, see Camic, Rhodes and Yardley, Qualitative Research, 2003; Colaizzi, Reflection, 1973; Embree, Reflective Analysis, 2006; Giorgi, Descriptive Phenomenological Method, 2009; Frie, Understanding, 2003; Denzin and Lincoln, Handbook. For the classic phenomenological method, see Steinbock, "Generativity"; Steinbock, Home. For the phenomenological analysis of religious experience, see Steinbock, Phenomenology. 34 For the experience-oriented theology of Philokalia, see William, "The Theological World". For the connections between Philokalia and oral tradition, see Zecher, "Tradition". For the direct experience of God in the PH, see Liester, "Hesychasm"; Toti, "Anthropological Significance"; Louchakova, "Ontopoiesis and Union"; Louchakova, "Prayer of the Heart"; Louchakova, "Spiritual Heart". On the anthropological significance of the Prayer of the Heart, see Toti, "Anthropological Significance". Unauthenticated Download Date | 2/29/16 5:15 PM Theophanis and Monoimus the Arab 59 of which emphasize the importance of the direct internal experience of God. The Ladder is the means for the ascension to "heaven's vault" (21) in ten steps which "strangely vivify the soul" (22) and are the "fruit of all the books" (29). The seventy one lines of the poem circle around the importance of this experience in preparation for dying: the Ladder is the path to heaven (30) and finding "life" (25), i.e., immortality, while still in this world (26), in which one's "...heart can inwardly experience it [the Ladder] " (33). Theophanis calls on the reader to identify what step he/she is on (40). From line 49 on, the antithesis is presented: "He who has no foothold on this Ladder" (50), will experience "terrible fear, terrible dread" (53) at the time of dying. This is completed with the traditional coda of humility, in which Theophanis condemns himself and states his "utter fruitlessness" (71). On the cusp of the Pre-Christian and Common Era various symbolic ladders were posited all over the Mediterranean35. Ladders and ascents are extremely polymorphic36, including the difference in the amount of steps, from three to twelve, to many more. Origen, who lived ca.184/185 – 253/254, indicated that according to Celsus' account of Mithraic initiation, the soul travels the celestial regions of the planets, symbolically depicted as a ladder with seven gates and an eighth gate on top37. St. Jerome, who lived ca. 347-420 C.E., attests a broad use of the metaphor of the ladder in the Christian milieu: "Since many people have surmised that angels travel this ladder daily as they go about the Lord's business, it has become a symbol of the comings and goings between heaven and earth of people, angels, and messages or prayers"38. St. Jerome probably had in mind the Jacob's Biblical dream of the Ladder in the Old Testament39. Jacob's Ladder probably served as a prototype for various later imageries of the Ladder because, as explained by the 15th century St. John of the Cross, the symbolism of the Ladder is especially appropriate for the "secret wisdom" of internal spiritual formation via "secret contemplation"40. In cases when the Ladders are not only symbolic, but metaphoric of some internal process, the underlying processes are also polymorphic. For example, the Ladder of St. John Klimakos designates the internal transformation which is not mentioned in context of the Ladders in the earlier Apophthegmata41. In turn, the psychological character of Klimakos's Ladder is different from the one in Theophanis' Ladder. Klimakos' Ladder is, so to speak, clinical, while the separation of modalities of experience in Theophanis' Ladder could have been a part of the decomposition of experience in cognitivist research. However, modalities of experience are not arranged in such an orderly mannera. If the prayer (line 1) in question is indeed the PH42 , the latter is known to produce messy and irregular experiences of thought and imagination with bursts of feeling and seemingly random instances of unitive consciousness, which do not accord with the neat geometrical beauty of the poem. The poem stands out from the rest of Philokalia in its nearly abstract quality, which suggests that the poem is either a mere literary composition, or embodies an uncommon form of experience which we do not know about. 35 For the diffusion of Ladders in the Mediterranean in Greek, Egyptian and Mithraic sources, see Ogawa, "Mithraic Ladder Symbols"; Foley, "Order Question". For the ladder as a physical object in rituals in the Hekhalot texts of late antiquity see Van Uchelen, "Ethical Terminology"; for more abstract ladders of moral virtues, i.e. purification, the ladder of beauty, i.e. Eros, or the dialectical ladder of wisdom, see the three Platonic ladders mentioned by Socrates, in Dorter, "Three Disappearing Ladders", 282. 36 For polymorphism of the ladders, see Kuntz and Kuntz, Jacob's Ladder; Idel, Ascensions. 37 Ulansey, The Origins, 18; Ogawa, "Mithraic Ladder Symbols". 38 As mentioned in Hardy, "Saint Gregory". 39 For Jacob's dream, see Genesis xxviii, 12. 40 St. John of the Cross, Complete Works. Volume 1, Dark Night, 432-434. 41 For Klimakos' Ladder, see St. John Climacus, Ladder; for comments see Johnsén, Reading John Climacus. 42 Under Prayer of the Heart (PH) we need to understand a complex of methods such as "sobriety" (mindfulness), an internal repetition of the words of prayer, and a cultivation of internal stillness. The inner prayer consists of a devotional repetition of the name of Jesus or other suitable names of divinity, or of a full formula of the prayer "Lord Jesus Christ, the Son of God alive, have mercy on me, the (a) sinner". PH is accompanied by somatic focusing in the chest, with mental motion inward, to the center of the inner space of the chest. The latter area is known in Hesychasm as the Spiritual Heart, see Špidlík, Prayer; Špidlík, Spirituality, 104-105. Unauthenticated Download Date | 2/29/16 5:15 PM 60 O. Louchakova-Schwartz The overarching "root metaphor" of the Ladder works in favor of this last proposition. In general, the metaphor indicates an ontic structure in the substratum of reference43. The metaphor of the ladder was omitted in Pepper's analysis of the epistemological function of metaphors, because of its being open-ended, theistic and not supported by scientific data44. However, the ladder metaphor works well for psychological processes which have a stage-like development. The metaphor enables the reader to understand and predict a psychological process not available in immediate observation45. In the structural organization of this metaphor, the processes signified by the Ladder are predicated on four main characteristics: the presence of a discrete phase or quantum conditions which are linear (homologous) within themselves; a step-like shift to the next condition; a hierarchical ordering of steps; and a teleological progression. In Patristic literature we can expect a close correspondence between the symbolism, borrowed or inherited, in the literary form, and the author's subjective contemplative experience46. Whether the characteristics of the metaphor pertain to the experience, or to the matrix of historical ideas, should become clear in the process of analysis47. Ideas in Theophanis' Poem Structural thematic analysis of the poem shows the diversity of sources which have been influential in the composition of the poem (Table 2). Table 2. Structural Thematic Analysis of Ideas in Theophanis' Poem4849 Theme Complex Source Stratification. Stratification and the travels of the soul are closely connected47. Stratification, travels of the soul and preparation for dying are themes belonging to various Ladders. Soul travels in steps already in the early material described Bousset, Die Himmelresie der Seele, later, in the Mi'rāj journey48; cf. The Nag Hammadi Apocalypse of Paul (see further below). Travels of the soul. Preparation for dying. The idea of preparation for dying can be traced back to Mithraic Ladder, Jewish, later Christian Ladders and St. John Klimakos. Mystiques of the number ten. Likely of Pythagorean origin. Central to an early influential Kabbalistic text Sephir Yetzirah. Strongly expressed by the Neoplatonizing Gnostic Monoimus the Arab; also, by Celsus (in Origen's Contra Celsum) with regard to the Ophite diagram. Ten heavens are mentioned in the Nag Hammadi Apocalypse of Paul. Perfection compassed into a single line. The same idea in Monoimus the Arab. Reference to the Heart. Connects to embodiment and requires research in oral tradition. Additional consideration: literary style of the time. Imitates the laconic style of St. Paul. 43 For root metaphors see Pepper, World Hypotheses; Pepper, "Root Metaphor". 44 For example, in St. Bonaventura's symbolism of the universe as a Ladder with God being on top or beyond the last step, there is no possibility of establishing any structural connection between the metaphor and some kind of realistic substratum. In Kuntz and Kuntz, Jacob's Ladder, 113. 45 For metaphor in developmental psychology see Super and Harkness, "Metaphor". 46 Golitzin, "Theophaneia"; Lourié, "Theophaneia School". 47 For perspectives concerning the relationship between the structures of experience and the cultural matrix, see Katz, Comparative Mysticism; Forman, Mysticism. 48 Collins and Fishbane, Death; Idel, Ascensions. 49 Hämeen-Anttila, "Descent". Unauthenticated Download Date | 2/29/16 5:15 PM Theophanis and Monoimus the Arab 61 Some of the ideas intrinsic to Theophanis' metaphor of the Ladder, such as the mystique of the number ten, or the final "perfection which is endless", appear to come from sources not associated with the Ladder or travels of the soul. Below, in the section entitled The Ladder and Related Ideas, I present a detailed analysis of each section of the compound. The much criticized historical critical analysis is absolutely necessary here because it helps to retrieve the reflective layer of constitution and uncover the pre-reflective experience with a stable structure. Without this analysis, it would also be impossible to understand how the poet conceptualizes introspective experience, and make inferences about how the same experience would contribute to the formation of ideas in other traditions (Tables 3 and 4, see p. 68 and 69). The Ladder and Related Ideas The morphology of the poem identifies it as a subtype of otherworldly journeys of the ascent and/or descent of the soul into the otherworld50; the Ladder-like structure for those journeys is frequent but not obligatory. The Ladders serve in Christian and Jewish literature as a symbol of a structured and often narrow and restricted passage. In Theophanis' poem, the Ladder is narrow in a sense that it suggests the presence of a very specific experience (see below the section entitled "Purest Prayer" and Asymmetry in Introspective Experience). The specifics of experience becomes clear in comparison with Klimakos' famous Ladder. While Klimakos enumerates thirty steps equal to the thirty years of the hidden life of Christ, Theophanis stresses ten steps without any reference to Christ. The difference between the mental contents of the steps was already indicated in the section above entitled Poetic Metaphor and the Structure of Experience. The thirty steps of Klimakos' Ladder include other structures of numerical symbolism related to Trinitarian theology and the developmental stages of psychospiritual monastic life, as well as elements of literary concentric parallelism51; the steps in Theophanis' Ladder do not have numeric substructures; the only number symbolism is that of the decad. Klimakos' writing includes imaginal aspects, such as his treatment of monastic life emulating the life of angels52. Theophanis' laconic and concrete writing follows the literary style of the Apostle Paul. Paul's style was turned into a literary canon by St. Gregory the Theologian (fourth century C.E.), who said: "to write laconically is not to write a few syllables, but to say much in few words... I measure a poem by its contents, and not by the number of letters"53. This style features clarity and plainness, hidden (implicit) character of depth and height, laconicism, ecclesiastical beauty and liveliness of the divine word. This form also implies humility54; cf. Theophanis' claim to "utter fruitlessness" in line 71. The closeness of Theophanis' themes to those of St. Gregory is also visible in the repetitive emphasis on the need of first-person experience (of the Deity) for knowledge to occur, such as in (19), (46), (33), in resonance with a famous quotation from Gregory, "Practice is the way to knowledge"55. Theophanis' poem demonstrates connection to early Christian sources. Decadology and Unification Theophanis says: (21) "Ten steps that strangely vivify the soul. (22) Ten steps that herald the soul's life... (27) Ten steps: a wisdom born of God. (28) Ten steps: fruit of all books. (29) Ten steps that point towards perfection. 50 Culianu, Psychanodia, 10-11. 51 For concentric parallelism, see Lawrence, "Three-Fold Structure", and most comprehensively, Douglas, Thinking in Circles. 52 For more on "angelic life" in Klimakos, see Zecher, Angelic Life. 53 Gregory the Theologian, Epistle 3, 769, as mentioned in Ullmann and Cox, Gregory, 287, footnote 1. 54 Bezarashvili, "Michael Psellos". 55 Gregory the New Theologian, as mentioned in Ullmann and Cox, Gregory, v. Unauthenticated Download Date | 2/29/16 5:15 PM 62 O. Louchakova-Schwartz (30) Ten steps that lead one up to heaven. (31) Ten steps through which man knows God... (36) This ten-graced Ladder is the best of masters. The number ten figures in the pre-Platonic cosmology as one of the perfect numbers operating in the cosmic harmony of rational proportions56. In the numeric symbolism of antiquity, the numbers often become synonymous with ideas,57 whereby ten indicates the unification of multiplicity in the central monad58. Unlike Theophanis' Ladder, in which the steps are clearly identified and numbered, in ancient Greek thought the stages of the soul's return to unity are hard to count and define sequentially59; ten appears as a cosmological, not intra-psychic, principle. A closer connection between numbers and psychological states appears e.g. in Mithraism, which had absorbed Greek ideas merged60; in this manner, the Mithraic journey of the soul through the seven planets (with the eighth step on top) possibly inspired a similar idea in non-orthodox Christian groups influenced by Greek thought, whereby a journey through the seven planets might have turned into a journey through ten spheres with the goal of the soul reaching God while still alive, translatio ad deos61. Such merging might have taken place in a classic account of the Gnostic initiatory soul's ascent in the Ophite diagram62. Similarly, a ten-stage ascent is found in the Nag Hammadi Apocalypse of Paul63: it sends Paul traveling to the tenth heaven64, which is, notably, not a living place of the righteous but The Throne of God (i.e., a place of monadic unification with phenomenological connotations similar to the "endless perfection" in Theophanis' poem). The count to ten often appears in Gnostic calculations as an event of unification, that is, an emergence of the monad in a series of preceding manifestation of other numbers, which, while either in natural sequence or added to one another, generate the number ten65. Ten is signified by the Greek letter iota, which is also tenth letter of the Byzantine Greek alphabet; iota is also the first letter of the name Jesus. This appeals to the imagination: iota-ten becomes a symbol of Jesus, who is the Alpha and Omega, i.e. the beginning and the end, of all things, i.e. a metaphysical principle of unification66. Hippolytus' extract from Monoimus the Arab is another non-canonical example of cosmology built around the number ten signified by iota. The iota's single line symbolizes a single stroke by which creation is made. Monoimus associates many Biblical decadological ideas with this single line, which is viewed by him both as an axis of creation and an axis of unification67. Using the language of the Gospels, e.g. "for the whole Pleroma was pleased to reside in the Son of Man in a bodily form"68, Monoimus avoids mentioning 56 For an example of numbers operating in cosmic harmony, see Posidonius, a Stoic philosopher with Platonizing tendencies, as mentioned in Burkert, Lore, 54-6. For perfect numbers in Pythagoreanism, see Kahn, Pythagoras, 25; Hopper 1938, 11; for the same in Gnosticism, see Kalvesmaki, Theology, 53-54. Valentinus' disciple Ptolemy (late second century) shows special appreciation for the number ten. Delatte (apud Culianu, Psychanodia, 27) assigns the celestial eschatology of to an uninterrupted Pythagorean underground tradition (see Platner, Topographical Dictionary). According to Huffman, the cosmos of Pythagoreans (ca. fourth century B.C.E) included ten orbits belonging to ten divine celestial bodies (Huffman, "Philolaus"). 57 Kahn, Pythagoras, 81. 58 For the number ten as a symbol of unification in Pythagoreanism, see Hopper, Medieval Number Symbolism, 44-45. For the number ten in early Christian writers, see ibid., 69-70. 59 For the stages of the soul in Neoplatonism, see Bennett, Syzygy; Dodds, Proclus, 313ff. 60 Culianu, Psychanodia. 61 Bianchi and Vermaseren, La soteriologia. The concept of theosis, which is the soul's union with God, in Eastern Orthodox theology is an experiential condition: Kharlamov, "Introduction"; Lossky, The Mystical Theology; Pelikan, Christian Tradition; Popov, "Idea", 74-75. 62 The Ophite diagram is found in Origen, Contra Celsum, vi, 24-38. For the research of ritual in Ophite diagram, see Welburn, "Reconstructing the Ophite Diagram"; DeConick, "The Road". 63 MacRae, Murdock and Parrot, "The Apocalypse of Paul". 64 Tabor, Things, 118-119. 65 In the Valentinian Gospel of Truth, a multiplicity of decads resolves into unity. See Kalvesmaki, Theology, 56-57. 66 The Greek iota is written as a vertical stroke, cf. lines 10-12 of Theophanis' poem. The Revelation to Marcus (in Irenaeus of Lyon) symbolizes Jesus as iota, i.e., unification. 67 Foerster and Wilson, Gnosis, 249. 68 Col 2:9; same in Matthew 5.18; also Luke 16.17. Unauthenticated Download Date | 2/29/16 5:15 PM Theophanis and Monoimus the Arab 63 Jesus directly but the symbolic language he uses points to the Jesus of the Gospels. Analogously, the lines 10 – 12 of Theophanis' poem state: (10) "Last comes –– a step that has no limit (11) Though compassed in a single line –– (12) Perfection that is endless. Jesus emerges in the poem as the tenth step of the Ladder, symbolized by single line in which all manifestations unite in the limitless monadic perfection. This is a common theme of both fragments, expressed by similar language. Theophanis might have been familiar with Hippolytus; he might have also encountered Monoimus' ideas via reading Clement of Alexandria, the first in the line of famous Alexandrian theologians, who argued extensively against Monoimus. For Clement, Christianity was gnosis, i.e. true knowledge, in the full and absolute meaning of the world [as distinct from Gnosticism used as an umbrella name for various syncretic cults which flourished on the territory of the border between the Greco-Roman world and the East between the last century B.C.E. and during the first three centuries C.E.]. Relevantly for our discussion, both Monoimus and Clement of Alexandria emphasized direct, immediate i.e. introspective, self-knowledge and knowledge of God. Clement says: "if the Gnostics were offered a choice between the salvation of the soul and the knowledge of God, supposing that these two things were distinct (although they are identical), he would chose knowledge of God"69. Shmeman, one of the leading traditional Eastern Orthodox Christian historians, further says: "Gnosis is the vision of God face to face, the mystical illumination of His truth; the Christian prefers this knowledge of God to all else and sees the purpose of his whole life in it"70. Monoimus' fragment ends with a passionate call to seek God by means of introspective self-examination: "Omitting to seek after God, and creation, and things similar to these, seek for Him from thyself, and learn who it is that absolutely appropriates all things in thee, and says, "My God, my mind, my understanding, my soul, my body."71 And learn from whence are sorrow, and joy, and love, and hatred, and involuntary wakefulness, and involuntary drowsiness, and involuntary anger, and involuntary affection; and if you accurately investigate these, you will discover Himself, unity and plurality, in thyself, according to that tittle72, and that He finds the outlet to be from thyself"73; or, in another translation: "you will find yourself within yourself, being both one and many like that stroke, and will find the outcome of yourself"74. This call to experience resonates with Theophanis' fragment. When we consider all the above-mentioned sources together, there emerges not only a vision of a religious tradition gnoseologically rooted in introspection, but of an oral tradition with a specific form of introspective practice which possibly flourished in GrecoChristian and Gnostic thought long before the habit of writing down systematic instructions to inner practice was shaped in the written tradition. An Anthropological-Phenomenological Bridge to Experience: "Purest Prayer" Two themes, the theme of "experience teaches one, not words" (line 19, also repeated in line 33 and 46) and the theme of the heart in "...your heart can inwardly experience it [the Ladder]" (line 33) are typical for the actual contemplative practice of the PH. Instructions to prayer are have been passed traditionally 69 Clement of Alexandria, as mentioned in Shmeman, Historical Road, 51. 70 Shmeman, Historical Road, 51. 71 The five elements in Monoimus's self-analysis suggest a hierarchical order of categories which is interpreted by Kalvesmaki in Theology, 92, as a connection with the anthropology of the number five and the Pentateuch. Even though the general character of the text here is clearly self-referencing, there is not enough material to judge whether this ordering bears a connection with introspective contemplation, or whether it is purely analytic. 72 For the mysticism of serif in iota, see Kalvesmaki, Theology, 88. 73 Salmon in http://encyclopedia.thefreedictionary.com/Monoimos. I am quoting it exactly, with vacillation in the second person pronoun. 74 Monoimus in Foerster and Wilson, Gnosis, 248-250. Unauthenticated Download Date | 2/29/16 5:15 PM 64 O. Louchakova-Schwartz from the teacher to disciple, but most of the contemporary practitioners begin the practice on their own. All respondents in the in vivo trials resonated with the poem, but not all of them displayed the exact ladder-like phenomenological structure of experience. Below, I describe the conditions under which, according to the accounts of the practitioners of the PH, the ladder-like experience takes place. The temporal extension of Theophanis' experience is contained within the two delimiters, the "purest prayer (line 1)75 from which there comes the warmth of heart" (line 2), and the absorption in "perfection which is endless" (line 12). "Purest prayer" must be differentiated from the Prayer of the Heart in general. For example, the tenth century fragment "Three Methods of Attention and Prayer"76 and the fifth century treatise "On Watchfulness and Holiness"77 in Philokalia also refer to introspective experience in the PH, but these descriptions differ from what is described by Theophanis. Experience in the practice of the PH generally develops from vocal prayer, to mental prayer, the descent of focus into the chest, the switch from the verbal prayer to stillness and cultivation of presence resolving into "glorious nothingness"78, without the stratification noted by Theophanis. "Purest prayer" and Asymmetry in Introspective Experience A traditional authority in introspective prayer, St. Isaac the Syrian, indicates that internal prayer progresses from invocation to abidance in a spiritual state that is beyond the activities of the mind79. For this state, he uses the term "pure prayer", by which he means a state in which usual mental speech is suspended, the sense of free will ceases to be, and only "a certain divine vision remains and the mind does not pray a prayer. ... Intellect enters into spiritual movements..."80. The prayer then happens on its own accord, without a sense of individual agency, in a state which Ware describes as "the continuous action of Another in me.... no longer a prayer to Jesus but the prayer of Jesus himself... 'self-acting' prayer"81. This state is followed by various internal manifestations, in which "we experience and feel the activity of the Spirit directly and immediately"82. Two crucial factors in attaining this state are a correct somatic attentional focus83 which causes the inward absorption of attention, and self-referencing words of the PH; without absorption or self-referencing the process doesn't deepen and the pure prayer doesn't emerge. The condition of "purest prayer' is what causes the ordering of experience; unless pure prayer is reached, the mind wanders, the current of internal impressions will be chaotic, and the ladder-like organization of internal experience doesn't show up. The accounts of the practitioners show that self-referencing in the prayer directs attention to the body, i.e. brings out somatic self-awareness84. The egological awareness of the body-self associated with the chest is a necessary condition for the emergence of the ladder-like experience (Fig. 2, C). 75 I thank Metropolitan Nikitas (Lulias) for pointing to the fact that Theophanis refers not simply to the Prayer of the Heart, but to the "purest prayer", and also for the insight that the poem may have a secret code. In this case, the iota indicating Jesus. 76 This treatise is ascribed to St. Simeon the New Theologian. 77 St. Hesychios the Priest, "On Watchfulness". 78 For the developmental stages of prayer see Chirban, "Developmental Stages"; Dionysios cited by Chirban, "Developmental Stages", 307. 79 St. Isaac, The Ascetical Homilies, 115-124. 80 Ibid., 116. For an example of this switch of agency, see a remarkable passage in Ware, Power, 2, in which he summarizes the quotations from the New Testament about "God within" taking over the prayer. 81 Ware, "Introduction", 19. 82 Ibid., 3. 83 For more on correct focusing, see St. Simeon the New Theologian, "Three Methods", 70-71; Ware, Power, 18. 84 Benedetti et al., "Mind Operational Semantics", 2010; Hampe and Grady, "From Perception to Meaning"; Louwerse and Jeuniaux, "The Linguistic and Embodied Nature"; Talmy, Toward a Cognitive Semantics. Volume 1, Concept, and Volume 2, Typology. Unauthenticated Download Date | 2/29/16 5:15 PM Theophanis and Monoimus the Arab 65 ! Figure 2. Configurations of somatic self-awareness. The attentional strategies of practicing focus attention in the interiority of the chest85; this intentional focus becomes automatic after the PH shifts to pure prayer. If in the beginning of practicing the PH, accounts of practitioners show self-awareness on the spectrum identified by Zahavi as minimal and maximal self-awareness86, by the time of the "pure prayer" self-awareness is further intensified, is somatic, and is condensed. Among the three kinds of configurations of self-awareness which we observed in the participants and termed non-egological or distributed (Fig. 2, A), egological distributed (Fig. 2, B), or egological condensed (Fig. 2, C), the ladder is associated only with the condition in which self-awareness is condensed in the chest (Fig 2, C). The shift to "pure prayer" also involves an emerging sense of presence. Multiple phenomenological investigations have demonstrated that "presence" is de facto a refined felt sense of embodiment, a hyletic (sensory) impression related to the materiality (tactility or density, or in cognitive terms, a haptic component of the body-sense) and co-constituting the sense of the sacred87. These hyletic impressions in PH shape the matrix of gradually decreasing density which creates the continuity uniting the steps of the Ladder (Fig 3.) ! Figure 3. A gradual "thinning out" of hyletic sense-data in the process of absorption in the PH. As a sense-datum in the somatic sense of the self, it serves as an organizing matrix for the Ladder. 85 For an example of attentional strategy focusing attention in the chest, see St. Simeon the New Theologian, "Three Methods", 72: "Rest your beard on your chest...and search inside yourself with your intellect so as to find the place of the heart, where all the powers of the soul reside". 86 According to Zahavi, Subjectivity, minimal self-awareness is a natural self-awareness in pre-reflective experience, and maximal self-awareness is the condition in which self is brought into awareness reflectively; the intensity of self-awareness in two conditions differs. For more on the distribution of the center of self-awareness in the body, see Spiegelberg, "On the Motility". 87 For more on presence, see Henry, Material Phenomenology; Louchakova, "Ontopoiesis and Union"; Louchakova, "Reconstitution". For embodiment of the sense of the sacred, see Ales Bello, "Husserlian Approach"; Ales Bello & Calcagno, "Sense". Unauthenticated Download Date | 2/29/16 5:15 PM 66 O. Louchakova-Schwartz The accounts of participants show a major asymmetry in the process between the left and the right side of the body88. On the left side, experience sustains an ordinary character of absorption, with some mental imagery and diffusion of the sense of self-awareness as attention moves inward. The ladder-like organization of experience appears only on the right side, where the sense of self extends inward in a condensed manner, like a root, directly to the introspectively perceived center of the chest (Fig. 3 and 4). ! Figure 4. Constitution of the introspective somatic self-awareness in the process of the "purest" Prayer of the Heart on the right side of the chest. This inward extension of the condensed hyletic sense of self is what organizes the sequencing of internal impressions; those, remarkably, emerge in a step-by-step progression, exactly as described by Theophanis89. Not all the layers appear in one single session, as the mind may become "stationed" in a single condition before the next shift, but the general tendency is very clear. Finally, the sense of self becomes absorbed in the perceived center of the chest, with an increase in the sense of internal expanse paradoxically combined with the sharpened absorptive focus. The practitioners usually interpret this internal landscape along the lines of one's belief system90. The sense of the sacred at the core of the chest appears regardless of the personal structure of belief91. Above we've shown the diachronic stability, from Theophanis to modern practitioners, and the synchronic stability, among practitioners, of a distinct phenomenological structure. As against a view on introspection as a disparate phenomenon92, we've demonstrated that introspection can have stable structure in the presence of the organizing matrix related to embodiment, i.e. spatiality, engagement of the body-schema, and hyletic component of self-awareness. The asymmetry in phenomenology of introspection may have further interesting connections to the ancient arithmetic, such as in the Gnostic version of Neopythagorean cosmology in which the decad involves finger calculus and thereby switches the 88 Similarly, the Indian saint Sri Ramana Maharshi, Words of Grace, termed the sense of self on the right side of the chest "ahaṃsphuraṇa" (Sanskrit, 'the radiance of the I'), and used this somatic impression as a platform for his method of religious introspection. 89 For more details on the process of absorption in the PH, see Louchakova, "Ontopoiesis and Union". 90 For the spatial organization of impressions in introspection, i.e. internal "landscape", see Louchakova-Schwartz, "Self and World". For the constitutive influences of belief see Nescolarde-Selva and Usó-Doménech, "Topological Structures". 91 For the invariability of religious sense to consciousness, see Ales Bello, "Husserlian Approach." 92 For the randomness of introspection, see the seminal paper by Nisbett and Wilson, "Telling More than We Can Know". For critical reassessment of this, see Froese, "Interactively Guided Introspection"; Jack, "Introspection"; Pasquali, Timmermans and Cleeremans, "Know Thyself"; Petitmengin et al., "Gap". Unauthenticated Download Date | 2/29/16 5:15 PM Theophanis and Monoimus the Arab 67 count between the left and the right side of the body each time the count reaches the tenth step93. A possible impact of such switch on introspective states requires further research. Agency In phenomenology of introspection in live subjects, the shift to the "pure prayer" happens in a context of the relationship with a deity posited within, with reallocation of the sense of agency from a praying individual to the deity94. In the guided PH (in quasiexperiment), in which the inter-subjectivity was divided between the external other and introspective Other, same reallocation of agency takes place, and the same structure of stratified, asymmetrical experience showed up. The Heart Both non-canonic and canonic Christian literature refers to the heart as a locus of the divine95. In line 33, Theophanis promises that "Your heart will inwardly experience it [i.e. the Ladder]". The term "heart" is used in the textual evidence in two senses, as a reference to the internal organ on the left side of the body, and as a signifier of a phenomenological core of the embodied self-awareness and an embodied correlate of pure subjectivity96. The compiler of Philokalia, Nikodemos of the Holy Mountain, clarifies the confusion between the physical heart and the perceived center of the chest, stressing the importance of a "para-natural" or "super-natural" center, i.e. what is given in perception97. Theophanis' "perfection which is endless" phenomenologically corresponds with the absorption of the mind in this center98. In our study of live subjects, this core is accessible for attention through the right and left sides of the body, as well as through the center of the chest, but the stratification in the process of absorption appears only on the right side. The Orthodox idea of Spiritual Heart as a junction between human being and God is similar to the Gnostic cosmological function of iota as a bridge between Man (the ontological ground of creation) and Son of Man (manifestation)99. This inspires further exploration of possible commonalities of introspective experience in Christian and Gnostic contexts. Attentional Strategies of Theophanis the Monk and Monoimus the Arab Like Theophanis, Monoimus the Arab stresses the need of specific introspective experience; however, on the basis of the extract alone, it is impossible to say what kind of experience he refers to. The analysis of shared themes between two authors (Table 3) helps to identify the leading commonality between Monoimus' introspection and Theophanis' introspection, which is the shift in the sense of agency like the one in "purest prayer". 93 Kalvesmaki, Theology 2013, 80; also, the body decadology in Sefer Yetzirah. 94 For more on the structuring of the introspective intersubjectivity, see Louchakova, "Ontopoiesis". 95 For the knowledge of The Farther in the heart in The Valentinian Gnosis, see Kalvesmaki, Theology, 30. For the Spiritual Heart in Christian mysticism see St. Nicodemus, Handbook, 154-155; For the phenomenology of experience of the Spiritual Heart see Louchakova, "Ontopoiesis"; Louchakova, "Spiritual Heart". 96 On the embodied constitution of transcendental subjectivity, see Henry, Material Phenomenology; Tito, Logic. 97 St. Nicodemus, Handbook, 154-155. 98 Louchakova, "Ontopoiesis"; Louchakova, "Spiritual Heart". 99 For heart as a junction between human being and God, see Špidlík, Spirituality, 104-105. For cosmological function of iota in Monoimus see Kalvesmaki, Theology. For the phenomenology of embodiment in PH, see Depraz, "Pratiquer la Réduction", 87. Unauthenticated Download Date | 2/29/16 5:15 PM 68 O. Louchakova-Schwartz Table 3. References to Introspective Experience in Monoimus' Fragment in comparison with Theophanis' poem Monoimus' fragment Correlation with Theophanis' poem Correlation with experience in live subjects "Omitting to seek after God, and creation, and things similar to these, seek for Him from thyself..." General introspective orientation of the traditional Prayer of the Heart. Turning attention inward, onto and into the egological somatic self-awareness. "...[L]earn who it is that absolutely appropriates all things in thee, and says, "My God, my mind, my understanding, my soul, my body." And learn from whence are sorrow, and joy, and love, and hatred, and involuntary wakefulness, and involuntary drowsiness, and involuntary anger, and involuntary affection..." [Stratification of experience in descending layers] "... He finds the outlet to be from thyself." Purest prayer, with its phenomenology of self-acting prayer. Ladder-like stratification. Sense of experience belonging to internal Other. "...[Y]ou will discover Himself, unity and plurality, in thyself, according to that tittle...", whereby tittle is a reference to the stroke of iota. (10) "Last comes – a step that has no limit, (11) Though compassed in a single line – (12) Perfection that is endless", whereby the "single line" is a reference to the line of iota. When the introspection culminates in full absorption in the core of the chest, participants feel as if absorbed into a "point infinitely extending inward"(from a report). I.e., the simile of a line. The correlations in Table 3 suggest that Monoimus also introspected into the embodied egological selfawareness in the chest, which might have been a predecessor of the fully developed Orthodox PH. Both in the accounts of practitioners, and in Monoimus' fragment, the internal Other who appropriates all things in the self, i.e. a sense of disowned intentionality, appears predictably in response to the mental motion by which the introspective absorbing divests itself from mental qualities. Reading mystical theology through phenomenological lenses100, we see the same mental motion in apophasis and in the Gnostic ideal of kenosis, emptying oneself. Another group of practitioners reported experiences which resonate more with the mental motion of absorption which culminates in the Christian ideal of theosis, with the corresponding cataphatic theology of Divine Names. Both mental modes emerge in the introspective reversal (Fig. 5) of attention in the prayer, in which the mind can either become absorbed in its own origin (as in Theophanis), or can divest itself from its qualities, negating them till the awareness has no contents (as in Monoimus)101. In its asymmetry and hierarchical constitution of the internal landscape, egological introspection is spatial, and phenomenological spatiality, especially in the case of asymmetry, contributes to the formation of abstract thought.102 The correlations between the ladder-like structure of introspection and selective religious ideas are indicated in Table 4. 100 For the phenomenological analysis of the mental disposition in generation of mystical philosophy, see Louchakova, "Seal"; Louchakova-Schwartz, "Phenomenological Approach". 101 Further connections can be established between the modes of introspection and religious ides, such as e.g. for yoga as an absorptive mode, vs. Vedanta as a mode of negation (cf. Lakṣmidhāra, "Advaita"; or for the connection between the egological mode of self-awareness and the Vedantic ideal of Brahman-self, or non-egological awareness and the Buddhist concepts of no-self, etc. 102 Gattis, Spatial Schemas. For arguments on whether theology can emerge out of experience, see Webb, "Religious Experience". Unauthenticated Download Date | 2/29/16 5:15 PM Theophanis and Monoimus the Arab 69 ! Figure 5. Introspective absorption of attention Table 4. Forms of Embodied Introspective Transcendence and Corresponding Religious Ideas and Psychological Changes. 103 Form of Introspective Experience Religious Ideas Psychological Changes Shifts between the stratified organized layers of introspective experience. Idea of a stages in spiritual ascent or descent (see Table 2). Gradual reconstitution of character.102 Gradual "thinning out" of haptic perception and an increase of the sense of internal space. Idea of ascent to heaven. The concept of subtle bodies in Hinduism. Gradual integration of the dissociative identities and "split-off" aspects of the self (unpublished observations). Flip-flop of identity and non-spatial, non-symbolic consciousness at the core. Idea of the Spiritual Heart in Hesychasm; ideas such as Atman is Brahman. Loosening of obsessive mental habits, positive impact on the posttraumatic stress disorder (Author's observations, unpublished). Conclusions, and Back to Cognitive Science Theophanis' text helped us to identify and describe a specific introspective experience with a temporalspatial stratification of mental faculties associated with the right side of the chest. This is a form of experience previously unnoticed by researchers. This experience appears in conjunction with the introspective shift in agency in internal prayer. Besides the historical considerations above, a conclusion can be made that embodied introspection can have phenomenologically stable spatio-temporal structures which can co-constitute religious ideatio104. On a cognitive level, specific attentional strategy, such as the one involved in the Prayer of the Heart, can also be a factor supporting religious ideation105. Because of the presence of this identifiable structure, this form of introspection is reproducible, generalizable, and can be researched scientifically106. Novel approaches such as neural computing and human neuroscience, and brain chaos, suggest that mental states can be correlated directly with brain states; these approaches depend on the phenomenology of experience107. The spatio-temporal organization of mental experience can be directly correlated with the patterns 103 Louchakova, "Ontopoiesis"; Louchakova, "Reconstitution". 104 For the generative analysis, see Steinbock, "Generativity". 105 For the role of attention in cognitive formation of religious beliefs, see Chasteen, Burdzy and Pratt, "Thinking of God". 106 For a perspective on scientific research of neurocorrelation in a similar situation with shifts in experience, see Bob and Louchakova, "States". 107 Dulany, "What Should Be the Roles?"; Josipovic, "Neural Correlates"; Lutz and Thompson, "Neurophenomenology"; Manna et al., "Neural Correlates". For the complex informational relationship in the subjective dimension of experience which can be mapped onto the brain without the intermediacy of a computer metaphor, see Oizumi, Albantakis, and Tononi, "From the Phenomenology to the Mechanisms"; Lagzi and Rotter, "Markov Model". Unauthenticated Download Date | 2/29/16 5:15 PM 70 O. Louchakova-Schwartz in spatio-temporal activity of the brain108. The stratification of experience such as the ladder-like shifts can be linked to brain chaos109, or measures of the nested fractal organization of non-linear brain patterns110. Non-linear brain activation patterns have been linked to the dynamics of large-scale neuronal networks111, including both networks with finite size and global adaptations of brain architecture112. It has been reported that such adaptations also have layered temporal profiles linked to hierarchical feedback modules113. In the insula, which connects subjective emotional states to the interoception114, the modules can underlie the phenomenology of the continuous sense of self organizing the shifts in internal impressions in Theophanis' introspective experience. In Theophanis' Ladder, mental states function as vectors predicting the emergence of the next state115, whereby the quanta of states in the ladder may be described through a quantum cognitive model116. Asymmetry may isomorphically mirror the geometry of space in neuronal populations117. The metaphor of the ladders conveys the paradoxical images of horizontal discreteness and vertical linearity118, which, however, work very well toward visioning the functioning of the brain. The resolution of the Ladder into a single line of endless perfection resonates with the "least effort" agenda of the brain119. Realized through a series of shifts in order and disorder in the transition from complexity to a determinate condition, the Ladder presents a special case of deterministic tendencies in cognitive aspects of chaos120. Even briefly outlined, these perspectives of the direct neurocorrelation of the phenomenological structure of introspection with the intrinsic functional architecture of the brain and its computational organization121 show that introspection with religious contexts, found in texts of antiquity, presents unique possibilities for scientific research122. Further studies of textual evidence are necessary to bring the phenomenological mapping of religious introspection to completion. Acknowledgements: This study was supported by the grants from the Zimmer and Spitzer Family Foundations and by faculty mini-grants (2004-2011) from the Institute of Transpersonal Psychology, and by a research grant from Sofia University. 108 Fingelkurts, Fingelkurts, and Neves, "Natural World Physical, Brain Operational, and Mind Phenomenal Space-Time". 109 Bob and Louchakova, "States". 110 For more on the correlations between experience and non-linear network dynamics, see Di Leva et al. "Fractals", 2013; Ibáñez-Molina and Iglesias-Parro, "Fractal Characterization"; Noda et al., "Graph Structure"; Pereda, Quiroga and Bhattacharya, "Nonlinear Multivariate Analysis"; Sporns, "Network Analysis". 111 For the dynamics of large scale networks, see Block, "Two Neural Correlates"; Block, Flanagan, and Güzeldere, Nature; Freeman, Societies; Dehaene and Christen, Characterizing; Dehaene and Dehaene-Lambertz, Apprendre. 112 For the networks of finite size see Lagzi and Rotter, "A Markov Model". For global adaptations of brain functional architecture, see Freeman, Neurodynamics; Freeman, Imaging Brain Function; Raffone and Pantani, "Global Workspace Model". For more on the correlation between the phase shifts in fMRI or EEG and subjectively experienced cognitive events, see Baerentsen et al., "An Investigation of Brain Processes Supporting Meditation"; Naruse et al., "Statistical Method"; Schwab et al., Time-Course; Luciani et al., "Neural Correlate"; Brockmeier et al., "Spatial Patterns "; Panagiotides et al., "Behavioral States". For fMRI, see Jang et al., "Increased Default Mode". 113 Xu and Lan, "Hierarchical Feedback Modules". For modularity, see Stanley et al., "Changes"; in cognition, see Barrett and Kurzban, "Modularity". 114 For the links between cortex, insula and limbic system, see Dijkerman and de Haan, "Somatosensory Processes"; Iacoboni and Lenzi, "Mirror Neurons"; Taylor, Seminowicz and Davis, "Two Systems". For connections between subjective emotional states and interoception see Terasawa, Fukushima and Umeda, "How Does Interoceptive Awareness Interact?"; Wiebking et al., "Interoception". 115 Cf. 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BOOKS 3 The study of this volume has much to offer the reader who is willing to see philosophical approaches amid the contributions of other disciplines rather than in isolation and apart. Fordham University Joseph W. Koterski, S.J. Kant and the Meaning of Religion. By Terry F. Godlove. New York NY: Columbia Univ. Press, 2014. Pp. 245. $90.00 cloth. Despite the alluring title, this book has very little to say about Immanuel Kant's theory of religion. Rather, it is a thoroughgoing study of his theory of how words acquire and retain their meanings that uses the word "religion" to illustrate the various aspects of Kant's position. The book does comment in some detail on theories of religion in general, inasmuch as many of the author's interlocutors are writers concerned with issues relating to how religious language and/or experience can be meaningful. But a discussion of Kant's official definition of religion is the closest that Terry Godlove comes to discussing Kant's multifaceted views on the nature and purpose of religious beliefs and practices. The introduction and the conclusion place the book's six chapters in a religious setting by discussing an issue that receives relatively little explicit attention within the book itself: Nietzsche's reference to Kant as a "cunning Christian" seems to entail the accusation that the Critical philosophy itself covertly stacks the deck in favor of Kant's childhood religion. According to Godlove, a careful examination of Kant's theory of meaning in the first Critique reveals the vacuity of Nietzsche's claim: although he admits that some of Kant's basic epistemological distinctions are "walking along the edge of religious reflection" (p. 9), Godlove's aim in writing this book is to demonstrate that such distinctions have "nothing to do with Christianity." He gives us clear forewarning: "The Kant I wish to portray has no substantive connection with Christianity," at least insofar as Christianity is understood in terms of Protestant individualism (p. 10). Rather, Kant's epistemology points in some rather surprising directions, often highly relevant to contemporary discussions. Godlove notes, for example, that it would be interesting to examine how Kant's theory of meaning "plays out in the arena of ritual studies" (p. 11), but this is not a lead followed within the present book. Chapter 1 explores Kant's theory of concept-formation, which Godlove calls "the spatial theory of concepts" (p. 18). According to this theory, every concept that we employ is part of a hierarchically interrelated network of concepts that intersect with each other like (potentially) endless collections of overlapping spheres. Their interrelations are so complex that, at least in the case of empirical concepts, we can never fully determine how a concept relates to specific individuals: "No amount of conceptual detail can guarantee univocal reference" (p. 19). The three main features of a concept, for Kant-namely, "generality, rule-governedness, and the denial of an infima [i.e., lowest] species" (p. 24)-are all commonly discussed in "contemporary portrayals of concepts." As such, Kant can be regarded as "an early, and probably the first, proponent of a strongly inferentialist theory of conceptual content," of the sort now associated with the work of Michael Dummett or Robert Brandom (p. 28). What sets a priori concepts apart from such empirical conceptual content is that the former, through Kant's special procedure of transcendental deduction, can be fully determined. While Godlove fully recognizes this important distinction, he unfortunately pays little attention to this special Kantian form of philosophical justification, because it does not apply to "such empirical concepts as 'religion'" (p. 18; see also p. 33). Had he delved further into Kant's International Philosophical Quarterly Vol. 55, No. 4, Issue 220 (December 2015) pp. 3–91 doi: 4 BOOKS actual theory of religion, especially as elaborated in Religion within the Bounds of Bare Reason, Godlove would have had to make more of the fact that Kant explicitly distinguishes between empirical religions (called "historical faiths") and his own special (a priori!) concept of rational religion. But such an extension is beyond this book's intended scope. While Godlove admits that Kant is an essentialist "as applied to concepts," he emphasizes the fact that Kant's epistemological distinctions require one to be an anti-essentialist "as applied to objects" (p.35). That is, insofar as we apply concepts to know any given object (including, for example, a particular religious tradition), our knowledge is always provisional. Chapter 2 (on Kant's theory of definition) begins by citing Kant's official definition of religion "as 'recognition of all duties as divine commands'" (p. 37). After reviewing the spatial theory of concepts, Godlove shows how various aspects of that theory can be illustrated by Kant's definition. As for the status of Kant's definition, he argues that Kant is attempting to isolate the analytic essence of what the concept "religion" entails. The further distinction made in Religion between "revealed" and "natural" religion demarcates two species of the genus "religion." Those who criticize Kant's definition as not being applicable to concrete religious traditions, Godlove points out, are not really criticizing his theory of religion but his theory of definition. "Concepts are general all the way down" (p. 57), and "Kant is well aware" that "his definition does not fit . . . even a single historical tradition" (p. 59). For Kant, the philosopher's job is to set such definitions; testing them through "experiment" is the task of the social sciences, whose creation was largely due to Kant's influence (pp. 61ff). Here Godlove launches into an interesting discussion of the Kantian influence on Durkheim and others, but unfortunately he makes no mention of the fact that in Religion Kant does claim to be conducting two "experiments" of his own, the second being close (if not identical) to the one that Godlove claims Kant bequeathed to social science. In chapter 3 Godlove analyzes Kant's understanding of the function of reason. Once we appreciate Kant's view of the role of reason (the unity-seeking faculty) in guiding the progress of science, Godlove insightfully points out, we can see that the function of concepts and definitions is no different when applied to the study of religion than it is when applied in any other scientific context. Here he defends the controversial position that regulative uses of ideas "make experience possible" as much as constitutive ones do (p. 86). He admits, however, that this is a strange form of necessity indeed. Here Godlove would have done well to consider an option that I have proposed: that reason's regulative use conveys an analytic a posteriori kind of necessity. If its necessity is synthetic a priori, then it becomes indistinguishable from what is constitutive of human knowledge. Chapters 4 and 5 offer what seem like a parenthetical discussion, respectively, of the role of experience in human cognition (i.e., how non-conceptual content can be conveyed meaningfully) and of how our experience of ourselves (i.e., as an inferred "self") illustrates this aspect of human experience. In the former Godlove persuasively demonstrates that Schleiermacher was assuming an essentially Kantian view that "the deliverances of receptivity are immediate and non-discursive" (p. 114) but when applying this fact about human experience to the understanding of religion, he committed errors that Kant avoided. In the latter he similarly argues that William James commits errors that Kant avoids, to the extent that in both cases "Kant can offer . . . a way to transcend a bare naturalism but one that stops short of dualistic theism." A possibility that Godlove does not consider is that Nietzsche's tantalizing reference to Kant's cunning might indicate a revolution that Kant was seeking to implement, whereby Christianity becomes a form of theistic monism. Chapter 6 aims to complete the foregoing analysis of Kant's spatial theory of concepts by exploring its implications for a more general theory of meaning. In this case, however, BOOKS 5 Godlove strays further from Kant than anywhere else in the book. After introducing Michael Williams's tri-partite theory of meaning in terms of its "inferential," "epistemic," and "functional" components (pp. 152–53), he considers a string of examples that are meant to provide "a broadly Kantian approach to illuminating the content of particular vocabulary items." God-talk, according to Godlove's Kant, must have a function that "is earth-bound" (p. 164). As such, theological language becomes one part of Kant's overall "humanizing program" (e.g., pp. 166,177). Unfortunately, as the chapter progresses, the anticipated climax never arrives, for Godlove's reflections on Durkheim, Weber, Davidson, and a host of other recent thinkers end up taking center stage. No doubt, this is because he thinks that these recent theorists aptly illustrate what Kant was trying to do with (and perhaps also to?) the concept "religion." Nevertheless, some treatment of Kant's position in its own right would have been very helpful at this point. While it is difficult to fault the author's detailed and insightful treatment of Kant's theory of meaning-surely the best treatment of its kind to date that focuses on its application to the thorny issue of conceptualizing religion-this book does leave the reader pining for what James aptly calls something more. If this was part of the author's intent, then he succeeded admirably. In any case, the book underplays the role of Kant's transcendental idealism as a backdrop (albeit, a hidden one) throughout Kant's Religion. Moreover, Godlove never mentions that what Kant purports to study on the empirical or historical side (i.e., what Kant calls his "second experiment") falls under the concept of faith, not under "religion." As a result, while Godlove convincingly refutes Nietzsche's characterization of Kant as a "cunning Christian" when it comes to his conceptualizing of religion, it is not at all clear that Nietzsche's remark can be faulted, if it is taken as a reference to Christian faith. Hong Kong Baptist University Stephen R. Palmquist The Problem of Evil. By Daniel Speak. Cambridge UK: Polity Press, 2015. Pp. 149. $64.95 cloth. Daniel Speak's The Problem of Evil has two primary goals. The first is to introduce thinkers to the family of issues that in the world of analytic philosophy is known as the "problem of evil." The second is to defend the rationality of theism against those who would argue that the existence of evil makes religious belief irrational. The book has six chapters. In the first, Speak broadly introduces the problem and the primary strategies with which theists have responded to it. In the second, third, and fourth he presents and discusses the specific argument that best captures the concern expressed in one of the primary formulations of the problem of evil-the "logical," "evidential," and "hiddenness" problems respectively-and presents the strongest theistic replies to it that he knows. In the fifth chapter he briefly explains the "project of theodicy." The sixth chapter serves both as a conclusion to his reflections and an introduction to aspects of the problem of evil that Speak has not directly addressed in his book. Each of the chapters is carefully articulated and accessible. Speak is mindful of the fact that logico-philosophical reasoning can seem to the neophyte to be nothing more than a series of variations on the same phrase. He intersperses logical reasoning with humor and stories. Above all, his clarity makes the ideal of philosophical precision attractive. International Philosophical Quarterly Vol. 55, No. 4, Issue 220 (December 2015) pp. 5–91 doi:

co m m ent review s repo rts depo sited research refereed research interactio ns info rm atio n Open Access2005Smithet al.Volume 6, Issue 5, Article R46Method Relations in biomedical ontologies Barry Smith*†, Werner Ceusters‡, Bert Klagges§, Jacob Köhler¶, Anand Kumar*, Jane Lomax¥, Chris Mungall#, Fabian Neuhaus*, Alan L Rector** and Cornelius Rosse†† Addresses: *Institute for Formal Ontology and Medical Information Science, Saarland University, D-66041 Saarbrücken, Germany. †Department of Philosophy, University at Buffalo, Buffalo, NY 14260, USA. ‡European Centre for Ontological Research, Saarland University, D-66041 Saarbrücken, Germany. §Department of Genetics, University of Leipzig, D-04103 Leipzig, Germany. ¶Rothamsted Research, Harpenden, AL5 2JQ, UK. ¥European Bioinformatics Institute, Hinxton, CB10 1SD, UK. #HHMI, Department of Molecular and Cellular Biology, University of California, Berkeley, CA 94729, USA. **Department of Computer Science, University of Manchester, M13 9PL, UK. ††Department of Biological Structure, University of Washington, Seattle, WA 98195, USA. Correspondence: Barry Smith. E-mail: [email protected] © 2005 Smith et al. ; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Relations in biomedical ontologies<p>T enhance the treatment of relations in biomedical ontologies we advance a methodology for providing consistent and unambiguous formal defi it ns of the relational expre s ons used n such ntologies in a way designe to assist developers a d users i avoiding err rin coding a d annota ion. The resulting Relation Ontology can promote interoperab lity of ntol gies an support new types f utomatedreasoni g about t e sp ial nd t mpo al dime sions of biological and medic l phenomena.</p> Abstract To enhance the treatment of relations in biomedical ontologies we advance a methodology for providing consistent and unambiguous formal definitions of the relational expressions used in such ontologies in a way designed to assist developers and users in avoiding errors in coding and annotation. The resulting Relation Ontology can promote interoperability of ontologies and support new types of automated reasoning about the spatial and temporal dimensions of biological and medical phenomena. Background Controlled vocabularies in bioinformatics The background to this paper is the now widespread recognition that many existing biological and medical ontologies (or 'controlled vocabularies') can be improved by adopting tools and methods that bring a greater degree of logical and ontological rigor. We describe one endeavor along these lines, which is part of the current reform efforts of the Open Biomedical Ontologies (OBO) consortium [1,2] and which has implications for ontology construction in the life sciences generally. The OBO ontology library [1] is a repository of controlled vocabularies developed for shared use across different biological and medical domains. Thus the Gene Ontology (GO) [3,4] consists of three controlled vocabularies (for cellular components, molecular functions, and biological processes) designed to be used in annotations of genes or gene products. Some ontologies in the library for example the Cell and Sequence Ontologies, as well as the GO itself contain terms which can be used in annotations applying to all organisms. Others, especially OBO's range of anatomy ontologies, contain terms applying to specific taxonomic groups such as fly, fungus, yeast, or zebrafish. Controlled vocabularies can be conceived as graph-theoretical structures consisting on the one hand of terms (which form the nodes of each corresponding graph) linked together by means of edges called relations. The ontologies in the OBO library are organized in this way by means of different types of relations. OBO's Mouse Anatomy ontology, for example, uses just one type of edge, labeled part_of. The GO currently uses two, labeled is_a and part_of. The Drosophila Anatomy ontology includes also a develops_from link. Other OBO Published: 28 April 2005 Genome Biology 2005, 6:R46 (doi:10.1186/gb-2005-6-5-r46) Received: 28 October 2004 Revised: 3 February 2005 Accepted: 31 March 2005 The electronic version of this article is the complete one and can be found online at http://genomebiology.com/2005/6/5/R46Genome Biology 2005, 6:R46 R46.2 Genome Biology 2005, Volume 6, Issue 5, Article R46 Smith et al. http://genomebiology.com/2005/6/5/R46ontologies include further links, for example (in the Sequence Ontology) position_of and disjoint_from. The National Cancer Institute (NCI) Thesaurus adds many additional links, including has_location for anatomical structures and different part_of relations for structures and for processes. The problem is that when OBO and similar ontologies incorporate such relations they typically do so in informal ways, often providing no definitions at all, so that the logical interconnections between the various relations employed are unclear, and even the relations is_a and part_of are not always used in consistent fashion both within and between ontologies. Our task in what follows is to rectify these defects, drawing on the requirements analysis presented in [5]. Of the criteria that ontologies must currently satisfy if they are to be included in the OBO library, the most important for our purposes are: first, inclusion of textual definitions or descriptions designed to ensure that the precise meanings of terms as used within particular ontologies will be clear to a human reader; second, employment of a standard syntax, such as the OWL or OBO flatfile syntax; third, orthogonality to the other ontologies already included in the library. These criteria are designed to support the integration of OBO ontologies, above all by ensuring the compatibility of ontologies pertaining to an identical subject matter. OBO has now added a fourth criterion to assist in achieving such compatibility, namely that the relations (edges) used to connect terms in OBO ontologies should be applied in ways consistent with their definitions as set forth in this paper. The Relation Ontology offered here is designed to put flesh on this criterion. How, exactly, should part_of or located_in be defined in order to ensure maximally reliable curation of each single ontology while at the same time guaranteeing maximal leverage in building a solid base for life-science knowledge integration in general? We describe a rigorous methodology for providing an answer to this question and illustrate its use in the construction of an easily extendible list of ten relations of a type familiar to those working in the bio-ontological field. This list forms the core of the new OBO Relation Ontology. What is distinctive about our methodology is that, while the relations are each provided with rigorous formal definitions, these definitions can at the same time be formulated in such a way that the underlying technical details remain invisible to ontology authors and curators. Shortcomings of biomedical ontologies While considerable effort has been invested in the formulation and definition of terms in biomedical ontologies, too little attention has been paid in the ontological literature to the associated relations. A number of characteristic types of shortcomings of controlled vocabularies can be traced back especially to the neglect of issues of formal structure in the treatment of relations [5-10]. To take just one example, the pre-2004 versions of GO allowed at least three different readings of the expression 'part of' as representing simultaneously: inclusion relations between vocabularies; a relation of possible parthood between biological entities; a relation of necessary parthood between biological entities. As was shown in [6], this coexistence of conflicting readings meant that three of the four rules given in the then effective documentation for reasoning with GO's hierarchies were logically incorrect. Another characteristic family of problems turns on the paucity of resources for expressing relations in ontologies like GO. For example, because GO has no direct means of asserting location relations, it must capture such relations indirectly by constructing new terms involving syntactic operators such as 'site of', 'within', 'extrinsic to', 'space', 'region', and so on. It then simulates assertions of location by means of 'is_a' and 'part_of' statements involving such composites, for example in: extracellular region is_a cellular component extrinsic to membrane part_of membrane both of which are erroneous. Additional problems arise from the fact that GO's extracellular region and extracellular space are both specified in their definitions as referring to the space (how large a space?) external to the outermost structure of a cell. Another type of problem turns on the failure to distinguish relational expressions which, though closely related in meaning, are revealed to be crucially distinct when explicated in the formally precise way that is demanded by computer implementations. An example is provided by the simultaneous use in OBO's Cell Ontology of both derives_from and develops_from while no clear distinction is drawn between the two [11]. This problem is resolved in the treatment of derivation and transformation below, and has been correspondingly corrected in versions 1.14 and later of the Cell Ontology. Efforts to improve GO from the standpoint of increased formal rigor have thus far been concentrated on re-expressing the existing GO schema in a description logic (DL) framework. This has allowed the use of a DL-reasoner that can identify certain kinds of errors and omissions, which have been corrected in later versions of GO [12]. DLs, however, can do no more than guarantee consistent reasoning according to the definitions provided to them. If the latter are themselves problematic, then a DL can do very little to identify or resolve the problems which result. Here, accordingly, we take a more radical approach, which consists in re-examining the basic definitions of the relations used in GO and in related ontologies in an attempt to arrive at a methodology which will lead to the construction of ontologies which are more fundamentally sound and thus more secure against errors and more amenable to the use of powerful reasoning tools.Genome Biology 2005, 6:R46 http://genomebiology.com/2005/6/5/R46 Genome Biology 2005, Volume 6, Issue 5, Article R46 Smith et al. R46.3 co m m ent review s repo rts refereed research depo sited research interactio ns info rm atio n This approach is designed also to be maximally helpful to biologists by avoiding the problems which arise by virtue of the fact that the syntax favored in the DL-community is of a type which can normally be understood only by DL-specialists. A theory of classes and instances The relations in biological ontologies connect classes as their relata. The term 'class' here is used to refer to what is general in reality, or in other words to what, in the knowledge-representation literature, is typically (and often somewhat confusingly [13]) referred to under the heading 'concept' and in the literature of philosophical ontology under the headings 'universal', 'type' or 'kind'. Biological classes are in first approximation those classes which have been implicitly sanctioned through usage of the corresponding general terms in the biological literature, for example cell or fat body development. Our task is to develop a suite of coherently defined bio-ontological relations that is sufficiently compact to be easily learned and applied, yet sufficiently broad in scope to capture a wide range of the relations currently coded in standard biomedical ontologies. Unfortunately the realization of this task is not a trivial matter. This is because, while the terms in biomedical ontologies refer exclusively to classes to what is general in reality we cannot define what it means for one class to stand to another, for example in the part_of relation, without taking the corresponding instances into account [6]. Here the term 'instance' refers to what is particular in reality, to what are otherwise called 'tokens' or 'individuals' entities (including processes) which exist in space and time and stand to each other in a variety of instance-level relations. Thus we cannot make sense of what it means to say cell nucleus part_of cell unless we realize that this is a statement to the effect that each instance of the class cell nucleus stands in an instance-level part relation to some corresponding instance of the class cell. This dependence of class-relations on relations among corresponding instances has long been recognized by logicians, including those working in the field of description logics, where the (all some) form of definition we utilize below has been basic to the formalism from the start [14]. Definitions of this type were incorporated also into the DL-based GALEN medical ontology [15], though the significance of such definitions, and more generally of the role of instances in defining class relations, has still not been appreciated in many user communities. It is also characteristically not realized that talk of classes involves in every case a more-or-less explicit reference to corresponding instances. When we assert that one class stands in an is_a relation to another (that is, that the first is a subtype of the second), for example, that glucose metabolism is_a carbohydrate metabolism, then we are stating that instances of the first class are ipso facto instances of the second. When we are dealing exclusively with is_a relations there is little reason to take explicit notice of this two-sided nature of ontological relations. When, however, we move to ontological relations of other types, then it becomes indispensable, if many characteristic families of errors are to be avoided, that the implicit reference to instances be taken carefully into account. Types of relations We focus here exclusively on genuinely ontological relations, which we take to mean relations that obtain between entities in reality, independently of our ways of gaining knowledge about such entities (and thus of our experimental methods) and independently of our ways of representing or processing such knowledge in computers. A relation like annotates is not ontological in this sense, as it links classes not to other classes in nature but rather to terms in a vocabulary that we ourselves have constructed. We focus also on general-purpose relations relations which can be employed, in principle, in all biological ontologies rather than on those specific relations (such as genome_of or sequence_of employed by OBO's Sequence Ontology) which apply only to biological entities of certain kinds. The latter will, however, need to be defined in due course in accordance with the methodology advanced here. The ontologies in OBO are designed to serve as controlled vocabularies for expressing the results of biological science. Sentences of the form 'A relation B' (where 'A' and 'B' are terms in a biological ontology and 'relation' stands in for 'part_of' or some similar expression) can thus be conceived as expressing general statements about the corresponding biological classes or types. Assertions about corresponding instances or tokens (for example about the mass of this particular specimen in this particular Petri dish), while indispensable to biological research, do not belong to the general statements of biological science and thus they fall outside the scope of OBO and similar ontologies as these are presented to the user as finished products. Yet such assertions are still relevant to ontologies. For it turns out that it is only by means of a detour through instances that the definitions and rules for coding relations between classes can be formulated in an intuitive and unambiguous and thus reliably applicable way. We can distinguish, in fact, the following three kinds of binary relations: <class, class>: for example, the is_a relation obtaining between the class SWR1 complex and the class chromatin remodeling complex, or between the class exocytosis and the class secretion; <instance, class>: for example, the relation instance_of obtaining between this particular vesicle membrane and theGenome Biology 2005, 6:R46 R46.4 Genome Biology 2005, Volume 6, Issue 5, Article R46 Smith et al. http://genomebiology.com/2005/6/5/R46class vesicle membrane, or between this particular instance of mitosis and the class mitosis; <instance, instance>: for example, the relation of instancelevel parthood (called part_of in what follows), obtaining between this particular vesicle membrane and the endomembrane system in the corresponding cell, or between this particular M phase of some mitotic cell cycle and the entire cell cycle of the particular cell involved. Here classes and the relations between them are represented in italic; all other relations are picked out in bold. Continuants and processes The terms 'continuant' and 'process' are generalizations of GO's 'cellular component' and 'biological process' but applied to entities at all levels of granularity, from molecule to whole organism. Continuants are those entities which endure, or continue to exist, through time while undergoing different sorts of changes, including changes of place. Processes are entities that unfold themselves in successive temporal phases [16]. The terms 'continuant' and 'process' thus correspond to what, in the literature of philosophical ontology, are known respectively as 'things' (objects, endurants) and 'occurrents' (activities, events, perdurants) respectively. A continuant is what changes; a process is the change itself. The continuant classes relevant to biological ontologies include molecule, cell, membrane, organ; the process classes include ion transport, cell division, fat body development, breathing. To formulate precise definitions of the <class, class> relations which form the target of ontology construction in biology we will need to employ a vocabulary that allows reference both to classes and to instances. For this we take advantage of the machinery of logic, and more specifically of the standard device of variables and quantifiers [17], using different sorts of variables to range across the classes and instances of continuants and processes, spatial regions and temporal instants, respectively. For the sake of intelligibility we use a semi-formal syntax, which can, however, be translated in a simple way into standard logical notation. We use variables of the following sorts: C, C1, ... to range over continuant classes; P, P1, ... to range over process classes; c, c1, ... to range over continuant instances; p, p1, ... to range over process instances; r, r1, ... to range over three-dimensional spatial regions; t, t1, ... to range over instants of time. In an expanded version of our formal machinery we will need also to incorporate further variables, ranging for example over temporal intervals, biological functions, attributes and values. Note that continuants and processes form non-overlapping categories. This means in particular that no subtype or parthood relations cross the continuant-process divide. The tripartite structure of the GO recognizes this categorical exclusivity and extends it to functions also. Continuants can be material (a mitochondrion, a cell, a membrane), or immaterial (a cavity, a conduit, an orifice), and this, too, is an exclusive divide. Immaterial continuants have much in common with spatial regions [18]. They are distinguished therefrom, however, in that they are parts of organisms, which means that, like material continuants, they move from one spatial region to another with the movements of their hosts. The three-dimensional continuants that are our primary focus here typically have a top and a bottom, an anterior and a posterior, an interior and an exterior. Processes, in contrast, have a beginning, a middle and an end. Processes, but not continuants, can thus be partitioned along the time axis, so that, for example, your youth and your adulthood are temporal parts of that biological process which is your life. As child and adult are continuants, so youth and adulthood are processes. We are thus clearly dealing here with two complementary space-focused and time-focused views of the same underlying subject matter, with determinate logical and ontological connections between them [16]. The framework advanced below allows us to capture these connections by incorporating reference to spatial regions and to temporal instants, both of which can be thought of as special kinds of instances. We shall also need to distinguish two kinds of instance-level relations: those (applying to continuants) whose representations must involve a temporal index, and those (applying to processes) which do not. Note that the drawing of this distinction is still perfectly consistent with the fact that processes themselves occur in time, and that processes may be built out of successive subprocesses instantiating distinct classes. Primitive instance-level relations We cannot, on pain of infinite regress, define all relations, and this means that some relations must be accepted as primitive. The relations selected for this purpose should be self-explanatory and they should as far as possible be domain-neutral, which means that they should apply to entities in all regions of being and not just to those in the domain of biology. Our choice of primitive relations is as follows:Genome Biology 2005, 6:R46 http://genomebiology.com/2005/6/5/R46 Genome Biology 2005, Volume 6, Issue 5, Article R46 Smith et al. R46.5 co m m ent review s repo rts refereed research depo sited research interactio ns info rm atio n c instance_of C at t a primitive relation between a continuant instance and a class which it instantiates at a specific time p instance_of P a primitive relation between a process instance and a class which it instantiates holding independently of time c part_of c1 at t a primitive relation between two continuant instances and a time at which the one is part of the other p part_of p1, r part_of r1 a primitive relation of parthood, holding independently of time, either between process instances (one a subprocess of the other), or between spatial regions (one a subregion of the other) c located_in r at t a primitive relation between a continuant instance, a spatial region which it occupies, and a time r adjacent_to r1 a primitive relation of proximity between two disjoint continuants t earlier t1 a primitive relation between two times c derives_from c1 a primitive relation involving two distinct material continuants c and c1 p has_participant c at t a primitive relation between a process, a continuant, and a time p has_agent c at t a primitive relation between a process, a continuant and a time at which the continuant is causally active in the process This list includes only those <instance-instance> relations, together with one <instance-class> relation, which are needed for defining the <class, class> relations which are our principal target in this paper. The items on the list have been selected because they enjoy a high degree of intelligibility to the human authors and curators of biological ontologies. For purposes of supporting computer applications, however, the meanings of the corresponding relational expressions must be specified formally via axioms, for example in the case of 'part_of' by axioms of mereology (the theory of part and whole: see below), and in the case of 'earlier' by axioms governing a linear order [17]. The relation located_in will satisfy axioms to the effect that for every continuant there is some region in which it is located; instance_of will satisfy axioms to the effect that all classes have (at some stage in their existence) instances, and that all instances are instances of some class. The formal machinery for reasoning with such axioms is in place, and a comprehensive set of axioms is being compiled. For the typical human user of biological ontologies, however, the listed primitive relations and associated axioms are designed to work invisibly behind the scenes. That is, they serve as part of the background framework that guides the construction and maintenance of such ontologies. Results Methodology We employed a multi-stage methodology for the selection of the relations to be included in this ontology and for the formulation of corresponding definitions. First, a sample of researchers involved in ontology construction in the life sciences, representing different groups and including the coauthors of this paper, was asked to prepare lists of principal relations in light of their own specific experience but focusing on relations which would be: 'ontological' in the sense introduced above; 'general-purpose' in the sense that they apply across all biological domains; and also such as to manifest a high degree of universality (in the sense explained in the section 'Types of relational assertions' below). The submitted lists manifested a significant degree of overlap, which allowed us to prepare a core list in whose terms a large number of the remaining relations on the list could be simply defined. A further constraint on the process was the goal of providing a simple formal definition for each included <class-class> relation. Those relations for which an appropriate simple definition could not be agreed upon were not included in this interim list. This includes most conspicuously relations involving analogs of the GO notion of molecular function. The relation has_agent was, however, included in light of a common understanding that the notion of agency would be involved in whatever candidate definition of function in biology is eventually accepted for use in OBO. This further constraint was chosen in light of the fact that our capacity to provide simple formal definitions definitions which will at one and the same time be intelligible to ontology authors and curators and also able to support logic-based tools for automatic reasoning and consistency-checking is the primary rationale for the methodology here advanced. The two relations is_a and part_of were unproblematic candidates for inclusion in the resulting list (though providing simple definitions even for these relations was not, as we shall see, a simple matter). Is_a and part_of have established themselves as foundational to current ontologies. They have a central role in almost all domain ontologies, including the Foundational Model of Anatomy (FMA) [19,20], GO and other ontologies in OBO, as well as in influential top-level ontologies such as DOLCE [21] and in digitalized lexical resources such as WordNet [22]. In preparing our sample lists we drew on representatives not only of the OBO consortium but also of GALEN and the FMA (itself a candidate for inclusion in OBO). Our temporal relations draw on existing OBO practice (where transformation_of is a generalization of the develops_fromGenome Biology 2005, 6:R46 R46.6 Genome Biology 2005, Volume 6, Issue 5, Article R46 Smith et al. http://genomebiology.com/2005/6/5/R46relation used in OBO's cell and anatomy ontologies) and our participation relations draw on current work addressing the need to provide relations that link entities in different ontologies (for example entities in GO's process, function and component ontologies) and on an evolving Physiology Reference Ontology that is being developed in conjunction with the FMA [23], from which our spatial relations were extracted. The OBO Relation Ontology The first proposed version of the OBO Relation Ontology is shown in Table 1. We shall deal here with each of the ten relations listed in Table 1 in turn, providing rigorous yet easily understandable definitions. Is_a It is commonly assumed in the literature of knowledge representation that the relation is_a (meaning 'is a subtype of') can be identified with the subset or set inclusion relation with which we are familiar from mathematical set theory [17]. Instance_of functions on this reading as a counterpart of the usual set-theoretic membership relation, yielding a definition of A is_a B along the lines of: for all x, if x instance_of A, then x instance_of B. Unfortunately, this reading provides at best a necessary condition for the truth of A is_a B. It falls short of providing a sufficient condition for two reasons. The first is because it admits cases of contingent inclusion such as: bacterium in 90 mm × 18 mm glass Petri dish is_a bacterium, and the second is because it fails to take account of time, so that when applied to classes of continuants it yields false positives such as adult is_a child (because every instance of adult was at some time an instance of child). We resolve the first problem by admitting as is_a links only assertions that reflect truths of biological science assertions involving genuine biological class names (such as 'enzyme' or 'apoptosis') rather than, for example, commercial or indexical names (such as 'bacterium in this Petri dish'). The second problem we resolve by exploiting our machinery for taking account of time in the assertion of is_a relations involving continuants. We can then define: C is_a C1 = [definition] for all c, t, if c instance_of C at t then c instance_of C1 at t. P is_a P1 = [definition] for all p, if p instance_of P then p instance_of P1. Note how the device of logical quantifiers (for all ..., for some ...) allows us to refer to instances 'in general' which means without the need to call on the proper names or indexical expressions (such as 'this' or 'here') which we use when referring to instances 'in specific'. Note also how instantiation for continuants involves a temporal argument. This reflects the fact that continuants, but not processes, can instantiate different classes in the course of their existence and yet preserve their identity. For simplicity of expression we shall henceforth write 'Cct' and 'Pp', as abbreviations for: 'c instance_of C at t ' and 'p instance_of P ', respectively. Part_of Parthood as a relation between instances. The primitive instance-level relation p part_of p1 is illustrated in assertions such as: this instance of rhodopsin mediated phototransduction part_of this instance of visual perception. This relation satisfies at least the following standard axioms of mereology: reflexivity (for all p, p part_of p); anti-symmetry (for all p, p1, if p part_of p1 and p1 part_of p then p and p1 are identical); and transitivity (for all p, p1, p2, if p part_of p1 and p1 part_of p2, then p part_of p2). Analogous axioms hold also for parthood as a relation between spatial regions. For parthood as a relation between continuants, these axioms need to be modified to take account of the incorporation of a temporal argument. Thus for example the axiom of transitivity for continuants will assert that if c part_of c1 at t and c1 part_of c2 at t, then also c part_of c2 at t. Table 1 First version of the OBO Relation Ontology Foundational relations is_a part_of Spatial relations (connecting one entity to another in terms of relations between the spatial regions they occupy) located_in contained_in adjacent_to Temporal relations (connecting entities existing at different times) transformation_of derives_from preceded_by Participation relations (connecting processes to their bearers) has_participant has_agentGenome Biology 2005, 6:R46 http://genomebiology.com/2005/6/5/R46 Genome Biology 2005, Volume 6, Issue 5, Article R46 Smith et al. R46.7 co m m ent review s repo rts refereed research depo sited research interactio ns info rm atio n Parthood as a relation between classes. To define part_of as a relation between classes we again need to distinguish the two cases of continuants and processes, even though the explicit reference to instants of time now falls away. For continuants, we have C part_of C1 if and only if any instance of C at any time is an instance-level part of some instance of C1 at that time, as for example in: cell nucleus part_ of cell. Formally: C part_of C1 = [definition] for all c, t, if Cct then there is some c1 such that C1c1t and c part_of c1 at t. Note the 'all-some' structure of this definition, a structure which will recur in almost all the relations treated here. C part_of C1 defines a relational property of permanent parthood for Cs. It tells us that Cs, whenever they exist, exist as parts of C1s. We can also define in the obvious way C temporary_part_of C1 (every C exists at some time in its existence as part of some C1) and also C initial_part_of C1 (every C is such that it begins to exist as part of some instance of C1). For processes, we have by analogy, P part_of P1 if and only if any instance of P is an instance-level part of some instance of P1, as for example in: M phase part_of cell cycle or neuroblast cell fate determination part_of neurogenesis. Formally: P part_of P1 = [definition] for all p, if Pp then there is some p1 such that: P1p1 and p part_of p1. An assertion to the effect that P part_of P1 thus tells us that Ps in general are in every case such as to exist as parts of P1s. P1s themselves, however, may exist without having Ps as parts (consider: menopause part_of aging). Note that part_of is in fact two relations, one linking classes of continuants, the other linking classes of processes. While both of the mentioned relations are transitive, this does not mean that part_of relations could be inferred which would cross the continuant-process divide. Located_in Location as a relation between instances. The primitive instance-level relation c located_in r at t reflects the fact that each continuant is at any given time associated with exactly one spatial region, namely its exact location [24]. Following [25] we can use this relation to define a further instance-level location relation not between a continuant and the region which it exactly occupies, but rather between one continuant and another. c is located in c1, in this sense, whenever the spatial region occupied by c is part_of the spatial region occupied by c1. Formally: c located_in c1 at t = [definition] for some r, r1, c located_in r at t and c1 located_in r1 at t and r part_of r1. Note that this relation comprehends both the relation of exact location between one continuant and another which obtains when r and r1 are identical (for example, when a portion of fluid exactly fills a cavity), as well as those sorts of inexact location relations which obtain, for example, between brain and head or between ovum and uterus. Location as a relation between classes. To define location as a relation between classes represented by sentences such as ribosome located_in cytoplasm, intracellular located_in cell we now set: C located_in C1 = [definition] for all c, t, if Cct then there is some c1 such that C1c1t and c located_in c1 at t. Note that C located_in C1 is an assertion about Cs in general, which does not tell us anything about C1s in general (for example, that they have Cs located in them). Contained_in If c part_of c1 at t then we have also, by our definition and by the axioms of mereology applied to spatial regions, c located_in c1 at t. Thus, many examples of instance-level location relations for continuants are in fact cases of instancelevel parthood. For material continuants location and parthood coincide. Containment is location not involving parthood, and arises only where some immaterial continuant is involved. To understand this relation, we first define overlap for continuants as follows: C1 overlap c2 at t = [definition] for some c, c part_of c1 at t and c part_of c2 at t. The containment relation on the instance level can then be defined as follows: c contained_in c1 at t = [definition] c located_in c1 at t and not c overlap c 1 at t. On the class level this yields: C contained_in C1 = [definition] for all c, t, if Cct then there is some c1 such that: C1c1t and c contained_in c1 at t. Containment obtains in each case between material and immaterial continuants, for instance: lung contained_in thoracic cavity; bladder contained_in pelvic cavity. Hence containment is not a transitive relation. Adjacent_to We can define additional spatial relations by appealing to the primitive adjacent_to, a relation of proximity between disjoint continuants. Adjacent_to satisfies some of the axiomsGenome Biology 2005, 6:R46 R46.8 Genome Biology 2005, Volume 6, Issue 5, Article R46 Smith et al. http://genomebiology.com/2005/6/5/R46governing the relation referred to in the literature of qualitative topology as 'external connectedness' [26]. Analogs of other mereotopological relations (qualitative relations between spatial regions involving parthood, boundary and connectedness) (Figure 1) can also be defined, and these too can be applied to the material and immaterial continuants which occupy such regions on the instance level. We define overlap for spatial regions as follows: r1 overlap r2 = [definition] for some r, r part_of r1 and r part_of r2. We then assert axiomatically that r1 adjacent_to r2 implies not r1 overlap r2 We can then define the counterpart relation of adjacency between classes as follows: C adjacent_to C1 = [definition] for all c, t, if Cct, there is some c1 such that: C1c1t and c adjacent_to c1 at t. Note that adjacent_to as thus defined is not a symmetric relation, in contrast to its instance-level counterpart. For it can be the case that Cs are in general such as to be adjacent to instances of C1 while no analogous statement holds for C1s in general in relation to instances of C. Examples are: nuclear membrane adjacent_to cytoplasm seminal vesicle adjacent_to urinary bladder ovary adjacent_to parietal pelvic peritoneum. We can, however, very simply define a symmetric relation of co-adjacency on the class level as follows: C1 co-adjacent_to C2 = [definition] C1 adjacent_to C2 and C2 adjacent_to C1. Examples are: inner layer of plasma membrane co-adjacent_to outer layer of plasma membrane right pulmonary artery co-adjacent_to right principal bronchus urinary bladder of female co-adjacent_to parietal peritoneum of female pelvis. Transformation_of When an embryonic oenocyte (a type of insect cell) is transformed into a larval oenocyte, one and the same continuant entity preserves its identity while instantiating distinct classes at distinct times. The class-level relation transformation_of obtains between continuant classes C and C1 wherever each instance of the class C is such as to have existed at some earlier time as an instance of the distinct class C1 (see Figure 2). This relation is illustrated first of all at the molecular level of granularity by the relation between mature RNA and the pre-RNA from which it is processed, or between (UV-induced) thymine-dimer and thymine dinucleotide. At coarser levels of granularity it is illustrated by the transformations involved in the creation of red blood cells, for example, from reticulocyte to erythrocyte, and by processes of development, for example, from larva to pupa, or from (post-gastrular) embryo to fetus [27] or from child to adult. It is also manifest in pathological transformations, for example, of normal colon into carcinomatous colon. In each such case, one and the same continuant entity instantiates distinct classes at different times in virtue of phenotypic changes. As definition for this relation we offer: C transformation_of C1 = [definition] C and C1 for all c, t, if Cct, then there is some t1 such that C1ct1, and t1 earlier t, and there is no t2 such that Cct2 and C1ct2. That is to say, the class C is a transformation of the class C1 if and only if every instance c of C is at some earlier time an instance of C1, and there is no time at which it is an instance of both C and C1. (The final clause, which asserts that C and C1 Standard mereotopological relations between spatial regionsFigure 1 Standard mereotopological relations between spatial regions. Separation Adjacency Partial overlap Tangential proper part Nontangential proper part Identity TransformationFigure 2 Transformation. Time C c at t C 1 c at t 1 Genome Biology 2005, 6:R46 http://genomebiology.com/2005/6/5/R46 Genome Biology 2005, Volume 6, Issue 5, Article R46 Smith et al. R46.9 co m m ent review s repo rts refereed research depo sited research interactio ns info rm atio n do not share instances at a time, is inserted in order to rule out, for example, adult human transformation_of human.) Note that C transformation_of C1 is a statement about Cs in general. It does not tell us of C1s in general that each gives rise to some C which stands to it in a transformation_of relation. Derives_from Derivation as a relation between instances. The temporal relation of derivation is more complex. Transformation, on the instance level, is just the relation of identity: each adult is identical to some child existing at some earlier time. Derivation on the instance-level is a relation holding between non-identicals. More precisely, it holds between distinct material continuants when one succeeds the other across a temporal divide in such a way that at least a biologically significant portion of the matter of the earlier continuant is inherited by the later. Thus we will have axioms to the effect that from c derives_from c1 we can infer that c and c1 are not identical and that there is some instant of time t such that c1 exists only prior to and c only subsequent to t. We will also be able to infer that the spatial region occupied by c as it begins to exist at t overlaps with the spatial region occupied by c1 as it ceases to exist in the same instant. Three simple kinds of instance-level derivation can then be distinguished (Figure 3): first, the succession of one single continuant by another single continuant across a temporal threshold (for example, this blastocyst derives from this zygote); second, the fusion of two or more continuants into one continuant (for example, this zygote derives from this sperm and from this ovum); and third, the fission of an earlier single continuant to create a plurality of later continuants (for example, these promyelocytes derive from this myeoloblast). In all cases we have two continuants c and c1 which are such that c begins to exist at the same instant of time at which c1 ceases to exist, and at least a significant portion of the matter of c1 is inherited by its successor c. Derivation of the first type is still essentially weaker than transformation, for the latter involves the identity of the continuant instances existing on either side of the relevant temporal divide. In derivation of the second type, the successor continuant takes the bulk of its matter from a plurality of precursors, where in cases of the third type, the bulk of the matter of a single precursor continuant is shared among a plurality of successors. We can also represent more complex cases where transformation and an analog of derivation are combined, for example in the case of budding in yeast [27], where one continuant continues to exist identically through a process wherein a second continuant floats free from its host; or in absorption, where one continuant continues to exist identically through a process wherein it absorbs another continuant, for example through digestion. Derivation as a relation between classes. To avoid troubling counter-examples, the relation of derivation we are seeking on the class level must be defined in two steps. First, the class-level counterpart of the relation of derivation on the instance level is identified as a relation of immediate derivation: C derives_immediately_from C1 = [definition] for all c, t, if Cct, then there is some c1,t1, such that: t1 earlier t and C1c1t1 and c derives_from c1. The more general class level derivation relation must then be defined in terms of chains of immediate derivation relations, as follows: C derives_from C1 = [definition] there is some sequence C = Ck, Ck-1, ..., C2, C1, such that for each Ci (1 ≤ i < k), Ci+1 derives_immediately_from Ci. In this way we can represent cases of derivation involved in the formation of lineages where there occurs a sequence of cell divisions or speciation events. Preceded_by With the primitive relations has_participant and earlier at our disposal we can define the instance-level relation p occurring_at t as follows: p occurring_at t = [definition] for some c, p has_participant c at t. Three simple cases of derivationFigure 3 Three simple cases of derivation. (a) Continuation; (b) fusion; (c) fission. C1 c1 at t1 C1′ c1′ at t1 C1 c1 at t1 C1 c1 at t1 C c at t C c at t C c at t C ′ c ′ at t (a) (b) (c)Genome Biology 2005, 6:R46 R46.10 Genome Biology 2005, Volume 6, Issue 5, Article R46 Smith et al. http://genomebiology.com/2005/6/5/R46We can then define: c exists_at t = [definition] for some p, p has_participant c at t p preceded_by p1 = [definition] for all t, t1, if p occurring_at t and p1 occurring_at t1, then t1 earlier t t first_instant p = [definition] p occurring_at t and for all t1, if t1 earlier t, then not p occurring_at t1 t last_instant p = [definition] p occurring_at t and for all t1, if t earlier t1, then not p occurring_at t1 p immediately_preceded_by p1 = [definition] for some t, t first_instant p and t last_instant p1. At the class level we have: P preceded_by P1 = [definition] for all p, if Pp then there is some p1 such that P1p1and p preceded_by p1. An example is: translation preceded_by transcription; aging preceded_by development (not however death preceded_by aging). Where derives_from links classes of continuants, preceded_by links classes of processes. Clearly, however, these two relations are not independent of each other. Thus if cells of type C1 derive_from cells of type C, then any cell division involving an instance of C1 in a given lineage is preceded_by cellular processes involving an instance of C. The assertion P preceded_by P1 tells us something about Ps in general: that is, it tells us something about what happened earlier, given what we know about what happened later. Thus it does not provide information pointing in the opposite direction, concerning instances of P1 in general; that is, that each is such as to be succeeded by some instance of P. Note that an assertion to the effect that P preceded_by P1 is rather weak; it tells us little about the relations between the underlying instances in virtue of which the preceded_by relation obtains. Typically we will be interested in stronger relations, for example in the relation immediately_preceded_by, or in relations which combine preceded_by with a condition to the effect that the corresponding instances of P and P1 share participants, or that their participants are connected by relations of derivation, or (as a first step along the road to a treatment of causality) that the one process in some way affects (for example, initiates or regulates) the other. Has_participant Has_participant is a primitive instance-level relation between a process, a continuant, and a time at which the continuant participates in some way in the process. The relation obtains, for example, when this particular process of oxygen exchange across this particular alveolar membrane has_participant this particular sample of hemoglobin at this particular time. To define the class-level counterpart of the participation relation we set: P has_participant C = [definition] for all p, if Pp then there is some c, t such that Cct and p has_participant c at t. Examples are: cell transport has_participant cell death has_participant organism breathing has_participant thorax. Once again, P has_participant C provides information only about Ps in general (that is, that they require instances of C as bearers). Has_agent Special types of participation can be distinguished according to whether a continuant is agent or patient in a process (for a survey see [28].) Here we focus on the factor of agency, which is involved, for example, when an adult engages in adult walking behavior. It is not involved when the same adult is the victim of an infection. Synonyms of 'is agent in' include: 'actively participates in', 'does', 'executes', 'performs', and so forth. We introduce the primitive instance-level relation has_agent, which obtains between a process, a continuant and a time whenever the continuant is a participant in the process and is at the same time directly causally responsible for its occurrence. Thus we have an axiom to the effect that agency implies participation: for all p, c, t, if p has_agent c at t, then p has_participant c at t. In addition we will have axioms to the effect that only material continuants can fill the agent role, that if c fills the agent role at t, then c must have existed at times earlier than t, that it must exercise its agent role for an interval of time including t, and so on. We can then define the class-level relation has_agent by stipulating: P has_agent C = [definition] for all p, if Pp then there is some c, t such that Cct and p has_agent c at t This relation gives us the means to capture the directionality (the from-to) nature of biological processes such as signaling, transcription, and expression, via assertions, for example, to the effect that in an interaction between molecules of types m1 and m2 it is molecules of the first type that play the role of agent.Genome Biology 2005, 6:R46 http://genomebiology.com/2005/6/5/R46 Genome Biology 2005, Volume 6, Issue 5, Article R46 Smith et al. R46.11 co m m ent review s repo rts refereed research depo sited research interactio ns info rm atio n One privileged type of agency consists in the realization of a biological function. To say that a continuant has a function is to assert, in first approximation, that it is predisposed (has the potential, the casual power) to cause (to realize as agent) a process of a certain type. Thus to say that your heart has the function: to pump blood is to assert that your heart is predisposed to realize as agent a process of the type pumping blood [29]. Regulation, promotion, inhibition, suppression, activation, and so forth, are among the varieties of agency that fall under this heading. On the other hand, many processes such as metabolic reactions involving enzymes, cofactors, and metabolites involve no clear factor of agent participation, but rather require more nuanced classifications of the roles of participants as acceptors or donors, for example. Hence the has_agent relation should be used in curation with special care. It should be borne in mind in this connection that agency is in every case a matter of the imposition of direct causal influence of a continuant in a process (a constraint that is designed to rule out inheritance of agency along causal chains), and also that (by our definition) only continuants can be agents. Where biologists describe processes as agents, for example, in talking about the effects of diffusion in development and differentiation, such phenomena are of a type that call for an expansion of our proposed Relation Ontology in the direction, again, of a treatment of the factor of causality. Discussion The logic of biological relations Inverse and reciprocal relations The inverse of a relation R is defined as that relation which obtains between each pair of relata of R when taken in reverse order. Inverses can be unproblematically defined for all instance-level relations. What, then, of inverses for class-level relations? The inverse relation for is_a can be defined trivially as follows: A has_subclass B = [definition] B is_a A. For the remaining class-level relations on our list, in contrast, the issue of corresponding inverses is more problematic [7]. Thus, while we have the true relational assertion human testis part_of human which means that all instances of human testis are part of instances of some human there is no corresponding true relational assertion linking instances of human to instances of human testis as their parts. For these remaining relations we need to work not with inverses but rather with what, following GALEN, we can call reciprocal relations. These are defined using the same family of instance-level primitives we introduced earlier. As reciprocal relations for the two varieties of part_of we have: C has_part C1 = [definition] for all c, t, if Cct then there is some c1 such that C1c1t and c1 part_of c at t P has_part P1 = [definition] for all p, if Pp then there is some p1 such that P1p1 and p1 part_of p Note that from A part_of B we cannot infer that B has_ part A; similarly, from A has_ part B we cannot infer that B part_of A. Thus cell nucleus part_of cell, but not cell has_part cell nucleus; running has_ part breathing, but not breathing part_of running. A third significant relation conjoining part_of and has_part can be defined as [6,30]: C integral_part_of C1 = [definition] C part_of C1 and C1 has_part C. For contained_in we have similarly the reciprocal relation: C contains C1 = [definition] for all C, t, if Cct then there is some c1 such that: C1c1t and c1 contained_in c at t For participation we can usefully define two alternative reciprocal relations: C sometimes_ participates_in P = [definition] for all c there is some t and some p such that Cct and Pp and p has_participant c at t C always_participates_in P = [definition] for all c, t, if Cct then there is some p such that Pp and p has_participant c at t We can also define, for example, what it is for continuants of a given type to participate at every stage in a process of a given type. Thus if a sperm participates in the penetration of an ovum, then it does so throughout the penetration. Types of relational assertions In light of the above, we can now observe certain differences in what we might call the relative universality of class-level relational assertions. There are many cases, above all involving is_a relations, where relational assertions hold with a maximal degree of universality, which means that they hold for every instance of the classes in question because they are a matter of analytic connections, that is, connections resting on the compositional nature of the class terms involved [10], as, for example, in: eukaryotic cell is_a cell, or adult walking behavior has_participant adult. (Contrast, adult participates_in adult walking behavior.) There are also other kinds of statements enjoying a high degree of universality, for example: penetration of ovum has_participant sperm. The first of our two corresponding reciprocal statements sperm participates_in penetration of ovum is in contrast true only in relation to certain isolated instances of sperm, and the second of our reciprocal statements sperm always_participates_in penetration of ovum is true in relation to no instances at all.Genome Biology 2005, 6:R46 R46.12 Genome Biology 2005, Volume 6, Issue 5, Article R46 Smith et al. http://genomebiology.com/2005/6/5/R46Table 2 Definitions and examples of class-level relations Relations and relata Definitions Examples C is_a C1; Cs and C1s are continuants Every C at any time is at the same time a C1 myelin is_a lipoprotein serotonin is_a biogenic amine mitochondrion is_a membranous cytoplasmic organelle protein kinase is_a kinase DNA is_a nucleic acid P is_a P1; Ps and P1s are processes Every P is a P1 endomitosos is_a DNA replication catabolic process is_a metabolic process photosynthesis is_a physiological process gonad development is_a organogenesis intracellular signaling cascade is_a signal transduction C part_of C1; Cs and C1s are continuants Every C at any time is part of some C1 at the same time mitochondrial matrix part_of mitochondrion microtubule part_of cytoskeleton nuclear pore complex part_of nuclear membrane nucleoplasm part_of nucleus promotor part_of gene P part_of P1; Ps and P1s are processes Every P is part of some P1 gastrulation part_of embryonic development cystoblast cell division part_of germ cell development cytokinesis part_of cell proliferation transcription part_of gene expression neurotransmitter release part_of synaptic transmission C located_in C1; Cs and C1s are continuants Every C at any given time occupies a spatial region which is part of the region occupied by some C1 at the same time 66s pre-ribosome located_in nucleolus intron located_in gene nucleolus located_in nucleus membrane receptor located_in cell membrane chlorophyll located_in thylakoid C contained_in C1; Cs are material continuants, C1s are immaterial continuants (holes, cavities) Every C at any given time is located in but shares no parts in common with some C1 at the same time thoracic aorta contained_in posterior mediastinal cavity cytosol contained_in cell compartment space thylakoid contained_in chloroplast membrane synaptic vesicle contained_in neuron C adjacent_to C1; Cs and C1s are continuants Every C at any time is proximate to some C1 at the same time Golgi apparatus adjacent_to endoplasmic reticulum intron adjacent_to exon cell wall adjacent_to cytoplasm periplasm adjacent_to plasma membrane presynaptic membrane adjacent_to synaptic cleft C transformation_of C1; Cs and C1s are material continuants Every C at any time is identical with some C1 at some earlier time facultative heterochromatin transformation_of euchromatin mature mRNA transformation_of pre-mRNA hemosiderin transformation_of hemoglobin red blood cell transformation_of reticulocyte fetus transformation_of embryoGenome Biology 2005, 6:R46 http://genomebiology.com/2005/6/5/R46 Genome Biology 2005, Volume 6, Issue 5, Article R46 Smith et al. R46.13 co m m ent review s repo rts refereed research depo sited research interactio ns info rm atio n It then seems reasonable to insist that biomedical ontologies should reflect those sorts of biological assertions that enjoy a high degree of universality (typically assertions involving just one of each pair of reciprocal relations). Tools for ontology curation We hope that, by providing clear and unambiguous specifications of what the class-level relational expressions used in biological ontologies mean, our formal definitions will assist curators engaged in ontology creation and maintenance. The corresponding definitions are summarized in Table 2, which also contains representative examples for each of the relations distinguished. Our definitions are designed to ensure that the corresponding general-purpose relational expressions are used in a uniform way in all biological ontologies. In this way we shall be in a position to contribute to the realization of the goal of bringing about a high degree of interoperability even where ontologies are produced by different groups and for different purposes. These definitions are designed also to enable the automatic detection of errors in biomedical ontologies, for example by allowing the construction of extensions of OBO-Edit and similar tools with the facility to test whether given relations are employed in an ontology in such a way as to involve relata of the appropriate types [31] or in such a way as to have the formal characteristics, such as transitivity or reflexivity, dictated by the definitions (Table 3). The framework can also support reasoning applications designed to enable the automated derivation of information from existing bodies of knowledge for example to infer the parts of a given cell continuant via the traversal of a part_of hierarchy including instance-based knowledge derived from the clinical record. Conclusion The Relation Ontology outlined above arose through collaboration between formal ontologists and biologists in the OBO, FMA and GALEN research groups and also incorporates suggestions from a number of other authors and curators of biomedical ontologies. It is designed to be large enough to overcome some of the problems arising in GO and similar systems as a result of the paucity of resources available hitherto for expressing relations between the classes in such ontoloC derives_from C1; Cs and C1s are material continuants Every C is such that in the first moment of its existence it occupies a spatial region which overlaps the spatial region occupied by some C1 in the last moment of its existence plasma cell derives_from B lymphocyte fatty acid derives_from triglyceride triple oxygen molecule derives_from oxygen molecule Barr body derives_from X-chromosome mammal derives_from gamete P preceded_by P1; Ps and P1s are processes Every P is such that there is some earlier P1 translation preceded_by transcription meiosis preceded_by chromosome duplication cytokinesis preceded_by DNA replication apoptotic cell death preceded_by nuclear chromatin degradation digestion preceded_by ingestion P has_participant C; Ps are processes, Cs are continuants Every P involves some C as participant mitochondrial acetylCoA formation has_participant pyruvate dehydrogenase complex translation has_participant amino acid photosynthesis has_participant chlorophyll apoptosis has_participant cell cell division has_participant chromosome P has_agent C; Ps are processes, Cs are material continuants Every P involves some C as agent (the C is involved in and is causally responsible for the P) gene expression has_agent RNA polymerase signal transduction has_agent receptor pathogenesis has_agent pathogen transcription has_agent RNA polymerase translation has_agent ribosome Table 2 (Continued) Definitions and examples of class-level relationsGenome Biology 2005, 6:R46 R46.14 Genome Biology 2005, Volume 6, Issue 5, Article R46 Smith et al. http://genomebiology.com/2005/6/5/R46gies [32]. It is this paucity of resources, above all, which gives rise to cases of multiple inheritance in GO as presently constructed, and we note here that multiple inheritance often goes hand in hand with errors in ontology construction not least because it encourages a relaxed reading of is_a (often a reading which involves the assertion of is_a relations which erroneously cross the divide between different ontological categories) [5,33]. Our present framework can contribute to error resolution not only by dictating a common interpretation of is_a which can serve as orientation for ontology authors and curators in their future work, but also by providing richer resources for the assertion of class-class relations within and between ontologies in such a way that the appeal to contrived and error-prone is_a relations can be more easily avoided. At the same time our suite of relations has been designed to be sufficiently small to attract wide acceptance in a range of different types of life-science communities. Where the latter use further, general-purpose or domain-specific relations of their own, we plan in due course to subject such relations to the same kind of analysis as presented here in order to preserve interoperability. The Relation Ontology has been incorporated into the OBO ontology library [34] and curators of the GO and FMA ontologies and also of the ChEBI chemical entities vocabulary [35] are already applying the relevant parts of the ontology in their work. The ontology has already been used to find errors not only in GO but also in SNOMED [36]. It is also being applied systematically in evaluations of the NCI Thesaurus [37] and the UMLS (Unified Medical Language System) Semantic Network of the National Library of Medicine. We are currently testing methodologies to obtain reliable quantitative evaluations of the utility of the proposed framework for purposes of ontology authoring and also for use in annotation and reasoning. We are also testing ways in which the framework can be expanded through the admission of pre-coordinated disjunctions (for example: either derivation or transformation), which can allow the coding of information in those cases where the precise nature of the relations involved is insufficiently clear to allow unique assignment. The Relation Ontology will be evaluated on two levels. First, on whether it succeeds in preventing those characteristic kinds of errors which have been associated with a poor treatment of relations in biomedical ontologies in the past. Second, and more important, on whether it helps to achieve greater interoperability of biomedical ontologies and thus to improve reasoning about biological phenomena. Acknowledgements Work on this paper was carried out under the auspices of the Wolfgang Paul Program of the Alexander von Humboldt Foundation, the EU Network of Excellence in Medical Informatics and Semantic Data Mining, the Project 'Forms of Life' sponsored by the Volkswagen Foundation, and the DARPA Virtual Soldier Project. Thanks go to Michael Ashburner, Fabrice Correia, Maureen Donnelly, Kai Hauser, Win Hyde, Ingvar Johansson, Janet Kelso, Suzanna Lewis, Katherine Munn, Maria Reicher, Alan Ruttenberg, Mark Scala, Stefan Schulz, Neil Williams, Lina Yip, Sumi Yoshikawa, and anonymous referees for valuable comments. References 1. OBO: Open Biomedical Ontologies [http://obo.source forge.net] 2. Mungall C: OBOL: integrating language and meaning in bioontologies. Comp Funct Genomics 2004, 5:509-520. 3. Gene Ontology Consortium: Creating the Gene Ontology resource: design and implementation. Genome Res 2001, 11:1425-1433. 4. Bada M, Stevens R, Goble C, Gil Y, Ashburner M, Blake JA, Cherry JM, Harris M, Lewis S: A short study on the success of the GeneOntology. J Web Semantics 2004, 1:235-240. 5. Smith B, Köhler J, Kumar A: On the application of formal principles to life science data: a case study in the Gene Ontology. DILS 2004: Data Integration in the Life Sciences. Lecture Notes in Computer Science 2994 2004:124-139. 6. Smith B, Rosse C: The role of foundational relations in the alignment of biomedical ontologies. In Proceedings Medinf 2004 Amsterdam: IOS Press; 2004:444-448. 7. Smith B, Kumar A: On controlled vocabularies in bioinformatics: a case study in the Gene Ontology. BioSilico: Inform Technol Drug Discovery 2004, 2:246-252. 8. Smith B, Williams J, Schulze-Kremer S: The ontology of the Gene Ontology. Proc AMIA Symp 2003:609-13. 9. Ogren PV, Cohen KB, Acquaah-Mensah GK, Eberlein J, Hunter L: The compositional structure of Gene Ontology terms. Pac Symp Biocomput 2004:214-225. 10. Ogren P, Bretonnel Cohen K, Hunter L: Implications of compositionality in the Gene Ontology for its curation and usage. Pac Symp Biocomput 2005:174-185. 11. Bard J, Rhee SY, Ashburner M: An ontology for cell types. Genome Biol 2005, 6:R21. 12. Wroe C, Stevens R, Goble CA, Ashburner M: An evolutionary methodology to migrate the Gene Ontology to a Description Logic environment using DAML+OIL. Pac Symp Biocomput 2003:624-635. 13. Smith B: Beyond concepts: ontology as reality representation. In Formal Ontology and Information Systems 2004 Amsterdam: IOS Press; 2004:73-84. 14. Levesque HJ, Brachman RJ: A fundamental tradeoff in knowledge representation and reasoning. In Readings in Knowledge Representation San Francisco: Morgan Kaufman; 1985:41-70. 15. Rogers J, Rector AL: The GALEN ontology. In Medical Informatics Europe 1996 Amsterdam: IOS Press; 1996:174-178. Table 3 Some properties of the relations in the OBO Relation Ontology Relation Transitive Symmetric Reflexive Antisymmetric is_a + - + + part_of + - + + located_in + - + contained_in - - - adjacent_to - - - transformation_of + - - derives_ from + - - preceded_by + - - has_participant - - - has_agent - - - -Genome Biology 2005, 6:R46 http://genomebiology.com/2005/6/5/R46 Genome Biology 2005, Volume 6, Issue 5, Article R46 Smith et al. R46.15 co m m ent review s repo rts refereed research depo sited research interactio ns info rm atio n 16. Grenon P, Smith B, Goldberg L: Biodynamic ontology: applying BFO in the biomedical domain. In Ontologies in Medicine Amsterdam: IOS Press; 2004:20-38. 17. Stoll R: Set Theory and Logic New York: Dover Publications; 1979. 18. Casati R, Varzi AC: Holes and Other Superficialities Cambridge, MA: MIT Press; 1994. 19. Rosse C, Mejino JLV Jr: A reference ontology for bioinformatics: the Foundational Model of Anatomy. J Biomed Inform 2003, 36:478-500. 20. Rogers J, Rector AL: GALEN's model of parts and wholes: experience and comparisons. In Proceedings AMIA Symposium 2000 Bethesda, MD: American Medical Informatics Association; 2000:819-823. 21. Gangemi A, Guarino N, Masolo C, Oltramari A: Sweetening WordNet with DOLCE. AI Magazine 2003, 24:13-24. 22. Fellbaum C, Ed: Wordnet. An Electronic Lexical Database Cambridge, MA: MIT Press; 1998. 23. Cook DL, Mejino JLV Jr, Rosse C: Evolution of a Foundational Model of Physiology: symbolic representation for functional bioinformatics. In Proceedings MedInfo 2004 Amsterdam: IOS Press; 2004:336-340. 24. Bittner T: Axioms for parthood and containment relations in bio-ontologies. In KR-MED 2004: Workshop on Formal Biomedical Knowledge Representation Aachen: University of Aachen; 2004:4-11. 25. Donnelly M: Layered mereotopology. In Proceedings 18th Joint International Conference on Artificial Intelligence San Francisco: Morgan Kaufman; 2003:1269-1274. 26. Smith B: Mereotopology: a theory of parts and boundaries. Data Knowledge Eng 1996, 20:287-303. 27. Smith B, Brogaard B: Sixteen days. J Med Philos 2003, 28:45-78. 28. Smith B, Grenon P: The cornucopia of formal-ontological relations. Dialectica 2004, 58:279-296. 29. Johansson I, Smith B, Munn K, Tsikolia N, Elsner K, Ernst D, Siebert D: Functional anatomy: a taxonomic proposal. Acta Biotheoret 2005 in press. 30. Schulz S, Hahn U: Towards a computational paradigm for biomedical structure. In KR-MED 2004: Workshop on Formal Biomedical Knowledge Representation Aachen: University of Aachen; 2004:63-71. 31. dos Santos MC, Dhaen C, Fielding M, Ceusters W: Philosophical scrutiny for run-time support of application ontology development. In Formal Ontology and Information Systems Amsterdam: IOS Press; 2004:342-352. 32. Kumar A, Smith B, Borgelt C: Dependence relationships between Gene Ontology terms based on TIGR gene product annotations. In Proceedings CompuTerm 2004 Geneva: COLING; 2004:31-38. 33. Bouaud J, Bachimont B, Charlet J, Zweigenbaum P: Acquisition and structuring of an ontology within conceptual graphs. Proceedings 2nd International Conference on Conceptual Structures: Workshop on Knowledge Acquisition using Conceptual Graph Theory. Lecture Notes Computer Sci 1994, 835:1-25. 34. OBO Relationship Ontology [http://obo.sourceforge.net/rela tionship] 35. ChEBI: Chemical Entities of Biological Interest [http:// www.ebi.ac.uk/chebi] 36. Ceusters W, Smith B, Kumar A, Dhaen C: Ontology-based error detection in SNOMED-CT. In Proceedings Medinfo 2004 Amsterdam: IOS Press; 2004:482-486. 37. Ceusters W, Smith B, Goldberg L: A terminological and ontological analysis of the NCI Thesaurus. Meth Inform Medicine. 2005, in press.Genome Biology 2005, 6:R

teorema Vol. XXXI/2, 2012, pp. 5-19 ISSN: 0210-1602 [BIBLID 0210-1602 (2012) 31:2; pp. 5-19] How Fundamental is the Fundamental Assumption? Nils Kürbis RESUMEN El supuesto fundamental de la justificación de la deducción basada en la teoría de la demostración al que se adhieren Dummett y Prawitz consiste en que "si tenemos un argumento válido para un enunciado complejo, podemos construir para él un argumento válido que termina con una aplicación de una de las reglas de introducción que gobieran su operador principal". Defiendo aquí que este supuesto, en su versión general, es defectuoso y que debería restringirse de manera que no se aplique a argumentos en general, sino sólo a demostraciones. Argumento también que el proyecto de Dummett y Prawitz de proporcionar una base lógica para la metafísica descansa sólo en el argumento restringido. PALABRAS CLAVE: semántica de teoría de la demostración, Michael Dummett, Dag Prawitz, teorías verificacionistas del significado, realismo versus antirrealismo. ABSTRACT The fundamental assumption of Dummett's and Prawitz' proof-theoretic justification of deduction is that 'if we have a valid argument for a complex statement, we can construct a valid argument for it which finishes with an application of one of the introduction rules governing its principal operator'. I argue that the assumption is flawed in this general version, but should be restricted, not to apply to arguments in general, but only to proofs. I also argue that Dummett's and Prawitz' project of providing a logical basis for metaphysics only relies on the restricted assumption. KEYWORDS: Proof-theoretic Semantics, Michael Dummett, Dag Prawitz, Verificationist Theories of Meaning, Realism vs. Anti-Realism. I. INTRODUCTION The fundamental assumption of Dummett's and Prawitz' prooftheoretic justification of deduction has largely been ignored in the literature, even though Dummett and Prawitz assign great importance to it. It originates in Dummett's The Logical Basis of Metaphysics (henceforth LBM). As the 5 6 Nils Kürbis name indicates, Dummett places it at the foundations of the proof-theoretic justification of deduction. Prawitz agrees that it is of central importance to their project, where the meanings of sentences are specified by what count as their direct verifications, so that the meanings of logical constants are specified in terms of their use in arguments: if the fundamental assumption fails, 'it probably means a failure also of verificationism' [Prawitz (1994), p. 375] and 'it is the whole verificationist project that is in danger when the fundamental assumption cannot be upheld' [Prawitz (2006), p. 523)]. Thus either a viable justification of the practice of ignoring the fundamental assumption is mandatory or a revision is called for. The aim of this paper is to justify the practice by arguing that the fundamental assumption should be dropped, as it is flawed. However, I shall argue that this is not a disaster for the prooftheoretic justification of deduction, as the assumption need not be made. As it is hardly ever discussed in the literature, workers in the field seem to agree implicitly with my result, so it is worth making it explicit. I shall first give some background and introduce the fundamental assumption together with the notions of direct and indirect verifications, compositionality and the complexity condition. I'll then give four objections to the fundamental assumption. They are not all wholly original. Some of them can be found in Dummett's writings, whose own examination leaves the fundamental assumption 'very shaky' [LBM p. 277]. Next I'll observe that Dummett and Prawitz make very little use of the assumption in the development of their theory. In particular, it is not used in formulating the notion of harmony and an argument against classical logic can be given solely on the basis of compositionality and the complexity condition. I conclude that the assumption is unnecessary to establish the main conclusions of Dummett's and Prawitz' programme. Finally, I shall argue that this does not matter for Dummett's and Prawitz' larger project of providing a logical basis for metaphysics: for this, only purely logical reasoning needs to be taken into account. Restricted to this realm, something very similar in content to the fundamental assumption reappears as a theorem on the forms of proofs in systems of natural deduction, i.e. as a consequence, not a prerequisite, of the proof-theoretic justification of deduction. II. DUMMETT'S FUNDAMENTAL ASSUMPTION According to Dummett, sentences have direct or canonical as well as indirect verifications. To illustrate, suppose Brutus desires a fig and Porcia knows that he does and Brutus knows that Porcia knows this: then, if Porcia offers to Brutus what he desires, Brutus can either verify directly that Porcia offers him a fig by waiting and seeing what she has for him, or he can verify this indirectly by deducing from his knowledge what Porcia offers him. How Fundamental is the Fundamental Assumption? 7 Direct verifications are linked to core uses of the expressions occurring in the sentence, whereas indirect verifications are further removed from them. '[Indirect verifications] will include deductive arguments involving sentences of an unbounded degree of complexity. It is this that requires us to distinguish between direct and indirect verifications of a statement, or, in mathematics, between canonical proofs and demonstrations of a more general kind. A direct verification of a statement is one which proceeds in accordance with the composition of the sentence by means of which it is expressed [...] When direct verification involves deductive reasoning, this reasoning must always proceed from less complex premises to a more complex conclusion' [LBM p. 229]. Indirect verifications of sentences have to be shown to be valid relative to direct ones: for any indirect verification, there must be a direct one. If this is not the case, it interferes with the partial ordering that dependence of meaning imposes on the language, as then the indirect verification licenses a use of the sentence not in accordance with its place in the ordering. Dependence of meaning is a relation Dummett takes to hold between sets of expressions based on the observation that 'the understanding of a word consists in the ability to understand characteristic members of a particular range of sentences containing that word' [LBM p. 225]. This leads to the principle of compositionality, which 'requires that the relation of dependence between [sets of] expressions and [sets of] sentence-forms be asymmetric' [LBM p. 223]. The qualification 'sets of' is needed because there may be collections of expressions that can only be learned simultaneously - Dummett mentions simple colour words like 'yellow', 'red', 'green', 'blue'. These however must form surveyable sets. Dependence of meaning has to have an end somewhere and can neither proceed ad infinitum nor in a circle. A compositional meaningtheory employs the relation of dependence to impose a partial ordering on the expressions of the language, which exhibits how the language is learnable step by step. It is not easy to give a precise general characterisation of the distinction between direct and indirect verifications. But some such distinction is certainly motivated by compositionality. We may grant at least the following: even though there may not be obvious positive criteria for what counts as a direct verification of a sentence, there are reasonable and workable negative criteria for when something does not. We can agree with Dummett that a very elaborate argument for a very simple sentence, for instance, counts as an indirect verification, even though we may not be so sure what its direct verification is. 'A Case of Identity' in The Adventures of Sherlock Holmes contains a good example. I'm not sure what counts as a direct verification of the claim 'Mr Hosmer Angel is Mr James Windibank in disguise', but Holmes' later explanation to Watson surely does not: '"Well, of course it was obvious from the first that this Mr. Hosmer Angel must have some strong object for his curious conduct [of proposing to Windibank's step-daughter, exacting vows of 8 Nils Kürbis fidelity from her, come what may, and disappearing the morning of the wedding after having alluded that something might happen to him], and it was equally clear that the only man who really profited by the incident, as far as we could see, was the stepfather. Then the fact that the two men were never together, but that the one always appeared when the other was away, was suggestive. So were the tinted spectacles and the curious voice, which both hinted at a disguise, as did the bushy whiskers. My suspicions were all confirmed by his peculiar action in typewriting his signature, which, of course, inferred that his handwriting was so familiar to her that she would recognise even the smallest sample of it. You see, all these isolated facts, together with many minor ones, all pointed in the same direction." – "And how did you verify them?" – "Having once spotted my man, it was easy to get corroboration. I knew the firm for which this man worked. Having taken the printed description, I eliminated everything from it which could be the result of a disguise - the whiskers, the glasses, the voice, and I sent it to the firm, with a request that they would inform me whether it answered to the description of any of their travellers. I had already noticed the peculiarities of the typewriter, and I wrote to the man himself at his business address asking him if he would come here. As I expected, his reply was typewritten and revealed the same trivial but characteristic defects. The same post brought me a letter from Westhouse & Marbank, of Fenchurch Street, to say that the description tallied in every respect with that of their employee, James Windibank. Voilà tout!"' The verification can only be indirect, as understanding the concept of being in disguise it is not necessary to know anything about family relations, typewriters or wine merchants of Fenchurch Street. Be that as it may, the difficulties surrounding the general case need not deter Dummett in the proof-theoretic justification of deduction. Restricted to the special case of the logical constants, the distinction between direct and indirect verification seems reasonably clear: 'the introduction rules for a logical constant c represent the direct or canonical means of establishing the truth of a sentence with principal operator c' [LBM p. 252]; verifications proceeding otherwise are indirect. The fundamental assumption now is that 'if we have a valid argument for a complex statement, we can construct a valid argument for it which finishes with an application of one of the introduction rules governing its principal operator' [LBM p. 254]. Employing the distinction between direct and indirect verifications as applied to logical constants, it says that for every indirect verification of a sentence !AB we can find a direct one. Accordingly, Dummett presents the fundamental assumption as a requirement following from the thesis that the introduction rules for a logical constant define its meaning. Another motivation for making the fundamental assumption is a relation between it, compositionality and the complexity condition. 'The minimal demand we should make on an introduction rule intended to be self-justifying How Fundamental is the Fundamental Assumption? 9 is that its form be such as to guarantee that, in any application of it, the conclusion will be of higher logical complexity than any of the premises and than any discharged hypothesis' [LBM p. 258]. The complexity condition ensures that the compositionality of the logic-free fragment of the language is kept intact after adding the logical constants. Thus, given an argument for A, by the fundamental assumption there always is an argument for A that proceeds according to its composition in a trivial, meaning-theoretically uninteresting sense––meaning no more than that it proceeds according to the way A is composed from its sub-sentential expressions. But this does not exclude that sentences are invoked in the argument that are of larger complexity than A –– the fundamental assumption alone does not exclude introduction rules from having premises more complex than the conclusion. However, if the rules governing the logical constants satisfy the complexity condition, there is a verification that proceeds according to its composition in a substantial sense: this argument for A does not invoke sentences of higher complexity than A in any of its subarguments. I need to dispose of a possible misunderstanding. It might be objected that the fundamental assumption is ridiculous, if it is an assumption about the form of deductions, as it would then entail that if " #– A $ B, then " #– A or " #– B. For if there is a deduction of A $ B from ", then, by the fundamental assumption, there is a deduction of A $ B from " which ends with a step by disjunction introduction. This rule has two forms: 'from A to infer A $ B' and 'from B to infer A $ B'. Hence by removing the last step from the deduction, either a deduction of A or a deduction of B from " could be constructed. But this cannot be correct. For there is a deduction of A $ B from A $ B, and there should be neither a deduction of A from A $ B nor one of B from A $ B. So the fundamental assumption is absurd. The misunderstanding is that the fundamental assumption is not applied to deductions, but to supplementations of arguments which are 'the result of replacing any complex initial premise by a canonical (sub)argument having that premise as its final conclusion' [LBM p. 255]. Here 'argument' is a notion slightly more general than 'deduction' as ordinarily used. The details need not concern us here, but arguments may contain steps by 'boundary rules' which allow the derivation of atomic sentences from other atomic sentences [LBM pp. 254f]. It is worth noting at this point, though, that two versions of the fundamental assumption can be extracted from Dummett's writings. Initially Dummett introduces the version given a few paragraphs earlier. Soon afterwards, however, he writes that 'the fundamental assumption is that, whenever we are entitled to assert a complex statement, we could have arrived at it by means of an argument terminating with at least one of the introduction rules governing its principal operator' [LBM p. 257], which seems to restrict it to arguments with premises we know to be true. Employing this second version of the fundamental assumption, what counts as an 10 Nils Kürbis 'argument' has to be compared to #– A rather than to " #– A. Although this may just be a slip of the pen, I shall suggest at the end of this paper that it is best to apply the fundamental assumption only to proofs of theorems, rather than deductions in general. When correctly applied, the fundamental assumption does not lead to logical absurdities. There are, however, other, substantial reasons against making it, as I shall show in the next section. III. FOUR OBJECTIONS TO THE FUNDAMENTAL ASSUMPTION 1. As the fundamental assumption favours introduction rules as giving the meanings of logical constants, applied to the case of disjunction this introduces a genuinely anti-realist thought: that a disjunction can only be asserted if we could also assert one of its disjuncts. This should be an illegitimate move if the aim is to establish which logic is the correct one independently of any prior postulations in favour of realism or anti-realism. Dummett concedes that for some constants it is more natural to view their meanings as determined by their elimination rules –– arguably, disjunction is one of them. So the neutral approach is to reserve judgement as long as possible regarding the question whether it is the introduction or the elimination rules which define the meaning of a constant and to see whether neutral grounds can be given to decide this question. Dummett has not formulated a suitable converse of the fundamental assumption in case it is the elimination rules that determine the meanings of logical constants. Although he acknowledges this to be an option, Dummett does not consider it in detail, and what he has to say about such an approach is much less specific than the development of his favoured one. It is even less clear how the fundamental assumption and its converse would have to be applied in case we adopt an approach in which the meanings of some constants are determined by their introduction rules and of others by their elimination rules, but again, Dummett does not exclude this option. 2. There is a problem with the generality of the fundamental assumption. The argument for why it is considered necessary if the meanings of the logical constants are defined by their introduction rules is the following. Suppose a sentence !AB cannot be verified by the application of an introduction rule for !, but only by the application of an elimination rule for another logical constant %. In order to command a full understanding of ! – to be able to make all the uses of it one is entitled to make – the speaker has to understand %. Hence the meaning of !AB is not determined completely by the meanings How Fundamental is the Fundamental Assumption? 11 of A, B and the introduction rule for !, which accordingly does not determine the meaning of ! completely. This argument is cogent if A and B are atomic sentences. It also generalises to complex A and B if % does not occur in them. It is not plausible, however, that it generalises any further than that. If % occurs in !AB, then a specification of the meaning of !AB appeals to a specification of the meaning of % and understanding !AB requires an understanding of %. But then a verification of !AB ending with an application of an elimination rule for % would only employ the conceptual resources already required for understanding !AB. Understanding a logical constant is a general capacity. So if I understand a logical constant, I can be expected to apply the rules governing it in all cases where I also understand all the other expressions occurring in the sentences it connects. Thus the partial ordering dependence of meaning imposes on the language would not be endangered, if, by hypothesis, I understand all the subsentential expressions of !AB, even if its canonical verification ended with an application of an elimination rule for %. If this correct, then the fundamental assumption is badly motivated because it is too general. 3. The fundamental assumption is only as clear as is the distinction between direct and indirect verifications. At first glance, in contrast to its application to other expressions of the language, the distinction seems admirably clear when applied to the logical constants. But Dummett concedes that we are sometimes justified in asserting A $ B without having any means of retrieving which of A or B is assertible. Dummett gives two examples. We may be able to infer 'That is either a boy or a girl over there' from 'That is a child over there', where 'the disjunctive conclusion was not arrived at by "or"introduction, and may well not have been able to be on the basis of the observation actually made. [...] Hardy may simply not have been able to hear whether Nelson said, "Kismet, Hardy" or "Kiss me, Hardy", though he heard him say one or the other: once we have the concept of disjunction, our perceptions themselves may assume an irremediably disjunctive form' [LBM p. 267]. Thus, not only does Dummett concede that we may derive disjunctive sentences without applying disjunction introduction, he goes as far as to acknowledge that it is possible to verify a disjunction directly without verifying either of its disjuncts. Dummett does not discuss any other logical constants in this context, but something analogous may plausibly be the case with negation: can't I infer 'This is not red' from 'This is green' or perceive directly that the wine is not in the fridge without having to derive an absurdity from contrary assumptions? The upshot of this is that it is not the case that, for any logical constant, every direct verification of a sentence A&B needs to end with the application of an introduction rule for &. This makes the distinction between direct and 12 Nils Kürbis indirect verifications problematic even in the special case of the logical constants, and obviously this undermines the fundamental assumption: if it is not the case that only those verifications of a complex sentence count as direct which end with an application of an introduction rule for its principal connective, then the fundamental assumption is false. Dummett's response to this problem is to claim that we are not concerned with actual verifications, but only with potential ones by a suitably placed observer. This response leads to my next objection to the fundamental assumption. 4. According to Dummett, the fundamental assumption needs to be interpreted in a suitable way. He concedes that 'what underpin the fundamental assumption are considerations that are not themselves proof-theoretic but are in a broad sense semantic: we are driven to invoke some notion of truth' [LBM p. 269]. Thus the fundamental assumption leads away from a purely proof-theoretic justification of deduction, which is not a bad thing, as it is implausible that no notion of truth enters it at all. Dummett needs to ensure that even if we are not actually in a position to proceed according to the fundamental assumption, an argument with a complex conclusion may nonetheless be valid. So the fundamental assumption needs to be interpreted in a suitable way: every complex sentence could have been verified by an application of an introduction rule for its principal connective by a suitably placed observer, who then could have produced a canonical argument for the sentence in question, even though we are not actually in a position to do so. In general, such an observer need not only be suitably placed, but also endowed with suitable powers, as such an observer would have to be able to, for instance, decide whether a very large number is either odd or even, so as to be able to produce a canonical argument for the statement that it is. Let's call such an observer 'ideal'. Then, if an argument is valid, an ideal observer could have produced a canonical argument for its conclusion. A realist, who accepts classical logic, would have to hold that an ideal observer can verify every proposition or its negation. But according to Dummett, this cannot be justified merely by appeal to the powers of an ideal observer. The laws validated by the proof-theoretic justification of deduction 'remain invariant under considerable variation in the interpretation of the fundamental assumption, because it will still serve to validate them by the proof-theoretic justification procedure, without the need for further assumptions. When the strong realist interpretation is adopted, however, the situation changes: not all laws can any longer be validated by proof-theoretic means, because their validity depends not only on the fundamental assumption but on the further assumption of bivalence' [LBM p. 271]. Whether a proposition can be verified does not depend on anyone's capacities, but only on the proposition and what it is about. Take Dummett's example of Jones, who's How Fundamental is the Fundamental Assumption? 13 dead now and has never been in a situation where he could have shown bravery. If we don't assume bivalence, we have no reason to assume that even the ideal observer will come up with an answer to the question whether Jones was brave. If Jones' actions leave that question open, we have no reason to believe that 'Jones is brave' is determinately either true or false, unless we assume the principle of bivalence. Similarly for Goldbach's Conjecture, for instance. Unless we accept bivalence, why should we exclude the option that mathematical reality leaves it undetermined whether every even number is the sum of two primes? Thus, even if we introduce the notion of an ideal observer to give a suitable interpretation of the fundamental assumption, according to Dummett the proof-theoretic justification of deduction still does not validate classical logic. This line of argument, however, is dependent on what capacities we allow the ideal observer to have. Classicists and intuitionists agree that there are only two truth values, i.e. true or false. They disagree over whether every proposition determinately is one of the two. We can follow Dummett and exclude the option of there being more than two truth values as of relatively minor significance: 'A meaning-theory which substitutes, for the two-valued semantics, a finitely many-valued one represents a very trivial variation of this: we have merely been provided with a slightly more complicated mechanism for determining the truth or otherwise of a complex sentence in accordance with its composition from the subsentences. In such a semantic theory, truth, as we have been using this notion, corresponds to having a designated value' [LBM p. 305]. The difference between realists and anti-realists relies on there being propositions to which we are in no position to attach a truth value. For the anti-realist, the world is underdetermined: there are, in a sense, gaps in reality. But we are in no position to recognize any gaps: to assume that we can implies a contradiction, because we would then be in a position to recognise that it can never be verified and never be falsified, in which case it would be neither true nor false, which is impossible. All we can do is note that, as things stand, certain propositions have as yet neither been verified nor falsified. We cannot exclude the possibility that we will be in a position to decide these propositions. The question now arises what the ideal observer is able to do with a proposition to which we cannot attach a truth value. Assuming that no proposition can be neither true nor false, he wouldn't be able to recognise a gap in reality either. But why should we go for the option that the ideal observer is in the same position that we are in? In the case of Goldbach's Conjecture, why can't we allow the ideal observer to be able to complete the infinite task of checking each even number whether it is the sum of two primes? In the Jones case, why can't we allow the ideal observer to have scientia media or sufficient powers of counterfactual reasoning to be able to determine what Jones would have done had he been in a situation where he could have acted bravely? A realist may not have a problem with attributing 14 Nils Kürbis such powers to the ideal observer; the anti-realist obviously would. But how are we to decide the question which position to take? It seems as if each position favours an account of the powers of the ideal observer suitable to its metaphysics. Put slightly differently, both the realist and the anti-realist agree that what the ideal observer could have verified extends our own recognitional capacities ad infinitum. But they disagree over what this means. The anti-realist insists we restrict ourselves to the potential infinite: the ideal observer's capacities extend our actual capacities in an arbitrarily large finite way, so that even the ideal observer cannot complete infinite tasks. The realist insists that we can allow the extension of our capacities by an actually infinite amount, so that the ideal observer can complete infinite tasks. The antirealist will say that we cannot conceive of an actual infinite, the realist will say that we can. But the proof-theoretic justification of logical laws is not the place to decide that question, and so introducing the ideal observer doesn't help. It is worth noting that in a later book, Dummett entertains the possibility that intuitionist logic is, as it were, the logic of our limitations: 'we cannot consistently envisage there being any such gap in a particular case; this would be to envisage a proposition's being neither true nor false, and this would be a contradictory supposition. [...] God can know where a gap in reality occurs, by knowing neither the truth nor the falsity of some proposition; he has available to him a negation which is not available to us. It might therefore be urged that the logic of God's thought, a logic to whose application we cannot attain, is a three-valued, rather than a classical, one' [Dummett (2004), p.96]. Thus, if we allow the ideal observer to have the powers of verification Dummett here attributes to God, and we stick to the assumption that no proposition is neither true nor false, it follows that every proposition is either true or false. It is not so much the principle of bivalence that is at issue, rather, it is the principle both classicists and intuitionists agree on, that no sentence can be neither true nor false, plus which capacities of surveying what can be verified we can attribute to the ideal observer. The appeal to an ideal observer in order to interpret the fundamental assumption, then, is problematic. Realists and anti-realists will assume the ideal observer to have capacities suitable to their respective notions of truth, i.e. metaphysics. We have not been given any criteria for deciding which notion of an ideal observer is the correct one. Evoking the ideal observer in the interpretation of the fundamental assumption results in placing metaphysical and epistemological issues at the foundations of the proof-theoretic justification of deduction, which, as the aim is to provide a logical basis for metaphysics, is detrimental to the project. The fact that the fundamental assumption stands in need of interpretation, and the kind of interpretation required according to Dummett, makes it counterproductive, given the larger aim of Dummett's project. How Fundamental is the Fundamental Assumption? 15 IV. DO WE NEED THE FUNDAMENTAL ASSUMPTION? Thus, as the fundamental assumption favours introduction rules, it introduces anti-realist prejudices; it is too general to be plausible; Dummett himself admits that it is possible to know that a disjunction is true without being in a position to verify either disjunct; and the appeal to an ideal observer to remedy the last point is not workable as classical and intuitionist logicians won't agree on the capacity of such an observer. On the one hand, Dummett considers the fundamental assumption to be a prerequisite of the prooftheoretic justification of deduction, and he gives an initial definition of the notion of valid canonical argument on the basis of it [LBM pp. 259ff]. On the other hand, Dummett admits that the fundamental assumption points away from a purely proof-theoretic justification of deduction, and he is forced to relax the requirements of his definition of validity, as the fundamental assumption is strictly speaking false when applied to some constants [LBM pp. 265ff]. Hence it is worth asking if the proof-theoretic justification of deduction can do without the fundamental assumption. In fact, the fundamental assumption has not attracted much attention in the literature, which suggests that the interest of the proof-theoretic justification of deduction is independent of it. I think this is so for the following reasons. The proof-theoretic justification of deduction consists of a formal project with a philosophical motivation. The formal project is to establish the normalisation of deductions: it is a requirement on a proof-theoretically justified logic that its deductions normalise. But normalisation proofs are independent of the fundamental assumption, because it does not introduce any new formal concepts and does not impose any restrictions on the forms of rules of inference. Although Prawitz thinks that the fundamental assumption is important philosophically, there is no formal equivalent of it in his work on normalization of proofs [Prawitz (1965)] and the fundamental assumption is not appealed to there. The philosophical motivation of normalisability is the demand that the rules of inference governing the logical constants be in harmony, i.e. that the grounds for asserting a proposition are in harmony with the consequences of accepting it. This requirement has the effect of excluding connectives like Prior's tonk, but is of more general importance. The motivation for demanding that deductions normalise and that detours through maximal formulas may be removed is the thought that applying logic to premises to derive conclusions should not result in more information than we already had, and all information contained in the premises should remain extractable from the conclusions drawn. As Dummett puts it, 'the requirement that this criterion for harmony be satisfied conforms to our fundamental conception of what deductive inference accomplishes. An argument or proof convinces us because we construe it as showing that, given the premises hold good according to our ordinary criteria, the conclusion must also hold ac16 Nils Kürbis cording to the criteria we already have for its holding' [LBM p. 219]. This view of the nature of deduction is again independent of the fundamental assumption. Harmony is a feature of rules of inference. The requirement of harmony imposes restrictions on the form of rules of inference: the introduction and elimination rules for a logical constant must match in such a way that the grounds for asserting a proposition by application of an introduction rule match the consequences that can be drawn from the proposition by an application of an elimination rule. Harmony between introduction and elimination rules is the basis of normalization of proofs, as harmony between introduction and elimination rules is shown to hold by establishing that there are procedures for removing maximal formulas from deductions. Thus this aspect of the proof-theoretic justification of deduction is independent of the fundamental assumption. Contrast the fundamental assumption with the complexity condition. The complexity condition is the formal equivalent of compositionality and imposes restrictions on the forms of rules of inference. Although the rules for classical negation seem to exhibit a somewhat different shape from the intuitionist ones, that does not change the fact that classical logic normalises, if suitably formalised. To exclude classical logic from being a justified logic, some other considerations are needed. This can be achieved by appealing to compositionality. By compositionality, the meaning of '' A is dependent on the meaning of A and negation. It may happen that a sentence of a language where double negation elimination is employed can be verified only via its double negation. In such a case the move from '' A to A would contribute to the meaning of A, because it licenses uses of A not otherwise possible, as ex hypothesi no other verification is available. Hence the meaning of A would depend on the meaning of '' A. This is a circular dependence of meaning and hence, Dummett would claim, A cannot have a stable meaning at all. A speaker could not break into the circle and learn the meaning of A, which could have no place in the partial ordering that dependence of meaning imposes on the language. Of course, classical logic can be formulated with rules other than double negation elimination. But any such formulation will violate either the complexity condition or the restriction motivated by compositionality that there should be no circular dependence of meaning amongst the logical constants. Besides, it will remain the case that the verification of A will somehow make use of ' A, hence the meaning of A depends on the meaning of ' A, which in turn depends on the meaning of ' and A, so that the circular dependence of meaning has not been avoided. For the argument to go through it is sufficient that double negation elimination is assumed to be valid, no matter whether it is a derived or primitive rule of inference. Again, the fundamental assumption has not been appealed to. Compositionality suffices. So the major result of the proof-theoretic justification of deduction, that How Fundamental is the Fundamental Assumption? 17 only intuitionist, but not classical logic, turns out to be justified can be established without appeal to the fundamental assumption.1 V. TOWARDS A LOGICAL BASIS OF METAPHYSICS WITHOUT THE FUNDAMENTAL ASSUMPTION According to Dummett, objections 3 and 4 arise when the fundamental assumption is applied to arguments in ordinary discourse for empirical propositions. The problems do not arise if applied to purely logical reasoning, i.e. proofs of theorems, because in this case it can be proved that something that looks very much like the fundamental assumption holds for suitably formulated systems: #– A&B if and only if there is a proof of A&B that ends with an application of an introduction rule for &. This holds, for instance, for the systems of intuitionist and classical logic in [Prawitz (1965)]. Furthermore, this proof will only use rules of inference for connectives contained in A&B, so that objection 2 cannot arise either. Of course, this is not the fundamental assumption, as it is not an assumption at all. Rather, it is a consequence of a suitably formulated logic. It is also possible to formulate a converse of the fundamental assumption in case it is the elimination rules that specify the meanings of the logical constants. In that case, we should be interested in showing whether something is a contradiction, i.e. not in the case where something follows from the empty set, #– A, but rather in the case where the empty set follows from something, A #– , or rather A #– (, as this is what corresponds in a calculus of natural deduction to the A #– of a sequent calculus. We can call such a construction a refutation of a contradiction. In a suitably formulated system of classical or intuitionist logic, it can be proved that A #– ( if and only if there is a proof in which begins with an application of an elimination rule to A. Dummett thinks that it is necessary to make the fundamental assumption and apply it to arguments in general, not just proofs, as it is arguments for empirical propositions that we are normally engaged in in everyday reasoning. Dummett is concerned about the actual use speakers make of expressions, including the logical ones. Intuitionist implication and disjunction, for instance, do not accord well with the use of 'or' and 'if then' in ordinary discourse. We don't assert 'If A then B' just in case we are in possession of an argument that shows how an argument for B may be constructed from an argument for A, as reflection on, e.g., Dummett's own example 'If you enter this room, you'll die before nightfall' shows. Similarly we don't just assert 'A or B' if we are in possession of a way of finding either an argument for A or one for B. To match the use of English 'or' and 'if then' with the use of intuitionist $ and ), the fundamental assumption needs to be interpreted in the 18 Nils Kürbis way that gives rise to objections 3 and 4. It is, however, plausible that this concern with the use of 'or' and 'if then' in ordinary discourse is not well motivated. It may of course be an interesting question in how far the meanings of the logical constants of a system of natural deduction reflect the meanings of 'or', 'all', 'if-then' in natural language. But it is plausible that we do not need to take these complications into account: if the thesis is that the meanings of the logical constants are completely specified by the rules of inference governing them, then they may be given from scratch by those rules of inference. The logical constants of formal systems do not need to match up with any expressions of natural language at all. If the meanings of the logical symbols are given by their rules of inference, we do not need to look at ordinary discourse for a specification of their meaning and use: the rules are enough. The complications with the fundamental assumption arise from taking into account arguments for empirical propositions. But metaphysics is not an empirical subject and it is not one that needs to be tied to everyday, ordinary reasoning. Of course, it would be desirable if metaphysics accords with untutored intuitions. But Dummett is a revisionist. He countenances the possibility that metaphysics will go against initial intuitions. Despite his revisionism, metaphysics should accord with untutored intuition, where there is no other basis for explaining what the terms of the metaphysician mean. For instance, if the metaphysician uses primitive terms in his theory that cannot be defined and have no analogue in ordinary language, the charge that we cannot understand him seems justified. But this problem cannot possibly arise in the case of a logical basis of metaphysics: the primitives, i.e. the logical constants, are defined entirely in terms of rules of inference. To provide a logical basis for metaphysics, it should suffice to account for purely logical reasoning: the complications arising from non-empirical reasoning are irrelevant to the project. The aim is achieved by the demand for harmony between introduction and elimination rules, normalisation of proofs and the complexity condition on rules of inference. Indeed, the metaphysical conclusion Dummett derives from his account of the justification of deduction is based on, not that certain inferences derived from some premises are invalid, but that A $ ' A is not a theorem of intuitionist logic: anti-realism corresponds to the failure of that proposition being a theorem. This is not to suggest that all there is to metaphysics is what we can read off the justified logic. If a system of metaphysics were formalised, it would presumably contain axioms of an extra-logical character. Whether the formal results that hold for the logic still hold for the whole system cannot be decided without actually having that system, except that any metaphysical axioms would have to be conservative over the logic. But as the prooftheoretic justification of deduction only gives the logical basis of metaphysics, the broad outline within which metaphysics is to be pursued, the latter requirement suffices to ensure the metaphysics is still anti-realist. How Fundamental is the Fundamental Assumption? 19 If we restrict consideration to purely logical reasoning, the content of the fundamental assumption reappears as something which is neither fundamental nor an assumption, but a theorem about the form of normalised proofs in systems of natural deduction. It is provable that in the case of both, classical and intuitionist logic as formulated by Prawitz, for any theorem, there is a proof of it which ends with an application of an introduction rule for its main connective. So instead of demanding that the fundamental assumption hold for any kind of argument, restricting consideration to purely logical reasoning we can observe that every theorem has a direct verification proceeding in accordance with its composition. This is a result that is formally tractable, as opposed to what Dummett originally had in mind. It is a consequence of the proof-theoretic justification of deduction rather than a precondition for it. But it is plausible that it is all we need if we are interested in a logical basis of metaphysics: for then we only need to take into account purely logical reasoning, and so all we need to take into account are proofs not Dummett's wider notion of an argument and its supplementation.2 Department of Philosophy University of Sheffield 45 Victoria Street, Sheffield, S3 7QB, United Kingdom E-mail: [email protected] NOTES 1 In [Kürbis (n.d.)] I consider possible responses to this argument on behalf of the classicist and the costs at which they come to the proof-theoretic justification of deduction. 2 I would like to thank the two referees for teorema, whose constructive criticisms helped to improve this paper. REFERENCES DUMMETT, M. (1993), The Logical Basis of Metaphysics, Cambridge, MA, Harvard University Press. –– (2004), Truth and the Past, New York, Columbia University Press. KÜRBIS, N. (n.d.), 'What is Wrong with Classical Negation?', unpublished manuscript. PRAWITZ, D. (1965), Natural Deduction, Stockholm, Göteborg, Uppsala, Almquist and Wiksell. –– (1994), 'Review of Michael Dummett: The Logical Basis of Metaphysics', Mind, vol. 103, pp. 373-376. –– (2006), 'Meaning Approached via Proofs', Synthese, vol. 148, pp. 507-524.

Linford, D 2016 Early-Modern Irreligion and Theological Analogy: A Response to Gavin Hyman's A Short History of Atheism. Secularism and Nonreligion, 5: 3, pp. 1–8, DOI: http://dx.doi.org/10.5334/snr.50 * Thomas Nelson Community College, US [email protected] RESEARCH ARTICLE Early-Modern Irreligion and Theological Analogy: A Response to Gavin Hyman's A Short History of Atheism Daniel Linford* Historically, many Christians have understood God's transcendence to imply God's properties categorically differ from any created properties. For multiple historical figures, a problem arose for religious language: how can one talk of God at all if none of our predicates apply to God? What are we to make of creeds and Biblical passages that seem to predicate creaturely properties, such as goodness and wisdom, of God? Thomas Aquinas offered a solution: God is to be spoken of only through analogy (the doctrine of analogy). Gavin Hyman argues Aquinas's doctrine of analogy was neglected prior to the early-modern period and the neglect of analogy produced the conception of a god vulnerable to atheistic arguments. Contra Hyman, in this paper, I show early-modern atheism arose in a theological context in which there was an active debate concerning analogy. Peter Browne (1665–1735) and William King (1650–1729) offered two competing conceptions of analogical predication that were debated through the 19th century, with interlocutors such as the freethinker Anthony Collins (1676–1729), theologian/philosopher George Berkeley (1685–1753), and skeptic David Hume (1711–1776). Lastly, I discuss the 18th century debate over theological analogy as part of the background relevant to understanding Hume's Dialogues Concerning Natural Religion. 1. Introduction According to Thomas Aquinas (1225–1274), God transcends all created categories (Turner 2004, 187–190). The terms native to human languages fail to describe God because human languages developed to discuss matters subsumed by created categories. God can only be referred to by analogy: the doctrine of analogy (herein: DOA) (see, for example, ST I, Q13, A5). Some theologians link a neglect of DOA to modern secularization. For example, Radical Orthodox theologians present a history according to which theological changes after Duns Scotus (1266–1308), in which God's being came to be seen as univocal – as opposed to analogical –with that of creatures, eventually resulted in modern secularization (for an outline of this historical narrative, see chapter 3 in (Smith 2004)). Richard Muller describes a related historico-theological narrative as follows: The Lortz thesis [. . .] claimed [. . .] Luther's theological revolt was precipitated by the decadent theology and philosophy of the later Middle Ages, specifically the developments that took place after Duns Scotus [. . .]. With the Radical Orthodox writers [. . .] the thesis has become more focused on Scotus and his understanding of the univocity of being. According to their theses, Scotus' understanding of univocity created a profound problem, identifying the being of God with the being of creatures but nonetheless placed at an infinite distance from them, undermining traditional teaching concerning divine transcendence. [. . .] This problematic theological and philosophical understanding then carried over wholesale into the Reformation, rendering Protestant theology highly flawed from the outset and, in the version of the thesis espoused by [historian Brad] Gregory, yielding a defective understanding of the relationship of God and world, reason and theology, ultimately bringing about a new and highly secularized worldview as an unintended result of the Reformation (Muller 2012). Or, as James K.A. Smith describes, "Scotus's shift away from a metaphysics of participation to an ontology predicated on the univocity of being rent the cords of suspension that hooked the immanent to the transcendent, the material to the more than material. The result [. . .] was modernity's 'flattened ontology', which eventually issued in nihilism" (Smith 2004, 93). Gavin Hyman takes Radical Orthodoxy's historical explanation of secularization one step further and argues early modern atheism arose because of DOA's prior neglect (Hyman 2007; 2010). Linford: EarlyModern Irreligion and Theological AnalogyArt. 3, page 2 of 8 Other authors agree that the redescription of God as a being ushered in atheism: "The bringing about of God as a being means the bringing about of one who can also be declared to be dead" (Hemming 1999, 95). Still other theologians posit a neglect in God's transcendence, as conceptualized in terms of theological analogy, resulted in an idolatrous conception of God atheists rightfully reject. As Paul Tillich describes, God, properly conceived, transcends the created order of beings and is not a being at all (Tillich 1951, 235–237). Common idolatrous conceptions of God, Tillich argues, were the cause of modern atheism because they cannot serve our existential needs, lead to illiberal politics, and are susceptible to atheistic attacks (Tillich 1952, 182–185). Popular religious apologists, such as Karen Armstrong (2009), follow Tillich in asserting the modern rejection of God was due to the ascendancy of a false idol and the neglect of a pre-modern God concept. For both Hyman and Armstrong, specifically modern God concepts were prerequisite for the rise of New Atheism (e.g. the post-9/11 movement of aggressive atheist authors as typified by authors like Richard Dawkins, Sam Harris, and Christopher Hitchens) over the past two decades.1 For Hyman, DOA's neglect rendered atheism "almost inevitable" (Hyman 2007, 40). According to Hyman, prior to early modernity, God was reconceived as part of the creaturely realm instead of wholly Other: "God is not only liable to appear incredible or unbelievable [. . .] but [. . .] as the world becomes more self-explanatory and self-sufficient, increasingly superfluous". Reconceiving God as part of the creaturely realm, and finding God does not fit the creaturely realm, resulted in atheism (Hyman 2007, 43). In this paper, I put aside philosophical, normative, and theological questions and show, contra Hyman's historical narrative, (a) anglophone theologians prior to and during the eighteenth century did not neglect DOA and (b) at least some of the eighteenth-century thinkers skeptical of religion responded to the debates over theological analogy. Furthermore, eighteenth-century critics of religion incorporated concerns about theological analogy in their irreligious arguments. First, I distinguish between two forms of DOA as they appear in Aquinas's works: the analogy of proportion (AOP1) and the analogy of proportionality (AOP2). Aquinas sided with AOP2. As I show, AOP2 was not neglected in the anglophone context in early modernity. I examine a debate over the nature of DOA and show the debate was directed to and influential on freethinkers and skeptics such as philosopher/freethinker Anthony Collins (1676–1729) and Scottish philosopher David Hume (1711–1776). 2. Aquinas's DOA Aquinas distinguishes three modes of predication: equivocal, univocal, and analogical. Creaturely predicates are those predicates that apply to objects or beings in the created order, i.e., "good", "bad", "blue", "right-handed", "square", and so on. In equivocal predication, creaturely predicates do not apply to God. But if creaturely predicates do not apply to God, one cannot use human languages – which, according to Aquinas, contain only creaturely predicates – to speak meaningfully about God. On the other hand, if one applies creaturely predicates to God (as in univocal predication between God and creatures), then one is anthropomorphizing God and the result is idolatrous (ST I, Q13, A5 6). Aquinas suggests we should apply predicates to God analogically as a mean between the two (ST I, Q13, A5). The two kinds of analogy – AOP1 and AOP2 – result from two corresponding conceptions of proportion. Sometimes, we say that there is a proportion between two objects in virtue of, for example, their relative sizes. A one centimeter by one centimeter portrait of George Washington can be said to be in proportion to a five meter by five meter poster of George Washington. Both pictures, while of different sizes, are not of different kinds. If creatures are related to God in virtue of AOP1, then God is an infinitely amplified version of a creature. Thus, if DOA is understood in terms of AOP1, then God's intellect (for example) is an infinitely amplified intellect of the same kind as those of creatures. However, another way of taking about proportion relates objects of two fundamentally different kinds. For example, there is a correspondence between a painting of a pipe and a pipe. For AOP2, God's properties are of a different kind from those of creatures. Aquinas rejected AOP1, arguing that there can be no proportion between the created intellect and God's uncreated intellect because the former is finite while the latter infinite (SS, IV, d49, q49, q2, a1). More generally, Aquinas maintained there is no proportion between creaturely and Divine properties. However, there can be an analogy between God and creatures in another way: AOP2, where a comparison between two things is identified with a comparison between two other things (SS, IV, d49, q2, a1). For example: feet : shoes :: hands : gloves Similarly, Aquinas argued the relationship between creatures and their properties can be identified with the relationship between God and His properties: creature : intellect :: God : Intellect The difference in meaning of the term "Intellect" when applied to God and when applied to creatures has been signified with the use of the capital /I/. In Hyman's account, early modern philosophers and theologians reified God and reduced God's transcendence "to such an extent that [God] becomes a 'thing' himself" (Hyman 2007, 38–39.). According to Hyman, philosophers and theologians committed the ontotheological error: mistakenly understanding God to have the same kind of being (ens) as creatures, when God transcends all created categories, including being (Adams 2014, 1–12; Turner 2004, 26–29, 187–190). Aquinas argues instead for the analogia entis, or the analogy of being, according to which being cannot be predicated univocally of God and creatures (Muller 2012, 135). For Hyman, ontotheology leaves theism vulnerable to Linford: Early -Modern Irreligion and Theological Analogy Art. 3, page 3 of 8 atheological attacks. Failing to find either God or justification for God in the world, atheism appeared "almost irresistible" (Hyman 2007, 43). Hyman's claim is contrary to Richard Muller's recent scholarship on early-modern theology (Muller 2012). Muller has identified 20 reformed theologians from across Europe who denied the univocity of being between 1590 and 1700, many of whom either affirmed or responded to the analogia entis. Five of the 20 reformed theologians Muller identifies lived in the anglophone context (Richard Crakanthorpe (1567– 1624), William Twisse (1578–1646), Thomas Barlow (1607–1691), Theophilus Gale (1628–1678), and Robert Baron (1593?–1639) (Muller 2012, 129). Furthermore, orthodox theologians engaging freethinking authors in the anglophone context continued to reference DOA into the eighteenth century. In the next section, I show that an eighteenth-century debate over DOA in the anglophone context references the analogia entis and was directed to and influential on freethinking authors. 3. The DOA in the Eighteenth-Century Anglophone Context The Spanish Jesuit theologian Francisco Suárez (1548–1617) revived the seventeenth-century debate over theological analogy by arguing for the univocity of being (Muller 2012, 129; Armogothe 2012, 309). Several theologians throughout Europe responded critically to Suárez (Muller 2012). By the start of the eighteenth century, the debate over theological analogy had become widespread. French Huguenot philosopher Pierre Bayle (1647–1706) states in his Dictionnaire Historique et Critique he "formerly examined this Dispute [over the analogy of being], which is very famous in the Schools" and "[t]hey, who deny the Univocation of Being, have the Crowd, the Many, on their Side" (Bayle 1734, 488. English translation). In 1728, English encyclopedist Ephraim Chambers (1680–1740) wrote (emphasis in the original) "[t]he schoolmen have long disputed about the univocaton of being, i.e. whether the general idea of being agree in the same manner, and in the same sense, to the substance and the accident; to God, and the creature?" (Chambers 1728, 325). For Bayle and Chambers, the dispute over theological analogy and its relation to being was live, ongoing, widespread, active throughout the first part of the eighteenth century, and most importantly for my purposes, hardly neglected. Furthermore, while Bayle's status as a clandestine atheist has long been disputed, Bayle's influence on early modern critics of religion is not contested (Lennon 2014; Heyd 1977, 157–165; Berman 2013, 159–162.). At the start of the eighteenth century, the anglophone theological debate over DOA centered on two Irish theologians: William King (1650–1729) and Peter Browne (166?–1735). From the beginning of the eighteenth century to at least the mid nineteenth century, the theological positions first articulated by King and Browne formed two competing theological conceptions of DOA (Buchanan 1864, 10). In what follows, I describe King and Browne's respective theological programs. King's Theological Program The Problem of Evil asks how God, if all powerful and perfectly good, could create and maintain a world containing evil. Bayle composed a dialogue in the footnotes of his Historical Dictionary in which one character argues Manichaeism – the view that there exist two gods, equally powerful, one of which is evil and the other good – is a better explanation of our world's mixture of good and evil than traditional theism. On a literal reading, Bayle uses his character's argument as justification to doubt the human mind's ability to reason about the Divine. Manichaeism is false, Bayle says, but one would have concluded Manichaeism were true if one incorrectly attempted to use Reason in place of Faith (Bayle 1734, 95). Radical fideism is the view that one should rely on faith in place of reason. Logically, the arguments Bayle presents for radical fideism entail atheism or other unorthodox views (i.e. Manichaeism); for this reason, Bayle's response to the Problem of Evil did not sit well with the theologically conservative King. King delivered his 1709 sermon, Divine Predestination and Foreknowledge, Consistent with the Freedom of Man's will, partially in response to Bayle. In the sermon, King uses his conception of theological analogy to sidestep a number of arguments he thought threatening to Christendom (King, 1709). For King, God's properties bear an analogous similarity to those of humans, while atheistic arguments mistakenly assume God's properties to be univocal with those of humans. For example, the problem of evil assumes God's goodness is an infinitely magnified version of creaturely goodness, so that we can know God would be unlikely to make a universe with evil by reflecting on creaturely goodness. However, as King points out, God's properties are unlikely to be like creaturely properties. On King's view, when the Bible describes God as having various limbs, the Bible should not be understood as saying the same as is meant when we say humans have limbs. Biblical limb-talk should be understood analogically. The same is true for other properties of God; just as God lacks limbs, so too God literally lacks foreknowledge, goodness, and a number of other attributes one might have otherwise literally ascribed to God. God possesses properties merely analogous to those of creatures which one might call divine foreknowledge and goodness. Since God's properties only bear an analogical similarity to those of humans, we should not expect God to behave as a perfectly good human, who possessed foreknowledge, would behave; humanly goodness – even if infinitely perfect – would remain categorically distinct from God's. Thus, King concludes, Manichaeism, and other unorthodox positions, do not explain our world better than traditional Christian theism. On King's account, any apparent inconsistency between the appearance of the world – such as the existence of suffering – and God's attributes – such as God's goodness – is illusory. Thus, according to King, the thought that God would not allow suffering in the world because God is infinitely good is mistaken (King 1709, 4–10). God does not literally possess goodness and whatever property God possesses, analogous to goodness, may or may not allow Linford: EarlyModern Irreligion and Theological AnalogyArt. 3, page 4 of 8 for suffering. We cannot understand God's properties, so any argument opposing theism on the basis of God's properties is, on King's account, mistaken. As Collins summarized King's position, "[...] no Man [sic] can object to he knows not what, all Objections supposing a meaning to the Proposition objected against" (Collins 1710, 11). From the perspective of many of his contemporaries, King confused analogy with metaphor.2 As Browne explained, metaphor is purely the result of the human mind, while analogy "is the Result of Reason viewing the True Nature of Beings". For Browne, when we say God possesses limbs, we are speaking metaphorically but not analogically: metaphors are "an Appearing or Imaginary Resemblance and Correspondency" (Browne 1734). Thus, to say God is analogically good, in King's sense, involves denying God is good at all. Collins responded directly to King, and in defense of those skeptical or critical of religion,3 in his Vindication of Divine Attributes in 1710. As Collins maintains, King's version of analogy renders natural theology impossible. A consequence of the impossibility of natural theology is the impossibility of proving the existence of God through evidence of design in nature.4 For Collins, King's account of analogy leaves us no conception at all of God's properties, rendering "religion" impossible.5 Collins concludes King's sermon was nothing more than a tacit acceptance of defeat. On one interpretation of Bayle's arguments, if God exists then either God does not have foreknowledge or humans do not possess free will. In King's response, God possessed a property merely "analogous", in King's sense, to foreknowledge; to Collins, this read as King conceding God's lack of foreknowledge. From Collins's perspective, we should not call a being without foreknowledge "God". Likewise for goodness: if King's God is not literally good, then in what sense is King's deity God? Collins proceeds to argue King's conception of God destroys both the project of natural theology and of religion generally, leaving the Christian little room in which to stand. For Collins, King's conception of God destroyed natural theology because theism could no longer be proven from evidence of design in nature. King is unable to prove the existence of God, Collins argued, due to the radical semantic underdetermination of "God"; all King could possibly mean by "God" is a "General Cause of Effects" (Collins 1710, 13). Collins states: "if that be all that is meant by the term ['God'], I see not why Atheists should not come into the Belief of such a Deity; for they, equally with Theists, allow some general Cause of all Effects to have eternally existed; but [. . .] differ from them in the Attributes of that general Cause" (Collins 1710, 13–14). According to Collins, King, and other theists, cannot provide any further "King's account seemed to entail that any attempt to refer to God's attributes would fail, leaving theism without substance. King's conception of God destroys the project of religion generally, Collins maintained, because one cannot prove, from such vague conceptions of God, that one should worship God, that there is an afterlife, or that there was once a human who was fully God and died for our sins. Browne's Theological Program Browne advocated a view of theological analogy in which humans and God have more in common than they do in King's view. Nonetheless, Browne, like King, asserts we cannot know what the term "goodness" means when applied to God (Browne 1733, 82) and cites Aquinas for support (Browne 1733, 84). For Browne, God's properties are inexpressible in creaturely languages and this finds its clearest expression in Aquinas's works: "But of all whom I have yet met with, the Angelic Doctor [Aquinas] hath set this whole Matter in the truest Light" (Browne 1733, 93). After explaining Aquinas's distinction between univocal and equivocal predication, denying that each of these hold between God and creatures, Browne explicates Thomistic analogical predication (i.e. AOP2) and asserts God's properties may only be predicated analogically of God and creatures. Browne then discusses analogical predication in relation to Aquinas's distinction between essence and existence (Browne 1733, 93–96). Browne's fame was owed to a response he wrote to deist John Toland (1670–1722) entitled A Letter in Answer to a Book Entitled Christianity not Mysterious (published in 1692). Later, Browne responded to other heretical views, each time utilizing DOA in defense of Christian orthodoxy.6 Browne's student, philosopher and Bishop George Berkeley (1685–1753), answered the debate between King and Collins in chapter IV of his Alciphron or: The Minute Philosopher A Defence of the Christian Religion against the So-called Free-thinkers (1732), in which the titular character represents Collins.7 For Berkeley, analogy had come to be seen as a weapon of the atheists. Berkeley argues that the atheistic weaponization of analogy is based on a misunderstanding of analogy and the use of AOP2 would disarm the atheists. The admission that God exists, while failing to admit any of God's properties, is the admission only that there is some object or other that one calls "God" and not the admission that the object has any particular description. But, without any particular description, the object in question could be any object whatsoever, including objects the atheist readily admits to exist. One of Berkeley's characters is a religious skeptic who advances Collins's argument in order to show that admitting the existence of God is not admitting much at all. An admission to God's existence grants the existence of God in only an "indefinite sense", in which God is understood to "properly speaking, [have] no knowledge or wisdom at all". That is, following Collins, the skeptic maintains only that there is some object one could arbitrarily call "God" (if one so chose) and not that the object possesses any of the divine attributes. Berkeley's skeptic goes on to explain that such conceptions entail disastrous consequences for natural theology, as very little can be shown from attributes that are either unknown or possessed in an unknown sense. Berkeley's skeptic concludes: "Since, therefore, nothing can be inferred from such an account of God, about conscience, or worship, or religion, you may even make the best of it. And, not to be singular, we will use the name too, and so at once there is an end of atheism" (Berkeley 1732, 248–249). Linford: Early -Modern Irreligion and Theological Analogy Art. 3, page 5 of 8 One of Berkeley's theistic characters responds that skeptics have misunderstood DOA. According to the character, if DOA is understood as Aquinas described – using proportionality – then the semantic underdetermination evaporates. Berkeley's theist proceeds through a history of DOA referencing both Aquinas and Suárez. Berkeley's theist notes, in a proper theological understanding of analogy, being – or existence – is analogical and not univocal between God and creatures (the analogia entis): At that time the scholastics generally held that even Being should be attributed to God and to created things only analogically. That is, they held that God- the supreme, independent, self-causing cause and source of all beings-mustn't be supposed to exist in the same sense of "exist" as that in which created beings exist; not that he exists less truly or properly than they do, but only that he exists in a more eminent and perfect manner (Berkeley 1732, 255). Lastly, against those who, like King, confuse analogical predication and metaphor, Berkeley distinguishes "metaphorical analogy" (e.g. equivocal predication) and "proper analogy" (AOP2), noting that DOA should be understood in terms of the latter (Berkeley 1732, 257). Hume Enters Stage Left While Hume has been variously interpreted as an agnostic (Noxon 1966), irreligious (Russell 2008), a skeptic (Price 1965), and a deist (Gaskin 1978), Hyman maintains Hume "dispens[ed] with God altogether" (Hyman 2010, 36). At times, in agreement with Noxon, Hyman describes Hume as an agnostic and not an atheist. Nonetheless, Hyman is explicit that, whatever Hume's views were, Hume was not a theist. In what follows, I follow Hyman in assuming Hume was a non-theist whose arguments lend themselves to atheistic conclusions. Unlike Hyman, I will examine Hume's engagement with theological analogy and religious language. Importantly, as I demonstrate, Hume's engagement with the King/Browne/Berkeley exchange was important in Hume's irreligiosity. I will conclude, contra Hyman, the neglect of analogy cannot explain the rise of early-modern anglophone atheism. Hume maintains a tension throughout his Dialogues Concerning Natural Religion between "anthropomorphite" theology, according to which God's properties differ only in magnitude, and not in kind, from those of creatures, and a theology according to which God is completely beyond human understanding and categorically distinct from creatures. The former seems necessary for inferring God's existence from evidence of design in nature. After all, when one infers nature was produced by an infinitely wise being for particular ends, one compares God to humans. However, the latter seems necessary for maintaining a traditional notion of God's transcendence. The tension can be understood in terms of its implications for religious language: the god inferred through design arguments is one whose properties are univocal with creaturely properties. To render God more transcendent is to render the predicates applied to God and creatures ever more equivocal. Hume's Dialogues involve three characters: Philo, the skeptic (often understood to represent Hume), Demea, the mystical theist and pietist, and Cleanthes, who presents design arguments for God's existence. Demea and Philo accuse Cleanthes of "anthropomorphitism" while Cleanthes and Philo accuse Demea of presenting God as so radically transcendent as to render theism indistinct from atheism. Towards the close of part III, Cleanthes presents a design argument, citing the intricate way in which each part of nature is fit for another, so that even the sexes were designed for each other (Hume 1947, 154). Demea responds that Cleanthes presents a strong argument, but at the cost of reducing God's transcendence: "it must be acknowledged, that, by representing the Deity as so intelligible and comprehensible, and so similar to a human mind, we are guilty of the grossest and most narrow partiality" (Hume 1947, 156). Demea explains God forms an incomprehensible unity, so that God cannot be subdivided in terms of His properties (divine Simplicity). Thus, none of God's properties are univocal with those of humans, whose properties, unlike God's, are not identical to their essence (Hume 1947, 158). In response, Cleanthes argues divine simplicity is tantamount to atheism because an incomprehensible timeless unity is not capable of performing acts or having sentiments, successive ideas, thoughts, reason, will, love, hatred, or even a mind because all of these properties require time and constitution. To describe an incomprehensible, timeless unity as God would be an "abuse of terms" (Hume 1947, 159). However, divine Simplicity was important for Aquinas's conception of theological analogy. For Aquinas, one reason we require theological analogy to talk about God is that creatures possess their properties in a fundamentally different way than God possesses His properties. Aquinas argues that God is Simple, by which he means that all of God's properties are identical to God's essence. There is no distinction between God's Being (ens) and God's Essence (esse) (ST P1 Q3, especially article 4). However, humans', and other creatures', properties are distinct from their essence, so, in creatures, there is a distinction between ens and esse. On Aquinas's view, God's essence is incomprehensible to the created intellect in the present life and, consequently, the manner of God's existence is incomprehensible to the created intellect in the present life. Hume's Demea agrees and notes that, "the manner of [God's] existence" is "mysterious" to "[f]inite, weak, and blind creatures" (Hume 1947, 141). Thus, by rendering incoherent one reason that Godtalk might be analogical, Cleanthes's argument contra Simplicity indirectly undermines theological analogy. As Philo indicates, Cleanthes painted all of the "orthodox divines" as atheists and has painted himself as the only orthodox individual in the world (Hume 1947, 159). In any case, if, as Hyman argues, we interpret Hume as a nontheist, then Hume's rejection of God involved Hume's Linford: EarlyModern Irreligion and Theological AnalogyArt. 3, page 6 of 8 reaction to theological analogy and not, as Hyman contends, because theological analogy had never been made available to him due to prior neglect. Elsewhere, Hume continues the theme of identifying a radically transcendent God with no god at all. Several commentators have noticed a strong similarity between Hume's argument in chapter XII of Dialogues and chapter IV of Berkeley's Alciphron.8 In that chapter, Philo argues that there is only a verbal distinction between atheism and theism. Philo begins by posing a question for theists: do they allow a "great and immeasurable, because incomprehensible difference between the human the divine mind"? The more pious the theist wishes to be, the more they will commit themselves to God's radical transcendence. Turning next to the atheist, Philo asks whether atheists disallow that the "rotting of a turnip, the generation of an animal, and the structure of human thought" could "bear some remote analogy to each other". Philo imagines atheists will answer in the affirmative without hesitation. Philo asks whether there could not be "some remote inconceivable analogy" between what Unknowable Thing caused the universe and "the other operations of nature", including "human mind and thought". Because any two things have some similarity or other between them, to deny that whatever created the universe possessed something or other analogous to a mind would be absurd. Thus, the dispute between theists and atheists has been dissolved: there is no distinction between theism and atheism after all (Hume 1947, 217). Philo goes on to consider those who believe the "whole of Natural Theology" has been reduced to "one simple, though somewhat ambiguous" or "at least undefined proposition": "That the cause or causes of order in the universe probably bear some remote analogy to human intelligence". Philo states the conclusion is inevitable and cannot be avoided, but does not amount to much. Assenting to some remote Something or Other bearing a vague analogy to a mind may be "the first and most essential step towards being a sound, believing Christian", but the Christianity assented to is not obviously distinct from atheism.9 4. Conclusion Gavin Hyman argues early modern atheism arose due to a neglect of theological analogy. I have argued that in fact religious skeptics broadly discussed and debated theological analogy. Browne, King, and Berkeley, each a prominent theologian of the period, used the doctrine in their defense of what they understood as orthodoxy. Even the irreligious Hume addressed theological analogy, arguing that it rendered atheism and theism only verbally distinct. Thus, Hyman's explanation of the appearance of atheism in modernity as a result of a prior neglect of DOA is without support. Several questions remain. In what follows, I offer several brief suggestions for future work. Although early-modern irreligious figures did not neglect theological analogy, twentieth-century analytic philosophers have largely neglected the analogy of being. Philosopher Kris McDaniel describes the "apparent consensus among contemporary analytic metaphysicians is that believing [. . .] things can exist in different ways [e.g. the analogy of being] is silly or confused" (McDaniel 2010, 689). As McDaniel notes, analytic philosophers after Quine have generally maintained existence is whatever the existential quantifier denotes and so is univocal. Religious disbelief is common among analytic philosophers (Bourget & Chalmers 2014) and those pursuing a revised version of Hyman's historical narrative may ask whether analytic philosophy's anglophone hegemony, with the associated univocity of being, is responsible for the rise of atheism among contemporary philosophers. Moreover, one may ask how the rise of atheism among anglophone philosophers affects the rise of the nonreligious and the secular in the broader culture. (I take no position on this issue here.) Future work may be done to explore the relationship between analytic philosophy's anglophone ascendance and the genealogy of the univocity of being. For example, Kant's response to Anselm's ontological argument – that existence is not a predicate – was incorporated into Frege's Foundations of Arithmetic as the existential quantifier (Labenz 2006). Frege's definition of the existential quantifier influenced Quine's work and thereby late twentieth-century analytic metaphysicians. How did Quine's work on existential quantification influence analytic philosophy of religion or the cultural debates over God's existence? Few authors have engaged these questions. The influence of Quine and other analytic metaphysicians on most New Atheist authors is far from obvious (Daniel Dennett excepted). However, some debate over the efficacy of Richard Dawkins's central atheistic argument in The God Delusion has centered on Dawkins's supposed lack of theological sophistication. Dawkins considers Creationist arguments that ask whether a random process – like a tornado plowing through a junkyard – would be likely to produce a Boeing 747. As Creationists point out, the odds are vastly opposed to junkyard tornados spontaneously assembling aircraft; so, the argument continues, natural processes are even more unlikely to produce living things, themselves vastly more complex than a Boeing 747. Dawkins agrees; natural processes are unlikely to produce life, but God is even less likely to exist since God must be even more complex than His Creation (Dawkins 2008, 137–139). Critics say Dawkins has misunderstood (or failed to respond to) the most sophisticated conceptions of God, in which God is Simple. While Dawkins neglects divine simplicity, Hume did not. Analytic philosopher Erik Wielenberg argues atheists and theists alike should put down Dawkins and pick up Hume's Dialogues, in which, as I explained in section 3, Hume argues against the coherency of a Simple God (Wielenberg, 2009). Nonetheless, whether Dawkins should have considered a Simple God is unclear; God's Simplicity is notoriously difficult to make sense of and, if McDaniel is correct, analytic philosophers are likely to consider conceptions of God that utilize Simplicity (especially if taken to entail the doctrine of analogy) "silly or confused" (McDaniel 2010, 689). Daniel Dennett, a more philosophically sophisticated Linford: Early -Modern Irreligion and Theological Analogy Art. 3, page 7 of 8 New Atheist than Dawkins, considers the view that God is beyond being tantamount to atheism (Dennett 2010). Similarly, as Mikael Stenmark observes, analytic philosophers of religion, whether atheists or theists, commonly retort that a God beyond being is nonsense (Stenmark 2015, 5); however, Stenmark (2015) makes inroads towards bringing into dialogue those who endorse and those who deny the analogy of being. Whether Stenmark's attempt is successful is, as yet, unclear. Competing Interests The author declares that they have no competing interests. Acknowledgements I thank Joseph C Pitt, Matthew Goodrum, and Benjamin Jantzen for supporting the MA thesis that resulted in this paper. I'd also like to thank the attendees at the 2013 Eastern Division Meeting of the Society of Christian Philosophers for their helpful comments on an earlier draft. Notes 1 Theologian Michael Buckley offers another view according to which the early modern neglect of important theological doctrines resulted in atheism. For Buckley, the neglect of Christology, instead of analogy, resulted in the rejection of Christianity (Buckley 1987). An analysis of Buckley's thesis is beyond the scope of this paper. 2 Aquinas covers this issue in ST 1 Q13 A3 and his comments mirror Browne's. Analogical predication is literal, not metaphorical, even though the meaning of terms when applied to God is incomprehensible to the created intellect. 3 Whether Collins himself was an atheist is unclear, but see chapter 3 in Berman (1988). 4 However, Collins's argument is more general – and damning for King's theism – than simply showing that natural theology is impossible. The implication of Collins's pamphlet seems to be that it is impossible to prove anything about King's God (including through the use of a priori reasoning) due to the kind of radical underdetermination in King's view. 5 In this context, the term "religion" is used in actor's categories (Collins 1710). 6 The Procedure, Extent, and Limits of Human Understanding in 1728 and Divine Analogy, or Things Divine and Supernatural Conceived by Analogy with Things Natural and Human in 1733. 7 The claim that Berkeley's character Alciphron should be identified with Collins is from Berman (1993, 10). 8 See, for example, Berman's (1993, 5–6). 9 Paul Russell describes Hume's assent to theism, as a vague assent to Something or Other, "thin theism", as distinguished from the theologically thick theism of orthodox religion (2010, 282–283). 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Secularism and Nonreligion, 5: 3, pp. 1–8, DOI: http://dx.doi.org/10.5334/snr.50 Published: 06 January 2016 Copyright: © 2016 The Author(s). This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 Unported License (CC-BY 3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. See http://creativecommons.org/licenses/by/3.0/. OPEN ACCESS Secularism and Nonreligion is a peer-reviewed open access journal published by Ubiquity Press.

Abstracta 5 : 1, pp. 16 – 29, 2009 NAGEL ON CONCEIVABILITY Terence Rajivan Edward Abstract In the sixth chapter of The View from Nowhere, Thomas Nagel aims to identify a form of idealism, to isolate the argument for it and to counter this argument. The position that Nagel takes to be idealist is that what there is must be possibly conceivable by us. In this paper, I show that Nagel has not made a convincing case against this position. I then present an alternative case. In light of this alternative case, we have reason to reject an important example that Nagel offers of a contemporary idealist, namely Donald Davidson. 1. In the sixth chapter of The View from Nowhere, Thomas Nagel draws our attention to a thesis that I shall refer to as the conceivability thesis. According to this thesis, what there is must be possibly conceivable by us. He claims that this thesis is a form of idealism and declares that a number of contemporary philosophers are idealists because they accept it (1986: 90). Nagel seeks to justify rejecting the thesis. First he presents what he takes to be the argument for it. Then he attempts to counter the argument. This paper has three aims. The first is to show that Nagel does not make a convincing case against the argument he uncovers for the conceivability thesis. The second is to show that there is an alternative case that he could have made. The third is to show that, in light of this alternative case, we have reason to reject an important example Nagel gives of a contemporary philosopher who is an idealist, namely Donald Davidson. Before pursuing these aims, I clarify the conceivability thesis. 2. There are at least two notions which are in need of clarification in order to understand the thesis that what there is must be possibly conceivable by us. One of these is the notion of 'us'. Who counts as one of us and who does not? The other notion in need of clarification is that of being possibly conceivable. Nagel contrasts being possibly conceivable by us with being actually conceived of by us and being currently conceivable by us. For instance, after briefly criticizing certain forms of idealism, he writes: T.R. Edward 17 But the form of idealism with which I am concerned isn't based on this mistake: it is not the view that what there is must be actually conceived or even currently conceivable. Rather it is the position that what there is must be possibly conceivable by us, or possibly something for which we could have evidence. (1986: 93) But what does it mean to say that something is merely possibly conceivable by a person as opposed to actually conceived by them or currently conceivable by them? In this section I seek to clarify the notion of us and the notion of being possibly conceivable. Let us begin with the notion of 'us'. Nagel does not define this notion, despite suggesting that a grasp of the criteria for counting someone as us is crucial for understanding the position he has in mind (1986: 90-1). However, while arguing against the conceivability thesis, he does clearly indicate that certain kinds of people are not to be counted as us. People with the permanent mental age of a nine-year-old are not us, for Nagel. His argument involves contrasting us with such people (1986: 95). According to Nagel, we can conceive things that they cannot conceive. Nagel also contrasts us with people he imagines whose mental faculties are superior to his to the extent that the gulf between them and him is comparable to that between him and people with a permanent mental age of nine (1986: 95). According to him, they might be able to conceive things that we cannot conceive. From these two contrasts, one might form the impression that a person only counts as one of us, for Nagel, if they have mental faculties that are not dramatically superior or inferior to his own. This is the understanding of 'us' that I shall work with in this paper. It may be that it is in need of refinement, though. It might not perfectly capture how Nagel uses the term 'us'. Nevertheless, it is adequate for the purposes of this paper. Let us turn now to the notion of being possibly conceivable by a person. The distinction between being actually conceived by a particular person and being currently conceivable by that person is straightforward. Someone might not actually conceive that snow is falling in the vicinity. If they do not think that it is, then they do not actually conceive that it is. Nevertheless, they might have the ability to conceive that snow is falling in the vicinity. If so, then it is currently conceivable to them that snow is falling. What though does it mean for something to be possibly conceivable by a particular person? There Nagel on Conceivability 18 are certain things that a person might not at present have the ability to conceive yet might one day be able to conceive. For example, a person who has never seen snow before might not have the ability to conceive of snow falling. If they are one day shown snow falling and are taught to think of this happening as snow falling, they might then acquire this ability. It is part of our commonsense outlook that in the future a particular person might acquire the ability to conceive of certain things which they cannot currently conceive. Such things are not currently conceivable by the person but they are possibly conceivable by them. All the things that the person can already conceive are also possibly conceivable by the person; it is just that with these things the possibility is already realized. In the quotation above, Nagel attempts to formulate the view that concerns him by writing not just of what must be possibly conceivable by us but also of what we could possibly have evidence for. However, prior to this point in the text, he discusses the view at length without doing this and it is not clear why he mentions possible evidence at all. I do not think he means to add anything significant. The position that Nagel focuses on is that what there is must be possibly conceivable by us. For understanding this thesis, it is important that the following point be kept in mind. If something can only be conceived by a being with mental faculties that are dramatically superior to our own, then it is not possibly conceivable by us. Perhaps a person who is currently one of us can undergo the kind of improvement which enables them to conceive of such a thing. But then they would not be one of us, on Nagel's understanding of 'us'. The conceivability thesis involves denying that there could be superior beings who are able to conceive things that beings with our mental faculties could never conceive. 3. Once Nagel has explained the position that concerns him, he makes an assertion about what an argument in favour of it must show: An argument for this general form of idealism must show that the notion of what cannot be thought about by us or those like us makes no sense. (1986: 93, his emphasis) T.R. Edward 19 The reason why Nagel thinks that such an argument would secure the conceivability thesis is as follows. The only seeming rival to this thesis is the view that there could be some things that are not possibly conceivable by us. This is a genuine rival view if it is intelligible. If it simply does not make sense, then it is not a genuine rival view. The view is only intelligible if there is an intelligible notion of what is not possibly conceivable by us. (In the quotation, Nagel writes of the notion of what cannot be thought about by us. More precisely, he means the notion of what is not possibly conceivable by us.) As such, an argument which shows that there is no intelligible notion of this kind provides us with a good reason to assert that what there is must be possibly conceivable by us. For the putative rival view, that there could be some things which are not possibly conceivably by us, is not a genuine option. Even if it appears at first to be an intelligible view, the argument reveals otherwise. Nagel does not just assert that the conceivability thesis would be supported by an argument which shows that a certain notion does not make sense. He also asserts that it has to be argued for in this way. In the quotation above, he writes as if no other line of argument could ever justify it. I shall not contest this. What I wish to present in this section is the argument that Nagel ascribes to philosophers who endorse the conceivability thesis, before evaluating Nagel's response to it. Here is Nagel's statement of this argument: The argument is this. If we try to make sense of the notion of what we could never conceive, we must use general ideas like that of something existing, or some circumstance obtaining, or something being the case, or something being true. We must suppose that there are aspects of reality to which these concepts that we do possess apply, but to which no other concepts that we could possess apply. To conceive simply that such things may exist is not to conceive of them adequately; and the realist would maintain that everything else about them might be inconceivable to us. The idealist reply is that our completely general ideas of what exists, or is the case, or is true, cannot reach any further than our more specific ideas of kinds of things that can exist, or be the case, or be true. (1986: 93-4, his emphasis) Nagel depicts the justification for the conceivability thesis as hinging on the claim that there is the following entailment relation: if something can be correctly represented using at least one general concept of ours, this entails that every property of this thing can be Nagel on Conceivability 20 represented using only concepts that are within our grasp. To illustrate the thought of such a relation, suppose that a being has a concept that we do not have which is used to identify a type of thing that we have not encountered before, for instance a kind of animal. The being uses this concept to declare that an instance of this type of thing exists. In doing so, they represent the world in a way that uses at least one general concept of ours, namely the concept of existence. Let us suppose that the representation is correct. If so, something has been correctly represented using at least one general concept of ours. For Nagel's opponents, this entails that every property of the thing can be represented using only concepts that are within our grasp, that is, either concepts we currently have or concepts that we could one day acquire without undergoing a dramatic improvement in any of our mental faculties. On the basis of this supposed entailment relation, they claim that what there is must be possibly conceivable by us. When Nagel explains the argument for the conceivability thesis, he does not present any case in favour of this supposed entailment relation. This point will play an important role in my evaluation of Nagel's response. Before moving onto this evaluation, here is the argument in steps. (1) The conceivability thesis is correct if and only if the following view does not make sense: there might be some things which are not possibly conceivable by us. (2) This view only makes sense if the notion of something which is not possibly conceivable by us makes sense. (3) This notion only makes sense if the following claim is true: if something can be correctly represented using at least one general concept of ours, it does not follow that all the properties of this thing can be correctly represented by using only concepts that are within our grasp. (4) But this claim is false. If something can be correctly represented by using at least one general concept of ours, it follows that all the properties of this thing can be correctly represented by using only concepts that are within our grasp. From (3) and (4): T.R. Edward 21 (5) The notion of something which is not possibly conceivable by us does not make sense. From (2) and (5): (6) The view that there could be some things which are not possibly conceivable by us does not make sense. From (6) and (1): (7) What there is must be possibly conceivably by us. Nagel does not himself break the argument down into steps, before specifying any premises or inferences which he is against. From examining what he says though, it becomes clear that the principal clash between himself and his opponents is over (4). After presenting the argument for the conceivability thesis, Nagel gives us a reason to reject (4). It is part of our commonsense outlook that some people cannot understand certain truths because their mental development has not advanced to the point where they can grasp certain concepts. For example, most little children cannot understand many mathematical truths because they do not have the relevant mathematical concepts nor are they at a stage in their mental development when they can be successfully taught these concepts. In light of such cases, it seems that there could be some truths that we too cannot understand because our mental faculties are not advanced enough to grasp certain concepts. Nagel presents cases of this kind in order to support the view that some of what there is might not be possibly conceivable by us (1986: 95). This way of supporting the view involves rejecting the inference that if something can be correctly represented by using at least one general concept of ours, it follows that all the properties of this thing can be correctly represented by using only concepts that are within our grasp. We suppose that there might be aspects of the world that can only be correctly represented by using at least one general concept of ours, such as the concept of existence, but along with some concepts that are simply not within our grasp. Hence Nagel's justification for his view involves rejecting (4). Nagel considers an objection to this justification. It appears to be an objection that he devised himself. He anticipates that someone might make this objection but he does not Nagel on Conceivability 22 present any philosopher as actually making it (1986: 96). I shall not introduce this objection or his response to it, because without doing so it can be shown that Nagel has not done enough to secure his view. Why does any philosopher endorse (4) to begin with? When one turns to the paragraph in Nagel which explains the argument for the conceivability thesis, no grounds are given for this premise. Advocates of the conceivability thesis are depicted as simply asserting that there is a certain entailment relation, even though it is not obvious that there is this entailment relation. This depiction is suspect. Consider the following remark which Nagel makes while introducing the conceivability thesis: The idea that the contents of the universe are limited by our capacity for thought is easily recognized as a philosophical view, which at first sight seems crazily selfimportant given what small and contingent pieces of the universe we are. It is a view that no one would hold except for philosophical reasons that seem to rule out the natural picture. (1986: 92) Given that (4) is glaringly in need of justification, a philosophical reason for the conceivability thesis cannot involve a bald assertion of this premise. It seems then that Nagel has left out a crucial piece of information when telling us the argument for the conceivability thesis. He has left out the reasons that have actually been given for (4). Without this information, we ought not to be convinced by Nagel. Why not? One might think that if Nagel has made a good case against (4), then we can know that any reasons given in favour of (4) are bad ones prior to being made aware of them. But consider again Nagel's case in light of what is said in the quotation above. In this quotation, philosophers who hold the conceivability thesis are depicted as knowingly departing from an intuitive view, or 'natural picture' to use Nagel's term, because they have reasons which they find compelling enough to warrant a departure. Now the case that Nagel makes against (4) is ultimately an explanation of why it is intuitive to reject (4). If someone wanted an explanation of why it is intuitive, one could say what he does: given that there are truths that others cannot understand but we can, surely there might be truths that we too cannot understand. (Note that the claim in the quotation that we are small and contingent pieces of the universe, rather than gods say, is not a different explanation; it just explains the use of 'surely' here.) But there is still room to wonder whether advocates of the conceivability T.R. Edward 23 thesis have a compelling reason in favour of (4), that is, a reason strong enough to warrant departing from the intuitive view. If some philosophers purport to offer reasons of this kind, Nagel needs to show that the reasons offered are not compelling, but all he does is imply that reasons are offered without telling us what they are and without revealing their inadequacy. This is why Nagel's case is unconvincing. It is analogous to a case against determinism which merely explains why it is intuitive to believe in free will. In the final section of this paper, which focuses on Davidson, I shall support the suspicion that Nagel has not considered the reasons that are actually given for (4). Before this section, I shall contest another premise of the argument that Nagel presents for the conceivability thesis. 4. The argument that Nagel isolates for the conceivability thesis is a bad one even if (4) is true. Consider (3) instead: The notion of something which is not possibly conceivable by us only makes sense if the following claim is true: if something can be correctly represented using at least one general concepts of ours, it does not follow that all the properties of this thing can be correctly represented by using only concepts that are within our grasp. Let us grant, for the sake of argument, that advocates of the conceivability thesis are right to assert that the claim following the colon is false. Contrary to (3), we can still form the notion of something that is not possibly conceivable by us. In order to see that this is the case, imagine that two people are playing a game of chess and a child who knows how to play chess is watching. A player resigns. The child asks why. The child is told that it was within the other player's power to achieve checkmate in four moves. Both players then try to explain to the child how. But the child cannot understand the explanation. Demonstrations are given using the board. Still the child cannot understand. The chess players then check to see whether the child can ever understand how it is within a player's power to achieve checkmate. Their tests reveal that the child can sometimes understand how, but only when the scenarios are relatively straightforward. There is something about Nagel on Conceivability 24 the child's mind which means that although they can understand the general idea of it being within a player's power to achieve checkmate in four moves, they cannot ever understand the details of how this is so in a particular match. What is this something about the child's mind? Even if we cannot specify exactly what it is – after all, the relevant psychological knowledge is not commonplace – we can say that the problem is not that the child lacks some concept or other. The child knows how to play chess and so it is possible to construct an explanation of how checkmate can definitely be achieved in four moves using only concepts that the child has grasped, such as the concept of a king, the concept of a move and so on. But the child cannot understand the explanation. From this example, we can see that there is a potential gap between all the correct representations that can be formed using a particular person's repertoire of concepts and all the correct representations that this person can understand. We can thus imagine that superior beings construct some correct representations that we cannot understand even though these representations are constructed using concepts that we have. Suppose then that we grant the entailment relation that Nagel rejects. Contrary to (3), we can still form the notion of something that is not possibly conceivable by us. For even if all the properties of a thing can be correctly represented using concepts that are within our grasp, it may nevertheless be the case that without an improvement in our mental faculties we cannot understand some of these correct representations. And so, there is still room for the thought that there are features of reality which are not possibly conceivable by us. Leaving aside whether or not (4) is acceptable, (3) is false. In the previous section, I claimed that we would expect Nagel to explain how various philosophers have attempted to justify (4) when he presents the argument for the conceivability thesis. Given that (3) is false, and therefore not self-evident, should we not also expect an explanation of how such philosophers have attempted to justify (3)? I think that we should not. Whereas (4) is glaringly in need of justification, it is understandable for a person to not register anything controversial about (3). Examples that can be used to contest (3) do not come to mind so easily. It is understandable then for proponents of the conceivability thesis to simply assert or assume (3), whereas it would be bewildering for them to simply assert (4), since this premise is patently controversial. T.R. Edward 25 5. Nagel regards the conceivability thesis as a form of idealism. Furthermore, he regards it as presupposed by all other forms of idealism. The other forms are characterised by him as specific forms, whereas what he is interested in is characterised as a general form of idealism, since it is presupposed by the specific forms (1986: 91). It is reasonable to doubt whether the conceivability thesis is a form of idealism, whether each form of idealism involves a commitment to it and whether Nagel is right to think that realism involves rejecting this thesis. But so far I have not engaged in these debates. I have remained neutral on these issues and continue to do so below. Nevertheless, the alternative counter that I have presented in the previous section can be used to dispute an important example that Nagel offers of a philosopher who is an idealist. When Nagel makes his case against the conceivability thesis, he refers to Davidson as an example of a thinker who espouses this supposed form of idealism. But he does not present Davidson as ever claiming that what there is must be possibly conceivable by us. Rather he quotes Davidson making a claim that he treats as amounting to an assertion of (4). The claim is this: it is impossible for there to be a truth which can be stated in another language but cannot be stated in our language (1986: 194). But even if we grant that the conceivability thesis is a genuine form of idealism, is Davidson an idealist just because he makes this claim? Nagel has convinced some philosophers to regard Davidson as an idealist (McGinn 1987: 268; Avramides 2006: 237). What I shall show is that we ought to reject this charge of idealism. The claim that Nagel quotes is made by Davidson in his renowned essay 'On the Very Idea of a Conceptual Scheme'. In that essay, Davidson aims to show that the idea of a conceptual scheme does not make sense (1984: 183). He thinks that the intelligibility of this idea depends on the intelligibility of the thought that different groups of people might have different conceptual schemes (1984: 198). Davidson considers four attempted explanations of what it would be for different groups to have completely different schemes, each of which he deems unintelligible (1984: 192-5). He then sets out to show that we cannot even make it intelligible to ourselves how different groups could have partially different schemes. The claim that Nagel quotes emerges from Davidson's treatment of two of the attempts to explain what it is to have a completely different scheme (1984: 193-4). Nagel on Conceivability 26 According to one attempt, a conceptual scheme consists of a set of beliefs that fit with the data of sensation. According to the other attempt, a conceptual scheme consists of a set of beliefs that fit with reality. Both attempted explanations hold that two groups of people have completely different schemes if and only if the beliefs which comprise the conceptual scheme of one group cannot be translated into the language of the other group and vice versa. In response to these attempts, Davidson proposes that to speak of beliefs fitting with either the data of sensation or reality is to say, in a metaphorical way, that the beliefs are true (1984: 194). Consequently, in Davidson's eyes the attempted explanations depend for their intelligibility on the intelligibility of the following thought: there might be truths which are expressible in one language but not in another. Davidson denies the intelligibility of this thought. He does this in the passage that Nagel quotes: The criterion of a conceptual scheme different from our own becomes: largely true but not translatable. The question whether this is a useful criterion is just the question of how well we understand the notion of truth, as applied to language, independent of the notion of translation. The answer is, I think, that we do not understand it independently at all. (Davidson, quoted in Nagel 1986: 94) For Davidson, it cannot be the case that there are truths which can be expressed in a language unfamiliar to us but cannot be expressed in our language. Note that Davidson's use of the term 'our language', which does not feature in this quotation but does elsewhere in his essay, is apt to appear obscure in light of Nagel's use of 'our'. For the people whom Nagel counts as us do not all speak the same language. Which language then does 'our language' refer to? It is tempting to say English, since this is the language that Davidson writes in. However, he clearly does not want to accord a special status to English. He does not want to say that English can be used to express any truth that is expressed in another language, but other languages might not be able to express certain truths that can be expressed in English. It seems that he is happy for 'our language' to be thought of as any natural language (Case 1997: 11). Whichever natural language is taken as 'our language', Davidson believes that this language can be used to express all truths that can be expressed in other languages. T.R. Edward 27 On the basis of his claim that there cannot be truths that are only expressible in certain other languages, Nagel interprets Davidson as asserting the entailment relation proposed in (4): if something can be correctly represented using at least one general concept of ours, it follows that all the properties of this thing can be correctly represented by using only concepts that are within our grasp. Nagel says that Davidson simply has another way of putting this point, in terms of language (1986: 94). Now, before proceeding to defend Davidson against the charge of idealism, it is worth noting how Davidson supports his claim. He appeals to Tarski as providing us with our best intuition about the concept of truth (1984: 194-5). According to him, an implication of this intuition is that every truth can be translated into our language. Some philosophers have discussed this appeal to Tarski (Hacker 1996: 300-1; Soames 2003: 324-330), but Nagel never does. I do not want to go into more detail about it here, only to make the following point. The fact that Nagel does not discuss Davidson's appeal to Tarski supports the suspicion raised in the third section of my paper that Nagel simply does not specify the reasons actually given for (4). If Davidson is asserting (4), he is doing so for a philosophical reason and Nagel does not address this reason. Let us return now to the purpose of this section: to dispute Nagel's charge of idealism against Davidson. The example in the previous section enables us to see how one can respond on behalf of Davidson. Using a vocabulary that the child in the example already has, it is possible to explain how it is within the power of one player to achieve checkmate in four moves. There is no individual piece of terminology involved in this explanation whose meaning eludes the child. Nevertheless, the child cannot understand the explanation. As such, there is a potential gap between which truths can be expressed in a language spoken by a particular person and which of those truths can be grasped by that person. This allows us to envisage the following possibility. There are beings of superior intelligence who speak a language that we do not know and sometimes make true statements in this language that we could never understand without a dramatic improvement in our mental faculties. Nevertheless, it is possible to translate the sentences that are used to express such truths into our language. It is just that we cannot understand these sentences. If there are beings of superior intelligence who speak our language, they Nagel on Conceivability 28 might be able to understand what is being said, but we cannot. When Davidson denies that there might be truths that can only be expressed in other languages, he leaves room for this possibility. Hence he can admit that there could be things that we cannot possibly conceive. He can say, 'We cannot understand some, or all, true statements about such things. Nevertheless, any truth that can be stated in another language can also be stated using our language.' This stance does not occur to Nagel. Nagel tells us that what he takes to be a form of idealism was popular at the time when he was writing. But while making his case against it, he only refers to Davidson as an example of this sort of idealist. Later on in the chapter, Strawson and Wittgenstein are also identified as idealists. The latter is presented as an important source of contemporary idealism (1986: 105). Whether or not Nagel is right about this or right to label these two philosophers as idealists, for now we should reject the charge that Davidson is one. Idealism is not regarded as an attractive metaphysical position within the philosophical culture from which Davidson's work emerges. This means that we should suppose that Davidson too is opposed to idealism unless we encounter evidence that indicates otherwise. There may be evidence of this kind, but Nagel does not provide us with it. Terence Rajivan Edward The University of Manchester [email protected] T.R. Edward 29 References Avramides, A. (2006) 'Thomas Nagel: The View from Nowhere', IN J. Shand (ed.), Central Works of Philosophy, Volume 5: The Twentieth Century: Quine and After, Chesham: Acumen. Case, J. (1997) 'On the Right Idea of a Conceptual Scheme', The Southern Journal of Philosophy, 35, 1-18. Davidson, D. (1984) 'On the Very Idea of a Conceptual Scheme', IN Inquiries into Truth and Interpretation, Oxford: Clarendon Press. Hacker, P. (1996) 'On Davidson's Idea of a Conceptual Scheme', The Philosophical Quarterly, 47, 285-305. McGinn, C. (1987) 'Critical Notice of The View from Nowhere by T. Nagel', Mind, 96, 263-272. Nagel, T. (1986) The View from Nowhere, New York: Oxford University Press. Soames, S. (2003) Philosophical Analysis in the Twentieth Century. Volume 2, The Age of Meaning, Princeton: Princeton University Press.

Kant's post-1800 disavowal of the highest good argument for the existence of God I have two main goals in this paper. The first is to argue that Kant gave up on his highest good argument for the existence of God around 1800. The second is to revive a dialogue about this thesis that died out in the 1960s: I believe that this dialogue, which seems to have concluded with the naysayers in ascendence, ended prematurely. The paper is divided into three sections. In the first, I reconstruct Kant's highest good argument in order to introduce the first piece of evidence in favor of my thesis: Kant's deep-seated ambivalence about the various premises in this argument. In the second, I turn to Kant's Opus postumum in order to canvass the second piece of evidence in favor of my thesis: Kant's many claims to the effect that there is only one way to argue for the existence of God, a way which resembles the highest good argument only in taking the moral law as its starting point. In the third, I examine the counterarguments to my thesis as they were introduced in the 1960s, and I explain why I do not find them persuasive: although they undercut some of the evidence mustered by Adickes (the original proponent of my thesis), they leave the evidence on which I build my argument unimpugned. In so doing, I introduce the third and final piece of evidence in favor of my thesis: the continuity between Kant's Opus postumum argument and a line of thought from his earlier work. Section 1. Kant's highest good argument I offer the following enumerated reconstruction of Kant's highest good argument: 1. The highest good is a world in conformity with all moral laws and in which happiness is united to virtue. 2. There is a duty to promote the highest good. 3. Ought implies can. 4. It is possible to promote the highest good only if God exists and we have immortal souls. 5. There are no theoretical proofs for or against God and immortality. 6. In the absence of theoretical proof one way or another [premise 5], the practical grounds from the highest good [premises 1-4] justify belief in God and immortality for the purposes of morality. 7. Therefore, belief in God and immortality is justified for the purposes of morality. Page ! of !1 19 In this section, I defend this reconstruction and then introduce evidence of Kant's ambivalence about each premise. Kant's definition of the highest good can be drawn from his Critique of pure reason discussion of the question: what may I hope? He begins by noting that a moral world is one "in conformity with all moral laws" (KrV A808/B836). But according to Kant, "all hope concerns happiness" (KrV A805/1 B833), so the idea of a moral world does not suffice to answer the question of what I may hope; rather, because conforming to moral laws renders one worthy of happiness, Kant maintains that "everyone has cause to hope for happiness in the same measure as he has made himself worthy of it in his conduct" (KrV A809/B837). Thus, the ideal of the highest good is a world in conformity with all moral laws and in which happiness is united to virtue. Kant is less clear on the second premise of the argument in the Critique of pure reason. But it comes out explicitly in the Critique of practical reason: "[i]t is a duty to realize the highest good to the utmost of our capacity" (KpV AA 5:143n). Kant does not have an argument for this duty in the way that 2 he does for other duties. Moreover, it is unclear how such an argument could go given that the moral 3 world exhausts the commands of morality and the highest good is something over and above the moral world. I shall return to this below. For now, the point is that Kant's use of the premise that agents have a 4 duty to promote the highest good in the Critique of pure reason may be inferred from the third step of the argument: his appeal to "ought implies can" (OIC). In the Critique of pure reason, Kant appeals to OIC in summarizing his reasoning about the question of hope: ...all hope concerns happiness, and with respect to the practical and the moral law it is the very same as what knowledge and the natural law is with regard to theoretical cognition of things. The former finally comes down to the inference that something is (which determines the ultimate final end) because something ought to happen... (KrV A805f/ B833f) Except where otherwise noted, all translations are from the Guyer/Wood Cambridge blue series. All citations follow the stan1 dard Academy pagination except for those from the Critique of pure reason (which follow A/B pagination). Similarly, see VNAEF AA 8:419n: "it is a duty to work toward a certain purpose (the highest good)."2 Beck contends that "it is seriously misleading to say that there is a command to seek the highest good which is different from 3 the command to fulfil the requirements of duty" (1963, 245). This point can be sharpened: it is also unclear how there could be a duty to realize a moral world, for if an agent realizes his/her 4 own moral perfection, s/he has exhausted the commands of morality directed at him/her, and the moral world is something over and above the moral perfection of any single agent. Page ! of !2 19 This is sometimes referred to as the "capacity-expanding" application of OIC. That is, Kant infers from 5 (1) OIC and (2) one ought to promote the highest good to (3) it is possible to promote the highest good, thereby appealing to OIC and modus ponens to expand our capabilities. 6 Kant's appeal to OIC works together with the fourth premise of my reconstruction to generate practical grounds for belief in the practical postulates. In the Critique of pure reason, for instance, Kant claims that the highest good argument leads to the concept of a will that is "omnipotent,...omniscient,... omnipresent,...eternal..., etc." (KrV A815/B843). Similarly, in the Critique of practical reason Kant claims that this argument enables him to infer the "omniscience, all-beneficence, omnipotence, and so forth" of the author of the world (KpV AA 5:139). 7 The fifth premise in my reconstruction might seem unnecessarily strong. On the one side, it might be thought that theoretical proofs could work in concert with practical grounds. On the other side, it might be thought that practical grounds have the potential to trump theoretical proofs to the contrary. Nonetheless, Kant does insist on this idea. Indeed, in the Critique of practical reason Kant suggests that theo8 retical proofs for God and immortality would undermine morality because, armed with such proofs, humans, being what they are, would not pursue moral ends from duty but from fear of punishment. And 9 Kant maintains that theoretical proofs against God and immortality would trump practical ones because practical reason takes primacy over theoretical reason only when its object is theoretically possible. 10 The sixth and final premise in the argument, latent in the passages cited above, is that in the absence of theoretical proof one way or another, the practical grounds from the highest good justify belief in God and immortality. However, it is important not to misunderstand this key step in the argument. Kant is not arguing that morality makes true claims about the world and thereby provides truth-conducive evidence for God and immortality. Rather, Kant's idea is that morality lowers the evidentiary bar required for one to be justified in believing. Thus, Kant cautions us that although through this argument "reality is 11 See Martin (2009, 110). Timmermann thinks that Kant uses OIC only in its capacity-expanding (and never in its duty-restrict5 ing) form (2003, 118). But in On the common saying Kant argues that because humans are unable to renounce their happiness, they have no duty to do so (TP AA 8:276-8). As reconstructed here, the argument in the Critique of pure reason is enthymematic: the second premise is unexpressed. See n24 6 below and the paragraph to which it is appended for a complication regarding the interpretation of this enthymematic premise. Förster claims that it is first in the Critique of the power of judgment that Kant infers an omniGod from the highest good argu7 ment (2005, 130). As the paragraph to which this note is appended shows, this is incorrect. Perhaps most famously when asserting the need to "deny knowledge in order to make room for faith" (KrV Bxxx). But see 8 also, e.g., KpV AA 5:142 and VNAEF AA 8:418n. KpV AA 5:146-8. 9 The claim that practical reason presupposes the theoretical possibility of its object is baked into Kant's appeal to OIC (see KpV 10 AA 5:143). In saying this I am ascribing to Kant a moral encroachment theory of justification like that developed in Pace (2011). For help11 ful discussion of different kinds of warranted assent in Kant, see Chignell (2007). However, I am wary of the behaviorist connotations of assent in this context. Page ! of !3 19 given to the...ideas of God, freedom, and immortality," nonetheless, this is "always only with reference to the practice of the moral law" (KpV AA 5:138). Kant takes his highest good argument quite seriously, not only stating it repeatedly in multiple works but also claiming that without the doctrine of the highest good, "we must...regard the moral laws as empty figments of the brain" (KrV A811/B839). However, as noted above and as I shall try to substantiate now, Kant was ambivalent about the premises of this argument. 12 That Kant has different conceptions of the highest good has been explored extensively in the secondary literature. Indeed, some argue that Kant's different conceptions of happiness (not documented 13 here) are due partly to ambiguity about whether the highest good must be realized in this world or another. This is related to the fact, noted above, that in the Critique of pure reason the highest good is intro14 duced as something for which to hope, something to give motivational force to the laws of morality; in the Critique of practical reason, by way of contrast, these ideas about motivation are jettisoned (only to be picked up again in later works). Instead, the highest good is introduced as the "unconditioned totality 15 of the object of pure practical reason" (KpV AA 5:108). 16 Issues surrounding the nature of the highest good bleed into issues surrounding the second premise in the above reconstruction, for the conception of the highest good will determine the nature of the duty to promote it. But there are also other issues regarding the second premise. For example, in the 17 Critique of practical reason, Kant remarks that because "the highest good...is an a priori necessary object of our will and inseparably bound up with the moral law, the impossibility of the first must also prove the falsity of the second" (KpV AA 5:114). But in the Critique of the power of judgment, Kant suggests that 18 someone who cannot convince herself of the existence of God would not have to surrender the moral law: rather, "[a]ll that would have to be surrendered in that case would be the aim of realizing the final end in Kant's ambivalence extends beyond the substance of the premises: "Kant uses the word 'postulate' rather loosely and...his list 12 of practical postulates varies from place to place...even within the Critique of Practical Reason" (Beck, 1963, 259). See, e.g., (Reath, 1988).13 Consider: "It is not by chance that there is ambiguity in the definition of happiness and ambiguity surrounding the location of 14 the highest good" (Wike, 1994, 25). I follow Wood (2001, 281n11) in being suspicious of Wike's explanation of Kant's conceptions of happiness. See, for example, RGV AA 6:5, where Kant asserts that the idea of the highest good "meets our natural need, which would 15 otherwise be a hindrance to moral resolve." It might be helpful to consider this in light of the remarks in n4 above and the paragraph to which it is appended. I return to it below. For a fuller discussion of this point (and much else), see Pasternack and Rossi (2014, section 3.5) or Düsing (2002, 105).16 Beck points out that if the highest good is conceived as a world in which happiness is proportionate to virtue (as Kant is wont 17 to conceive it) rather than as a world in which everyone is maximally virtuous and maximally happy (as in the above reconstruction), then the Critique of practical reason immortality argument is rendered void: the highest good so conceived would not require holiness or, thus, an infinite temporal progression toward holiness (1963, 268f). Echoing the claim from the Critique of pure reason (reproduced above in the paragraph to which n12 is appended) about 18 morality being an empty figment of the brain without the highest good. Page ! of !4 19 the world" (KpV AA 5:451). Both of these passages evince Kant's commitment to OIC. But whereas the 19 first suggests a strong commitment to a duty to promote the highest good (bound up with the legitimacy of morality as a whole), the second suggests a weaker commitment to the pursuit of the highest good, a pursuit that even may be foresworn (without rejecting morality writ large) in case the conditions of its realization are rejected. 20 The preface to the first edition of the Religion within the boundaries of mere reason reveals a yet more complicated picture of Kant's commitment to both the second and third premises. Kant says there that the command to pursue the highest good "does not increase the number of morality's duties" but rather arises from "our natural need" to have an end at which to aim when considering the commands of morality, a need the repudiation of which would be a "hindrance to moral resolve" (RGV AA 6:5). Kant 21 reiterates this in an extended footnote. In these passages, to promote the highest good is enjoined by rea22 son, but it exceeds and is not contained in the moral law. Nonetheless, he goes on to assert that because 23 human capacity does not suffice to bring about the highest good, "an omnipotent moral being must be assumed...under whose care this would come about" (RGV AA 6:8n). This bears on Kant's commitment to OIC (premise 3), for the "ought" in this version of the argument is not the ought of duty but something else. 24 Kant's ambivalence regarding the fourth premise manifests in his varying explanations of why God and immortality are necessary for the highest good. For instance, in the Critique of practical reason the postulate of immortality flows "from the practically necessary condition of a duration befitting the complete fulfillment of the moral law" (KpV AA 5:132). But in part two of the Religion within the boundaries of mere reason Kant seems to repudiate this, arguing that a change in disposition is sufficient for moral goodness and, therefore, that "notwithstanding his permanent deficiency, a human being can still expect to be generally well-pleasing to God, at whatever point in time his existence be cut This point is also made in Pasternack and Rossi (2014, sections 3.5.4-5).19 This poses a problem for the Düsings' claim that the "essential contents" of the Critique of practical reason highest good ar20 gument are preserved in the Critique of the power of judgment (2002, 112): on their account of the former, one cannot relinquish the highest good without giving up all other moral ends, for all other moral ends are subordinate to the highest good (2002, 108). This might be taken to ground an indirect duty to promote the highest good. However, Kant does not say so, and detailed dis21 cussion is beyond the scope of this paper. RGV AA 6:7n. Similarly, in an important footnote in On the common saying Kant remarks that "the need for a final end as22 signed by pure reason...is a need of an unselfish will extending itself beyond observance of the formal law" (TP AA 8:280n). However, he then asserts that "there is...the duty to bring it about as far as we can that such a relation (a world in keeping with the moral highest ends) exists." Kant's argument for this rests on the assertion that the laws of morality "command absolutely...and thereby...make of duty an 23 object of the highest respect, without proposing to us...an end...such as would constitute some sort of inducement" (RGV AA 6:7n). This might be read as an assertion that there are no ends that are also duties. If so, it is contradicted in the Metaphysics of morals. Whether Kant was aware of this is unclear; note that in the Critique of pure reason passage reproduced above, Kant appeals to 24 OIC "with respect to the practical and the moral law" (my emphasis). Page ! of !5 19 short" (RGV AA 6:67). Similarly, Förster catalogs no fewer than four separate roles attributed to God in 25 these texts. And it is also notable that freedom, omitted in my reconstruction above, is sometimes men26 tioned and sometimes overlooked in Kant's various statements of the highest good argument. All of these shifts suggest Kant's dissatisfaction with his articulation and defense of premise four. Kant's changing attitude toward the fifth premise is evidenced by his changing attitude toward physicotheology. For example, in the Critique of pure reason Kant argues that "neither in speculative nor in natural theology...do we find even a single significant ground for assuming a single being to set before all natural causes" (KrV A814f/B842f): the highest good argument is what leads (inexorably) to physicotheology (KrV A815f/B843f). But in the Critique of practical reason, Kant claims that even if physicotheology cannot justify belief in an omniGod, "we can well infer from...[the] order, purposiveness, and magnitude [of the world] a wise, beneficent, powerful, and so forth author of it" (KpV AA 5:139). More, 27 although Kant does not think that physicotheology can justify belief in an omniGod by the time of the Critique of the power of judgment, he does think that it provides the highest good argument with "desired confirmation" (KU AA 5:479). Note the change in evidential relations: from highest good leading to 28 physicotheology to the two being independent to physicotheology providing confirmation. With all of this (that is, Kant's repeated and substantive reformulations of the highest good argument) in mind, I submit that it should be unsurprising to find Kant eschewing the highest good argument for the existence of God in the Opus postumum. 29 Section 2. Kant's Opus postumum rejection of the highest good argument In order to find support for my thesis in the Opus postumum, one must look at the first and seventh convolutes of the text (in volumes 21 and 22 of the Academy edition), which are dated to 1800 or thereafter. 30 Kant's discontent with his immortality argument might be borne out in his claim that belief in immortality "is only a belief in 25 the second rank" (Refl AA 19:644; translation from Wood (1970, 182n)). However, Kant continues unambiguously to support the immortality postulate in published work as late as 1796 (see VNAEF AA 8:418). See (2005, esp. 134f).26 This poses a problem for the Düsings, who maintain that, according to Kant, theoretical philosophy yields only negative theol27 ogy (2002, 101). At KU AA 5:445 it provides "incidental confirmation."28 It might be objected that these reformulations evince Kant's increasing acuity regarding the highest good. Length constraints 29 prevent me from addressing this objection, which I owe to an anonymous referee, in full. But I think it worth pointing out that Kant goes back and forth repeatedly on these issues (see n15 above and the paragraph to which it is appended) and does not seem always to be aware of either his vacillations or their implications (see n24 above and the paragraph to which it is appended). See Förster's discussion of Adickes' dating of the manuscript in the translator's introduction to the Cambridge blue series Opus 30 postumum (xxiv-xxix). See also Adickes (1920, part 1). Page ! of !6 19 There are four main ways in which the Opus postumum supports my thesis: (1) Kant articulates (repeatedly) a different argument for the existence of God, one which also falls within the purview of moral theology; (2) Kant claims (repeatedly) that this different argument is the only possible way to justify belief in God; (3) Kant says that the concept of God just is the concept of the author of the laws of duty (a concept that encapsulates the argument referred to in (1)); and (4) the highest good argument has all but disappeared. I shall return to (4) in the next section. For now, I focus on (1), (2) and (3). In support of (1), consider the following 8 passages: a) The categor. Imper. and the thereupon grounded knowledge of all man's duties as divine commands is the practical proof of God's being. (OP AA 21:74.8-10) 31 b) Categorial imperative which our reason expresses through the divine. Freedom under laws, duties as divine commands. There is a God (OP AA 22:104.16-18) c) There is a God: for there is in moral-practical reason a categorical imperative, which extends to all rational world-beings and through which all world-beings are united. (OP AA 22:105.1-3) d) Reason proceeds according to the categorical imperative, and the legislator is God.- There is a God because there is a categorical imperative. (OP AA 22:106.17-19) e) Moral-practical reason in the categorical imperative in freedom under laws shows that such a being [as God] exists in the knowledge of all duties as divine commands. (OP AA 22:108.22-25) 32 f) The sum of all duties as divine commands exterminates atheism, pantheism and the denial of God. (OP AA 22:113.21-23) 33 g) The existence of such a being, however, can only be postulated in a practical respect: Namely, the necessity of acting in such a way as if I stood under such a fearsome-but yet, at the same time, salutary-guidance and also guarantee, in the knowledge of all my duties as divine commands (tanquam non ceu)... (OP AA 22:116.20-24) h) A command, to which everyone must absolutely give obedience, is to be regarded by everyone as from a being which rules and governs over all. Such a being, as moral, however, is called God. So there is a God. (OP AA 22:127.1-4) In these passages, Kant articulates a new argument for God's existence. This argument, like the highest good argument, begins from the moral law. However, it does not proceed through the highest good: it appeals directly to the necessity of regarding duties as divine commands, whence it follows that My translation.31 My translation.32 My translation.33 Page ! of !7 19 God exists (in order to command accordingly). I shall return to this argument toward the end of section 3 below to suggest one way in which it might be fleshed out. For now, I take these 8 passages to establish (1), that in the Opus postumum Kant articulates (repeatedly) a different argument for the existence of God (different from the highest good argument), one which also falls within the purview of moral theology. 34 Now consider the following 3 passages in support of (2) (which also support (1)): i) However, there still seems to be the question as to whether this idea, the product of our own reason, has reality or whether it is a mere thought-object (ens rationis), and there remains to us nothing but the moral relationship to this object [namely, God]-which is merely problematic, and which leaves only the formula of the knowledge of all human duties as (tanquam) divine commands, whenever the iron voice of the categorical imperative of duty resounds between all siren temptations of the senses and threatening deterrents. (OP AA 22:117.7-15) j) Only as hypothetical, however, can such an ens constitute a principle-not as given, but only as thought...but only for the sake of the recognition of our duties as divine commands. (OP AA 22:125.30-126.3) k) There is only one practically sufficient argument for faith in one God, which is theoretically insufficient-knowledge of all human duties as (tanquam) divine commands. (OP AA 22:127.12-14) In these passages, Kant says that the only way to justify belief in God is through the moral relationship referred to in passages a-h ("there remains to us nothing but the moral relationship to this object"); that God can be thought as hypothetical "only for the sake of the recognition of our duties as divine commands"; that there is "only one practically sufficient argument" for God, the one occurring in passages a-h, not the highest good argument. To put it bluntly: according to the post-1800 Opus postumum Kant, there exists exactly one argument to justify belief in God, an argument beginning with the idea that agents are compelled to regard their duties as divine commands. Finally, I reproduce the following 4 passages in support of (3): l) God is...the personified idea of justice and benevolence... (OP AA 22:108.11-12) m) ...he who is justified through the categorical imperative alone to give pronouncements of the same for all rational beings, is God... (OP AA 22:109.7-8) 35 Kant's Reflexionen from this time period are also revealing. Consider the following excerpt from a reflection dated to 1800: 34 "the concept of God arises from morality. The observance of all moral duties as (instar) divine commands" (Refl AA 19:650.19-20, my translation). My translation.35 Page ! of !8 19 n) ...a highest being commanding supremely through the categorical imperative, setting all rational beings in the world in the unity of ethical relations-God. (OP AA 22:113.17-19) 36 o) A being, who is capable of and entitled to command a1l rational beings according to laws of duty (the categorical imperative) of moral-practical reason, is God...(OP AA 22:116.12-14) In these passages, Kant introduces the concept of God as if derived from the argument in passages a-k: God is justified by virtue of regarding one's duties as divine commands, whence it follows that the concept of God is the concept of a being "who is capable of and entitled to command all rational beings according to laws of duty." Thus, I think passages l-o provide confirmation for the lesson I am trying to draw from passages a-k. Many more passages could be cited in support of (1), (2) and (3). However, I would like to say 37 only two more things before turning to the next section. First, one might make finer distinctions than I have in setting out the Opus postumum argument for God. The Opus postumum is a set of notes Kant made for himself, not for publication, and he tries 38 and retries diverse lines of argument, sometimes with subtle (and sometimes with not so subtle) variations. For instance, at OP AA 22:120.1-2 Kant says that "[t]o prescribe all human duties as divine commands is already contained in every categorical imperative," which suggests that he might take this new argument to be analytic. Another variation is suggested at the end of OP AA 22:121.13-21 when Kant remarks that the knowledge of duties as divine commands engendered through his argument is authorized "as a principle of practical reason, in which there is a valid inference from ought to can." It is unclear what work OIC is doing here, and this is one of the only places in the Opus postumum where OIC is appealed to in this context. One might conjecture that it is a vestige of the highest good argument. However, I shall not pursue such conjectures here: for my purposes, it suffices to point out that the theme on which these variations are made is the disappearance of the highest good. Second, Kant's new argument for God in the Opus postumum is connected with his Opus postumum argument for the uniqueness of God, one in which he maintains that the existence of more than one My translation.36 For instance, OP AA 21:17.2-3, 17.12-15, 22.28-29, 22.30-31, 25.10-21, 28.11-17, 30.4-11, 37.11-18, 50.20-22, 60.31-32, 37 79.11-12, 113.9-10, 118.16, 144.24-25, 145.4-5, 146.25-28, 152.18-21 and 157.16-17. See also OP AA 22:49.21-22, 49.23-26, 51.18-52.2, 53.3-6, 54.7-8, 56.1-2, 57.16-18, 57.22-25, 58.3-9, 58.30-32, 59.1-2, 64.21-29, 104.8-12, 105.6-9, 109.20-25, 112.3-10, 116.10-11, 116.27-117.5, 118.11-13, 118.14-15, 119.20-22, 120.8-15, 120.24-26, 122.3-8, 122.22-23, 122.24-25, 126.18-23, 127.5-11, 127.23-26, 128.11-21, 128.22-24 and 129.3-6. Adickes distinguishes four versions (1920, 802-11). Smith collapses Adickes' first two versions but otherwise follows him in 38 this (1962, 638-40). The distinction between the third and fourth versions also might be collapsable given Kant's Opus postumum contention (perhaps articulated under the influence of Lichtenberg (see Smith, 1962, 637f)) that to have an idea of God is to believe in God. (It is on the basis of this contention that some maintain that Kant renounces his critical stance toward the ontological argument in the Opus postumum.) Page ! of !9 19 being who is all-obliging but never obliged is self-contradictory. Moreover, Kant often seems to want to 39 make both arguments in one fell swoop: p) The being whose will is a practical law for all rational beings, is the highest moral being...the highest intelligence which is distinguished from all world-beings and which is law-giving by one principle, that is, it is God. There therefore is one God. (OP AA 22:114.15-19) 40 This connection is more notable because Kant's preoccupation with proving the uniqueness of God in the Opus postumum seems to be connected (sometimes explicitly) with his efforts to prove the uniqueness of the world. This indicates that Kant's interest in his new argument for God is bound up with a larger re41 search program aimed at a suite of problems he was trying to solve at the time. It thus lends further legitimacy to my thesis. Section 3. Why I do not find the counterarguments persuasive As noted in the introduction to this paper, my thesis originally was articulated by Adickes in his 1920 commentary on the Opus postumum. Adickes' commentary, which came out before the Opus postumum 42 was more generally available, makes extensive use of Kant's text and was widely influential. For example, in the second edition of his commentary on the Critique of pure reason, Smith generally follows Adickes' lead in interpreting the Opus postumum, including Adickes' claim regarding Kant's rejection of the highest good argument for the existence of God. Dakin then follows both Adickes and Smith in an 43 essay on Kant's philosophy of religion. 44 See, e.g., OP AA 22:124.22-26.39 My translation. See also OP AA 22:61.8-11. The claim that there is (only) "one principle" might be an attempt to rule out the 40 possibility of two all-but-one-obliging (or three all-but-two-obliging, etc.) but never-obliged beings. But if there can be multiple sources for a single principle, the challenge remains. I think this comes out especially in the first convolute. See OP AA 21:37.7-8, 40.9-11, 48.24-25, 54.18-20, 55.8, 57.25-26, 41 71.18-20, 79.10-12, 91.4-5, 140.19-20, 141.1-2, 143.4-6, 143.14, 143.26-27, 144.16-17, 144.22-23, 145.6-8, 151.24-26, 151.27-29, 152.1-2, 152.23-24 and 157.16-18. But compare OP AA 22:49.27-28, 53.7-10, 59.10-12, 64.1, 64.17-18 and 125.1-2 from the seventh convolute. Adickes (1920, part 4, section 2, chapter 4).42 Smith (1962 [originally 1923], appendix C). 43 Dakin (1962 [originally 1939], 413-6).44 Page ! of !10 19 However, after the Opus postumum became more generally available, Schrader published a criticism of Adickes' argument and the dialogue seems to have died off. In support of this (i.e., that the dialogue died off), I offer three pieces of evidence. First, the Stanford Encyclopedia of Philosophy entry on "Kant's Philosophy of Religion" does not cite Adickes in any of its (extensive) bibliographies, and although it mentions (and repudiates) commentators who think Kant rejected the highest good argument for immortality, it devotes only a short paragraph to consideration of God in Kant's Opus postumum and does not suggest that such consideration might reveal Kant's rejection of the highest good argument for God. 45 Second, although in his recent monograph on the Opus postumum Förster devotes chapter five to discussion of "The Subject as Person and the Idea of God" and gives (therein) an admirable account of some of the differences in Kant's various highest good arguments for God, Förster does not refer to my thesis or to Adickes anywhere in that chapter or its associated endnotes, and he ends up coming to a different conclusion about Kant on God in the Opus postumum. Third, prominent commentators writing in the immedi46 ate wake of Schrader's criticism (to which I shall advance forthwith) cite this criticism as a decisive rebuttal of Adickes. 47 Schrader begins by pointing out the difficulties inherent in ascertaining Kant's post-1800 views. Schrader argues that "[t]he Opus Postumum is quite similar in character to the Reflexionen" (1951, 230). He continues in a footnote: "One can imagine the difficulties that would be involved in trying to ascertain Kant's views if it were necessary to substitute his Reflexionen for the Critique of Pure Reason" (1951, 230n2). Schrader concludes that because there is "no indication" of a shift in Kant's thinking with regard to the highest good argument prior to 1800, any case for a shift after 1800 "is initially weakened" and must be tenuous, indeed. I would like to say two things about this. First, there are multiple shifts in Kant's thinking on the highest good argument prior to 1800. As noted in the first section of this paper, Kant tergiversates on various parts of this argument, and throughout his pre-1800 work he continued to experiment with different versions in ways that often fly in the face of earlier ones. Thus, although there is no clear evidence of Kant's having rejected the highest good argument as a whole prior to 1800, his texts do contain clear evidence of his dissatisfaction with the argument and, thus, pave the way for his rejection of it. Pasternack and Rossi (1994, sections 3.6.3 and 4). The neglect of Adickes is made more noticeable by the inclusion in their 45 second bibliography of Greene's essay on Kant's Religion within the boundaries of mere reason, an essay in which Greene, writing (like Smith) before the full text of the Opus postumum became generally available, accepts Adickes' contention about Kant's abandonment of the highest good argument for God (1960 [originally 1934], lxvf). Förster (2005, chapter 5, esp. 137-47). Given that Förster refers to Adickes elsewhere in the text and even includes Adickes in 46 the index, his silence regarding Adickes in chapter 5 is especially telling. It is also worth pointing out that Förster makes no reference to Adickes or to my thesis in the section on "Practical self-positing and the idea of God" in the translator's introduction to the Cambridge blue series Opus postumum. Consider: "After a careful examination of [the Opus postumum]...Professor G.A. Schrader demonstrated the inadequacy of 47 Adickes' interpretation" (Greene and Silber, 1960, cxl). See also Beck (1963, 274n35). Beck's post-Schrader disavowal of Adickes' thesis is the more remarkable because he previously had subscribed to it (1950 [originally 1949], 47-9). Page ! of !11 19 Second, Schrader's analogy is a bad one. It is true that there are ways in which the Opus postumum is like Kant's Reflexionen and that it would be well nigh impossible to reconstruct the Critique of Pure Reason from the Reflexionen. But on the one side, I am not attempting to reconstruct Kant's post-1800 views writ large: my thesis is about only one specific part of Kant's post-1800 views. And on the other side, I am not attempting to reconstruct Kant's views on this particular question by appeal to a single or even a small number of fragmentary and merely suggestive excerpts. On the contrary, Kant revisits this issue regularly throughout the Opus postumum, and as may be seen from the previous section, his texts are as clear and unequivocal on this score as could be desired. Schrader next points out that Adickes based his advocacy of my thesis on perceived deficiencies in Kant's highest good argument. For example, Adickes thought that the highest good argument is too objective, and when Kant says in the Opus postumum that faith in God is purely subjective, Adickes takes Kant to have recognized the flaw that he, Adickes, diagnosed. Against this, Schrader complains that Kant characterizes the highest good argument as subjective in his earlier work. Similarly, Adickes thought 48 that the highest good argument introduces a hedonistic principle into Kant's ethics and that Kant came to recognize this in the Opus postumum, whereas Schrader points out that there is no evidence that Kant ever took his highest good argument to introduce hedonism into his moral philosophy and, in fact, there is ample evidence that he did not. 49 I would like to steer a middle course between Adickes and Schrader on these issues. Against Schrader I note that Kant's characterization of the highest good argument as subjective prior to 1800 is no evidence that he did not come to see it as objective later, and the same goes for Adickes' point about hedonism. But against Adickes, I agree with Schrader that there is no evidence that Kant came to view the highest good argument as objectionably objective or hedonistic: there is ample evidence that Kant rejected the highest good argument, but why he did so is a matter of conjecture. Moreover, I do not share Adickes' diagnosis of the argument's deficiencies. I do share Adickes' view that Kant was aware of some of the problems with the highest good argument. But I take these problems to lie elsewhere than in be50 ing too objective or hedonistic. However, Schrader also points out that the highest good argument has not disappeared from Kant's post-1800 philosophy. For example, consider the following two passages from the first and seventh convolutes, respectively: Now since wisdom, in a strict sense, only can be attributed to God and such a being at the same time must be endowed with all power; because without this the final end (the highSee (1951, 231-3 and 236f). Schrader is followed here by Silber (Greene and Silber, 1960, cxl).48 See (1951, 234f). Silber argues also that hedonism has a positive place in Kant's philosophy (Greene and Silber, 1960, cxlf).49 For evidence I point to places where Kant seems explicitly to disavow earlier versions of the argument (see n20 and n25 above 50 and the paragraphs to which they are appended). Page ! of !12 19 est good) would be an idea without reality; thus the proposition: There is a God becomes an existential proposition. (OP AA 21:149.20-24) 51 The first question is: is there a moral, practical reason and together with this a concept of duty as a principle of freedom under laws, finally if there is a Substance which, according to these laws...judges them [men] to be worthy or unworthy of happiness and makes it possible for them to participate in it. (OP AA 22:125.23-27) 52 In these two passages, Kant makes an argument like the one canvassed in section one of this paper: the existence of God is justified by appeal to the necessity of God for fulfilling the duty to promote the highest good. In fact, Adickes himself mentions passages in the post-1800 convolutes of the Opus postumum in which Kant rehearses the highest good argument. Adickes does not take these passages to undermine his 53 thesis because he adopts the sound interpretive principle of considering the convolutes of the Opus postumum holistically: rather than "tear out individual passages and consider them in isolation," Adickes insists that "one must...interpret one passage in terms of others and in terms of the whole background of thought which one is able to infer from their totality" (1920, 772). Indeed, this is the method that Kant 54 recommends for understanding his work. 55 Schrader is cognizant of Adickes' interpretive method and, in fairness to Schrader, does not take the comparatively few passages from the first and seventh convolutes in which Kant affirms the highest good argument by themselves to invalidate my thesis. Neither do I. Following in Adickes' footsteps with 56 regard to method, it seems to me that the weight of the (many) passages in which Kant attempts to infer God directly from the Categorical Imperative (taken as a divine command), of the (many) passages in which Kant states that this is the only way to justify belief in God, of the (many) passages in which Kant cashes out the concept of God in terms of divine commands, and of the (many) passages in which Kant appeals to his parallel argument to prove the uniqueness of God-the weight of all of this easily tips the scales against the weight of the passages cited by Schrader (and acknowledged by Adickes). My translation.51 I have used Schrader's translation (1951, 236). However, Schrader cites the passage as being from OP AA 22:126.52 See (1920, 801n2). Indeed, I owe the first of the two passages above to Adickes, who labels it C415 (1920, 783). This might 53 come as a surprise to Silber, who attributes to Adickes the assertion that "Kant fails to restate the [highest good] argument in the Opus postumum," an assertion that (says Silber) can be dismissed in the wake of Schrader's investigation (Greene and Silber, 1960, cxlif). This passage is quoted in Schrader (1951, 230). I have modified slightly Schrader's translation. Silber also is aware of Adickes' 54 interpretive principle (Greene and Silber, 1960, cxl). See (KrV Bxliv).55 See (1951, 236): "I would not want to make too much of this passage, and certainly would not attempt to refute Adickes' inter56 pretation solely on the basis of it." Page ! of !13 19 In fact, I think these putatively counter-passages speak in favor of my thesis rather than against it. That is, rather than merely take these passages as counterweight to my thesis which is nonetheless anchored in place by others, I take the presence (and relative scarcity) of these passages to make more impressive the (mass of) passages in which Kant makes his divine command argument, for their presence renders it more difficult to deny that Kant was alive to the difference between these arguments. The presence of these passages shows what Kant could have done but did not do, namely: continue to tinker with the highest good argument, or even just let the highest good argument lie fallow and work on others. Instead of pursuing either of these strategies, however, Kant struggled over and over to articulate a different argument, one which bypasses the highest good, and he claims repeatedly that this is the only viable argument, clearly thereby eschewing the highest good argument. To put the point formulaically, the exception proves the rule. I shall confront one last criticism from Schrader: he complains that the passages in the Opus postumum in which Kant speaks of the Categorical Imperative as the voice of God "are perfectly consistent with his critical position" (Schrader, 1951, 240). Schrader substantiates this by appeal to excerpts like this one: ...the moral law leads through the concept of the highest good, as the object and final end of pure practical reason, to religion, that is, to the recognition of all duties as divine commands... (KpV AA 5:129) 57 In this excerpt (published before the Opus postumum passages considered in the previous section were written), Kant suggests that the Categorical Imperative be regarded as a divine command on the basis of the highest good argument. Schrader takes the continuity between this excerpt and the claims made in the Opus postumum to indicate that Kant has neither repudiated nor modified his earlier position. I maintain otherwise. Although there is continuity between this excerpt and the Opus postumum argument (in the idea of regarding duties as divine commands), there is a crucial difference: the stepping stone of the highest good all but disappears in the first and seventh convolutes. The reason this is so crucial is that it manifests and, I think, magnifies a difference in the structure of the arguments. That is, the point of conflux between pre-1800 and post-1800 passages (regarding duty as divine command) foregrounds the fact that they diverge in the justificatory structure of the connection between this regarding and belief in God: pre-1800 justifies regarding duty as divine command on belief in God (in turn justified Schrader reproduces this passage on (1951, 239), however he cites it as coming from 5:140 (1951, 239n5). I have used the 57 Cambridge blue series translation rather than Schrader's except that I have unitalicized the word 'to' in both of its instances. Page ! of !14 19 on the basis of the highest good); post-1800, by way of contrast, justifies belief in God immediately on the inexorable regarding of duty as divine command. 58 However, there is an even more important continuity (than the one alleged by Schrader) between Kant's pre-1800 work and the Opus postumum existence of God argument. This more important continuity can be found in Kant's Metaphysics of morals and, in particular, in the theory of conscience Kant began to articulate in that work. For instance, consider the following two passages: 59 Such an ideal person (the authorized judge of conscience) must be a scrutinizer of hearts, since the court is set up within the human being. But he must also...[be] a person in relation to whom all duties whatsoever are to be regarded as...his commands...Now since such a moral being must also have all power...in order to give effect to his laws...and since such an omnipotent moral being is called God, conscience must be thought of as the subjective principle of being accountable to God for all one's deeds. (MS AA 6:439) 60 The formal aspect of all religion, if religion is defined as "the sum of all duties as (instar) divine commands," belongs to philosophic morals, since this definition expresses only the relation of reason to the idea of God which reason makes for itself...we cannot very well make obligation (moral constraint) intuitive for ourselves without thereby thinking of another's will, namely God's (of which reason in giving universal laws is only the spokesman). (MS AA 6:487) 61 In the first of these two passages, Kant argues that (1) the authorized judge of conscience is (a) a moral being (qua imposer of obligation) and (b) omnipotent (qua judge able to give effect to the laws of obligation); (2) a moral omnipotent being is called God; and therefore (3) conscience should be thought of as an agent's being accountable to God. Similarly, in the second passage Kant argues that the only way in which It is important to Adickes that Kant's post-1800 God argument came from Kant himself, that Kant was not, say, merely writing 58 under the influence of Lichtenberg (1920, section 344). I take the fact that Kant clearly had thought about regarding duty as divine command before 1800 to lend support to this. In fact there is a plethora of pre-1800 texts showing that this idea is not new to the post-1800 Kant, although it is not generally deployed in these texts to justify belief in God. For example, see KU AA 5:481; RGV AA 6:84, 99, 110, 153f, 192; MS AA 6:227, 440, 443, 487; SF AA 7:36; VNAEF AA 8:418; VAMS AA 23:401.14-15; and Br AA 10:192.24-32. I owe the reference to 6:227 to Beck (1963, 280n55), who also cites Br AA 11:137.7-12. As far as I know, this continuity was first noted by Reinhard (1927, section I.2).59 Reinhard refers to this passage on (1927, 24).60 Reinhard refers to this passage on (1972, 18), interpreting it as an alternate solution to the problem with which Kant is grap61 pling at MS AA 6:417, that "The concept of a duty to oneself contains (at first glance) a contradiction." However, in the passage to which this footnote is appended, Kant is talking about all duties, not only duties to oneself. Page ! of !15 19 obligation can be made intuitive is by thinking of it as having been generated by another's will and, in particular, by the will of God. 62 For current purposes, what is most important about these two passages is that both justify belief in God on the basis of regarding one's duties as divine commands without appeal to the highest good. This gives more grist to grind against Schrader's first point considered above about there being "no indication" of a shift in Kant's thinking with regard to the highest good argument prior to 1800. As already 63 remarked, there is abundant evidence of Kant's pre-1800 dissatisfaction with the various parts of the highest good argument. Moreover, as can be seen from these two passages, there is also evidence of Kant's pre-1800 attempts to spell out new grounds for belief in God, grounds which are (prima facie) independent of the highest good and which are based on the very idea that seems so to captivate Kant in the first and seventh convolutes of the Opus postumum: believing in God on the basis of regarding one's duties as divine commands (rather than the other way around). 64 These passages from the Metaphysics of morals also can be used to give more structure to the Opus postumum argument explored in the previous section. Perhaps Kant's idea is that the voice of conscience, the "iron voice" of the Categorical Imperative (see passage i above), should be regarded as the voice of God because its pronouncements are those of a moral being and because the being is a judge, one with omniscience (or at least knowledge of the heart) and various other attributes that render it most appropriate to regard it as divine. However, this must be admitted to be conjectural extrapolation. 65 Conclusion In section 1 of this paper, I reconstructed Kant's highest good argument and gave evidence of Kant's dissatisfaction with the various premisses of this argument; in section 2, I gave evidence to support my thesis that Kant gave up on this argument around 1800; and in section 3, I explained why I do not find the existing counterarguments to my thesis persuasive, introducing along the way evidence of continuity between Kant's pre-1800 and post-1800 non-highest-good remarks on the existence of God. Perhaps my arguments and my responses to the existing counterarguments will not stand the test of time. Perhaps better counterarguments will appear. But as noted in the introduction to this paper, my goal is not only to argue for my Despite the similarity between these passages, it should be noted that in the second there is no mention of omnipotence or of 62 scrutinizing the heart. It also lends further support to Adickes' thesis that Kant's post-1800 God argument came from Kant himself (see n58 above).63 Reinhard finds evidence of this line of thought also in Kant's Lectures on ethics (1927, 16f).64 Webb, writing before Schrader and following Adickes, finds a strong continuity between the Opus postumum argument and 65 other pre-1800 lines of thought: "I...do not find any really new doctrine in the Opus postumum" (1926, 196). I am more cautious: I am unsure whether the Opus postumum argument is new. The point I am most concerned about is not whether a genuinely novel and unprecedented line of argument appears beginning around 1800 but rather whether a genuinely old and thoroughly explored line of argument is given up around 1800. Page ! of !16 19 thesis: it is also to revive a dialogue initiated by Adickes in 1920. And even if my thesis is ultimately overthrown, this second goal, which is probably the worthier one, thereby will have been accomplished. 66 I would like to express my deepest gratitude and indebtedness to Allen Wood, who generously provided me with feedback on 66 an earlier (and significantly worse) version of this paper. Page ! of !17 19 Bibliography Erich Adickes (1920): Kants Opus postumum, Reuther & Reichard. Lewis White Beck (1950): Introduction, in: Kant: Critique of practical reason and other writings in moral philosophy (trans. and ed. Beck), The University of Chicago Press. Lewis White Beck (1963): A commentary on Kant's critique of practical reason, The University of Chicago Press. Andrew Chignell (2007): Belief in Kant, in: The Philosophical Review, 116.3, pp. 323-360. A. Hazard Dakin (1962): Kant and religion, in: G. Whitney and D. Bowers (ed.): The Heritage of Kant, Russel & Russell, pp. 405-420. Edith and Klaus Düsing (2002): Negative und positive Theologie bei Kant, in: D. Hüning, G. Stiening, U. Vogel (ed.): Festschrift für B. Tuschling zum 65. Geburtstag, Berlin, pp. 85-118. Eckart Förster (2000): Kant's Final Synthesis, Harvard University Press. Theodore Greene (1960): The historical context and religious significance of Kant's Religion, in: Kant: Religion within the limits of Reason alone (trans. T. Greene and H. Hudson), First Harper Torchbook, pp. ix-lxxviii. Thomas Greene and John Silber (1960): Preface to the second edition of this translation, in: Kant: Religion within the limits of Reason alone (trans. T. Greene and H. Hudson), First Harper Torchbook, pp. cxxxix-cxlii. Wayne Martin (2009): Ought but cannot, in: Proceedings of the Aristotelian Society New Series 109, pp. 103-128. Michael Pace (2011): The Epistemic Value of Moral Considerations, in: Noûs 45.2, pp. 239-268. Lawrence Pasternack and Philip Rossi (2014): Kant's Philosophy of Religion, in: E. Zalta (ed.): The Stanford Encyclopedia of Philosophy, URL = <https://plato.stanford.edu/archives/fall2014/entries/kant-religion/>. Page ! of !18 19 Andrews Reath (1988): Two conceptions of the highest good in Kant, in: Journal of the history of philosophy 26.4, pp. 593-619. Walter Reinhard (1927): Ueber das Verhältnis von Sinnlichkeit und Religion bei Kant, Paul Haupt Akademische Buchhandlung vorm. Max Drechsel. George Schrader (1951): Kant's Presumed Repudiation of the "Moral Argument" in the "Opus Postumum," in: Philosophy 26.98, pp. 228-241. Norman Kemp Smith (1962): A Commentary to Kant's 'Critique of Pure Reason', Humanities Press. Jens Timmermann (2003): Sollen und können, in: Philosophiegeschichte und logische Analyse 6, pp. 113122. Clement Webb (1926): Kant's philosophy of religion, Oxford University Press. Victoria Wike (1994): Kant on happiness in ethics, State University of New York Press. Allen Wood (1970): Kant's moral religion, Cornell University Press. Allen Wood (2001): Kant versus Eudaemonism, in: P. Cicovacki (ed.): Kant's Legacy, University of Rochester Press, pp. 261-82. Page ! of !19

SARTRE ON EMBODIMENT, TOUCH, AND THE "DOUBLE SENSATION" Dermot Moran No phenomenology of life, of body and the flesh, can be constituted without basing itself on a phenomenology of touch. Jean-Louis Chrétien 1 The chapter titled "The Body" in Being and Nothingness offers a ground-breaking, if somewhat neglected, philosophical analysis of embodiment.2 Written in Sartre's customary dialectical style, it is dense, difficult, confused, original, insightful, brilliant. As part of his "essay on phenomenological ontology," he is proposing a new multi-dimensional ontological approach to the body. For Sartre, traditional philosophy has misunderstood the body because the orders of knowing and being have been conflated or inverted.3 Sartre begins from but creatively develops phenomenological discussions of embodiment found in Husserl (without direct access to Ideas II),4 Scheler,5 and Heidegger.6 In the background, of course, is an established-and predominantly French-tradition of physiological/psychological discussion of the body in relation to consciousness found in Descartes, Condillac, Maine de Biran, Comte, Bergson, Brunschwicg, Pradines,7 Marcel,8 Bachelard, and others, authors with whom Sartre was familiar. Indeed, Sartre provisionally maps out much of the ground later retraced by Merleau-Ponty's Phenomenology of Perception (1945) and posthumous The Visible and the Invisible (1964).9 For instance, Sartre discusses the artificiality of the psychological concept of sensation, the intrinsic temporality of experience, the Müller-Lyer illusion, the "double sensation" (one hand touching the other), Gestalt figure-ground structures, and so on. But in many ways, especially in his discussion of fleshly intercorporeity, he goes beyond Merleau-Ponty. Indeed, Sartre introduces the notion of "flesh" (la chair), now more usually associated with Merleau-Ponty. For Sartre, flesh is the locus of contingency and intercorporeity. Flesh is "the pure contingency of presence" (BN 343/410).10 More importantly, my flesh constitutes the other's flesh, especially in the acts of touching and caressing: The caress reveals the Other's flesh as flesh to myself and to the Other. . . . It is my body as flesh which causes the Other's flesh to be born (quit fait naître la chair d'autrui). (BN 390/ 459–60) I have one kind of knowledge of the body within my experience and another experience of the body given from the perspective of the other: the body as it is "for me" and the body as it is "for others" or "for the other" (pour l'autrui). These two dimensions are, according to Sartre, "incommunicable" and "irreconcilable": Either it [the body] is a thing among other things, or else it is that by which things are revealed to me. But it cannot be both at the same time. (BN304/366) The first "ontological dimension" addresses the way, as Sartre puts it, "I exist my body" (J'existe mon corps) (BN 351/428), the body as non-thing, as medium for my experience of the world, but also as somehow surpassed towards the world. This is le corps-existé, the body as lived, as opposed to le corps-vu, the body as seen from the perspective of the other (BN 58/426). The second dimension is the manner in which my body is experienced and indeed utilized by the other (and utilized by myself occupying the role of third-person observer of my body). This includes my ready-to-hand equipmental engagement with the world and my body as the "tool of tools." The third dimension is more complicated: it is the manner in which "I exist for myself as a body known by the other" (BN 351/419), what Martin C. Dillon has characterized as "the body-for-itself-for-others."11 This captures the PHILOSOPHY TODAY SPEP SUPPLEMENT 2010 135©2010 DePaul University intersubjective dimension: the manner I experience my body as experienced by others, the dialectics of which Sartre has explored more than any other phenomenologist (with the possible exception of Levinas). This is the body as I experience it under the gaze of the other, as in the case of shame or embarrassment. I experience how the other sees me, even in the physical absence of the other. I am, Sartre says, "imprisoned in an absence" (BN 363/430). This is a contested domain: "Conflict is the original meaning of being-for-others" (Le conflit est le sens original de l'être-pour-autrui) (BN 364/ 431). Sartre begins from the concrete unity of body and consciousness, with the body as lived and experienced from within (although that spatial metaphor is shown to be inadequate), from the "first-person" perspective. This is neither pure consciousness nor physical thing. The lived, experienced body-corresponding to Husserl's Leib-can never be construed as a transcendent object, and certainly not something purely physical. In fact, Sartre paradoxically asserts: The body is the psychic object par excellence- the only psychic object. (BN 347/414) The body dominates the psyche; it is present even in dreams, and the body we experience from within is itself psychically constituted. The material, objective body, as idealized in the sciences (physics, biology, physiology), on the other hand, is, in Sartre's pithy phrase, the "body of others" (le corps d'autrui), the body of the anonymous other. Sartre distinguishes between this body understood as object in the world, seen from "the physical point of view," the "point of view of the outside, of exteriority" (le point de vue du dehors, de l'extériorité) (BN 305/367), and the body as experienced from within. From within, the body as lived is invisible, impalpable, "ineffable" (BN 354/ 421). I do not know experientially that I have a brain or endocrine glands (BN303/365) - that is something I learn from others. Likewise, I don't know the inner anatomy of my body. I have, as it were, a "folk anatomy"-where I "think" my stomach is. This can be more or less well informed by science, more or less accurate, but this scientific map, superimposed on the felt body, does not necessarily coincide with the body as felt. I can visualise my ulcerous stomach but I live its discomfort in a different way (BN 355–56/423).12 There is an immediately intuited or felt body (Merleau-Ponty's phenomenal body). However, most of the time, this felt body is non-objectified and experienced in a diffuse, amorphous and almost invisible manner (which is precisely its mode of appearing). It becomes obtrusive in illness (I become dizzy), failure (the stone is too heavy to lift), disability (the anorexic experiences her body as too gross), or, as Sartre emphasises, in the look of the other.13 Furthermore, and this is Sartre's originality, even when I see and touch my body, I am in these situations experiencing my body from without, from the point of view of an "other": "I am the other in relation to my eye." I can see my eye as a sense organ but I cannot, contra Merleau-Ponty, "see the seeing" (BN 304/ 366). I see my hand, Sartre acknowledges, but only as an external thing. I cannot see the sensitivity of the hand, even its mineness: For my hand reveals to me the resistance of objects, their hardness or softness, but not itself. Thus I see this hand only in the way that I see this inkwell. I unfold a distance between it and me. (BN 304/366; his italics) I see my hand as another object in the world. In other words, my sight (and indeed my touch) manifests my body in the same way as it is available to another. Here Sartre and Merleau-Ponty disagree. Merleau-Ponty emphasizes the feeling body as a continuing presence in cases of seeing and touching; whereas Sartre maintains that our perceivings objectify what we perceive and displace the feeling onto the felt. Thetic consciousness is objectifying or reifying. Physicians and others have an experience of my body, but they experience it as a piece of the world, "in the midst of the world" (au milieu du monde) (BN 303/365). This is the body in its "being for others" (être-pourautrui) (BN 305/367). Sartre claims that my own body is primarily present to me in this "for-others" (pour-autrui) way most of the time. Despite this dominance of the "for-others" body, Sartre strongly rejects the view that our ontology of the body should begin from the third-person, "externalist" (du dehors) view (BN 303/365). PHILOSOPHY TODAY SPEP SUPPLEMENT 2010 136 ©2010 DePaul University This is "to put the corpse at the origin of the living body" (BN 344/411). The "category mistake" of previous philosophy has been its absurd attempt to unite the first-person experienced body with the "the body of others" (corps des autres) (BN 303/365). Rejecting this externalist approach as hopeless, Sartre maintains one must start from the recognition that, first and foremost, our experience is not of the body at all, but rather, of the world, or the situation: Our being is immediately "in situation;" that is, it arises in enterprises and knows itself first in so far as it is reflected in those enterprises. (BN 39/ 76) And again: the body is identified with the whole world inasmuch as the world is the total situation of the for-itself and the measure of its existence. (BN 309/372) We are completely in the world: The concrete is man within the world in that specific union of man with the world which Heidegger, for example, calls "being-in-theworld." (BN 3/38) It is because of our intentional directedness to the world that we have to overcome, surpass, transcend the body. The whole thrust of human subjectivity is to overcome or cancel itself, to "nihilate" (néantiser) itself by intending towards the world. Intentionality is world-directed. The embodied consciousness has to "surpass" i tsel f . This "surpassing" (dépassement) constitutes the essence of intentionality understood as self-transcendence. This surpassing of the body, however, does not mean its elimination: The body is necessary again as the obstacle to be surpassed in order to be in the world; that is, the obstacle which I am to myself. (BN 326/ 391) For Sartre, our transcendence towards the world is part of what he takes to be our original "upsurge in the world." But it is we ourselves who decide these very dimensions by our very upsurge (notre surgissement) into the world and it is very necessary that we decide them, for otherwise they would not be at all. (BN 308/370) Sartre frequently speaks of the "upsurge" (surgissement) of the pour-soi towards the world, of the "upsurge" of the other in my world, and so on. In a sense, this upsurge is the primal situation: consciousness and world emerging together in one blow. MerleauPonty also speaks of the "unmotivated upsurge" (le jaillissement immotivé du monde) (PP xiv/viii) of the world. For Sartre, this "upsurge" has both a certain necessity and a certain contingency, this combination he calls "facticity." For Sartre, paradoxically, while the body is that which necessarily introduces the notion of perspective and point of view, at the same time the body is a contingent viewpoint on the world. Our body exemplifies the very contingency of our being: it is a body in pain, or whatever. To apprehend this contingency, is to experience "nausea": "A dull and inescapable nausea perpetually reveals my body to my consciousness" (BN 338/404). Being embodied brings ontological un-ease (disease). For Sartre, as for Husserl, consciousness requires incarnation, which situates and locates consciousness, gives it a point of view, and makes it possible as consciousness. Sartre writes: the very nature of the for-itself demands that it be body, that is, that its nihilating escape from being should be made in the form of an engagement in the world. (BN 309/372) Moreover, the world in which we are embodied is a world that has been humanized by us: "the world is human" (BN 218/270): The body is the totality of meaningful relationships to the world . . . The body in fact could not appear without sustaining meaningful relations with the totality of what is. (BN 344/411) Sartre insists on the synthetic union between body and world. On the other hand, he rejects the deep significance that Husserl and Merleau-Ponty accord the phenomenon of the "intertwining" in the double sensation. EMBODIMENT, TOUCH, AND THE "DOUBLE SENSATION" 137©2010 DePaul University The "Double Sensation" Sartre clearly distinguishes between my body as experienced (ambiguously and nonobjectively) by me and the body as it is for myself occupying the perspective of another. These different "bodies" underpin different, even irreconcilable ontologies. Sartre's analysis of the phenomenon of the double sensation reinforces this irreconcilability between these opposing "ontological" dimensions. Although many philosophers think the phenomenon of the "double sensation" is a discovery of Husserl or Merleau-Ponty, in fact it is a recurrent theme in nineteenth-century psychology (found, for instance, in E. H. Weber14 and David Katz15). Husserl employs the term "double sensation" (Doppelempfindung) in Ideas II §36 (152–54; Hua IV 144–47), and, indeed, had already discussed the phenomenon in his Thing and Space (1907).16 For Husserl, when one hand touches the other, the sensations of touching can be reversed into sensations of being touched. Husserl calls this "intertwining" (Verflechtung). Likewise in Ideas II §36 Husserl is interested in the manner in which the lived-body (Leib) is constituted as a "bearer of localized sensations." These "localized sensations" or "sensings" (Empfindisse) are not directly but only indirectly sensed by a "shift of apprehension."17 The touching hand must make movements in order to feel the smooth and soft texture of the touched hand. Husserl says that the "indicational sensations" of movement and the "representational" sensations of smoothness to the touch in fact belong to the touching right hand but they are "objectivated" in the touched left hand. Husserl speaks of the sensation being "doubled" when one hand touches or pinches the other. Each hand experiences this "double sensation." Furthermore, for Husserl "double sensation" belongs essentially to touch but not vision (Ideas II, §37); there are no comparable visual sensings. We see colors but there is no sensing color: "I do not see myself, my body, the way I touch myself" (Ideas II, §37, 155; HUA IV 148). All Husserl allows is that the eye is a center for touch sensations (the eyeball can be touched, we can feel the movement of the eye in the eye-socket through "muscle sensations," and so on). Overall, in these discussions, Husserl's employs the double sensation to distinguish touch from vision. For Husserl (following Aristotle), it is primarily touch that anchors us in the body. He writes: Everything that we see is touchable and, as such, points to an immediate relation to the body, though it does not do so in virtue of its visibility. A subject whose only sense was the sense of vision could not at all have an appearing body. . . . The body as such can be constituted originally only in tactuality. (Ideas II §37, 158; HUA IV 150) Touch localizes us in the world in a way that seeing does not. Merleau-Ponty discusses the phenomenon of the "double sensation" most fully in The Visible and the Invisible.18 Since his account is well known, I will not summarize it but only say that it follows Husserl closely, except that Merleau-Ponty emphasizes the continuities between seeing and touching and their interconnection. In contrast to Merleau-Ponty, however, Sartre claims that the phenomenon of double sensation does not reveal something essential about embodiment. It is contingent. It can be removed by morphine, making my leg numb and insensitive to being touched (BN 304/366). To touch and be touched reflect different orders or "levels" of being. When one hand touches the other hand, I directly experience the hand that is being touched first. It is only with a certain reflection that I can turn back and focus on the sensation in the touching hand. Sartre maintains that this constitutes ontological proof that the body-for-me and the body-for-the-other are entirely separate intentional objectivities. Merleau-Ponty's metaphysical use of the double sensation, then, is the opposite of Sartre's. Merleau-Ponty claims that both vision and touch have this doubleness. Sartre, on the other hand, wants to prioritize not one hand touching the other, but one body touching or caressing the other's body. Primacy is given to the other, not to selfexperience. Intercorporeity is the source and ground of self-experience. Conclusion Sartre's account of the body is subtle, complex, and many layered. While not as deeply informed by psychological studies as Merleau-Ponty's,19 Sartre's account of PHILOSOPHY TODAY SPEP SUPPLEMENT 2010 138 ©2010 DePaul University intersubjective embodied relations (e.g., the erotic caress) is equally original. MerleauPonty and Sartre disagree concerning the role of bodily consciousness in perception. Whereas Merleau-Ponty, following Husserl, emphasizes the ineliminability of the felt body in all perceiving; Sartre maintains that our perceivings objectify what we perceive. Hence, for Sartre, the phenomenon of "double sensation" or "touching-touched" is irrelevant and indeed falsely described in psychology, whereas for Merleau-Ponty, especially in his late Visible and Invisible, it becomes the very essence of flesh and our "entwinement" in the world. For Sartre, on the other hand, it is in in touching the other that we encounter ourselves as flesh. ENDNOTES EMBODIMENT, TOUCH, AND THE "DOUBLE SENSATION" 139 1. Jean-Louis Chrétien, L'Appel et la réponse (Paris: Minuit, 1992); The Call and the Response, trans. Anne Davenport (New York: Fordham University Press, 2004), 86. 2. Jean-Paul Sartre, L'Etre et le Néant. Essai d'Ontologie Phénoménologique (Par is : Gallimard, 1943); Being and Nothingness: An Essay on Phenomenological Ontology, trans. Hazel Barnes (London: Routledge, 1995). Hereafter cited in my text as BN, followed by English pagination and then the pagination of the French original. Of course, one should not assume that everything Sartre says about the body is to be found in the chapter bearing that title. In fact, the body pervades the whole of Being and Nothingness. In particular, his discussion of hunger and desire, for instance, in the chapter on "Concrete Relations with Others," continues the analysis of the experience of one's own body and of the flesh of the other. 3. In Being and Nothingnes, Sartre speaks variously of the "order of being" (l'ordre de l'être) (305/ 367), "orders of reality" (ordres de réalité) (304/ 366) , and "ontological levels" (plans ontologiques) (305/367). 4. Edmund Husserl, Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie. Zweites Buch: Phänomenologische Untersuchungen zur Konstitution, Husserliana IV, ed. Marly Biemel (Dordrecht: Kluwer, 1991); Ideas pertaining to a Pure Phenomenology and to a Phenomenological Philosophy, Second Book, trans. R. Rojcewicz and A. Schuwer (Dordrecht: Kluwer, 1989). Hereafter cited in my text as Ideas II, followed by English pagination, Husserliana (hereafter Hua) volume and German pagination. Sartre of course read Husserl's published writings, but had little access to the unpublished drafts, except through conversation with his close friend Maurice Merleau-Ponty, who was receiving material from Herman Leo Van Breda, Director of the Husserl Archives in Leuven, even during the German occupation. See Herman Leo Van Breda, History of the Husserl-Archives (Dordrecht: Springer, 2007). 5. For an interesting survey of the role of the body in Scheler's writings, see Daniela Vallega-Neu, "Driven Spirit: On the Body in Max Scheler's Phenomenology," Epoche 9 (2004): 19–36, reprinted in idem, The Bodily Dimension in Thinking (Albany: SUNY Press, 2005): 43–58. 6. Before writing Being and Nothingness (while in the detention camp), Sartre had read Heidegger's Sein und Zeit (1927), ( Being and Time, trans. John Macquarrie and Edward Robinson, [Oxford: Basil Blackwell, 1962]), the 1929 essay "What is Metaphysics?" as well as some of Heidegger's later essays of the 1930s and early 1940s. Although, strictly speaking, the body hardly makes an appearance in Being and Time, Sartre interprets the facticity and contingency of Dasein's "being-inthe-world" as referring primarily to our embodiment. 7. Maurice Pradines (1874–1958), a follower of Bergson, taught Levinas at Strasbourg. See his Philosophie de la Sensation, vol. I: Le Problème de la sensation (Paris: Les Belles Lettres, 1928), listed by Merleau-Ponty in the bibliography to his Phenomenology of Perception. 8. Gabriel Marcel (1918–1933), Être et avoir (Paris, Aubier, 1935); Being and Having (London: Fontana, 1965). 9. Maurice Merleau-Ponty, Phénoménologie de la perception (Paris: Gallimard, 1945); Phenomenology of Perception, trans. Colin Smith (London: Routledge and Kegan Paul, 1962). Henceforth cited in my text as PP, followed by page number of English translation and then pagination of French edition. Le Visible et l'invisible, texte établi par Claude Lefort (Paris: Gallimard, 1964); The Visible and the Invisible, trans. Alphonso Lingis (Evanston: Northwestern University Press, 1968). ©2010 DePaul University PHILOSOPHY TODAY SPEP SUPPLEMENT 2010 140 Hereafter cited in my text as VI, followed by the pagination of the English translation. 10. Sartre develops the notion of the "flesh" (la chair) from Husserl's conception of Leibhaftigkeit, the bodily presence of the object in perception. Indeed, Sartre already talks about the "flesh of the object in perception" in his earlier 1940 study, L'Imaginaire, see Jean -Paul Sartre, The Psychology of Imagination (London: Methuen, 1972), 15. The French translation of leibhaftig in Husserlian texts (as also cited by Merleau-Ponty and Levinas) is en chair et en os, meaning literally "in flesh and bone." 11. See Martin C. Dillon, "Sartre on the Phenomenal Body and Merleau-Ponty's Critique," in Jon Stewart, ed., The Debate Between Sartre and Merleau-Ponty (Evanston: Northwestern University Press, 1998), 121–43, see especially 126. 12. Or, for example, in challenging Freudian psychoanalytic accounts of the child's fascination with holes, Sartre claims that the child could never experience his own anus as a hole (as part of the objective structure of the universe). The child learns this through another (Being and Nothingness, 612–13/704). 13. In Ideas II Husserl too had already distinguished between "normal" or optimal cases of experiencing, and impaired ones, e.g., touching a surface with a blistered finger. 14. E. H. Weber (1795–1879), published two studies of touch: De Tactu (1834) and Tastsinn und Gemeingefühl, which first appeared in 1846 in R. Wagner, ed., Handwörterbuch der Physiologie, vol. III, Part 2, 481–588. Both works have now been translated in E. H. Weber on the Tactile Senses, ed. and trans. Helen E. Ross and David J. Murray, 2nd ed. (Hove, East Sussex: Erlbaum, Taylor and Francis, 1996). Weber and Gustav T. Fechner (1801–1887) were founders of psychophysics, the attempt to systematically relate physical phenomena, e.g., sound or weight, with the perception of them. Psychophysics can be considered the earliest form of experimental psychology in the modern sense. Weber carefully documented the different sensitivities to touch in various parts of the body, the perception of weight, heat, cold, etc., and the ability of the perceiver to distinguish when being touched by two points of a compass at the same time. In Der Tastsinn, for instance, Weber discusses the issue of whether two sensations arise when sensitive areas of the body touch each other. He claimed that the two sensations do not merge into one: a cold limb touching a warm limb (e.g., a hand touching the forehead) reveals both heat and cold. 15. David Katz, Der Aufbau der Tastwelt (Darmstadt: Wissenschaftliche Buchgesellschaft, 1969) (first published in 1925); The World of Touch, trans. Lester E. Krueger (Hillsdale, N.J: Lawrence Erlbaum Publishers, 1989). The German psychologist David Katz (1884–1953) studied at Göttingen under the renowned psychologist Georg Elias Müller (1850–1934) and Husserl, who was one his doctoral examiners in 1907 and whose seminars he continued to attend. Katz worked on experimental and developmental psychology at Göttingen until 1919 when he moved to Rostok. He was close to the Gestalt psychologists but was forced to leave Germany in 1933, going first to England and then, in 1937, to the University of Stockholm, where he remained. He was a major influence on the work of J. J. Gibson. See the obituary of R. Arnheim, "David Katz, 1884–1953," American Journal of Psychology 66 (1954): 638– 42. See also Lester Krueger, "Tactual Perception in Historical Perspective: David Katz's World of Touch," in Tactual Perception: A Sourcebook, ed. William Schiff, Emerson Foulke (New York: Cambridge University Press, 1982). For Katz's relations with Husserl, see Herbert Spiegelberg, Phenomenology in Psychology and Psychiatry: A Historical Introduction (Evanston: Northwestern University Press, 1972), 42–44. By his own admission Katz attended Husserl's lectures and seminars and learned the phenomenological method of unprejudiced description from him; see Katz's autobiography, in Edwin Boring, ed., History of Psychology in Autobiography (New York: Russell and Russell, 1952), 4:189–211, esp. 194. Katz also acknowledges the influence of Scheler. MerleauPonty relies heavily on Katz's World of Touch for his account of touch in Phenomenology of Perception, see esp. 315–18/364–68. 16. Edmund Husserl, Ding und Raum. Vorlesungen 1907, Hua XVI, ed. Ulrich Claesges (The Hague: Nijhoff, 1973); Thing and Space: Lectures of 1907, trans. by R. Rojcewicz (Dordrecht: Kluwer, 1997). Hereafter cited in my text as DR, with English and then German pagination. The reference here is to §47, 137/162. 17. Husserl famously distinguishes between "sensations" (Empfindungen) that are interpreted as properties of the object and the "sensings" (Empfindnisse) themselves which he speaks of as "indicational or presentational" (Ideas II, 154/ 146); see Elizabeth A. Behnke, "Edmund ©2010 DePaul University University College, Dublin, Ireland EMBODIMENT, TOUCH, AND THE "DOUBLE SENSATION" 141 Husserl's Contribution to Phenomenology of the Body in Ideas II," in Issues in Husserl's "Ideas II," ed. Thomas Nenon and Lester E. Embree (Dordrecht: Kluwer Academic Publishers, 1996), 135–60. 18. See Merelau-Ponty, "The Intertwining-The Chiasm," in The Visible and the Invisible, 130–55. 19. Merleau-Ponty is deeply influenced, as we have seen, by David Katz's studies of vision and touch, and also by studies such as Jean Lhermitte, L'Image de notre corps (Paris: Editions de la Nouvelle Revue Critique, 1939), which introduces the idea of the "body image," which Merleau-Ponty refers to as le schéma corporel (translated by Colin Smith as "body image"). For further discussion of this concept, see Shaun Gallagher, How the Body Shapes the Mind (Oxford: Oxford University Press, 2005), who explains Merleau-Ponty's schéma corporel as the "dynamic functioning of the body in its environment" (20). ©2010 DePaul University

Raphael F. Alvarenga Dialética negativa e radicalismo negro: Angela Davis nos anos 1960 https://blogdaboitempo.com.br/2018/05/10/dialetica-negativa-e-radicalismo-negro-angela-davis-nos-anos-1960/ Dialética negativa e radicalismo negro: Angela Davis nos anos 1960 O pensamento de Angela Davis extrai sua força da combinação original de duas tradições de pensamento crítico radical que não costumam andar juntas: a frankfurtiana e a afroamericana. Publicado em 10/05/2018 Por Raphael F. Alvarenga. Raphael F. Alvarenga Dialética negativa e radicalismo negro: Angela Davis nos anos 1960 https://blogdaboitempo.com.br/2018/05/10/dialetica-negativa-e-radicalismo-negro-angela-davis-nos-anos-1960/ No final do século passado, Angela Davis publicou um livro sobre o legado do blues feminino nos Estados Unidos, no qual toma de empréstimo a Herbert Marcuse, seu antigo professor, a noção de "dimensão estética", propondo uma reconceituação da mesma, no sentido de fundamentá-la histórica e coletivamente. Ela discerne, por exemplo, na dimensão estética da obra de Billie Holiday uma espécie de "simbiose" com a comunidade negra americana: sua música contribui para a mesma história social e musical afro-americana de que se impregna, no interior da qual a práxis política feminina nutre e é nutrida pela práxis estética 1 . Não era a primeira vez que a famosa ativista – mais conhecida por seu feminismo radicalmente anticapitalista, por sua luta pela abolição do sistema carcerário e por seu suporte à causa palestina – se debruçava sobre questões relativas às relações "simbióticas" ou dialéticas entre arte e sociedade, cultura e política. Não custa lembrar que, com um ano de intercâmbio universitário passado na França, durante o qual lera Balzac, Flaubert, Baudelaire, Rimbaud, a Recherche de Proust de cabo a rabo, Sartre praticamente todo, Davis graduou-se em Literatura Francesa com grande distinção na Universidade Brandeis, em Massachusetts, com uma dissertação sobre a obra de Alain Robbe-Grillet 2 . Marcado por sintaxe intricada e trama esquartejada, a estudante de 21 anos vislumbrava no nouveau roman do autor uma expressão potencialmente desmistificadora do nó existencial da realidade contemporânea: a carência de referentes, o apagamento das personagens e a ausência de profundidade apontariam para o predomínio de relações cada vez mais anônimas, definidas pela crescente mecanização, burocratização e fragmentação da existência na era da bomba 3 . Sem deixar de lado o interesse pela literatura, que via como uma sondagem em profundidade da realidade humana e social, Davis foi iniciada à Filosofia na mesma instituição pelo ainda não muito conhecido Professor Marcuse, com quem, antes mesmo de se graduar, passou a ler e discutir semanalmente os clássicos do pensamento ocidental, dos pré-socráticos à filosofia transcendental alemã. O estudo desta última deveria ser aprofundado, segundo combinado com o mestre, na Universidade Goethe, na Alemanha, onde lecionavam Max Horkheimer e Theodor W. Adorno. Kant, Schiller e o conhecimento para mudar o mundo Frankfurt nos anos 1960 era uma espécie de Meca da Filosofia, principalmente se o intuito fosse o estudo rigoroso da constelação formada por Kant, Hegel e Marx. No entanto, no verão de 65 a jovem Angela Davis tomara o navio para a Europa com a consciência intranquila – as ruas de Watts, distrito pobre de Los Angeles, estavam ardendo. Estaria trilhando o caminho correto, ou se afastando do que realmente interessava? Passado o tempo, não parece ter se arrependido, embora viesse a pôr um fim na estada frankfurtiana dois anos depois, bem antes do previsto, atraída pelo recém-formado Black Panther Party na Califórnia. Ultramar, sob a orientação do disputado Professor Adorno, suas pesquisas girariam em torno de um tema à primeira vista alheio às convulsões do período: a liberdade como categoria estética nas obras de Kant e Schiller. Olhando mais atentamente veremos que Davis não somente dava continuidade ao trabalho precedente sobre o nouveau roman – buscando no idealismo crítico alemão a possibilidade de abordar de modo mais radical a realidade da humanidade danificada pelos dilaceramentos da civilização capitalista –, como colocava no centro de suas preocupações teóricas a questão da liberdade e da transformação social. Resumindo bastante, podemos dizer que na filosofia de Kant o juízo estético vem fazer a ponte entre o conhecimento científico das determinações causais do mundo natural (a razão pura teórica) e a esfera algo etérea dos imperativos morais (a razão pura prática). Kant se opunha a Baumgarten, considerado o fundador da estética moderna, para quem o juízo de gosto seria uma forma inferior de cognição. A Crítica do juízo – publicada apenas um ano após a tomada da Bastilha – deitava por terra tal hierarquia, e conferia plena cidadania à esfera dos sentimentos e do desejo, que conjugados com a imaginação entram num jogo sem conceito e desinteressado com a inteligência. Coube a Schiller, poucos anos depois, nas Cartas sobre a educação estética do homem, dar à empresa crítica kantiana um contorno mais explicitamente político, em que o potencial libertador da função estética ressurge como suporte de uma revolta social e política contra a sociedade repressiva em vistas da humanidade concreta, capaz de traduzir em razão a força mobilizadora do Raphael F. Alvarenga Dialética negativa e radicalismo negro: Angela Davis nos anos 1960 https://blogdaboitempo.com.br/2018/05/10/dialetica-negativa-e-radicalismo-negro-angela-davis-nos-anos-1960/ desejo 4 . Desencantado porém com os rumos tomados pela Revolução Francesa, avesso ao Terror revolucionário, Schiller defendia que a educação estética (e não uma revolução) seria o meio mais adequado para transformar as leis da liberdade em práxis cotidiana. Por outras palavras, caberia a uma nova humanidade, a um tempo sensível e autônoma, ela mesma fruto de uma tal educação, criar um estado de coisas em acordo com os postulados morais da razão. De recurso e elo mediador entre dois mundos antagônicos (físico e moral), o "terceiro estado" (estético ou lúdico), único âmbito onde o homem é plenamente humano, passa no decurso do livro a fim último e ideal máximo a ser realizado, no qual vida e forma se tornam uma só e mesma coisa – na carta XV Schiller fala em "forma viva" (lebende Gestalt), conceito que resume "tudo que entendemos no mais amplo sentido por beleza" 5 . Interessa sublinhar o seguinte: a capacidade de ser tocado por uma obra de arte apontaria para a possibilidade de usar a imaginação de uma forma que não é limitada por nossas práticas cognitivas habituais: aqui vislumbramos uma liberdade que não tem nada de vago, porque vem atrelada à possibilidade de produzir um conhecimento de tipo novo, necessário para mudar o mundo. Acresce que o juízo de gosto, a um tempo subjetivo e generalizável, apela a (ou parece supor) uma "comunalidade" entre os homens, quiçá algo como uma "comunidade imaginada" 6 . Na esteira de Kant e Schiller, Adorno e Marcuse enxergariam na experiência estética de obras importantes, concebida como jogo livre e imaginativo das faculdades humanas, como que uma prefiguração sem conceito da utopia de um mundo liberto do trabalho alienado. O efeito de estranhamento (Verfremdungseffekt), na expressão de Brecht, produzido pelo contato com a obra de arte funciona como uma alienação da existência alienada (entfremdetes Dasein). Ao lançar uma luz sobre desajustes e inconsistências que geralmente não se dão a ver no banal do cotidiano, sobre aspectos da realidade social não determinados pelas representações correntes, a obra consequente permite vislumbrar o que nunca foi, o que poderia ser, ou poderia ter sido, impasses e bloqueios reais, rumos diversos que poderiam ser tomados. Mensagens numa garrafa e os chineses no Reno Dividida entre a leitura dos clássicos do pensamento alemão e a participação em protestos contra a guerra do Vietnã, Davis chegou a intervir – a despeito da barreira da língua – no concorrido seminário organizado conjuntamente por Adorno e Horkheimer para discutir a Dialética negativa. O seminário em questão, realizado no verão de 1966, era frequentado em grande parte por alunos ligados ao movimento estudantil – no seio do qual Adorno mais adiante veria germinando uma forma de totalitarismo ("os chineses no Reno" 7 ) – como o promissor Hans-Jürgen Krahl, o favorito de Adorno, bem como por colegas expatriados como Davis, e que viriam a se tornar conhecidos scholars e tradutores do pensamento frankfurtiano em língua inglesa, como Irving Wohlfarth e Samuel M. Weber (os quais, diga-se de passagem, alguns anos antes, em Yale, foram introduzidos ao marxismo e à Escola de Frankfurt por um brilhante jovem mestrando brasileiro de origem austro-judaica). Esquematicamente, digamos que a dialética negativa oscila por assim dizer em permanência entre desespero conceitual e possibilidade objetiva. A tarefa árdua de conduzir o pensamento para além do conceito sem para tanto abdicar do conceito tinha por finalidade indicar em diferentes âmbitos e contra diversas tendências de pensamento positivo (ontologia existencial, positivismo lógico, filosofias da história) a falsa identidade de razão e realidade. Diferentemente do positivismo em suas mais variadas vertentes, que subordina a razão à autoridade dos fatos do mundo existente, a dialética negativa se coloca inequivocamente do lado do possível, mais precisamente da possibilidade de uma racionalidade intensamente vital, em conexão com a qual as noções de liberdade e subjetividade devem ser pensadas. Neste sentido, o livro de 1966 era solidário da crítica estética praticada por Adorno ao longo de toda a vida, uma vez que aquilo que escapa tanto da teoria tradicional como da ciência objetiva – a não-identidade, a não-liberdade – costuma vir à tona na obra de arte de qualidade. Discurso propositalmente aporético, a aposta do livro era de que a insistência no negativo – naquilo que não se encerra em nenhuma determinação – pudesse desprender possibilidades bloqueadas pela esperança de se encontrar uma saída no interior das presentes coordenadas de pensamento e ação. Não muito diferente de Frantz Fanon, que parecia encontrar no desespero a fonte mesma da vitalidade revolucionária, também em Raphael F. Alvarenga Dialética negativa e radicalismo negro: Angela Davis nos anos 1960 https://blogdaboitempo.com.br/2018/05/10/dialetica-negativa-e-radicalismo-negro-angela-davis-nos-anos-1960/ Adorno há a ideia de que a paixão do negativo possa, mesmo se a longo prazo, enriquecer a imaginação política e revitalizar a práxis social 8 . Segundo consta, Davis e Wohlfarth ficaram encarregados de apresentar o capítulo introdutório do livro, ao que se seguiu uma discussão portando sobre a questão do enraizamento histórico-social do pensamento dialético 9 . Seria a dialética, enquanto experiência da contradição, um gênero histórico imanente à consciência, ou seria ao contrário o caso de dizer que o impulso à unidade da consciência provém dos antagonismos objetivos da sociedade burguesa? A própria ideia de um estado reconciliado, ou verdadeiro, em oposição ao estado falso vigente, dependeria da resposta que se dê a esta pergunta. Em réplica, Adorno se referiu primeiramente ao que ele entendia como o materialismo negativo de Marx, no qual a própria dialética deveria ser absorvida e superada (aufgehoben) na prática, uma vez que estaria ligada à sociedade burguesa em termos de conteúdo e forma, como a experiência que tal sociedade tem de si mesma. Em seguida Adorno afirmou ter desenvolvido na Dialética negativa "modelos" de comportamento intelectual, os quais teriam consequências para o pensamento emancipado, no sentido de que um pensamento livre da compulsão do sistema possui uma maior força de resistência com relação ao existente, pois pensa o nãoidêntico sem hipostasiá-lo, e tampouco se põe a si mesmo como Filosofia Primeira. Embora tal pensamento não possa prescindir de categorias lógicas, é importante frisar que estas mudam no decorrer da reflexão crítica. Na objeção levantada haveria implicitamente uma dicotomia inconciliável entre a felicidade humana e a racionalidade, oposição que Adorno via como um traço irracionalista. A crítica justificada à razão puramente particular, à razão instrumental, não deveria redundar numa negação abstrata ou num apelo ao salto no abismo do reprimido no inconsciente. A libertação da repressão não implica jogar fora o pensamento conceitual, o que abriria a porta para todo tipo de regressão e extremismo. Os estudantes, entretanto, insistiam no que viam como uma contradição na abordagem do professor. "A dialética", asseverava ele no capítulo em discussão, "é a consciência consequente da não-identidade", razão pela qual "não assume antecipadamente um ponto de vista" 10 . Isso quer dizer que ao mesmo tempo em que a experiência intelectual é responsável pelo seu ponto de vista teórico ela não deve ser deixada à sua própria dinâmica. Em que se fundamenta então a crítica? Para Davis, Wohlfarth e muitos colegas ali presentes, a crítica dialética não poderia adotar uma posição neutra; careceria ao contrário de assumir explicitamente o ponto de vista social dos oprimidos, como sugerira Walter Benjamin. Adorno redarguiu que tal tomada de partido é insuficientemente dialética, a começar porque morte, opressão e injustiça constituíam o oposto daquilo com o que o pensamento deveria realmente se identificar. Assumir o ponto de vista do oprimido – ou qualquer outro ponto de vista particular – como lugar privilegiado do pensamento a seu ver corria o risco de levar à instrumentalização deste, "servir-se da dialética ao invés de perder-se nela" 11 . Além do mais, é sabido que diversas teorias que no passado pretenderam falar em nome do oprimido e da justiça social acabaram sendo utilizadas para justificar a exploração e a opressão. O pensamento dialético teria ao contrário que ver com a dissolução mesma de todo e qualquer ponto de vista. Contra o relativismo, Adorno sustentava que os fenômenos investigados pela dialética não devem ser examinados de fora, a partir de uma perspectiva exterior, qualquer que seja, mas julgados de acordo com seu próprio conceito. Tudo bem, insistiam os estudantes, mas como o movimento da crítica imanente é controlado? Através de um modo de conhecimento que quer o conteúdo, respondeu Adorno, que acabou reconhecendo que o máximo que a dialética negativa pode evocar é um interesse cognitivo específico, que aponta para além da crítica imanente, embora não possa dela ser separado, e que diz respeito à tomada de partido em prol da possibilidade do indivíduo. A crítica dialética teria então um ponto de vista – o individual, o singular, o concreto, o múltiplo –, mas um ponto de vista negativo, que é pressuposto e não fundamento 12 . Pondo de lado a dialética, alguns fariam carreira em cima da questão da necessidade de uma fundamentação normativa sólida para a Teoria Crítica, que de crítica mesmo acabaria não tendo mais muita coisa... Seja como for, no fundo a exigência de uma tomada explícita de partido não dizia respeito apenas ao livro debatido, o que Raphael F. Alvarenga Dialética negativa e radicalismo negro: Angela Davis nos anos 1960 https://blogdaboitempo.com.br/2018/05/10/dialetica-negativa-e-radicalismo-negro-angela-davis-nos-anos-1960/ resultou numa espécie de diálogo de surdos. Os estudantes queriam que o professor – que durante anos os encorajara a resistir à transição suave a uma sociedade totalmente administrada – tirasse as devidas consequências práticas da radicalidade de seu pensamento, que afirma aliás sem ambiguidade que a "necessidade de dar voz ao sofrimento é condição de toda a verdade", e que "o especificamente materialista [a exigência de se pôr um fim ao sofrimento] converge com aquilo que é crítico, com uma práxis socialmente transformadora" 13 . O desejo geral parecia ser de que o professor seguisse o exemplo do amigo Marcuse, que naquele momento já dava o que falar: "[...] a conjuntura histórica que ligou seu próprio desenvolvimento intelectual com a busca de um novo vocabulário político no final da década de 1960 permitiu a muitos de nós entender até que ponto ele [Marcuse] tomou a sério o encargo da teoria crítica de desenvolver abordagens interdisciplinares, ancoradas na promessa emancipatória da tradição filosófica no interior da qual ele trabalhou, que sinalizaria a possibilidade e a necessidade de intervenções transformadoras no mundo social real." 14 Adorno, entretanto, chamava atenção para o perigo da falsa imediatidade, para o seu poder ofuscante, e sublinhava que a forma com que pensamos os problemas já é ela mesma de certa maneira política, repisando que a promessa de um mundo melhor hiberna na atenção ao conteúdo sensível, ao dado particular, ao não imediatamente existente, a tudo o que não se enquadra imediatamente no âmbito do conceito, do pensamento identitário etc. etc. Davis salvo engano não deixou de assimilar algo da discussão toda, do risco que comporta a teoria crítica se deixar engolfar pelo imediatismo das questões práticas. Nem por isso seguiu à risca o conselho que lhe dera o professor de não tentar juntar os interesses aparentemente discrepantes pela filosofia e o ativismo social 15 . A seu ver uma teoria crítica consequente da sociedade não nos livra da necessidade de lutar pela mudança social assinalada e exigida pela mesma teoria, logo de ter de descer ao plano da práxis possível – que não tem nada que ver com a pseudo-atividade criticada por Adorno no movimento estudantil alemão no fim dos anos 60. Apartada da práxis modificadora solicitada pela crítica dialética, a própria teoria passa a girar em falso, levando à "atrofia da imaginação política estratégica" 16 . Na condição de mulher negra norte-americana, o que estava acontecendo no país natal – "a ascensão coletiva do meu povo à consciência" – não somente não a deixava indiferente como exigia formas de intervenção crítica bem diversas de mensagens numa garrafa: "A luta era um nervo vital, nossa única esperança de sobrevivência. Eu me decidi. A jornada começava." 17 A liberdade começa com a libertação De volta aos Estados Unidos, premida pelas exigências do momento, que era de luta, quer dizer, próxima aos Panteras e bastante envolvida no Coletivo Che-Lumumba, sem dúvida com os Damnés de Fanon na cabeça, a dissertação de doutorado, desta vez supervisionada por Marcuse, teria agora por tema a força (e a violência) na filosofia de Kant, focando em particular na reação do filósofo à Revolução Francesa. Davis não chegou ao tema arbitrariamente, note-se de passagem. Ainda em Frankfurt havia escrito um artigo – elogiado por Adorno e por outros professores, como Oskar Negt, de quem se sentia mais próxima – sobre a noção de interesse na estética e na filosofia moral kantianas. No texto em questão ela procurava mostrar que o problema dialético da imbricação de forma e conteúdo no tratamento dado por Kant a conceitos como interesse, dignidade, caráter inteligível etc. é sistematicamente obnubilado por sua construção aporética. Hegel já havia notado que a terceira Crítica, ao mesmo tempo em que minaria a fundação epistemológica do edifício crítico arduamente construído nas duas outras, como que abria a porta e conduzia de certa forma ao idealismo especulativo que seria a marca de seu próprio sistema. Sem recorrer diretamente a Hegel, Davis procurou mostrar, mediante uma crítica imanente dos teoremas kantianos, como estes já apontam para a dialética 18 . Raphael F. Alvarenga Dialética negativa e radicalismo negro: Angela Davis nos anos 1960 https://blogdaboitempo.com.br/2018/05/10/dialetica-negativa-e-radicalismo-negro-angela-davis-nos-anos-1960/ A pesquisa que deveria levar a cabo na Califórnia daria continuidade ao que vinha desenvolvendo já em Frankfurt, ainda que desta vez com um tom político mais acentuado. É que nos escritos mais explicitamente políticos de Kant vem à tona uma contradição entre os comandos morais, de acordo com os quais todo sujeito racional deveria agir de forma incondicional, e o princípio segundo o qual se julga. Assim, por exemplo, na concepção kantiana, a Revolução Francesa de 1789 seria a um tempo ilegítima – do ponto de vista da Constituição em vigor e até que uma nova fosse promulgada – e um acontecimento formidável, porque a despeito da violência o entusiasmo que suscitou nos espíritos esclarecidos da época dava prova de uma disposição moral da humanidade; tal simpatia inspirava esperança numa sociedade futura em que as capacidades humanas pudessem se desenvolver livremente 19 . Sublinhe-se de passagem que a experiência estética, tal como a concebe Kant na Crítica do juízo, não somente permite a mediação entre o campo da razão teórica e a esfera do agir moral, como é impregnada pela ideia de uma articulação de força e liberdade (cf. por ex. a Analítica do sublime). Também numa nota do tratado sobre a religião, diz Kant que não se pode amadurecer para a liberdade, porque "há que ser livre para alguém se poder servir convenientemente das próprias forças na liberdade" 20 . Por intermédio de Rousseau, Davis encontrava na noção kantiana de força nada menos que a possibilidade de repensar de forma radical o vínculo interno unindo teoria e práxis 21 . O que estava em jogo, em suma, era nada menos que a realização prática do conceito de liberdade, ou, mais precisamente, para retomar a formulação de um bom leitor de Rousseau, a coincidência da imaginação e do desejo com um poder real sobre o tempo e a vida presentes 22 . Infelizmente Davis não teve como levar adiante a tese; como temia Adorno, foi levada de roldão pelo tumultuado contexto da luta em que mergulhou de corpo e alma, sobretudo pelas consequências do seu engajamento, que não tardaram a bater em sua porta: demissão em 1969, por razões políticas, do corpo docente da Universidade da Califórnia em Los Angeles, onde era professora assistente de Filosofia; aparição no ano seguinte na lista das dez pessoas mais procuradas do FBI; a prisão, o julgamento e a absolvição no caso do sequestro e assassinato de um juiz da Califórnia; subsequente pesquisa sobre o complexo industrial penitenciário como um desdobramento das relações de dominação racial do período colonial, advocacia de um novo abolicionismo, empenho pela causa palestina... Muito embora o projeto tenha sido abandonado, não deixa de ser notável a postura intelectual que nele se delineava: a teoria mais exigente se deixava orientar pelo interesse emancipatório do qual está imbuída a práxis transformadora, cujos problemas decisivos eram iluminados pela radicalidade da análise que não se deixa instrumentalizar. Justificando o engajamento político da ex-aluna quando esta se encontrava na prisão, Marcuse notava uma ligação essencial com o comprometimento intelectual manifesto no curto tempo em que ela atuara como docente: "Ela se recusava a tratar as ideias libertárias da civilização ocidental como mero material apostilado, como matéria para provas ou para obter diplomas – para ela, elas estavam vivas e tinham que se tornar realidade, aqui e agora, não em dias remotos no tempo, não em eternas promessas e expectativas." 23 Davis por sua vez aprendera com o exemplo de Marcuse que agir com os outros (no mundo) e julgar (a realidade) por si mesma não eram necessariamente coisas inconciliáveis. Estava fora de cogitação colocar o cérebro de molho em nome de uma causa, por nobre e legítima que fosse a seus olhos. Antes pelo contrário, foram sua sólida formação filosófica e as convicções políticas bem pesadas que lhe permitiram se situar acima de qualquer partidarismo fácil. Embora próxima dos Panteras, incomodava à jovem marxista o discurso nacionalista de alguns de seus membros, sem falar no papel auxiliar reservado às mulheres no seio do movimento. A preocupação em diferenciar, o cuidado com o uso correto das palavras estavam sempre presentes, e adquiriam sentido estratégico. Aos que afirmavam, por exemplo, que os Estados Unidos eram um país fascista, propunha cautela, pois embora houvesse de fato tendências fascistas, ou protofascistas, muito claramente delineadas, a diferença entre a existência destas no seio de uma democracia – ainda que concebida em termos puramente burgueses – e uma sociedade realmente fascista é fundamental, uma vez que determina os objetivos da luta, contra o quê ou quem lutar, com quem se aliar e que táticas adotar. Raphael F. Alvarenga Dialética negativa e radicalismo negro: Angela Davis nos anos 1960 https://blogdaboitempo.com.br/2018/05/10/dialetica-negativa-e-radicalismo-negro-angela-davis-nos-anos-1960/ Numa carta de 1970, Marcuse admitiu sem rodeios ter aprendido algo novo a respeito da própria natureza da liberdade à luz do empenho teórico e político da aluna: "O abstrato conceito filosófico de liberdade que nunca deve sair de cena de repente ganha vida e revela sua própria verdade concreta: a liberdade não é apenas o objetivo da libertação; ela começa com a libertação; ela existe para ser „praticada‟. Isso, confesso, eu aprendi com você!" 24 Política e(m) sala de aula De acordo com o lugar-comum retomado por Mefistófeles no Fausto de Goethe, verde é a árvore da vida, cinzenta a teoria. Sobre ele se funda em parte o anti-intelectualismo que infelizmente ainda hoje se nota em muitos meios militantes, bem como em setores da sociedade que, excluídos da vida do espírito, transformam, contra seu próprio interesse, a necessidade em virtude. Corrigindo Mefistófeles, Adorno lembrava naqueles mesmos anos que a árvore da vida no mundo do capitalismo administrado estava longe de ser verde, do mesmo modo que o caráter grisalho da teoria é ele mesmo resultado e função da desqualificação geral da vida na ordem burguesa 25 . Recusando a chantagem da alternativa, Davis cedo percebeu que a vida do pensamento residia precisamente na luta teórica e prática contra tudo que diminui o ser humano, reduzindo-o a uma vida que não vive. Adorno achava, não sem alguma razão, que era preciso aproveitar ao máximo os espaços e o tempo ainda disponíveis para o pensamento no mundo totalmente administrado do capitalismo tardio. Ocorre que a própria conservação dos nichos ecológicos onde a reflexão crítica possa florescer em liberdade – sem falar em sua ampliação e na criação de outros nichos – depende cada vez mais da luta coletiva. Davis foi precocemente forçada a compreender isso: a liberdade intelectual – da qual faz parte a acadêmica, que de lá pra cá naufragou completamente – deve ser disputada, reivindicada numa luta constante contra o avanço do capitalismo, que não cessa de abafá-la. Adepta avant la lettre da ideologia tosca da "escola sem partido", a administração Reagan (então governador do estado da Califórnia) tratou logo de destituir a jovem professora negra de suas funções letivas na UCLA em razão de uma visita a Cuba e de sua filiação ao Partido Comunista dos Estados Unidos. Em vez de recorrer à Quinta Emenda da Constituição, como era praxe, Angela Davis não somente reivindicou abertamente o fato de ser comunista como chamou a atenção para a impossibilidade de se separar política e processo pedagógico. Ideias e opiniões políticas, dizia, tinham de ser levadas à sala de aula; a educação é inerentemente política, está fadada a ser política se o seu objetivo for formar seres humanos que se preocupem genuinamente com seus semelhantes e que usem o conhecimento adquirido para conquistar a natureza com o objetivo de libertar a humanidade de necessidades escravizadoras. Davis dizia ainda ter aprendido com Marcuse que o conhecimento, para ser conhecimento de verdade, tem de ser relevante para a realidade humana. Para ela, a escola e a universidade deveriam ser lugares onde a consciência se torna explícita e é impelida numa direção transformadora. Nas aulas proferidas em 1969, Davis decidira abordar o problema da liberdade – questão maior para os venerados "Pais Fundadores" da nação estadunidense (Thomas Jefferson e Cia.) – sob o prisma da mais radical experiência da não-liberdade, da mais brutal forma de alienação, a escravidão 26 . Mas ao invés de analisar esta à luz das teorias correntes ou em voga sobre a liberdade, ela passava estas em revista à luz da literatura negra, em cujo epicentro agiria uma verdadeira dialética, não da liberdade, mas da libertação, entendida como processo inserido numa duração histórica específica. A própria questão da identidade negra, geralmente tratada de forma estática, deveria ser repensada em conexão com tal dialética. A par das aulas dadas sobre Frederick Douglass, o curso incluiria leituras de W.E.B. DuBois, Jean Toomer, Richard Wright e John A. Williams, intercaladas pelas teorias de Fanon e a poesia de vários períodos da história dos negros nos Estados Unidos, a qual seria comparada com textos selecionados de autores africanos e poemas de Nicolás Guillén, poeta cubano negro. À luz de uma formação histórica peculiar, bem como da própria experiência, tais autores teriam colocado em evidência não apenas o formalismo das democracias Raphael F. Alvarenga Dialética negativa e radicalismo negro: Angela Davis nos anos 1960 https://blogdaboitempo.com.br/2018/05/10/dialetica-negativa-e-radicalismo-negro-angela-davis-nos-anos-1960/ liberais ocidentais, mas igualmente a promessa (e os impasses) de uma liberdade em contraste com a qual as liberdades existentes na sociedade burguesa aparecem como mancas: "A história da literatura negra fornece [...] uma descrição da natureza da liberdade, de sua extensão e seus limites, muito mais elucidativa que todos os discursos filosóficos sobre o tema na história da sociedade ocidental. [...] Os negros expuseram, por sua própria existência, as inadequações não apenas da prática [corrente] da liberdade, mas de sua própria formulação teórica." 27 Um gesto nada anódino, diga-se de passagem. E não somente por mostrar que as teorias filosóficas em questão (como o existencialismo) seriam no fundo irrelevantes diante de problemas reais e arraigados historicamente. A força e a vitalidade do programa de pesquisa e ensino de Davis naquele momento – resumido na "necessidade de se estabelecer uma continuidade entre passado e presente a fim de descobrir a gênese de problemas que continuam a existir hoje, [e] descobrir como nossos antepassados lidaram com eles" 28 – provinha da retomada crítica de toda uma tradição de pensamento periférica (muito embora elaborada em sua maior parte no coração do sistema), entendida "não como peso morto, mas como elemento dinâmico e irresolvido, subjacente às contradições contemporâneas", no intuito de recolher "as forças em presença" e solicitar "o passo adiante" 29 . Raça e gênero na luta de classes Muitas teorias contemporâneas partem da premissa de que a igualdade democrática na sociedade presente (pelo menos no ocidente) já é um fato consumado, faltando apenas a inclusão de alguns grupos marginais ou refratários em seu âmbito. No melhor dos casos, a assimilação ou o reconhecimento no seio de um Estado de direito torna-se o ideal, o objetivo a ser alcançado. Mas se a premissa não resiste à prova da realidade, então a inclusão não pode ser o objetivo principal da luta de grupos marginalizados. Como não cessa de frisar Davis, não há como tornar efetivamente igualitária uma sociedade estruturalmente racista e sexista, diante de cuja lógica excludente a inclusão social-democrática de negros, mulheres, homossexuais etc. se torna escárnio. O fato, por exemplo, de se ter um negro na presidência da maior potência militar do mundo não alterou em nada o dado calamitoso de os negros formarem 40% da população carcerária dos EUA, quando não passam de 13% da população do país. Desde a leitura do Manifesto comunista ainda adolescente, Davis passou progressivamente a enxergar os problemas enfrentados pela população negra nos Estados Unidos no interior do contexto mais amplo do movimento da classe trabalhadora. Já nos anos 60, quando voltara de Frankfurt, via como insuficientes tanto o marxismo sem crítica do racismo quanto o black power sem análise de classe. Porque é preciso reconhecer que o racismo no capitalismo sempre funcionou como instrumento de controle, a nível mundial, dos produtores diretos, servindo para justificar as desigualdades de renda que acompanham inevitavelmente a hierarquização da força de trabalho, jogando por conseguinte os oprimidos uns contra os outros, moldando, limitando e reprimindo seus desejos 30 . Além de manter dividida a classe trabalhadora, a raça é ademais, como notou Fanon, o sítio por excelência onde se instala o ressentimento de classe. Após a publicação de Mulheres, raça e classe, de 1981, Davis passou a rechaçar igualmente o feminismo burguês branco – que abarca de Hillary Clinton a um movimento como o europeu Femen, em que adquire traços embaraçosamente caricaturais – por ignorar os problemas maiores da exploração econômica e do racismo estrutural. Assim como o racismo, o sexismo (misoginia, homofobia) e o patriarcalismo são constitutivos da civilização capitalista, tendo as mulheres desde o início sido relegadas à esfera do trabalho doméstico e improdutivo (com todos os sentimentos, atitudes e afetos correspondentes), forçadas a se ocupar do conjunto de atividades não valorizadas muito embora indispensáveis à valorização do capital. Na sociedade de classes a mulher de cor, por ser mulher e de cor, é duplamente prejudicada e oprimida. A situação da mulher, em particular da mulher pobre, da mulher negra, da imigrante, bem como de membros da comunidade LGBTQ, sobretudo os mais pobres, se torna ainda mais vulnerável no contexto do que Robert Kurz chamou de "colapso da modernização" – quer dizer, da deterioração do mundo do trabalho, da erosão da família nuclear Raphael F. Alvarenga Dialética negativa e radicalismo negro: Angela Davis nos anos 1960 https://blogdaboitempo.com.br/2018/05/10/dialetica-negativa-e-radicalismo-negro-angela-davis-nos-anos-1960/ burguesa, em que o homem, séculos a fio, fora o principal provedor, etc., sem que novas relações e formas de socialização venham ocupar o lugar das antigas em crise –, colapso que de modo algum significa o fim do patriarcado, mas lhe impinge ao contrário uma feição particularmente ferina. Em 2015, Hillary Clinton fez um discurso numa histórica igreja afro-americana do estado do Missouri, durante o qual afirmou que "toda vida conta" (all lives matter). Verdadeira boutade, uma vez que cartazes do lado de fora traziam justamente a divisa entoada em Ferguson e outras cidades, ela mesma nome de uma campanha internacional contra a violência e o racismo sistêmico contra negros: Black Lives Matter. No ano seguinte, a infeliz declaração da já então presidenciável viria a se tornar um slogan explicitamente contra tal movimento, o qual segundo os proponentes de All Lives Matter tende a focar em injustiças específicas propagadas contra negros, obnubilando contudo outros tipos de injustiça, como se fossem secundários, como se outras vidas não tivessem a mesma importância. All Lives Matter surgiu portanto para denunciar o suposto "racismo invertido" de Black Lives Matter. Poucos meses após a declaração de Clinton, numa conferência em Estocolmo, Angela Davis não deixou barato. Num discurso inspirado, após chamar a atenção significativamente para a Crítica do juízo de Kant, insistindo tanto no alargamento da imaginação que podem suscitar grandes obras de arte como na negatividade originária imbricada na dimensão estética, que permitem criticar e reformular de maneira radical as condições de subordinação existentes e que por esta razão figuram na base da Teoria Crítica de tradição marxista 31 , Davis ressaltou o caráter a um tempo banal e ideológico da asserção de que "toda vida conta": se nas presentes condições toda vida fosse realmente digna de respeito, argumentou, não haveria necessidade de colocar tanta ênfase em relembrar que vidas negras contam. Para ilustrar o raciocínio, Davis evocou a Revolução de São Domingo e a Constituição haitiana de 1805, cujo parágrafo 14 asseverava que todo cidadão da república livre do Haiti, qualquer que fosse a raça, seria considerado negro. Isso significa que os negros – vistos e tratados então e ainda hoje por muita gente como uma humanidade de segunda classe, e cuja liberdade (no caso haitiano) não viera na forma de um presente dos de cima, mas teve de ser arrancada à força – seriam a medida não apenas da cidadania (haitiana), mas da própria humanidade emancipada: por representarem a parte dos que não têm parte na sociedade existente, é precisamente a sua não-identidade que figura como lugar-tenente (o termo Statthalter aparece com frequência em Adorno) de uma universalidade mais plena e verdadeira; lugar-tenente (negativo) de um "excedente ineliminável, que escapa a qualquer captura e fixação num estatuto social e jurídico" 32 , e que por isso mesmo prefigura um estado de coisas – igualmente vislumbrado por Fanon – em que "raça" deixaria de ser cimento de construção identitária, não seria sequer uma questão. Angela Davis não para aí, e propõe levar adiante a ideia contida no artigo da Constituição haitiana, igualmente à base de Black Lives Matter: se nossos sonhos de liberdade podem ser enriquecidos por tal proposição, por que não imaginarmos mulheres negras como medida da humanidade? E por que não particularizar ainda mais, estendendo o raciocínio para mulheres negras pobres, trabalhadoras, imigrantes, mães solteiras, suburbanas, sem-teto, LGBTQ? 33 Como nas melhores obras de arte, a manifestação de configurações e formas de opressão particulares é aqui mais imediata e sensível do que propriamente simbólica, e o mais individual como que incarna diretamente o universal, tornando-se "o precursor de uma verdade universal que irrompe em seu destino e lugar únicos" 34 . Ao contrário do que se passa no pensamento humanista ou iluminista europeu, a humanidade aqui não é um conceito vazio; não é mero fundamento ideológico de práticas que no mais das vezes contradizem ou distorcem a formulação original, mas antes algo pressuposto. De novo, não estamos longe da dialética negativa adorniana, para a qual o homem "não é apenas o que foi e é, mas também aquilo que pode vir a ser" 35 , ou de Fanon, na bela conclusão de Pele negra, máscaras brancas, onde estima que não deveríamos "fixar o homem, pois o seu destino é ser solto" 36 . O que está em jogo por conseguinte não é um retorno às raízes, a afirmação de uma cultura originária, de uma essência negra ou pan-africana, pois é grande o risco de ver toda uma política de libertação reduzida a uma moda inofensiva (penteado afro ou rasta, batuque afro-reggae, trajes etíopes), não somente aceita mas Raphael F. Alvarenga Dialética negativa e radicalismo negro: Angela Davis nos anos 1960 https://blogdaboitempo.com.br/2018/05/10/dialetica-negativa-e-radicalismo-negro-angela-davis-nos-anos-1960/ encorajada pelos poderes estabelecidos, além de apreciada e consumida pela burguesia branca. Não dá para lutar pelo reconhecimento da particularidade negada nos próprios termos da sociedade patriarcal de classes que a nega, procurando dar uma expressão afirmativa a uma suposta autenticidade (feminidade, negritude, cultura queer, cultura proletária) num quadro de tolerância multicultural de identidades plenamente postas, enquanto que o universo mais e mais degradado do trabalho no capitalismo em fim de linha – o anacronismo das formas capitalistas de socialização, tanto mais agudo agora que a integração da classe trabalhadora se dá no mundo todo sob condições cada vez mais miseráveis e precárias, na forma de desemprego tecnológico e empregos improdutivos de toda sorte, bem como completamente supérfluos do ponto de vista das necessidades reais e do nível já alcançado das forças produtivas da sociedade – passa sem um arranhão. "O capitalismo", dizia Davis num texto escrito na prisão, "pouco importa a cor, corre na contramão dos interesses da humanidade e da racionalidade." 37 Recentemente disse ainda acreditar "que o capitalismo é o tipo mais perigoso de futuro que possamos imaginar" 38 . Por tudo isso tinha razão Jean Genet, num texto que escreveu sobre Davis e seus "irmãos": "[Desde que] os negros compreenderam que eram perfeitamente capazes de resolver eles mesmos seus próprios problemas, de evoluir muito à vontade na política mais complexa, de elaborar teses revolucionárias audaciosas e as aplicar [...] eles renunciaram aos trajes, aos enfeites e às quimeras que queriam fazê-los crer que não passavam de africanos. [...] Auxiliados pelas ideias de DuBois, de Richard Wright, de Fanon, de Malcolm X, de Newton e Seale, eles [os Panteras Negras] compreenderam que um povo cortado muito tempo de sua verdadeira tradição corre o risco de se perder na que ele crê ter reencontrado e que se apresenta, na verdade, sob forma de um folclore muito tranquilizador para a nação opressora. Os Panteras, contra isso, escolheram, deliberadamente, o projeto revolucionário." 39 Dando continuidade à rica tradição do pensamento radical negro 40 , à qual procurou integrar a perspectiva crítico-emancipatória da dialética negativa frankfurtiana – em particular a confiança no testemunho e na força negativa da arte, a atenção à experiência sensível como algo pré-formado social e historicamente e a concepção de uma pedagogia emancipatória, visando ao alargamento social da capacidade de julgar e imaginar outros mundos –, Angela Davis sempre viu como indispensável entender como a estrutura econômica da sociedade determina os modos como esta mesma sociedade define as categorias de raça e de gênero. Pensando a intersecção necessária das diferentes lutas por justiça social – haja vista que hierarquias de raça e de gênero têm a ver, intrinsecamente, com a hierarquia de classe, quer dizer, com a posição que se ocupa dentro das relações de produção dominantes –, nunca deixou de buscar uma solução radicalmente democrática e socialista que se inscrevesse num plano de solidariedade de luta internacionalista, no interior do qual o combate às estruturas racistas, patriarcais, sexistas e de classe dentro do próprio país pudessem ganhar alento, um maior significado e uma ressonância mais ampla. NOTAS 1 Angela Y. Davis, Blues Legacies and Black Feminism. Gertrude "Ma" Rainey, Bessie Smith, and Billie Holiday, New York: Vintage, 1998, p. 164. 2 Salvo indicação diversa, os dados biográficos no presente texto são tirados da autobiografia da autora: An Autobiography [1974], New York: International Publishers, 1996. 3 Cf. Alice Kaplan, Dreaming in French. The Paris Years of Jacqueline Bouvier Kennedy, Susan Sontag, and Angela Davis, Chicago/London: University of Chicago, 2012, p. 179. 4 Cf. Herbert Marcuse, Eros e civilização. Uma interpretação filosófica do pensamento de Freud [1955], trad. Á. Cabral, Rio de Janeiro: Zahar, 1975, cap. 9. 5 Friedrich Schiller, Cartas sôbre a educação estética do homem [1794], trad. R. Schwarz, São Paulo: Herder, 1963, p. 79. A próposito, veja-se na mesma edição a excelente introdução de Anatol Rosenfeld. 6 Cf. Paulo E. Arantes, "Nação e reflexão" [2001], em Zero à esquerda, São Paulo: Conrad, 2004, pp. 79-108. 7 Cf. Theodor W. Adorno & Herbert Marcuse, "As últimas cartas" [1969], em Marcuse, A grande recusa hoje, Raphael F. Alvarenga Dialética negativa e radicalismo negro: Angela Davis nos anos 1960 https://blogdaboitempo.com.br/2018/05/10/dialetica-negativa-e-radicalismo-negro-angela-davis-nos-anos-1960/ trad. I. Loureiro e R. de Oliveira, Petrópolis: Vozes, 1999, pp. 87-101. 8 Na estrada do desespero, o filósofo frankfurtiano e o psiquiatra martinicano andariam lado a lado, segundo Robyn Marasco, The Highway of Despair. Critical Theory after Hegel, New York: Columbia University, 2015. No Brasil, a ideia de desespero conceitual como tarefa do pensamento crítico foi retomada por Vladimir Safatle, Cinismo e falência da crítica, São Paulo: Boitempo, 2008, p. 204. 9 Para a discussão que segue, cf. Alex Demirović, Der nonkonformistische Intellektuelle. Die Entwicklung der Kritischen Theorie zur Frankfurter Schule, Frankfurt/M.: Suhrkamp, 1999, pp. 656-58. 10 Adorno, Dialética negativa [1966], trad. M. A. Casanova, Rio de Janeiro: Zahar, 2009, p. 13. 11 Adorno, Minima moralia. Reflexões a partir da vida lesada [1944-47], trad. G. Cohn, Rio de Janeiro: Beco do Azougue, 2008, § 152, p. 243. 12 Para tal distinção, cf. Ruy Fausto, "Sobre o jovem Marx"[1968/1979], Discurso, n° 13, 1983, pp. 7-52. 13 Dialética negativa, pp. 24 e 173. 14 Davis, "Marcuse‟s Legacies" [1998], em Marcuse, Collected Papers, vol. 3: The New Left and the 1960s, New York: Routledge, 2005, p. viii [Nota do Editor: Uma tradução saiu recentemente na revista da Boitempo: "Os legados de Marcuse", Margem Esquerda #30, 2018]. 15 "Numa de minhas últimas reuniões com ele [...] [Adorno] sugeriu que meu desejo de trabalhar diretamente nos movimentos radicais do período era algo semelhante a um estudante de mídia que decide se tornar um técnico de rádio" (ob. cit., p. xi). 16 Arantes, "Zero à esquerda: uma coleção da hora" [1998], em Zero à esquerda, p. 248. 17 An Autobiography, p. 145. 18 Cf. Oskar Negt, "Gutachten über die Kant-Arbeit von Angela Davis" [1972], Stiftung Ethik & Ökonomie, Berlin: Dossier Blue Planet Award, 2011, pp. 39-40. 19 Cf. Hannah Arendt, Lições sobre a filosofia política de Kant [1970], Rio de Janeiro: Relume Dumará, 1993. 20 Immanuel Kant, A religião nos limites da simples razão [1793], trad. A Morão, Covilhã: Universidade da Beira, 2008, p. 214. 21 É possível que Davis tivesse em mente uma passagem do Contrato social em que o filósofo defende a necessidade paradoxal de forçar à liberdade todo aquele que se recusar a obedecer à vontade geral do povo – entendido como sujeito moral coletivo livre e soberano –, de modo a evitar a recaída em relações de abuso e dependência pessoal. Cf. Jean-Jacques Rousseau, Do Contrato social, ou Princípios do direito político [1757/1762], trad. L. S. Machado, São Paulo: Abril Cultural, 1978, livro I, cap. VII, p. 36. 22 Cf. Bento Prado Jr., A retórica de Rousseau [1974], trad. Cristina Prado, São Paulo: Cosac Naify, 2008, pp. 233 e 237. 23 Marcuse, "NBC, Jan. 31, 1961", em Collected Papers, vol. 6: Marxism, Revolution and Utopia, London/New York: Routledge, 2014, p. 215. 24 Marcuse, "Dear Angela" [carta de 18 de novembro de 1970], em Collected Papers, vol. 3, p. 49. 25 Adorno, "Notas marginais sobre Teoria e Práxis" [1969], em Modelos Críticos, vol. 2: Palavras e Sinais, trad. M. H. Ruschel, Petropólis: Vozes, 1995, pp. 202-29. 26 Embora não o cite, a maneira de abordar o problema é próxima da de Adorno, que sustentava que, "na era da opressão social universal, é somente nos traços do indivíduo massacrado e violado que sobrevive a imagem da liberdade contra a sociedade" (Dialética negativa, p. 222). 27 Davis, Lectures on Liberation [1969], New York: Committee to Free Angela Davis, 1971, p. 4. 28 Ob. cit., p. 13. 29 Roberto Schwarz, "Nacional por subtração" [1986], em Que horas são?, São Paulo: Companhia das Letras, 2002, p. 31. O autor, no caso, se referia a Machado de Assis, Mário de Andrade e Antonio Candido, escritores cuja qualidade se prendia ao fato de terem sabido "retomar criticamente e em larga escala o trabalho dos predecessores". 30 Cf. Immanuel Wallerstein, Capitalismo histórico [1983], trad. R. Aguiar, Rio de Janeiro: Contraponto, 2001, cap. 3. 31 Ela faz questão de frisar, sem dúvida a fim de demarcar a diferença com relação aos herdeiros oficiais da Escola de Frankfurt, os quais, como se sabe, avessos à reflexão estética e tendo feito as pazes com o capitalismo, costumam se contentar em fornecer uma legitimação sociológica para o desalentado reformismo do compromisso socialdemocrata do pós-guerra, trocando emancipação por integração, quer dizer, reduzindo a política ao processo de reconhecimento de afirmações genéricas dos diversos grupos minoritários. Raphael F. Alvarenga Dialética negativa e radicalismo negro: Angela Davis nos anos 1960 https://blogdaboitempo.com.br/2018/05/10/dialetica-negativa-e-radicalismo-negro-angela-davis-nos-anos-1960/ 32 Achille Mbembe, Crítica da razão negra [2013], trad. M. Lança, Lisboa: Antígona, 2014, p. 88. 33 Davis retoma a ideia no prefácio que escreveu para A. T. Lamas, T. Wolfson & P. N. Funke (orgs.),Herbert Marcuse and Contemporary Social Movements, Philadelphia: Temple University, 2017, pp. vii-xii. 34 Marcuse, Contra-revolução e revolta, trad. Á. Cabral, Rio de Janeiro: Zahar, 1973, p. 88. 35 Dialética negativa, p. 51. 36 Pele negra, máscaras brancas [1952], trad. R. da Silveira, Salvador: Edufba, 2008, p. 190. 37 "Rhetoric Vs. Reality", Ebony, Jul. 1971, pp. 115-20, p. 116. 38 Entrevista a Pat Morrison, "Angela Y. Davis on what‟s radical in the 21st century", Los Angeles Times, 06 de maio de 2014. 39 "Angela et ses frères"[1970], em L'Ennemi déclaré. Textes et entretiens, Paris: Gallimard, 1991, pp. 74-75. 40 Para uma boa síntese, cf. Cedric J. Robinson, Black Marxism. The Making of the Black Radical Tradition [1983], Chapel Hill/London: University of North Carolina, 2000. Para desenvolvimentos mais recentes, cf. Gaye T. Johnson & Alex Lubin (orgs.), Futures of Black Radicalism, London/New York: Verso 2017. *** Raphael F. Alvarenga é doutor em Filosofia pela Universidade de Louvain (Bélgica), autor de Desejo de ruptura (2012) e coeditor da revista Sinal de Menos (sinaldemenos.org).

UNIVERSIDADE DE TURIM DEPARTAMENTO DE FILOSOFIA E CIÊNCIAS EDUCACIONAIS Artigo revisado pela Universidade de Turim – 10 de Maio de 2019 Michelangelo Bovero O ENSINO DE SOCIOLOGIA (CIÊNCIAS SOCIAIS) Por: Emanuel Isaque Cordeiro da Silva1 Contato: [email protected] WhatsApp: (82)9.8143-8399 O ensino da Sociologia no Ensino Médio, ao considerar as indicações dos Parâmetros Curriculares Nacionais (PCN), tem como objetivo introduzir o aluno nos principais saberes referentes às questões conceituais e metodológicas que fundamentam a Sociologia, a Antropologia e a Ciência Política2 O contexto de transformação social inaugurado nos séculos XVIII e XIX e a busca pela compreensão científica deste processo vieram estruturar as grandes questões que permeiam este campo do saber. Os paradigmas fundantes da Sociologia, em seus esforços de interpretar o curso das transformações sociais advindas das revoluções industriais e político-sociais, são, portanto, produtos culturais deste processo. Radicado, sobretudo, na busca por estruturar um saber que pudesse oferecer respostas conscientes para dotar a práxis de um sentido capaz de se impor sobre as questões da nova dinâmica social, o campo do conhecimento sociológico, ao chegar às escolas do século XXI, deve ser capaz de promover a reflexão sobre suas próprias bases operacionais. Isso significa que seus parâmetros teóricos e metodológicos fundantes 1 Técnico em Agropecuária pelo IFPE-BJ. Técnico em Biologia, Filosofia e Sociologia pelo Colégio de Aplicação da UFPE. Professor substituto e de reforço do Colégio de Aplicação da UFPE e do Colégio Santa Maria. Bacharelando em Zootecnia pela UFRPE. 2 No portal do MEC podem ser encontrados os documentos sobre as bases legais dos Parâmetros Curriculares Nacionais para o Ensino Médio (2000), Lei de Diretrizes e Bases da Educação Nacional – 9 394/96 (LDB), as Diretrizes Curriculares Nacionais para o Ensino Médio (DCNEM) e Orientações Educacionais Complementares aos Parâmetros Curriculares Nacionais para a área de Ciências Humanas e suas Tecnologias no Ensino Médio (PCN+). Artigo revisado pela Universidade de Turim – 10 de Maio de 2019 Michelangelo Bovero UNIVERSIDADE DE TURIM DEPARTAMENTO DE FILOSOFIA E CIÊNCIAS EDUCACIONAIS Artigo revisado pela Universidade de Turim – 10 de Maio de 2019 Michelangelo Bovero necessitam ser postos a prova constantemente, sendo confrontados com a complexidade do mundo atual. A tradição da Sociologia jamais negou a importância de seu desenvolvimento enquanto ciência crítica de si mesma. Devido a este incessante processo de autocrítica, a reflexão sociológica avalia constantemente de que modo as questões colocadas pelas teorias fundantes de seu saber – as chamadas sociologias clássicas – se comportam diante de um mundo em constante transformação. A inserção da Sociologia como disciplina na grade curricular do Ensino Médio – que veio a realizar-se a partir do Parecer 38/2006, que alterou as Diretrizes Curriculares Nacionais do Ensino Médio tornando a Filosofia e a Sociologia disciplinas obrigatórias, o que efetivou-se com a Lei no 11 684/08 – vai ao encontro dos objetivos das mudanças propostas pela Lei de Diretrizes e Bases da Educação Nacional (LDBEN) de 1996. Tais mudanças procuraram vincular os pilares do Ensino Médio ao mundo do trabalho e à prática social e objetivaram orientar o papel da educação para capacitar o aprendizado contínuo e autônomo e para o exercício da cidadania. Tais reformas, inclusive, atenderam as propostas da Unesco, que visavam estrutrar a educação em torno de quatro princípios: aprender a conhecer, aprender a fazer, aprender a conviver e aprender a ser.3 Ao postular que as atribuições básicas do conhecimento sociológico são a investigação, a identificação, a descrição e a interpretação/explicação de todos os fatos relacionados à vida social, os PCN enfatizam o papel desse saber para proporcionar ao aluno os instrumentos necessários para decodificar a complexidade da realidade social. Assim, a Sociologia se apresenta, na grade curricular do Ensino Médio, como instrumento necessário à construção da cidadania. Tal compreensão fortalece os laços da Sociologia com as finalidades do Ensino Médio, defnidas pelas mudanças da LDB, sendo respaldada especialmente pela Lei 9 394/96, que estabelece como meta da educação a construção da cidadania. 3 DELORS, Jacques. Educação: um tesouro a descobrir; relatório para a Unesco da Comissão Internacional sobre Educação para o século XXI. São Paulo: Cortez, 1996. UNIVERSIDADE DE TURIM DEPARTAMENTO DE FILOSOFIA E CIÊNCIAS EDUCACIONAIS Artigo revisado pela Universidade de Turim – 10 de Maio de 2019 Michelangelo Bovero O ensino da Sociologia, desta forma, ocupa relevan-te papel na construção de uma consciência crítica e reflexiva diante das questões do mundo contemporâneo. Rompendo com as barreiras do senso comum, espera-se que o conjunto sistematizado do conhecimento próprio da Sociologia forneça um aparato teórico que torne o estudante capaz de compreender a dinâmica e as contradições da sociedade em que vive. Voltada à realização do exercício pleno da cidadania, a Sociologia esclarece que a construção de uma sociedade mais justa e solidária é tarefa que exige tanto compreender a complexidade social como as formas de responder e agir em sociedade. Para alcançar esse objetivo em sala de aula, segundo as Orientações Curriculares para o Ensino Médio – Conhecimentos de Sociologia4, o ensino da Sociologia deve se basear em dois princípios epistemológicos fundamentais: estranhamento e desnaturalização. O estranhamento é posto como uma forma de duvidar, que a nada outorga normalidade e nem se conforma diante dos fatos, exercício necessário à problematização dos fenômenos sociais. No momento em que nada se torna óbvio, nem pressuposto, nem simplesmente aceito, se abre o caminho para o educando romper com as amarras do senso comum e construir uma reflexão sistematizada da realidade. Já o momento de desnaturalização é aquele que procura romper com toda e qualquer forma de compreensão das relações sociais como "imutáveis no tempo e no espaço". Os fenômenos sociais que vivenciamos no presente são, em geral, apreendidos pelo senso comum como simplesmente preestabelecidos, causando o entendimento de uma origem natural das relações sociais. Cabe ao ensino da Sociologia superar esse entendimento e promover a dessacralização e a desnaturalização da realidade, rompendo com seu imediatismo ao submetêla a critérios científicos de análise. Desta forma, seja qual for o conteúdo trabalhado pela Sociologia, a Antropologia e a Ciência Política, mais do que oferecer um conhecimento dos fatos, o que se privilegia é o desenvolvimento das perspectivas sociológicas, antropológicas e políticas; a história e o contexto de suas produções e as relações estabelecidas com as realidades que contemplam e com as quais dialogam continuamente. A contribuição das Ciências Sociais reside, pois, justamente na formação humana, ao promover constantemente a problematização da realidade, 4 Disponível em: <http://portal.mec.gov.br/seb/arquivos/pdf/cienciah.pdf>. Acesso em: Março de 2019. UNIVERSIDADE DE TURIM DEPARTAMENTO DE FILOSOFIA E CIÊNCIAS EDUCACIONAIS Artigo revisado pela Universidade de Turim – 10 de Maio de 2019 Michelangelo Bovero sempre confrontada pelo olhar inquieto e crítico, não apenas do que se encontra ao redor, mas de si próprio e de sua própria perspectiva. Trata-se, portanto, de desenvolver um distinto modo de pensar a vida em sociedade. BIBLIOGRAFIA CONSULTADA DELORS, Jacques. Educação: um tesouro a descobrir; relatório para a Unesco da Comissão Internacional sobre Educação para o século XXI. São Paulo: Cortez, 1996. MORAES, Amaury César (Org.). Sociologia: Ensino Médio. Coleção Explorando o Ensino; v. 15. Brasília: Ministério da Educação, Secretaria de Educação Básica, 2010. __________________________. Por que Sociologia e Filosofia no Ensino Médio? Revista de Educação, 10: 50-52, abr., São Paulo: Apeoes, 1999. SILVA, I. F. A. Sociologia no Ensino Médio: os desafios institucionais e epistemológicos para a consolidação da disciplina. Natal: Cronos, 2007. Disponível em: <www.cchla.ufrn.br/cronos/pdf/8.2/d3.pdf>. Acesso em: Março de 2019. Enviado por: Emanuel Isaque Cordeiro da Silva, em 12 de Abril de 2019 – (GOOGLE MAPS: Belo Jardim-PE. – 12:56:23 PM). Revisado e editado por: TORINO EDITIONS DELL'ITALIA – Michelangelo Bovero Dr. Sc. em 14 de Maio de 2019. TURIM-ITÁLIA (19:45:58 PM). Orcid: 1234-544.3/19T Doi:23B.984/019IT UNIVERSIDADE DE TURIM DEPARTAMENTO DE FILOSOFIA E CIÊNCIAS EDUCACIONAIS Artigo revisado pela Universidade de Turim – 10 de Maio de 2019 Michelangelo Bovero

Vol. 15 No. 1 Debates in Aesthetics is a peer-reviewed, open-access journal for articles, interviews and book reviews. The journal's principal aim is to provide the philosophical community with a dedicated venue for debate in aesthetics and the philosophy of art. Vol. 15 No. 1 March 2020 Edited by Claire Anscomb and Eleen M. Deprez Published by The British Society of Aesthetics Typesetting Eleen M Deprez Proofreading Laura Harris Typeface The Brill, designed by John Hudson Avenir, designed by Adrian Frutiger Cover Eduard Isaac Asser, Stilleven met Foto's, Lens, Beeld en andere Objecten [Still Life with Photographs, Lens, Sculpture, and other Objects] 1855. Salted paper print. Collection Rijksmuseum Amsterdam, reproduced with permission: CC0 1.0 Universal. Contact www.debatesinaesthetics.org [email protected] ISSN 2514-6637 SENSORY AUGMENTATION AND THE TACTILE SUBLIME This paper responds to recent developments in the field of sensory augmentation by analysing several technological devices that augment the sensory apparatus using the tactile sense. First, I will define the term sensory augmentation, as the use of technological modification to enhance the sensory apparatus, and elaborate on the preconditions for successful tactile sensory augmentation. These are the adaptability of the brain to unfamiliar sensory input and the specific qualities of the skin lending themselves to be used for the perception of additional sensory information. Two devices, Moon Ribas' Seismic Sense and David Eagleman's vest, will then be discussed as potential facilitators of aesthetic experiences in virtue of the tactile sensory augmentation that these devices allow. I will connect the experiences afforded by these devices to the Kantian categories of the mathematical and the dynamical sublime, and to existing accounts of tactile sublimity. Essentially, the objects these devices make sensible, earthquakes for the Seismic Sense and digital information for the vest, produce pleasurable feelings of potential danger, awe, and respect. The subsequent acclimation to this new way of sensing and the aim to comprehend its sensed object are then discussed as possible objections to the interpretation of these experiences as sublime, and as aesthetic in general. To exemplify these issues and concretise my thesis of tactile sensory augmentation as a trigger of the sublime, I will outline an experiment to use the vest as an aid for faster decision making on the stock market. Yorick Berta Independent Scholar 12 Yorick Berta Introduction 1 On the concept of cyborgism see Grenville (2001) and Kirkup (2010). 2 For an overview on augmented reality technologies, see Dey et al. (2018); a review of literature on virtual reality can be found in Berntsen et al. (2016). 3 Our tactile sense is thus a subcategory of the sense of touch, which includes tactile, haptic, proprioceptive, kinaesthetic, vestibular, and cutaneous sensations. For an overview and definition of these different 'senses of touch' see Paterson 2007, IX. 4 Even on Ribas' own various web presences, there is conflicting information on the Seismic Sense. On the project's website (ThoughtWorks Arts 2016), it is described as a wearable, an exteriorly worn device, and illustrated with a picture of it worn on Moon Ribas' ankle. Later, it is described as an implant into Ribas' arm, however. On another webpage co-founded by Ribas, (Cyborg Arts 2019) the Seismic Sense is described as an implant in the artist's feet. Since the function of the Seismic Sense is not affected by these differences, I will address it as an ankle bracelet, which, as a wearable, seems its potentially most popular form. Interestingly, Ribas also uses her device as a prop in various performances, whose choreography depends on the seismic activity felt by her. Here, the sensorally augmented artist functions as a medium for the creative power of 'nature' itself. For a summary of Ribas' artistic activities and strategies, see her TED-Talk in Munich in 2014 (TEDx Talks, 2015). 5 This, too, has been presented in a TED-Talk, in Vancouver in March 2015 (TEDx Talks, 2015a). From the figure of the cyborg, an ambiguous merger of flesh and technology1, to the recent innovations in the field of virtual and augmented reality,2 the technological mediation of our embodied experience of the world elicits constant debate about its potential, implication, and value. In addition, phenomena such as wearables or body modifications pose interesting questions to the field of aesthetics. Virtual reality goggles or smart glasses, for example, transform, denaturalise, and deconstruct the way we sensually experience the world. Such devices challenge the very basis of aesthetic experience. In this paper, I want to examine how technological modification of the senses can alter or facilitate aesthetic experiences. I will discuss two devices that exploit our tactile sense (the registration of pressure on the skin).3 The first, Seismic Sense, is a vibrating ankle bracelet, invented by the artist Moon Ribas.4 The second, the Versatile Extra-Sensory Transducer (vest), is a vibratory device worn as a vest developed by neuroscientist David Eagleman 5 The ankle brace13Sensory Augmentation and the Tactile SublimeVol 15 No 1 let makes earthquakes perceptible at a distance, by vibrating whenever there is an earthquake happening somewhere in the world. The device is connected to an online seismograph, causing the vibrations to vary in strength according to the earthquake's magnitude (Boukhris and Almiron 2017, 203). The vest is equipped with small vibratory sensors on its back which can be controlled via Bluetooth. Depending on the programming, the vest can receive and automatically transduce different kinds of digital information into vibratory patterns. This way, digital information, such as data from the stock market, can be translated into tactile sensations. I will argue that both facilitate the aesthetic experience of the sublime. In other words the experiences afforded by these devices can be understood in terms of the Kantian dynamical and mathematical sublime In Section One, I will explain what sensory augmentation is and how it works. Important here are the premises of brain plasticity (its functional and structural changeability) and the possibility of tactile sensory augmentation. In Section Two, I discuss the two aforementioned devices, which are able to transduce abstract information into immediately felt tactile sensations. Section Three develops an understanding of the experience afforded by the devices using the Kantian categories of the dynamical and the mathematical sublime. Section Four examines a potential worry: how can we render sensory augmentation devices as sublime, when their aim for sensual comprehension of abstract phenomena stands contrary to the sublime as an aesthetic of failed comprehension? To further explore this worry, in Section Five I discuss a particular example of sensory augmentation technology facilitating an experience of the sublime, namely the use of the vest in stock trading experiments. This will lead me to the conclusion that the sublime experiences manifested with these devices are only possible during the early stages of use before the inevitable normalisation of the tactile sensations, although the desired effect of sensual comprehension is open to question. 14 Yorick Berta 1 Sensory Augmentation and Neuroplasticity 6 The authors distinguish between 'compensatory prostheses', which corresponds to my definition of a) 'within-sense referral devices', which expands b) onto relocations of the sensor with mirrors etc.; 'between-sense referral devices', which, as the authors state, is largely synonymous with sensory substitution; and 'novel-sense referral devices', which corresponds to my usage of the term sensory augmentation. 7 The term 'sensory extrapolation' is borrowed from Humphreys (2004, 4). One way of understanding sensory augmentation is an alteration of the way we sensually experience the world. On this understanding a mirror which relocates the sense of vision, or a white cane which extends the tactile sense, or even temporary sensory alterations such as hallucinogenic drugs can be seen as sensory augmentations. This ambiguity necessitates a distinction between four related concepts of sensory alteration for the purpose of this essay, a distinction which roughly follows four definitions proposed by Jamie Ward and Thomas Wright (2014, 9-10):6 a) Sensory Restoration – repairing a damaged sense modality with, for example, prescription lenses or cochlear implants b) Sensory Extrapolation7 – increasing the range of perception of an existing sense modality, with devices such as night vision goggles or microscopes c) Sensory Substitution – replacing a certain sense by translating sensory stimuli of one modality into information for another modality, for example braille d) Sensory Augmentation – using technology to make something formerly imperceptible known to the sensory apparatus, such as x-ray radiography (Ward and Wright 2014, 9-10) Although the line of distinction between (c) and (d) is rather thin, it is important to separate them to account for the different underlying 15Sensory Augmentation and the Tactile SublimeVol 15 No 1 motivations. Whilst sensory substitution (c) – and also sensory restoration (a) for that matter – simulates the sensing of another modality, restores previous sensory abilities, or establishes normative sensory abilities for differently abled people, sensory augmentation (d) accesses information which lies completely outside the human sensory realm.8 Particularly, sensory augmentation (d) does not understand the substituting sensory modality as a mere compensation for the substituted modality, but embraces the specific way of each modality to sense the world. The tactile augmentations discussed here are therefore no mere substitutions for vision, but augmentations in their own right. However, sensory augmentation and substitution are inevitably linked. The attempt to augment the sensory apparatus was only made possible by innovations in the field of sensory substitution. The same device can often be used to both substitute and augment a sensory modality, as the history of tactile sensory substitution shows. Therefore, although my main interest is sensory augmentation (d), it is so closely connected to sensory substitution that a prior discussion is required. Crucial for the development of sensory substitution and augmentation devices was the research by neuroscientist Paul Bach-y-Rita, starting in the late 1960s. His experiments made two important discoveries: (1) the ability of the brain to interpret new sensory stimuli, and (2) the affordance of the skin. Bach-y-Rita developed a method for Tactile-Visual-Sensory-Substitution (tvss). This enabled blind people to perceive objects at a distance using their tactile sense. In Bach-y-Rita's original setup, an array of vibratory solenoid plates was mounted to a dentist's chair and connected to a camera (1969).9 The blind participants controlled a camera to scan various items placed on a table in front of them. The camera image was then transduced into vibratory responses, 8 For an ideological and technological critique of sensory restoration as a reproduction of ableism, see Campbell (2009, 79-96). 9 For a more recent review of the progressings in the field of sensory substitution using different sensory modalities, see Bach-y-Rita and Kercel (2003). 16 Yorick Berta corresponding to the image in intensity and position on the array By interpreting the vibrations on their backs, the participants were able to discriminate between the objects placed in front of the camera. In comparison with previous technological inventions which can be seen as tvss-devices, such as the white cane or the braille writing system, Bach-y-Rita's setup works without directly or indirectly touching the object and performs a greater range of functions than identifying obstacles or reading text – it makes possible the simultaneous perception of several objects, including non-haptic features such as their shadows, their potential movement, and their spatial relation (Bach-y-Rita et al. 1969, 963-4). The participants in this experiment reported a fast process of adaptation, as they learnt this new means of perception. With time, users became less aware of the tactile stimulation itself and instead began to perceive the objects directly, i.e., without being conscious of the technological mediation facilitating the sensual experience (Bach-y-Rita et al. 1969, 964; Bach-y-Rita 2004, 86). The experiment demonstrates two factors which are essential for the success of sensory augmentation devices discussed in the next section: neuroplasticity and the affordance of the skin for additional sensory input. Neuroplasticity denotes the theory that the connections between neurons in the brain are not fixed but get established and consolidated by frequent use. (Hebb 1949, 62; Bachy-Rita 1967). According to neuroplasticity the brain can transform its anatomical or functional structure according to changing requirements, i.e., it is malleable. Bach-y-Rita's experiments with sensory substitution demonstrate this. After an initial learning phase, the brain is able to interpret new sensory stimuli. This ability, which is necessary for successful sensory substitution, also enables sensory augmentation. Instead of transducing the sensory information of one modality into another, information that is completely alien to the sensory apparatus is translated into tactile stimuli and is perceived directly 17Sensory Augmentation and the Tactile SublimeVol 15 No 1 Another finding of Bach-y-Rita's experiment is the potential of the skin to process additional sensory input. Unlike the eyes or the ears, the skin is not permanently occupied with discriminating unfamiliar sensory information. At least on the back, most passive stimulations on the skin are processed unconsciously – if we do not concentrate on it, the sensations of the air or of clothes touching our skin go largely unnoticed. Although subliminal sensations occur within all sense modalities, the threshold for the skin on the back to consciously perceive sensory stimuli is comparatively high (Weinstein 1968). Therefore, it offers a large, typically underutilised area for additional sensory information in the form of vibrations. Although the tongue or the fingertips have far more sensory receptors, Bach-y-Rita's original experiment has shown that the sensibility of the skin of the back is effective enough to discriminate the sensations of four hundred vibratory rods on an array. Keeping these discoveries in mind – (1) the ability of the brain to interpret new sensory stimuli, and (2) the affordance of the skin, especially on the back, for sensory augmentation – I would now like to introduce two technical devices, both of which have popularised sensory augmentation using the tactile sense 2 Tactile Sensory Augmentation Devices In this section, I will introduce two devices which engage with the findings of Bach-y-Rita's research to transit from sensory substitution to sensory augmentation: the Seismic Sense and the vest In 2013, the artist and co-founder of the Cyborg Foundation Moon Ribas started wearing a vibratory ankle bracelet with a wireless connection to the US government web service for earthquake data (Boukhris and Almiron 2017, 203). Whenever it receives information about an earthquake happening somewhere in the world, the ankle bracelet vibrates. The vibration's strength matches the earthquake's magnitude on the Richter scale. Ribas thus carries a tactile augmentation which functions as a 'Seismic Sense'. In an interview from 2016, the artist stressed the 18 Yorick Berta feeling of closeness this device facilitates, when she described the effect of a particularly strong earthquake: "It felt very weird, like I was there [...] I feel connected to the people who suffer through an earthquake." (Quito 2016). The Seismic Sense thus combines an affective quality ("I feel connected") and sensual immediacy ("like I was there") with the safe distance to the actual object of perception, the earthquake. The second example of sensory augmentation is a tvss-device developed by the team of neuroscientist David Eagleman. This device is called the vest (Versatile Extra-Sensory Transducer). It functions as a portable and flexible version of Bach-y-Rita's dentist chair, which I discussed in Section One.10 The vest has several vibratory motors on its back which are controlled by an implemented microcontroller. With a specifically designed interface – a smartphone app, for example – information is compressed and sent to the microcontroller via Bluetooth Here, the information is transduced into vibratory patterns, which can be felt as tactile sensations on the back. If a working interface to compress the incoming information is provided, any kind of information can be transduced into tactile sensations. Given the plasticity of the brain, it is possible that information will eventually be perceived directly as an additional sense The vest still remains in a highly experimental state with no established ways of usage. There are, however, many contexts it could be introduced to. The vest could be connected to a camera, for instance, through which the user scans their surroundings and transduces the appearance of objects into tactile sensations on the back. However, we can imagine many other ways to apply the vest It could be used to ease navigation in space by vibrating when the user has to turn to a certain 10 Bach-y-Rita himself developed several portable applications of his TVSS system, including an array of tactile stimulators grid worn as a belt on the abdomen and a small array placed on the tongue, connected to a camera mounted within spectacles. See: Paterson (2016, 168-170); Chirimuuta and Paterson (2015, 416); Renier and Volder (2013, 855). 19Sensory Augmentation and the Tactile SublimeVol 15 No 1 direction, for example; to get direct bodily feedback whilst controlling complex machinery; as a notification system for upcoming appointments or social media activities; to notify the user of changes of internal bodily states they might otherwise be unaware of (e.g. ovulation or hypoglycaemia); or it could increase synaesthetic experience, enhancing, for example, the enjoyment of music. When sitting in a concert hall, the vest could be connected to microphones and, depending on the chosen programming, vibrate according to the music's volume or timbre, or the different vibrators on the vest could match different instruments. Or when going on a run, the vest could warn against dehydration, give feedback on the heart rate or keep the wearer on a pre-programmed path by vibrating on crossways. Again, theoretically, the relayed information via the vest could be of any kind as long as it is translatable into vibrations via Bluetooth. These two devices suggest that the brain adapts itself to new sensory stimuli, particularly to those utilising the skin, which is a sensory medium without many interfering conscious stimuli In addition, the tactile sensations seem to be able to produce feelings of immediacy and closeness. These feelings are important for the next section of this paper, in which I will argue for the Seismic Sense and the vest as facilitators of sublime experiences – the latter not in a certain area of application, but its general incentive to make abstract information tactilely sensible. While the association of the Seismic Sense with the sublime might seem obvious, given the preoccupation of both with natural phenomena, the relationship between abstract information and the sublime will require some elaboration. 3 The Tactile Sublime I will first briefly discuss the sublime as conceptualised by Immanuel Kant and describe existing accounts of a tactile sublime. I will then relate the sublime to the devices introduced in the previous section In the Critique of Judgement, Kant develops two distinct categories of sublime experiences (1987, 101 §24). On the one hand, the dynamical 20 Yorick Berta sublime describes the merely theoretical fear triggered by overwhelming or threatening phenomena in nature, such as thunderclouds or waterfalls. To experience the dynamical sublime in nature, the subject has to reassure herself that the danger imposed by nature is not harming them, yet re-enact the fear it would cause as a form of amazement.11 On the other hand, the mathematical sublime describes the incapability of human imagination to comprehend magnitudes in the same way that mathematical concepts can frame them. Magnitude is a relative term, denoting objects which are large beyond comparison, or 'in comparison with which everything else is small.' (Kant 1987, 106 §25). Their totality exceeds imaginative abilities and curbs the urge for aesthetic comprehension (Kant (1987, 97-8 §23). Similar to the dynamical sublime, this failure of the aesthetic faculty entails a certain threat for the subject, a momentary questioning of his sovereignty. It opens up 'an abyss in which the imagination is afraid to lose itself.' (Kant 1987, 115 §27) The momentary trembling, facing a danger or the limits of aesthetic comprehension, arouses a feeling which is 'not so much a positive pleasure as rather admiration and respect ' (Kant 1987, 98 §23 and 114 § 27) Kant's account of the sublime is based solely on the sense of vision. Yet, recent work by Alan McNee and less directly by Carolyn Korsmeyer explore the possibility of a tactile sublime (McNee 2015; Korsmeyer 2014 and 2016). In his work on 19th-century mountaineering as a multisensory aesthetic experience, McNee coined the term 'haptic sublime', which is, unlike the ocularcentric,12 distanced aesthetic theory by Kant based on 11 '[...] any spectator who beholds massive mountains climbing skyward, deep gorges with raging streams in them, wastelands lying in deep shadow and inviting melancholy meditation, and so on is indeed seized by amazement bordering on terror, by horror and a sacred thrill; but, since he knows he is safe, this is not actual fear: it is merely our attempt to incur it with our imagination, in order that we may feel that very power's might and connect the mental agitation this arouses with the mind's state of rest' (Kant 1987, 129 §24). 12 I borrow the term 'ocularcentrism' from Martin Jay (1988). 21Sensory Augmentation and the Tactile SublimeVol 15 No 1 the proximity of tactile encounter.13 According to McNee, it is the sensually experienced closeness to the mountains which evokes sublime feelings in the touching subject: the sensations of climbing a mountain, feeling the tension in the whole body, its slow movement in the scenery, the exhilarating feeling of danger, and the contact of the skin to the rock or ice wall all come together to make the mountaineer aware of their bodily presence while simultaneously inducing a feeling of transcendence. McNee stresses that the sublimity of mountaineering cannot only be attributed to the overwhelming nature of mountains, but to the human urge to conquer them: 'The sublime [...] is to some extent an aesthetic of mastery, of overcoming a threat or difficulty.' (2014, 15) This threat is, unlike the Kantian dynamical sublime, not theoretical but real; it intensifies the threat for human sovereignty and the accompanying feeling of respect (McNee 2014, 15-7). In Korsmeyer's writing on genuineness and the sense of touch, the category of the sublime is repeatedly implied, when she references David Lowenthal's notion of the 'shiver of contact' David Hume on the sublimity of cross-temporal contact (Korsmeyer 2016, 219) and the tactile encounter of ruins (Korsmeyer 2014, 434). In touching something genuinely old, the touching subject is directly confronted with a temporal scale far beyond the human lifespan. Touch, in this case, establishes the perceptual link necessary for an aesthetic experience and elicits the feeling of respect. Both these models of tactile sublimity replace the distance necessary for experiencing the Kantian dynamical sublime with a moment of physical contact. Ribas' seismic device, then, can be understood as a reconciliation of this tactile sublimity with the distanced dynamical sublime. The Seismic Sense conforms to the dynamical sublime firstly because earthquakes as the given object of perception are a traditional 13 McNee uses the term 'haptic' instead of 'tactile' to account for the synaesthetic experience of mountaineering, including proprioceptive and kinaesthetic sensations (2014, 14). 22 Yorick Berta trigger of sublime experiences. For example, in his treatise on the dynamical sublime, Kant discusses natural phenomena such as 'tempests, storms, earthquakes.' (1987,122 §28) Secondly, the Seismic Sense elicits a sense of possible danger while maintaining a safe distance. It is exactly this combination of danger and distance that makes the Seismic Sense eligible for the aesthetic category of the sublime. In the accounts of McNee and Korsmeyer, the sublime experience is brought about by physical presence. In the case of the Seismic Sense, the physical presence becomes telepresence, which denotes 'the presentation of perceptual information that claims to correspond to a remote physical reality' (Goldberg 1998, 33). The ankle bracelet as a device able to facilitate telepresence reconciles the tactile sublime and the Kantian dynamical sublime: it establishes an immediate contact with the destructive powers of nature but secures a safe distance, which enables the feeling of respect while emphasising the subject's independence. The Vest, on the other hand, responds to the mathematical sublime This is because it establishes a perceptual link with phenomena across multiple spatio-temporal dimensions, questioning human sovereignty. Specifically, in this case, the overwhelming phenomenon is the sheer amount of information gathered in the 'information society'. The post-industrial paradigm of the information society has been, among others, postulated by Theodore Roszak. In his book The Cult of Information, Roszak analyses the impact of a 1948 paper on communication theory by Claude Shannon. In this paper, Shannon proposes a new definition of information, not as a carrier of meaning, but as a 'purely quantitative measure of communicative exchanges' (Roszak 1986, 11) which is independent from both sender and receiver. Information gains value on its own and effectively reduces human agency and sovereignty. Roszak argues that this new definition of information prefigures the paradigm of the information society, as a society organised in decentralised global networks, in which humans become mere 23Sensory Augmentation and the Tactile SublimeVol 15 No 1 participants (1986, 15–16).14 Information itself, organising and controlling human interaction, becomes a sublime phenomenon – something that is increasingly hard to grasp in its totality and is less and less subordinate to human command This relation of information and sublimity has in recent years been further popularised by 'developments across a range of technologies, from digital sensors, computer networks, data storage, cluster computer systems and cloud computing facilities' (Bryant et al 2008, 2-3 in McCosker and Wilken 2014, 156), developments that have led to the algorithmic aggregation of information in nearly every aspect of life. This is subsumed under the term 'big data', which denotes amounts of data that are too big to be intelligible by humans Moreover, due to the logic of total data accumulation, which lead to the phenomenon of big data, humans themselves became profitable sources of information. This logic increases the subjugation denoted in Shannon's model of information. Thus, the concept of big data as a relative magnitude, the challenge it poses for human sovereignty and the problem of representability or comprehension lead back to the mathematical sublime. The Vest not only allows users to experience phenomena such as big data sensually, but makes this experience a potentially aesthetic one, as a feeling of the sublime. Although the category of the sublime was originally used in relation to predominantly visually experienced phenomena, the work of McNee and Korsmeyer have paved the way for an understanding of a tactilely experienced sublime. The physical presence, which is essential to both accounts of tactile sublimity, is, in the devices discussed here, transformed into telepresence. While the Seismic Sense refers to the traditionally sublime phenomenon of earthquakes, the Vest is able to make digital information sensible through vibrations, facilitating an equal feeling of sublimity. 14 See also Poster (1994, 173). 24 Yorick Berta Yet, framing tactile devices as sublime facilitators poses a problem. The Vest proposes an alternative way of knowing, an immediate, sensual form of comprehension. If, however, the vest is able to aptly comprehend big data for its user, a conflict with my interpretation of the feelings it produces as sublime would arise. In the following section, I will explore this conflict between the vest's aim for comprehension and my interpretation of it as aesthetic facilitator further. Observing a process of normalisation in the sensual experience afforded by the Vest, which reinforces the aim for comprehension, I will argue that sublime experiences are only possible in the early stages of the use of such devices. 15 See Manovich (2002); Jevbratt (2004); Stalbaum (2006); Sack (2011); McCosker and Wilken (2014). 4 Sensual Comprehension and Normalisation The relationship between sensual comprehension of abstract information and the sublime has been discussed in detail for the topic of data visualisation 15 I will relate this discussion to the given case of tactile sensory augmentation devices, with regard to Bach-y-Rita's experiments, as well as existing work on somaesthetics – the aesthetic experience of one's own body. For the case of data visualisation, Lev Manovich argues that visualisation is indeed anti-sublime, since it attempts to represent overwhelming masses of data in perceptible, and what are frequently referred to as 'beautiful', graphs (Manovich 2002, 8). Anthony McCosker and Rowan Wilken, on the other hand, plea for data visualisation as a trigger of sublime experiences in its own right, which rather 'reinforces the sense of 'unknowability' that has become associated with sublime experiences of phenomena connected with 'big data'' (2014, 156). For the case of the vest, I will take the view of McCosker and Wilken. The vest primarily establishes a sensual relationship with the represented object, a sublime anticipation. Similar to imaginatively encoun25Sensory Augmentation and the Tactile SublimeVol 15 No 1 tering centuries of history with a single touch, the venerating gesture Korsmeyer sees in the tactile encounter of ruins, I propose that the vibrations of the vest can produce a feeling of respect for the excessive amounts of information channelled through it. In the beginning, when the vibrations seem erratic and incomprehensible, the wearer of the vest might feel overwhelmed by the constant stream of data, and the sheer vastness of the informational world they are now exposed to. Here, the 'urge for mastery' (McNee) is paired with the 'sense of unknowability' (McCosker and Wilken). It was reported that the participants of Bach-y-Rita's original experiment became accustomed to the vibrations and with time came to perceive objects directly. This indicates that the initial sublime 'shiver of contact' may weaken. This process of normalisation is the exact reversal of what Richard Shusterman and Sherri Irvin describe as a criterion for somaesthetics - the unusual attention one has to give to bodily sensations to perceive them aesthetically (Shusterman 1997; Irvin 2008). Due to the malleability of the brain, the user of the vest will give less and less attention to the sensation itself, and start to feel them as a direct manifestation of the data the vest has been connected with. Following this line of thought, neuroplasticity hinders aesthetic experience: the tactile devices discussed here start out as facilitating sublime aesthetic experiences, which, with increasing normalisation, eventually fade into ordinary sensory perception. The aesthetic appreciation of ordinary objects thus requires a sense of unfamiliarity and an unusual attention to the sensual experience itself, both of which are challenged by constant exposure I will end my paper by exemplifying this tension between the category of the sublime and the process of normalisation with a particularly remarkable experiment performed with the vest. This experiment, aiming for a tactile sensing of the fluctuations of the stock market, combines the characteristics of the mathematical and the dynamical 26 Yorick Berta sublime and, notwithstanding that this aesthetic effect might dissipate after repeated use, illustrates the device's processing of big data. 16 David Eagleman reports on the ongoing experiments in a TED-talk in Vancouver, March 2015 (TEDx Talks, 2015a). A scientific publication of the results is yet to be published. 17 See, for example, Shiller (2005), Tversky and Kahneman (1974), and Brunnermeier (2001). 18 Research has shown that traders with a highly developed sense of interoception, a visceral feeling for the internal state in the body, are significantly more successful at high-frequency-trading. See Kandasamy et al (2016). 5 Sensing the Stock Market In a separate experiment carried out by Eagleman's team, the vest was linked to data from the stock market.16 The stock market is an electronic network of shareholders, which has been described by economists such as Shiller or Kahnemann as behaving like a herd, yet without any common or higher purpose 17 The fluctuations in the stock market, which the buying and selling of stocks produce, are in their entirety mostly erratic, unforeseeable, and in times of crisis highly destructive for the economy In this context, the vest is promoted as a novel approach to the stock market. It promises an intuitive, visceral understanding of the market fluctuations and a considerable advantage in the virtual trade of buying and selling stocks, in which only the fastest and most knowledgeable participants profit.18 Eagleman explains the rationale as follows: We stream 5 seconds of real-time data from the internet to a person wearing the vest. Then two buttons appear on a screen, and the person has to make choice A second later they get a smiley face or frowny face telling them whether their choice was the right one. The person has no idea that what they're feeling is real-time stock market data, and that the buttons represent buy or sell decisions. [...] Eventually we'll tell the participants what is really going on, and we are interested to know what the expe27Sensory Augmentation and the Tactile SublimeVol 15 No 1 rience will be like for a person who wears this stock market vest for long time. Are they suddenly going to feel a tightening in their stomach and think, "Oh gosh, the oil price is about to crash"? (Eagleman in Thomson 2015, 27) In this setup, the vest enables its wearer to adapt to the rapid fluctuations of the stock market and to integrate its, at times, devastating instabilities directly into the sensory apparatus In particular, the vest offers a new way to interpret the constant, overwhelming and often hardly intelligible flow of information in real-time, and to sensually experience the devastations caused by ruptures in the stock prices. This way, the tactile sensing of the stock market responds to the mathematical as well as the dynamical sublime Eagleman does not try to conceal the fact that the experiment is highly speculative and might not provide a substitute or even a supplement for traditional stock market analysis. However, the erratic vibrations indicating changes on the incomprehensible scale of the world economy will eventually cease to produce sublime experiences and dissipate into ordinary tactile sensations that the wearer of the vest will get accustomed to – whether they can make sense out of them or not. 6 Conclusion In this article, I discussed two tactile sensory augmentation devices and their potential for aesthetic experiences, in particular experience of the sublime. Following Jamie Ward and Thomas Wright, 'sensory augmentation' has been defined as a way to exploit an existing sense modality in order to make something that was formerly imperceptible known to the sensory apparatus (Ward & Wright 2014, 9-10). Consequently, I have proposed that this is distinct from sensory restoration, extrapolation and substitution I have suggested that sensory substitution is the precursor for sensory augmentation, given that the sensory substitution experiments by Bach28 Yorick Berta y-Rita have demonstrated that the plasticity of the brain and the affordance of the skin for sensory substitution are key to successful tactile sensory augmentations. The devices I have examined, in this paper, make use of these preconditions to enable new aesthetic experiences of the world - earthquakes in the case of the Seismic Sense and digital information in the case of the vest Moreover, I have demonstrated that these experiences fit the Kantian concept of the dynamical and mathematical sublime. I have singled out the problem of comprehension as a possible objection to my account. The sublime, as an aesthetic of failed comprehension, stands contrary to a possible motivation of the vest, which is to comprehend phenomena, such as big data, tactilely. I have argued that instead of a feeling of comprehension, the vest produces a feeling of unknowability, which triggers the sublime. An aesthetic feeling such as the sublime, however, requires that attention be given to the sensation, which will probably fade with constant use of tactile sensory augmentation devices. The facilitation of sublime experiences is therefore an effect confined to the initial stage of the usage of such devices. In order to concretise this issue and the abstract feature of the vest to give a sublime sense of overwhelming amounts of data, I have examined an experiment to make stock data tactilely sensible. 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www.ssoar.info Rationalität und Normativität Marx, Johannes; Tiefensee, Christine Veröffentlichungsversion / Published Version Zeitschriftenartikel / journal article Zur Verfügung gestellt in Kooperation mit / provided in cooperation with: Verlag Barbara Budrich Empfohlene Zitierung / Suggested Citation: Marx, J., & Tiefensee, C. (2015). Rationalität und Normativität. ZPTh Zeitschrift für Politische Theorie, 6(1), 19-37. https://doi.org/10.3224/zpth.v6i1.19901 Nutzungsbedingungen: Dieser Text wird unter einer CC BY-SA Lizenz (NamensnennungWeitergabe unter gleichen Bedingungen) zur Verfügung gestellt. Nähere Auskünfte zu den CC-Lizenzen finden Sie hier: https://creativecommons.org/licenses/by-sa/4.0/deed.de Terms of use: This document is made available under a CC BY-SA Licence (Attribution-ShareAlike). For more Information see: https://creativecommons.org/licenses/by-sa/4.0 Diese Version ist zitierbar unter / This version is citable under: https://nbn-resolving.org/urn:nbn:de:0168-ssoar-60208-v2-8 Johannes Marx, Christine Tiefensee: Rationalität und Normativität, ZPTh Jg. 6, Heft 1/2015, S. 19–37 Rationalität und Normativität Johannes Marx / Christine Tiefensee* Schlüsselwörter: Rationalität, Normativität, Ethik, normative Politische Theorie, positive Politische Theorie Abstract: In den Sozialwissenschaften nimmt der Rationalitätsbegriff eine zentrale Rolle ein. Hierbei werden rationalitätsbasierte Theorien, insbesondere die Rational-Choice-Theorie, zumeist als Teil der positiven Politikwissenschaft verstanden. Gleichzeitig greifen jedoch auch normative Theorien der Politischen Theorie verstärkt auf Rational-Choice-Argumente zurück. In diesem Artikel untersuchen wir die Frage, wie ein solcher angeblich positiver rationalitätsbasierter Ansatz Verwendung in normativen Theorien der Politischen Theorie finden kann. Um diese Frage zu beantworten, unterscheiden wir zwischen einer empirischen und einer normativen Interpretation von Rationalitätszuschreibungen. Da, wie wir zeigen, das empirische Verständnis des Rationalitätsbegriffs den Einsatz von Rationalitätsargumenten in der normativen Politikwissenschaft nicht überzeugend erklären kann, argumentieren wir für eine normative Interpretation der Rationalitätsannahme. Der Artikel endet mit einer Betrachtung der Konsequenzen eines so verstandenen Rationalitätsbegriffs für die Verwendung von Rational-Choice-Argumenten in der normativen wie auch der positiven Politikwissenschaft. Abstract: The concept of rationality, predominantly in the guise of rational choice theory, plays a key role in the social sciences. Yet, whilst rational choice theory is usually understood as part of positive political science, it is also widely employed within normative political theories. In this paper, we examine how allegedly positive rational choice arguments can find application within normative political theories. To this effect, we distinguish between two interpretations of rationality ascriptions, one empirical, the other normative. Since, as we demonstrate, empirical readings of the rationality assumption cannot convincingly explain the role of rational choice arguments in normative theories, we argue that the rationality assumption should be given a normative interpretation. We conclude by considering what this result implies for the use of rational choice arguments in normative and positive political science. 1. Einleitung Rational Choice nimmt seit Langem eine zentrale Stellung in der normativen Politischen Theorie ein. So sind Rechtfertigungen staatlicher Herrschaft (Hobbes 1991), Begründungen von Gerechtigkeitsprinzipien (Rawls 1980) und Argumente für die Begrenzung * Prof. Dr. Johannes Marx, Otto-Friedrich-Universität Bamberg Kontakt: [email protected] Jun.-Prof. Dr. Christine Tiefensee, Frankfurt School of Finance & Management Kontakt: [email protected] 20 Zeitschrift für Politische Theorie, Heft 1/2015 staatlicher Regulierung (Nozick 1974) nur einige Beispiele normativer Theorien, die Rational-Choice-Argumente an entscheidenden Stellen einsetzen. Doch obwohl eine solche Verwendung des Rational-Choice-Ansatzes1 in Anbetracht seiner weiten Verbreitung vertraut erscheinen mag, wirft sie grundlegende, noch ungeklärte Fragen zum Verhältnis zwischen Rational Choice und normativen Begründungen auf. Schliesslich wird Rational Choice in der modernen Politischen Theorie klassischerweise als ein Instrument positiver Politikwissenschaft interpretiert, das der wertfreien Erklärung und Vorhersage empirischer Phänomene dienen soll (vgl. Amadae/Bueno de Mesquita 1999). Wie kann es nun aber sein, dass ein solcher vermeintlich wertneutraler Ansatz zur Lösung normativer Fragestellungen beitragen kann? Wir werden im Folgenden argumentieren, dass die Verwendung von Rational-ChoiceArgumenten in normativen Theorien dann am verständlichsten wird, wenn das Rationalitätskonzept entgegen der gängigen politikwissenschaftlichen Meinung nicht als wertneutral angesehen, sondern selbst als normatives Konzept interpretiert wird. Demnach wird sich die oben genannte Frage im Laufe unseres Artikels geradezu umkehren: Am Ende unserer Diskussion wird nicht die Frage stehen, wie Rational Choice als wertneutraler Ansatz innerhalb normativer Theorien zum Einsatz kommt, sondern wie Rational Choice als normativer Ansatz in positiven Theorien Verwendung finden kann. Unser Beitrag unterteilt sich in vier Abschnitte. Zunächst werden wir das Rationalitätskonzept begrifflich näher spezifizieren und zwischen zwei möglichen Statusinterpretationen dieses Konzeptes unterscheiden: einer empirischen und einer normativen. Im zweiten Teil werden wir untersuchen, ob und wie eine empirische Deutung des Rationalitätskonzepts Eingang in normative Theorien finden könnte. Nachdem wir Zweifel an dieser empirischen Interpretation aufgezeigt haben, werden wir im dritten Teil für die normative Interpretation des Rationalitätsbegriffs argumentieren und das Zusammenspiel zwischen normativer Rationalität und moralischen Konklusionen beleuchten. Erste Überlegungen zu der Frage, inwiefern sich dieses normative Verständnis auf den Einsatz von Rational Choice in der positiven Politikwissenschaft auswirkt, schliessen unseren Artikel ab. 2. Bedeutung und Status des Rationalitätskonzepts Der Begriff der Rationalität ist ein zentrales Konzept in den Sozialund Geisteswissenschaften. Zugleich ist ‚Rationalität' ein schillernder Begriff, der keineswegs einheitlich verwendet wird.2 Die für unsere Zwecke wichtigste Unterscheidung betrifft hierbei die Trennung zwischen einem gehaltvollen, substanziellen Begriff der Rationalität und einem prozeduralen, formalen oder auch instrumentellen Begriff.3 Gemäss dem substanziellen 1 Wir werden im Folgenden die Begriffe ‚Rational-Choice-Theorie', ‚Rational-Choice-Ansatz' und ‚Rational Choice' gleichbedeutend verwenden. 2 Zur Begriffsgeschichte und zum ideengeschichtlichen Kontext des Rationalitätsbegriffs vergleiche Schnädelbach (1984); Jonas (1981); Gosepath (1992). 3 Darüber hinaus lässt sich theoretische von praktischer Rationalität unterscheiden. Wohingegen theoretische Rationalität auf die Rationalität von Überzeugungen und meinungsbeziehungsweise theoriebildenden – das heisst kognitiven – Prozessen abzielt, stehen bei der praktischen Rationalität Handlungen und Intentionen im Mittelpunkt. Obwohl diese Unterscheidung auch für unsere Fragestellung relevant ist, wird sie nicht im Mittelpunkt unserer Untersuchungen stehen. Johannes Marx, Christine Tiefensee: Rationalität und Normativität 21 Rationalitätsbegriff, der eng mit dem der Vernunft verknüpft ist, besteht Rationalität in der Verfolgung von gewissen substanziell rationalen Zielen. Wer zum Beispiel danach strebt, die Anzahl der Grashalme in seinem Vorgarten zu zählen oder den intrinsischen Wunsch nach einer Tasse Schlamm verspürt, kann gemäss dem substanziellen Verständnis aufgrund der Unvernünftigkeit dieser Ziele nicht als rational gelten (Anscombe 1979; Quinn 1993; Hooker/Streumer 2004). Substanzielle Rationalität steht den Zielen von Akteuren4 daher nicht neutral gegenüber, sondern bewertet sie, indem sie eine gehaltvolle Begrenzung derjenigen Ziele vornimmt, die rational verfolgt werden können. Im Gegensatz zu diesem substanziellen Rationalitätsbegriff strebt der formale, instrumentelle Rationalitätsbegriff keine Bewertung der von dem Akteur verfolgten Ziele an. Vielmehr handelt dem instrumentellen Rationalitätsverständnis zufolge derjenige rational, welcher adäquate Mittel zum Erreichen seiner Ziele wählt. Rational-ChoiceTheorien formalisieren sodann dieses instrumentelle Verständnis, indem Rationalität als die Maximierung des Erwartungswerts einer Nutzenfunktion vor dem Hintergrund subjektiver Wahrscheinlichkeitsverteilungen spezifiziert wird (Binmore 1998: 360–361).5 Hierbei werden die Ziele des Akteurs anders als im Fall der substanziellen Rationalität keiner materialen Bewertung unterzogen, unterliegen jedoch gewissen formalen Rationalitätsstandards wie die der Konsistenz und Transitivität.6 Es ist dieser formale, instrumentelle Rationalitätsbegriff, der in sozialwissenschaftlichen Disziplinen wie auch in vielen normativen Theorien der Politischen Theorie im Vordergrund steht. Im Folgenden werden wir den Rationalitätsbegriff daher immer im Sinne von instrumenteller Rationalität verwenden. Welche Art von Aussage treffen wir nun, wenn wir einen Akteur oder eine Handlung als rational bezeichnen? Wir werden hier zwei mögliche Interpretationen betrachten.7 Eine erste Lesart interpretiert Aussagen über die Rationalität von Akteuren als empirische Behauptungen. Wird einem Akteur Rationalität zugeschrieben, handelt es sich demnach um eine empirische Beschreibung oder Hypothese über die Psychologie des handelnden Akteurs.8 Diese empirische Eigenschaft der Rationalität muss sodann weiter spezifiziert 4 Aus Gründen der Lesbarkeit verwenden wir das generische Maskulinum, welches auch immer die weibliche Form impliziert. 5 Spezifische Interpretationen der Rationalitätsannahme, zum Beispiel in Bezug auf Fragen zu maximising versus satisficing, können innerhalb des Rational-Choice-Ansatzes variieren. Da unser Argument nicht von einer bestimmten Interpretation der formalen Rationalitätsannahme abhängig ist, sondern verschiedenen Spezifizierungen neutral gegenübersteht, ist es für unsere Zwecke nicht nötig, Position zu den technischen Details des Rational-Choice-Ansatzes zu beziehen. 6 Für eine Auseinandersetzung mit den notwendigen Eigenschaften einer Präferenzordnung vergleiche Nida-Rümelin (1994: 5 f.). 7 Rational-Choice-Theorie wird in der Politikwissenschaft darüber hinaus oft als analytisches Instrument verstanden, das in theoretischen Modellen Einsatz findet (vgl. etwa Friedman 1953). Diesem Verständnis zufolge kommt Rationalitätsannahmen daher der Status stipulierter Axiome zu. Wir werden im Folgenden diese analytische, modelltheoretische Interpretation nicht näher betrachten, da die oben vorgetragenen Überlegungen auch auf sie zutreffen: Werden Rationalitätszuschreibungen rein modelltheoretisch verstanden, werfen diese bei der Verwendung in normativen Theorien die gleichen Probleme wie die empirische Lesart auf. Wird die analytische Interpretation hingegen als Begriffsschärfung des normativen Rationalitätskonzepts verstanden, finden dieselben Argumente Anwendung, die auch im Kontext der normativen Deutung vorgebracht werden. Für einen Überblick über unterschiedliche Verwendungsweisen vergleiche etwa Johnson (2010) und Lovett (2006). 8 Eine solche empirische Verwendungsweise ist primär im soziologischen Zweig des ökonomischen Ansatzes angesiedelt (Kiser/Hechter 1998; Kunz 2004; Opp 1989). 22 Zeitschrift für Politische Theorie, Heft 1/2015 werden, wie zum Beispiel als ein Bündel von Dispositionen D eines Akteurs, bestimmte Handlungsoptionen vor dem Hintergrund gegebener Umweltrestriktionen und Präferenzen zu wählen (vgl. Hempel 1962: 13). Wenn im Folgenden auf die empirische Interpretation des Rationalitätskonzeptes Bezug genommen wird, wird dieses dispositionelle Verständnis zugrunde gelegt.9 Im Gegensatz dazu schreibt die zweite Interpretation Rationalitätsaussagen normativen Status zu. Gemeint ist damit nicht die Verwendung eines andernfalls nichtnormativen Rational-Choice-Ansatzes für normative Zwecke, wie sie zum Beispiel in der Wohlfahrtsökonomie anzutreffen ist, die Ergebnisse von empirisch-analytischen Rational-ChoiceAnalysen in normative Untersuchungen zur Institutionengestaltung einfliessen lässt (Mashaw 1997; Petrick 2004; Hausman/McPherson 2006). Vielmehr beinhaltet die normative Interpretation, dass das Rationalitätskonzept und damit Rationalitätszuschreibungen selbst irreduzibel normativ sind. Wenn wir einen Akteur oder seine Handlungen als rational bezeichnen, nehmen wir demnach keine empirische Beschreibung des Akteurs vor, sondern bewerten seine Handlungen.10 Wie Rational-Choice-Argumente in normativen Theorien eingesetzt werden können, hängt nun entscheidend von dem Status von Rationalitätszuschreibungen ab. Welche Lesart am besten mit dem Einsatz von Rational Choice in normativen Theorien zu vereinbaren ist, wird in einem nächsten Schritt untersucht. Wir beginnen mit der empirischen Interpretation. 3. Empirisch interpretierte Rationalitätsannahmen in normativen Theorien Um die Rolle empirisch gedeuteter Rationalitätsannahmen in normativen Theorien beurteilen zu können, muss zunächst zwischen verschiedenen möglichen Verwendungsformen unterschieden werden. Am Beispiel von Legitimitätstheorien wollen wir hier zwei mögliche Einsatzweisen diskutieren, eine definitorische und eine begründende.11 9 Wie schon von Hempel (1962) angedeutet und von Davidson (1984) weiter ausführt, sind ernsthafte Probleme mit diesem dispositionellen Verständnis von Rationalität verbunden. So kann argumentiert werden, dass Rationalität nicht anhand von Dispositionen spezifiziert werden kann, da Rationalitätsannahmen bereits in die Individuierung der entsprechenden Dispositionen einfliessen (vgl. die Überlegungen in 4.2 zum konstitutiven Charakter von Rationalitätszuschreibungen). Wir glauben, dass dieses Problem schwerwiegend ist, werden diese externe Kritik an der empirischen Interpretation von Rationalität hier jedoch nicht weiter verfolgen. 10 Die Entscheidungswie auch die Spieltheorie werden zum Teil normativ gedeutet (vgl. Hands 2011; 2012; Grüne-Yanoff/Lehtinen 2012; Rubinstein 1991). Die normative Interpretation des Rationalitätsbegriffs ist zudem in der Philosophie, insbesondere in der Ethik und Metaethik, weit verbreitet (vgl. stellvertretend Gibbard 1990; 2003; Southwood 2008; Hubin 2001). 11 Weitere Verwendungsweisen von Rational Choice wären denkbar, wie zum Beispiel ein kriterieller Einsatz, bei dem der Verweis auf rationale Wahl ein Kriterium bereitstellen würde, mit dessen Hilfe Verteilungsordnungen als gerecht oder ungerecht eingestuft werden können (Nida-Rümelin 1999: 37). Da ein solcher Einsatz sich mit ähnlichen Argumenten konfrontiert sieht wie die begründende Verwendungsweise, werden wir diese Funktion hier nicht gesondert diskutieren. Johannes Marx, Christine Tiefensee: Rationalität und Normativität 23 3.1 Definitorische Verwendung Gemäss der ersten Einsatzweise könnte das Konzept der rationalen Wahl zur Definition von Legitimität herangezogen werden: Ein legitimer Staat ist ein Staat, der von allen Betroffenen rational gewählt würde. (DEFINITION) ‚legitim' =def ‚rational (von Akteuren mit Disposition D) gewählt' Da ‚rational gewählt' in diesem Fall gemäss dem dispositionellen Verständnis empirisch gedeutet werden muss, wird allerdings in Anbetracht des von G. E. Moore (1960) propagierten Naturalistischen Fehlschlusses schnell klar, dass eine solche definitorische Verwendungsweise ausgeschlossen werden muss. In leicht abgewandelter Form besteht der Naturalistische Fehlschluss darin, einen normativen Term – nehmen wir zum Beispiel den moralischen Term ‚gut' – durch einen nichtnormativen Term – wie zum Beispiel den der Nutzenmaximierung – definieren zu wollen.12 Den Grund, weswegen Moore solche Definitionsversuche als Fehlschlüsse brandmarkt, finden wir in seinem Argument der offenen Frage. Demnach sei es immer eine nichttriviale Frage, ob eine Handlung, die Nutzen maximiert, auch gut sei. Wären ‚gut' und ‚nutzenmaximierend' jedoch bedeutungsgleich, müsste diese Frage geschlossen sein: Kompetente Sprecher dürften nicht anzweifeln können, ob eine nutzenmaximierende Handlung auch gut ist.13 Die Offenheit der Frage signalisiere sodann, dass ‚gut' nicht mit ‚nutzenmaximierend' synonym sein könne und die Definition entsprechend abgelehnt werden müsse. Dasselbe Argument trifft nun auch auf Definitionen des normativen, moralischen Begriffes ‚legitim' durch den empirisch, dispositionell gedeuteten Begriff der rationalen Wahl zu. Denn die Frage ‚Dieser Staat würde von Akteuren mit einer gewissen Disposition D gewählt, aber ist er legitim?' ist offen: Wer behauptet, dass ein Staat zwar von Akteuren mit einer solchen empirischen Eigenschaft gewählt würde, aber dennoch illegitim ist, verstrickt sich in keinen Widerspruch. Demnach muss mit Moore die definitorische Verwendung eines empirisch gedeuteten Begriffes der Rationalität innerhalb normativer Theorien ausgeschlossen werden. 3.2 Begründende Einsatzweisen Anstatt sich auf definitorische Einsatzweisen zu konzentrieren, scheint es vielversprechender zu sein, rationaler Wahl eine begründende Funktion in normativen Theorien zuzuschreiben. In diesem Fall wäre ein Staat legitim, weil er von rationalen Akteuren gewählt würde: 12 Moore bezieht sich primär auf das Konzept ‚gut'. Der Naturalistische Fehlschluss kann aber auch auf andere normative und moralische Konzepte ausgeweitet werden. Ausserdem gilt zu beachten, dass die von Moore vorgeschlagene Bezeichnung des Fehlschlusses als ‚naturalistisch' wie auch als ‚Fehlschluss' unglücklich gewählt ist, da der Fehlschluss erstens nicht nur auf naturalistische Begriffe begrenzt ist, sondern alle nichtnormativen Begriffe miteinschliesst, und zweitens auch kein falsches Schliessen beinhaltet. Zur Abgrenzung dieses Fehlschlusses zur Sein-Sollen-Problematik und der Fact-value-Dichotomie vergleiche exemplarisch Dodd/Stern-Gillet (1995); Frankena (1939); Putnam (2003). 13 Zum Vergleich: Die Frage ‚Sissi hat einen Bruder und eine Schwester, aber hat sie Geschwister?' ist für kompetente Sprecher geschlossen – wer die Bedeutung von ‚Geschwister' kennt, kann nicht sinnvoll anzweifeln, dass Sissi Geschwister hat. Gegen eine Definition von ‚Geschwister' durch die Begriffe ,Bruder' und ,Schwester' ist daher nichts einzuwenden. 24 Zeitschrift für Politische Theorie, Heft 1/2015 (BEGRÜNDUNG1) (P1) Staat X würde rational (von Akteuren mit Disposition D) gewählt. (K) Also ist Staat X legitim. Doch auch hier ist offensichtlich, dass die begründende Funktion empirisch dispositionell interpretierter Rationalitätsaussagen in dieser Form zunächst nicht stehen bleiben kann. Denn wie David Hume (1978) im Rahmen des Sein-Sollen-Problems erklärt hat, können normative Konklusionen nicht aus rein nichtnormativen Prämissen folgen. Die normative Konklusion ‚Staat X ist legitim' kann demnach nicht aus der empirischen Aussage ‚Staat X würde rational (von Akteuren mit Disposition D) gewählt' abgeleitet werden.14 Vielmehr kann diese Ableitung nur dann in einen gültigen Schluss verwandelt werden, wenn sie um ein normatives Brückenprinzip ergänzt wird, wie zum Beispiel: (BEGRÜNDUNG2) (P1) Staat X würde rational (von Akteuren mit Disposition D) gewählt. (P2) Ein Staat, der rational (von Akteuren mit Disposition D) gewählt würde, ist legitim. (K) Also ist Staat X legitim. Da nun die empirisch gedeutete Rationalitätsannahme in Prämisse (P1) mit der moralischen Prämisse (P2), die letztlich den Ansatz der Vertragstheorie expliziert, kombiniert ist, tritt das Sein-Sollen-Problem nicht mehr auf:15 Die moralische Konklusion kann logisch aus den Prämissen (P1) und (P2) abgeleitet werden. Folglich kann die empirisch gedeutete Rationalitätsannahme nur dann in normativen Theorien begründend eingesetzt werden, wenn sie um normative Prinzipien wie (P2) ergänzt wird, die eine Brücke zwischen der empirischen, dispositionellen Eigenschaft D und normativen Bewertungen schlagen. In Anbetracht der bedeutungsvollen Rolle von (P2) stellt sich nun allerdings die Frage: Wie ist wiederum dieses moralische Brückenprinzip begründet? 3.3 Begründung des moralischen Brückenprinzips Die gängigste Begründung dieses Moralprinzips, und damit des vertragstheoretischen Ansatzes, findet sich im Wertefundament des Liberalismus, das den Begriffen der Freiheit, Autonomie und Selbstbestimmung einen zentralen Stellenwert beimisst. Demzufolge liegt der Ursprung staatlicher Legitimität beziehungsweise der Gültigkeit moralischer Normen in der Zustimmung von Akteuren, die sich in deren freier und rationaler Wahl staatlicher Ordnungen beziehungsweise moralischer Prinzipien manifestiert. Das heisst also, dass 14 Die empirische Interpretation von (P1) ist hier nicht zu verwechseln mit einer empirischen, historischen Interpretation von Vertragstheorien. Das heisst, es wird hier nicht behauptet, dass ein Vertrag tatsächlich geschlossen wird, sondern dass Akteure mit einer gewissen empirischen Eigenschaft – der empirischen Eigenschaft der Rationalität – einen solchen Vertrag schliessen würden. Rationalität bleibt daher ein empirisches Merkmal von Akteuren, wobei Vertragstheorien weiterhin hypothetisch gedeutet werden. 15 Es könnte argumentiert werden, dass (P2) keine moralische, sondern eine analytische Prämisse ist, in welchem Falle das Sein-Sollen-Problem widerlegt wäre, da eine normative Konklusion aus nichtnormativen – nämlich ausschliesslich empirischen und analytischen – Prämissen folgen würde (vgl. z. B. Searle 1964 für ein analoges Argument). Dieses Gegenargument ist jedoch nicht erfolgreich, da die analytische Interpretation von (P2) dem Naturalistischen Fehlschluss zum Opfer fallen würde. Für weitere Argumente gegen die analytische Interpretation von Brückenprinzipien wie (P2) vergleiche auch Hare (1964). Johannes Marx, Christine Tiefensee: Rationalität und Normativität 25 staatliche Ordnungen nur dann als legitim und Verteilungsordnungen nur dann als gerecht gelten können, wenn sie von freien, rationalen Akteuren gewählt würden. Diese Begründung des Brückenprinzips ist weit verbreitet und soll von unserer Argumentation auch nicht angezweifelt werden. Dennoch greift sie für die Ziele unserer Forschungsfrage zu kurz. Denn, obwohl diese Begründung die zentrale Rolle der Wahl in Vertragstheorien rechtfertigen mag, begründet sie dennoch nicht, weswegen diese Wahl ausgerechnet rational, also durch Akteure mit einer bestimmten Disposition D, erfolgen muss. Welche Erklärung gibt es also dafür, dass nicht nur Wahl, sondern ausgerechnet rationale Wahl im Fokus normativer Theorien stehen sollte? Anders ausgedrückt: Was macht die Zustimmung von Akteuren mit einer bestimmten Disposition D so besonders, dass sie zu einem Kernstück normativer Theorien wird? Um diese Frage beantworten zu können, muss geklärt werden, worin die normative Relevanz dieser dispositionellen Eigenschaft D besteht. Das heisst, genauso wie auch die normative Relevanz von Zustimmung anhand des Werts der Autonomie erklärt wird, muss ebenso verdeutlicht werden, weswegen die Disposition D eine empirische, aber normativ relevante Eigenschaft sein soll, ohne die staatliche Legitimität oder die Gerechtigkeit von Verteilungsordnungen nicht hergestellt werden kann. Die Unterscheidung zwischen den beiden folgenden Szenarien bietet hierbei einen hilfreichen Ausgangspunkt, um dieser Frage auf den Grund zu gehen: (SZENARIO1) Akteure stimmen der Einrichtung eines Staates frei zu. Diese Akteure besitzen zudem Disposition D: Sie haben das Ziel, ihre eigene Sicherheit zu schützen, erachten die Etablierung staatlicher Gewalt als das beste Mittel, um dieses Ziel zu erreichen, und entscheiden sich dementsprechend für die Bildung eines Staates. (SZENARIO2) Akteure stimmen der Einrichtung eines Staates frei zu. Diese Akteure besitzen nicht die Disposition D: Sie haben zwar das Ziel, ihre eigene Sicherheit zu schützen, erachten die Etablierung staatlicher Gewalt allerdings nicht als das beste Mittel, um dieses Ziel zu erreichen. Dennoch entscheiden sie sich für die Bildung eines Staates. Die Forderung, dass staatliche Herrschaft aufgrund von rationaler Wahl legitimiert wird, hat zur Folge, dass die Bildung eines Staates im ersten Szenario legitimiert, im zweiten Szenario jedoch nicht legitimiert ist. Wieso sollte ausgerechnet die Disposition D zu einer unterschiedlichen moralischen Einschätzung dieser Szenarien führen? Eine mögliche Erklärung hierfür könnte beispielsweise darauf abzielen, dass es sich evolutionär bewährt habe, über die Disposition D zu verfügen. Ziele können schliesslich in der Regel nur dann erreicht werden, wenn die richtigen Mittel zur Erreichung dieser Ziele eingesetzt werden. Diesem Verständnis zufolge wäre die empirische Eigenschaft, über die Disposition D zu verfügen, also deswegen instrumentell wünschenswert, weil sie ein zentrales Instrument zur Verwirklichung unserer Ziele darstellt. Kurzum: Disposition D wäre normativ relevant, weil es gut ist, unsere Ziele zu erreichen, und Disposition D für diese Zielerreichung unabdingbar ist. Obwohl diese Ausführungen zum Wert der Disposition D nicht unplausibel sind, können sie unserer Meinung nach dennoch nicht zwischen den moralischen Implikationen der beiden Szenarien überzeugend differenzieren. Die beiden Szenarien unterscheiden sich weder im Hinblick auf die Freiheit der Akteure, ihre Zustimmung zur Errichtung eines Staates zu erteilen, noch hinsichtlich ihrer Ziele: Akteuren steht es frei, sich ihre eigene Meinung zu bilden und Überlegungen zu den verfügbaren Handlungsoptionen anzustellen, ohne hierbei in irgendeiner Form bevormundet oder in ihren Einstellungen mani26 Zeitschrift für Politische Theorie, Heft 1/2015 puliert zu werden. Das moralische Kernstück der Vertragstheorie ist demnach intakt: In beiden Fällen wird die Autonomie der Akteure gewahrt und staatliche Gewalt durch Konsens der Akteure etabliert. Der einzige Unterschied zwischen diesen Szenarien besteht also darin, ob Akteure die in ihrer Meinung effizientesten Mittel zur Zielerreichung wählen. Doch selbst wenn es generell wünschenswert ist, Ziele effizient zu verfolgen, ist nicht ersichtlich, weswegen moralische Legitimität in Zweifel gezogen werden sollte, wenn Akteure in dieser Situation Effizienzüberlegungen nicht gerecht werden. Anders ausgedrückt: Disposition D ist sicherlich wünschenswert, weil sie uns generell hilft, unsere Ziele zu erreichen. Eine Begründung, weswegen moralische Legitimität oder Gerechtigkeitsüberlegungen jedoch an diese Eigenschaft gekoppelt sein sollen, steht nicht nur aus, sondern würde nach unserer Meinung zudem einer Überhöhung des Werts dieser Disposition gleichkommen. Der Grund, weswegen das erste Szenario staatliche Gewalt legitimiert, das zweite jedoch trotz identischer Ziele und freier Zustimmung der Akteure nicht, kann daher nicht überzeugend auf den instrumentellen Wert der Disposition D zurückgeführt werden. Zusammenfassend ist damit festzuhalten, dass vor dem Hintergrund der empirischen Deutung der Rationalitätsannahme nicht plausibel begründet werden kann, weswegen ausgerechnet die dispositionell gedeutete Eigenschaft, rational zu sein, ein essenzieller Bestandteil normativer Theorien sein soll. Im Gegensatz dazu werden wir uns im nächsten Abschnitt für die These stark machen, dass eine normative Interpretation der Rationalitätsannahme eben diese Begründungsleistung erbringen kann. So werden wir argumentieren, dass die Zustimmung von Akteuren nicht (primär) deswegen rational erfolgen muss, weil es gut ist, seine eigenen Ziele effizient zu verfolgen, sondern weil Rationalitätsüberlegungen die Gründe der handelnden Akteure offenlegen, die in moralische Begründungen einfliessen müssen. Bevor wir uns diesem Argument zuwenden, fasst Abbildung 1 unsere bisherigen Argumente nochmals zusammen. Johannes Marx, Christine Tiefensee: Rationalität und Normativität 27 Abb. 1: Zusammenfassung der bisherigen Argumentation (eigene Darstellung) 28 Zeitschrift für Politische Theorie, Heft 1/2015 4. Normativ interpretierte Rationalitätsannahmen in normativen Theorien Wie bereits angedeutet, ist ein normatives Verständnis von Rationalität in der Philosophie weitverbreitet, wenn auch nicht unumstritten. Obwohl sich diese Position in den Wirtschaftsund Sozialwissenschaften dagegen seltener findet, stellt beispielsweise auch Harsanyi (1976: 90) unumwunden fest, dass ‚Rationalität' ein normatives Konzept sei: „In everyday life, when we speak of ,rational behavior', in most cases we are thinking of behavior involving a choice of the best means available for achieving a given end. This implies that, already at a common-sense level, rationality is a normative concept: it points to what we should do in order to attain a given end or objective." Worin genau diese Normativität besteht, soll in einem ersten Schritt geklärt werden. In einem zweiten Schritt werden wir dann erläutern, wie diese normative Interpretation die Rolle von Rationalitätsargumenten in normativen Theorien begründen kann. 4.1 Die Normativität des Rationalitätskonzepts Normativität zu definieren, ist ein bekanntermassen schwieriges Unterfangen. Wir werden hier davon ausgehen, dass ein Konzept normativ ist, wenn Folgendes gilt: (NORMATIVITÄT) Ein Konzept K ist dann normativ, wenn (i) die Kategorisierung eines Objekts als K aufgrund von Standards erfolgt, die (ii) mit einer positiven oder negativen Bewertung des Objekts einhergehen, und (iii) die Kategorisierung eines Objekts als K mit einem starken Handlungsbezug verbunden ist. Das Rationalitätskonzept erfüllt alle drei Kriterien. Erstens beruhen Rationalitätszuschreibungen auf der Anwendung von Rationalitätsstandards und -normen. Mit List/Pettit (2011: 24) lassen sich hier drei solcher Normen unterscheiden: Attitude-to-fact-Standards, die vorgeben, in welcher Beziehung die Einstellungen eines Akteurs zu seiner Umwelt stehen müssen, Attitude-to-attitude-Standards, die das Verhältnis der Einstellungen des Akteurs untereinander betreffen, und Attitude-to-action-Standards, die festlegen, wie diese Einstellungen in Handlungen münden müssen. Die Konformität einer Einstellung beziehungsweise Handlung mit diesen Standards führt zur Bewertung als rational, wohingegen deren Verletzung eine Bewertung als irrational nach sich führt. Zweitens sind Rationalitätszuschreibungen, wie Southwood (2008: 11) bemerkt, nicht einfach nur normenbasiert, sondern auch in einem tieferen Sinne normativ: „However, the normativity of rationality does not seem to consist merely in the fact that it is constituted by requirements. English grammar is also constituted by requirements. But rational requirements and local grammatical requirements do not seem to be remotely normatively on par. Local grammatical requirements are merely constitutive rules or conventions that do not possess any kind of intrinsic normative status. In violating a local grammatical requirement, we are guilty of nothing more than a conventional breach. Rational requirements, by contrast, seem to be normative in a deeper sense. If we fail to comply with them, it seems that we've necessarily gone wrong in some deeper way." Johannes Marx, Christine Tiefensee: Rationalität und Normativität 29 Kategorisierungen von Handlungen und Akteuren als rational oder irrational sind demnach nicht normativ neutral, sondern besitzen eine positive beziehungsweise negative Valenz: Eine Handlung, die als rational beurteilt wird, entspricht nicht lediglich einem wertneutralen Standard, so wie beispielsweise die Kategorisierung einer Partei als konservativ oder sozialdemokratisch aufgrund von wertneutralen Standards erfolgt, die die jeweiligen politischen Einstellungen klassifizieren. Bewerten wir eine Handlung als rational, geht dies vielmehr mit einer positiven Bewertung der Handlung einher: Die Handlung ist auf eine gewisse Art und Weise gelungen; sie ergibt Sinn, ist nachvollziehbar und kann durch Gründe unterstützt werden. Gleichzeitig sind irrationale Handlungen nicht nur nicht regelkonform, sondern grundlos. Kategorisieren wir eine Handlung als irrational, implizieren wir damit, dass der irrationale Akteur einen Fehler begangen hat, und können nicht den Sinn der Handlung erkennen. Rationale Handlungen werden befürwortet, irrationale Handlungen ernten Kritik. Kategorisierungen einer Handlung oder eines Akteurs als rational oder irrational sind daher nicht normativ gleichwertig, sondern sind mit positiven beziehungsweise negativen Bewertungen verbunden. Drittens besitzen Rationalitätszuschreibungen das für normative Aussagen typische Merkmal der Praktikalität (Smith 1995).16 Normative Aussagen zeichnen sich dadurch aus, dass wir erwarten würden, dass jemand, der ein normatives Urteil akzeptiert, auch seine Handlungen an diesem Urteil ausrichtet.17 Kommt jemand zum Beispiel zu dem Schluss, dass es moralisch geboten ist, zur Bekämpfung einer Hungersnot an karitative Einrichtungen zu spenden, würden wir ceteris paribus erwarten, dass er bei dem nächsten Auftreten einer solchen Hungersnot eine ebensolche Spende tätigt. Dieser Handlungsbezug lässt sich nun auch bei Rationalitätszuschreibungen feststellen. Will jemand zum Beispiel schnellstmöglich nach Berlin reisen und stimmt dieser Akteur mit der Einschätzung überein, dass es am rationalsten wäre, im Zug zu reisen, würden wir davon ausgehen, dass dieser Akteur auch tatsächlich den Zug nimmt und nicht ins Auto steigt. Gleichzeitig würden wir von einem Akteur erwarten, dass er sein Verhalten korrigiert, wenn ihm durch gute Gründe plausibel gemacht werden kann, dass sein Verhalten irrational ist. Während die Bewertung einer Handlung als rational ein gewisses ‚Gütesiegel' bereitstellt, rechtfertigt Irrationalität also nicht nur Kritik, sondern verlangt auch Verhaltensänderungen auf Seiten des irrationalen Akteurs.18 Diese Überlegungen zeigen folglich, dass Rationalität ein normatives Konzept ist: Rationalitätszuschreibungen nehmen keine Beschreibung von Akteuren oder Handlungen vor, sondern bewerten sie. Diese Überlegungen implizieren jedoch nicht, dass Rationalität auch ein moralisches Konzept ist. Moralische Aussagen stellen eine Klasse normativer Aussagen dar, zeichnen sich jedoch u. a. durch ihren spezifischen Bezug auf zwischen- 16 Dies ist nicht zu verwechseln mit Praktikabilität im Sinne von Durchführbarkeit einer Handlung. Gemeint ist stattdessen, dass normative Urteile einen Handlungsbezug haben. 17 Wie eng dieser Handlungsbezug ist – ob er, wie Internalisten behaupten, intrinsisch gedeutet werden sollte (Blackburn 1998) oder aber, wie Externalisten argumentieren, extrinsisch zu erklären ist (Brink 1989) – kann und soll hier nicht geklärt werden. 18 In diesem Kontext sind die Resultate von spieltheoretischen Verhaltensexperimenten mit Studierenden interessant: So scheinen Studierende der Ökonomie, die sich am Anfang ihres Studiums befinden, in Gefangenendilemmata verstärkt kooperative Strategien zu wählen, wohingegen fortgeschrittene Studenten zur Defektion tendieren (Frank et al. 1993). Eine mögliche Erklärung könnte lauten, dass Studierende, die in ihrem Studium auf die Irrationalität der Kooperation in Situationen, die dem Gefangenendilemma strukturell entsprechen, hingewiesen werden, ihr Verhalten korrigierend ändern, um Rationalitätsnormen gerecht zu werden. 30 Zeitschrift für Politische Theorie, Heft 1/2015 menschliches Handeln aus. Der instrumentelle Rationalitätsbegriff teilt diesen Fokus nicht. Rationalität ist dementsprechend ein normatives, jedoch kein moralisches Konzept. 4.2 Rationalität und Gründe Neben ihrer Normativität ist eine zweite Beobachtung zu Rationalitätszuschreibungen zentral für unser Argument. Diese betrifft die Rolle von Rationalitätsannahmen in der Zuschreibung von Einstellungen und Handlungen, die in Davidsons (1984; 2004a) holistischer Theorie der Bedeutung und des Mentalen herausgearbeitet wurde. Davidson zufolge werden Akteuren mentale Zustände aufgrund von Interpretationen zugeschrieben, die es zum Ziel haben, die Handlungen und Äusserungen dieser Akteure möglichst verständlich zu machen. Der Drehund Angelpunkt in diesem Interpretationsprozess bestehe in dem sogenannten ‚principle of charity', welches besagt, dass Akteuren nur diejenigen mentalen Zustände zugeschrieben werden dürfen, die in Anbetracht der Situation, in welcher sich diese Akteure befinden, vor dem Hintergrund unserer eigenen Überzeugungen den grössten Sinn ergeben, das heisst, am verständlichsten – am rationalsten – sind. Kurz gesagt, schreiben wir einem Akteur beispielsweise den Wunsch zu, ein Glas Wasser zu trinken, wenn diese Zuschreibung die Handlung, ein Glas Wasser zu trinken, rational erscheinen lässt (vorausgesetzt, der Akteur glaubt, dass das Glas Wasser enthält). Ebenso unterstellen wir dem Akteur nur dann die Überzeugung, dass das Glas Wasser enthält, wenn diese Überzeugung die rationalste Erklärung dafür ist, dass er das Wasser trinkt (vorausgesetzt, der Akteur hat den Wunsch, Wasser zu trinken). Schliesslich gilt Wassertrinken nur dann als eine Handlung, wenn sein Wunsch, Wasser zu trinken und die Überzeugung, dass das Glas Wasser enthält, die Gründe für diese Handlung liefern. Der Interpretationsprozess wird folglich geleitet durch Rationalitätsnormen: Wir müssen bewerten, welche Zuschreibung von mentalen Zuständen und Handlungen vor dem Hintergrund der gegebenen Situation den grössten Sinn ergeben, oder anders ausgedrückt, für welche mentalen Zustände und Handlungen es die besten Gründe gibt. Interpretationen sind folglich keine blossen empirischen Beschreibungen unserer Psychologie, sondern erfordern die Einordnung unserer mentalen Zustände und Handlungen in „the space of reasons" (Sellars 1997: 160).19 In den Worten von Davidson (2004a: 130): „The entire theory [of thought, meaning and action] is built on the norms of rationality; it is these norms that suggested the theory and give it the structure it has. [...] However, norms or considerations of rationality also enter with the application of the theory to actual agents, at the stage where an interpreter assigns his own sentences to capture the contents of another's thoughts and utterances. The process necessarily involves deciding which pattern of assignments makes the other intelligible (not intelligent, of course!), and this is a matter of using one's own standards of rationality to calibrate the thoughts of the other. [...] Norms are being employed as a standard of norms." 20 19 Für weitere Details zum Verhältnis von Psychologie und der Normativität der Rationalitätsannahme vergleiche Davidson (2004b). 20 Da Davidson zufolge eine Handlung, gemessen am Primärgrund der Handlung, in einem schwachen Sinn immer rational ist, ergibt sich die Frage, ob es vor dem Hintergrund dieser Annahme überhaupt irrationales Handeln geben kann. Nach Spitzley (2008) ist dies eine Frage des Massstabes. Denn selbst wenn eine Handlung im schwächeren Sinne rational wäre, so könnte sie dennoch strengeren Rationalitätsstandards, wie zum Beispiel die Forderung nach globaler Kohärenz der Präferenzen oder strikt evidenzbasierten Überzeugungen, nicht genügen. In diesem Sinne ist auch Davidsons Äusserung zu verstehen, dass bei irrationalen Handlungen die Bedingung der Kohärenz verletzt wird (Davidson 1974; 2004a). Johannes Marx, Christine Tiefensee: Rationalität und Normativität 31 Werden mentale Zustände und Handlungen Akteuren zugeschrieben, wirken also Normen der Rationalität implizit mit, da solche Zuschreibungen immer auf normativen Urteilen über Handlungsgründe beruhen.21 4.3 Die Rolle rationaler Wahl in normativen Theorien Diese beiden Betrachtungen zu Rationalitätsannahmen – ihre Normativität sowie ihre konstitutive Rolle für die Zuschreibung von Einstellungen und Handlungen – versetzen uns nun in die Lage, zu unserer Ausgangsfrage zurückzukehren und zu begründen, weswegen nicht nur freie Wahl, sondern freie rationale Wahl im Zentrum normativer Theorien stehen sollte. In aller Kürze lautet die Antwort mit Davidson: Die Wahl muss rational sein, nicht etwa, weil Rationalität eine instrumentell wertvolle empirische Eigenschaft wäre, sondern weil die Rationalität der Wahl diese Handlung erst nachvollziehbar macht. Nur wenn die Wahl rational ist, werden die Gründe der Akteure, der Bildung eines Staates oder einer gewissen Verteilungsordnung zuzustimmen, verständlich. Näher ausgeführt lässt sich das Verhältnis zwischen normativer Rationalität und moralischen Schlussfolgerungen durch die folgenden Überlegungen beleuchten. Erstens muss festgestellt werden, dass auch die normative Interpretation des Rationalitätskonzepts die Verwendung eines moralischen Brückenprinzips, das die Verbindung zwischen rationaler Wahl und der Gerechtigkeit einer Ordnung oder Legitimität eines Staates herstellt, nicht überflüssig macht. Da Rationalität, wie oben erläutert, ein normatives, aber kein moralisches Konzept ist, ist die Frage nach der Verbindung zwischen rationaler Wahl und moralischer Konklusion nach wie vor eine substanzielle, die von Legitimationsbeziehungsweise Gerechtigkeitstheorien beantwortet werden muss. Auch die Normativität von Rationalität ändert folglich nichts daran, dass moralische Begründungen unter Rekurs auf Werte wie Autonomie und Freiheit unabdingbar sind, um die moralische Relevanz von freier Zustimmung zu begründen. Zweitens ist festzuhalten, dass der Verweis auf die Freiheit der Wahl und die Autonomie der Akteure nicht ausreicht, um die Errichtung staatlicher Herrschaft zu legitimieren oder die Gerechtigkeit von Verteilungsordnungen zu begründen. Vielmehr muss diese Zustimmung aus nachvollziehbaren Gründen erfolgen. Das ist der Punkt, an dem die Rationalität der Wahl ins Spiel kommt.22 Denn Vertragstheorien argumentieren nicht nur, dass Akteure die Errichtung eines Staats wählen würden, sondern sie explizieren, aus welchen Gründen sie ihre Zustimmung zu staatlicher Gewalt geben. Staatliche Herrschaft ist demnach nur dann legitimiert beziehungsweise eine Verteilungsordnung nur dann gerecht, wenn sie aus nachvollziehbaren Gründen gewählt würde. Es ist genau dieser Bezug 21 An dieser Stelle könnte der Einwand vorgebracht werden, dass es bei Einstellungszuschreibungen nicht um die Interpretation von Handlungen geht, die normative Rationalitätsunterstellungen erfordert, sondern um die empirische Erklärung von Handlungen, die auf kausale, neuronale Abläufe abzielt. Dieser Einwand setzt offensichtlich voraus, dass Letzteres von Ersterem getrennt werden kann. In welcher Beziehung Handlungsursachen und Handlungsgründe zueinander stehen, werden wir in unserem Schlusskapitel betrachten. 22 Die Frage, weswegen eine Wahl ausgerechnet rational sein muss, stellt sich nicht nur im Kontext von Gerechtigkeitsoder Legitimationstheorien, sondern auch in Bezug auf alle anderen politikwissenschaftlichen Theorien, die auf rationale Wahl rekurrieren (vgl. zum Beispiel die Verwendung der Rationalitätsannahme in Downs 1957). Davidsons Antwort auf diese Frage lautet immer gleich: Die Wahl muss rational sein, um als sinnvoll zu erscheinen. 32 Zeitschrift für Politische Theorie, Heft 1/2015 auf die Gründe der Akteure, der den normativen Unterschied zwischen den beiden oben eingeführten Szenarien erklären kann: (SZENARIO1) Akteure stimmen der Einrichtung eines Staates frei zu. Diese Akteure besitzen zudem Disposition D: Sie haben das Ziel, ihre eigene Sicherheit zu schützen, erachten die Etablierung staatlicher Gewalt als das beste Mittel, um dieses Ziel zu erreichen, und entscheiden sich dementsprechend für die Bildung eines Staats. (SZENARIO2) Akteure stimmen der Einrichtung eines Staates frei zu. Diese Akteure besitzen nicht die Disposition D: Sie haben zwar das Ziel, ihre eigene Sicherheit zu schützen, erachten die Etablierung staatlicher Gewalt allerdings nicht als das beste Mittel, um dieses Ziel zu erreichen. Dennoch entscheiden sie sich für die Bildung eines Staats. Der Grund, weswegen Szenario1 als Legitimation der Errichtung eines Staates anerkannt wird, Szenario2 jedoch nicht, liegt nicht etwa in dem instrumentellen Wert der Disposition D, sondern in der Angabe von Gründen, aufgrund derer Akteure handeln. Szenario1 expliziert diese Gründe: Da die Wahl hier rational erfolgt, indem Akteure vor dem Hintergrund ihrer Ziele das beste Mittel zur Zielerreichung wählen, werden die Handlungsgründe der Akteure verständlich. In Szenario2 hingegen bleiben diese Handlungsgründe obskur: Da Akteure in diesem Fall zwar auch das Ziel haben, Sicherheit zu maximieren, allerdings nicht rational wählen – sie stimmen der Errichtung staatlicher Gewalt zu, obwohl sie glauben, dass der Staat nicht das beste Mittel zur Zielerreichung darstellt – bleibt völlig unklar, aus welchem Grund Akteure die Errichtung des Staates überhaupt wählen. Ohne die Rationalität der Wahl ist die Zustimmung der Akteure nicht nachvollziehbar. Das Versprechen der Vertragstheorie, Gründe für die Bildung eines Staates zu liefern, ist folglich trotz der freien Zustimmung der Akteure in Szenario2 wegen der Irrationalität der Akteure nicht eingelöst. Der Verweis auf Rationalität ersetzt demnach substanzielle moralische Überlegungen nicht, sondern ergänzt sie durch eine gewisse normative Arbeitsteilung: Die Frage, weswegen staatliche Herrschaft nur dann legitim ist, wenn ihr freie Akteure zustimmen, wird durch die Vertragstheorie mit dem Verweis auf den Wert der Autonomie beantwortet. Die Frage, weswegen diese Zustimmung rational erfolgen muss, wird durch die normative Interpretation von Rationalität gelöst, die die Rationalität von Handlungen in den Handlungsgründen der Akteure verankert, wodurch Handlungen erst verständlich werden. Die Kombination dieser beiden Argumentationsstränge führt zu dem vertragstheoretischen Ziel, Legitimität an die autonome, durch Handlungsgründe verständliche und damit rationale Zustimmung von Akteuren zu koppeln. Die Beschäftigung mit der Frage, weshalb dem angeblich positiven Ansatz der rationalen Wahl ein solch zentraler Stellenwert innerhalb normativer politischer Theorien zukommt, führt also zu dem Resultat, dass diese Rolle am besten erklärt werden kann, wenn Rationalitätszuschreibungen selbst normativer Status zugeschrieben wird. Die Frage nach der Einsatzweise des Rational-Choice-Ansatzes kehrt sich vor dem Hintergrund dieser Argumentation also geradezu um: Die Frage lautet nun nicht mehr, wie positive Rationalitätszuschreibungen in normativen politischen Theorien verwendet werden können, sondern wie normative Rationalitätszuschreibungen in positiven politischen Theorien zum Einsatz kommen. Dieser Frage werden wir uns abschliessend zuwenden. Johannes Marx, Christine Tiefensee: Rationalität und Normativität 33 5. Normative Rationalität in positiver Politikwissenschaft? Es scheint offensichtlich, dass insbesondere in weiten Teilen der Ökonomie ein normatives Verständnis des Rationalitätsbegriffs auf grossen Vorbehalt stösst. Folgende Zitate klassischer ökonomischer Positionen bestätigen diesen Eindruck: „[...] science is science and ethics is ethics; it takes both to make a whole man; but only confusion, misunderstanding and discord can come from not keeping them separate and distinct, from trying to impose the absolutes of ethics on the relatives of science" (Friedman 1955: 405). „Unfortunately it does not seem logically possible to associate the two studies in any form but mere juxtaposition. Economics deals with ascertainable facts; ethics with valuations and obligations. The two fields of enquiry are not on the same plane of discourse. Between the generalizations of positive and normative studies there is a logical gulf fixed which no ingenuity can disguise and no juxtaposition in space or time bridge over. [...] Propositions involving the word ,ought' are different in kind from propositions involving the verb ,is'" (Robbins 2007: 132, 133). „Der ökonomische Ansatz [gemeint ist hier die Verwendung des Rationalitätsbegriffs; die Autoren] innerhalb der Sozialwissenschaften ist ein positiver, erklärender Ansatz, der zunächst keine Wertungen enthält. Es geht ausschliesslich um kognitive Aussagen, darum, wie sich Individuen unter welchen Bedingungen verhalten" (Kirchgässner 2008: 304). Doch wenn auch befürchtet wird, dass eine normative Interpretation des Rationalitätsbegriffs dem Einsatz in der positiven Politischen Theorie entgegenstehe, denken wir, dass die Normativität von Rationalität kein grundsätzlicher Einwand gegen die Verwendung dieses Begriffs in positiven Ansätzen darstellt. Dies ausführlich zu zeigen, würde eine detailliertere Antwort erfordern, als wir sie hier geben können. Um unseren Standpunkt zu skizzieren, seien allerdings die folgenden kurzen Überlegungen angeführt. Zentrales Anliegen der positiven Politischen Theorie ist es, kausale Erklärungen für empirische Prozesse zu geben. Elster (1989:3) expliziert den Erklärungsbegriff folgendermassen: „To explain an event is to give an account of why it happened. Usually, and always ultimately, this takes the form citing an earlier event as the cause of the event we want to explain, together with some account of the causal mechanism connecting the two events". Dieses Erklärungsziel scheint auf den ersten Blick mit einem normativen Verständnis von Rationalität nicht vereinbar zu sein. Wie kann dann aber ein normatives Konzept für diese Zielsetzung verwendet werden? Die Lösung dieses Rätsels liegt in der Verbindung zwischen normativen Handlungsgründen und kausalen Handlungsursachen. Wir folgen in dieser Frage der Position, dass der Verweis auf Handlungsgründe nicht ausschliesst, dass diese Handlungsgründe auch kausal wirksam sind. Vielmehr können Handlungsgründe Handlungen nur dann rationalisieren wie auch erklären, wenn diese Handlungsgründe die entsprechenden Handlungen auch verursachen (vgl. Elster 1985). Das Argument hierzu, das wieder von Davidson (1963; 1984) erarbeitet wurde, ist komplex, kann aber folgendermassen zusammengefasst werden: Handlungsgründe beziehen sich auf die propositionalen Einstellungen oder mentalen Zustände – also die Überzeugungen und Präferenzen – der handelnden Akteure. Aktivierte Einstellungen sind wiederum identisch mit kausal wirksamen, neuronalen Ereignissen.23 Da Handlungsgründe also die mentalen Zustände der Akteure betreffen, welche 23 Es handelt sich hierbei um eine Token-Identität, nicht um eine Type-Identität, da Davidson zufolge mentale Sprache nicht auf physikalische Sprache reduziert werden kann. Für eine ausführlichere Darstellung der Position Davidsons und ihrer Implikationen für ein positives Verständnis von Rational Choice vergleiche Marx (2010). 34 Zeitschrift für Politische Theorie, Heft 1/2015 wiederum identisch sind mit kausal wirksamen neuronalen Ereignissen, sind Gründe auch Ursachen. Wie genau Handlungsgründe und Einstellungen miteinander in Beziehung stehen, müsste näher beleuchtet werden, als Davidson dies tut. Dennoch ist die Brücke zwischen normativen Gründen und einer kausalen Welt geschlagen: Da Gründe Handeln verursachen, können normative Rationalitätsüberlegungen zu Handlungsgründen auch in die positive, an Erklärungen interessierte Politikwissenschaft einfliessen.24 Mehr noch, ist es eben diese Kombination aus Handlungsgründen und Handlungsursachen, die dafür sorgt, dass politikwissenschaftliche Erklärungen, die auf Rational Choice basieren, nicht lediglich Ursachen bestimmen, sondern diese Ursachen auf eine Art und Weise identifizieren, die uns das Handeln der Akteure erst verstehen und nachvollziehen lässt (Coleman 1991: 17).25 Interessanterweise trifft dieselbe Einschätzung auch auf die Verwendung von normativen Rationalitätszuschreibungen in modelltheoretischen Argumentationen zu. Diese in der Politikwissenschaft sehr weit verbreiteten Modelle zeichnen sich dadurch aus, auf der Basis möglichst geringer Annahmen fruchtbare Hypothesen generieren zu wollen. In analytischen Modellen werden hierbei die Präferenzordnungen der Modellakteure wie auch die Entscheidungsregel, nach der Handlungsoptionen gewählt werden, analytisch gesetzt, um Aussagen über das Verhalten der Modellakteure zu treffen. Doch auch hier werden implizit Handlungsgründe als Handlungsursachen herangezogen. Zur Illustration dieser These sollen zwei interessante Strategien betrachtet werden, die häufig in RationalChoice-Analysen angewandt werden: (a) Fokus auf die Aggregatebene. Handlungsziele lassen sich zumeist nicht direkt erreichen, sondern nur über eine Kette von Zwischengütern (zum Beispiel Geld, Bildung etc.), die sozial definiert sind (Lindenberg 1989). Selbst wenn man die individuellen Handlungsziele nicht kennt, kann in aller Regel davon ausgegangen werden, dass diese Zwischengüter von nahezu allen Akteuren angestrebt werden. „If money is needed for buying whatever one may wish to acquire we do not need a ,taste' for money to predict money-seeking behavior. What has instrumental value is contextually determined, not motivationally" (Zintl 2001: 43). Eine Analyse der Verfügbarkeit und der Änderung der Kosten der Zwischengüter kann hier an die Stelle der Interpretation der jeweiligen Handlungsgründe treten, da mit guten Gründen davon ausgegangen wer- 24 Gleichzeitig zieht die normative Interpretation von Rationalitätsannahmen als konstitutiver Aspekt von Einstellungszuschreibungen wichtige Änderungen nach sich. Wird Rationalität als konstitutiv für Handlungen und propositionale Einstellungen angesehen, ist zum Beispiel die Frage nach der Rationalität von Akteuren geschlossen: Da eine Person nur dann als Akteur gilt und ihr Einstellungen nur dann zugeschrieben werden können, wenn sie einen Minimalstandard an Rationalität erfüllt, ist es keine empirische Frage, ob Akteure wirklich rational sind oder nicht. Rationalitätszuschreibungen sind vielmehr bereits in den Akteursstatus eingebettet. Für die Frage, wie sich diese Einsicht auf die Debatte zwischen realistischen und instrumentalistischen Interpretationen von Rational Choice auswirkt, vergleiche Tiefensee (2015). 25 Vor diesem Hintergrund lässt sich die Entwicklung der Rational-Choice-Theorien der letzten Jahrzehnte auch aus einer anderen Perspektive betrachten. Üblicherweise wird diese als eine zunehmende Abkehr von der Idee der Rationalität individueller Handlungen gesehen. Man kann diese Entwicklung jedoch auch als Rationalisierungsversuch interpretieren. Die Modifizierung der Rationalitätsannahme im Verlauf der Rational-Choice-Debatte von einer ursprünglich sehr strikten Maximierungsregel, über das Konzept der ‚bounded rationality' hin zur Verhaltensökonomik (vgl. Blaug 1994; Camerer et al. 2004), könnte so als Versuch gewertet werden, möglichst viele Handlungen als rational gelten zu lassen und Irrationalitätszuschreibungen zu minimieren. Johannes Marx, Christine Tiefensee: Rationalität und Normativität 35 den kann, dass die Maximierung von Zwischengütern individuelle Handlungsziele darstellen. (b) Hochkostensituationen. Eine ähnliche Argumentation gilt auch für die Analyse von Hochkostensituationen. Solche Situationen zeichnen sich durch starke Mechanismen aus, die dem Anwender von Rational Choice die Interpretation von Handlungsgründen erleichtern. Akteure, deren Handlungen nicht der exogen vorgegebenen Anreizstruktur entsprechen, werden aus dem Spiel ausgeschlossen. In solchen Handlungskontexten kann somit ebenfalls auf die individuelle Interpretation von Handlungsgründen verzichtet werden, da auch hier davon ausgegangen werden kann, dass die individuellen Präferenzen strukturell determiniert sind (Zintl 2001: 44). Offensichtlich ist dies für das Verhalten auf Märkten: Produzenten oder Konsumenten, die sich nicht marktkonform verhalten, werden auf Märkten nicht überlebensfähig sein und über kurz oder lang verschwinden. Hier sorgt die Institution Markt dafür, dass das Verhalten von Produzenten und Konsumenten verstehbar wird, selbst wenn wir die individuellen Präferenzstrukturen der Akteure nicht kennen. Anstatt tief in idiosynkratische Beweggründe von Akteuren einzusteigen, ist demzufolge das Ziel vieler Rational-Choice-basierter Analysen, vereinfachte und verallgemeinerte Annahmen über die Präferenzordnung von Akteuren zu treffen. Individuelle Handlungsgründe spielen in diesen Anwendungen zumeist keine explizite Rolle. Entscheidend für unsere Zwecke ist aber, dass auch solche modelltheoretischen Einsätze von RationalChoice-Argumenten und abstrahierten Präferenzordnungen mit der normativen Interpretation von Rationalitätszuschreibungen kompatibel sind und sie sogar voraussetzen. Denn obwohl analytische Modelle Präferenzordnungen und Entscheidungsregeln setzen, so tun sie dies doch innerhalb des allgemeinen Kontextes der Einbettung von Modellhandlungen in das Rahmenwerk von Gründen: Dass es beispielsweise sinnvoll ist, Akteuren in gewissen Situationen ein Interesse an Zwischengütern zuzuschreiben, muss auf der Einschätzung von Handlungsgründen beruhen. Genauso kann auch die Interpretation und Wirkung starker Mechanismen auf idealtypische Akteure nur vor dem Hintergrund erfolgen, dass konkrete Annahmen über den Zusammenhang von Struktur und Handlungsgründen getroffen werden. Analytische Modelle greifen normative Rationalitätszuschreibungen daher auf, indem sie Handlungsgründe explizieren. Ob diese Modelle Aspekte der Realität korrekt wiedergeben oder nicht, bleibt wiederum eine Frage der Interpretation, das heisst die Frage, ob die in dem Modell vorgeschlagene Bestimmung von Handlungsgründen die Handlungsursachen realer Akteure widerspiegelt. Obwohl diese Überlegungen lediglich skizzenhaft sind, spricht somit einiges dafür, dass auch eine normativ zu verstehende Rationalitätsannahme in positiven Untersuchungen Verwendung finden kann. Wie genau sich das Verhältnis zwischen der normativen Rationalitätsannahme und positiver Politikwissenschaft gestaltet, muss offensichtlich erheblich präziser untersucht werden, als wir es hier tun konnten. Wir hoffen jedoch, dass durch unsere Argumente die Grundlage für eine solche weitere Untersuchung gelegt wurde, indem wir gezeigt haben, dass nur die normative Interpretation der Rationalitätsannahme die Relevanz der Rationalität in normativen wie auch positiven Theorien erklären kann. 36 Zeitschrift für Politische Theorie, Heft 1/2015 Literatur Amadae, Sonja / Bueno De Mesquita, Bruce, 1999: The Rochester School: The Origins of Positive Political Theory. In: Annual Review of Political Science 2, 269–295. Anscombe, Gertrude, 1979 [1957]: Intention, 2. Auflage, Oxford. 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Lightning in a Bottle Jonathan Lawhead Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2014 2014 To the extent possible under law, Jonathan Lawhead has waived all copyright and related or neighboring rights to Lightning in a Bottle. No rights reserved. ABSTRACT Lightning in a Bottle Jonathan Lawhead Climatology is a paradigmatic complex systems science. Understanding the global climate involves tackling problems in physics, chemistry, economics, and many other disciplines. I argue that complex systems like the global climate are characterized by certain dynamical features that explain how those systems change over time. A complex system's dynamics are shaped by the interaction of many different components operating at many different temporal and spatial scales. Examining the multidisciplinary and holistic methods of climatology can help us better understand the nature of complex systems in general. Questions surrounding climate science can be divided into three rough categories: foundational, methodological, and evaluative questions. "How do we know that we can trust science?" is a paradigmatic foundational question (and a surprisingly difficult one to answer). Because the global climate is so complex, questions like "what makes a system complex?" also fall into this category. There are a number of existing definitions of 'complexity,' and while all of them capture some aspects of what makes intuitively complex systems distinctive, none is entirely satisfactory. Most existing accounts of complexity have been developed to work with information-theoretic objects (signals, for instance) rather than the physical and social systems studied by scientists. Dynamical complexity, a concept articulated in detail in the first third of the dissertation, is designed to bridge the gap between the mathematics of contemporary complexity theory (in particular the formalism of "effective complexity" developed by Gell-Mann and Lloyd [2003]) and a more general account of the structure of science generally. Dynamical complexity provides a physical interpretation of the formal tools of mathematical complexity theory, and thus can be used as a framework for thinking about general problems in the philosophy of science, including theories, explanation, and lawhood. Methodological questions include questions about how climate science constructs its models, on what basis we trust those models, and how we might improve those models. In order to answer questions about climate modeling, it's important to understand what climate models look like and how they are constructed. Climate model families are significantly more diverse than are the model families of most other sciences (even sciences that study other complex systems). Existing climate models range from basic models that can be solved on paper to staggeringly complicated models that can only be analyzed using the most advanced supercomputers in the world. I introduce some of the central concepts in climatology by demonstrating how one of the most basic climate models might be constructed. I begin with the assumption that the Earth is a simple featureless blackbody which receives energy from the sun and releases it into space, and show how to model that assumption formally. I then gradually add other factors (e.g. albedo and the greenhouse effect) to the model, and show how each addition brings the model's prediction closer to agreement with observation. After constructing this basic model, I describe the so-called "complexity hierarchy" of the rest of climate models, and argue that the sense of "complexity" used in the climate modeling community is related to dynamical complexity. With a clear understanding of the basics of climate modeling in hand, I then argue that foundational issues discussed early in the dissertation suggest that computation plays an irrevocably central role in climate modeling. "Science by simulation" is essential given the complexity of the global climate, but features of the climate system--the presence of non-linearities, feedback loops, and chaotic dynamics--put principled limits on the effectiveness of computational models. This tension is at the root of the staggering pluralism of the climate model hierarchy, and suggests that such pluralism is here to stay, rather than an artifact of our ignorance. Rather than attempting to converge on a single "best fit" climate model, we ought to embrace the diversity of climate models, and view each as a specialized tool designed to predict and explain a rather narrow range of phenomena. Understanding the climate system as a whole requires examining a number of different models, and correlating their outputs. This is the most significant methodological challenge of climatology. Climatology's role contemporary political discourse raises an unusually high number of evaluative questions for a physical science. The two leading approaches to crafting policy surrounding climate change center on mitigation (i.e. stopping the changes from occurring) and adaptation (making post hoc changes to ameliorate the harm caused by those changes). Crafting an effective socio-political response to the threat of anthropogenic climate change, however, requires us to integrate multiple perspectives and values: the proper response will be just as diverse and pluralistic as the climate models themselves, and will incorporate aspects of both approaches. I conclude by offering some concrete recommendations about how to integrate this value pluralism into our socio-political decision making framework. Table of Contents List of Figures ii Acknowledgements iii-iv Dedication v Prelude Doing Better 1-14 Chapter One Who Are You, and What Are You Doing Here? 15-50 Chapter Two What's the Significance of Complexity? 51-77 Chapter Three Dynamical Complexity 78-101 Chapter Four A Philosopher's Introduction to Climate Science 102-146 Chapter Five Complexity, Chaos, and Challenges in Modeling Complex Systems 147-186 Chapter Six Why Bottle Lightning? 187-219 Coda Modeling and Public Policy 220-232 Works Cited 233-240 i LIST OF FIGURES Fig. 2.1 70 Fig. 2.2 71 Fig. 4.1 120 Fig. 4.2 123 Fig. 5.1 167 Fig. 5.2 176 Fig. 6.1 190 ii ACKNOWLEDGEMENTS This work is the sum result of the effort of an innumerable list of people, only one of which (me) actually wrote the thing. There are too many contributors to mention, but I'll highlight a few of the most significant, in order of the product of contribution and temporal appearance. First, my mother Peggy Polczinski who taught me the value of education and the value of humor, and who has been my biggest supporter and cheerleader since the day I was born. In a very literal sense, I would not be here without her, and she deserves top billing. Second, my stepfather Peter Polczinski, who taught me the value of hard work and perseverance, despite my innate aversion to both those virtues. Long days and pleasant nights to both of you. I get by with a little help from my friends. Philip Kitcher, who has been a tireless, patient, kind, and inspirational advisor during my time at Columbia. Professor Kitcher published five books (and counting!) while I was his graduate student, and was responsible for suggesting the topic of this work to me one rainy December New York morning on the steps of Low Library on the Columbia University campus. Every aspiring thinker should be so lucky as to have a mentor like you, Philip, and your encouragement, prodding, criticism, and advice have been indispensable and unforgettable. It's a debt and an honor that I will never be able to repay. Allegra Pincus who has been along for the last and possibly most bumpy part of this project, and who has been a steadfast partner, friend, companion, and supporter. I love you very much, my dear. Thank you for being in my life. I've always been the kind of person who really only produces anything of value when forced into argument with someone else, so all of the various people who have spoken to and debated with me over the years deserve a mention. Chief among these are two individuals: Daniel Estrada and Quee Nelson. Dan and Quee (who would be horrified at being included in the same exclusive set) have both functioned as steadfast critics and interlocutors over the course of years. They've both read early drafts of things I've written, offered arguments that challenged my assumptions, and stayed up late into the night debating with me. Dan and Quee, the friendship of both of you has been invaluable, and you've both functioned as mentors in your own way. I can't wait to keep working with you both. iii The members of my dissertation examination committee: David Albert, John Collins, Mark Cane, and Michael Strevens, who pushed me to produced a stronger final draft of this work, and who provided excellent, insightful, and appreciated criticism, deserve special mention. In particular, Mark and John did their best to ensure that I wasn't totally uninformed about my topics. I owe them a huge debt. Finally, all of my other family, friends, and colleagues at Columbia University (and elsewhere) who have read my work, spoken with me about my ideas, and influenced me in ways they'll never know deserve a mention. There are far too many to list, but some of the most significant (still in order of appearance) are: Chelsea Canon (who taught me to sing out and be free), Matt Royal and Jason Euren (who have been better brothers than I could have ever gotten by blood), Adam See (who is one of the kindest people I've ever known), Katie MacIntyre (who taught me that maybe Foucault isn't that bad), Mark Berger (who dogged me about social responsibility), Porter Williams (who could beat me up if he wanted to), Timothy Ignaffo (who has supported this project in ways that defy mentioning here), and Rebecca Spizzirri, (who has suffered living with me and hearing about these ideas for two years). Thanks to all of you, and to those whom I haven't specifically mentioned; this meatbrain is fallible and weak. Thank you, everyone. iv DEDICATION To Peggy (of course) v "The sciences, each straining in its own direction, have hitherto harmed us little; but some day the piecing together of dissociated knowledge will open up such terrifying vistas of reality, and of our frightful position therein, that we shall either go mad from the revelation or flee from the light into the peace and safety of a new dark age." -H.P. Lovecraft Prelude Doing Better 0.0 Motivation and Preliminaries The world is messy, and science is hard. These two facts are, of course, related: science seeks to understand a messy world, and that's a difficult task. Scientists have a variety of tools at their disposal to cope with this messiness: the creation of idealized models, the scientific division of labor, and the proliferation of increasingly elaborate pieces of technology all serve to help us predict and control a complex world. Not all tasks call for the application of the same tools, though, and so the scientific project takes all kinds: there's room for a variety of contributions, and we must be willing to change tactics as new problems present themselves. Adaptation, flexibility, and collaboration are at the heart of scientific progress. This dissertation is intended not to be a work in the philosophy of science precisely, but neither is it, strictly speaking, a work of "pure science" (whatever that might mean). Rather, it is a philosophical contribution to science itself: I will attempt to employ the methods and tools of the philosopher to engage with a concrete issue in contemporary science-the problem of global climate change. In March of 2010, Dr. Jon Butterworth of University College, London's high energy physics group published a short piece in The Guardian titled "Come on, 'philosophers of science,' you must do better than this, " in which he called upon philosophers of science to make a real 1 1 Butterworth (2010) 1 contribution to the emerging (and increasingly important) climate science debate. Butterworth's call for philosophers of science to "do better" was inspired by another contribution to The Guardian from a few days earlier, this one written by Nicholas Maxwell, a philosopher at University College, London. Maxwell's piece, "Scientists should stop deceiving us ," criticizes 2 scientists generally (and climate scientists in particular) for producing what he calls "incomprehensible gobbledygook" that (he suggests) is to blame for the public's rejection of scientific insights. Going even further, Maxwell suggests that underlying this problem is an even deeper one-an insistence on the part of scientists (especially physicists) that scientific theories be "unified"-capable of applying to all parts of the world in their domain-and that more explanatorily satisfying theories are rejected on the basis of disunity, leading to a thicket of incomprehensible theories that make little contact with the values of contemporary society. As Butterworth points out, there is surely some truth to Maxwell's criticism: Science often falls short of its ideals, and the climate debate has exposed some shortcomings. Science is done by people, who need grants, who have professional rivalries, limited time, and passionately held beliefs. All these things can prevent us from finding out what works. This is why the empiricism and pragmatism of science are vital, and why when scientific results affect us all, and speak against powerful political and financial interests, the openness and rigour of the process become ever more important. 3 Science (to recapitulate the point from above) is hard, and indeed does often fall short of its goal of predicting what will happen in the world. The reasons for these failures are varied and complicated, but Maxwell is surely right to say that some of them have to do with the attitudes of some scientists themselves. With Butterworth, though, I have a hard time seeing the force of the claim that much of the blame for this problem is to be laid at the feet of specialization: the 2 Maxwell (2010) 3 Butterworth (ibid.) 2 division of scientific labor is a natural, reasonable, and deeply effective response to a messily complex world. The "gobbledygook" that Maxwell decries is (as Butterworth notes) a kind of sophisticated short-hand meant for communication between experts themselves, not between experts and the public; the problem there, then, is less with the science itself and more with the communication of science. The problem, to put the point another way, is that it is difficult for working scientists themselves to take a high-level view of the project as a whole, and to see the scientific forest for the experimental trees. This, perhaps, is where a philosopher might help. Butterworth closes his article with a few distinctly philosophical-sounding assertions. Science is a form of systematised pragmatism: it finds out what works, and in the process we increase our understanding of the universe in which we live. I have no objection to philosophers watching, and trying to understand and improve the processes. It might even work. But they really ought to (and often do) have an understanding of what they are watching. ... This is worth discussing, and I sincerely hope philosophers of science can do better than Maxwell in contributing to a debate of huge significance for the future of our species. I agree whole-heartedly with this sentiment. Philosophers of science do indeed need to do better with regard to climate science-it is a real, pressing issue: perhaps the most pressing contemporary scientific issue facing us. To a very great extent, this means doing something: the degree to which philosophers have engaged with climate science at all is minimal even compared to the general paucity of philosophical contact with applied contemporary social issues. While some people in philosophy departments have begun to take notice of this (more on this later), it is high time that more followed suit, and that this became a topic of wide-spread discussion among philosophers. It is in this spirit that this project is conceived; my hope here is not to solve the climate change problem (that is not my job), nor is it simply to provide the kind of abstract theoretical criticism that Butterworth rightly calls down Maxwell (as a representative of philosophy of science generally) for being obsessed with. Rather, it is to sketch the lay of the 3 land. My hope is that this dissertation will open the door to contributions by my peers (many of whom are, I am sure, far better equipped to deal with these issues than I am) to begin to have a conversation about this pressing social and scientific problem. My hope is that this will be the beginning of philosophers of science at least trying to do better. 0.1 Outline and General Structure The somewhat unusual nature of this project, though, means that the structure and methodology of this dissertation will be somewhat different from most works both in philosophy and science. Before beginning the project proper, then, I want to say a bit about why I chose to structure things as I have, and why I have focused on the issues that I chose. My hope is that in flagging some of the unorthodox aspects of this work as intentional, I might short-circuit a few lines of objection to my project that would (I think) serve only to distract from the real work to be done. To get the ball rolling, let me lay out a sketch of how this work will proceed. First, there are foundational questions. These questions concern the structure of science generally, the relationship between the various branches of science, climate science's continuity (or lack thereof) with the rest of science, and other issues that don't seem to be investigated directly by any other branch of science. Foundational questions include those that are traditionally thought of as the purview of the philosopher: "how do we know that we can trust science?" is a paradigmatic foundational question (and a surprisingly difficult one to answer, at that). Chapters One, Two, and Three of this work will focus on foundational questions. Specifically, Chapter One outlines a novel approach to philosophy of science based on recent advances in information theory, and lays the groundwork for applying that approach to the 4 problem of climate science. Chapters Two and Three review some contemporary work being done in complexity theory, with a particular focus on attempts to define and quantify the notion of "complexity" itself, then sketch an account of complexity that builds on the work done in Chapter One. Second, there are methodological questions. These questions are more specifically concerned with the structure and operation of a particular branch of science; the methodological questions that will concern (say) a fundamental physicist will be different from the methodological questions that will concern a climate scientist. Questions about how climate science makes its predictions, on what basis we ought to trust those predictions, how we might use the tools of climate science to make better predictions, how to interpret the climate data on record, and how to best make use of our limited computing resources are all methodological questions. "How should we decide which factors to include in our climate model?" is a paradigmatic methodological question. Chapters Four, Five, and Six will focus on methodological questions. Chapter Four consists in a general introduction to the project of climate modeling, with a focus on the limitations of simple climate models that are solvable in the absence of pure computer simulations. In Chapter Five, I examine the challenges of building more complex climate models, with special attention to the problems posed by non-linearity and chaos in the climate system. In Chapter Six, I examine the role that computational simulation plays in working with climate models, and attempt to reconcile the novel problems posed by "science by simulation" with the results of climate science. The answers to questions in each of the categories will (of course) be informed by answers to 5 questions in the other categories; how we ought to react to a rapidly changing climate (an evaluative question) will clearly depend in part on how much we trust the predictions we've generated about the future (a foundational question), and that trust will depend in part on how we design and implement our climate models (a methodological question). My purpose in delineating these categories, then, is not to suggest that this division corresponds to essentially different spheres of inquiry-rather, this way of carving up the complicated and multi-faceted problems in the philosophy of climate science is just a pragmatic maneuver. Indeed, it is one of the principal theses of my project that none of these groups of questions can be effectively dealt with in isolation: they need to be tackled as a package, and a careful examination of that package is precisely what I am concerned with here. With this general structural outline in mind, then, let me say a bit more about what I intend to do in each chapter. Chapter One is the most traditionally philosophical, and deals with general questions in the philosophy of science. In particular, I focus on the question of how philosophy can make a contribution to the scientific project. I offer an apocryphal quotation attributed to Richard Feynman, viz., " Philosophy of scientists is about as useful to scientists as ornithology is to birds," as my primary target, and attempt to see how a philosopher of science might respond to Feynman's charge. I argue that none of the accounts of science on offer in the literature can easily meet this challenge, in large part because they're often concerned with questions that are of little real consequence to practicing scientists. Drawing on concepts in information theory, I construct a novel account of structure of the scientific project that (I hope) skirts some of the stickier (but, I argue, less important) issues in which 20th century philosophy of science often became mired. With that account of science in hand, I argue that philosophy has a real 6 contribution to make to the scientific project as a whole-I argue, that is, that there are issues with which the scientific project ought to be concerned that are not precisely scientific issues, and that philosophers are in a good position to tackle those issues. In offering this account of the structure of science, I also give a novel way of understanding what it means to say that the scientific project is "unified." This is not merely an abstract point, but has real consequence for what will and will not count as a legitimate scientific theory: as we saw, one of the criticisms Maxwell offers is that scientists reject what he considers perfectly good theories on the basis of disunity. Is this true? In what sense is the unity of science an important guide to scientific theory, and how should we evaluate the relative unity of different theories? Does the unity of science conflict with the obvious methodological division of labor across the different branches of science? In addressing these questions, I hope to set the stage for a more fruitful examination of climate science's place in the scientific project overall. Chapters Two and Three taken together are primarily a contribution to the foundations of complex-systems theory. Building on the account of science from Chapter One, I argue that the traditional bifurcation of science into physical and social sciences is, at least sometimes, misleading. I suggest that we should also see some scientific problems in terms of a distinction that cuts across the physical/social science division: the distinction between complex-systems sciences and simple-systems sciences. After reviewing some of the attempts to define "complexity" in the (relatively nascent) field of complex-systems theory (and arguing that none of the attempts fully succeeds in capturing the relevant notion), I use the machinery assembled in Chapter One to construct a novel account of complexity that, I argue, unifies a few of the most plausible definitions in the literature. This concept, which I will call dynamical complexity gives 7 us a theoretical tool to help us think about the difference between systems that seem intuitively "simple" (e.g. a free photon in a vacuum) and systems that seem intuitively "complex" (e.g. the global climate) more clearly, and to begin to get a grasp on important differences between the methods of sciences that study systems with high dynamical complexity and those of sciences that study systems with low dynamical complexity. I then argue that, based on this definition, climate science is a paradigmatic complex-systems science, and that recognition of this fact is essential if we're to bring all our resources to bear on solving the problems posed by climate change. In Chapter Four, we turn from explicitly foundational issues in the philosophy of science and complexity theory to more concrete methodological questions. I introduce the basics of climate science, and construct a very simple climate model from first principles. This chapter closes with a consideration of the limitations of the methods behind this basic model, and of the general principles that inform it. This paves the way for the discussion of deeper challenges in Chapter Five. Chapter Five describes some of the specific problems faced by scientists seeking to create detailed models of complex systems. After a general introduction to the language of dynamical systems theory, I focus on two challenges in particular: non-linearity and chaotic dynamics. I discuss how these challenges arise in the context of climatology. We'll then focus on a more concrete examination of a particular methodological innovation that is characteristic of complex-systems sciences: computer-aided model-building. Because of the nature of complexity (as described in Chapter Three) and the various special difficulties 8 enumerated in Chapter Five, many of the techniques that simple-systems sciences rely on to make progress are unavailable to climate scientists. Like economists and evolutionary biologists, climatologists' most potent weapon is the creation of complex mathematical models that underlie a host of computer simulations. In Chapter Six, I examine some of the widespread criticisms of this "science by simulation," and argue that they are either misinformed or not fatal to the project of climate science. Drawing further on the resources of complex-systems theory, I argue that the function of computational models is not exactly to predict, but rather to act as "tools for deciding," helping us coordinate and organize our more detailed investigation of the global climate. 0.2 Methods and Problems The relative paucity of philosophical literature dealing with issues in the foundations of climate science puts me in the somewhat unusual position of having to cover an enormous amount of territory in order to mark out the lay of the land. In order to do what I want to do, then, I need to sacrifice a certain amount of depth in the name of achieving a certain amount of breadth. This is a deliberate move, but it does not come without consequences. Before beginning the actual project, I want to take a few pages to review some of these issues, flag them as problems that I have considered, and offer a few justifications for why I have chosen the approach that I have. There is some risk that in trying to speak to everyone with this dissertation, I will end up satisfying no one at all. I suspect that individual readers will find my discussions of their particular areas of specialization somewhat unsatisfying: philosophers of science operating in the 9 tradition of the profession-those who have inherited their methods and problems down from Hempel, Kuhn, Popper, van Fraassen, and so on-will likely find my discussion of the structure of the scientific project in Chapter One unsatisfying in virtue of the fact that it makes very little contact with the classic literature in the field. Mathematicians and physicists working in dynamical systems theory will likely find my discussion of dynamical complexity unsatisfying in virtue of its relatively informal and non-mathematical presentation. Practicing climatologists will likely find my discussion of Mann's work in particular (and the methods of climate science in general) unsatisfying in virtue of the fact that I am not myself a climatologist, and thus lack the kind of sensitivity and feel for the scientific vernacular that comes from years of graduate school spent simmering in the relevant scientific literature. Ethicists and political philosophers will likely find my discussion of the moral and social issues surrounding climate science's predictions unsatisfying in virtue of the fact that I (quite admittedly) know very little about the state of the ethics literature today, and thus will be presenting largely what I see as common-sense approaches to solving these problems that are as devoid of ethical theory as possible. In short, no matter who you are, you're probably going to be deeply suspicious of what I have to say, particularly about the topic in which you specialize. Why, then, have I chosen to approach this project in the way that I have? Instead of leaving everyone upset, why not try to please a small number of people and make a deep contribution to just one of the issues I discuss here? There are a few answers to this that are, I think, related. Perhaps primarily, I'm concerned with philosophy's treatment of climate science generally, and a highly general approach is (I think) the best way to express this concern. As I've said, while there has been a not-insignificant 10 amount of value theory done on the topic of environmental ethics, there's been very little philosophical contribution to the actual science of climate change. In effect, then, one of the principal goals of this dissertation is to jump up and down, wave my arms, and shout "over here!" As I inevitably get some (many) of the details wrong in my discussion, I hope others will be inspired to step in and correct things, point out what I've done incorrectly, and do better than I am capable of doing. If I can inspire enough controversy to get the philosophical community involved in the climate change debate, then I will count this as a success, irrespective of whether or not my own views are accepted or rejected. Relatedly, part of my intention here is to stake out a large amount of territory all at once to suggest how those with expertise in specific problems might make deeper contributions than I make here. In discussing philosophy of science, complexity theory, model-building, and value theory all in a single work, I hope to sketch the general shape that a fully-fledged "philosophy of climate science" literature might take, and to open the door for more systematic contributions to that literature by those who are best equipped to make them. In order to make this goal achievable in only a few hundred pages of writing, I'm forced to make a number of simplifying assumptions in some places, and to ignore significant problems entirely in other places. Whenever possible, I will offer a footnote flagging the fact that I'm doing this deliberately, and suggesting what a more careful elaboration of the topic might look like. If I were to give each topic here the full attention it deserves, this work would be thousands of pages in length (not to mention beyond my ability). I far prefer to leave the project of elaborating and expanding most of what I'm trying to start here to my betters. To facilitate this, I will close each chapter with a series of questions for further exploration, or a brief discussion of the shape that future research 11 might take. I intend to take up at least some of this research myself in the future (particularly work in the foundations of complexity theory and information theory as they relate to climate science and the scientific project as a whole), but I am equipped with neither the time nor the ability to take all of it up; climate change is a pressing issue that demands our immediate attention, and we'll need to work together if we're to solve this problem. If nothing else, this dissertation is a sustained argument for precisely this point. Finally, it is worth highlighting that this dissertation is motivated by an explicitly pragmatic approach to philosophy and science. I think that Butterworth is precisely correct when he says that "science is a form of systematized pragmatism," and I suspect that most scientists (insofar as they think about these things at all) would, given the chance, assent to that statement. The largest consequence of this is that I wish, whenever possible, to remain totally neutral as to how what I'm saying makes contact with more traditionally philosophical questions-particularly those in mainstream metaphysics. Chapter One places a great deal of weight on facts about patternhood, and there is a temptation to attempt to read what I'm saying as making a claim about the metaphysical status of patterns-a claim relating to the emerging metaphysical position that some have termed "ontic structural realism." I will say a bit more about this in Chapter One 4 when the issue comes up directly, but this is worth mentioning here by way of one last methodological preliminary: while I do indeed have a position on these issues, I think the point I am making here is independent of that position. I'm inclined to agree with something like the structural realist position the James Ladyman and Don Ross have pioneered-that is, I'm inclined to agree that, if we're to take science seriously as a metaphysical guide, we ought to take 4 See, canonically, Dennett (1991) and Ladyman et. al., (2007) 12 something like patterns (in a robust, information-theoretic sense) as the primary objects in our ontology-but this is a highly controversial claim in need of defense on its own terms. This is neither the time nor the place for me to enter into that debate . When I couch my discussion in 5 terminology drawn from the structural realist literature-when I speak, for instance, of "real patterns,"-it is merely for the sake of convenience. Nothing in my project turns on taking this language as anything but a convenience, though-if you prefer to take the Humean view, and think of patterns as the sort of things that supervene on purely local facts about spatio-temporal particulars, that will do no violence to the story I want to tell in this dissertation. Conversely, if you wish to read parts of this (particularly the first three chapters) as the preliminaries of a contribution to the metaphysics of patterns, or as a sketch of how such a metaphysics might be tied to issues in the foundations of complex systems theory, this also will not impact the larger point I want to make. Indeed, I will suggest at the close of Chapter Three that such an exploration might be one of the future research programs suggested by this project. I take it as one of the strengths of this approach that it is neutral between these two interpretations-whether or not you are sympathetic to the Dennett/Ladyman account of patterns as primary metaphysical objects or not, my discussion of patternhood turns exclusively on patterns understood in the (relatively) uncontroversial information-theoretic sense. That's the sense in which I want to maintain metaphysical neutrality here-some of my discussion adopts conventions from the structural realist camp, but this is strictly a matter of convenience and clarity (they have developed this vocabulary more than any other area of philosophy). I'm confident that the points I make could be translated into more obviously neutral terms without 5 I do intend to develop the kind of framework I deploy in Chapter One into a robust metaphysical theory at some point. That is simply not the project with which I am concerned here. 13 any significant problems. With these preliminaries out of the way, then, let's begin. 14 Chapter One Who Are You, and What Are You Doing Here? 1.0 Cooperate or Die The story of science is a story of progress through collaboration. The story of philosophy, on the face of it, is a story of neither: it is an academic cocktail party cliché that when an area of philosophy starts making progress, it's time to request funds for a new department. If this observation is supposed to be a mark against philosophy, I'm not sure I understand the jibe-surely it's a compliment to say that so much has sprung from philosophy's fertile soil, isn't it? Whether or not the joke contains a kernel of truth (and whether or not it does indeed count as a black mark against the usefulness of the discipline) is not immediately important. This project is neither a work in philosophy as traditionally conceived, nor a work in science as traditionally conceived: it is, rather, a work on a particular problem. I'll say a bit more about what that means below, but first let's start with an anecdote as a way into the problem we'll be tackling. In 2009, Columbia University's Mark Taylor, a professor of Religion, wrote an Op-Ed for the New York Times calling for a radical restructuring of academia. Among the controversial changes proposed by Taylor was the following: "Abolish permanent departments, even for undergraduate education, and create problem-focused programs. These constantly evolving programs would have sunset clauses, and every seven years each one should be evaluated and either abolished, continued or significantly changed. " This suggestion drew a lot of fire from 6 other academics. Brian Leiter, on his widely-circulated blog chronicling the philosophy 6 Taylor (2009) 15 profession, was particularly scathing in his rebuke: "Part of what underlies this is the fact that Taylor has no specialty or discipline of his own, and so would like every other unit to follow suit, and 'specialize' in intellectual superficiality across many topics. " Ouch. Professor John 7 Kingston of the University of Massachusetts, Amherst's linguistics department was a bit more charitable in his response, which appeared in the published reader comments on the New York Times' website: Rather than looking inward as [Taylor] claims we all do, my colleagues and I are constantly looking outward and building intellectual bridges and collaborations with colleagues in other departments. In my department's case, these other departments include Psychology, Computer Science, and Communications – these collaborations not only cross department boundaries at my institution but college boundaries, too. Moreover, grants are increasingly collaborative and interdisciplinary. 8 This seems to me to be a more sober description of the state of play today. While some of us might cautiously agree with Taylor's call for the radical restructuring of university departments (and, perhaps, the elimination of free-standing disciplines), virtually all of us seem to recognize the importance and power of collaboration across existing disciplines, and to recognize that (contra what Leiter has said here) generality is not necessarily the same thing as superficiality. The National Academies Press' Committee on Science, Engineering, and Public Policy recognized the emerging need to support this kind of collaborative structure at least as far back as 2004, publishing an exhaustive report titled Facilitating Interdisciplinary Research. The report describes the then-current state of interdisciplinary research in science and engineering: Interdisciplinary thinking is rapidly becoming an integral feature of research as a result of four powerful "drivers": the inherent complexity of nature and society, the desire to explore problems and questions that are not confined to a single discipline, the need to solve societal problems, and the power of new technologies. 9 7 Leiter (2009) 8 Kingston (2009) 9 Committee on Science, Engineering, and Public Policy (2004), p. 3 16 The times, in short, are a-changing; the kinds of problems facing science today increasingly call for a diverse and varied skill-set-both in theory and in practical application-and we ignore this call at our peril. This is true both inside traditional disciplines and outside them; in that sense, Taylor's call was perhaps not as radical as it first appears-the kind of collaborative, problem-focused research that he advocates is (to a degree) alive and well in the traditional academic habitat. Research in quantum mechanics, to take one example on which my background allows me to speak at least semi-intelligently, might incorporate work from particle physicists doing empirical work with cloud chambers, high-energy particle physicists doing other empirical work with particle accelerators, and still other particle physicists investigating the mathematics behind spontaneous symmetry breaking. Progress will come as a result of a synthesis of these approaches to the problem. This is hardly earth-shattering news: science has long labored under an epistemic and methodological division of labor. Problems in physics (for instance) have long-since become complex to such a degree that no single physicist can hope to understand all the intricacies (or have the equipment to perform all the necessary experiments), so physicists (and laboratories) specialize. The results that emerge are due to the action and work of the collective-to the institutional practices and structures that allow for this cooperative work-as much as to the work of individual scientists in the laboratories. Each branch supports all the others by working on more-or-less separable problems in pursuit of a common goal-a goal which no one branch is suited to tackle in isolation. In the case of elementary particle physics, that goal is (roughly) to understand patterns in the behavior of very, very small regions of the physical world; every relevant tool (from mathematical manifolds to particle accelerators) is recruited in pursuit of that 17 goal. More recently, however, a more sweeping collaborative trend has begun to emerge; increasingly, there have been meaningful contributions to quantum mechanics that have come not just from particle physicists, nor even just from physicists: the tool box has been enlarged. The work of W.H. Zurek on the relationship between quantum mechanics and classical mechanics, for instance, has been informed by such diverse fields of science as Shannon-Weaver information theory, mathematical game theory, and even Darwinian evolutionary biology . 10 "Pure" mathematics has contributions to make too, of course; much of the heavy-lifting in General Relativity (for example) is done by differential geometry, which was originally conceived in the purely theoretical setting of a mathematics department. Philosophy too has been included in this interdisciplinary surge. The particular tools of the philosopher-the precise nature of which we shall examine in some detail in the coming sections-are well-suited to assist in the exploration of problems at the frontiers of human knowledge, and this has not gone unappreciated in the rest of the sciences. Gone are the days when most physicists shared the perspective apocryphally attributed to Richard Feynman, viz., "Philosophy of science is about as useful to scientists as ornithology is to birds." There are real conceptual problems at the heart of (say) quantum mechanics, and while the sort of scientifically-uninformed speculation that seems to have dominated Feynman's conception of philosophy is perhaps of little use to working scientists, the interdisciplinary turn in academia has begun to make it safe for the careful philosopher of science to swim along the lively reef of physical inquiry with the physicist, biologist, and chemist. Science is about collaboration, and 10 See Zurek (2002), Zurek (2003), and Zurek (2004), respectively. 18 there is room for many different contributions. No useful tool should be turned away. So this call for radical collaboration is hardly new or revolutionary, despite the minor uproar that Taylor and his critics caused. The problem with which this project is concerned-the use to which I'll be putting my own tools here-is not a new one either. It is one about which alarm bells have been ringing for at least 60 years now, growing steadily louder with each passing decade: the problem of rapid anthropogenic global climate change. I shall argue that what resources philosophy has to offer should not be ignored here, for every last bit of information that can be marshaled to solve this problem absolutely must be brought to bear. This is a problem that is more urgent than any before it, and certainly more than any since the end of the nuclear tensions of the Cold War. While it likely does not, as some have claimed, threaten the survival of the human species itself-short of a catastrophic celestial collision, few things beyond humanity's own weapons of mass destruction can claim that level of danger-it threatens the lives of millions, perhaps even billions, of individual human beings (as well as the quality of life for millions more), but only if we fail to understand the situation and act appropriately. I shall argue that this is quite enough of a threat to warrant an all-out effort to solve this problem. I shall argue that philosophy, properly pursued, has as real a contribution to make as any other branch of science. I shall argue that we must, in a very real sense, cooperate or die. 1.1 What's a Philosopher to Do? Of course, we need to make all this a good deal more precise. It's all well and good for philosophers to claim to have something to add to science in general (and climate science in particular), but what exactly are we supposed to be adding? What are the problems of science 19 that philosophical training prepares its students to tackle? Why are those students uniquely prepared to tackle those questions? What is it about climate science specifically that calls out for philosophical work, and how does philosophy fit into the overall project of climate science? Why (in short) should you care what I have to say about this problem? These are by no means trivial questions, and the answers to them are far from obvious. Let's start slowly, by examining what is (for us) perhaps the most urgent question in the first of the three categories introduced in Chapter Zero : the question of how philosophy relates to the scientific project, and how 11 philosophers can contribute to the advancement of scientific understanding . 12 The substance of the intuition lurking behind Feynman's quip about ornithology is this: scientists can get along just fine (thank you very much) without philosophers to tell them how to do their jobs. To a point, this intuition is surely sound-the physicist at work in the laboratory is concerned with the day-to-day operation of his experimental apparatus, with experiment design, and (at least sometimes) with theoretical breakthroughs that are relevant to his work. Practicing scientists-with a few very visible exceptions like Alan Sokal-paid little heed to the brisk "science wars" of the 1980s and 1990s. On the other hand, though, the intuition behind Feynman's position is also surely mistaken; as I noted in Section 1.0, many of those same practicing physicists often acknowledge (for example) that people working in philosophy departments have made real contributions to the project of understanding quantum mechanics. It seems reasonable to suppose that those (living) scientists ought to be allowed to countermand 11 I suggested that questions we might ask about climate science could be roughly divided into three categories: foundational questions, methodological questions, and evaluative questions. This chapter and the following one will deal with foundational questions. See Section 0.1 for more detail. 12 The sense in which this is the most urgent question for us should be clear: the chapters that follow this one will constitute what is intended to be a sustained philosophical contribution to the climate change debate. On what basis should this contribution be taken seriously? Why should anyone care what I have to say? If we can't get a clear answer to this question, then all of what follows will be of suspect value. 20 Feynman who, great a physicist as he was, is not in a terribly good position to comment on the state of the discipline today; as James Ladyman has observed, "the metaphysical attitudes of historical scientists are of no more interest than the metaphysical opinions of historical philosophers ." I tend to agree with this assessment: primacy should be given to the living, and 13 (at least some) contemporary scientists are happy to admit a place for the philosopher in the scientific project. Still, it might be useful to pursue this line of thinking a bit further. We can imagine how Feynman might respond to the charge leveled above; though he's dead we might (so to speak) respond in his spirit. Feynman might well suggest that while it is true that genuine contributions to quantum mechanics (and science generally) have occasionally come from men and women employed by philosophy departments, those contributions have come about as a result of those men and women temporarily leaving the realm of philosophy and (at least for a time) doing science. He might well suggest, (as John Dewey did) that, "...if [philosophy] does not always become ridiculous when it sets up as a rival of science, it is only because a particular philosopher happens to be also, as a human being, a prophetic man of science. " That is, he might well side 14 with the spirit behind the cocktail party joke mentioned in Section 1.0-anything good that comes out of a philosophy department isn't philosophy: it's science. How are we to respond to this charge? Superficially, we might accuse the spirit of Feynman of simply begging the question; after all, he's merely defined science in such a way that it includes (by definition!) any productive work done by philosophers of science. Given that 13 Ladyman, Ross, Spurrett, and Collier (2007) 14 Dewey (1929), p.408 21 definition, it is hardly surprising that he would consider philosophy of science qua philosophy of science useless-he's defined it as the set of all the work philosophers of science do that isn't useful! 'Philosophy of science is useless to scientists,' on that view, isn't a very interesting claim. By the same token, though, we might think that this isn't a very interesting refutation; let's give the spirit of Feynman a more charitable reading. If there's a more legitimate worry lurking behind the spirit of Feynman's critique, it's this: philosophers, on the whole, are not qualified to make pronouncements about the quality of scientific theories-they lack the training and knowledge to contribute non-trivially to any branch of the physical sciences, and while they might be well-equipped to answer evaluative questions, they ought to leave questions about the nature of the physical world to the experts. If philosophers occasionally make genuine progress in some scientific disciplines, cases like that are surely exceptional; they are (as Dewey suggests) the result of unusually gifted thinkers who are able to work both in philosophy and science (though probably not at the same time). What's a philosopher of science to say here? How might we justify our paychecks in the face of the spirit of Feynman's accusations? Should we resign ourselves to life in the rich (if perhaps less varied) world of value theory and pure logic, and content ourselves with the fact that condensed-matter physicists rarely attempt to expound on the nature of good and evil? Perhaps, but let's not give up too quickly. We might wonder (for one thing) what exactly counts as "science," if only to make sure that we're not accidentally trespassing where we don't belong. For that matter, what counts as philosophy and (in particular) what is it that philosophers of science are doing (useful or not) when they're not doing science? Surely this is the most basic of all foundational questions, and our answers here will color everything that follows. With that in 22 mind, it's important to think carefully about how best to explain ourselves to the spirit of Feynman. 1.2 What's a Scientist to Do? Let's start with a rather banal observation: science is about the world . Scientists are in the 15 business of understanding the world around us-the actual world, not the set of all possible worlds, or Platonic heaven, or J.R.R Tolkien's Middle Earth . Of course, this isn't just limited 16 to the observable, or visible world: science is interested in the nature of parts of the world that have never been directly observed and (in at least some cases) never will be. Physicists, for instance, are equally concerned that their generalizations apply to the region of the world inside the sun as they are that those generalizations apply to their laboratory apparatuses. There's a 17 more important sense in which science is concerned with more than just the observed world, though: science is not just descriptive, but predictive too-good science ought to be able to make predictions, not just tell us the way the world is right now (or was in the past). A science that 15 The philosophically sophisticated reader might well be somewhat uncomfortable with much of what follows in the next few pages, and might be tempted to object that the observations I'll be making are either fatally vague, fatally naïve, or both. I can only ask this impatient reader for some patience, and give my assurance that there is a deliberate method behind this naïve approach to philosophy of science. I will argue that if we start from basic facts about what science is-not as a social or professional institution, but as a particular attitude toward the world- how it is practiced both contemporarily and historically, and what it is supposed to do for us, we can short-circuit (or at least sneak by) many of the more technical debates that have swamped the last 100 years of the philosophy of science, and work slowly up to the tools we need to accomplish our larger task here. I ask, then, that the philosophically sophisticated reader suspend his sense of professional horror, and see if the result of our discussion here vindicates my dialectical (and somewhat informal) methodology. I believe it will. See Section 0.2for a more comprehensive defense of this naive methodology. 16 Though it is worth mentioning that considerations of possible worlds, or even considerations of the happenings in Tolkien's Middle Earth might have a role to play in understanding the actual world. Fiction authors play a central role in the study of human culture: by running detailed "simulations" exploring elaborate hypothetical scenarios, they can help us better understand our own world, and better predict what might happen if certain facets of that world were different than they in fact are. This, as we will see, is a vital part of what the scientific enterprise in general is concerned with doing. 17 Some philosophers of science (e.g. van Fraassen) have argued that there is a sense in which we observe what goes on inside the sun. This is an example of the sort of debate that I do not want to enter into here. The question of what counts as observation is, for our purposes, an idle one. I will set it to the side. 23 consisted of enumerating all the facts about the world now, as useful as it might be, wouldn't seem to count as a full-fledged science by today's standard, nor would it seem to follow the tradition of historical science; successful or not, scientists since Aristotle (at least!) have, it seems, tried to describe the world not just as it is, but as it will be. This leads us to another (perhaps) banal observation: science is about predicting how the world changes over time. Indeed, a large part of how we judge the success (or failure) of scientific theories is through their predictive success; the stock example of Fresnel's success with the wave theory of light, as demonstrated by the prediction (and subsequent observation) of a bright spot at the center of the shadow cast by a round disk is a stock example for good reason-it was a triumph of novel predictive utility. General relativity's successful prediction of the actual orbit of the planet Mercury is another excellent paradigm case here; Mercury's erratic orbit, which was anomalous in Newton's theory of gravity, is predicted by Einstein's geometric theory. This success, it is important to note, is not in any sense a result of "building the orbit in by hand;" as James Ladyman and John Collier observe, though Einstein did (in some sense) set out to explain Mercury's orbit through a general theory of gravitation, he did this entirely by reference to general facts about the world-the empirically accurate prediction of Mercury's orbit followed from his theory, but nothing in the theory itself was set with that particular goal in mind. The history of science is, if not exactly littered with, certainly not lacking in other examples of success like this; indeed, having surprising, novel, accurate predictions "pop out" of a particular theory is one of the best markers of that theory's success . 18 18 The Aharnov-Bohm effect, a surprising quantum mechanical phenomenon in which the trajectory of a charged particle is affected by a local magnetic field even when traversing a region of space where both the magnetic field and the electric fields' magnitudes are zero, is another excellent example here. This particular flavor of non-locality implies 24 It is not enough, then, to say that science is about prediction of how the world will change over time. Science doesn't just seek to make any predictions, it seeks to make predictions of a particular sort-predictions with verifiable consequences-and it does this by attempting to pick out patterns that are in evidence in the world now, and projecting them toward the future. That is to say: science is the business of identifying genuine patterns in how the world changes over 19 time. It is precisely this projectability that makes a putative pattern genuine rather than ersatz; this is why science is of necessity concerned with more than just enumerating the facts about the way the world is now-just given the current state of the world, we could hypothesize a virtually infinite number of "patterns" in that state, but only some of those putative patterns will let us make accurate predictions about what the state of the world will be in (say) another hour. 1.3 Toy Science and Basic Patterns Let's think more carefully about what it means to say that science is in the business of identifying genuine patterns in the world. Consider a simple example-we'll sharpen things up as we go along. Suppose we're given a piece of a binary sequence, and asked to make predictions about what numbers might lie outside the scope of the piece we've been given: S1: 110001010110001 Is there a genuine pattern in evidence here? Perhaps. We might reasonably suppose that the that the classical Maxwellian formulation of the electromagnetic force as a function of a purely local electrical field and a purely local magnetic field is incomplete. The effect was predicted by the Schrodinger equation years before it was observed, and led to the redefinition of electromagnetism as a gauge theory featuring electromagnetic potentials, in addition to fields. See Ahranov and Bohm (1959). Thanks to Porter Williams for suggesting this case. 19 The sense of "genuine" here is something like the sense of "real" in Dennett's "real patterns" (Dennett 1991). I wish to delay questions about the metaphysics of patterns for as long as possible, and so opt for "genuine" rather than the more ontologically-loaded "real." What it means for a pattern to be "genuine" will become clearer shortly. Again, see Section 0.2 for more on the underlying metaphysical assumptions here. 25 pattern is "two 'ones,' followed by three 'zeros' followed by 'one, zero, one, zero,' and then repeat from the beginning." This putative pattern R is empirically adequate as a theory of how this sequence of numbers behaves; it fits all the data we have been given. How do we know if this is indeed a genuine pattern, though? Here's an answer that should occur to us immediately: we can continue to watch how the sequence of numbers behaves, and see if our predictions bear out. If we've succeeded in identifying the pattern underlying the generation of these numbers, then we'll be able to predict what we should see next: we should see a 'zero' followed by a 'one,' and then another 'zero,' and so on. Suppose the pattern continues: S2: 0101100010101 Ah ha! Our prediction does indeed seem to have been born out! That is: in S2, the string of numbers continues to evolve in a way that is consistent with our hypothesis that the sequence at large is (1) not random and (2) is being generated by the pattern R. Of course, this is not enough for us to say with certainty that R (and only R) is the pattern behind the generation of our sequence; it is entirely possible that the next few bits of the string will be inconsistent with R; that is one way that we might come to think that our theory of how the string is being generated is in need of revision. Is this the only way, though? Certainly not: we might also try to obtain information about what numbers came before our initial data-set and see if R holds there, too; if we really have indentified the pattern underlying the generation of S, it seems reasonable to suppose that we ought to be able to "retrodict" the structure of sub-sets of S that come before our initial data-set just as well as we can predict the structure of sub-sets of S that come after our initial data-set. Suppose, for example, that we find that just before our initial set comes the 26 string: S0: 00001000011111 The numbers in this string are not consistent with our hypothesis that all the numbers in the sequence at large are generated by R. Does this mean that we've failed in our goal of identifying a pattern, though? Not necessarily. Why not? There's another important question that we've been glossing over in our discussion here: for a pattern in some data to be genuine must it also be global ? That is, for us to say reasonably that 20 R describes the sequence S, must R describe the sequence S everywhere? Here's all the data we have now: S0-2: 000010000111111100010101100010101100010101 It is clear that we can no longer say that R (or indeed any single pattern at all) is the pattern generating all of S. This is not at all the same thing as saying that we have failed to identify a pattern in S simpliciter, though. Suppose that we have some reason to be particularly interested in what's going on in a restricted region of S: the region S1-2. If that's the case, then the fact that R turns out not to hold for the totality of S might not trouble us at all; identifying a universal pattern would be sufficient for predicting what sequence of numbers will show up in S1-2, but it is by no means necessary. If all we're interested in is predicting the sequence in a particular region of S, identifying a pattern that holds only in that region is no failure at all, but rather precisely 21 20 The sense of 'global' here is the computer scientist's sense-a global pattern is one that holds over the entirety of the data set in question. 21 Of course, it might not be true that R holds only in S1-2. It is consistent with everything we've observed about S so far to suppose that the sub-set S0 and the sub-set S1-2 might be manifestations of an over-arching pattern, of which R is only a kind of component, or sub-pattern. 27 what we set out to do to begin with! It need not trouble us that the pattern we've identified doesn't hold everywhere in S-identifying that pattern (if indeed there is one to be identified) is another project entirely. When we're investigating a sequence like S, then, our project is two-fold: we first pick a region of S about which we want to make predictions, and then attempt to identify a pattern that will let us make those predictions. When we have a candidate pattern, we can apply it to heretofore unobserved segments of our target region and see if the predictions we've made by using the pattern are born out. That is: we first identify a particular way of carving up our target data-set and then (given that carving) see what patterns can be picked out. That any patterns identified by this method will hold (or, better, that we have good reason to think they'll hold) in a particular region only is (to borrow the language of computer programmers) a feature rather than a bug. It's no criticism, in other words, to say that a putative pattern that we've identified relative to a particular carving of our subject-matter holds only for that carving; if our goal is just to make predictions about a restricted region of S, then identifying a pattern that holds only in that region might well make our jobs far easier, for it will give us license to (sensibly) ignore data from outside our restricted region, which might well make our task significantly easier . 22 Let's think about another potentially problematic case. Suppose now that we're given yet another piece of S: S3: 0010100100010 S3 is almost consistent with having been generated by R-only a single digit is off (the bolded 22 For more discussion of approximate pattern and their role in science, see Lawhead (2012) 28 zero ought to be a one if R is to hold)-but still, it seems clear that it is not an instance of the pattern. Still, does this mean that we have failed to identify any useful regularities in S3? I will argue that it most certainly does not mean that, but the point is by no means an obvious one. What's the difference between S3 and S0 such that we can say meaningfully that, in picking out R, we've identified something important about the former but not the latter? To say why, we'll have to be a bit more specific about what counts as a pattern, and what counts as successful identification of a pattern. Following Dennett and Ladyman et. al. , we might begin by thinking of patterns as being 23 24 (at the very least) the kinds of things that are "candidates for pattern recognition. " But what 25 does that mean? Surely we don't want to tie the notion of a pattern to particular observers-whether or not a pattern is in evidence in some dataset (say S3) shouldn't depend on how dull or clever the person looking at the dataset is. We want to say that there at least can be cases where there is in fact a pattern present in some set of data even if no one has yet (or perhaps even ever will) picked it out. As Dennett notes, though, there is a standard way of making these considerations more precise: we can appeal to information theoretic notions of compressibility. A pattern exists in some data if and only if there is some algorithm by which the data can be significantly compressed. This is a bit better, but still somewhat imprecise. What counts as compression? More urgently, what counts as significant compression? Why should we tie our definition of a pattern to those notions? Let's think through these questions using the examples we've been looking at 23 Dennett (1991) 24 Ladyman, Ross, Spurrett, and Collier (2007) 25 Dennett (op. cit.), p. 32, emphasis in the original 29 for the last few pages. Think, to begin with, of the sequence : S1-2: 1100010101100010101100010101 This, recall, was our perfect case for R: the pattern we identified holds perfectly in this data-set. What does it mean to say that R holds perfectly in light of the Dennettian compressibility constraint introduced above, though? Suppose that we wanted to communicate this string of digits to someone else-how might we go about doing that? Well, one way-the easiest way, in a sense-would just be to transmit the string verbatim: to communicate a perfect bit map of the data. That is, for each digit in the string, we can specify whether it is a 'one' or a 'zero,' and then transmit that information (since there are 28 digits in the dataset S1-2, the bit-map of S1-2 is 28 bits long). If the string we're dealing with is truly random then this is (in fact) the only way to transmit its contents : we have to record the state of each bit individually, because (if the string 26 is random) there is no relationship at all between a given bit and the bits around it. Now we're getting somewhere. Part of what it means to have identified a pattern in some data-set, then, is to have (correctly) noticed that there is a relationship between different parts of the data-set under consideration-a relationship that can be exploited to create a more efficient encoding than the simple verbatim bit-map. The sense of 'efficiency' here is a rather intuitive one: an encoding is more efficient just in case it is shorter than the verbatim bit map-just in case it requires fewer bits to transmit the same information. In the case of S1-2, it's pretty easy to see what this sort of encoding would look 26 Citing Chaitin (1975), Dennett (op. cit.) points out that we might actually take this to be the formal definition of a random sequence: there is no way to encode the information that results in a sequence that is shorter than the "verbatim" bit map. 30 like-we specify R, then specify that the string we're passing consists in two iterations of R. Given a suitable way of encoding things, this will be much shorter than the verbatim bit map. For example, we might encode by first specifying a character to stand for the pattern, then specifying the pattern, then specifying the number of times that the pattern iterates. It might look something like this: R:110001010:RRR This string is 15 bits long; in just this simple encoding scheme, we've reduced the number of characters required to transmit S1-2 by almost 50%. That's a very significant efficiency improvement (and, given the right language, we could almost certainly improve on it even further) . 27 This compressibility criterion is offered by Dennett as a necessary condition on patternhood: to be an instance of a (real) pattern, a data-set must admit of a more compact description than the bitmap. However, as a number of other authors have pointed out , this cannot be the whole 28 story; while compressibility is surely a necessary condition on patternhood, it cannot be both necessary and sufficient, at least not if it is to help us do useful work in talking about the world (recall that the ultimate point of this discussion is to articulate what exactly it is that science is doing so that we can see if philosophy has something useful to contribute to the project). Science cannot simply be in the business of finding ways to compress data sets; if that were so, then every new algorithm-every new way of describing something-would count as a new 27 All of this can be made significantly more precise given a more formal discussion of what counts as a "good" compression algorithm. Such a discussion is unnecessary for our current purposes, but we will revisit information theory in significantly more detail in Chapter Two. For now, then, let me issue a promissory note to the effect that there is a good deal more to say on the topic of information-content, compression, and patternhood. See, in particular, Section 2.1.3. 28 Collier (1999) and Ladyman, Ross, Spurrett, and Collier (2007) 31 scientific discovery. This is manifestly not the case; whatever it is that scientists are doing, it is not just a matter of inventing algorithm after algorithm. There's something distinctive about the kinds of patterns that science is after, and about the algorithms that science comes up with. In fact, we've already identified what it is: we've just almost lost sight of it as we've descended into a more technical discussion-science tries to identify patterns that hold not just in existing data, but in unobserved cases (including future and past cases) as well. Science tries to identify patterns that are projectable. How can we articulate this requirement in such a way that it meshes with the discussion we've been having thus far? Think, to begin, of our hypothetical recipient of information once again. We want to transmit the contents of S1-2 to a third party. However, suppose that (as is almost always the case) our transmission technology is imperfect-that we have reason to expect a certain degree of signal degradation or information loss in the course of the transmission. This is the case with all transmission protocols available to us; in the course of our transmission, it is virtually inevitable that a certain amount of noise (in the information-theoretic sense of the dual of signal) will be introduced in the course of our message traveling between us and our interlocutor. How can we deal with this? Suppose we transmit the bitmap of S1-2 and our recipient receives the following sequence: S1-2: 1100010101100010101100??0?01 Some of the bits have been lost in transmission, and now appear as question marks-our interlocutor just isn't sure if he's received a one or a zero in those places. How can he correct for this? Well, suppose that he also knows that S1-2 was generated by R. That is, suppose that we've 32 also transmitted our compressed version of S1-2. If that's the case, then our interlocutor can, by following along with R, reconstruct the missing data and fill in the gaps in his signal. This, of course, requires more transmission overall-we have to transmit the bitmap and the pattern-encoding-but in some cases, this might well be worth the cost (for instance, in cases where there is a tremendous amount of latency between signal transmission and signal reception, so asking to have specific digits repeated is prohibitively difficult). This is in fact very close to how the Transmission-Control Protocol (TCP) works to ensure that the vast amount of data being pushed from computer to computer over the Internet reaches its destination intact. Ok, but how does this bear on our problem? Next, consider the blanks in the information our interlocutor receives not as errors or miscommunication, but simply as unobserved cases. What our interlocutor has, in this case, is a partial record of S1-2; just as before, he's missing some of the bits, but rather than resulting from an error in communication, this time we can attribute the information deficit to the fact that he simply hasn't yet looked at the missing cases. Again, we can construct a similar solution-if he knows R, then just by looking at the bits he does have, then our interlocutor can make a reasonable guess as to what the values of his unobserved bits might be. It's worth pointing out here that, given enough observed cases, our interlocutor need not have learned of R independently: he might well be able to deduce that it is the pattern underlying the data points he has, and then use that deduction to generate an educated guess about the value of missing bits. If an observer is clever, then, he can use a series of measurements on part of his data-set to ground a guess about a pattern that holds in that data set, and then use that pattern to ground a guess about the values of unmeasured parts of the data set. At last, then, we're in a position to say what it is that separates S3 from S0 such that it is 33 reasonable for us to say that R is informative in the former case but not in the latter, despite the fact that neither string is consistent with the hypothesis that R is the pattern underlying its generation. The intuitive way to put the point is to say that R holds approximately in the case of S3 but not in the case of S0, but we can do better than that now: given R, and a restricted set of S3, an observer who is asked to guess the value of some other part of the set will do far better than we'd expect him to if R was totally uninformative-that is, he will be able to make predictions about S3 which, more often than not, turn out to be good ones. In virtue of knowing R, and by measuring the values in one sub-set of S3, he can make highly successful predictions about how other value measurements in the set will turn out. The fact that he will also get things wrong occasionally should not be too troubling; while he'd certainly want to work to identify the exceptions to R-the places in the sequence where R doesn't hold-just picking out R goes a very long way toward sustained predictive success. Contrast that case to the case in S0: here, knowledge of R won't help an observer make any deductions about values of unobserved bits. He can learn as much as he wants to about the values of bits before and after a missing bit and he won't be any closer at all to being able to make an educated guess about the missing data. 1.4 Fundamental Physics and the Special Sciences It might be worth taking a moment to summarize the rather lengthy discussion from the last section before we move on to considering how that discussion bears on the larger issue at hand. We started by observing that science is "about the world" in a very particular sense. In exploring what that might mean, I argued that science is principally concerned with identifying patterns in how the world around us changes over time . We then spent some time examining some basic 29 29 A similar view of scientific laws is given in Maudlin (2007). Maudlin argues that scientific laws are best understood 34 concepts in information theory, and noted that many of the insights in the philosophy of information theory first articulated by Dennett (1991) and later elaborated by other authors fit rather neatly with a picture of science as the study of patterns in the world. We looked at a few problem cases in pattern identification-including patterns that hold only approximately, and data-sets with partial information loss-and argued that even in cases like that, useful information can be gleaned from a close search for patterns; patterns neither need to be universal nor perfect in order to be informative. We tried to give an intuitive picture of what we might mean when we say that science looks for patterns that can be projected to unobserved cases. I'd like to now drop the abstraction from the discussion and make the implicit parallel with science that's been lurking in the background of this discussion explicit. We should be able to draw on the machinery from Section 1.3 to make our earlier discussion of science more concrete, and to examine specific cases of how this model actually applies to live science. Here's the picture that I have in mind. Scientists are in the business of studying patterns in how the world changes over time. The method for identifying patterns varies from branch to branch of science; the special sciences differ in domain both from each other and from fundamental physics. In all cases, though, scientists proceed by making measurements of certain parts of the world, trying to identify patterns underlying those measurements, and then using those patterns to try to predict how unobserved cases-either future measurements or as what he calls LOTEs-"laws of temporal evolution." This is largely consistent with the picture I have been arguing for here, and (not coincidentally) Maudlin agrees that an analysis of scientific laws should "take actual scientific practice as its starting point" (p. 10), rather than beginning with an a priori conception of the form that a law must take. Our point of departure from Maudlin's view, as we shall see, lies in our treatment of fundamental physics. While Maudlin wants to distinguish "FLOTEs" (fundamental laws of temporal evolution) from normal LOTEs on the basis of some claim of "ontological primacy" (p. 13) for fundamental physics, the view I am sketching here requires no such militantly reductionist metaphysics. My view is intended to be a description of what working scientific laws actually consist in, not a pronouncement on any underlying metaphysics. 35 measurements in a novel spatial location-might turn out. Occasionally, they get a chance to compare those predictions to observed data directly. This is more common in some branches of science than in others: it is far more difficult to verify some of the predictions of evolutionary biology (say, speciation events) by observation than it is to verify some of the predictions of quantum mechanics (say, what state our measurement devices will end up in after a Stern-Gerlach experiment). More frequently, they are able to identify a number of different patterns whose predictions seem either agree or disagree with one another. Evolutionary biology is a well-confirmed science in large part not because large numbers of speciation events have been directly observed, but because the predictions from other sciences with related domains (e.g. molecular biology)-many of which have been confirmed through observation-are consistent with the predictions generated by evolutionary biologists. Just as in the case of our toy science in Section 1.3, it seems to me that science generally consists in two separate (but related) tasks: scientists identify a domain of inquiry by picking out a way of carving up the world, and then identify the patterns that obtain given that way of carving things up. This is where the careful discussion from Section 1.3 should be illuminating: not all scientists are interested in identifying patterns that obtain everywhere in the universe-that is, not all scientists are interested in identifying patterns that obtain for all of S. Indeed, this is precisely the sense in which fundamental physics is fundamental: it alone among the sciences is concerned with identifying the patterns that will obtain no matter where in the world we choose to take our measurements. The patterns that fundamental physics seeks to identify are patterns that will let us predict the behavior of absolutely any sub-set of the world-no matter how large, small, or oddly disjunctive-at which we choose to look; it strives 36 to identify patterns that describe the behavior of tiny regions of space-time in distant galaxies, the behavior of the interior of the sun, and the behavior of the Queen of England's left foot. This is a fantastically important project, but it is by no means the only scientific project worth pursuing . The special sciences are all, to one degree or another, concerned with identifying 30 patterns that hold only in sub-sets of the domain studied by physics. This is not to say that the special sciences all reduce to physics or that they're all somehow parasitic on the patterns identified by fundamental physics. While I want to avoid engaging with these metaphysical questions as much as possible, it's important to forestall that interpretation of what I'm saying here. The special sciences are, on this view, emphatically not second-class citizens-they are just as legitimate as fields of inquiry as is fundamental physics. Again (and contra Maudlin), the sense of "fundamental" in "fundamental physics" should not be taken to connote anything like ontological primacy or a metaphysically privileged position (whatever that might mean) within the general scientific project. Rather (to reiterate) it is just an indicator of the fact that fundamental physics is the most general part of the scientific project; it is the branch of science that is concerned with patterns that show up everywhere in the world. When we say that other sciences are concerned with restricted sub-sets of the physical world, we just mean that they're concerned with picking out patterns in some of the systems to which the generalizations of fundamental physics apply . 31 30 It is worth pointing out that it is indeed possible that there just are no such patterns in the world: it is possible that all laws are, to a greater or lesser extent, parochial. If that were true, then it would turn out that the goal underlying the practice of fundamental physics was a bad one-there just are no universal patterns to be had. Because of this possibility, the unity of science is an hypothesis to be empirically confirmed or disconfirmed. Still, even its disconfirmation might not be as much of a disaster as it seems: the patterns identified in the course of this search would remain legitimate patterns, and the discovery that all patterns are to some extent parochial would itself be incredibly informative. Many advances are made accidentally in the course of pursuing a goal that, in the end, turns out to not be achievable. 31 Ladyman, Ross, Spurrett, and Collier (2007) put the point slightly differently, arguing that fundamental physics is 37 In contrast to fundamental physics, consider the project being pursued by one of the special sciences-say, molecular biology. Molecular biologists are certainly not interested in identifying patterns that hold everywhere in the universe; biologists have relatively little to say about what happens inside the sun (except perhaps to note that the conditions would make it difficult for life to prosper there). They are, instead, concerned with the behavior of a relatively small sub-set of regions of the universe. So far, the patterns they've identified have been observed to hold only on some parts of Earth, and that only in the last few billion years. It's 32 clearly no criticism of molecular biology to point out that it has nothing to say on the subject of what happens inside a black hole-that kind of system is (by design) outside molecular biology's domain of interest. Just as in the case of S1-2 above, this restriction of domain lets molecular biologists focus their efforts on identifying patterns that, while they aren't universal, facilitate predictions about how a very large class of physical systems behave. What exactly is the domain of inquiry with which molecular biology is concerned? That is, how do molecular biologists carve up the world so that the patterns they identify hold of systems included in that carving? It is rather unusual (to put it mildly) for the creation of a domain in this sense to be a rapid, deliberate act on the part of working scientists. It is unusual, that is, for a group of people to sit down around a table (metaphorical or otherwise), pick out a heretofore fundamental in the sense that it stands in an asymmetric relationship to the rest of science: generalizations of the special sciences are not allowed to contradict the generalizations of fundamental physics, but the reverse is not true; if the fundamental physicists and the biologists disagree, it is the biologist who likely has done something wrong. They call this the "Primacy of Physics Constraint" (PPC). It seems to me that while this is certainly true-that is, that it's certainly right that the PPC is a background assumption in the scientific project-the way I've put the point here makes it clear why the PPC holds. 32 It's worth noting, though, that the search for habitable planets outside our own solar system is guided by the patterns identified by biologists studying certain systems here on Earth. This is an excellent case of an application of the kind of projectability we discussed above: biologists try to predict what planets are likely to support systems that are relevantly similar to the systems they study on Earth based on patterns they've identified in those terrestrial systems. It remains to be seen whether or not this project will prove fruitful. 38 unexplored part of the world for empirical inquiry, and baptize a new special science to undertake that inquiry. Rather, new sciences seem most often to grow out of gaps in the understanding of old sciences. Molecular biology is an excellent illustration here; the isolation of DNA in 1869-and the subsequent identification of it as the molecule responsible for the heritability of many phenotypic traits-led to an explosion of new scientific problems: what is the structure of this molecule? How does it replicate itself? How exactly does it facilitate protein synthesis? How can it be damaged? Can that damage be repaired? Molecular biology is, broadly speaking, the science that deals with these questions and the questions that grew out of them-the science that seeks to articulate the patterns in how the chemical bases for living 33 systems behave. This might seem unsatisfactory, but it seems that it is the best answer we're likely to get: molecular biology, like the rest of science, is a work-in-progress, and is constantly refining its methodology and set of questions, both in light of its own successes (and failures) and in light of the progress in other branches of the scientific project. Science is (so to speak) alive. This is an important point, and I think it is worth emphasizing. Science grows up organically as it attempts to solve certain problems-to fill in certain gaps in our knowledge about how the world changes with time-and is almost never centrally planned or directed. Scientists do the best they can with the tools they have, though they constantly seek to improve those tools. The fact that we cannot give a principled answer to the question "what parts of the world does molecular biology study?" should be no bar to our taking the patterns identified by molecular biology seriously. Just as we could not be sure that R, once identified, would hold in any 33 This includes not just the bases in the technical sense-nucleic acids-but also other chemical foundations that are necessary for life (e.g. proteins). 39 particular segment of S that we might examine, we cannot be sure of precisely what regions of the world will behave in ways that are consistent with the patterns identified by molecular biologists. This is not to say, though, that the molecular biologists have failed to give us any interesting information-as we saw, universality (or even a rigidly defined domain of applicability) is no condition on predictive utility. To put the point one more way: though the special sciences are differentiated from one another in part by their domains of inquiry, giving an exhaustive account of exactly what parts of the world do and don't fall into the domain of a particular science is likely an impossible task. Even if it were not, it isn't clear what it would add to our understand of either a particular science or of science as a whole: the patterns identified by molecular biology are no less important for our not knowing if they do or don't apply to things other than some of the systems on Earth in the last few billion years; if molecular biology is forced to confront the problem of how to characterize extraterrestrial living systems, it is certainly plausible to suppose that its list of patterns will be revised, or even that an entirely new science will emerge from the realization that molecular biology as thus far conceived is parochial in the extreme. Speculating about what those changes would look like-or what this new special science would take as its domain-though, is of little real importance (except insofar as such speculation illuminates the current state of molecular biology). Like the rest of the sciences, molecular biology takes its problems as they come, and does what it can with the resources it has. If we can't say for any given special science what exactly its domain is, then, perhaps we can say a bit more about what the choice of a domain consists in-that is, what practical activities of working scientists constitute a choice of domain? How do we know when a formerly singular 40 science has diverged into two? Perhaps the most important choice characterizing a particular science's domain is the choice of what measurements to make, and on what parts of the world. That is: the choice of a domain is largely constituted by the choice to treat certain parts of the world as individuals, and the choice of what measurements to make on those individuals. Something that is treated as an individual by one special science might well be treated as a composite system by another ; the distinction between how human brains are treated by 34 cognitive psychology (i.e. as the primary objects of prediction) and how they're treated by neurobiology (i.e. as aggregates of individual neural cells) provides an excellent illustration of this point. From the perspective of cognitive psychology, the brain is an unanalyzed individual object-cognitive psychologists are primarily concerned with making measurements that let them discern patterns that become salient when particular chunks of the physical world (that is: brain-containing chunks) are taken to be individual objects. From the perspective of neurobiology, on the other hand, brains are emphatically not unanalyzed objects, but are rather composites of neural cells-neurobiologists make measurements that are designed to discern patterns in how chunks of the physical world consisting of neural cells (or clusters of neural cells) evolve over time. From yet another perspective-that of, say, population genetics-neither of these systems might be taken to be an individual; while a population geneticist might well be interested in brain-containing systems, she will take something like alleles to be her primary objects, and will discern patterns in the evolution of systems from that perspective. We should resist the temptation to become embroiled in an argument about which (if any) of 34 We'll explore this point in much more depth in Chapter Two. 41 these individuals are real individuals in a deep metaphysical sense. While it is certainly right to point out that one and the same physical system can be considered either as a brain (qua individual) or a collection of neurons (qua aggregate), this observation need not lead us to wonder which of these ways of looking at things (if either) is the right one. Some patterns are easier to discern from the former perspective, while others are easier to discern from the latter. For the purposes of what we're concerned with here, it seems to me, we can stop with that fact-there is no need to delve more deeply into metaphysical questions. Insofar as I am taking any position at all on questions of ontology, it is one that is loosely akin to Don Ross' "rainforest realism: " a systematized version of Dennett's "stance" stance toward ontology. Ross' picture, 35 like the one I have presented here, depicts a scientific project that is unified by goal and subject matter, though not necessarily by methodology or apparatus. It is one on which we are allowed to be frankly instrumentalist in our choice of objects-our choice of individuals-but still able to be thoroughly realists about the relations that hold between those objects-the patterns in how the objects change over time. This metaphysical position is a natural extension of the account of science that I have given here, and one about which much remains to be said. To engage deeply with it would take us too far afield into metaphysics of science, though; let us, then, keep our eye on the ball, and content ourselves with observing that there is at least the potential for a broad metaphysical position based on this pragmatically-motivated account of science. Articulating that position, though, must remain a project for another time. 1.5 Summary and Conclusion: Exorcising Feynman's Ghost The story of science is a story of progress through collaboration: progress toward a more 35 See Ross (2000) and Chapter Four of Ladyman et. al. (2007), as well as Dennett (1991) 42 complete account of the patterns in how the world evolves over time via collaboration between different branches of science, which consider different ways of carving up the same world. Individual sciences are concerned with identifying patterns that obtain in certain subsets of the world, while the scientific project in general is concerned with the overarching goal of pattern-based prediction of the world's behavior. Success or failure in this project is not absolute; rather, the identification of parochial or "weak" patterns can often be just as useful (if not more useful) as the identification of universal patterns. Scientists identify patterns both by making novel measurements on accessible regions of the world and by creating models that attempt to accurately retrodict past measurements. The scientific project is unified in the sense that all branches of science are concerned with the goal of identifying patterns in how the physical world changes over time, and fundamental physics is fundamental in the sense that it is the most general of the sciences-it is the one concerned with identifying patterns that will generate accurate predictions for any and all regions of the world that we choose to consider. Patterns discovered in one branch of the scientific project might inform work in another branch, and (at least occasionally) entirely novel problems will precipitate a novel way of carving up the world, potentially facilitating the discovery of novel patterns; a new special science is born. We might synthesize the discussions in Section 1.3 and Section 1.4 as follows. Consider the configuration space D of some system T-say, the phase space corresponding to the kitchen in 36 36 That is, consider the abstract space in which every degree of freedom in T is represented as a dimension in a particular space D (allowing us to represent the complete state of T at any given time by specifying a single point in D), and where the evolution of T can be represented as a set of transformations in D. The phase space of classical statistical mechanics (which has a dimensionality equal to six times the number of classical particles in the system), the Hilbert space of standard non-relativistic quantum mechanics, and the Fock space of quantum field theory (which is the direct sum of the tensor products of standard quantum mechanical Hilbert spaces) are all prime examples of spaces of this sort, but are by no means the only ones. Though I will couch the discussion in terms of phase space for the sake of concreteness, this is not strictly necessary: the point I am trying to make is abstract enough that it should stand for any of these cases. 43 my apartment. Suppose (counterfactually) that we take Newtonian dynamics to be the complete fundamental physics for systems like this one. If that is the case, then fundamental physics provides a set of directions for moving from any point in the phase space to any other point-it provides a map identifying where in the space a system whose state is represented by some point at t0 will end up at a later time t1. This map is interesting largely in virtue of being valid for any point in the system: no matter where the system starts at t0, fundamental physics will describe the pattern in how it evolves. That is, given a list of points [a0,b0,c0,d0...z0], the fundamental physics give us a corresponding list of points [a1,b1,c1,d1...z1] that the system will occupy after a given time interval has passed. In the language of Section 1.3, we can say that fundamental physics provides a description of the patterns in the time-evolution of the room's bit map: given a complete specification of the room's state (in terms of its precise location in phase space) at one time, applying the algorithm of Newtonian mechanics will yield a complete specification of the room's state at a later time (in terms of another point in phase space). This is surely a valuable tool, but it is equally surely not the only valuable tool. It might be (and, in fact, is) the case that there are also patterns to be discerned in how certain regions of the phase space evolve over time. That is, we might be able to describe patterns of the following sort: if the room starts off in any point in region P0, it will, after a given interval of time, end up in another region P1. This is, in fact, the form of the statistical-mechanical explanation for the Second Law of Thermodynamics. This is clearly not a description of a pattern that applies to the "bit map" in general: there might be a very large number (perhaps even a continuous infinity) of points that do not lie inside P0, and for which the pattern just described thus just has nothing to say. This is not necessarily to say that the project of identifying patterns like P0 P1 isn't one 44 that should be pursued, though. Suppose the generalization identified looks like this: if the room is in a region corresponding to "the kitchen contains a pot of boiling water and a normal human being who sincerely intends to put his hand in the pot " at t0, then evolving the system (say) 10 37 seconds forward will result in the room's being in a region corresponding to "the kitchen contains a pot of boiling water and a human being in great pain and with blistering skin." Identifying these sorts of patterns is the business of the special sciences. Not all regions will admit of interesting patterns in this way. This is the sense in which some ways of "carving up" a system's space seem arbitrary in an important way. In a system with a relatively high degree of complexity-very roughly, a system with a relatively high-dimensional configuration space -there will be a very large number of ways of specifying regions such that 38 we won't be able to identify any interesting patterns in how those regions behave over time. This is the sense in which some objects and properties seem arbitrary in problematic ways: carvings corresponding to (for example) grue-like properties (or bizarre compound objects like "the conjunction of the Queen of England's left foot and all pennies minted after 1982") just don't support very many interesting patterns. Regions picked out by locutions like that don't behave in ways that are regular enough to make them interesting targets of study. Even in cases like this, though, the patterns identified by fundamental physics will remain reliable: this (again) is the sense in which fundamental physics is fundamental. The behavior of even arbitrarily-specified regions-regions that don't admit of any parochial patterns-will be 37 We can think of the "sincerely intends to put his hand in the pot" as being an assertion about location of the system when its state is projected onto a lower-dimensional subspace consisting of the configuration space of the person's brain. Again, this location will (obviously) be a regional rather than precise one: there are a large number of points in this lower-dimensional space corresponding to the kind of intention we have in mind here. 38 This is only a very rough gesture at a definition of complexity, but we're not yet in a position to do better than this. For a more precise discussion of the nature (and significance) of complexity, see Section 2.2. 45 predictable by an appeal to the bit-map level patterns of fundamental physics. More precisely, then, the business of a particular special sciences consists in identifying certain regions of a system's configuration space as instantiating enough interesting patterns to be worth considering, and then trying to enumerate those patterns as carefully as possible. A new special science emerges when someone notices that there exist patterns in the time-evolution of regions which have heretofore gone unnoticed. The borders of the regions picked out by the 39 special sciences will be vaguely-defined; if the special scientists were required to give a complete enumeration of all the points contained in a particular region (say, all the possible configurations corresponding to "normal human observer with the intention to stick his hand in the pot of boiling water"), then the usefulness of picking out patterns of those regions would be greatly reduced. To put the point another way, there's a very real sense in which the vagueness of the carvings used by particular sciences is (to borrow from computer science yet again) a feature rather than a bug: it lets us make reliable predictions about the time-evolution of a wide class of systems while also ignoring a lot of detail about the precise state of those systems. The vagueness might lead us to occasionally make erroneous predictions about the behavior of a system, but (as I argued in Section 1.3) this is not at all a fatal criticism of a putative pattern. The progress of a particular special science consists largely in attempts to make the boundaries of its class of carvings as precise as possible, but this notion of progress need not entail that the ultimate goal of any special science is a set of perfectly defined regions. To be a pattern is not necessarily to be a perfect pattern, and (just as with compression algorithms in information theory) we might be happy to trade a small amount of error for a large gain in utility. The 39 It might be appropriate to remind ourselves here that the regions under discussion here are regions of configuration space, not space-time. 46 scientific project consists in the identification of as many of these useful region/pattern pairings as possible, and individual sciences aim at careful identification of patterns in the evolution of particular regions . 40 With this understanding of science (and the scientific project more generally) in hand, then, we can return to the question we posed near the beginning of this chapter: how are we to respond to the spirit of Richard Feynman? What's a philosopher to say in his own defense? What do we bring to the scientific table? It should be clear from what we've said thus far that philosophy is not, strictly speaking a science; philosophy (with a very few exceptions) does not seek to make measurements of the world around us , use those measurements to identify patterns in that 41 world, and construct models under which those patterns are projected to future unobserved cases. That is, philosophy is not a science in the way that chemistry, biology, economics, climate science, or (a fortiori) fundamental physics are sciences; there is no set of configuration-space carvings with which philosophy is concerned. However, this does not mean that philosophy is not a part of Science in the sense of contributing to the overall scientific project. How does that relationship work? An analogy might help here. Consider the relationship between commercial airline pilots and the air-traffic controllers working at major metropolitan airports around the world. The kind of specialized knowledge required to operate (say) a Boeing 747 safely-as 40 There will often be overlap between the regions studied by one science and the regions studied by another. The "human with his hand in a pot of boiling water" sort of system will admit of patterns from (for example) the perspectives of biology, psychology, and chemistry. That is to say that this sort of system is one that is in a region whose behavior can be predicted by the regularities identified by all of these special sciences, despite the fact that the unique carvings of biology, psychology, and chemistry will be regions with very different shapes. Systems like this one sit in regions whose time-evolution is particularly rich in interesting patterns. 41 Of course, this is not to dismiss experimental philosophy as a legitimate discipline. Rather, on the view that I am advocating here, traditional experimental philosophy would count as a special science (in the sense described above) in its own right-a special science with deep methodological, historical, and conceptual ties to philosophy proper, but one which is well and truly its own project. 47 well as the rather restricted vantage point from which an individual pilot can view the airspace surrounding a port-of-call-leaves little room for coordination between planes themselves. While some communication is present between pilots, most of the direction comes from the ground-from people who, though they lack the incredibly technical know-how required to fly any one of the planes they support, fulfill a vital role, both in virtue of their position as outsiders with (so to speak) a bird's eye view on the complicated and fast-paced project of moving people in and out of cities via air travel and in virtue of their specialized training as managers and optimizers. Philosophers, I suggest, play a role similar to that of air traffic controllers while scientists play the role of pilots: while it is the pilots who are directly responsible for the success or failure of the project, their job can be (and is) made significantly easier with competent support and direction from the ground. The air traffic controllers cooperate with the pilots to further a shared goal: the goal of moving people about safely. Likewise, philosophers cooperate with scientists to further a shared goal: the goal of identifying genuine projectable patterns in the world around us. If this example strikes you as over inflating the philosophers' importance-who are we to think of ourselves as controlling anything?-then consider a related case. Consider the relationship between highway transportation qua vehicles and highway transportation qua broad system of technology-a technology in the fourth and last of the senses distinguished by Kline . 42 Think of the system of highway system in the United States : while the vehicles-cars, 43 trucks, motorcycles, bicycles, and so on-are in some sense the central components of the highway system (without vehicles of some sort, there would be no system to speak of at all), they 42 Kline (1985) 43 I owe this example to conversation with my friend and colleague Daniel Estrada. 48 by no means exhaust the vital components of the system. The highway system as a whole consists of a highly designed, standardized, well-maintained, incredibly diverse set of objects and practices that are just as essential for the smooth transportation of the people using the system as are the vehicles that traverse it: the traffic lights, the signs, the rest stops, the paint on the road, the safety-rails, the traffic cones, and so on are as vital as the cars themselves. Even more saliently for the purposes of our discussion, consider all the knowledge that went into conceptualizing, constructing, and maintaining that system, and of the skills and knowledge that must be imparted to each driver before he or she is competent to control a ton of metal and plastic moving at 75 miles per hour: these skills (and the tens of thousands of man-hours behind their conceptualization and implementation) are likewise essential. Think of the actual production and maintenance of those roads, the hundreds of thousands of miles of concrete, construction, and cleanup- as well as the hours of political negotiations and legal regulations and labor disputes that sit behind every mile of that road. Only through the smooth operation of this system as a whole is actual use of the road-the sitting behind the wheel, listening to terrible music, with only some destination in mind-made possible. If the previous comparison of philosophers to air-traffic controllers seems to elevate philosophy beyond its rightful station, then we might take comfort in the fact that, though we might play the role of the lowly dotted yellow line, this role is still deeply essential to the functioning of the whole. Philosophers are not scientists in just the same way that dotted yellow lines are not cars, or that air-traffic controllers are not pilots, or that traffic engineers are not commuters trying to get to work on time. Like our transportation analogues, though, philosophers have a vital role to play in the scientific project as a whole: a role of coordination, 49 general analysis, optimization, and clarification. We are suited to play this role precisely in virtue of not being scientists: we are uniquely suited (to both carry the transportation theme and echo a famous metaphor of Wilfred Sellars') "build bridges" between the activities of individual scientists, and between different branches of the scientific project as a whole. Philosophers are trained to clarify foundational assumptions, note structural similarities between arguments (and problems) that at first glance could not seem more disparate, and to construct arguments with a keen eye for rigor. These skills, while not necessarily part of the scientist's tool-kit, are vital to the success of the scientific project as a whole: if we're to succeed in our goal of cataloging the interesting patterns in the world around us, we need more than just people directly looking for those patterns. We might take this as a special case of Bruno Latour's observation that "the more non-humans share existence with humans, the more humane a collective is, " and note that the 44 more non-scientists share in the scientific project, the more scientific the project becomes. Now, let us turn to that project in earnest. 44 Latour (1999) 50 Chapter Two What's the Significance of Complexity? 2.0 Introduction and Overview In Chapter One, I presented a general theory about the nature of the scientific project, and argued that this general theory suggests a natural way of thinking about the relationship between (and underlying unity of) the different branches of science. This way of looking at science is instructive but (as I said), doing abstract philosophy of science is not really my goal here. Eventually, we will need to turn to consider climate science specifically and examine the special problems faced by those studying the Earth's climate system. Before we get down into the nitty-gritty concrete details, though, we'll need a few more theoretical tools. Here's how this chapter will go. In 2.1 I will introduce a distinction between "complex systems" sciences and "simple systems" sciences, and show how that distinction very naturally falls out of the account of science offered in Chapter One. I will draw a distinction between "complex" and "complicated," and explore what it is that makes a particular system complex or simple. We'll think about why the distinction between complex and simple systems is a useful one, and discuss some attempts by others to make the notion of complexity precise. In 2.2, we will attempt to construct our own definition using the framework from the last chapter. Finally, in 2.3, I'll set up the discussion to come in Chapter Three, and suggest that climate science is a paradigmatic 51 complex systems science, and that recognizing that fact is essential if we're to make progress as rapidly as we need to. More specifically, I'll argue that the parallels between climate science and other complex systems sciences-particularly economics-have been largely overlooked, and that this oversight is primarily a result of the tradition of dividing the sciences into physical and social sciences. This division, while useful, has limitations, and (at least in this case) can obfuscate important parallels between different branches of the scientific project. The complex/simple systems distinction cuts across the physical/social science distinction, and serves to highlight some important lessons that climate science could learn from the successes (and failures) of other complex systems sciences. This is the second (and last) chapter that will be primarily philosophical in character; with the last of our conceptual tool-kit assembled here, we'll be ready to move on to a far more concrete discussion in Chapter Three and beyond. 2.1 What is "Complexity?" Before we can actually engage with complex systems theories (and bring those theories to bear in exploring the foundations of climate science), we'll need to articulate what exactly makes a system complex, and examine the structure of complex systems theories generally. Just as in Chapter One, my focus here will be primarily on exploring the actual practice of contemporary, working science: I'm interested in what climate scientists, economists, and statistical physicists (as well as others working in the branches of science primarily concerned with predicting the behavior of complex systems) can learn from one another, rather than in giving a priori pronouncements on the structure of these branches of science. With that goal in mind, we will anchor our discussion with examples drawn from contemporary scientific theories whenever possible, though a certain amount of purely abstract theorizing is unavoidable. Let's get that over 52 with as quickly as possible. It is important, first of all, to forestall the conflation of "complex/simple" and "complicated/simplistic." All science is (to put the point mildly) difficult, and no branch of contemporary science is simplistic in the sense of being facile, superficial, or easy. In opposing complex systems to simple systems, then, I am not claiming that some branches of science are "hard" and some are "soft" in virtue of being more or less rigorous-indeed, the hard/soft science distinction (which roughly parallels the physical/social science distinction, at least most of the time) is precisely the conceptual carving that I'm suggesting we ought to move beyond. There are no simplistic sciences: all science is complicated in the sense of being difficult, multi-faceted, and messy. Similarly, there are no simplistic systems in nature; no matter how we choose to carve up the world, the result is a set of systems that are decidedly complicated (and thank goodness for this: the world would be incredibly boring otherwise!). This point should be clear from our discussion in Chapter One. If all systems are complicated, then, what makes one system a complex system, and another a simple system? This is not an easy question to answer, and an entirely new academic field-complex systems theory-has grown up around attempts to do so. Despite the centrality of the concept, there's no agreed-upon definition of complexity in the complex systems theory literature. We'll look at a few different suggestions that seem natural (and suggest why they might not be entirely satisfactory) before building our own, but let's start by trying to get an intuitive grasp on the concept. As before, we'll tighten up that intuitive account as we go along; if all goes well, we'll construct a natural definition of complexity piece by piece. 53 Rather than trying to go for a solid demarcation between complex and simple systems immediately, it might be easier to start by comparing systems. Here are some comparisons that seem intuitively true : a dog's brain is more complex than an ant's brain, and a human's brain is 45 more complex still. The Earth's ecosystem is complex, and rapidly became significantly more complex during and after the Cambrian explosion 550 million years ago. The Internet as it exists today is more complex than ARPANET-the Internet's progenitor-was when it was first constructed. A Mozart violin concerto is more complex than a folk tune like "Twinkle, Twinkle, Little Star." The shape of Ireland's coastline is more complex than the shape described by the equation x2 + y2 = 1. The economy of the United States in 2011 is more complex than the economy of pre-Industrial Europe. All these cases are (hopefully) relatively uncontroversial. What quantity is actually being tracked here, though? Is it the same quantity in all these cases? That is, is the sense in which a human brain is more complex than an ant brain the same sense in which a Mozart concerto is more complex than a folk tune? One way or another, what's the significance of the answer to that question-if there's an analogous sense of complexity behind all these cases (and I shall argue that there is, at least in most cases), what does that mean for the practice of science? What can we learn by looking at disparate examples of complex systems? Let's look at a few different ways that we might try to make this notion more precise. We'll start with the most naïve and intuitive paths, and work our way up from there . Once we have a few 46 45 I'm going to rely quite heavily on our intuitive judgments of complexity in this chapter; in particular, I'll argue that some of the definitions we consider later on are insufficient because they fail to accord with our intuitive judgments about what counts as a complex system. Since constructing a more rigorous definition is precisely what we're trying to do here, this doesn't seem like much of a problem. We've got to start somewhere. 46 For an even more exhaustive survey of different attempts to quantify "complexity" in the existing literature, see Chapter 7 of Mitchell (2009). We will not survey every such proposal here, but rather will focus our attention on a few of the leading contenders-both the most intuitive proposals and the proposals that seem to have gotten the most mileage-before offering a novel account of complexity that attempts to synthesize these contenders. 54 proposals on the table, we'll see if there's a way to synthesize them such that we preserve the strengths of each attempt while avoiding as many of their weaknesses as possible. 2.1.1 Complexity as Mereological Size One simple measure tends to occur to almost everyone when confronted with this problem for the first time: perhaps complexity is a measure of the number of independent parts that a system has-a value that we might call "mereological size." This accords rather well with complexity in the ordinary sense of the word: an intricate piece of clockwork is complex largely in virtue of having a massive number of interlocking parts-gears, cogs, wheels, springs, and so on-that account for its functioning. Similarly, we might think that humans are complex in virtue of having a very large number of "interlocking parts" that are responsible for our functioning in the way we do -we have a lot more genes than (say) the yeast microorganism . Something like 47 48 this definition is explicitly embraced by, for example, Michael Strevens: "A complex system, then, is a system of many somewhat autonomous, but strongly interacting parts ." Similarly, 49 Lynn Kiesling says, "Technically speaking, what is a complex system? It's a system or arrangement of many component parts, and those parts interact. These interactions generate outcomes that you could not necessarily have predicted in advance. " 50 There are a few reasons to be suspicious of this proposal, though. Perhaps primarily, it will 47 It's interesting to point out that this is precisely the intuition that many proponents of the "intelligent design" explanation for biological complexity want to press on. See, canonically, Paley (1802). 48 Even still, the amount of information encoded in the human genome is shockingly small by today's storage standards: the Human Genome Project has found that there are about 2.9 billion base-pairs in the human genome. If every base-pair can be coded with two bits, this corresponds to about 691.4 megabytes of data. Moreover, Christley et. al. (2009) point out that since individual genomes vary by less than 1% from each other, they can be losslessly compressed to roughly 4 megabytes. To put that in perspective, even a relatively cheap modern smartphone has about 16 gigabytes of memory-enough to store almost 5,000 complete human genomes. 49 Strevens (2003), p. 7 50 Kiesling (2011) 55 turn the question "how complex is this system?" into a question that's only answerable by making reference to what the system is made out of. This might not be a fatal issue per se, but it suggests that measuring complexity is an insurmountably relativist project-after all, how are we to know exactly which parts we ought to count to define the complexity of a system? Why, that is, did we choose to measure the complexity of the human organism by the number of genes we have? Why not cells (in which case the blue whale would beat us handily), or even atoms (in which case even the smallest star would be orders of magnitude more complex than even the most corpulent human)? Relatedly, how are we to make comparisons across what (intuitively) seem like different kinds of systems? If we've identified the gene as the relevant unit for living things, for instance, how can we say something like "humans are more complex than cast-iron skillets, but less complex than global economies ?" 51 Even if we waive that problem, though, the situation doesn't look too good for the mereological size measure. While it's certainly true that a human being has more nucleotide base pairs in his DNA than a yeast microbe, it's also true that we have far fewer base pairs than most amphibians, and fewer still than many members of the plant kingdom (which tend to have strikingly long genomes) . That's a big problem, assuming we want to count ourselves as more 52 51 Whether or not these comparisons are accurate is another matter entirely. That is, whether you think it's actually true to say that humans are less complex than the 21st century global economy, it seems clear that the comparison is at least sensible. Or, at least, it seems clear that it ought to be sensible if we're to succeed in our goal of finding a notion of "complexity" that is widely-applicable enough to be useful. I'll argue in 2.2 that there is sense to the comparison and (moreover) that the global economy is more complex than an individual human. For now, though, it's enough to point out that even having that discussion presupposes a wide notion of complexity that renders the mereological size measure suspect. 52 Most amphibians have between 109 and 1011 base-pairs. Psilotum nudum, a member of the fern family, has even more: something on the order of 2.5 x 1011 base-pairs. The latter case is perhaps the most striking comparison, since P. nudum is quite primitive, even compared to other ferns (which are among the oldest plants still around): it lacks leaves, flowers, and fruit. It closely resembles plants from the Silurian epoch (~443 million years ago – 416 million years ago), which are among the oldest vascular plants we've found in the fossil record. 56 complex than frogs and ferns. This isn't going to do it, then: while size certainly matters somewhat, the mereological size measure fails to capture the sense in which it matters. Bigger doesn't always mean more complex, even if we can solve the all-important problem of defining what "bigger" even means. In the case of Strevens' proposal, we might well be suspicious of what Wikipedia editors would recognize as "weasel words" in the definition: a complex system is one that is made up of many parts that are somewhat independent of one another, and yet interact strongly. It's difficult to extract anything very precise from this definition: if we didn't already have an intuitive grasp of what 'complex' meant, a definition like this one wouldn't go terribly far toward helping us get a grasp of the concept. How many parts do we need? How strongly must they interact? How autonomous can they be? Without a clear and precise answer to these questions, it's hard to see how a definition like this can help us understand the general nature of complexity. In Strevens' defense, this is not in the least fatal to his project, since his goal is not to give a complete analysis of complexity (but rather just to analyze the role that probability plays in the emergence of simple behavior from the chaotic interaction of many parts). Still, it won't do for what we're after here (and Kiesling can claim no such refuge, though her definition does come from an introductory-level talk). We'll need to find something more precise. 2.1.2 Complexity as Hierarchical Position First, let's try a refinement of the mereological size measure. The language of science (and, to an even greater degree, the language of philosophy of science) is rife with talk of levels. It's natural to think of many natural systems as showing a kind of hierarchical organization: lakes are 57 made out of water molecules, which are made out of atoms, which are made out of quarks; computers are made out of circuit boards, which are made out of transistors and capacitors, which are made out of molecules; economies are made out of firms and households, which are made out of agents, which are made out of tissues, which are made out of cells &c.. This view is so attractive, in fact, that a number of philosophers have tried to turn it into a full-fledged metaphysical theory . Again, I want to try to avoid becoming deeply embroiled in the 53 metaphysical debate here, so let's try to skirt those problems as much as possible. Still, might it not be the case that something like degree of hierarchy is a good measure for complexity? After all, it does seem (at first glance) to track our intuitions: more complex systems are those which are "nested" more deeply in this hierarchy. It seems like this might succeed in capturing what it was about the mereological size measure that felt right: things higher up on the hierarchy seem to have (as a general rule) more parts than things lower down on the hierarchy. Moreover, this measure might let us make sense of the most nagging question that made us suspicious of the mereological size measure: how to figure out which parts we ought to count when we're trying to tabulate complexity. As attractive as this position looks at first, it's difficult to see how it can be made precise enough to serve the purpose to which we want to put it here. Hierarchy as a measure of complexity was first proposed by Herbert Simon back before the field of complex systems theory diverged from the much more interestingly named field of "cybernetics." It might be useful to actually look at how Simon proposed to recruit hierarchy to explain complexity; the difficulties, I think, are already incipient in his original proposal: 53 See, e.g., Morgan (1923), Oppenheim & Putnam (1958), and (to a lesser extent) Kim (2002) 58 Roughly, by a complex system I mean one made up of a large number of parts that interact in a non-simple way. In such systems, the whole is more than the sum of the parts, not in an ultimate, metaphysical sense, but in the important pragmatic sense that, given the properties of the parts and the laws of their inter-action, it is not a trivial matter to infer the properties of the whole. In the face of complexity, an in-principle reductionist may be at the same time a pragmatic holist... 54 This sounds very much like the Strevens/Kiesling proposal that we looked at in 2.1.1, and suffers from at least some of the same problems (as well as a few of its own). Aside from what I flagged above as Wikipedian "weasel words," the hierarchical proposal suffers from some of the same subjectivity issues that plagued the mereological proposal: when Simon says (for instance) that one of the key features of the right sort of hierarchical composition is "near-decomposability," exactly what is it that's supposed to be decomposable? Again, the hierarchical position seems to be tracking something interesting here-Simon is right to note that it seems that many complex systems have the interesting feature of being decomposable into many (somewhat less) complex subsystems, and that the interactions within each subsystem are often stronger than interactions between subsystems. This structure, Simon contends, remains strongly in view even as the subsystems themselves are decomposed into sub-subsystems. There is certainly something to this point. Interactions between (say) my liver and my heart are relatively "weak" compared to interactions that the cells of my heart (or liver) have with each other. Similarly, the interactions between the mitochondria and the Golgi body of an individual cell in my heart are stronger than the interactions between the individual cells. Or, to move up in the hierarchy, the interactions between my organs seem stronger than the interactions between my body as a whole and other individual people I encounter on my daily commute to Columbia's campus. Still, a problem remains. What's the sense of "stronger" here? Just as before, it seems like 54 Simon (1962) 59 this proposal is tracking something, but it isn't easy to say precisely what. We could say that it is easier for the equilibrium of my body to be disturbed by the right (or, rather, wrong) sort of interaction between my liver and heart than it is for that same equilibrium to be disturbed by the right kind of interaction between me and a stranger on the subway, but this still isn't quite correct. It might be true that the processes that go on between my organs are more fragile-in the sense of being more easily perturbed out of a state where they're functioning normally-than the processes that go on between me and the strangers standing around me on the subway as I write this, but without a precise account of the source and nature of this fragility, we haven't moved too far beyond the intuitive first-pass account of complexity offered at the outset of Section 2.1. Just as with mereological size, there seems to be a nugget of truth embedded in the hierarchical account of complexity, but it will take some work to extract it from the surrounding difficulties. 2.1.3 Complexity as Shannon Entropy Here's a still more serious proposal. Given the discussion in Chapter One, there's another approach that might occur to us: perhaps complexity is a measure of information content or degree of surprise in a system. We can recruit some of the machinery from the last chapter to help make this notion precise. We can think of "information content" as being a fact about how much structure (or lack thereof) exists in a particular system-how much of a pattern there is to be found in the way a system is put together. More formally, we might think of complexity as being a fact about the Shannon entropy in a system. Let's take a moment to remind ourselves 55 55 See Shannon (1948) and Shannon & Weaver (1949) 60 of what exactly that means, and see if it succeeds in capturing our intuitive picture of complexity. "Amount of surprise" is a good first approximation for the quantity that I have in mind here, so let's start by thinking through a simple analogy. I converse with both my roommate and my Siamese cat on a fairly regular basis. In both cases, the conversation consists in my making particular sounds and my interlocutor responding by making different sounds. Likewise, in both cases there is a certain amount of information exchanged between my interlocutor and me. In the case of my roommate, the nature of this information might vary wildly from conversation to conversation: sometimes we will talk about philosophy, sometimes about a television show, and sometimes what to have for dinner. Moreover, he's a rather unusual fellow-I'm never quite sure what he's going to say, or how he'll respond to a particular topic of conversation. Our exchanges are frequently surprising in a very intuitive sense: I never know what's going to come out of his mouth, or what information he'll convey. My Siamese cat, on the other hand, is far less surprising. While I can't predict precisely what's going to come out of her mouth (or when), I have a pretty general sense: most of the time, it's a sound that's in the vicinity of "meow," and there are very specific situations in which I can expect particular noises. She's quite grandiloquent for a cat (that's a Siamese breed trait), and the sight of the can opener (or, in the evening, just someone going near the can opener) will often elicit torrent of very high-pitched vocalizations. I'm not surprised to hear these noises, and can predict when I'll hear them with a very high degree of accuracy. The difference between conversing with these two creatures should be fairly clear. While my cat is not like a recording-that is, while I'm not sure precisely what she's going to say (in the way that, for instance, I'm precisely sure what Han Solo will say in his negotiations with Jabba 61 the Hutt), there's far less variation in her vocalizations than there is in my roommate's. She can convey urgent hunger (and often does), a desire for attention, a sense of contentment, and a few other basic pieces of information, but even that variation is expressed by only a very narrow range of vocalizations. My roommate, on the other hand, often surprises me, both with what kind of information he conveys and how he conveys it. Intuitively, my roommate's vocalizations are the more complex. We can also think of "surprise" as tracking something about how much I miss if I fail to hear part of a message. In messages that are more surprising (in this sense), missing just a small amount of data can make the message very difficult to interpret, as anyone who has ever said expressed incredulity with "What?!" can attest; when a message is received and interpreted as being highly surprising, we understand that just having misheard a word or two could have given us the wrong impression, and request verification. Missing just two or three words in a sentence uttered by my roommate, for instance, can render the sentence unintelligible, and the margin for error becomes more and more narrow as the information he's conveying becomes less familiar. If he's telling me about some complicated piece of scholarly work, I can afford to miss very little information without risking failing to understand the message entirely. On the other hand, if he's asking me what I'd like to order for dinner and then listing a few options, I can miss quite a bit and still be confident that I've understood the overall gist of the message. My cat's communications, which are less surprising even than the most banal conversation I can have with my roommate, are very easily recoverable from even high degrees of data loss; if I fail to hear the first four "meows," there's likely to be a fifth and sixth, just to make sure I got the point. Surprising messages are thus harder to compress in the sense described in Chapter One, as the 62 recovery of a missing bit requires a more complex pattern to be reliable. Shannon entropy formalizes this notion. In Shannon's original formulation, the entropy (H) of a particular message source (my roommate's speech, my cat's vocalizations, Han Solo's prevarications) is given by an equation, the precise details of which are not essential for our 56 purposes here, that specifies how unlikely a particular message is, given specifications about the algorithm encoding the message. A particular string of noises coming out of my cat are (in general) far more likely than any particular string of noises that comes out of my roommate; my roommate's speech shows a good deal more variation between messages, and between pieces of a given message. A sentence uttered by him has far higher Shannon entropy than a series of meows from my cat. So far, then, this seems like a pretty good candidate for what our intuitive sense of complexity might be tracking: information about complex systems has far more Shannon entropy than information about simple systems. Have we found our answer? Is complexity just Shannon entropy? Alas, things are not quite that easy. Let's look at a few problem cases. First, consider again the "toy science" from Section 1.3. We know that for each bit in a given string, there are two possibilities: the bit could be either a '1' or a '0.' In a truly random string in this language, knowing the state of a particular bit doesn't tell us anything about the state of any other bits: there's no pattern in the string, and the state of each bit is informationally independent of each of the others. What's the entropy of a string like that-what's the entropy of a 56 H = ∑PiHi This equation expresses the entropy in terms of a sum of probabilities pi(j)for producing various i symbols j such that the message in question is structured the way it is. Thus, the more variation you can expect in each bit of the message, the higher the entropy of the total message. For a more detailed discussion of the process by which this equation can be derived, see Shannon (1948) and Shannon & Weaver (1964). 63 "message" that contains nothing but randomly generated characters? If we think of the message in terms of how "surprising" it is, the answer is obvious: a randomly-generated string has maximally high Shannon entropy. That's a problem if we're to appeal to Shannon entropy to characterize complexity: we don't want it to turn out that purely random messages are rated as even more complex than messages with dense, novel information-content, but that's precisely what straight appeal to Shannon entropy would entail. Why not? What's the problem with calling a purely random message more complex? To see this point, let's consider a more real-world example. If we want Shannon entropy to work as a straight-forward measure for complexity, it needs to be the case that there's a tight correlation between an increase (or decrease) in Shannon entropy and an increase (or decrease) in complexity. That is: we need it to be the case that complexity is proportional to Shannon entropy: call this the correlation condition. I don't think this condition is actually satisfied, though: think (to begin) of the difference between my brain at some time t, and my brain at some later time t1. Even supposing that we can easily (and uncontroversially) find a way to represent the physical state of my brain as something like a message, it seems clear that we can construct 57 a case where measuring Shannon entropy isn't going to give us a reliable guide to complexity. Here is such a case. Suppose that at t, my brain is more-or-less as it is now-(mostly) functional, alive, and doing its job of regulating the rest of the systems in my body. Now, suppose that in the time 57 Mitchell (op. cit.) points out that if we're to use any measure of this sort to define complexity, anything we wish to appropriately call "complex" must be put into a form for which Shannon entropy can be calculated-that is, it has to be put into the form of a message. This works just fine for speech, but it isn't immediately obvious how we might go about re-describing (say) the brain of a human and the brain of an ant messages such that we can calculate their Shannon entropy. This problem may be not be insurmountable (I'll argue in 2.2 that it can indeed be surmounted), but it is worth noting still. 64 between t and t1, someone swings a baseball bat at my head. What happens when it impacts? If there's enough force behind the swing, I'll die. Why is that? Well, when the bat hits my skull, it transfers a significant amount of kinetic energy through my skull and into my brain, which (among other things) randomizes large swaths of my neural network, destroying the 58 correlations that were previously in place, and making it impossible for the network to perform the kind of computation that it must perform to support the rest of my body. This is (I take it) relatively uncontroversial. However, it seems like we also want to say that my brain was more complex when it was capable of supporting both life and significant information processing than it was after it was randomized-we want to say that normal living human systems are more complex than corpses. But now we've got a problem: in randomizing the state of my brain, we've increased the Shannon entropy of the associated message encoding its state. A decrease in complexity here is associated with an increase in Shannon entropy. That looks like trouble, unless a system with minimal Shannon entropy is a system with maximal complexity (that is, unless the strict inverse correlation between entropy and complexity holds). But that's absurd: a system represented by a string of identical characters is certainly not going to be more complex than a system represented by a string of characters in which multiple nuanced patterns are manifest . The correlation condition between entropy and complexity fails. 59 58 The sense of "randomizes" here is a thermodynamic one. By introducing a large amount of kinetic energy into my brain, my assailant (among other things) makes it the case that the volume of the region of configuration space associated with my brain is wildly increased. That is, the state "Jon is conscious and trying to dodge that baseball bat" is compatible with far fewer microstates of my brain than is the state "Jon has been knocked out by a baseball bat to the face." The bat's impacting with my skull, then, results in a large amount of information loss about the system-the number of possible encodings for the new state is larger than the number of possible encodings for the old state. The Shannon entropy has thus increased. 59 To see this point, think of two pieces of DNA-one of which codes for a normal organism (say, a human being) and one of similar length, but which consists only in cytosine-guanine pairs. Each DNA string can be encoded as a message consisting entirely of the letters A, C, G, and T. The piece of DNA that codes for a functional organism will be associated with a message with far higher Shannon entropy than the piece of DNA associated with a message that 65 Shannon entropy, then, can't be quite what we're looking for, but neither does it seem to miss the mark entirely. On the face of it, there's some relationship between Shannon entropy and complexity, but the relationship must be more nuanced than simple identity, or even proportionality. Complex systems might well be those with a particular entropic profile, but if that's the case, then the profile is something more subtle than just "high entropy" or "low entropy." Indeed, if anything, it seems that there's a kind of "sweet spot" between maximal and minimal Shannon entropy-systems represented by messages with too much Shannon entropy tend not to be complex (since they're randomly organized), and systems represented by messages with too little Shannon entropy tend not to be complex, since they're totally homogenous. This is a tantalizing observation: there's a kind of Goldilocks zone here. Why? What's the significance of that sweet spot? We will return to this question in Section 2.1.5. For now, consider one last candidate account of complexity from the existing literature. 2.1.4 Complexity as Fractal Dimension The last candidate definition for complexity that we'll examine here is also probably the least intuitive. The notion of a fractal was originally introduced as a purely geometric concept by French mathematician Benoit Mandelbrot , but there have been a number of attempts to connect 60 the abstract mathematical character of the fractal to the ostensibly "fractal-like" structure of certain natural systems. Many parts of nature are fractal-like in the sense of displaying a certain degree of what's sometimes called "statistical self-similarity." Since we're primarily interested in real physical systems here (rather than mathematical models), it makes sense to start with that consists entirely of the string 'CG' repeated many times. Surely DNA that codes for a functional organism, though, is more complex than a non-coding DNA molecule. Again, the correlation condition fails. 60 Mandelbrot (1986) 66 sense of fractal dimension before considering the formal structure of mathematical fractals. Let's begin by getting a handle on what counts as statistical self-similarity in nature, then, to begin with. Consider a stalk of broccoli or cauliflower that we might find in the produce section of a supermarket. A medium-sized stalk of broccoli is composed of a long smooth stem (which may be truncated by the grocery store, but is usually still visible) and a number of lobes covered in what look like small green bristles. If we look closer, though, we'll see that we can separate those lobes from one another and remove them. When we do, we're left with several things that look very much like our original piece of broccoli, only miniaturized: each has a long smooth stem, and a number of smaller lobes that look like bristles. Breaking off one of these smaller lobes reveals another piece that looks much the same. Depending on the size and composition of the original stalk, this process can be iterated several times, until at last you're removing an individual bristle from the end of a small stalk. Even here, though, the structure looks remarkably similar to that of the original piece: a single green lobe at the end of a long smooth stem. This is a clear case of the kind of structure that generally gets called "fractal-like." It's worth highlighting two relevant features that the broccoli case illustrates nicely. First, fractal-like physical systems have interesting detail at many levels of magnification: as you methodically remove pieces from your broccoli stem, you continue to get pieces with detail that isn't homogenous. Contrast this with what it looks like when you perform a similar dissection of (say) a carrot. After separating the leafy bit from the taproot, further divisions produce (no pun intended) pieces that are significantly less interesting: each piece ends up looking more-or-less 67 the same as the last one-smooth, orange, and fibrous. That's one feature that makes fractal-like parts of the world interesting, but it's not the only one. After all, it's certainly the case that there are many other systems which, on dissection, can be split into pieces with interesting detail many times over-any sufficiently inhomogeneous mixture will have this feature. What else, then, is the case of fractals tracking? What's the difference between broccoli and (say) a very inhomogeneous mixture of cake ingredients? The fact that (to put it one more way) a stalk of broccoli continues to evince interesting details at several levels of magnification cannot be all that makes it fractal-like, so what's the second feature? Recall that the kind of detail that our repeated broccoli division produced was of a very particular kind-one that kept more-or-less the same structure with every division. Each time we zoomed in on a smaller piece of our original stalk, we found a piece with a long smooth stem and a round green bristle on the end. That is, each division (and magnification) yielded a structure that not only resembled the structure which resulted from the previous division, but also the structure that we started with. The interesting detail at each level was structurally similar to the interesting detail at the level above and below it. This is what separates fractal-like systems from merely inhomogeneous mixtures-not only is interesting detail present with each division, but it looks the same. Fractal-like systems (or, at least the fractal-like systems we're interested in here) show interesting details at multiple levels of magnification, and the interesting details present at each level are self-similar. With this intuitive picture on the table, let's spend a moment looking at the more formal definition of fractals given in mathematics. Notice that we've been calling physical systems "fractal-like" all along here-that's because nothing in nature is actually a fractal, in just the 68 same sense that nothing in nature is actually a circle. In the case of circles, we know exactly what it means to say that there are no circles in nature: no natural systems exist which are precisely isomorphic to the equation that describes a geometric circle: things (e.g. basketball hoops) are circular, but on close enough examination they turn out to be rough and bumpy in a way that a mathematical circle is not. The same is true of fractals; if we continue to subdivide the broccoli stalk discussed above, eventually we'll reach a point where the self-similarity breaks down-we can't carry on getting smaller and smaller smooth green stems and round green bristles forever. Moreover, the kind of similarity that we see at each level of magnification is only approximate: each of the lobes looks a lot like the original piece of broccoli, but the resemblance isn't perfect-it's just pretty close. That's the sense in which fractal-like physical systems are only statistically self-similar-at each level of magnification, you're likely to end up with a piece that looks more-or-less the same as the original one, but the similarity isn't perfect. The tiny bristle isn't just a broccoli stalk that's been shrunk to a tiny size, but it's almost that. This isn't the case for mathematical fractals: a true fractal has the two features outlined above at every level of magnification-there's always more interesting detail to see, and the interesting details are always perfectly self-similar miniature copies of the original Here's an example of an algorithm that will produce a true fractal: 1. Draw a square. 2. Draw a 45-45-90 triangle on top of the square, so that the top edge of the square and the base of the triangle are the same line. Put the 90 degree angle at the vertex of the triangle, opposite the base 3. Use each of the other two sides of the triangle as sides for two new (smaller) squares. 4. Repeat steps 1-4 for each of the new squares you've drawn. Here's what this algorithm produces after just a dozen iterations: 69 Fig. 2.1 Look familiar? This shape is starting to look suspiciously like our stalk of broccoli: there's 61 a main "stem" formed by the first few shapes (and the negative space of later shapes), "lobes" branching off from the main stem with stems of their own, and so on. If you could iterate this procedure an infinite number of times, in fact, you'd produce a perfect fractal: you could zoom in on almost any region of the shape and find perfect miniaturized copies of what you started with. Zooming in again on any region of one of those copies would yield even more copies, ad infinitum. This is a neat mathematical trick, but (you might wonder) what's the point of this discussion? How does this bear on complexity? Stay with me just a bit longer here-we're almost there. To explain the supposed connection between fractal-like systems and complexity, we have to look a bit more closely at some of the mathematics behind geometric fractals; in particular, we'll have to introduce a concept called fractal dimension. All the details certainly aren't necessary for what we're doing here, but a rough grasp of the concepts will be helpful for what follows. Consider, to begin with, the intuitive notion of "dimension" that's taught in high school math classes: the dimensionality of a space is just a specification of how many numbers need to be 61 The shape generated by this procedure is called the Pythagoras Tree. 70 given in order to uniquely identify a point in that space. This definition is sufficient for most familiar spaces (such as all subsets of Euclidean spaces), but breaks down in the case of some more interesting figures . One of the cases in which this definition becomes fuzzy is the case of 62 the Pythagoras Tree described above: because of the way the figure is structured, it behaves in some formal ways as a two-dimensional figure, and in other ways as a not two-dimensional figure. The notion of topological dimensionality refines the intuitive concept of dimensionality. A full discussion of topological dimension is beyond the scope of this chapter, but the basics of the idea are easy enough to grasp. Topological dimensionality is also sometimes called "covering dimensionality," since it is (among other things) a fact about how difficult it is to cover the figure in question with other overlapping figures, and how that covering can be done most efficiently. Consider the case of the following curve : 63 Fig. 2.2 62 Additionally, it's difficult to make this definition of dimensionality more precise than the very vague phrasing we've given it here. Consider a curve embedded in a two-dimensional Euclidean plane-something like a squiggly line drawn on a chalkboard. What's the dimensionality of that figure? Our intuitions come into conflict here: for each point on the curve, we have to specify two numbers (the Cartesian coordinates) in order to uniquely pick it out. On the other hand, this seems to just be a consequence of the fact that the curve is embedded in a two-dimensional space, not a fact about the curve itself-since it's just a line, it seems like it ought to just be one-dimensional. The intuitive account of dimensionality has no way to resolve this conflict of reasoning. 63 This figure is adapted from one in Kraft (1995) 71 Suppose we want to cover this curve with a series of open (in the sense of not having a precisely-defined boundary) disks. There are many different ways we could do it, three of which are shown in the figure above. In the case on the bottom left, several points are contained in the intersection of four disks; in the case in the middle, no point is contained in the intersection of more than three disks; finally, the case on the right leaves no point contained in the intersection of more than two disks. It's easy to see that this is the furthest we could possibly push this covering: it wouldn't be possible to arrange open disks of any size into any configuration where the curve was both completely covered and no disks overlapped . We can use this to define 64 topological dimensionality in general: for a given figure F, the topological dimension is defined to be the minimum value of n, such that every finite open cover of F has a finite open refinement in which no point is included in more than n+1 elements. In plain English, that just means that the topological dimension of a figure is one less than the largest number of intersecting covers (disks, in our example) in the most efficient scheme to cover the whole figure. Since the most efficient refinement of the cover for the curve above is one where there is a maximum of two disks intersecting on a given point, this definition tells us that the figure is 1-dimensional. So far so good-it's a line, and so in this case topological dimensionality concurs with intuitive dimensionality . 65 There's one more mathematical notion that we need to examine before we can get to the punch-line of this discussion: fractal dimensionality. Again, a simple example can illustrate 66 64 Why not? Remember that the disks are open, so points just at the "boundary" are not contained in the disks. Thus, a series of very small disks that were very near each other without intersecting would necessarily leave at least some points uncovered: those in the tiny region between two open disks. The only way to cover the whole figure is to allow the disks to overlap slightly. 65 This also lets us move beyond our problem case from above: we can say why it is that a curve on a plane can be one-dimensional even though it is embedded in a two-dimensional space. 66 This exceedingly clear way of illustrating the point is due to Mitchell (op. cit), though our discussion here is 72 this point rather clearly. Consider a Euclidean line segment. Bisecting that line produces two line segments, each with half the length of the original segment. Bisecting the segments again produces four segments, each with one-quarter the length of the original segment. Next, consider a square on a Euclidean plane. Bisecting each side of the square results in four copies, each one-quarter the size of the original square. Bisecting each side of the new squares will result in 16 squares, each a quarter the size of the squares in the second step. Finally, consider a cube. Bisecting each face of the cube will yield eight one-eighth sized copies of the original cube. These cases provide an illustration of the general idea behind fractal dimension. Very roughly, fractal dimension is a measure of the relationship between how many copies of a figure are present at different levels of magnification and how much the size of those copies changes between levels of magnification . In fact, we can think of it as a ratio between these two 67 quantities. The fractal dimension d of an object is equal to log(a)/log(b), where a = the number of new copies present at each level, and b is the factor by each piece must be magnified in order to have the same size as the original. This definition tells us that a line is one-dimensional: it can be broken into n pieces, each of which is n-times smaller than the original. If we let n = 2, as in our bisection case, then we can see easily that log(2)/log(2) = 1. Likewise, it tells us that a square is two-dimensional: a square can be broken into n2 pieces, each of which must be somewhat more technically precise than the discussion there; Mitchell hides the mathematics behind the discussion, and fails to make the connection between fractal dimension and topological dimension explicit, resulting in a somewhat confusing discussion as she equivocates between the two senses of "dimension." For a more formal definition of fractal dimensionality (especially in the case of Pythagoras Tree-like figures), see Lofstedt (2008). 67 In the illustration here, we had to build in the presence of "copies" by hand, since a featureless line (or square or cube) has no self-similarity at all. That's OK: the action of bisecting the figure is, in a sense, a purely abstract operation: we're not changing anything about the topology of the figures in question by supposing that they're being altered in this way. In figures with actual self-similarity (like fractals), we won't have to appeal to this somewhat arbitrary-seeming procedure. 73 magnified by a factor of n to recover the size of the original figure; again, let n = 2 as in our bisection case, so that the bisected square contains 22 = 4 copies of the original figure, each of which must be doubled in size to recover the area of the original figure. Log(4)/log(2) = 2, so the square is two-dimensional. So far so good. It's worth pointing out that in these more familiar cases intuitive dimension = topological dimension = fractal dimension. That is not the case for all figures, though. Finally, consider our broccoli-like fractal: the Pythagoras Tree. The Pythagoras Tree, as you can easily confirm, has a fractal dimension of 2: at each step n in the generation, there are 2n copies of the figure present: 1 on the zeroth iteration, 2 after a single iteration, 4 after two iterations, 8 after three, 16 after four, and so on. Additionally, each iteration produces figures that are smaller by a factor of √2/2. Following our formula from above, we can calculate log(2)/log(√2/2), which is equal to 2. This accords with our intuitive ascription of dimensionality (the Pythagoras Tree looks like a plane figure) but, more interestingly, it fails to accord with the topological dimension of the figure. Perhaps surprisingly, the Pythagoras Tree's topological dimension is not 2 but 1-like a simple curve, it can be covered by disks such that no point is in the intersection of more than two disks . Topologically, the Pythagoras Tree behaves 68 like a simple one-dimensional line, while in other ways it behaves more like a higher dimensional figure. Fractal dimension lets us quantify the amount by which these behaviors diverge: in fact, this is a characteristic that's common to many (but not all) fractals. In addition to the two-pronged "fine detail and self-similarity" definition given above, Mandelbrot, in his 68 The math behind this assertion is, again, beyond the scope of what we're concerned with here. For a detailed discussion of why the topological dimension of fractal canopies-the class of figures to which the Pythagoras Tree belongs-is 1, see Mandelbrot (1986), Chapter 16. 74 original discussion of fractals, offers an alternative definition: a fractal is a figure where the fractal dimension is greater than the topological dimension . 69 At last, we're in a position, then, to say what it is about fractals that's supposed to capture our notion of complexity. Since fractal dimension quantifies the relationship between the proliferation of detail and the change in magnification scale, an object with a higher fractal dimension will show more interesting detail than an object with a lower fractal dimension, given the same amount of magnification. In the case of objects that are appropriately called "fractal-like" (e.g. our stalk of broccoli), this cascade of detail is more significant than you'd expect it to be for an object with the sort of abstract (i.e. topological) structure it has. That's what it means to say that fractal dimension exceeds topological dimension for most fractals (and fractal-like objects): the buildup of interesting details in a sense "outruns" the buildup of other geometric characteristics. Objects with higher fractal dimension are, in a sense, richer and more rewarding: it takes less magnification to see more detail, and the detail you can see is more intricately structured. So is this measure sufficient, then? You can probably guess by now that the answer is 'no, not entirely.' There are certainly cases where fractal dimension accords very nicely with what we mean by 'complex:' it excels, for instance, at tracking the rugged complexity of coastlines. Coasts-which were among Mandelbrot's original paradigm cases of fractal-like objects-are statistically self-similar in much the same way that broccoli is. Viewed from high above, coastlines look jagged and irregular. As you zoom in on a particular section of the coast, this kind of jaggedness persists: a small segment of shore along a coast that is very rugged in general 69 Mandelbrot offered these two definitions as equivalent. It has since been discovered, though, that there are a number of fractals (in the first sense) for which the latter definition does not hold. See Kraft (1995) for more on this. 75 is likely to be very rugged itself. Just as with the broccoli, this self-similarity is (of course) not perfect: the San Francisco bay is not a perfect miniaturization of California's coastline overall, but they look similar in many respects. Moreover, it turns out that the more rugged a coastline is, the higher fractal dimension it has: coasts with outlines that are very complex have higher fractal dimension than coasts that are relatively simple and smooth. The most serious problem with using fractal dimension as a general measure of complexity is that it seems to chiefly be quantifying a fact about how complex an object's spatial configuration is: the statistical self-similarity that both broccoli and coastlines show is a self-similarity of shape. This is just fine when what we're interested in is the structure or composition of an object, but it isn't at all clear how this notion might be expanded. After all, at least some of our judgments of complexity seem (at least at first glance) to have very little to do with shape: when I say (for instance) that the global economy is more complex today than it was 300 years ago, it doesn't look like I'm making a claim about the shape of any particular object. Similarly, when I say that a human is more complex than a fern, I don't seem to be claiming that the shape of the human body has a greater fractal dimension than the shape of a fern. In many (perhaps most) cases, we're interested not in the shape of an object, but in how the object behaves over time; we're concerned not with relatively static properties like fractal dimension, but with dynamical ones too. Just as with Shannon entropy, there seems to be a grain of truth buried in the fractal dimension measure, but it will takes some work to articulate what it is; also like Shannon entropy, it seems as though fractal dimension by itself will not be sufficient. 2.2 Moving Forward We have spent the majority of this chapter introducing some of the concepts behind 76 contemporary complexity theory, and examining various existing attempts to define 'complexity.' I have argued (convincingly, I hope) that none of these attempts really captures all the interesting facets of what we're talking about when we talk about complex physical systems (like the Earth's climate). I have not yet offered a positive view, though-I have not yet told you what I would propose to use in place of the concepts surveyed here. In Chapter Three, I shall take up that project, and present a novel account of what it means for a physical system to be complex in the relevant sense. This concept, which I will call dynamical complexity, is presented as a physical interpretation of some very recent mathematical advancements in the field of information theory. The central problem that shall occupy us in the next chapter, then, is how to transform a discussion of complexity that seems to work very well for things like messages into an account that works well for things like climate systems. My hope is that dynamical complexity offers this bridge. Once this final conceptual tool is on the table, we can start applying all of this to the problem of understanding the Earth's climate. 77 Chapter Three Dynamical Complexity 3.0 Recap and Survey Let's take a moment to summarize the relative strengths and weaknesses of the various approaches to defining complexity we considered in the last section; it will help us build a satisfactory definition if we have a clear target at which to aim, and clear criteria for what our definition should do. Here's a brief recap, then. The mereological size and hierarchical position measures suffered from parallel problems. In particular, it's difficult to say precisely which parts we ought to be attending to when we're defining complexity in terms of mereological size or (similarly) which way of structuring the hierarchy of systems is the right way (and why). Both of these approaches, though, did seem to be tracking something interesting: there does seem to be a sense in which a system's place in a sort of "nested hierarchy" seems to be a reliable guide to its complexity. All other things being equal, a basic physical system (e.g. a free photon traveling through deep space) does indeed seem less complex than a chemical system (e.g. a combination of hydrogen and oxygen atoms to form H2O molecules), which in turn seems less complex than a biological system (e.g. an amoeba undergoing asexual reproduction), which seems less complex than a social system (e.g. the global stock market). The problem (again) is that it's difficult to say why this is the case: the hierarchical and mereological size measures take it as a brute fact that chemical systems are less complex than biological systems, but have trouble explaining that relationship. A satisfactory 78 theory of complexity must account for both the intuitive pull of these measures and deal with the troubling relativism lurking beneath their surfaces. The Shannon entropy measure suffered from two primary problems. First, since Shannon entropy is an information theoretic quantity, it can only be appropriately applied to things that have the logical structure of messages. To make this work as a general measure of complexity for physical systems, we would have to come up with an uncontroversial way of representing parts of the world as messages generally-a tall order indeed. Additionally, we saw that there doesn't seem to be a strict correlation between changes in Shannon entropy of messages and the complexity of systems with which those messages are associated. I argued that in order for Shannon entropy to function as a measure of complexity, a requirement called the correlation condition must be satisfied: it must be the case that a monotonic increase in complexity in physical systems is correlated with either a monotonic increase or a monotonic decrease in the Shannon entropy of the message associated with that system. The paradigm case here (largely in virtue of being quite friendly to representation as a string of bits) is the case of three strings of DNA: one that codes for a normal human, one that consists of randomly paired nucleotides, and one that consists entirely of cytosine-guanine pairs. In order for the correlation condition to obtain, it must be the case that the system consisting of either the randomly paired nucleotides (which has an associated message with maximal Shannon entropy) or the C-G pair molecule (which has an associated messages with minimal Shannon entropy) is more complex than the system consisting of the human-coding DNA molecule (which has an associated message with Shannon entropy that falls between these two extremes). This is not the case, though: any reasonable measure of complexity should rate a DNA strand that codes for a normal organism as 79 more complex than one that's either random or homogeneous. The correlation condition thus fails to hold. A successful measure of complexity, then, should account for why there seems to be a "sweet spot" in between maximal and minimal Shannon entropy where the complexity of associated systems seems to peak, as well as give an account of how in general we should go about representing systems in a way that lets us appropriately judge their Shannon entropy. Finally, fractal dimension suffered from one very large problem: it seems difficult to say how we can apply it to judgments of complexity that track characteristics other than spatial shape. Fractal dimension does a good job of explaining what we mean when we judge that a piece of broccoli is more complex than a marble (the broccoli's fractal dimension is higher), but it's hard to see how it can account for our judgment that a supercomputer is more complex than a hammer, or that a human is more complex than a chair, or that the global climate system on Earth is more complex than the global climate system on Mars. A good measure of complexity will either expand the fractal dimension measure to make sense of non-geometric complexity, or will show why geometric complexity is just a special case of a more general notion. 2.1 Dynamical Complexity With a more concrete goal at which to aim, then, let's see what we can do. In this section, I will attempt to synthesize the insights in the different measures of complexity discussed above under a single banner-the banner of dynamical complexity. This is a novel account of complexity which will (I hope) allow us to make sense of both our intuitive judgments about complexity and open the door to making those judgments somewhat more precise. Ultimately, remember, our goal is to give a concept which will allow us to reliably differentiate between complex systems and simple systems such that we can (roughly) differentiate complex systems 80 sciences from simple systems sciences, opening the door to more fruitful cross-talk between branches of science that, prior to the ascription of complexity, seemed to have very little in common with one another. I shall argue that such an understanding of complexity emerges very naturally from the account of science given in Chapter One. I'm going to begin by just laying out the concept I have in mind without offering much in the way of argument for why we ought to adopt it. Once we have a clear account of dynamical complexity on the table, then I'll argue that it satisfies all the criteria given above-I'll argue, in other words, that it captures what seems right about the mereological, hierarchical, information-theoretic, and fractal accounts of complexity while also avoiding the problems endemic to those views. Back in Section 1.5, I said, "In a system with a relatively high degree of complexity-very roughly, a system with a relatively high-dimensional configuration space-there will be a very large number of ways of specifying regions such that we won't be able to identify any interesting patterns in how those regions behave over time," and issued a promissory note for an explanation to come later. We're now in a position to examine this claim, and to (finally) cash that promissory check. First, note that the way the definition was phrased in the last chapter isn't going to quite work: having a very high-dimensional configuration space is surely not a sufficient condition for complexity. After all, a system consisting of a large number of non-interacting particles may have a very high-dimensional phase space indeed: even given featureless particles in a Newtonian system, the dimensionality of the phase space of a system with n particles will be (recall) 6n. Given an arbitrarily large number of particles, the phase space of a system like this will also be of an arbitrarily large dimensionality. Still, it seems clear that simply increasing the number of particles in a system like that doesn't really increase the 81 system's complexity: while it surely makes the system more complicated, complexity seems to require something more. This is a fact that the mereological size measure (especially in Kiesling's phrasing) quite rightly seizes on: complexity is (at least partially) a fact not just about parts of a system, but about how those parts interact. Let's start to refine Chapter One's definition, then, by thinking through some examples. As a reminder, let's remind ourselves of the example we worked through there: consider a thermodynamically-isolated system consisting of a person standing in a kitchen, deliberating about whether or not to stick his hand in the pot of boiling water. As we saw, a system like this one admits of a large number of ways of carving up the associated configuration space : 70 describing the situation in the vocabulary of statistical mechanics will yield one set of time-evolution patterns for the system, while describing it in the vocabulary of biology will yield another set, and so on. Fundamental physics provides the "bit mapping" from points in the configuration space representing the system at one instant to points in the same space at another instant; the different special sciences, then, offer different compression algorithms by which the state of a particular system can be encoded. Different compressions of the same system will evince different time-evolution patterns, since the encoding process shifts the focus from points in the configuration space to regions in the same space. All of this is laid out in significantly more detail in Chapter One. Now, consider the difference between the person-stovewater system and the same system, only with the person removed. What's changed? For one thing, the dimensionality of the 70 Equivalently, we might say that a system like this admits of a very large number of interesting configuration spaces; there are very many ways that we might describe the system such that we can detect a variety of interesting time-evolution patterns. 82 associated configuration space is lower; in removing the person from the system, we've also removed a very large number of particles. That's far from the most interesting change, though-in removing the human, we've also significantly reduced the number of interesting ways of carving up the configuration space. The patterns identified by (for instance) psychology, biology, and organic chemistry are no longer useful in predicting what's going to happen as the system evolves forward in time. In order to make useful predictions about the behavior of the system, we're now forced to deal with it in the vocabulary of statistical mechanics, inorganic chemistry, thermodynamics, or (of course) fundamental physics. This is a very significant change for a number of reasons. Perhaps paramount among them, it changes the kind of information we need to have about the state of the system in order to make interesting predictions about its behavior. Consider, for instance, the difference between the following characterizations of the system's state: (1) "The water is hot enough to cause severe burns to human tissue" and (2) "The water is 100 degrees C." In both cases, we've been given some information about the system: in the case of (1), the information has been presented in biological terms, while in the case of (2), the information has been presented in thermodynamic terms . Both of these characterizations will 71 let us make predictions about the time-evolution of the system, but the gulf between them is clear: (2) is a far more precise description of the state of the system, and requires far more 72 detailed information to individuate than does (1). That is, there are far more points in the system's configuration space that are compatible with (1) than with (2), so individuating cases of 71 That is, the information has been presented in a way that assumes that we're using a particular state-space to represent the system. 72 That is, there are far fewer possible states of the system compatible with (2) than there are states compatible with (1). 83 (2) from cases of not-(2) requires more data about the state of the system than does individuating cases of (1) from cases of not-(1). This is a consequence of the fact that (as we saw in Chapter 73 One) some special science compressions are more lossy (in the sense of discarding more information, or coarse-graining more heavily) than others: biology is, in general, a more lossy encoding scheme than is organic chemistry. This is (again) a feature rather than a bug: biology is lossy, but the information discarded by biologists is (ideally) information that's irrelevant to the patterns with which biologists concern themselves. The regions of configuration space that evolve in ways that interest biologists are less precisely defined than the regions of configuration space that evolve in ways that interest chemists, but the biologists can take advantage of that fact to (in a sense) do more work with less information, but that work will only be useful in a relatively small number of systems-those with paths that remain in a particular region of configuration space during the time period of interest. The significance of this last point is not obvious, so it is worth discussing in more detail. Note, first, that just by removing the human being from this system, we haven't necessarily made it the case that the biology compression algorithm fails to produce a compressed encoding of the original state: even without a person standing next to the pot of water, generalizations like "that water is hot enough to burn a person severely" can still be made quite sensibly. In other words, the set of points in configuration space that a special science can compress is not necessarily identical to the set of points in configuration space that the same special science can usefully 73 This does not necessarily mean that the associated measurements are operationally more difficult to perform in the case of (2), though-how difficult it is to acquire certain kinds of information depends in part on what measurement tools are available. The role of a thermometer, after all, is just to change the state of the system to one where a certain kind of information (information about temperature) is easier to discern against the "noisy" information-background of the rest of what's going on in the system. Measurement tools work as signal-boosters for certain classes of information. 84 compress; the information that (for instance) the inside of my oven is too hot for infants to live comfortably is really only interesting if there is an infant (or something sufficiently like an infant) in the vicinity of my oven. If there isn't, that way of describing my oven's state remains accurate, but ceases to be very relevant in predicting how the system containing the over will change over time; in order for it to become predicatively relevant, I'd need to change the state of the system by adding a baby (or something suitably similar). This is a consequence of the fact that (as we saw in 1.5), the business of the special sciences is two-fold: they're interested both in identifying novel ways of carving up the world and in applying those carvings to some systems in order to predict their behavior over time. Both of these tasks are interesting and important, 74 but I want to focus on the latter one here-it is analysis of the latter task that, I think, can serve as the foundation for a plausible definition of 'complexity.' By removing the person from our example system, we reduce the complexity of that system. This is relatively uncontroversial, I take it-humans are paradigmatic cases of complex systems. My suggestion is that the right way to understand this reduction is as a reduction in the number of predictively useful ways the system can be carved up. This is why the distinction just made between special-scientific compression and useful special-scientific compression is essential-if we were to attend only to shifts that changed a system enough for a particular special science's compression to fail entirely, then we wouldn't be able to account for the uncontroversial reduction of complexity that coincides with the removal of the human from our kitchen-system. After all, as we just saw, the fact that the compression scheme of biology is useless for predicting 74 Of course, these two interests are often mutually-reinforcing. For a particularly salient example, think of the search for extraterrestrial life: we need to both identify conditions that must obtain on extrasolar planets for life to plausibly have taken hold and, given that identification, try to predict what sort of life might thrive on one candidate planet or another. 85 the behavior of a system doesn't imply that the compression scheme of biology can't be applied to that system at all. However, removing the person from the system does render a large number of compression schemes predictively useless, whether or not they still could be applied: removing the person pushes the system into a state for which the patterns identified by (e.g.) biology and psychology don't apply, whether or not the static carvings of those disciplines can still be made. This fact can be generalized. The sense in which a system containing me is more complex (all other things being equal) than is a system containing my cat instead of me is just that the system containing me can be usefully carved up in more ways than the system containing my cat. My brain is more complex than my cat's brain in virtue of there being more ways to compress systems containing my brain such that the time-evolution of those states can be reliably predicted than there are ways to compress systems containing my cat's brain such that the same is true. The global climate today is more complex than was the global climate 1 billion years ago in virtue of there being more ways to usefully carve up the climate system today than there were 1 billion years ago . Complexity in this sense, then, is a fact not about what a system is made out 75 of, or how many parts it has, or what its shape is: it is a fact about how it behaves. It is a dynamical fact-a fact about how many different perspectives we can usefully adopt in our quest to predict how the system will change over time. One system is more dynamically complex than another if (and only if) it occupies a point in configuration space that is at the intersection of 75 If this assertion seems suspect, consider the fact that patterns identified by economists (e.g. the projected price of fossil fuels vs. the projected price of cleaner alternative energies) are now helpful in predicting the evolution of the global climate. This was clearly not the case one billion years ago, and (partially) captures the sense in which humanity's emergence as a potentially climate-altering force has increased the complexity of the global climate system. This issue will be taken up in great detail in Chapter Three. 86 regions of interest to more special sciences: a system for which the patterns of economics, psychology, biology, chemistry, and physics are predictively useful is more complex than one for which only the patterns of chemistry and physics are predictively useful. 2.2.1 Dynamical Complexity as a Unifying Definition I have now given a definition of dynamical complexity. Before we close this theoretical discussion and move on to consider the special problems faced by climate science as a complex science, it's worth briefly reviewing the attempted definitions of complexity we surveyed in Section 2.1 to see how dynamical complexity fares as a unifying definition of complexity. In this section, I will argue that dynamical complexity succeeds in cherry-picking the best features of the mereological size measure, the hierarchical position measure, the information-theoretic measure, and the fractal dimension measure, while avoiding the worst difficulties of each of them. Let's begin with the mereological size measure. As I mentioned above, one of the strongest virtues of the mereological size measure is that (at least in its better formulations) it attends to the fact that complexity is a concept that deals not with static systems, but with dynamic systems-with systems that are moving, changing, and exchanging information with their environments. Strevens , for instance, emphasizes not only 76 the presence of many parts in a complex system, but also the fact that those parts interact with one another in a particular way. This is an insight that is clearly incorporated into dynamical complexity: since dynamical complexity deals with the number of different ways of carving configuration space that yield informative time-evolution patterns for a given system, the presence of interacting constituent parts is indeed, on this view, a great contributor to 76Strevens (Ibid) 87 complexity. Why? Well, what does it mean to say that a system is "composed" of a large number of interacting parts? It means (among other things) that the system can be fruitfully redescribed in the language of another science-the one that carves configuration space in terms of whatever the parts for this particular system are. To say that the human body is composed of many interacting cells, for instance, is just to say that we can either treat the body as an individual (as, say, evolutionary biology might) and make use of the patterns that can be identified in the behavior of systems like that, or treat it as a collection of individual cells (as a cellular biologist might) and predict its behavior in terms of those patterns. Systems which can appropriately be said to be made out of many parts are often systems which can be treated by the vocabulary of multiple branches of the scientific project. Moreover, since we're tying dynamical complexity not to composition but behavior, we don't need to answer the uncomfortable questions that dog the avid proponent of the mereological size measure-we don't need to say, for instance, which method of counting parts is the right one. Indeed, the existence of many different ways to count the parts of a system is something that dynamical complexity can embrace whole-heartedly-the fact that the human body can be seen as a collection of organs, or cells, or molecules straightforwardly reflects its status as a complex system: there are many different useful ways to carve it up, and many interesting patterns to be found in its time-evolution. This leads directly into the hierarchical position measure. Here too the relationship to dynamical complexity is fairly clear. What does it mean to say that one system is "nested more 77 deeply in the hierarchy?" It means that the system can be described (and its behavior predicted) 77 Indeed, it was my reading of the canonical articulation of the hierarchical scheme-Oppenheim and Putnam (1954)-that planted the seed which eventually grew into the position I have been defending over the last 60 pages. 88 in the language of more branches of science. The central mistake of previous attempts to make this notion precise, I think, lies in thinking of this "nestedness" as hierarchical in the traditional linear sense: of there being strict pyramidal structure to the relationship between the various branches of science. In Oppenheim and Putnam's formulation, for instance, physics was at the 78 bottom of the pyramid, then chemistry, then biology, then psychology, then sociology. The assumption lurking behind this model is that all systems described by chemistry can also be described by physics (true enough, but only in virtue of the fact that the goal of physics is to describe all systems), all systems described by biology can also be described by chemistry (probably also true), that all systems that can be described by psychology can also be described by biology (possibly not true), and that all systems described by sociology can also be described by psychology (almost certainly not true). The last two moves look particularly suspect, as they rule out a priori the possibility of non-biological systems that might be usefully described as psychological agents, or the possibility of systems that cannot be treated by psychology, and yet 79 whose behavior can be fruitfully treated by the social sciences. 80 Dynamical complexity escapes from this problem by relaxing the pyramidal constraint on the relationship between the various branches of science. As I argued in Chapter One, the intersections between the domains of the various sciences are likely to be messy and complicated: while many psychological systems are in fact also biological systems, there may well be psychological systems which are not-the advent of sophisticated artificial intelligence, 78 Op. cit. 79 This is the worry that leads Dennett to formulate his "intentional stance" view of psychology. For more discussion of this point, see Dennett (1991). 80 Social insects-bees and ants, for instance-might even be an existing counterexample here. The fascinating discussion in Gordon (2010) of ant colonies as individual "superorganisms" lends credence to this view. Even if Earthly ants are not genuine counterexamples, though, such creatures are surely not outside the realm of possibility, and ought not be ruled out on purely a priori grounds. 89 for instance, would give rise to systems that might be fruitfully studied by psychologists but not by biologists. This is a problem for someone who wants to embrace position in a strict hierarchy as a measure of complexity: there may be no strict hierarchy to which we can appeal. Dynamical complexity cheerfully acknowledges this fact, and judges complexity on a case-by-case basis, rather than trying to pronounce on the relative complexity of all biological systems, or all psychological systems. What aspects of fractal dimensionality does dynamical complexity incorporate? To begin, it might help to recall why fractal dimensionality by itself doesn't work as a definition of complexity. Most importantly, recall that fractal dimensionality is a static notion-a fact about the shape of an object-not a dynamical one. We're interested in systems, though, not static objects-science deals with how systems change over time. On the face of it, fractal dimensionality doesn't have the resources to deal with this: it's a geometrical concept properly applied to shapes. Suppose, however, that think not about the geometry of a system, but about the geometry of the space representing the system. Perhaps we can at least recover self-similarity and see how complexity is a fractal-like concept. Start with the normal configuration space we've been dealing with all along. From the perspective of fundamental physics, each point in the space represents an important or interesting distinction: fundamental physics is a bit-map from point-to-point. When we compress the configuration space for treatment by a special science, though, not all point differences remain relevant-part of what it means to apply a particular special science is to treat some distinctions made by physics as irrelevant given a certain set of goals. This is what is meant by thinking of the special sciences as coarse-grainings of fundamental physics. 90 Suppose that instead of thinking of the special sciences as providing compressed versions of the space provided by fundamental physics, though, we take the view offered in Chapter One: we can think of a special science as defining a new configuration space for the system. What were formerly regions in the very high-dimensional configuration space defined by fundamental physics can now be treated as points in a lower dimensional space defined by the special science in question. It is tempting to think that both these representations-the special sciences as coarse-graining and the special sciences as providing entirely novel configuration spaces-are predicatively equivalent, but this is not so. The difference is that the second way of doing things actually makes the compression-the information loss-salient; it isn't reversible. It also (and perhaps even more importantly) emphasizes the fact that the choice of a state-space involves more than choosing which instantaneous states are functionally equivalent-it involves more than choosing which collections of points (microstates) in the original space to treat as macrostates. The choices of a state-space also constitutes a choice of dynamics: for a system with a high degree of dynamical complexity, there are a large number of state spaces which evince not only interesting static detail, but interesting dynamical detail as well. Thinking of (say) a conscious human as being at bottom a system that's only really completely describable in the state space of atomic physics eclipses not just the presence of interesting configurations of atomic physics' particles (interesting macrostates), but also the presence of interesting patterns in how those configurations change over time: patterns that might become obvious, given the right choice of state space. Choosing a new state space in which to describe the same system can reveal dynamical constraints which might otherwise have been invisible. 91 We can think of the compression from physics to (say) chemistry, then, as resulting in a new configuration space for the same old system-one where points represent regions of the old space, and where every point represents a significant difference from this new (goal-relative) perspective, with the significance stemming from both the discovery of interesting new macrostates and interesting new dynamics. This operation can be iterated for some systems: biology can define a new configuration space that will consist of points representing regions of the original configuration space. Since biology is more "lossy" than chemistry (in the sense of 81 discarding more state-specific information in favor of dynamical shortcuts), the space defining a system considered from a biological perspective will be of a still lower dimensionality that the space considering the same system from a chemical perspective. The most dynamically complex systems will be those that admit of the most recompressions-the ones for whom this creation of a predictively-useful new configuration space can be iterated the most. After each coarse-graining, we'll be left with a new, lower-dimensional space wherein each point represents an importantly different state, and wherein different dynamical patterns describe the transition from state to state. That is, repeated applications of this procedure will produce increasingly compressed bitmaps, with each compression also including a novel set of rules for evolving the bitmap forward in time. We can think of this operation as akin to changing magnification scale with physical objects that display fractal-like statistical self-similarity: the self-similarity here, though, is not in shape but in the structure and behavior of different abstract configuration spaces: there's interesting 81 Note that it isn't right to say "regions of chemistry's configuration space." That would be to implicitly buy into the rigid hierarchical model I attributed to Oppenheim and Putnam a few pages back, wherein all biology is a sub-discipline of chemistry, psychology is a sub-discipline of biology, and so on. That won't do. Many of the points might well correspond to regions of the "one step lower" space, but not all will. 92 detail, but rather than being geometrically similar, it is dynamically similar. Call this dynamical self-similarity. Still, there's a clear parallel to standard statistical self-similarity: fractal dimension for normal physical objects roughly quantifies how much interesting spatial detail persists between magnification operations, and how much magnification one must do move from one level of detail to another. Similarly, dynamical complexity roughly quantifies how much interesting detail there is in the patterns present in the behavior of the system (rather than in the 82 shape of the system itself), and how much coarse-graining (and what sort) can be done while still preserving this self-similar of detail. This allows us to recover and greatly expand some of the conceptual underpinnings of fractal dimensionality as a measure of complexity-indeed, it ends up being one of the more accurate measures we discussed. 2.2 Effective Complexity: The Mathematical Foundation of Dynamical Complexity Finally, what of Shannon entropy? First, notice that this account of dynamical complexity also gives us a neat way of formalizing the state of a system as a sort of message so that its Shannon entropy can be judged: the state of a system is represented by its position in configuration space, and facts about how the system changes over time are represented as patterns in how that system moves through configuration space. All these facts can easily be expressed numerically. The deeper conceptual problem with Shannon entropy remains, though: if the correlation condition fails (which it surely still does), how can we account for the fact that there does seem to be some relationship between Shannon entropy and dynamical complexity? That is, how do we explain the fact that where there is no strict, linear correlation between changes in dynamical complexity and changes in Shannon entropy, there does indeed seem to be 82 We will consider how this quantification works in just a moment. There is a mathematical formalism behind all of this with the potential to make things far more precise. 93 a "sweet spot"-middling Shannon entropy seems to correspond to maximal complexity in the associated system. In other words, identifying complexity with compressibility leads to an immediate conflict with our intuitions. A completely random string-a string with no internal structure or correlation between individual bits-will, on this account, said to be highly complex. This doesn't at all accord with our intuitions about what complex systems look like; whatever complexity is, a box of gas at perfect thermodynamic equilibrium sure doesn't have it. This 83 observation has led a number of information theorists and computer scientists to look for a refinement on the naïve information-theoretic account. A number of authors have been independently successful in this attempt, and have produced a successor theory called "effective complexity." Let's get a brief sense of the formalism behind this view (and how it resolves the problem of treating random strings as highly complex), and then examine how it relates to the account of dynamical complexity given above. The central move from the information-content account of complexity that's built on the back of Shannon entropy to the notion of effective complexity is analogous to the move from thinking about particular strings and thinking about ensembles of strings. One way of presenting the standard Shannon account of complexity associates the complexity of a string with the length of the shortest computer program that will print the string, and then halt. The incompressibility problem is clear here as well: the shortest computer program that will print a random string just is the random string: when we say that a string S is incompressible, we're saying (among other 83 A system like that could be appropriately represented as a random string, as part of what it means for a system to be at thermodynamic equilibrium is for it to have the maximum possible entropy for a system constituted like that. Translated into a bit-string, this yields a random sequence. 94 things) that "Print S" is the shortest possible program that will reproduce S. Thus, a maximally random (incompressible) string of infinite length is infinitely complex, as the shortest program that produces it is just the string itself. Suppose that rather than think of individual strings, though, we shift our attention to ensembles of strings that share certain common features. In the language of Gell-Mann and Lloyd, suppose that rather than think about the shortest program that would reproduce our target string exactly, we think about the shortest program that would reproduce the ensemble of strings which "best represents" the target string . Gell-Mann argues that the best representative of a 84 random string is the uniform ensemble-that is, the ensemble of strings that assigns all possible strings equal probability. This is supposed to resolve the compressibility issues in the traditional information-theoretic account of complexity. It's easy to see why: suppose we want to print a random string of length n. Rather than printing n characters directly, Gell-Mann proposes that we instead write a program that prints a random character n times. The program to do this is relatively short, and so the effective complexity of a random string will rate as being quite low, despite the fact that individual random strings are incompressible. Gell-Mann is capitalizing on a higher-order regularity: the fact that all random strings are, in a certain respect, similar to one another. While there's no pattern to be found within each string, this higher-order similarity lets us produce a string that is in some sense "typical" of its type with relative ease. Conversely, a string with a certain sort of internal structure-one with a large number of patterns-is a member of a far more restricted ensemble. The collected work of Shakespeare (to use one of Gell-Mann's own examples) rates as highly complex because it (considered as a 84 Gell-Mann and Lloyd (2003). See also Foley and Oliver (2011). 95 single string) is a member of a very small ensemble of relevantly similar strings. There is very little (if anything) in Shakespeare that is well-captured by the uniform ensemble; the information, to a very large degree, is specialized, regular, and non-incidental. In other words, the effective complexity of a string is the algorithmic information content of the ensemble that "best represents" the string. If the ensemble is easy to produce (as in the case of both a random string and an entirely uniform string), then any string belonging to that ensemble is itself is low in effective complexity. If the ensemble is difficult (that is, requires a lengthy program) to produce, then any string that is a member of that ensemble is high in effective complexity. This resolves the central criticism of the algorithmic information content (i.e. Shannon) approach to defining complexity, and seems to accord better with our intuitions about what should and should not count as complex. What, then, is the relationship between effective complexity and dynamical complexity? Moreover, if effective complexity is the right way to formalize the intuitions behind complexity, why is this the case? What's the physical root of this formalism? To answer these questions, let's look at one of the very few papers yet written that offers a concrete criticism of effective complexity itself. McAllister (2003) criticizes Gell-Mann's formulation on the grounds that, when given a physical interpretation, effective complexity is troublingly observer-relative. This is a massively important point (and McAllister is entirely correct), so it is worth quoting him at length here: The concept of effective complexity has a flaw, however: the effective complexity of a given string is not uniquely defined. This flaw manifests itself in two ways. For strings that admit a physical interpretation, such as empirical data sets in science, the effective complexity of a string takes different values depending on the cognitive and practical interests of investigators. For strings regarded as purely formal constructs, lacking a physical interpretation, the effective complexity of a given string is arbitrary. The flaw derives from the fact that any given string displays multiple patterns, each of which has a different algorithmic complexity and each of which can, in a suitable context, count as the regularity of the string. 96 [...] For an example, consider a data set on atmospheric temperature. Such a data set exhibits many different patterns (Bryant 1997). These include a pattern with a period of a day, associated with the earth's rotation about its axis; patterns with periods of a few days, associated with the life span of individual weather systems; a pattern with a period of a year, associated with the earth's orbit around the sun; a pattern with a period of 11 years, attributed to the sunspot cycle; a pattern with a period of approximately 21,000 years, attributed to the precession of the earth's orbit; various patterns with periods of between 40,000 and 100,000 years, attributed to fluctuations in the inclination of the earth's axis of rotation and the eccentricity of the earth's orbit; and various patterns with periods of between 107 and 109 years, associated with variations in the earth's rate of rotation, the major geography of the earth, the composition of the atmosphere, and the characteristics of the sun. Each of these patterns has a different algorithmic complexity and is exhibited in the data with a different noise level. Any of these patterns is eligible to be considered as the regularity of the data set. Depending on their cognitive and practical interests, weather forecasters, meteorologists, climatologists, palaeontologists, astronomers, and researchers in other scientific disciplines will regard different patterns in this series as constituting the regularity in the data. They will thus ascribe different values to the effective complexity of the data set . 85 McAllister's observations are acute: this is indeed a consequence of effective complexity . 86 I think McAllister is wrong in calling this a fatal flaw (or even a criticism) of the concept, though, for reasons that should be relatively obvious. The central thrust of McAllister's criticism is that it is difficult to assign a determinate value to the effective complexity of any physical system, as that system might contain a myriad of patterns, and thus fail to be best represented by any single ensemble. The question of what effective complexity we assign a system will depend on what string we choose to represent the system. That choice, in turn, will depend on how we carve the system up-it will depend on our choice of which patterns to pay attention to. Choices like that are purpose-relative; as McAllister rightly says, they depend on our practical and cognitive interest. Given the account of science I developed in Chapter One, though, this is precisely what we 85 Ibid. pp. 303-304 86 In addition, his choice to use climate science as his leading example here is very interesting, given the overall shape of the project we're pursuing here. Chapter Five will consider the ramifications of this discussion for the project of modeling climate systems, and Chapter Seven will deal with (among other things) the policy-making implications. For now, it is more important to get a general grasp on the notion of effective complexity (and dynamical complexity). 97 should expect out of a concept designed to describe the relationship between how different branches of science view a single physical system. There's no single correct value for a system's effective complexity, because there's no single correct way to carve up a system-no single way to parse it into a string of patterns. Far from making us think that effective complexity gets it wrong, then, this should lead us to think that effective complexity gets things deeply right: the presence of a plurality of values for the effective complexity of a system reflects the methodological plurality of the natural sciences. McAllister suggests that we might instead choose to sum different values to get a final value, but his proposal is limited to summing over the complexity as defined by algorithmic information content. Because McAllister believes his observation that effective complexity contains an observer-relative element to be a fatal flaw in the concept, he doesn't consider the possibility that we might obtain a more reliable value by summing over the effective complexity values for the system. My proposal is that dynamical complexity, properly formalized, is precisely this: a sum of the effective complexity values for the different strings representing the different useful carvings of the system. While there is no single value for effective complexity, we can perfectly coherently talk about summing all the useful ways given our goals and values. The value of this sum will change as we make new scientific discoveries-as we discover new patterns in the world that are worth paying attention to-but this again just serves to emphasize the point from Chapter One: the world is messy, and science is hard. Complexity theory is part of the scientific project, and so inherits all the difficulties and messiness from the rest of the project. Dynamical complexity, in other words, offers a natural physical interpretation for the 98 formalism of effective complexity, and a physical interpretation that takes the multiplicity of ways that physical systems can be described into account. It offers a natural way to understand how the abstraction described by Gell-Mann and others relates to the actual practice of scientists. The conceptual machinery underwriting the account of science that we developed in this chapter and the last helps us get an intuitive picture of complexity and its place in science. The formalism of effective complexity provides a formalism that can be used to underwrite this intuitive formulation, making the concepts described more precise. 2.3 Conclusion, Summary, and the Shape of Things to Come In the previous chapter, we examined several different ways that "complexity" might be defined. We saw that each attempt seemed to capture something interesting about complexity, but each also faced serious problems. After arguing that none of these definitions by itself was sufficient to yield a rigorous understanding of complexity, I introduced a new concept-dynamical complexity. This chapter has consisted in a sustained description of the concept, and an argument for its role as a marker for the kind of complexity we're after when we're doing science. The insight at the heart of dynamical complexity is that complexity, at least as it concerns science, is a feature of active, changing, evolving systems. Previous attempts to define complexity have overlooked this fact to one degree or another, and have tried to account for complexity primarily in terms of facts about the static state of a system. Dynamical complexity, on the other hand, tracks facts about how systems change over time, and (moreover) embraces the notion that change over time can be tracked in numerous different ways, even for a single system. If our account of science from Chapter One is right-if science is the business 99 of identifying new ways to carve up the world such that different patterns in how the world changes over time become salient-then dynamical complexity is a concept that should be of great interest to working scientists, since it captures (in a sense) how fruitful (and how difficult) scientific inquiry into the behavior of a given system is likely to be. Finally, we saw how the formalism of effective complexity very naturally dove-tails with the intuitive conceptual machinery developed here and in Chapter One. I argued that summing over the effective complexities of different representations of the same system offers a way to quantify the dynamical complexity of the system. This value will be a moving target, and will be observer (and goal) relative to some degree. This should concern us no more than the observation that the choice of what patterns we pay attention to in science is goal-relative should trouble us, as they stem from precisely the same features of the scientific project. In Chapter Four, we will leave foundational questions behind and move on to considering some methodological questions relevant to climate science. We'll introduce the basics of climatology and atmospheric science, and examine the difficulties involved in creating a working model of the Earth's climate. From there, we will consider the particular challenges that climate science faces, given that it explicitly deals with a system of high dynamical complexity, and think about and how have those challenges been met in different fields. We'll examine why it is that scientists care about dynamical complexity, and what can be learned by assessing the dynamical complexity of a given system. In Chapter Five, I'll synthesize the two threads that have, up to that point, been pursued more-or-less in parallel and argue the global climate is a paradigmatic dynamically complex system. We'll examine how that fact has shaped the methodology of climate science, as well as how it has given rise to a number of unique problems 100 for climatologists to tackle. I shall argue that the markedly high degree of dynamical complexity in the global climate system is best dealt with by strongly interdisciplinary scientific inquiry, and that a failure to recognize the role that dynamical complexity plays in shaping the practices of some branches of science is what has led to most of the general criticism faced by climate science. In Chapter Six, we'll look at one case in particular-Michael Mann's "hockey stick" prediction-and see how the criticisms levied at Mann often result from a failure to understand the special problems faced by those studying dynamically complex systems. Finally, in Chapter Seven, we'll examine the political controversy surrounding climate science, assess various recommended responses to anthropogenic climate change, and examine the role that complexity-theoretic reasoning should play in the policy-making process. Onward, then. 101 Chapter Four A Philosopher's Introduction to Climate Models 4.0 What Have We Gotten Ourselves Into? As usual, let's begin by briefly reviewing where we are in our overall discussion, with an eye toward how to proceed from here. The last two chapters have focused very heavily on the details of certain aspects of complexity theory, and it might be easy to lose sight of our overall goal. In Chapter Two, I presented a primer on complex systems theory and surveyed various attempts to reduce the notoriously slippery notion of complexity itself to various proxy concepts, including mereological size, chaotic behavior, algorithmic incompressibility, fractal dimension, Shannon entropy, and hierarchical position. I argued (convincingly, I hope) that none of these definitions precisely captures the intuition behind complexity and that moreover, the nature of complexity is such that it is likely that no single unifying definition is forthcoming. Rather, we should aim at a constellation of related notions of complexity, each of which is tailored to the different purposes toward which complexity theory might be used. I proposed the concept of dynamical complexity as best capturing the aspects of the varied proxy concepts we considered that are most relevant to scientists seeking to understand active, dynamical complex systems in the natural world (as opposed to, say, those interested in studying aspects of abstract signals), and argued effective complexity can plausibly be taken as a physical interpretation of the existing mathematical framework of effective complexity. A system's dynamical complexity, recall, is a fact about the pattern-richness of the system's location in the configuration space defined by fundamental physics. Equivalently, we can think of it as being a fact about how many predictively useful ways the system can be carved up. Formally, a system's dynamical complexity is the sum of the 102 effective complexity values for all relevant ways of representing the system. See Section 2.2.2 for more on this. In this chapter, I would like to narrow our focus and apply some of the concepts we've developed over the last hundred (or so) pages to more practical concerns. In Chapter Zero, I argued that the issue of global climate change is perhaps the most pressing scientific problem of our time, and suggested that the paucity of philosophical engagement with this problem is a travesty in need of serious attention. Chapter One consisted of a systematic description of the kind of contribution that philosophers can be expected to make to problems like this one, and Chapters Two and Three laid the groundwork for making some contributions of that kind. In this chapter, we will start to examine climate science itself. As I have repeatedly emphasized, philosophy is at its best when it makes contact with the social and scientific issues of the day, and it is difficult to imagine a more pressing social and scientific problem than that of global climate change. Here's how this chapter will go. In Section 4.1, I will offer a brief overview to some of the central concepts and terminology of climate science. The focus of this section will be not on the controversial aspects of climatology, but just on introducing some of the basic jargon and ideas behind the science; at this point, we will have very little to say about what makes climate science particularly difficult, or about the nature of the political dispute raging in the wake of the science. Rather, our goal shall be just to get enough of the basics on the table to allow for an intelligible discussion of some of the specifics that are of particular philosophical interest. We'll introduce these concepts by way of a concrete examination of the practice of model building in climate science. Sticking with the generally dialectical style we've been using so far, we'll begin with a 103 simple, intuitive observation about the relationship between the climate and incoming solar radiation and build up from there. As we run up against the short-comings of each candidate-model we consider, we'll introduce some more terminology and concepts, incorporating them into increasingly more sophisticated models. By the end of Section 4.1, we will have constructed a working (if still quite basic) climate model piece by piece. Section 4.2 will build from there (and will lay the groundwork for the next chapter). With a firm grasp on the basic model we've constructed in Section 4.1, we'll survey some of the considerations that guide climatologists in their construction of more elaborate models. We'll examine the notion of a "hierarchy of models" in climate science, and explore the connection between this hierarchy and the discussions of science and complexity theory we've had so far. We'll take a look at the diverse family of models (so-called "Earth models of intermediate complexity") that occupy the territory between the relatively simple model we've constructed here and the elaborate supercomputer-dependent models that we'll consider in Chapter Five. We'll think about what climate scientists mean when they say "intermediate complexity," and how that concept might relate to dynamical complexity. Finally, we'll consider some of the limitations to the scientific methodology of decomposing systems into their constituent parts for easier analysis. We'll explore the parallels between the development of complexity-theoretic reasoning in climate science and biology, two more striking examples of sciences which have begun to turn away from the old decompositionist-centered scientific method. This critique will lay the groundwork for Chapter Five, in which we'll examine the elaborate, holistic, complicated family of cutting-edge climate models, which seek to represent the climate as a unified complex system within a single comprehensive model. 104 4.1 Fundamentals of Climate Science Climate science is a mature science, with a large body of technically-sophisticated and specialized literature. The goal of giving a complete and substantive introduction to its fundamentals in anything as short as a single section of this dissertation is surely impossible to achieve. I'll refer the curious reader to a number of secondary sources for further clarification 87 of the terms I'll present here, as well as for elaboration on concepts I don't discuss. My objective here is just to present the bare minimum of terminology necessary to make the rest of our discussion comprehensible. I'll highlight some of the subtleties later on in this chapter (and the next), but many important details will necessarily be left out in the cold (so to speak), and some of the concepts I do discuss will be simplified for presentation here. Whenever possible I'll flag these simplifications in a footnote. Let's start with distinguishing between the study of the climate and the study of the weather. We can think of weather as a set of short-term, more-or-less localized facts about the prevailing atmospheric conditions in particular places. Questions about whether or not it will rain tomorrow, what tonight's low temperature will be, and so on are (generally speaking) questions about the weather. The study of climate, on the other hand, consists in studying both the long-term trends in the prevalence of certain weather events in particular places (is it, on average, raining more or less this century than it was last century?), and also in studying the factors that produce particular weather events (e.g. the interplay between ocean and atmosphere temperatures that produces hurricanes generally). Standard definitions used by climatologists 87 Dawson & Spannagle (2009) is perhaps the most comprehensive and accessible general reference; I'd recommend that as a first stop on a more detailed tour of the climate science literature. 105 resemble something like "the mean [weather] state together with measures of variability or fluctuations, such as the standard deviation or autocorrelation statistics for the period ." 88 Additionally (and perhaps more saliently), climate study includes the identification of factors that drive the evolution of these long-term trends, and this is the aspect of climatology that has drawn the most attention recently. The claim that the activity of human beings is causing the average temperature to increase, is a claim of this third kind. It's also worth emphasizing that since the study of climate is concerned with the factors that produce weather conditions, it is not necessarily limited to the study of atmospheric conditions. In particular, the relationship between the ocean and the atmosphere is a very significant sub-field of climate science , while 89 those who study the weather directly are significantly less concerned with exploring the dynamics of the ocean. Here's a question that might immediately occur to us: what exactly counts as "long-term" in the relevant sense? That is, at what time-scale does our attempt to predict facts about temperature, precipitation, &c. cease to be a matter of weather prediction (that is, the kind of forecasting you might see on the nightly news), and become a matter of climate prediction? By now, our answer to this question should be fairly easy to predict: there is no concrete line other than that of actual scientific practice. As with all other special sciences, the difference between weather forecasting and climatology is defined only by the research questions that drive scientists working in their respective disciplines. There are clear cases that fall into one or another discipline-the question of how likely it is that it will rain tomorrow is clearly a question 88 Schneider (2009), p. 6 89 For an obvious example, consider the importance of the El Nino-Southern Oscillation-a coupled atmosphere/ocean phenomenon that occurs cyclically in the Pacific ocean region (and has received significant media attention). 106 for weather forecasting, while the question of how the Earth's changing axis of rotation contributes to ice ages is clearly a question for climatology-but many questions will be of interest to both disciplines, and there is bound to be significant overlap in both topic and method. It is worth pointing out, as a brief historical aside, that this reunification is a relatively recent event. Until recently (as late as the middle of the 20th century), the study of climate fell into three largely independent camps: short-term weather forecasting, climatology, and theoretical meteorology. Practical forecasting and climatology were almost purely descriptive sciences, concerned solely with making accurate predictions without concern for the mechanisms behind those predictions. Weather forecasts in particular were devoid of any theoretical underpinnings until well into the 20th century. The most popular method for forecasting the weather during the first part of the 20th century involved the use of purely qualitative maps of past weather activity. Forecasters would chart the current state to the best of their ability, noting the location of clouds, the magnitude and direction of prevailing winds, the presence of precipitation, &c. Once the current state was recorded on a map of the region of interest, the forecasters would refer back to past charts of the same region until they found one that closely resembled the chart they had just generated. They would then check to see how that past state had evolved over time, and would base their forecast of the current situation on that past record. This turned forecasting into the kind of activity that took years (or even decades) to become proficient in; in order to make practical use of this kind of approach, would-be forecasters had to have an encyclopedic knowledge of past charts, as well as the ability to make educated guesses at how the current system might diverge from the most similar past cases . Likewise, climatology at the time was 90 90 For a detailed discussion of the evolution of the science of forecasting, see Edwards (2010) 107 more-or-less purely descriptive, consisting of the collection and analysis of statistical information about weather trends over long time-scales, and relying almost exclusively on graphical presentation. Although some inroads were being made in theoretical meteorology at the same time-mostly by applying cutting-edge work in fluid dynamics to the flow of air in the upper atmosphere-it wasn't until the advent of the electronic computer in the 1950s and 1960s, which made numerical approximation of the solutions to difficult-to-solve equations finally feasible on a large scale, that forecasting and climatology moved away from this purely qualitative approach. Today, the three fields are more tightly integrated, though differences in the practical goals of weather and climate forecasting-most significantly, the need for weather forecasts to be generated quickly enough to be of use in (say) deciding whether or not to take an umbrella to work tomorrow-still give rise to somewhat different methods. We will return to these issues in Chapter Five when we discuss the role of computer models in climate science. We can think of the relationship between weather and climate as being roughly analogous to the relationship between (say) the Newtonian patterns used to predict the behavior of individual atoms, and thermodynamics, which deals with the statistical behavior of collections of atoms. The question of exactly how many atoms we need before we can begin to sensibly apply patterns that make reference to average behavior-patterns like temperature, pressure, and so on-just isn't one that needs a clear answer (if this dismissive shrug of an answer bothers you, review the discussion of the structure of the scientific project in Chapter One). When we apply the patterns of thermodynamics and when we apply the dynamics of Newtonian mechanics to individual atoms is a matter of our goals, not a matter of deep metaphysics. Precisely the same is true of the line between weather forecasting and climatology: which set of patterns we choose to 108 pay attention to depends on our goals. For more on the question of how to individuate particular special sciences, see Section 1.4. For now, we will set this question aside and focus on climate science as it is practiced. As a general rule of thumb, weather forecasting is concerned with predicting particular events, and climatology is concerned with predicting trends. This definition is good enough for our purposes, at least for now. 4.1.1 Basic Energy Balance Models What, then, are the patterns of interest to climate scientists? In general, climate scientists are interested in predicting the long-term behavior of the Earth's atmosphere (as well as the systems that are tightly coupled to the atmosphere). A tremendous number of patterns turn out to play a role in this general predictive enterprise (indeed, this is part of what makes climate science a complex-systems science; more on this below), but not all of them are necessarily of immediate interest to us here . Since our ultimate goal is to focus our discussion in on anthropogenic 91 climate change, we can limit our attention to those factors that might play a significant role in understanding that problem. To begin, it might be helpful to get a very basic picture of how the Earth's climate works, with particular attention to temperature, since this is a feature of the climate that will be of great interest to us as we proceed. Like most contemporary science, climate science relies very heavily on the construction of models-artifacts which are supposed to represent interesting aspects of a physical system . 92 91 In particular, it's worth flagging that (at least recently) economic patterns have become very salient in the prediction of the time-evolution of the climate: as the activity of human civilization has become a more important factor in forcing the climate state, patterns that are relevant in predicting that activity have become relevant in predicting climate states as well. We will explore the connection with economic patterns more in the next two chapters. 92 I'm using "artifact" in a very broad sense here. Some models are themselves physical systems (consider a model airplane), while others are mathematical constructions that are supposed to capture some interesting behavior of the system in question. The main point of model-building is to create something that can be more easily manipulated and 109 The simplest climate model is the energy balance model, which is concerned with the amount of energy received and emitted by the Earth. All matter emits electromagnetic radiation, and the 93 wavelength (λ) of that emitted radiation straightforwardly varies with the temperature of the object. The Sun, a relatively hot object, emits E/M radiation across a very wide spectrum, from very short-wave gamma radiation (λ > 10-12 m) to very long-wave microwave and radio radiation (λ > 102 m). Some of the radiation emitted by the Sun, of course, is in the very narrow range of the E/M spectrum that is visible to the naked human eye (λ = ~.4-.8 x 10-6 m). The surface temperature of the sun is approximately 5,778K; this means that the sun's peak E/M emission-that is, the area of the E/M spectrum with the most intense emission-falls into this visible spectrum, at somewhere around λ = .5-.6 x 10-6 m. This corresponds to light that normal humans perceive as yellowish-green (the sun appears primarily yellow from Earth because of atmospheric scattering of light at the blue end of the visible spectrum). Similarly, the Earth emits electromagnetic radiation. However, the Earth is (thankfully) much cooler than the sun, so it radiates energy at a significantly different wavelength. Peak E/M emission wavelength is inversely proportional to the temperature of the radiator (this is why, for instance, the color of a heating element in a toaster progresses from red, to orange, to yellow as it heats up), and the Earth is sufficiently cold so that its peak E/M emission is somewhere around λ = 20 x 10-6 m. This means that the Earth's emission is mostly in the infrared portion of the spectrum, a fact which plays a very significant role in the dynamics of the greenhouse effect (see Section 4.1.3). studied than the object itself, with the hope that in seeing how the model behaves, we can learn something interesting about the world. There is a thicket of philosophical issues here, but a full exploration of them is beyond the scope of this project. The philosophical significance of one class of models in particular-computer simulations-will be the primary subject of Chapter Five, but for a more general contemporary overview of representation and model-building, see van Fraassen (2010). 93 Or, at least, all matter with temperature greater than absolute zero. 110 The input of energy from the sun and the release of energy (in the form of infrared radiation) by the Earth dominate the temperature dynamics of the planet. At the simplest level, then, understanding how the temperature of the Earth changes over time is just a matter of balancing an energy budget: if the Earth absorbs more energy than it emits, it will warm until it reaches thermal equilibrium . The simplest energy balance models, so-called "zero-dimensional energy 94 balance models," (ZDEBM) model the Earth and the Sun as point-like objects with particular temperatures, absorption characteristics, and emission characteristics. We can quantify the amount of energy actually reaching any particular region of the Earth (e.g. a piece of land, a layer of the atmosphere, or just the Earth simpliciter for the most basic ZDEBM) in terms of Watts per square meter (Wm-2). The amount of energy reaching a particular point at a given time is called the radiative forcing active on that point . Assuming that the Earth is in 95 equilibrium-that is, assuming that the radiated energy and the absorbed energy are in balance-the simplest possible ZDEBM would look like this: S = F (4a) Here, S represents the amount of solar energy input to the system (i.e. absorbed by the Earth), and F represents the amount of energy radiated by the Earth. How much solar energy does the 94 A very simple model of this sort treats the Earth as an "ideal black body," and assumes that it reflects no energy. Thus, the model only needs to account for the energy that's radiated by the Earth, so we can work only in terms of temperature changes. This is an obvious simplification, and the addition of reflection to our model changes things (perhaps even more significantly than we might expect). We'll discuss this point more in a moment. 95 The Intergovernmental Panel on Climate Change (IPCC) uses the term "radiative forcing" somewhat idiosyncratically. Since they are concerned only with possible anthropogenic influences on the climate system, they express radiative forcing values in terms of their deviation from pre-Industrial levels. In other words, their values for the amount of energy reaching certain points on the Earth "subtract out" the influence of factors that they have good reason to think are unrelated to human intervention on the climate. These radiative forcing values might be more properly called net anthropogenic radiative forcing; an IPCC value of (say) .2 Wm-2 represents a net increase of .2 Watts per square meter, over and above the radiative forcing that was already present prior to significant human impacts. Unless otherwise specified, I will use 'radiative forcing' in the standard (non-IPCC) sense. 111 Earth receive? Well, just however much of the sun's energy actually reaches as far as the Earth multiplied by the size of the area of the Earth that the sun is actually shining on. Filling in some values, we can expand that to: (4b)σTS = 4 So = 4p = F In this expanded equation, So is the solar constant (the amount of energy radiated by the sun which reaches Earth), which is something like 1367 Wm-2. Why is this value divided by four? Well, consider the fact that only some of the Earth is actually receiving solar radiation at any particular time-the part of the Earth in which it is day time. Without too much loss of accuracy, we can think of the Earth as a whole as being a sphere, with only a single disc facing the sun at any given time. Since all the surface areas we'll be dealing with in what follows are areas of circles and disks, they're all also multiplied by πr2; for the sake of keeping things as clean-looking as possible, I've just factored this out except when necessary, since it is a common multiple of all area terms. That's the source of the mysterious division by 4 in (4b), though: the area of the Earth as a whole (approximated as a sphere) is 4 πr2, while the area of a disk is just πr2. On the other side of the balance, we have σTp4 = F. The value σTp4 is obtained by applying the Stefan-Boltzmann law, which gives the total energy radiated by a blackbody (F) as a function of its absolute temperature (Tp), modified by the Stefan-Boltzmann constant (σ), which itself is derived from other constants of nature (the speed of light in a vacuum and Planck's constant). Filling in actual observed values, we get: [(5.670373 × 10 W m )K ](255K )4 [(1367 W m )−2 = −8 −2 −4 4 (4c) 112 Unfortunately, evaluating this leaves us with 341.75 Wm-2 = 240 Wm-2, which is (manifestly) not valid-though at least both sides come out on the same order of magnitude, which should suggest that we're on to something. What's the problem? In order to diagnose where things are going wrong here, we'll have to dig more deeply into the energy balance class of models, and start to construct a more realistic model-one which begins to at least approximately get things right. 4.1.2 Albedo The basic ZDEBM of the climate is roughly analogous to the simple "calorie balance" model of nutrition-if you consume more calories than you burn each day you will gain weight, and if you burn more calories than you consume you will lose weight. In both cases, while the model in question does indeed capture something accurate about the system in question, the real story is more complicated. In the case of nutrition, we know that not all calories are created equal, and that the source of the calories can make a difference: for instance, consuming only refined carbohydrates can negatively impact insulin resistance, which can affect the body's metabolic pathways in general, leading to systemic changes that would not have occurred as a result of consuming an equal amount of calories from protein . Analogously, the most simple 96 ZDEBM-in which the Earth and the sun are both featureless points that only absorb and radiate energy-doesn't capture all the factors that are relevant to temperature variation on Earth. 96 Even more strongly, it might be the case that calories in and calories out are not entirely independent of one another. That is, there might be interesting feedback loops at play in constructing an accurate calorie balance: a fact which is obfuscated in this simple presentation. For example, it might be the case that consuming a lot of calories leads to some weight gain, which leads to low self-esteem (as a result of poor body-image), which leads to even more calorie consumption, and so on. This sort of non-linear multi-level feedback mechanism will be treated in detail in Chapter Five, but will be ignored for the time being. 113 Adding some more detail, consider a slightly more sophisticated ZDEBM, the like of which actually represents the planet in enough detail to be of actual (though limited) predictive use. To begin, we might note that only some of the wide spectrum of E/M radiation reaching the Earth actually makes it to the planet's surface. This reflects the fact that our first approximation of the Earth as a totally featureless ideal black-body is, as we've seen, very inaccurate: in addition to radiating and absorbing, the Earth also reflects some energy. The value representing the reflectance profile of a particular segment of the planet (or the entire planet, in this simple model) is called the albedo. At the very least, then, our ZDEBM is going to have to take albedo into account: if we allow our model to correct for the fact that not all of the energy that reaches the Earth is actually absorbed by the Earth, then we can approach values that accurately represent the way things are. Earth's albedo is highly non-uniform, varying significantly over both altitude and surface position. In the atmosphere, composition differences are the primarily relevant factors, while on the ground color is the most relevant characteristic. Cloud cover is certainly the most significant factor for calculating atmospheric albedo (clouds reflect some energy back to space). On the ground, the type of terrain makes the most significant difference: the ocean reflects very little energy back to space, and snow reflects a great deal (dry land falls somewhere between these two extremes, depending on what's on it). However, we're getting ahead of ourselves: ZDEBMs don't take any of this variation into account, and operate on the simplifying assumption that albedo can be averaged for the planet (in much the same way that emission and absorption can be). In all cases, though, albedo is expressed as a dimensionless fraction, with a value between 0 and 1 (inclusive). 0 albedo represents total absorption (a perfectly black surface), and 1 albedo 114 represents a total reflection (a perfectly white surface). To get an idea of the relative values at play here, consider the following table. 97 Surface Albedo Equatorial oceans at noon 0.05 Dense forest 0.05-0.10 Forest 0.14-0.20 Modern city 0.14-0.18 Green crops 0.15-0.25 Grassland 0.16-0.20 Sand 0.18-0.28 Polar oceans with sea ice 0.6 Old snow 0.4-0.6 Fresh snow 0.75-0.95 Clouds 0.40-0.9 Spherical water droplet with low angle of incidence 98 0.99 Taking albedo into account will clearly affect the outcome of the model we've been working with. We were implicitly treating the Earth as if it were a perfect absorber-an object with albedo 0-which would explain why our final result was so far off base. Let's see how our result changes when we jettison this assumption. We will stick with the simplification we've been working with all along so far and give a single average albedo value for the Earth as a whole, a value which is generally referred to as the "planetary albedo." More nuanced energy 97 Adapted from Ricklefs (1993) 98 This explains why, in practice, the albedo of large bodies of water (e.g. oceans or very large lakes) is somewhat higher than the listed value. Choppy water has a layer of foam (whitecap) on top of it, which has an albedo value that's much closer to the value for a water droplet than to the value for calm water. The value of the oceans as a whole, then, is somewhere between the values of a water droplet and calm water. This is an example of the sort of small space-scale difficulty that causes problems for the more sophisticated general circulation model, discussed in more detail in Chapter Six. 115 balance models, which we will discuss shortly, might refine this assumption somewhat. Our modified model should decrease the value of S (the amount of energy absorbed by the Earth) by a factor that is proportional to the albedo: as the albedo of the planet increases it absorbs less energy, and as the albedo decreases it absorbs more. Let's try this, then: (4d)σT4 S (1−α)o = 4p In the special case where the Earth's albedo α is 0, (4d) reduces to (4c), since 1-α is just 1. OK, so once again let's fill in our observed values and see what happens. We'll approximate α as being equal to .3, so now we have: (4e) [(5.670373 × 10 ) W m K ](255K )4 [(1367 W m )(1−.3)]−2 = −8 −2 −4 4 Which gives us a result of: 239.225 Wm-2 = 240 Wm-2 (4f) This is far more accurate, and the remaining difference is well within the margin of error for our observed values. So now we're getting somewhere. We have a simple model which, given a set of observed values, manages to spit out a valid equality. However, as we noted above, the purpose of a model is to help us make predictions about the system the model represents, so we shouldn't be satisfied just to plug in observed values: we want our model to tell us what would happen if the values were different than they in fact are. In this case, we're likely to be particularly interested in Tp: we want to know how the temperature would change as a result of changes in albedo, 116 emitted energy, or received energy. Fortunately, it's only a trivial matter of algebraic manipulation to rearrange our last equation to solve for Tp: = Tp √4 4σ(S (1−α)o (4g) We're now free to plug in different values for incoming solar radiation and planetary albedo to see how the absolute temperature of the planet changes (try it!). But wait: something is still amiss here. By expressing the model this way, we've revealed another flaw in what we have so far: there's no way to vary the amount of energy the planet emits. Recall that we originally expressed F-the total energy radiated by Earth as a blackbody-in terms of the Stefan-Boltzmann law. That is, the way we have things set up right now, the radiated energy only depends on the Stefan-Boltzmann constant σ (which, predictably, is constant) and the absolute temperature of the planet Tp. When we set things up as we did just now, it becomes apparent that (since the Stefan-Boltzmann constant doesn't vary), the amount of energy that the planet radiates depends directly (and only) on the temperature. Why is this a problem? Well, we might want to see how the temperature varies as a result of changes in how much energy the planet radiates . That is, we might want to figure out how the temperature would change if we 99 were to add an atmosphere to our planet-an atmosphere which can hold in some heat and alter the radiation profile of the planet. In order to see how this would work, we need to understand how atmospheres affect the radiation balance of planets: we need to introduce the greenhouse effect and add a parameter to our model that takes it into account. 99 In fact, there's another clue that something's not right here. Solving the equation using the values we've got so far gives us a temperature of 255K, which is significantly below the freezing point of water (it's around 0 degrees F, or -18 degrees C). As you can easily verify, this is not the temperature of the planet's surface, at least most of the time. Something is wrong here. Hang in there: we'll see the explanation for this anomaly soon, in Section 4.1.3. 117 4.1.3 The Greenhouse Effect and Basic Atmospheric Physics So how does the greenhouse effect work? To begin, we should note that as some skeptics 100 of anthropogenic climate change have pointed out, the term "greenhouse effect" is somewhat misleading: the mechanics of the effect bear only a passing resemblance to the mechanics of man-made greenhouses. Artificial greenhouses are kept warmer than the ambient environment primarily through a suppression of convection: that is, the glass in the greenhouse prevents warm air-which is less dense than cold air, and so will tend to rise above it-from rising away from ground level, and thus keeps conditions warmer than they would be otherwise. A similar mechanism is at work when you leave your car parked in the sun on a warm day: the interior heats up, but because the cabin is air-tight (at least on the timescales of interest to you during your trip to the shopping mall or grocery store), the warmer air inside the car and the cooler air outside the car cannot circulate, so the temperature increase can build up over time. The planetary greenhouse effect operates very differently. The layers of the Earth's atmosphere are not closed systems in this sense, and while convection impediment can play a role in increasing radiative forcing felt on the ground-the fact that cloudy nights are generally warmer than clear nights is partially explained by this effect-it is not the driving factor in keeping the surface of the Earth warm. Rather than blocking the motion of air itself-convection-the greenhouse effect operates primarily by altering the balance of radiation that is emitted by the planet (conveniently, this is 100 Gerlich and Tscheuschner (2009). This paper should be taken with a very large grain of salt (a full shaker would perhaps be even better), as the arguments Gerlich and Tscheuschner make about the "falsification" of the greenhouse effect are highly suspect. Halpern et. al. (2010) argue convincingly that Gerlich and Tscheuschner fundamentally misunderstand much of the involved physics. Still, they are (at least) correct on this point: the atmospheric greenhouse effect is very different from the effect involved in glass greenhouses. 118 just what is missing from the model we've constructed so far). Up to this point, recall, we've been treating the Earth as if it is a naked point: the only feature we've added thus far is planetary albedo, which can be thought of as just preventing some energy from reaching the planet in the first place. This is reflected (no pun intended) in the fact that our albedo factor α modifies the value of the solar radiance term So directly: albedo comes in on the left side of the equation on our model. What we're looking for now, remember, is something that modifies the value on the right side of the equation. In order to do that, we have to tinker with the energy not before it is received, but as it is released back into space. This is what the greenhouse effect does. But how? Departing from our ZDEBM for a moment, consider the way the atmosphere of the Earth is actually structured. The Earth's atmosphere is highly non-uniform in several different ways. Most importantly for us right now, the atmosphere is an extremely heterogeneous mixture, containing significant amounts of several gasses, trace amounts of many more, and small airborne solids (e.g. specks of dust and soot) collectively called "aerosols." Ignoring aerosols for the moment (which are far more relevant to albedo calculation than to the greenhouse effect ), the composition of the atmosphere looks like this : 101 102 101 Aerosols like dust, soot, and sulfate aerosols (which are a byproduct of fossil fuel combustion) modify the albedo directly and indirectly. Direct modification comes as a result of radiation scattering (increasing the albedo of the atmosphere in which they are suspended, providing a kind of "miniature shade"). Indirect modification comes as a result of their action as nuclei of cloud condensation: they make it easier for clouds to form in the atmosphere by acting as "seeds" around which water vapor can condense into clouds. This leads to increased cloud formation and average cloud lifespan (increasing albedo), but also reduced precipitation efficiency (since less water vapor is needed to form clouds, so clouds that do form are less moisture-dense). Aerosols thus play an important (and complicated) role in climate forcing: a role which is beyond the scope of our current discussion. They will be discussed in more detail when we consider feedback mechanisms in Section 4.2. 102 Source for figures: Carbon dioxide: NOAA (2012), Methane: IPCC AR4 (2007). 119 Gas Volume Nitrogen (N2) 780,840 ppmv (78.084%) 103 Oxygen (O2) 209,460 ppmv (20.946%) Argon (Ar) 9,340 ppmv (0.9340%) Carbon dioxide (CO2) 393.65 ppmv (0.039365%) Neon (Ne) 18.18 ppmv (0.001818%) Methane (CH4) 1.77 ppmv (0.000177%) Helium (He) 5.24 ppmv (0.000524%) Krypton (Kr) 1.14 ppmv (0.000114%) Hydrogen (H2) 0.55 ppmv (0.000055%) Nitrous oxide (N2O) 0.3 ppmv (0.00003%) Carbon monoxide (CO) 0.1 ppmv (0.00001%) Xenon (Xe) 0.09 ppmv (0.000009%) Ozone (O3) 0.0 to 0.07 ppmv (0 to 0.000007%) 104 Nitrogen dioxide (NO2) 0.02 ppmv (0.000002%) Iodine (I2) 0.01 ppmv (0.000001%) Ammonia (NH3) trace Water vapor (H2O) ~0.40% over full atmosphere, typically 1%-4% at surface Fig. 4.1 Different gases have different absorption properties, and so interact differently with various wavelengths of radiation. Radiation of a given wavelength may pass almost unimpeded through relatively thick layers of one gas, but be almost totally absorbed by even small amounts of another gas. This is the source of the greenhouse effect: the composition of the atmosphere directly affects how much radiation (and of which wavelengths) is able to escape to space. Recall that the wavelength of the energy radiated by an object depends on its absolute 103 "ppmv" stands for "parts per million by volume." 104 Ozone composition varies significantly by vertical distance from the surface of the Earth, latitude, and time of year. Most ozone is concentrated in the lower-to-mid stratosphere (20-35 km above the surface of the Earth), and there is generally less ozone near the equator and more toward the poles. Ozone concentration is at its highest during the spring months (March-May and September-November for the Northern and Southern hemispheres, respectively). 120 temperature, and that this means that (contrary to the model we've been working with so far), the temperature of the Earth depends on the composition of the atmosphere. Here's a simple account of the physics behind all this. Molecules of different gases have different molecular structures, which (among other things) affects their size and chemical properties. As incoming radiation passes through the atmosphere, it strikes a (quite large) number of different molecules. In some cases, the molecule will absorb a few of the photons (quanta of energy for electromagnetic radiation) as the radiation passes through, which can push some of the electrons in the molecule into an "excited" state. This can be thought of as the electron moving into an orbit at a greater distance from the nucleus, though it is more accurate to simply say that the electron is more energetic. This new excited state is unstable, though, which means that the electron will (eventually) "calm down," returning to its previous ground state. Because energy is conserved throughout this process, the molecule must re-emit the energy it absorbed during the excitation, which it does in the form of more E/M radiation, which might be of different wavelengths than the energy originally absorbed . Effectively, the gas molecule 105 has "stored" some of the radiation's incoming energy for a time, only to re-radiate it later. More technically, the relationship between E/M radiation wavelength and molecular absorption depends on quantum mechanical facts about the structure of the gas molecules populating the atmosphere. The "excited" and "ground" states correspond to electrons transitioning between discrete energy levels, so the wavelengths that molecules are able to absorb and emit depend on facts about which energy levels are available for electrons to 105 Though, of course, this means that the number of photons will also have to be different, unless the energy difference is accounted for in some other way. 121 transition between in particular molecules. The relationship between the energy change of a given molecule and an electromagnetic wave with wavelength λ is: 106 ΔE = ħ/λ (4h) where ħ is the reduced Planck constant (h/2π), so larger energy transitions correspond to shorter wavelengths. When ΔE is positive, a photon is absorbed by the molecule; when ΔE is negative, a photon is emitted by the molecule. Possible transitions are limited by open energy levels of the atoms composing a given atom, so in general triatomic molecules (e.g. water, with its two hydrogen and single oxygen atoms) are capable of interesting interactions with a larger spectrum of wavelengths than are diatomic molecules (e.g. carbon monoxide, with its single carbon and single oxygen atoms), since the presence of three atomic nuclei generally means more open energy orbital states. 107 Because the incoming solar radiation and the outgoing radiation leaving the Earth are of very different wavelengths, they interact with the gasses in the atmosphere very differently. Most saliently, the atmosphere is nearly transparent with respect to the peak wavelengths of incoming radiation, and nearly opaque (with some exceptions) with respect to the peak wavelengths of outgoing radiation. In the figure below, the E/M spectrum is represented on the x-axis, and the absorption efficiency (i.e. the probability that a molecule of the gas will absorb a photon when it encounters an E/M wave of the given wavelength) of various molecules in Earth's atmosphere is represented on the y-axis. The peak emission range of incoming solar radiation is colored 106 All of what follows here holds for simple atoms as well, though free atoms are relatively rare in the Earth's atmosphere, so the discussion will be phrased in terms of molecules. 107 For details, see Mitchell (1989) 122 yellow, and the peak emission range of outgoing radiation is colored blue (though of course some emission occurs from both sources outside those ranges) .108 FIG. 4.2 Note the fact that incoming solar radiation is not absorbed efficiently by any molecule, whereas outgoing radiation is efficiently absorbed by a number of molecules, particularly carbon dioxide, nitrous oxide, water vapor, and ozone. This is the source of the greenhouse effect. A more apt metaphor for the effect, then, might be the "one-way mirror" effect. Rather than acting like a greenhouse (which suppresses convection), the presence of a heterogeneous atmosphere on Earth acts something like an array of very small one-way mirrors, permitting virtually all incoming radiation to pass relatively unimpeded, but absorbing (and later re-radiating) much of the energy emitted by the planet itself. Of course this too is just a metaphor, since true mirrors are reflective (rather than radiative), and changing the reflection profile of the system (as we've seen) changes the albedo, not the radiative values. Moreover, 108 Figure adapted from Mitchell (op. cit.) 123 while mirrors are directional, the reradiation of energy from greenhouse gasses is not: the emitted photons might travel in any direction in the atmosphere, possibly resulting in their reabsorption by another molecule. Still, it can be useful to keep this picture in mind: adding more greenhouse gasses to the atmosphere is rather like adding more of these tiny mirrors, trapping energy for a longer time (and thus allowing the same amount of energy to have a greater net radiative forcing effect) than it otherwise would be. The greenhouse effect explains, among other things, why the temperature of Earth is relatively stable during both the days and nights. On bodies without an atmosphere (or without an atmosphere composed of molecules that strongly interact with outgoing radiation), an absence of active radiative forcing (during the night, say) generally results in an extreme drop in temperature. The difference between daytime and nighttime temperatures on Mercury (which has virtually no atmosphere) is over 600 degrees C, a shift which is (to put it mildly) hostile to life. With an atmosphere to act as a heat reservoir, though, temporary removal of the active energy source doesn't result in such an immediate and drastic temperature drop. During the Earth's night, energy absorbed by the atmosphere during the day is slowly re-released, keeping surface temperatures more stable. A similar effect explains why land near large bodies of water (oceans or very large lakes) tends to have a more temperate climate than land that is isolated from water; large bodies of water absorb a significant amount of solar radiation and re-release it very slowly, which tends to result in less extreme temperature variation . 109 109 The clever reader will note that this implies that the water on Earth's surface plays a significant role in regulating the overall climate. This is absolutely true (aren't you clever?), and the most advanced climate models are, in effect, models of atmospheric and aquatic dynamics that have been "coupled" together. So far, though, this too is a detail that is beyond the scope of our discussion (and the simple model we've been considering). We'll return to this point in the next chapter. 124 How do we square this with the ZDEBM we've been working with so far? As we noted above, the model as we've expressed it suggests that the Earth's temperature ought to be somewhere around 255K, which is below the freezing point of water. The solution to this puzzle lies in recognizing two facts: first that the effective temperature of the planet-the temperature that the planet appears to be from space-need not be the same as the temperature at the surface, and second that we've been neglecting a heat source that's active on the ground. The second recognition helps explain the first: the greenhouse gasses which re-radiate some of the outgoing energy keep the interior of the atmosphere warmer than the effective surface. If this seems strange, think about the difference between your skin temperature and your core body temperature. While a healthy human body's internal temperature has to remain very close to 98.6 degrees F, the temperature of the body along its radiative surface-the skin-can vary quite dramatically (indeed, that's part of what lets the internal temperature remain so constant). At first glance, an external observer might think that a human body is much cooler than it actually is: the surface temperature is much cooler than the core temperature. Precisely the same thing is true in the case of the planet; the model we've constructed so far is accurate, but it has succeeded in predicting the effective temperature of the planet-the temperature that the planet appears to be if we look at it from the outside. What we need now is a way to figure out the difference between the planet's effective temperature Tp and the temperature at the surface, which we can call Ts. Let's think about how we might integrate all that into the model we've been building. It might be helpful to start with an explicit statement of the physical picture as it stands. We're still working with an energy balance model, so the most important thing to keep in mind is just the 125 location of radiative sources and sinks; we know that all the radiation that comes in has to go out eventually (we're still assuming things are in equilibrium, or rather close to it). So here's what we have. Incoming solar radiation reaches the Earth, passing mostly unimpeded through the atmosphere. It reaches the surface of the Earth, where some of it is immediately reflected, 110 which we've accounted for already by building in a term for albedo. The remainder is absorbed by the Earth. Later, it is reradiated, but at a very different wavelength than it was when it came in. On its way out, some of this radiation is absorbed by greenhouse gas molecules in the atmosphere, and the rest of it passes back out into space. The radiation that is absorbed by the atmosphere creates (in effect) a new source of radiation, which radiates energy both back toward the surface and out to space. Our picture, then, consists of three sources: the sun (which radiates energy to the surface), the surface (which radiates energy to the atmosphere and space), and the atmosphere (which radiates energy to the surface and space). The true temperature of the surface Ts, then, is a function of both the radiation that reaches it from the sun and the radiation that reaches it from the atmosphere after being absorbed and re-emitted. Let's see how to go about formalizing that. Recall that before we had the radiation balance of the planet, which predicts the effective temperature of the planet as seen from the outside: σT4 S (1−α)o = 4p (4d) OK, so how shall we find the actual surface temperature of the planet? To start, let's note that we can model the atmosphere and the surface of the Earth as two "slabs" that sit on top of one 110 For simplification, we'll just assume that all of it passes unimpeded; this is very close to being the case. 126 another, each with approximately the same area. The surface of the Earth radiates energy upward only (i.e. to the atmosphere and space), while the atmosphere radiates energy in both directions (i.e. back to the surface and to space). So far, recall, we've been treating the part of the Earth absorbing energy from the sun as a uniform disk with an area equal to the "shadow" of the sun (that is, 1⁄4 the area of the entire Earth's surface); this is a fairly good approximation, since we're already disregarding variations in albedo and emissivity across different latitudes and longitudes (that's part of what it means to be a zero-dimensional model). We can think of the atmosphere, then, as consisting of another slab with approximately the same surface area as the surface itself. This is not quite right, but it is also a fairly good approximation. Since the atmosphere, as we've seen, absorbs energy only from the surface of the Earth, but emits energy both back toward the Earth and to space, we have to adjust its surface area accordingly in our model. For the purposes of absorption, we can treat the atmosphere as having twice the area of the surface, since it radiates along both the inside and outside. Just as with the surface of the Earth, the atmosphere radiates energy in accord with the Stefan-Boltzmann law. That is, it radiates energy as a function of its surface area and temperature. We also stipulate that (since this is an energy balance model), the atmosphere emits exactly as much as it absorbs. We've already noted that the atmosphere isn't entirely transparent from the perspective of the Earth: it absorbs some (but not all) of the outgoing radiation. Let us add a term to our model to reflect the opacity of absorbing surfaces. Call this term γ. A surface that is totally opaque has γ = 1 (it absorbs all the energy that actually reaches it), and a surface that is totally transparent to incoming radiation has γ = 0. Note that this term is independent of α: a surface's opacity only comes into play with regard to the energy that isn't just reflected outright. 127 That is, γ represents how likely a surface is to absorb some radiation that tries to pass through it; reflected energy never makes this attempt, and so does not matter here. This behavior is intuitive if we think, to begin, about the surface of the planet: while it has a non-negligible albedo (it reflects some radiation), it is effectively opaque. The planet's surface does reflect some energy outright, but virtually all of the energy it doesn't reflect is absorbed. Very little E/M radiation simply passes through the surface of the planet. We can thus set γs = 1. We are interested in solving for γa-we're interested in figuring out just how opaque the atmosphere is. From all of this, we can deduce another equation: one for the energy emitted by the atmosphere (Fa). γ σTF a = a 4a (4e) We have to include γ in this equation, as (recall) the atmosphere is transparent (or nearly so) only with respect to incoming solar radiation. Radiation emitted both by the surface and by the atmosphere itself has a chance of being reabsorbed. At last, then, we're in a position to put all of this together. We have an equation for the energy emitted by the atmosphere and an equation for the energy reaching the ground from the sun. For the purposes of this model, this exhausts all the sources of radiative forcing on the surface of the Earth. If we hold on to the supposition that things are at (or near) equilibrium, we know that the energy radiated by the surface (which we can calculate independently from the Stefan-Boltzmann law) must be in balance with these two sources. The full balance for the surface at equilibrium, then, is: γ σT γ σT4 S (1−α)o + a 4a = s s4 (4f) 128 Moreover, we can deduce a second balance equation for the atmosphere alone. Recall that the atmosphere receives energy only from the surface, and that it radiates with twice the area that it receives-it is "heated" from below only, but radiates heat in two directions. With another application of the Stefan-Boltzmann law, then, we know that: γ σT γ σT2 a 4a = s s4 (4j) A bit of algebraic manipulation to solve this system of equation-by inserting (4j) into (4f) and solving the resulting equation for Ts-gives us a final solution to the whole shebang (as noted above, we shall assume that the Earth is opaque and that = 1):γs √4 S (1−α)o4σ(1− )2γa = T s (4k) With no atmosphere at all, γa = 0 and the equation above just reduces to our original equation, giving us an answer of 255K. By plugging in the observed temperature at the Earth's surface (288K) and solving for γa, we obtain a value of γ = .76. With that value in hand, then, we can actually use this model to explore the response of the planet to changes in albedo or greenhouse gas composition-we can make genuine predictions about what will happen to the planet if our atmosphere becomes more opaque to infrared radiation, more energy comes in from the sun, or the reflective profile of the surface changes. This is a fully-developed ZDEBM, and while it is only modestly powerful, it is a working model that could be employed to make accurate, interesting predictions. It is a real pattern. 129 4.2 The Philosophical Significance of the Hierarchy of Climate Models While the model we have just constructed is a working model, the like of which one might encounter in an introductory course on climate science, it still represents only a tiny slice of the myriad of processes which underlie the Earth's climate. We went through the extended derivation of the last section for two reasons: first, to provide some structure to the introduction of central concepts in climate science (e.g. albedo, the greenhouse effect, opacity) and second, to demonstrate that even the simplest models of the Earth's climate are incredibly complicated. The dialectical presentation (hopefully) provided an intuitive reconstruction of the thinking that motivated the ZDEBM, but things still got very messy very quickly. Let us now turn from this relatively comprehensible model to other more complicated climate models. As we've seen, the ZDEBM treats the entire planet as being completely uniform with respect to albedo, temperature, opacity, and so on. However, the real Earth is manifestly not like this: there is a significant difference between land, water, and atmosphere, as well as a significant difference between the composition of different layers of the atmosphere itself. Moreover, the shape and orientation of the Earth matters: the poles receive far less solar energy than the equator, and some of the energy that reaches the Earth is reflected in one location but not another, either by features of the atmosphere (clouds, for instance), or by the surface (white snow and ice is particularly reflective). Representing the Earth as a totally uniform body abstracts away from these differences, and while zero-dimensional energy balance models are useful as first approximations, getting a more accurate picture requires that we insert more detail into our model, but what kind of detail should we add? How do we decide which parts of the world are 111 111 It's important to note that increasing the sophistication of a model is a necessary but not sufficient condition for generating more accurate predictions. While it seems intuitively apparent that more sophisticated models should be 130 important enough to deserve inclusion in our models, and which can be ignored? These are incredibly deep questions-they represent some of the most difficult practical challenges that working scientists in any discipline face in designing their models-and giving a general answer to them is beyond the scope of our project here. Still, it is worth our time to briefly examine the plethora of climate models that have sprung up in the last few decades, and to think about the conceptual underpinnings of this highly diverse collection of scientific tools. Perhaps we can at least suggest the shape of an answer to these questions with respect to climate science in particular. In practice, climate scientists employ a large family of models for different purposes. Zero-dimensional energy balance models like the one we just constructed are the most basic models actually used in the real world, and form what can be thought of as a the "lowest level" of a kind of "model pyramid." The logic of energy balance models is sound, and more sophisticated energy balance models add more detail to account for some of the factors we just enumerated; with every addition of detail, the model becomes capable of generating more accurate predictions but also becomes more difficult to work with. For instance, we might move from the ZDEBM to a one-dimensional energy balance model, modeling the Earth not as a point but as a line, and expressing the parameters of the model (like albedo) not as single terms, but as differential equations whose value depends on where we are on the line. This allows us to take the latitudinal variation of incoming solar energy into account, for example: in general, areas better models, it is also the case that more sophisticated models generally leave more room for failure, either as a result of measurement error, because the model accounts for only half of an important feedback loop, or for some other reason. Recall the characterization of models as artifacts-in some ways, they are very like mechanical artifacts, and the old engineering adage that "anything that moves can break" applies here as well. We will revisit this point in Chapter Five when we discuss the special difficulties of modeling complex systems. 131 near the equator receive more energy, and the incoming energy drops off as we move north or south toward the poles. Alternatively, if we are interested in differences in radiation received by different levels of the atmosphere, we might implement a one-dimensional model that's organized vertically, rather than horizontally. Even more detailed models combine these approaches: two-dimensional models account for variation in incoming solar energy as a function of both height and latitude. Energy balance models, though, are fundamentally limited by their focus on radiation as the only interesting factor driving the state of the climate. While the radiative forcing of the sun (and the action of greenhouse gasses in the presence of that radiative forcing) is certainly one of the dominant factors influencing the dynamics of the Earth's climate, it is equally certainly not the only such factor. If we want to attend to other factors, we need to supplement energy balance models with models of a fundamentally different character, not just create increasingly sophisticated energy balance models. McGuffie & Herderson-Sellers (2005) list five different components that need to be considered if we're to get a full picture of the climate: radiation, dynamics, surface processes, chemistry, and spatio-temporal resolution. While I will 112 eventually argue that this list is incomplete, it serves as a very good starting point for consideration of the myriad of climate models living in the wild today. Radiation concerns the sort of processes that are capture by energy balance models: the transfer of energy from the sun to the Earth, and the release of energy back into space (in the form of infrared radiation) from the Earth. As we've seen, careful attention to this factor can produce a model that is serviceable for some purposes, but which is limited in scope. In 112 McGuffie & Herderson-Sellers (2005), p. 49 132 particular, pure radiative models (energy balance models, for instance) neglect the transfer of energy by non-radiative processes and are unable to model any of the other more nuanced the dynamical processes that govern both the climate and weather on Earth. A radiative model, for example, will be entirely silent on the question of whether or not increased greenhouse gas concentration is likely to change the behavior of ocean currents. Even if we were to devise an energy balance that is sophisticated enough to model radiative transfer between the ocean, land, and atmosphere as separate energy reservoirs, the inclusion of facts about currents is simply beyond the scope of these models. To include facts like those, we need to appeal to a new class of models-so-called "radiative-convective" (RC) models are designed to address these issues. These models incorporate many of the same insights about radiation balance that we saw in the ZDEBM, but with the addition of dynamical considerations. Basic RC models will treat the planet not just as a set of "lamps" which absorb and emit radiation, but rather will include enough detail to model the transfer of energy via convection-the movement of air-as well. We can think of RC models as presenting the Earth as a set of connected boxes of various sizes containing gas of various temperatures. While some energy is transferred between the boxes as a result of radiative forcing, the boundaries where one box meets another are equally important-there, the contents of the two boxes mix, and energy transfer as a result of convection becomes possible as well. A simple one-dimensional RC model might treat the surface of the Earth as consisting of regions of different temperature arrayed along a line, calculating the interaction of different regions at their boundary by employing a fixed lapse-rate to model convective energy transfer. This information might then be incorporated into a relatively sophisticated energy balance 133 model, yielding an increase in the accuracy of radiative process models as a result of more precise information about temperature gradients and exchanges of air . 113 While RC models offer an improvement in accuracy over simple radiative models (as a result of taking some dynamical processes into account), they are still far away from being robust enough to capture all the details of our complex climate. Beyond RC models, the field becomes increasingly differentiated and heterogeneous-in the last 30 years in particular, a large number of so-called "Earth models of intermediate complexity" (EMIC) have sprung up in the literature. It is impossible to characterize these models in any general way, as each is constructed for a very particular purpose-to model some very specific aspect of the global climate based on a parameterization that fixes other potentially relevant factors as (more or less) constant. As an example of the tremendous variability present in this class of models, EMICs include RC models that also model cloud formation (which is an important factor in determining albedo), sea-ice models that focus primarily on the surface processes that drive the formation (and break-up) of arctic and Antarctic ice, spatio-temporally constrained models of the short-term effect of volcanic aerosols on planetary albedo, and even ocean models that focus primarily on the procession of regular cycles of ocean temperatures and currents (e.g. the models used to predict the effects of the El Nino/Southern Oscillation on annual rainfall in the United States' west coast). The EMIC represent a veritable zoo of wildly different models developed for wildly different purposes. The fact that all these can (apparently) peacefully coexist is worthy of philosophical interest, and warrants some consideration here . 114 113 As we shall see, this practice of using the output of one kind of model as input for another model is characteristic of much of contemporary climate science. 114 In addition, the policy implications of this diverse zoo of important models will be the primary topic of Chapter 134 4.2.1 Climate Models and Complexity Earlier in the history of climate science, even textbooks within the field were willing to attempt to rank various climate models in terms of ascending "complexity ." While the sense 115 of the term 'complexity' doesn't exactly mirror the concept of dynamical complexity developed in Chapter Three, there are enough parallels to be worth remarking on, and I shall argue that the important aspects of the climate modeler's sense are, like the various approaches to complexity surveyed in Chapter Two, well-captured by the notion of dynamical complexity. Interestingly, there's at least some evidence that more recent work in climatology has backed off from the attempt to rank models by complexity. While the hierarchical "climate pyramid" reproduced below appears in all editions of McGuffie & Herderson-Sellers' work on climate modeling, by 2005 (and the publication of the third edition of the work), they had introduced a qualification to its presentation: This constructed hierarchy is useful for didactic purposes, but does not reflect all the uses to which models are put, nor the values that can be derived from them. The goal of developers of comprehensive models is to improve performance by including every relevant process, as compared to the aim of [EMIC] modelers who try to capture and understand processes in a restricted parameter space. Between these two extremes there is a large territory populated, in part, by leakage from both ends. This intermediate area is a lively and fertile ground for modeling innovation. The spectrum of models [included in EMICs] should not be viewed as poor cousins to the coupled models . 116 It is worth emphasizing that this egalitarian perspective on climate science-in which a multitude of perspectives (encoded in a multitude of models) are included without prejudice-fits nicely with the account of science in general we explored in Chapter One, and only serves to reinforce the view that contemporary scientific practice requires this multifarious Seven. 115 See, e.g., McGuffie and Herderson-Sellers (op. cit.), though this treatment is far from unique 116 Ibid. p. 117 135 foundation. Their observation that EMICs should not be viewed as "poor cousins" of more elaborate models similarly seems to support the view that we should resist the impulse to try to 117 decide which models are "more real" than others. Any model which succeeds in capturing a real pattern in the time-evolution of the world (and which is of consequent predictive use) should be given equal standing. The sense of "complexity" here also has more than a little in common with the notion we've been working with so far. McGuffie & Henderson-Sellers chose to illustrate the climate model hierarchy as a pyramid for good reason; while they say that the "vertical axis [is] not intended to be qualitative, " the pyramidal shape is intended to illustrate the eventual convergence of the 118 four different modeling considerations they give in a single comprehensive model. A complex model in this sense, then, is one which incorporates patterns describing dynamics, radiative processes, surface processes, and chemical processes. The parallels to dynamical complexity should be relatively clear here: a system that is highly dynamically complex will admit of a variety of different modeling perspectives (in virtue of exhibiting a plethora of different patterns). For some predictive purposes, the system can be treated as a simpler system, facilitating the identification of (real) patterns that might be obfuscated when the system is considered as a whole. I have repeatedly argued that this practice of simplification is a methodological approach that should not be underappreciated (and which is not overridden by the addition of complexity theory to mainstream science). EMIC are fantastic case-study in this fact, a diverse mixture of idealizations and simplifications of various stripes that have been developed to explore particular climate subsystems, but whose outputs frequently are of use in 117 We shall discuss these more elaborate models in detail in the next chapter. 118 Ibid., p. 51 136 more global analyses. We'll explore the role that EMICs play in more comprehensive models in the next chapter (when we explore cutting-edge global circulation models and the tools climate scientists employ to create and work with them). For now, though, I would like to end this chapter with a few words about the limitation of the analytic method that undergirds both the creation of EMICs and much of science in general. We've seen a number of reasons why this analytic approach is worth preserving, but there are also good reasons to think that it cannot take us as far as we want to go. 4.2.2 Limits of the Analytic Method It might help to begin by thinking about the traditional scientific paradigm as it has existed from the time of Newton and Galileo. The account that follows is simplified to the point of being apocryphal, but I think it captures the spirit of things well enough. For our purposes here, that's enough: I'm interested not in giving a detailed historical account of the progress of science (many who are more well-suited to that task have already done a far better job than I ever could), but in pointing to some general themes and assumptions that first began to take root in the scientific revolution. It will be helpful to have these themes clearly in mind, as I think complexity theory is best understood as an approach to science that fills in the gaps left by the approach I'm about to describe. If you are a historian of science, I apologize for the simplifying liberties that I take with this complicated story (see Chapter Zero for more on why you're probably not alone in being dissatisfied with what I have to say). The greatest triumph of the scientific revolution was, arguably, the advent of the kind of experimental method that still underlies most science today: the fundamental insight that we 137 could get a better handle on the natural world by manipulating it through experiment was, to a large degree, the most important conceptual leap of the era. The idea that science could proceed not just through abstract theorizing about ideal cases (as many ancients had) nor just through passive observation of the world around us, but by systematically intervening in that world, observing the results of those interventions, and then generalizing those results into theories about how systems outside the laboratory behaved was unbelievable fruitful. The control aspect of this is important to emphasize: the revolution was not primarily a revolution toward empiricism strictly speaking-people had been doing science by looking at the world for a long time-but a revolution toward empiricism driven by controlled isolation . 119 This kind of interventionist approach to science was vital to the later theoretical breakthroughs: while Newton's genius lay in realizing that the same patterns of motion lay behind the movement of bodies on Earth and in space, that insight wouldn't have been possible if Galileo hadn't first identified those patterns in terrestrial falling bodies. It was Galileo's genius to realize that by reducing a system of interest to its simplest form-by controlling the system to hold fixed as many variables as possible-patterns that might be obscured by the chaos and confusion of the unmodified natural world would become more apparent. All of this is very well-known and (I take it) uncontroversial-at least if you take my simplifications in stride. My purpose here is not to comment on the history of science per se but (in good classical scientific fashion) to isolate and emphasize a single thread in this narrative: that of isolated decomposition of systems. After the revolution that this approach precipitated in physics, the basic experimental method 119 For more on the role of intervention in science, see Woodward (2011) 138 of intervening in the natural world to isolate variables for testing came to dominate virtually all of the natural sciences for hundreds of years. Scientists in chemistry, biology, and even the social sciences attempted to copy (with varying degrees of success) the physics-inspired model of identifying single constituents of interesting systems, seeing how those constituents behaved when isolated from each other (and, a fortiori, from a complicated external environment), and using that information to deduce how collections of those constituents would behave in more realistic circumstances. This approach was enormously, earth-shatteringly, adverb-confoundingly successful, and gave us virtually all the scientific advances of the 18th, 19th, and 20th centuries, culminating in the triumph of physics that is quantum mechanics, as well as the more domain-specific (if no less impressive) advances of molecular biology (studying the gene to understand the organism), statistical mechanics (studying the particle to understand the thermodynamic system), and cognitive neuroscience (studying the neuron to understand the brain), just to name a few. Moreover, this way of thinking about things came to dominate the philosophy of science (and scientifically-informed metaphysics) too. Many of the influential accounts of science developed in the 19th and 20th centuries rely (more or less implicitly) on this kind of model of scientific work. The logical positivists, for whom science was a matter of deduction from particular observations and a system of formal axioms perhaps exemplify this approach, though (as Hooker [2011a] argues), the Popperian model of theory generation, experimental data collection, and theory falsification also relies on this decomposition approach to scientific work, as it assumes that theorists will proceed by isolating variables to such a degree that cases of direct falsification will (at least sometimes) be clearly discernible. The account of science developed in Chapter 139 One is intended to contribute to the beginning of a philosophy of science that moves beyond dogmatic clinging to decomposition, but it will likely still be some time before this thinking becomes part of the philosophical mainstream. Part of the problem is that the primary opponents of the decomposition approach to science (at least before the 1970s) were the vitalists and the strong emergentists. The common 120 criticism marshaled by these two camps was that the analytic approach championed by mainstream science was inevitably doomed to fail, as some aspect of the natural world (living things, for example) were sui generis in that their behavior was not governed by or deducible from the behavior of their parts, but rather anomalously emerged in certain circumstances. The last major stronghold for this view-life-was dealt a critical blow by the advent of molecular biology, though: the discovery of genetic molecules showed that living things were not anomalous, sui generis systems, but rather were just as dependent on the coordinated action of simpler constituents as any physical system. By the middle of the 20th century, vitalism had fallen far out of favor, and most mainstream scientists and philosophers held at least a vaguely reductionistic view of the world. While quantum mechanics was busy overthrowing other pillars of classical physics, it seemed to only reinforce this one: the whole is nothing more than the sum of its parts. While the behavior of that sum may be difficult (or even impossible) to predict sometimes just by looking at the parts, there's nothing fundamentally new to be learned by looking at systems; any higher-level scientific laws are just special cases, course-grainings, or simplifications of the story that fundamental physics has to tell. The moral of the science's success in the 20th century is that the mainstream scientists were 120 See, for instance, Morgan (1921) 140 right and the vitalists were wrong: living things (a fortiori, brains, democracies, economies) are really nothing over and above the sum of their parts-there is no vital spark, and no ghost in the machine, and no invisible hand. The progress of science seems to have born this out, and in a sense it has: in looking for (say) living things to behave in ways that were not determined by the behavior of their cells and genes, vitalists were chasing ghosts. Still, in the last few decades cracks have begun to appear in the hegemonic analytic approach: cracks that suggest not that the insights garnered by that approach were wrong, but that they were incomplete. This is where complexity theory enters our story. As an example, consider the highly computational theory of mind that's been developed by some cognitive psychologists and philosophers of mind . On this account, psychology as a 121 scientific practice is, in a very real sense, predicated on a very large misunderstanding: according to the most radical computationalists, what we take to be "psychological states" are really nothing more than formal computational operations being carried out by the firing of one or another set of neurons in our brain. It's worth emphasizing that this is a stronger thesis than the standard "metaphysical reduction" that's rather more common in the philosophy of mind literature, and it is certainly a stronger thesis than a generally physicalist view of psychology (where psychological states in some sense are realized by or depend on the action of neurons). The strongest adherents of computational neuroscience argue that not only do mental states depend on brain states, but that (as a methodological dictum) we ought to focus our scientific efforts on mapping neuronal firings only. That is, it's not just necessary to understand the brain in order to get a grip on psychology-understanding how neurons work just is understanding 121 See, for instance, Pinker (2000). This position is also there at times in the work of Paul and Patricia Churchland, though it is also moderated at times when compared to the fairly hard-line computationalism of Pinker. 141 psychology. There are no higher level patterns or processes to speak of. This is a very substantive methodological thesis-one which (if it were true) would have significant implications for how research time and money ought to be allocated. Increasingly, it is also a thesis that is being rejected by mainstream cognitive science. In the decades since Pinker's book was published, cognitive scientists have gradually come to recognize that neuronal firings, while surely central in determining the behavior of creatures like us, are far from the only things that matter. Rather, the neurons (and their accompanying chemical neurotransmitters, action potentials, &c.) function as one sub-system in a far more complicated web of interrelated interactions between the brain, the rest of the body, and various aspects of the external environment. While some cognitive mechanisms can be completely understood through the decompositionist approach, the higher-level cognition of complicated 122 organisms embedded in dynamic environments (humans engaged in complex, conscious reasoning, for example) certainly cannot. The gradual relaxation of the demand that all cognitive science be amenable to something like this radically eliminative computational hypothesis has produced an explosion of theoretical insights. The appreciation of the importance of embodied cognition-that is, the importance of non-neurological parts of the body in shaping cognitive states-exemplifies this trend, as does the work of Andy Clark in exploring the "extended mind" hypothesis, in which environmental props can be thought of as genuine components of higher level cognitive processes . 123 122 Simple reflex behavior like the snapping of carnivorous plants (as well as basic reflexes of human beings), for instance, can be understood as a very simple mechanism of this sort, where the overall behavior is just the result of individual constituent parts operating relatively independently of one another. See Moreno, Ruiz-Mirazo, & Barandiaran (2011) for more on this. 123 See Clark (2001) and (2003) 142 Similarly, contemporary biology has rejected the notion that the evolution of organism populations just is the evolution of individual genes in the organisms of the population. This move away from "selfish gene" type approaches to evolutionary theory might be thought of as mirroring the move away from strict eliminative computationalism in cognitive neuroscience; the appreciation of epigenetic influences on evolution exemplifies this trend in biology, as does 124 the proliferation of the "-omics" biological sciences (e.g. genomics, proteomics, biomics). In rejecting the decompositionist approach to cognition (or evolution), though, neuroscientists (or biologists) have not returned to the vitalist or emergentist positions of the 19th and early 20th centuries-it is certainly not the case that the only alternative to the Pinker/Churchland position about the mind is a return to Cartesian dualism, or the sort of spooky emergentism of Morgan (1921). Rejecting the notion that interesting facts about cognition are exhausted by interesting facts about neuronal firings need not entail embracing the notion that cognitive facts float free of physics and chemistry; rather, it just entails a recognition that neural networks (and the organisms that have them) are embedded in active environments that contribute to their states just as much as the behavior of the network's (proper) parts do, and that the decompositionist assumption that an understanding of the parts entails an understanding of the whole need not hold in all cases. In studying organisms as complex systems, we need not reject the vast and important insights of traditional decompositionist science (including biology, neuroscience, and 124 Epigenetics is the study of how factors other than changes in the underlying molecular structure of DNA can influence the expression and heritability of phenotypic traits, and encompasses everything from the study of how environmental changes can affect the expression of different genes to the exploration of how sets of genes can function as regulatory networks within an organism, affecting each others' behavior and expression in heritable ways without actually modifying genotypic code. As a simple example, consider the way in which restricted calorie diets have been shown to modulate the activity of the SIR2/SIRT1 genes in laboratory rats, resulting in longer life-spans without change to the actual structure of the genes in question. See Oberdoerffer et. al. (2008). The most important point here is that these changes can be heritable, meaning that any account of evolution that treats evolution as a process that works strictly on genes can't be the whole story. 143 others)-rather, we need only recognize that system-theoretic approaches supplement (but don't supplant) existing paradigms within the discipline. The recognition, to put the point another way, is not that Pinker was entirely wrong to think that neuronal computation played a central role in cognition, but only that his view was too limited-rather than evolution simply operating on an unconstrained string of genetic code, it operates in a "highly constrained (occasionally discontinuous) space of possible morphologies, whose formation requires acknowledging the environmental, material, self-organized and often random processes that appear at different scales." 125 The move from an exclusively decompositionist approach to one incorporating both decompositionist and holistic work is underway in disciplines other than biology and neuroscience. It's particularly important for our purposes to note that the peaceful coexistence of EMICs with more comprehensive, high-level models (to be discussed in the next chapter) requires an appreciation both of the power of decomposition and of its limits. Surveying all the areas in which this type of thinking has made an impact would require far more space than I have here, so I will let these two cases-the biological and the climatological-stand on their own, and refer the interested reader to the list of references provided here for further exploration of complexity theoretic approaches to cognitive science, economics, medicine, engineering, computer science, and others. 4.2.2 Next Steps This quiet conceptual revolution has proceeded more-or-less independently in these 125 Moreno, Ruiz-Mirazo, & Barandiaran (2011) 144 disciplines until fairly recently. Increasingly, though, the question of whether there might be general principles underlying these cases-principles that deal with how systems of many highly connected interactive parts behave, regardless of the nature of those parts-has started to surface in these discussions. This is precisely the question that complexity theory aims to explore: what are the general features of systems for which the decompositionist approach fails to capture the whole story? What rigorous methods might we adopt to augment traditional approaches to science? How can we integrate holistic and analytic understanding into a unified scientific whole? These are, I suspect, the questions that will come to define scientific progress in the 21st century, and they are questions that climate science-perhaps more than anything else-urgently needs to consider. The contribution of EMICs shouldn't be underestimated: they are very important tools in their own right, and they have much to contribute to our understanding of the climate. Still, though, they're highly specific tools, deliberately designed to apply to a very narrow range of circumstances. EMICs are intentionally limited in scope, and while this limitation can take different forms (e.g. spatio-temporal restriction vs. restriction to a single climate sub-system considered more-or-less in isolation), it is a defining characteristic of the class of models-perhaps the only defining characteristic. Such a narrow focus is a double-edged sword; it makes EMICs far easier to work with than their monstrously complicated big brothers, but it also limits the class of predictions that we can reasonably expect to get out of applying them. If we're going to get as complete a picture of the patterns underlying the time-evolution of the Earth's climate as possible, then we'll need as many tools as possible at our disposal: low-level energy balance models, EMICs, and high-level holistic models. 145 In the next chapter, we'll consider problems associated with these holistic models in detail, introducing a few of the more pressing puzzles that neither energy balance models nor EMICs are capable of resolving, and then surveying how more elaborate models are supposed to meet these challenges. However, high-level climate models (and the methods scientists employ to work with them) are not without problems of their own; while they are capable of meeting some of the challenges that EMICs cannot meet, they face other challenges that EMICs do not face. Let us now turn to the problems that force us to supplement EMICs and examine how high-level models are designed and employed. 146 Chapter Five Complexity, Chaos, and Challenges in Modeling the Complex Systems 5.0 A Road Map We concluded the last chapter with something of a cliff-hanger: I argued that while the classical scientific method of decomposing systems into their constituent parts and studying the behavior of those parts in isolation has been spectacularly successful in the history of science, a number of contemporary problems have forced us to look for tools to supplement that approach. We saw that both biology and climate science have begun to explore more holistic models, with the hope that those perspectives will shed some light on issues that have stymied the decompositionalist approach. The bulk of the last chapter was dedicated to exploring a simplified climate model-the zero-dimensional energy balance model-and to articulating the physical intuitions behind the mathematics of that model. Near the end, we discussed the highly heterogeneous family of models called "Earth models of intermediate complexity," and thought about the relationship between those models and the concept of dynamical complexity. I suggested that while EMICs shouldn't be thought of as inferior imitations of more comprehensive models, the project of getting a clear understanding of the patterns that underlie the global climate will involve recruiting all available tools. To that end, I would like to spend this chapter discussing cutting-edge, high-level climate models, with particular attention to the computer simulations in which many of these models are implemented. This chapter will be the first to engage with some of the more controversial aspects of climate science, and will constitute a direct response to the critique of climatology as a "cooked up" enterprise-a "science by 147 simulation." Here's how things will go. In Section 5.1, we'll begin to examine some of the more difficult points of climate science, with special attention to features of the global climate system that contribute to its high dynamical complexity. In particular, we'll focus on two aspects of the global climate which, while neither necessary nor sufficient for high dynamical complexity in themselves, are characteristic of complex systems: the presence of non-linear feedback mechanisms, and the presence of chaotic behavior. We'll think about what it means for a system to be chaotic, and how the presence of feedback mechanisms (which are represented as non-linearities in the mathematics describing the system's behavior) can contribute to chaos. I shall argue that careful attention to these two factors can shed a tremendous amount of light on some of the vagaries of climatology. We will see that the kind of model we constructed in 4.1 is incapable of handling these issues, and will survey some more robust models which attempt to come to terms with them. After describing some of the problems endemic to the study of the Earth's climate (and the models designed to solve them), we shall consider how climate scientists meet the methodological challenges they face in actually using more sophisticated models. In Section 5.2, we will discuss one of the defining tools in the climatologist's tool-kit: computer simulation. The construction of simulations-computer-solved models designed to be run repeatedly-is a methodological innovation common to many complex system sciences; we'll think about why this is the case, and consider the relationship between the challenges presented by non-linearity and chaos, and the unprecedented methodological opportunities presented by modern supercomputers. I will argue that while "science by simulation" is an absolutely indispensable 148 approach that climate science must take advantage of, it also comes with its own set of novel pitfalls, which must be carefully marked if they are to be avoided. More specifically, I argue that careful attention to the nature of chaos should force us to attend to the limitations of science by simulation, even in ideal conditions. It is worth emphasizing that these limitations are just that, though: limitations, and not absolute barriers. Popular dissatisfaction with the role that computational models play in climate sciences is largely a result of conflating these two notions, and even some people who ought to know better sometimes confuse the existence of chaos with the impossibility of any significant forecasting. We'll think about the nature of the limitations imposed by chaos (especially in light of the method of computational model building), and see how those general limitations apply to climate science. Finally, I'll argue that even with these limitations taken into account, the legitimate predictions made by climate science have serious implications for life on Earth. 5.1 The Challenges of Modeling Complexity Individual special sciences have been increasingly adopting the concepts and methods of complexity theory, but this adoption has been a piecemeal response to the failures of the decompositionalist method in individual domains. So far, there exists little in the way of an integrative understanding of the methods, problems, or even central concepts underlying the individual approaches. Given the highly practical nature of science, this should not be terribly surprising: science does the best with the tools it has, and creates new tools only in response to new problems. The business of science is to figure out patterns in how the world changes over time, and this business requires a degree of specialized knowledge that makes it natural to focus on the trees rather than the forest (unless you happen to be working in forestry science). As a 149 result, we're at one of those relatively unusual (so far) junctures where there is genuinely important multidisciplinary conceptual clarification waiting to be done. We've been in this situation before. The mechanistic revolution of the scientific enlightenment forced us to confront the question of how humanity might fit into a world that was fundamentally physical, leading to an explosion of new philosophical ideas about man and his place in nature. More recently, the non-classical revolution in the early 20th century forced us to refine concepts that we'd taken to be rock-solid in our conception of the world, and the philosophical implications of quantum mechanics and relativity are still being fought out in ways that are actually relevant to the progress of science. There is similar room for conceptual work 126 here. The time is ripe for philosophical analysis, which makes it all the more distressing that so little philosophical attention has been paid to the topic of complexity. One of the consequences of the piecemeal way in which complexity-theoretic considerations have taken hold in the special sciences is that there's a good deal of confusion about how to use some of the central concepts. It is instructive to note that many of the same terms (e.g. "emergence," "self-organized," "chaotic") show up in complexity-motivated discussions of very diverse sciences, and there's surely a sense in which most variations of those terms show a kind of family resemblance. Still, the fact that they are often defined with a specific context in mind means that it is not always easy to explicitly state the common core of these important terms as 126 The question of how to interpret the formalism of non-relativistic quantum mechanics, for instance, still hasn't been answered to the satisfaction of either philosophers or physicists. Philosophical attention to the measurement problem in the mid-20th century led directly to the overthrow of the Copenhagen Interpretation, and (more recently) to work on decoherence and einselection (e.g. Zurek [2003]). For an accessible survey of some of the ways in which philosophical thinking has contributed to physics in the 20th century, see Maudlin (2007). For examples of excellent current work in these areas, see Wallace (2011) and (2009), as well as Albert (2000). 150 they appear across disciplines. Articulating this common core in a systematic way is one of the most important foundational contributions that remains to be made, as it will provide a common language in which scientists interested in complexity (but trained in different disciplines) can come together to discuss their work. Doing this ground-clearing work is also a necessary precursor to the more daunting task of defining complexity itself. While I cannot hope to disentangle all the relevant concepts here, I would like to now turn to an examination of two of the most important for our purposes: non-linearity and chaos. Where our discussions of complexity have thus far been principally focused on defining complexity, this section focuses on the practical challenges of actually working with dynamically complex systems. We would do well to keep the distinction between these two lines of discussion clear in our minds, though-while the issues we'll be discussing in this chapter are characteristic of complex systems, they are not definitive of them. That is, neither non-linearity nor chaos (nor the conjunction of the two) is sufficient for dynamical complexity . 127 5.1.1 Non-Linearity Before we can tackle what it means to say that a system's behavior is non-linear, we need to get some basic terminology under our belt. Complex systems theory is built largely on the back of a more general approach to scientific modeling called dynamical systems theory, which deals 127 Whether or not either of these two features is a necessary feature of dynamically complex systems is a more complicated question. As we shall see, both non-linearity and chaos are best understood as properties of particular models rather than of systems themselves. Dynamically complex systems are (by definition) those which admit of sensible and useful consideration from a large variety of different perspectives; many interesting dynamically complex systems might exhibit chaotic behavior from some perspectives but not others. We should resist the temptation to even consider the question of whether systems like that are "really" chaotic or not in just the same way that we should resist the temptation to generally privilege one set of real patterns describing a system's time-evolution over the others. 151 with the creation of mathematical models describing change ("dynamics") in parts of the world ("systems") as time progresses. For our purposes, a few of the methods of dynamical systems theory (DyST) are particularly worth flagging. First, it's important to note that DyST takes change as its primary object of interest. This might seem obvious given the name of the field, but it is vital that we appreciate the degree to which this assumption colors the DyST approach to scientific model-building. Rather than focusing on particular instantaneous states of systems-say, the position and momentum of each particle in a box of gas, or particular weather-states (the like of which were the focus of the qualitative approach to weather forecasting discussed in Chapter Four)-DyST focuses on ensembles of states that describe a system over some time period, not just at a single instant. The central mathematical tool of DyST is an equation that describes how different physical quantities of a system (e.g. force, mass, and velocity in Newtonian physics; populations of predator animals and prey animals in ecology; presence and concentration of certain atmospheric chemicals and global temperature in climatology) vary in relation to one another over time. That is, DyST is concerned with modeling how physical quantities differ with respect to one another at different times in a system's lifetime-in most systems, this is accomplished through the use of differential equations, which describe how variables change in response to one another . The 128 familiar Newtonian equation of motion (F = ma) is a simple differential equation, as it relates the 128 Strictly speaking, differential equations are only applicable to systems in which the values in question can be modeled as varying continuously. In discrete-time systems, a separate (but related) mathematical tool called a difference equation must be used. For our purposes here, this distinction is not terribly important, and I will restrict the rest of the discussion to cases where continuous variation of quantities is present, and thus where differential equations are the appropriate tool. 152 change in velocity (acceleration) to other quantities of interest (force and mass) in physical 129 systems. We can think of a system of interest (for example, a box of gas) as being represented by a very large space of possible states that the system can take. For something like a box of gas, this space would be composed of points, each of which represents the specific position and velocity of each molecule in the system. For Newtonian systems like gasses, this space is called a 130 phase space. More generally, a space like this-where the complete state of a system at a particular time is represented by a single point-is called a configuration space or state space. Since DyST is concerned with modeling not just a system at a particular time (but rather over some stretch of time), we can think of a DyST model as describing a path that a system takes through its state space. The succession of points represents the succession of states that the system goes through as it changes over time. Given a configuration space and a starting point for a system, then, DyST is concerned with watching how the system moves from its starting position. The differential equations describing the system give a kind of "map"-a set of directions for how to figure out where the system will go next, given a particular position. The configuration space and the differential equations work together as a tool-kit to model the behavior of the system in question over time. The differential 129 Of course, velocity too is a dynamical concept that describes the change in something's position over time. The Newtonian equation of motion is thus a second order differential equation, as it describes not just a change in a basic quantity, but (so to speak) the change in the change in a basic quantity. 130 This means that for a system like that, the space would have to have 6n dimensions, where n is the number of particles in the system. Why six? If each point in our space is to represent a complete state of the system, it needs to represent the x, y, and z coordinates of each particle's position (three numbers), as well as the x, y, and z coordinates of each particle's velocity (three more numbers). For each particle in the system, then, we must specify six numbers to get a complete representation from this perspective. 153 equation describes how interesting quantities (e.g. position and velocity) of the system change, and the configuration space is a representation of all the different possible values those quantities can take. The advantage of this approach should be obvious: it lets us reduce difficult questions about how complicated systems behave to mathematically-tractable questions about tracing a path through a space according to a rule. This powerful modeling tool is the heart of DyST. Some systems can be modeled by a special class of differential equations: linear differential equations. Intuitively, a system's behavior can be modeled by a set of linear differential equations if: (1) the behavior of the system is (in a sense that we shall articulate more precisely soon) the sum of the behavior of the parts of the system, and (2) the variables in the model of the system vary with respect to one another at constant rates . (1) should be relatively familiar: 131 it's just the decompositionalist assumption we discussed back at the end of Chapter Four! 132 This assumption, as we saw, is innocuous in many cases. In the case of a box of gas, for example, we could take the very long and messy differential equation describing how all the trillions of molecules behave together and break it up into a very large collection of equations describing the behavior of individual molecules, and (hopefully) arrive at the very same predictions. There's no appreciable interaction between individual molecules in a gas, so 133 131 In mathematical jargon, these two conditions are called "additivity" and "degree 1 homogeneity," respectively. It can be shown that degree 1 homogeneity follows from additivity given some fairly (for our purposes) innocuous assumptions, but it is heuristically useful to consider the two notions separately. 132 Ladyman, Lambert, & Wiesner (2011) quite appropriately note that "a lot of heat and very little light" has been generated in philosophical treatments of non-linearity. In particular, they worry about Mainzer (1994)'s claim that "[l]inear thinking and the belief that the whole is only the sum of its parts are evidently obsolete" (p. 1). Ladyman, Lambert, & Wiesner reasonably object that very little has been said about what non-linearity has to do with ontological reductionism, or what precisely is meant by "linear thinking." It is precisely this sort of murkiness that I am at pains to dispel in the rest of this chapter. 133 Fans of Wikipedia style guidelines might call "appreciable" here a "weasel-word." What counts as an appreciable interaction is, of course, the really difficult question here. Suffice it to say that in practice we've found it to be the case that assuming no interaction between the molecules here gives us a model that works for certain purposes. A whole 154 breaking the system apart into its component parts, analyzing the behavior of each part, and then taking the system to be (in some sense) the "sum" of that behavior should yield the same prediction as considering the gas as a whole. It's worth briefly considering some of the technicalities behind this condition. Strictly speaking, the additvity condition on linearity makes no reference to "parts," as it is a condition on equations, not physical systems being modeled by equations. Rather, the condition demands that given any set of valid solutions to the equation describing the behavior of the system, the linear combination of those solutions is itself a solution. This formal statement, though more precise, runs the risk of obfuscating the physical (and philosophical) significance of linearity, so it is worth thinking more carefully about this condition with a series of examples. Linearity is sometimes referred to as "convexity," especially in discussions that are grounded in set-theoretic ways of framing the issue . In keeping with our broadly geometric approach to 134 thinking about these issues, this is perhaps the most intuitive way of presenting the concept. Consider, for instance, the set of points that define a sphere in Euclidean space. This set is convex (in both the ordinary sense and the specialized sense under consideration here), since if we take any two points that are inside the sphere, then the linear combination-the weighted average of the two points-is also inside the sphere. Moreover, the line connecting the two points will be inside the sphere, the triangle defined by connecting any three points will lie entirely inside the sphere, and so on. More formally, we can say that a set of points is convex if separate paper could be written on the DyST account of these ceteris paribus type hedges, but we shall have to set the issue aside for another time. 134 For a nice case-study in the benefits of framing discussions of non-linearity in terms of convexity, see Al-Suwailem (2005)'s discussion of non-linearity in the context of economic theory and preference-ranking. 155 for all points xi in the set, x ∑ ai i 5(a) is also in the set as long as 5(b) ∑ ai = 1 (2) is necessary to ensure that the summation in (1) is just a weighted average of the values of the points, otherwise we could always define sets that were outside the initial set just by multiplying the points under consideration by arbitrarily large values. It's easy to see that while the set of points defining a sphere is convex, the set of points defining a torus-a donut shape-is not. Two points can be inside the set, while their weighted average--the line connecting them--is outside the set (think of two points on either side of the "hole" in the middle of a donut, for instance). Why is this particular sort of geometric structure relevant to our discussion here? What is it about sets that behave like spheres rather than like donuts that make them more well-behaved mathematical representations of physical systems? We'll return to that question in just a moment, but first let's briefly examine the other way of articulating the linearity condition-(2) described above. Ultimately, we shall see that these two conditions are, at least in most cases of relevance to us, just different ways of looking at the same phenomenon. For the moment, though, it is dialectically useful to examine each of the two approaches on its own. 156 The second condition for linearity given above is a condition not on the relationship between the parts of the system, but on the relationship between the quantities described by the differential equation in question. (2) demands that the way that the quantities described by the equation vary with respect to one another remain constant. To get a sense of what that means, it's probably easiest to think about some cases where the requirement holds, and then think about some cases where the requirement doesn't hold. Suppose you're walking on a treadmill, and want to vary the speed at which the belt is moving so that you walk more quickly or more slowly. You can do this by pressing the up and down arrows on the speed control; each time you press one of the arrows, the speed of the belt will change by (say) .1 MPH. This is an example of a variation that satisfies condition (2). We could write down a simple differential equation relating two quantities: the number of times you've pressed each button, and the speed at which the treadmill's belt is moving. No matter how many times you press the button, though, the value of the button press will remain constant: the amount by which pressing the up arrow varies the speed doesn't depend on how many times you've pressed the button, or on how fast the treadmill is already turning. Whether you're walking slowly at one mile per hour or sprinting at 15 miles per hour, pressing that button will always result in a change of .1 mile per hour. Condition (2) is satisfied. 135 OK, with an understanding of what a system must look like in order to be linear, let's think about what sorts of systems might fail to satisfy these requirements. Let's return to the treadmill 135 Actually, this case satisfies both conditions. We've just seen how it satisfies (2), but we could also break the system apart and consider your "up arrow" presses and "down arrow" presses independently of one another and still calculate the speed of the belt. Treadmill speed control is a linear system, and this underscores the point that conditions (1) and (2) are not as independent as this presentation suggests. 157 example again, and think about how it might be designed so that it fails to satisfy (2). Suppose that we were designing a treadmill to be used by Olympic sprinters in training. We might decide that we need fine-grained speed control only at very high speeds, and that it's more important for the athletes to get up to sprint speed quickly than to have fine control over lower speeds. With that in mind, we might design the treadmill such that if the speed is less than (say) 10 MPH, each button press increments or decrements the speed by 2 MPH. Once the speed hits 10 MPH, though, we need more fine grained control, so each button press only changes the current speed by 1 MPH. At 15 MPH, things get even more fine grained, and each press once again changes things by .1 MPH. In this case, condition (2) is not satisfied: the relationship between the quantities of interest in the system (number of button presses and speed of the belt) doesn't vary at a constant rate. Just knowing that you've pressed the "up arrow" button three times in the last minute is no longer enough for me to calculate how much the speed of the belt has changed: I need to know what the starting speed was, and I need to know how the relationship between button presses and speed changes varies with speed. Predicting the behavior of systems like this is thus a bit more complicated, as there is a higher-order relationship present between the changing quantities of the system. 5.1.2 Two Illustrations of Non-Linearity The logistic function for population growth in ecology is an oft-cited example of a real-world non-linear system. The logistic function models the growth of a population of individuals as a function of time, given some basic information about the context in which the population exists (e.g. the carrying-capacity of the environment). One way of formulating the equation is: 158 N (1 )dt dN = r − K N 5(c) N represents the number of individuals in the population, r represents the relative rate at which the members of the population reproduce when unchecked, and K represents the carrying capacity of the environment. Though quite simple, the logistic equation displays quite interesting behavior across a wide spectrum of circumstances. When N is low-when there are relatively few members of a population-growth can proceed almost unchecked, as the first term on the right side of the equation dominates. As the population grows in size, though, the value of increases, making the carrying capacity of the environment-how many (say) deer the woodsK N can support before they begin to eat themselves out of house and home-becomes increasingly important. Eventually, the contribution of outpaces the contribution of rN, putting a check onK N population growth. More sophisticated versions of the logistic equation-versions in which, for instance, K itself varies as a function of time or even as a function of N-show even stronger non-linear behavior. It is this interrelationship between the variables in the equation that 136 makes models like this one non-linear. Just as with the Olympian treadmill we described above, the values of the relevant variables in the system of differential equations describing the system depend on one another in non-trivial ways; in the case of the treadmill, the value of a button-press varies with (and affects) the speed of the belt, and in the case of the logistic equation, the rate of population growth varies with (and affects) extant population. This general 136 Consider, for instance, a circumstance in which the carrying capacity of an environment is partially a function of how much food is present in that environment, and in which the quantity of food available is a function of the present population of another species. This is often the case in predator-prey models; the number of wolves an environment can support partially depends on how many deer are around, and the size of the deer population depends both on how much vegetation is available for the deer to eat and on how likely an individual deer is to encounter a hungry wolf while foraging. 159 behavior-the presence of feedbacks-is characteristic of non-linear systems. Let us consider a more realistic concrete example by way of illustration: the relationship between material wealth and subjective utility. On the face of it, we might assume that the relationship between these two quantities is linear, at least in most cases. It seems reasonable, that is, to think that getting $10 would not only leave you with more utility--make you happier--than getting $5 would, but also that it would leave you with twice as much utility. Empirical investigation has not supported this idea, though, and contemporary economic theory generally holds that the relationship between wealth and utility is non-linear. This principle, called the principle of diminishing marginal utility, was originally developed as a response to the St. Petersburg Paradox of decision theory. Consider a casino game in which the pot begins at a single dollar, and a fair coin is tossed repeatedly. After each toss, if the coin comes up heads the quantity of money in the pot is doubled. If the coin comes up tails, the game ends and the player wins whatever quantity is in the pot (i.e. a single dollar if the first toss comes up tails, two dollars if the second toss comes up tails, four if the third toss comes up tails, &c.). The problem asks us to consider what a rational gambler ought to be willing to pay for the privilege of playing the game. On the face of it, it seems as if a rational player ought to be willing to pay anything less than the expected value of a session of the game--that is, if the player wants a shot at actually making some money, she should be willing to pay the casino anything less than the sum of all the possible amounts of money she could win, each multiplied by the probability of winning that amount. The problem is that the value of this sum grows without bound: there is a probability of one-half that she will win one dollar, probability one-fourth that she'll win two dollars, probability one-eighth that she'll win four dollars, &c. 160 More formally, the probability of winning n dollars is and so the overall expected value ofn2n playing the game (assuming that the house has unlimited resources and will allow the game to continue until a flip comes up tails) is given by: ∑ ∞ 1 2 1 5(d) If the amount of money that our gambler should be willing to pay to play a game is constrained only by the demand that it be less than the expected return from the game, then this suggests that she should pay any finite amount of money for a chance to play the game just once. That seems very strange. While there are a number of solutions to this problem, the one of most immediate interest to us was proposed in Bernoulli (1738). Bernoulli suggested that we ought 137 to think of utility gained from the receipt of a quantity of some good (in this case money) as being inversely proportional to the quantity of that same good already possessed. He justifies this by pointing out that The price of the item is dependent only on the thing itself and is equal for everyone; the utility, however, is dependent on the particular circumstances of the person making the estimate. Thus there is no doubt that a gain of one thousand ducats is more significant to a pauper than to a rich man though both gain the same amount 138 Bernoulli's original suggestion of this fairly straightforward (albeit still non-linear) relationship between wealth and utility has been refined and expanded by a number of thinkers. The 139 137 Translation by Sommer (1954). 138 Op. cit., pp. 158-159 139 The principle of diminishing marginal utility was developed by a number of economists over the course of several decades, and continues to be refined to this day. See, for example, Menger (1950), Bohm-Bawerk (1955), and McCulloch (1977). While the originators of this principle (particularly Menger and Bohm-Bawerk) were associated with the Austrian school of economics, diminishing marginal utility has found its way into more mainstream neoclassical economic theories (Kahneman and Deaton, 2010). 161 failure of variations in utility to be tied linearly to variations in wealth, though, can be understood as a failure of condition (2) from Section 5.1.1-the wealth/utility relationship is like the Olympic treadmill. More recently, empirical work in the social sciences has gone even further. Kahneman and Deaton (2010) argue that utility (or, as they put it, "emotional well-being") increases with the logarithm of wealth, but only up to a point. On their account, plotting the relationship between utility and wealth yields a strongly concave function, which is what we ought to expect. However, they also argue that there is a leveling off point in the function, beyond which "there is no improvement whatever in any of the three measures of emotional well-being ." 140 Of course, it is worth noting that Kahneman and Deaton's investigation involved observation only of residents of the United States. Interestingly, as Kahneman and Deaton point out, the mean income in the United States at the time in which they conducted their research was just under $72,000: very close to the mark at which they observed the disappearance of any impact of increased income on emotional well-being. There is at least some reason to think that this is 141 not entirely a coincidence. McBride (2001) argues that the impact of changes in wealth on an agent's subjective utility depends not just on how much wealth the subject already possesses, but also on wealth possessed by others in the agent's social circles. That is, being wealthier than those around you might itself have a positive impact on your subjective utility--an impact that is at least partially independent of the absolute quantity of wealth you possess. McBride found that people are made happier by being the richest people in a poorer neighborhood, and that increasing their wealth (but moving them to a cohort where they'd be among the poorest 140 Kahneman and Deaton (2010), p. 16491 141 Op. cit., p. 16492 162 members) might result in a decrease in subjective utility! This hints at what might be partial explanation for the effect described by Kahneman and Deaton: being less wealthy than average is itself a source of negative subjective utility. This suggests that the relationship between wealth and utility also fails to satisfy condition (1) from Section 5.1.1. Given a group of people (neighbors, for instance), the differential equations describing the change in utility of members of the group relative to their changes in wealth will resist decomposition, because their utilities are a function not just of their own wealth, but of the wealth of other members of the community as well. By decomposing the system into component parts, we would miss this factor, which means that even if we took the principle of diminishing marginal utility into account in our calculations, the decompositionalist approach would still fail to capture the actual dynamics of the overall system. A more holistic approach is required. This suggests an important lesson for the study of natural systems in which non-linearities play a significant role: the presence of unexpected feedback and variable degrees of mutual influence between different components of a system might well mean that attempts to model the system's behavior by way of aggregating models of the components are, if not exactly doomed to failure, at least of very limited use. We must be extraordinarily careful when we attempt to tease general predictions about the future of the global climate out of families of EMICs for precisely this reason. We shall return to this point in Section 5.2, but first let us turn our attention to the other central challenge to be discussed here: chaotic behavior. 163 5.1.3 Chaos Like non-linearity, chaos is best understood as a dynamical concept-a feature of how systems changed over time that is represented by certain conditions on the DyST models of those systems. Chaos has played an increasingly central role in a number of sciences since the coinage of the term "butterfly effect" in the mid 20th century as a response to Lorenz (1963) . Indeed, 142 the evocative idea of the butterfly effect-that idea that the flapping of a butterfly's wings on one side of the world can lead to a hurricane on the other side of the world days later-has percolated so thoroughly into popular culture that the broad strokes of the concept are familiar even to many laypeople. Still, the specifics of the concept are often misunderstood, even by many philosophers of science. In particular, chaotic systems are sometimes thought to be indeterministic, a mistake which has the potential to create a great deal of confusion. Let's think things through slowly, and add on the formalism as we get a better handle on the concept. Let's start here: suppose that it is in fact true that the flapping of a butterfly's wings in Portugal can spawn a hurricane off the coast of Mexico days later. Here's a question that should immediately jump out at us: under what conditions does something like this happen? Clearly, it cannot be the case that every butterfly's flapping has this sort of catastrophic effect, as there are far more butterfly flaps than there are hurricanes. That is, just saying that a tiny change (like a flap) can cause a big change (like a hurricane) doesn't tell us that it will, or give us any information about what the preconditions are for such a thing to happen. This point is worth 142Lorenz (1963) never employs this poetic description of the effect, and the precise origin of the phrase is somewhat murky. In 1972, Lorenz delivered an address to the American Association for the Advancement of Science using the title "Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?" The resemblance between the Lorenz system's state space graph (Figure 2) and a butterfly's wings is likely not coincidental. 164 emphasizing: whatever a chaotic system is, it is not a system where every small change immediately "blows up" into a big change after a short time. We'll need to get more precise. Let's stick with the butterfly effect as our paradigm case, but now consider things from the perspective of DyST. Suppose we've represented the Earth's atmosphere in a state space that takes into account the position and velocity of every gas molecule on the planet. First, consider the trajectory in which the nefarious butterfly doesn't flap its wings at some time t1, and the hurricane doesn't develop at a later time t2. This is a perfectly well-defined path through the state space of the system that can be picked out by giving an initial condition (starting point in the space), along with the differential equations describing the behavior of the air molecules. Next, consider the trajectory in which the butterfly does flap its wings at t1, and the hurricane does develop at t2. What's the relationship between these two cases? Here's one obvious feature: the two trajectories will be very close together in the state space at t1-they'll differ only with respect to the position of the few molecules of air that have been displaced by the butterfly's wings-but they'll be very far apart at t2. Whatever else a hurricane does, it surely changes the position and velocity of a lot of air molecules (to say the least!). This is an interesting observation: given the right conditions, two trajectories through state space can start off very close together, then diverge as time goes on. This simple observation is the foundation of chaos theory. Contrast this case with the case of a clearly non-chaotic system: a pendulum, like the arm on a grandfather clock. Suppose we define a state space where each point represents a particular angular velocity and displacement angle from the vertical position for the pendulum. Now, look at the trajectory that the pendulum takes through the state space based on different initial 165 conditions. Suppose our initial condition consists in the pendulum being held up at 70 degrees from its vertical position and released. Think about the shape that the pendulum will trace through its state space as it swings. At first, the angular velocity will be zero (as the pendulum is held ready). As the pendulum falls, its position will change in an arc, so its angular displacement will approach zero until it hits the vertical position, where its angular velocity will peak. The pendulum is now one-quarter of the way through a full period, and begins its upswing. Now, its angular displacement starts to increase (it gets further way from vertical), while its angular momentum decreases (it slows down). Eventually, it will hit the top of this upswing, and pause for a moment (zero angular velocity, high angular displacement), and then start swinging back down. If the pendulum is a real-world one (and isn't being fed by some energy source), it will repeat this cycle some number of times. Each time, though, its maximum angular displacement will be slightly lower-it won't make it quite as high-and its maximum angular velocity (when it is vertical) will be slightly smaller as it loses energy to friction. Eventually it will come to rest. If we plot behavior in a two-dimensional state space (with angular displacement on one axis and angular momentum on the other), we will see the system trace a spiral-shaped trajectory ending at the origin. Angular velocity always falls as angular displacement grows (and vice-versa), so each full period will look like an ellipse, and the loss of energy to friction will mean that each period will be represented by a slightly smaller ellipse as the system spirals toward its equilibrium position of zero displacement and zero velocity: straight up and down, and not moving. See Figure 5.1 for a rough plot of what the graph of this situation would look like in a state-space for the pendulum. 166 Fig. 5.1 Now, consider the difference between this case and a case where we start the pendulum at a slightly smaller displacement angle (say, 65 degrees instead of 70). The two trajectories will (of course) start in slightly different places in the state space (both will start at zero angular velocity, but will differ along the other axis). What happens when you let the system run this time? Clearly, the shape it traces out through the state space will look much the same as the shape traced out by the first system: a spiral approaching the point (0,0). Moreover, the two trajectories should never get further apart, but rather will continue to approach each other more and more quickly as they near their point of intersection . The two trajectories are similar 143 enough that it is common to present the phase diagram like Figure 5.1: with just a single trajectory standing in for all the variations. Trajectories which all behave similarly in this way are said to be qualitatively identical. The trajectories for any initial condition like this are sufficiently similar that we simplify things by just letting one trajectory stand in for all the others 143 This is a defining characteristic of dissipative systems. Conservative systems-undamped pendulums that don't lose energy to friction-will feature trajectories that remain separate by a constant amount. 167 (this is really handy when, for instance, the same system can show several different classes of behavior for different initial conditions, and keeps the phase diagram from becoming too crowded) . 144 Contrast this to the butterfly-hurricane case from above, when trajectories that started very close together diverged over time; the small difference in initial conditions was magnified over time in one case, but not in the other. This is what it means for a system to behave chaotically: small differences in initial condition are magnified into larger differences as the system evolves, so trajectories that start very close together in state space need not stay close together. Lorenz (1963) discusses a system of equations first articulated by Saltzman (1962) to describe the convective transfer of some quantity (e.g. average kinetic energy) across regions of a fluid: (5e)dt dx σ(y )= − x (5f) x (ρ ) dt dy = − z − y (5g) xy βzdt dz = − In this system of equations, x, y, and z represent the modeled system's position in a three-dimensional state space represents the intensity of convective motion, while , , andσ ρ 145 are parameterizations representing how strongly (and in what way) changes in each of theβ 144 Indeed, even our pendulum is like this! There is another possible qualitatively identical class of trajectories that's not shown in Figure 1. Think about what would happen if we start things not by dropping the pendulum, but by giving it a big push. If we add in enough initial energy, the angular velocity will be high enough that, rather than coming to rest at the apex of its swing toward the other side and dropping back down, the pendulum will continue on and spin over the top, something most schoolchildren have tried to do on playground swings. Depending on the initial push given, this over-the-top spin may happen only once, or it may happen several times. Eventually though, the behavior of the pendulum will decay back down into the class of trajectories depicted here, an event known as a phase change. 145 Precisely what this means, of course, depends on the system being modeled. In Lorenz's original discussion, x represents the intensity of convective energy transfer, y represents the relative temperature of flows moving in opposite directions, and z represents the the degree to which (and how) the vertical temperature profile of the fluid diverges from a smooth, linear flow. 168 state variables are connected to one another. The important feature of Lorenz's system for our discussion is this: the system exhibits chaotic behavior only for some parameterizations. That is, it's possible to assign values to , ,σ ρ and such that the behavior of the system in some sense resembles that of the pendulumβ discussed above: similar initial conditions remain similar as the system evolves over time. This suggests that it isn't always quite right to say that systems themselves are chaotic. It's possible for some systems to have chaotic regions in their state spaces such that small differences in overall state not when the system is initialized, but rather when (and if) it enters the chaotic region are magnified over time. That is, it is possible for a system's behavior to go from non-chaotic (where trajectories that are close together at one time stay close together) to chaotic (where trajectories that are close together at one time diverge) . Similarly, it is possible for 146 systems to find their way out of chaotic behavior. Attempting to simply divide systems into chaotic and non-chaotic groups drastically over-simplifies things, and obscures the importance of finding predictors of chaos-signs that a system may be approaching a chaotic region of its state space before it actually gets there . 147 Another basic issue worth highlighting is that chaos has absolutely nothing to do with indeterminism: a chaotic system can be deterministic or stochastic, according to its underlying dynamics. If the differential equations defining the system's path through its state space contain 146 The Phillips curve in economics, which describes the relationship between inflation and unemployment, is a good real-world example of this. Trajectories through economic state space described by the Phillips curve can fall into chaotic regions under the right conditions, but there are also non-chaotic regions in the space. 147 A number of authors have succeeded in identifying the appearance of a certain structure called a "period-doubling bifurcation" as one predictor of chaotic behavior, but it is unlikely that it is the only such indicator. 169 no probabilistic elements, then the system will be deterministic. Many (most?) chaotic systems of scientific interest are deterministic. The confusion here stems from the observation that the behavior of systems in chaotic regions of their state space can be difficult to predict over significant time-scales, but this is not at all the same as their being non-deterministic. Rather, it just means that the more unsure I am about the system's exact initial position in state space, the more unsure I am about where it will end up after some time has gone by. The behavior of systems in chaotic regions of their state space can be difficult to forecast in virtue of uncertainty about whether things started out in exactly one or another condition, but that (again) does not make them indeterministic. Again, we will return to this in much greater detail in Section 3 once we are in a position to synthesize our discussions of chaos and path-dependence. Exactly how hard is it to predict the behavior of a system once it finds its way into a chaotic region? It's difficult to answer that question in any general way, and saying anything precise is going to require that we at least dip our toes into the basics of the mathematics behind chaotic behavior. We've seen that state space trajectories in chaotic region diverge from one another, but we've said nothing at all about how quickly that divergence happens. As you might expect, this is a feature that varies from system to system: not all chaotic behavior is created equal. The rate of divergence between two trajectories is given by a particular number-the Lyapunov exponent-that varies from system to system (and from trajectory to trajectory within the system ). The distance between two trajectories x0 → xt and y0 → yt at two different times can, for any 148 148 Because of this variation-some pairs of trajectories may diverge more quickly than others-it is helpful to also define the maximal Lyapunov exponent (MLE) for the system. As the name suggests, this is just the largest Lyapunov exponent to be found in a particular system. Because the MLE represents, in a sense, the "worst-case" scenario for prediction, it is standard to play it safe and use the MLE whenever we need to make a general statement about the behavior of the system as a whole. In the discussion that follows, I am referring to the MLE unless otherwise specified. 170 given system, be expressed as: x | e |x || t − yt = λt 0 − y0 5(h) where λ is the "Lyapunov exponent," and quantifies the rate of divergence. The time-scales at which chaotic effects come to dominate the dynamics of the system, then depend on two factors: the value of the Lyapunov exponent, and how much divergence we're willing to allow between two trajectories before we're willing to consider it significant. For systems with a relatively small Lyapunov exponent, divergence at short timescales will be very small, and will thus likely play little role in our treatment of the system (unless we have independent reasons for requiring very great precision in our predictions). Likewise, there may be cases when we care only about whether the trajectory of the system after a certain time falls into one or another region of state space, and thus can treat some amount of divergence as irrelevant. This point is not obvious but it is very important; it is worth considering some of the mathematics in slightly more detail before we continue on. In particular, let's spend some time thinking about what we can learn by playing around a bit with the definition of a chaotic system given above. To begin, let D be some neighborhood on Rn such that all pairs of points iff , y ∈ D< x0 0 > 5(i)x | ≤ ε| 0 − y0 That is, let D be some neighborhood in an n-dimensional space such that for all pairs of points that are in D, the distance between those two points is less than or equal to some small value epsilon. If Rn is the state space of some dynamical system S with Lyapunov exponent λ, then 171 combining (5) and (6) lets us deduce ∀(t > 0) : 5(j) , y ∈ D< xt t > x y | ≤ ε(e )| t − t λt In other (English) words, if the space is a state space for some dynamical system with chaotic behavior, then for all times after the initialization time, the size of the smallest neighborhood that must include the successors to some collection of states that started off arbitrarily close together will increase as a function of the fastest rate at which any two trajectories in the system could diverge (i.e. the MLE) and the amount of time that has passed (whew!). That's a mouthful, but the concepts behind the mathematics are actually fairly straightforward. In chaotic systems, the distance between two trajectories through the state space of the system increases exponentially as time goes by-two states that start off very close together will eventually evolve into states that are quite far apart. How quickly this divergence takes place is captured by the value of the Lyapunov exponent for the trajectories under consideration (with the "worst-case" rate of divergence defining the MLE). Generalizing from particular pairs of trajectories, we can think about defining a region in the state space. Since regions are just sets of points, we can think about the relationship between our region's volume at one time and the smallest region encompassing the end-state of all the trajectories that started in that region at some later time. This size increase will be straightforwardly related to the rate at which individual trajectories in the region diverge, so the size of the later region will depend on three things: the size of the initial region, the rate at which paths through the system diverge, and the amount of time elapsed . If our system is chaotic, then no matter how small we make our region the trajectories 149 149 If we have some way of determining the largest Lyapunov exponent that appears in D, then that can stand in for the global MLE in our equations here. If not, then we must use the MLE for the system as a whole, as that is the only way 172 followed by the states that are included in it will, given enough time, diverge significantly . 150 How much does this behavior actually limit the practice of predicting what chaotic systems will do in the future? Let's keep exploring the mathematics and see what we can learn. Consider two limit cases of the inequality in 5(j). First: 5(k)lim ε(e ) ε→0 λt = 0 This is just the limiting case of perfect measurement of the initial condition of the system-a case where there's absolutely no uncertainty in our first measurement, and so the size of our "neighborhood" of possible initial conditions is zero. As the distance between the two points in the initial pair approaches zero, then the distance between the corresponding pair at time t will also shrink. Equivalently, if the size of the neighborhood is zero-if the neighborhood includes one and only one point-then we can be sure of the system's position in its state space at any later time (assuming no stochasticity in our equations). This is why the point that chaotic dynamics are not the same thing as indeterministic dynamics is so important. However: 5(l)lim ε(e ) λ→0 λt = ε As the Lyapunov exponent λ approaches zero, the second term on the right side of the inequality in 5(j) approaches unity. This represents another limiting case-one which is perhaps even more interesting than the first one. Note that 5(k) is still valid for non-chaotic systems: the MLE is just set to zero, and so the distance between two trajectories will remain constant as those points are evolved forward in time . More interestingly, think about what things look like 151 of guaranteeing that the region at the later time will include all the trajectories. 150 Attentive readers will note the use of what Wikipedia editors call a "weasel word" here. What counts as "significant" divergence? This is a very important question, and will be the object of our discussion for the next few pages. For now, it is enough to note that "significance" is clearly a goal-relative concept, a fact which ends up being a double-edged sword if we're trying to predict the behavior of chaotic systems. We'll see how very soon. 151 If the Lyapunov exponent is negative, then the distance between two paths decreases exponentially with time. Intuitively, this represents the initial conditions all being "sucked" toward a single end-state. This is, for instance, the 173 if λ > 0 (the system is chaotic) but still very small. No matter how small λ is, chaotic behavior will appear whenever t ≫ : even a very small amount of divergence becomes significant on1 λ long enough time scales. Similarly, if t ≪ then we can generally treat the system as if it is1 λ non-chaotic (as in the case of the orbits of planets in our solar system). The lesson to be drawn is that it isn't the value of either t or λ that matters so much as the ratio between the two values. 5.1.4 Prediction and Chaos It can be tempting to conclude from this that if we know λ, ε, and t, then we can put a meaningful and objective "horizon" on our prediction attempts. If we know the amount of uncertainty in the initial measurement of the system's state (ε), the maximal rate at which two paths through the state space could diverge (λ), and the amount of time that has elapsed between the initial measurement and the time at which we're trying to make our prediction (t), then shouldn't we be able to design things to operate within the uncertainty by defining relevant macroconditions of our system as being uniformly smaller than ? If this were true, it(e )ε λt would be very exciting-it would let us deduce the best way to construct our models from the dynamics of the system under consideration, and would tell us how to carve up the state space of some system of interest optimally given the temporal scales involved. Unfortunately, things are not this simple. In particular, this suggestion assumes that the state space can be neatly divided into continuously connected macroconditions, and that it is not possible for a single macrostate's volume to be distributed across a number of isolated regions. It assumes, that is, that simple distance in state-space is always going to be the best measure of qualitative similarity between two states. This is manifestly not the case. Consider, for instance, case with the damped pendulum discussed above-all initial conditions eventually converge on the rest state. 174 the situation in classical statistical mechanics. Given some macrocondition M* at t0 , what are the constraints on the system's state at a later time t1? We can think of M* as being defined in terms of 5(j)-that is, we can think of M* as being a macrocondition that's picked out in terms of some neighborhood of the state space of S that satisfies 5(j). By Liouville's Theorem, we know that the total density ρ of states is constant along any trajectory through phase space. That is: 5(m)dt dρ = 0 However, as Albert (2000) points out, this only implies that the total phase space volume is invariant with respect to time. Liouville's theorem says absolutely nothing about how that volume is distributed; it only says that all the volume in the initial macrocondition has to be accounted for somewhere in the later macrocondition(s). In particular, we have no reason to expect that all the volume will be distributed as a single path-connected region at t1: we just know that the original volume of M* must be accounted for somehow. That volume could be scattered across a number of disconnected states, as shown in Figure 5.2. 175 Fig. 5.2 While the specifics of this objection are only relevant to statistical mechanics, there is a more general lesson that we can draw: the track that we started down a few pages ago-of using formal features of chaos theory to put a straight-forward cap on the precision of our predictions on a given system after a certain amount of time-is not as smooth and straight as it may have initially seemed. In particular, we have to attend to the fact that simple distance across a state-space may not always be the best measure of the relative "similarity" between two different states; the case of thermodynamics and statistical mechanics provides an existence proof for this claim. Without an independent measure of how to group regions of a state space into qualitatively similar conditions-thermodynamic macroconditions in this case-we have no way of guaranteeing that just because some collection of states falls within the bounds of the region defined by 5(j) they are necessarily all similar to one another in the relevant respect. This account ignores the fact that two states might be very close together in state space, and yet differ 176 in other important dynamical respects. Generalizing from this case, we can conclude that knowing λ, ε, and t is enough to let us put a meaningful cap on the resolution of future predictions (i.e. that they can be only as fine-grained as the size of the neighborhood given by ) only if we stay agnostic about the presence (and(e )ε λt location) of interesting macroconditions when we make our predictions. That is, while the inequality in 5(j) does indeed hold, we have no way of knowing whether or not the size and distribution of interesting, well-behaved regions of the state-space will correspond neatly with size of the neighborhoods defined by that inequality. To put the point another way, restricting our attention to the behavior of some system considered as a collection of states can distract us from relevant factors in predicting the future of the system. In cases where the dynamical form of a system can shift as a function of time, we need to attend to patterns in the formation of well-behaved regions (like those of thermodynamic macroconditions)-including critical points and bifurcations-with just as much acumen as we attend to patterns in the transition from one state to another. Features like those are obscured when we take a static view of systems, and only become obvious when we adopt the tools of DyST. 5.1.5 Feedback Loops In Section 5.1.2, we considered the relationship between non-linearities in the models of dynamical systems and the presence of feedback generally. Our discussion there, however, focused on an example drawn from economics. Moreover, we didn't discuss feedback mechanisms themselves in much detail. Let us now fill in both those gaps. While CGCMs are 177 breathtakingly detailed models in many respects, their detailed incorporation of feedback mechanisms into their outputs--a task that is impossible for EBMs and met by individual EMICs only for their narrow domains of application (if it is met at all). Since CGCMs are characterized as a group by their melding of atmospheric, oceanic, and land-based models, let's begin by considering a representative sample of an important feedback mechanism from each of these three domains. While feedback mechanisms are not definitive of complex systems like the climate, they are frequently the sources of non-linear behavior in the natural world, and so are often found in real-world complex systems. It's not difficult to see why this is the case; dynamically complex systems are systems in which interesting behavioral patterns are present from many perspectives and at many scales (see Chapter Three), and thus their behavior is regulated by a large number of mutually interacting constraints. Feedback mechanisms are a very common way for 152 natural systems to regulate their own behavior. Dynamically complex systems, with their layers of interlocking constraints, have ample opportunity to develop a tangled thicket of feedback loops. Jay Forrester, in his 1969 textbook on the prospects for developing computational models of city growth, writes that "a complex system is not a simple feedback loop where one system state dominates the behavior. It is a multiplicity of interacting feedback loops [the behavior of which is] controlled by nonlinear relationships. " The global climate is, in this respect, very 153 152 The fact that a particular complex system exhibits interesting behavior at many scales of analysis implies this kind of inter-scale regulation: the features of a given pattern in the behavior of the system at one scale can be thought of a constraint on the features of the patterns at each of the other scales. After all, the choice of a state space in which to represent a system is just a choice of how to describe that system, and so to notice that a system's behavior is constrained in one space is just to notice that the system's behavior is constrained period, though the degree of constraint can vary. 153 Forrester (1969), p. 9 178 similar to an active urban center. Feedback mechanisms are said to be either positive or negative, and the balance and interplay between these two different species of feedback is often the backbone of self-regulating dynamical systems: the global climate is no exception. Positive feedback mechanisms are those in which the action of the mechanism serves to increase the parameter representing the input of the mechanism itself. If the efficacy of the mechanism for producing some compound A depends (in part) on the availability of another compound B and the mechanism which produces compound B also produces compound A, then the operation of these two mechanisms can form a positive feedback loop-as more B is produced, more A is produced, which in turn causes B to be produced at a greater rate, and so on. Consider, for example, two teenage lovers (call them Romeo and Juliet) who are particularly receptive to each other's affections. As Romeo shows more amorous interest in Juliet, she becomes more smitten with him as well. In response, Romeo-excited by the attention of such a beautiful young woman-becomes still more affectionate. Once the two teenagers are brought into the right sort of contact-once they're aware of each other's romantic feelings-their affection for each other will rapidly grow. Positive feedback mechanisms are perhaps best described as "runaway" mechanisms; unless they're checked (either by other mechanisms that are part of the system itself or by a change in input from the system's environment), they will tend to increase the value of some parameter of the system without limit. In the case of Romeo and Juliet, it's easy to see that once the cycle is started, the romantic feelings that each of them has toward the other will, if left unchecked, grow without bound. This can, for obvious reasons, lead to serious instability in the overall system-most interesting systems cannot withstand the unbounded increase of any of their 179 parameters without serious negative consequences. The basic engineering principles underlying the creation of nuclear weapons exploit this feature of positive feedback mechanisms: the destructive output of nuclear weapons results from the energy released during the fission of certain isotopes of (in most cases) uranium or plutonium. Since fission of these heavy isotopes produces (among other things) the high-energy neutrons necessary to begin the fission process in other nearby atoms of the same isotope, the fission reaction (once begun) can-given the right conditions-become a self-sustaining chain reaction, where the result of each step in the cycle causes subsequent steps, which are both similar and amplified. Once the fission reaction begins it reinforces itself, resulting in the rapid release of energy that is the nominal purpose of nuclear weapons. Of course, in most real-world cases the parameters involved in positive feedback loops are not able to increase without bound. In most cases, that is, dynamical systems that include positive feedback loops also include related negative feedback loops, which provide a check on the otherwise-unbounded amplification of the factors involved in the positive feedback loops. While positive feedback loops are self-reinforcing, negative feedback loops are self-limiting; in the same way that positive loops can lead to the rapid destabilization of dynamical systems in which they figure, negative loops can help keep dynamical systems in which they figure stable. Consider, for instance, a version of the story of Romeo and Juliet in which the teenage lovers are somewhat more dysfunctional. In this version of the tale, Romeo and Juliet still respond to each others' affections, but they do so in the opposite way as in the story told above. Romeo, in this story, likes to "play hard to get:" the more he sees that Juliet's affections for him are growing, the less interested he is in her. Juliet, on the other hand, is responsive to 180 encouragement: the more Romeo seems to like her, the more she likes him. It's easy to see that the story's outcome given this behavior will be far different than the outcome in which their affections are purely driven by mutually reinforcing positive feedback loops. Rather than growing without bound, their affections will tend to stabilize at a particular level, the precise nature of which is determined by two factors: the initial conditions (how much they like each other to begin with), and the level of responsiveness by each teen (how much Juliet's affection responds to Romeo's reciprocity, and how much Romeo's affection responds to Juliet's enthusiasm). Depending on the precise tuning of these values, the relationship may either stabilize in a mutually congenial way (as both lovers are drawn toward a middle ground of passion), or it may stabilize in a way that results in the relationship ending (as Romeo's lack of interest frustrates Juliet and she gives up). In either case, the important feature of the example is its eventual movement toward a stable attractor. 154 5.2.2 The Role of Feedback Loops in Driving Climate Dynamics Similar feedback mechanics play central roles in the regulation and evolution of the global climate system. Understanding the dynamics and influence of these feedback mechanics is essential to understanding the limitations of basic models of the sort considered in Chapter Four. Some of the most important positive feedback mechanics are both obvious and troubling 154 Under some conditions, the situation described here might fall into another class of attractors: the limit cycle. It is possible for some combinations of Romeo and Juliet's initial interest in each other to combine with features of how they respond to one another to produce a situation where the two constantly oscillate back and forth, with Romeo's interest in Juliet growing at precisely the right rate to put Juliet off, cooling his affections to the point where she once again finds him attractive, beginning the cycle all over again. In either case, however, the stability of the attractor is the important feature is the attractor's stability. Both the two fixed-point attractors described in the text (the termination of the courtship and the stabilization of mutual attractiion) result in the values of the relevant differential equations "settling down" to predictable behavior. Similarly, the duo's entrance into the less fortunate (but just as stable) limit cycle represents predictable long-term behavior. 181 in their behavior. Consider, for instance, the relationship between planetary albedo and warming. Albedo, as you may recall from Chapter Four is a value representing the reflectivity of a given surface. Albedo ranges from 0 to 1, with higher values representing greater reflectivity. Albedo is associated with one of the most well-documented positive feedback mechanisms in the global climate. As the planet warms, the area of the planet covered by snow and ice tends to decrease. Snow and ice, being white and highly reflective, have a fairly high 155 albedo when compared with either open water or bare land. As more ice melts, then, the planetary (and local) albedo decreases. This results in more radiation being absorbed, leading to increased warming and further melting. It's easy to see that unchecked, this process could facilitate runaway climate warming, which each small increase in temperature encouraging further, larger increases. This positive feedback is left out of more basic climate models, which lack the formal structure to account for such nuanced behavior. Perhaps the most significant set of positive feedback mechanisms associated with the long-term behavior of the global climate are those that influence the capacity of the oceans to act as a carbon sink. The planetary oceans are the largest carbon sinks and reservoirs in the global 156 climate system, containing 93% of the planet's exchangeable carbon. The ocean and the 157 atmosphere exchange something on the order of 100 gigatonnes (Gt) of carbon (mostly as CO2) each year via diffusion (a mechanism known as the "solubility pump") and the exchange of 155 At least past a certain tipping point. Very small amounts of warming can (and have) produced expanding sea ice, especially in the Antarctic. The explanation for this involves the capacity of air of different temperatures to bear moisture. Antarctica, historically the coldest place on Earth, is often so cold that snowfall is limited by the temperature related lack of humidity. As the Antarctic continent has warmed slightly, its capacity for storing moisture has increased, leading to higher levels of precipitation in some locations. This effect is, however, both highly localized and transient. Continued warming will rapidly undo the gains associated with this phenomenon. 156 Feely et. al. (2007) 157 That is, 93% of the carbon that can be passed between the three active carbon reservoirs (land, ocean, and atmosphere), and thus is not sequestered (e.g. by being locked up in carbon-based minerals in the Earth's mantle). 182 organic biological matter (a mechanism known as the "biological pump), with a net transfer of approximately 2 Gt of carbon (equivalent to about 7.5 Gt of CO2) to the ocean. Since the industrial revolution, the planet's oceans have absorbed roughly one-third of all the anthropogenic carbon emissions. Given the its central role in the global carbon cycle, any 158 feedback mechanism that negatively impacts the ocean's ability to act as a carbon sink is likely to make an appreciable difference to the future of the climate in general. There are three primary positive warming feedbacks associated with a reduction in the oceans' ability to sequester carbon: (1) As anyone who has ever left a bottle of soda in a car on a very hot day (and ended up with an expensive cleaning bill) knows, liquid's ability to store dissolved carbon dioxide decreases as the liquid's temperature increases. As increased CO2 levels in the atmosphere lead to increased air temperatures, the oceans too will warm. This will decrease their ability to "scrub" excess CO2 from the atmosphere, leading to still more warming. (2) This increased oceanic temperature will also potentially disrupt the action of the Atlantic Thermohaline Circulation. The thermohaline transports a tremendous amount of water--something in the neighborhood of 100 times the amount of water moved by the Amazon river--and is the mechanism by which the cold anoxic water of the deep oceans is circulated to the surface. This renders the thermohaline essential not just for deep ocean life (in virtue of oxygenating the depths), but also an important component in the carbon cycle, as the water carried up from the depths is capable of absorbing more CO2 than the warmer water near the surface. The thermohaline is driven primarily by differences in water density, which in turn is a 158 Dawson and Spannagle (2007), p. 303-304 183 function of temperature and salinity . The heating and cooling of water as it is carried along by 159 the thermohaline forms a kind of conveyor belt that keeps the oceans well mixed through much the same mechanism responsible for the mesmerizing motion of the liquid in a lava lamp. However, the fact that the thermohaline's motion is primarily driven by differences in salinity and temperature means that it is extremely vulnerable to disruption by changes in those two factors. As CO2 concentration in the atmosphere increases and ocean temperatures increase accordingly, melting glaciers and other freshwater ice stored along routes that are accessible to the ocean can result in significant influxes of fresh (and cold) water. This alters both temperature and salinity of the oceans, disrupting the thermohaline and inhibiting the ocean's ability to act as a carbon sink. Teller et. al. (2002) argue that a similar large-scale influx of cold freshwater (in the form of the destruction of an enormous ice dam at Lake Agassiz) was partially responsible for the massive global temperature instability seen 15,000 years ago during the last major deglaciation . 160 (3) Perhaps most simply, increased acidification of the oceans (i.e. increased carbonic acid concentration as a result of CO2 reacting with ocean water) means slower rates of new CO2 absorption, reducing the rate at which excess anthropogenic CO2 can be scrubbed from the atmosphere. Examples like these abound in climatology literature. As we suggested above, though, perhaps the most important question with regard to climate feedbacks is whether the net 159 Vallis and Farnetti (2009) 160 In this case, the temporary shutdown of the thermohaline was actually responsible for a brief decrease in average global temperature--a momentary reversal of the nascent warming trend as the climate entered an interglacial period. This was due to differences in atmospheric and oceanic carbon content, and were a similar event to occur today it would likely have the opposite effect. 184 influence is positive or negative with respect to climate sensitivity. Climate sensitivity, recall, is the relationship between the change in the global concentration of greenhouse gases (given in units of CO2-equivalent impacts on radiative forcings) and the change in the annual mean surface air temperature (see Chapter Four). If the Earth were a simple system, free of feedbacks and other non-linearly interacting processes, this sensitivity would be a straightforwardly linear one: each doubling of CO2-e concentration would result in an increase of ~.30 , which would correspond to a mean surface temperature change of 1.2 degrees CK(W /m2) at equilibrium . 161 Unfortunately for climate modelers, things are not so simple. The net change in average surface air temperature following a CO2-e concentration doubling in the atmosphere also depends on (for instance) how the change in radiative forcing that doubling causes impacts the global albedo. The change in the global albedo, in turn, impacts the climate sensitivity by altering the relationship between radiative flux and surface air temperature. Just as with albedo, we can (following Roe & Baker [2007]) introduce a single parameter ᵱ� such that the net influence of feedbacks on the equation describing climate sensitivity: = ) 5(n) dt dT φ( dt dR In a feedback-free climate system, we can parameterize 5(n) such that ᵱ� = 1, and such that That is, we can assume that the net impact of positive and negative feedbacks on the. φ0 = φt total radiative flux is both constant and non-existent. However, just as with albedo, observations suggest that this simplification is inaccurate; ≠ . Discerning the value of is one of the φ0 φt φ 161 Roe & Baker (2007), p. 630 185 most challenging (and important) tasks in contemporary climate modeling. The presence of so many interacting feedback mechanisms is one of the features that makes climatology such a difficulty science to get right. It is also characteristic of complex systems more generally. How are we to account for these features when building high-level models of the global climate? What novel challenges emerge from models designed to predict the behavior of systems like this? In Chapter Six, we shall examine Coupled General Circulation Models (CGCMs), which are built to deal with these problems. 186 Chapter Six Why Bottle Lightning? 6.0 A Different Kind of Model We've now explored several significant challenges that climatologists must consider when attempting to create models of the global climate that even approach verisimilitude. The global climate is chaotic in the sense that very small perturbations of its state at one time lead to exponentially diverging sequences of states at later times. The global climate is also non-linear in the sense that equations describing its behavior fail both the additivity and degree-1 homogeneity conditions. They fail these conditions primarily in virtue of the presence of a number of distinct feedbacks between the subsystems of the global climate. In Chapter Four, we noted that while energy balance models in general are useful in virtue of their simplicity and ease of use, they fail to capture many of the nuances responsible for the behavior of the Earth's climate: while things like radiative balance are (generally speaking) the dominant features driving climate evolution, attending only to the most powerful influences will not always yield a model capable of precise predictive success. We saw how the more specialized EMIC-family of models can help ameliorate the shortcomings of the simplest models, and while the breadth and power of EMICs is impressive, there is surely a niche left to be filled in our modeling ecosystem: the comprehensive, high-fidelity, as-close-to-complete-as-we-can-get class of climate models. Coupled global circulation models (CGCMs) fill that niche, and strive for as much verisimilitude as possible given the 162 162 The term "coupled general circulation models" is also occasionally used in the literature. The two terms are generally equivalent, at least for our purposes here. 187 technological constraints. In contrast to the rough-and-ready simplicity energy balance models and the individual specialization of EMICs, CGCMs are designed to be both general and detailed: they are designed to model as many of the important factors driving the Earth's climate as well as they possibly can. This is a very tall order, and the project of crafting CGCMs raises serious problems that EBMs and EMICs both manage to avoid. Because of their comprehensiveness, though, they offer the best chance for a good all-things-considered set of predictions about the future of Earth's climate. The implementation of CGCMs is best understood as a careful balancing act between the considerations raised in Chapter Five. CGCMs deliberately incorporate facts about the interplay between atmospheric, oceanic, and terrestrial features of the global climate system, and thus directly confront many of the feedback mechanisms that regulate the interactions between those coupled subsystems of the Earth's climate. It should come as no surprise, then, that most CGCMs prominently feature systems of nonlinear equations, and that one of the primary challenges of working with CGCMs revolves around how to handle these non-linearities. While the use of supercomputers to simulate the behavior of the global climate is absolutely essential if we're to do any useful work with CGCMs, fundamental features of digital computers give rise to a set of serious challenges for researchers seeking to simulate the behavior of the global climate. The significance of these challenges must be carefully weighed against the potentially tremendous power of well-implemented CGCMs. In the end, I shall argue that CGCMs are best understood not as purely predictive models, but rather as artifacts whose role is to help us make decisions about how to proceed in our study of (and interaction with) the global climate. 188 6.1 Lewis Richardson's Fabulous Forecast Machine The dream of representing the world inside a machine--of generating a robust, detailed, real-time forecast of climate states--reaches all the way back to the early days of meteorology. In 1922, the English mathematician Lewis Richardson proposed a thought experiment that he called "the forecast factory." The idea is so wonderfully articulated (and so far-seeing) that it is worth quoting at length here: Imagine a large hall like a theatre, except that the circles and galleries go right round through the space usually occupied by the stage. The walls of this chamber are painted to form a map of the globe. The ceiling represents the north polar regions, England is in the gallery, the tropics in the upper circle, Australia on the dress circle, and the Antarctic in the pit. A myriad computers are at work upon the weather of the part of 163 the map where each sits, but each computer attends only to one equation or one part of an equation. The work of each region is coordinated by an official of higher rank. Numerous little 'night signs' display the instantaneous values so that neighboring computers can read them. Each number is thus displayed in three adjacent zones so as to maintain communication to the North and South on the map. From the floor of the pit a tall pillar rises to half the height of the hall. It carries a large pulpit on its top. In this sits the man in charge of the whole theatre; he is surrounded by several assistants and messengers. One of his duties is to maintain a uniform speed of progress in all parts of the globe. In this respect he is like the conductor of an orchestra in which the instruments are slide-rules and calculating machines. But instead of waving a baton he turns a beam of rosy light upon any region that is running ahead of the rest, and a beam of blue light upon those who are behindhand. Four senior clerks in the central pulpit are collecting the future weather as fast as it is being computed, and dispatching it by pneumatic carrier to a quiet room. There it will be coded and telephoned to the radio transmitting station. Messengers carry piles of used computing forms down to a storehouse in the cellar. In a neighboring building there is a research department, where they invent improvements. But there is much experimenting on a small scale before any change is made in the complex routine of the computing theatre. In a basement an enthusiast is observing eddies in the liquid lining of a huge spinning bowl, but so far the arithmetic proves the better way. In another building are all the usual financial, correspondence, and 164 163 At the time when Richardson wrote this passage, the word 'computer' referred not to a digital computer--a machine--but rather to a human worker whose job it was to compute the solution to some mathematical problem. These human computers were frequently employed by those looking to forecast the weather (among other things) well into the 20th century, and were only supplanted by the ancestors of modern digital computers after the advent of punch card programming near the end of World War II. 164 Here, Richardson is describing the now well-respected (but then almost unheard of) practice of studying what might be called "homologous models" in order to facilitate some difficult piece of computation. For example, Bringsjord and Taylor (2004) propose that observation of the behavior of soap bubbles under certain conditions might yield greater understanding of the Steiner tree problem in graph theory. The proposal revolves around the fact that soap bubbles, in order to maintain cohesion, rapid relax their shapes toward a state where surface energy (and thus area) is minimized. There are certain structural similarities between the search for this optimal low-energy state and the search for the shortest-length graph in the Steiner tree problem. Similarly, Jones and Adamatzsky (2013) show slime molds' growth and foraging networks show a strong preference for path-length optimization, a feature that can be used to compute a fairly elegant solution to the Traveling Salesman problem. 189 administrative offices. Outside are playing fields, houses, mountains, and lakes, for it was thought that those who compute the weather should breathe of it freely. Fig. 6.1 Artist's conception of Lewis Richardson's forecast factory 165 Richardson's forecast factory (Fig. 6.1) was based on an innovation in theoretical meteorology and applied mathematics: the first step toward integrating meteorology with atmospheric physics, and thus the first step toward connecting meteorology and climatology into a coherent discipline united by underlying mathematical similarities. Prior to the first decade of the 20th century, meteorologists spent the majority of their time each day charting the weather in their region--recording things like temperature, pressure, wind speed, precipitation, humidity, and so on over a small geographical area. These charts were meticulously filed by day and time, and when the meteorologist wished to make a forecast, he would simply consult the most current chart and then search his archives for a historical chart that was qualitatively similar. He would then examine how the subsequent charts for the earlier time had evolved, and would forecast something similar for the circumstance at hand. This qualitative approach began to fall out of favor around the advent of World War I. In the 165 Image by Francois Schuiten, drawn from Edwards (2010), p. 96 190 first years of the 20th century, a Norwegian physicist named Vilhelm Bjerknes developed the first set of what scientist today would call "primitive equations" describing the dynamics of the atmosphere. Bjerknes' equations, adapted primarily from the then-novel study of fluid dynamics, tracked four atmospheric variables--temperature, pressure, density, and humidity (water content)--along with three spatial variables, so that the state of the atmosphere could be represented in a realistic three-dimensional way. Bjerknes, that is, defined the first rigorous state space for atmospheric physics . 166 However, the nonlinearity and general ugliness of Bjerknes' equations made their application prohibitively difficult. The differential equations coupling the variables together were far too messy to admit of an analytic solution in any but the most simplified circumstances. Richardson's forecast factory, while never actually employed at the scale he envisioned, did contain a key methodological innovation that made Bjerknes' equations practically tractable again: the conversion of differential equations to difference equations. While Bjerknes' atmospheric physics equations were differential--that is, described infinitesimal variations in quantities over infinitesimal time-steps--Richardson's converted equations tracked the same quantities as they varied by finite amounts over finite time-steps. Translating differential equations into difference equations opens the door to the possibility of generating numerical approximation of answers to otherwise intractable calculus problems. In cases like Bjerknes' where we have a set of differential equations for which it's impossible to discern any closed-form analytic solutions, numerical approximation by way of difference equations can be a godsend: it allows us to transform calculus into repeated arithmetic. More importantly, it allows 166 Edwards (2010), pp. 93-98 191 us to approximate the solution to such problems using a discrete state machine--a digital computer. 6.2.0 General Circulation Models Contemporary computational climate modeling has evolved from the combined insights of Bjerknes and Richardson. Designers of high-level Coupled General Circulation Models (CGCMs) build on developments in atmospheric physics and fluid dynamics. In atmospheric circulation models, the primitive equations track six basic variables across three dimensions : 167 surface pressure, horizontal wind components (in the x and y directions), temperature, moisture, and geopotential height. Oceanic circulation models are considerably more varied than their atmospheric cousins, reflecting the fact that oceanic models' incorporation into high-level climate models is a fairly recent innovation (at least compared to the incorporation of atmospheric models). Until fairly recently, even sophisticated GCMs treated the oceans as a set of layered "slabs," similar to the way the atmosphere is treated in simple energy balance models (see Chapter Four). The simple "slab" view of the ocean treats it as a series of three-dimensional layers stacked on top of one another, each with a particular heat capacity, but with minimal (or even no) dynamics linking them. Conversely (but just as simply), what ocean modelers call the "swamp model" of the ocean treats it as an infinitely thin "skin" on the surface of the Earth, with currents and dynamics that contribute to the state of the atmosphere but with no heat capacity of its own. Early CGCMs thus incorporated ocean modeling only as a kind of adjunct to the more sophisticated atmospheric models: the primary focus was on impact that 167 Ibid., p. 178 192 ocean surface temperatures and/or currents had on the circulation of air in the atmosphere. Methodological innovations in the last 15 years--combined with theoretical realizations about the importance of the oceans (especially the deep oceans) in regulating both the temperature and the carbon content of the atmosphere (see Section 5.2.2)--have driven the creation of more sophisticated oceanic models fusing these perspectives. Contemporary general ocean circulation models are at least as sophisticated as general atmospheric circulation models--and often more sophisticated. The presence of very significant constant vertical circulation in the oceans (in the form of currents like the thermohaline discussed in 5.2.2) means that there is a strong circulation between the layers (though not as strong as the vertical circulation in the atmosphere). Moreover, the staggering diversity and quantity of marine life--as well as the impact that they have on the dynamics of both the ocean and atmosphere--adds a wrinkle to oceanic modeling that has no real analog in atmospheric modeling. Just as in Richardson's forecast factory, global circulation models (both in the atmosphere and the ocean) are implemented on a grid (usually one that's constructed on top of the latitude/longitude framework). This grid is constructed in three dimensions, and is divided into cells in which the actual equations of motion are applied. The size of the cells is constrained by a few factors, most significantly the computational resources available and the desired length of the time-step when the model is running. The first condition is fairly intuitive: smaller grids require both more computation (because the computer is forced to simulate the dynamics at a larger number of points) and more precise data in order to generate reliable predictions (there's no use in computing the behavior of grid cells that are one meter to a side if we can only 193 resolve/specify real-world states using a grid 1,000 meters to a side). The link between time-step length and grid size, though, is perhaps slightly less obvious. In general, the shorter the time-steps in the simulation--that is, the smaller ᵱ�t is in the difference equations underlying the simulation--the smaller the grid cells must be. This makes sense if we recall that the simulation is supposed to be modeling a physical phenomenon, and is therefore constrained by conditions on the transfer of information between different physical points. After all, the grid must be designed such that during the span between one time-step and the next, no relevant information about the state of the world inside one grid cell could have been communicated to another grid cell. This is a kind of locality condition on climate simulations, and must be in place if we're to assume that relevant interactions--interactions captured by the simulation, that is--can't happen at a distance. Though a butterfly's flapping wings might eventually spawn a hurricane on the other side of the world, they can't do so instantly: the signal must propagate locally around the globe (or, in the case of the model, across grid cells). This locality condition is usually written: ᵱ�t ≤ ᵱ�x / c 6(a) In the context of climate modeling, c refers not to the speed of light in a vacuum, but rather the maximum speed at which information can propagate through the medium being modeled Its value thus is different in atmospheric and oceanic models, but the condition holds in both cases: the timesteps must be short enough that even if it were to propagate at the maximum possible speed, information could not be communicated between one cell and another between one time step and the next. 194 One consequence of 6(a) is that smaller spatial grids also require shorter time steps. This means that the computational resources required to implement simulations at a constant speed increase not arithmetically, but geometrically as the simulation becomes more precise . 168 Smaller grid cells--and thus more precision--require not just more computation, but also faster computation; the model must generate predictions for the behavior of more cells, and it must do so more frequently . 169 Implementing either an atmospheric or oceanic general circulation model is a careful balancing act between these (and many other) concerns. However, the most sophisticated climate simulations go beyond even these challenges, and seek to couple different fully-fledged circulation models together to generate a comprehensive CGCM. 6.2.1 Coupling General Circulation Models We can think of CGCMs as being "meta-models" that involve detailed circulation models of the atmosphere and ocean (and, at least sometimes, specialized terrestrial and cryosphere models) being coupled together. While some CGCMs do feature oceanic, atmospheric, cryonic, and terrestrial models that interface directly with one another (e.g. by having computer code in the atmospheric model "call" values of variables in the oceanic model), this direct interfacing is incredibly difficult to implement. Despite superficial similarities in the primitive equations underlying both atmospheric and oceanic models--both are based heavily on fluid 168 In practice, halving the grid size does far more than double the computational resources necessary to run the model at the same speed. Recall that in each grid, at least six distinct variables are being computed across three dimensions, and that doubling the number of cells doubles the number of each of these calculations. 169 Of course, another option is to reduce the output speed of the model--that is, to reduce the ratio of "modeled time" to "model time." Even a fairly low-power computer can render the output of a small grid / short time step model given enough time to run. At a certain point, the model output becomes useless; a perfect simulation of the next decade of the global climate isn't much use if it takes several centuries to output. 195 dynamics--differences in surface area, mass, specific heat, density, and a myriad of other factors lead to very different responses to environmental inputs. Perhaps most importantly, the ocean and atmosphere have temperature response and equilibrium times that differ by several orders of magnitude. That is, the amount of time that it takes the ocean to respond to a change in the magnitude of some climate forcing (e.g. an increase in insolation, or an increase in the concentration of greenhouse gases) is significantly greater than the amount of time that it takes the atmosphere to respond to the same forcing change. This is fairly intuitive; it takes far more time to heat up a volume of water by a given amount than to heat up the same volume of air by the same amount (as anyone who has attempted to boil a pot of water in his or her oven can verify). This difference in response time means that ocean and atmosphere models which are coupled directly together must incorporate some sort of correction factor, or else run asynchronously most of the time, coupling only occasionally to exchange data at appropriate intervals. Were they to couple directly and constantly, the two models' outputs would 170 gradually drift apart temporally. In order to get around this problem, many models incorporate an independent module called a "flux coupler," which is designed to coordinate the exchange of information between the different models that are being coupled together. The flux coupler is directly analogous to the "orchestra conductor" figure from Richardson's forecast factory. In just the same way that Richardson's conductor used colored beams of light to keep the various factory workers synchronized in their work, the flux coupler transforms the data it receives from the component models, implementing an appropriate time-shift to account for differences in response time (and 170 McGuffie and Anderson-Sellers (2010), p. 204-205 196 other factors) between the different systems being modeled. A similarly named (but distinct) process called "flux adjustment" (or "flux correction") has been traditionally employed to help correct for local (in either the temporal or spatial sense) cyclical variations in the different modeled systems, and thus help ensure that the model's output doesn't drift too far away from observation. Seasonal temperature flux is perhaps the most significant and easily-understood divergence for which flux adjustment can compensate. Both the atmosphere and the ocean (at least the upper layer of the ocean) warm during summer months and cool during winter months. In the region known as the interface boundary--the spatial region corresponding to the surface of the ocean, where water and atmosphere meet--both atmospheric and oceanic models generate predictions about the magnitude of this change, and thus the fluctuation in energy in the climate system. Because of the difficulties mentioned above (i.e. differences in response time between seawater and air), these two predictions can come radically uncoupled during the spring and fall when the rate of temperature change is at its largest. Left unchecked, this too can lead to the dynamics of the ocean and atmosphere "drifting" apart, magnifying the error range of predictions generated through direct couplings of the two models. Properly designed, a flux adjustment can "smooth over" these errors by compensating for the difference in response time, thus reducing drift. 6.2.2 Flux Adjustment and "Non-Physical" Modeling Assumptions Flux adjustment was an early and frequent object of scrutiny by critics of mainstream climatology. The "smoothing over" role of the flux adjustment is frequently seized upon by critics of simulation-based climate science as unscientific or ad-hoc in a problematic way. The 197 NIPCC's flagship publication criticizing climate science methodology cites Sen Gupta et. al. (2012), who write that "flux adjustments are nonphysical and therefore inherently undesirable... [and] may also fundamentally alter the evolution of a transient climate response. " Even the 171 IPCC's Fourth Assessment Report acknowledges that flux adjustments are "essentially empirical corrections that could not be justified on physical principles, and that consisted of arbitrary additions of surface fluxes of heat and salinity in order to prevent the drift of the simulated climate away from a realistic state. " 172 What does it mean to say that flux adjustments are "non-physical?" How do we know that such adjustments shift the climate system away from a "realistic state?" It seems that the most plausible answer to this question is that, in contrast to the other components of climate simulations, the flux adjustment fails to correspond directly with quantities in the system being modeled. That is, while the parameters for (say) cloud cover, greenhouse gas concentration, and insolation correspond rather straightforwardly to real aspects of the global climate, the action of the flux adjustment seems more like an ad hoc "fudge factor" with no physical correspondence. The most forceful way of phrasing the concern suggests that by manipulating the parameterization of a flux adjustment, a disingenuous climate modeler might easily craft the output of the model to suit his biases or political agenda. Is the inclusion of a flux adjustment truly ad hoc, though? Careful consideration of what we've seen so far suggests that it is not. Recall the fact that the patterns associated with coarse-grained climate sensitivity have been well-described since (at least) Arrhenius' work in 171 Sen Gupta et. al. (2012), p. 4622, quoted in Lupo and Kininmonth and (2013), p. 19 172 IPCC AR4: 1.5.3 198 the late 19th century. Moreover, the advent of quantum mechanics in the 20th century has provided a succinct physical explanation for Arrhenius' observed patterns (as we saw in Chapter Four). Changes in the concentration of CO2-e greenhouse gases in the Earth's atmosphere have a deterministic impact on the net change in radiative forcing--an impact that is both well understood and well supported by basic physical theory. But what of the arguments from Chapter One, Two, and Three about the scale relative behavior of complex systems? Why should we tolerate such an asymmetrical "bottom-up" constraint on the structure of climate models? After all, our entire discussion of dynamical complexity has been predicated on the notion that fundamental physics deserves neither ontological nor methodological primacy over the special sciences. How can we justify this sort of implied primacy for the physics-based patterns of the global climate system? These questions are, I think, ill-posed. As we saw in Chapter One, there is indeed an important sense in which the laws of physics are fundamental. I argued there that they are fundamental in the sense that they "apply everywhere," and thus are relevant for generating predictions for how any system will change over time, no matter how the world is "carved up" to define a particular system. At this point, we're in a position to elaborate on this definition a bit: fundamental physics is fundamental in the sense that it constrains each system's behavior at all scales of investigation. 6.3.1 Constraints and Models The multiplicity of interesting (and useful) ways to represent the same system-the fact that precisely the same physical system can be represented in very different state spaces, and that 199 interesting patterns about the time-evolution of that system can be found in each of those state spaces-has tremendous implications. Each of these patterns, of course, represents a constraint on the behavior of the system in question; if some system's state is evolving in a way that is described by some pattern, then (by definition) its future states are constrained by that pattern. As long as the pattern continues to describe the time-evolution of the system, then states that it can transition into are limited by the presence of the constraints that constitute the pattern. To put the point another way: patterns in the time-evolution of systems just are constraints on the system's evolution over time. It's worth emphasizing that all these constraints can (and to some degree must) apply to all the state spaces in which a particular system can be represented. After all, the choice of a state space in which to represent a system is just a choice of how to describe that system, and so to notice that a system's behavior is constrained in one space is just to notice that the system's behavior is constrained period. Of course, it's not always the case that the introduction of a new constraint at a particular level will result in a new relevant constraint in every other space in which the system can be described. For a basic example, visualize the following scenario. Suppose we have three parallel Euclidean planes stacked on top of one another, with a rigid rod passing through the three planes perpendicularly (think of three sheets of printer paper stacked, with a pencil poking through the middle of them). If we move the rod along the axis that's parallel to the planes, we can think of this as representing a toy multi-level system: the rod represents the system's state; the planes represent the different state-spaces we could use to describe the system's position (i.e. by specifying its location along each plane). Of course, if the paper is intact, we'd rip the sheets as we dragged the pencil around. Suppose, then, that the rod 200 can only move in areas of each plane that have some special property-suppose that we cut different shapes into each of the sheets of paper, and mandate that the pencil isn't allowed to tear any of the sheets. The presence of the cut-out sections on each sheet represents the constraints based on the patterns present on the system's time-evolution in each state-space: the pencil is only allowed in areas where the cut-outs in all three sheets overlap. Suppose the cut-outs look like this. On the top sheet, almost all of the area is cut away, except for a very small circle near the bottom of the plane. On the middle sheet, the paper is cut away in a shape that looks vaguely like a narrow sine-wave graph extending from one end to another. On the bottom sheet, a large star-shape has been cut out from the middle of the sheet. Which of these is the most restrictive? For most cases, it's clear that the sine-wave shape is: if the pencil has to move in such a way that it follows the shape of the sine-wave on the middle sheet, there are vast swaths of area in the other two sheets that it just can't access, no matter whether there's a cut-out there or not. In fact, just specifying the shape of the cut-outs on two of the three sheets (say, the top and the middle) is sometimes enough to tell us that the restrictions placed on the motion of the pencil by the third sheet will likely be relatively unimportant-the constraints placed on the motion of the pencil by the sine-wave sheet are quite stringent, and those placed on the pencil by the star-shape sheet are (by comparison) quite lax. There are comparatively few ways to craft constraints on the bottom sheet, then, which would result in the middle sheet's constraints dominating here: most cutouts will be more restrictive than the top sheet and less restrictive than the middle sheet 173 The lesson here is that while the state of any given system at a particular time has to be 173 Terrance Deacon (2012)'s discussion of emergence and constraint is marred by this confusion, as he suggests that constraints in the sense of interest to us here just are boundary conditions under which the system operates. 201 consistent with all applicable constraints (even those resulting from patterns in the state-spaces representing the system at very different levels of analysis), it's not quite right to say that the introduction of a new constraint will always affect constraints acting on the system in all other applicable state spaces. Rather, we should just say that every constraint needs to be taken into account when we're analyzing the behavior of a system; depending on what collection of constraints apply (and what the system is doing), some may be more relevant than others. The fact that some systems exhibit interesting patterns at many different levels of analysis-in many different state-spaces-means that some systems operate under far more constraints than others, and that the introduction of the right kind of new constraint can have an effect on the system's behavior on many different levels. 6.3.2 Approximation and Idealization The worry is this: we've established a compelling argument for why we ought not privilege the patterns identified by physics above the patterns identified by the special sciences. On the other hand, it seems right to say that when the predictions of physics and the predictions of the special sciences come into conflict, the predictions of physics ought to be given primacy at least in some cases. However, it's that last clause that generates all the problems: if what we've said about the mutual constraint (and thus general parity) of fundamental physics and the special sciences is correct, then how can it be the case that the predictions of physics ever deserve primacy? Moreover, how on earth can we decide when the predictions of physics should be able to overrule (or at least outweigh) the predictions of the special sciences? How can we reconcile these two arguments? Here's a possible answer: perhaps the putative patterns identified by climate science in this 202 case are approximations or idealizations of some as-yet unidentified real patterns. If this is the case, then we have good reason to think that the patterns described by (for instance) Arrhenius deserve some primacy over the approximated or idealized erstaz patterns employed in the construction of computational models. What counts as an approximation? What counts as an idealization? Are these the same thing? It's tempting to think that the two terms are equivalent, and that it's this unified concept that's at the root of our difficulty here. However, there's good reason to think that this assumption is wrong on both counts: there's a significant difference between approximation and idealization in scientific model building, and neither of those concepts accurately captures the nuances of the problem we're facing here. Consider our solar system. As we discussed in Chapter Five, the equations describing how the planets' positions change over time are technically chaotic. Given the dynamics describing how the positions of the planets evolves, two trajectories through the solar system's state space that begin arbitrarily close together will diverge exponentially over time. However, as we noted before, just noting that a system's behavior is chaotic leaves open a number of related questions about how well we can predict its long-term behavior. Among other things, we should also pay attention to the spatio-temporal scales over which we're trying to generate interesting predictions, as well as our tolerance for certain kinds of error in those predictions. In the case of the solar system, for instance, we're usually interested in the positions of the planets (and some interplanetary objects like asteroids) on temporal and spatial scales that are relevant to our decidedly humanistic goals. We care where the planets will be over the next few thousand years, and at the most are interested in their very general behavior over times ranging from a few 203 hundred thousand to a few million years (to study the impact of Milankovitch cycles on the global climate, for instance). Similarly, we're usually perfectly comfortable with predictions that introduce errors of (say) a few thousand kilometers in the position of Mercury in the next century . The fact that we can't give a reliable prediction about where Mercury will be in its orbit at 174 around the time Sol ceases to be a main-sequence star--or similarly that we can't give a prediction about Mercury's position in its orbit in five years that gets things right down to the centimeter--doesn't really trouble us most of the time. This suggests that we can fruitfully approximate the solar system's behavior as non-chaotic, given a few specifications about our predictive goals. Norton (2012) argues that we can leverage this sort of example to generate a robust distinction between approximation and idealization, terms which are often used interchangeably. He defines the difference as follows: "approximations merely describe a target system inexactly" while "[i]dealizations refer to new systems whose properties approximate those of the target system." Norton argues that the important distinction here is one of reference, with "idealizations...carry[ing] a novel semantic import not carried by approximations." The 175 distinction between approximation and idealization, on Norton's view, is that idealization involves the construction of an entirely novel system, which is then studied as a proxy for the actual system of interest. Approximation, on the other hand, involves only particular parameterizations of the target system--parameterizations in which assigned values describe the 174 Of course, there are situations in which we might demand significantly more accurate predictions than this. After all, the difference between an asteroid slamming into Manhattan and drifting harmlessly by Earth is one of only a few thousand kilometers! 175 Norton (2012), pp. 207-208 204 original system inexactly in some sense. It's worth pointing out that Norton's two definitions will, at least sometimes, exist on a continuum with one another: in some cases, approximations can be smoothly transformed into idealizations. 176 This interconversion is possible, for instance, in cases where the limits used in constructing idealized parameterizations are "well-behaved" in the sense that the exclusive use of limit quantities in the construction of the idealized system still results in a physically realizable system. This will not always be the case. For example, consider some system S whose complete state at a time t is described by an equation of the form 6(b)(t) ( )S = α n 1 In this case, both ᵯ� and n can be taken as parameterizations of S(t). There are a number of approximations we might consider. For instance, we might wonder what happens to S(t) as ᵯ� and n both approach 0. This yields a prediction that is perfectly mathematically consistent; S(t) approaches a real value as both those parameters approach 0. By Norton's definition this is an approximation of S(t), since we're examining the system's behavior in a particular limit case. However, consider the difference between this approximation and the idealization of S in which ᵯ� = 0 and n = 0. Despite the fact that the approximation yielded by considering the system's behavior as ᵯ� and n both approach 0 is perfectly comprehensible (and hopefully informative as well), actually setting those two values to 0 yields a function value that's undefined. The limits involved in the creation of the approximation are not "well behaved" in 176 Norton (2012), p. 212 205 Norton's sense, and so cannot be used directly to create an idealization. Norton argues that qualitatively similar behavior is common in the physical sciences--that perfectly respectable approximations of a given system frequently fail to neatly correspond to perfectly respectable idealizations of the same system. Of course, we might wonder what it even means in those cases to say that a given system is an idealization of another system. If idealization involves the genesis of a novel system that can differ not just in parameterization values but in dynamical form the original target system, then how do idealizations represent at all? The transition from an approximation to its target system is clear, as such a transition merely involves reparameterization; the connection between target system and idealization is far more tenuous (if it is even coherent). Given this, it seems that we should prefer (when possible) to work with approximations rather than idealizations. Norton shares this sentiment, arguing that since true idealizations can incorporate "infinite systems" of the type we explored above and "[s]ince an infinite system can carry unexpected and even contradictory properties, [idealization] carries considerably more risk [than approximation]. [...] If idealizations are present, a dominance argument favors their replacement by approximations." 177 6.3.3 Idealization and Pragmatism It's interesting to note that the examples in Norton (2012) are almost uniformly drawn from physics and statistical mechanics. These cases provide relatively easy backdrops against which to frame the discussion, but it's not immediately apparent how to apply these lessons to the 177 Norton (2012), p. 227 206 messier problems in the "high level" special sciences--particularly those concerned with complex systems. Weisberg (2007) suggests a framework that may be more readily applicable to projects like climate modeling. Weisberg discusses a number of different senses of 'idealization,' but for our purposes the concept that he calls "multiple-model idealization" (MMI) is the most interesting. Weisberg defines MMI as "the practice of building multiple related but incompatible models, each of which makes distinct claims about the nature and causal structure giving rise to a phenomenon." He presents the model building practice of the United States' National 178 Weather Service (NWS) as a paradigmatic example of day-to-day MMI: the NWS employs a broad family of models that can incorporate radically different assumptions not just about the parameters of the system being modeled, but of the dynamical form being modeled as well. This pluralistic approach to idealization sidesteps the puzzle we discussed at the close of Section 6.3.2. On Norton's view, it's hard to see how idealizations represent in the first place, since the discussion of representation can't even get off the ground without an articulation of a "target system" and the novel idealized system cooked up to represent it. Weisberg-style pluralistic appeals like MMI are different in subtle but important ways. Weisberg's own formulation makes reference to a "phenomenon" rather than a "target system:" a semantic difference with deep repercussions. Most importantly, MMI-style approaches to modeling and idealization let us start with a set of predictive and explanatory goals to be realized rather than some putative target system that we may model/approximate/idealize more-or-less perfectly. By Norton's own admission, his view of approximation and idealization is one that grounds the distinction firmly in representational content. While this approach to the philosophy of 178 Weisberg (2007), p. 647 207 science is the inheritor of a distinguished lineage, the more pragmatically oriented approach sketched by Weisberg is more suitable for understanding contemporary complex systems sciences. As we saw in Section 6.3.2, the question of whether or not a non-chaotic approximation of our solar system's behavior is a "good" approximation is purpose-relative. There's no interesting way in which one or another model of the solar system's long-term behavior is "good" without reference to our predictive goals. Pragmatic idealization lets us start with a goal--a particular prediction, explanation, or decision--and construct models that help us reach that goal. These idealizations are good ones not because they share a particular kind of correspondence with an a priori defined target system, but because they are helpful tools. We will revisit this point in greater detail Section 6.4.2. 6.3.4 Pragmatic Idealization The solar system, while chaotic, is a system of relatively low dynamical complexity. The advantages of pragmatic MMI-style accounts of idealization over Norton-style hard-nosed realist accounts of idealization become increasingly salient as we consider more dynamically complex systems. Let's return now to the question that prompted this digression. How can we reconcile a strongly pluralistic view of scientific laws with the assertion that the greenhouse effect's explanatory grounding in the patterns of physics should give us reason to ascribe a strong anthropogenic component to climate change even in the face of arguments against the veracity of individual computational climate simulations? At the close of Section 6.3.1, I suggested that perhaps the resolution to this question lay in a consideration of the fact that models like the GISS approximate the dynamics of the global climate. In light of the discussion in Sections 6.3.2 and 6.3.3, though, this doesn't seem quite right. Computational models are not approximations of the 208 global climate in any interesting sense; they are not mere limit-case parameterizations of a single complete model. Neither, though, are they idealizations in Norton's sense. It seems far more accurate to think of general circulation models (coupled or otherwise) as pragmatic idealizations in the sense described above. More strongly, this strikes me as the right way to think about climate models in general--as tools crafted for a particular purpose. This lends further credence to the point that I've argued for repeatedly here: that the pluralistic and heterogeneous character of the climate model family reflects not a historical accident of development or a temporary waystation on the road to developing One Model to Rule them All. Rather, this pluralism is a natural result of the complexity of the climate system, and of the many fruitful perspectives that we might adopt when studying it. The project of modeling the global climate in general, then, is a project of pragmatic idealization. The sense of 'idealization' here is perhaps somewhere between Weisberg's and Norton's. It differs most strongly from Norton's in the sense that the values of parameters in a pragmatic idealization need not approximate values in the "target system" of the global climate at all. Some apsects of even the best models, in fact, will have explicitly non-physical parameters; this was the worry that kicked off the present discussion to begin with, since it seems that processes like flux adjustment have no direct physical analogues in the global climate itself. Rather, they are artifacts of the particular model--the particular approach to pragmatic idealization--under consideration. How problematic is it, then, that the flux adjustment has no direct physical analog in the 209 system being modeled? It seems to me that the implication is not so dire as Lupo and Kinimouth make it out to be. This is one sense in which the pragmatic idealization approach shares something in common with Norton's story--when we create any climate model (but especially a CGCM like the GISS), we have done more than approximate the behavior of the climate system. We've created a novel system in its own right: one that we hope we can study as a proxy for the climate itself. The objection that there are aspects of that novel system that have no direct analogue in the global climate itself is as misguided as the objection that no climate model captures every aspect of the climate system. The practice of model building--the practice of pragmatic idealization--involves choices about what to include in any model, how to include it, what to leave out, and how to justify that exclusion. These questions are by no means trivial, but neither are they insurmountable. 6.3.5 Ensemble Modeling and CGCMs Our discussion so far has focused on the advantages of studying feedback-rich nonlinear systems via computational models: numerical approximation of the solutions to large systems of coupled nonlinear differential equations lets us investigate the global climate in great detail, and through the use of equations derived from well-understood low-level physical principles. However, we have said very little so far about the connection between chaotic behavior and computational modeling. Before we turn to the criticisms of this approach to modeling, let's say a bit about how simulation is supposed to ameliorate some of the challenges of chaotic dynamics in the climate. Chaos, recall, involves the exponential divergence of the successors to two initial conditions 210 that are arbitrarily close together in the system's state space. The connection to climate modeling is straightforward. Given the difficulty--if not impossibility--of measuring the current (not to mention the past) state of the climate with anything even approaching precision, it's hard to see how we're justified in endorsing the predictions made by models which are initialized using such error-ridden measurements for their initial conditions. If we want to make accurate predictions about where a chaotic system is going, it seems like we need better measurements--or a better way to generate initial conditions . 179 This is where the discussion of the "predictive horizon" from Section 5.1.3 becomes salient. I argued that chaotic dynamics don't prevent us from making meaningful predictions in general; rather, they force us to make a choice between precision and time. If we're willing to accept a certain error range in our predictions, we can make meaningful predictions about the behavior of a system with even a very high maximal Lyapunov exponent out to any arbitrary time. This foundational observation is implemented in the practice of ensemble modeling. Climatologists don't examine the predictions generated by computational models in isolation--no single "run" of the model is treated as giving accurate (or even meaningful) output. Rather, model outputs are evaluated as ensembles: collections of dozens (or more) of runs taken as a single unit, and interpreted as defining a range of possible paths that the system might take over the specified time range. Climate modelers' focus is so heavily on the creation and interpretation of ensembles that the 179 This problem is compounded by the fact that we often want to initialize climate models to begin simulating the behavior of the climate at times far before comprehensive measurements of any kind--let alone reliable measurements--are available. While we can get some limited information about the climate of the past through certain "proxy indicators" (see Michael Mann's work with glacial air bubbles, for instance), these proxy indicators are blunt tools at best, and are not available at all for some time periods. 211 in most cases CGCMs aren't even initialized with parameter values drawn from observation of the real climate's state at the start of the model's run. Rather, GCMs are allowed to "spin up" to a state that's qualitatively identical to the state of the global climate at the beginning of the model's predictive run. Why add this extra layer of complication to the modeling process, rather than just initializing the model with observed values? The spin up approach has a number of advantages; in addition to freeing climate modelers from the impossible task of empirically determining the values of all the parameters needed to run the model, the spin up also serves as a kind of rough test of the proposed dynamics of the model before it's employed for prediction and ensures that parameter values are tailored for the grid-scale of the individual model. A typical spin up procedure looks like this. The grid size is defined, and the equations of motion for the atmospheric, oceanic, terrestrial, and cryonic models are input. In essence, this defines a "dark Earth" with land, sky, and water but no exogenous climate forcings. The climate modelers then input relevant insolation parameters--they flip on the sun. This (unsurprisingly) causes a cascade of changes in the previously dark Earth. The model is allowed to run for (in general) a few hundred thousand years of "model time" until it settles down into a relatively stable equilibrium with temperatures, cloud cover, and air circulation patterns that resemble the real climate's state at the start of the time period under investigation. The fact that the model does settle into such a state is at least a prima facie proof that it's gotten things relatively right; if the model settled toward a state that looked very little like the state of interest (if it converged on a "snowball Earth" covered in glaciers, for instance), we would take it as evidence that something was very wrong indeed. Once the model has converged on this equilibrium state, modelers can feed in hypothetical parameters and observe the impact. They can change the 212 concentration of greenhouse gases in the atmosphere, for instance, and see what new equilibrium the system moves to (as well as what path it takes to get there). By tinkering with the initial equations of motion (and doing another spin up), the length of the spin-up, and the values of parameters fed in after the spin up, modelers can investigate a variety of different scenarios, time-periods, and assumptions. The use of spin up and ensemble modeling is designed to smooth over the roughness and error that results from the demonstrably tricky business of simulating the long-term behavior of a large, complex, chaotic system; whether simple numerical approximations of the type discussed above or more sophisticated methods are used, a degree of "drift" in these models is inevitable. Repeated runs of the model for the same time period (and with the same parameters) will invariably produce a variety of predicted future states as the sensitive feedback mechanisms and chaotic dynamics perturb the model's state in unexpected, path-dependent ways. After a large number of runs, though, a good model's predictions will sketch out a well-grouped family of predictions--this range of predictions is a concrete application of the prediction horizon discussion from above. Considered as an ensemble, the predictions of a model provide not a precise prediction for the future of the climate, but rather a range of possibilities. This is true in spite of the fact that there will often be significant quantitative differences between the outputs of each model run. To a certain extent, the name of the game is qualitative prediction here. This is one respect in which the practices of climatology and meteorology have become more unified since Richardson's and Bjerknes' day. Meteorologists--who deal with many of the same challenges that climatologists tackle, albeit under different constraints --employ nearly 180 180 This too is a practical illustration of the concept of the predictive horizon. Weather prediction must be far more 213 identical ensemble-based approaches to weather modeling and prediction. In both cases, the foundation of the uncertainty terms in the forecast--that is, the grounding of locutions like "there is a 70% chance that it will rain in Manhattan tomorrow" or "there is a 90% chance that the global average temperature will increase by two or more degrees Celsius in the next 20 years"--is in an analysis of the ensemble output. The methods by which the output of different models (as well as different runs of the same model) are concatenated into a single number are worthy of investigation (as well as, perhaps, criticism), but are beyond the scope of this dissertation. 6.4 You Can't Get Struck By Lightning In a Bottle: Why Trust Simulations? How do we know that we can trust what these models tell us? After all, computational models are (at least at first glance) very different from standard scientific experiments in a number of different ways. Let us close this chapter with a discussion of the reliability of simulation and computational models in general. 6.4.1 Something Old, Something New Oreskes (2000) points out that some critics of computational modeling echo a species of hard-line Popperian verificationism. That is, some critics argue that our skepticism about computational models should be grounded in the fact that, contra more standard models, computational models can't be tested against the world in the right way. They can't be falsified, as by the time evidence proves them inadequate, they'll be rendered irrelevant in any case. The precise than climate prediction in order to be interesting. However, it also need only apply to a timeframe that is many, many order of magnitude shorter than climate predictions. Meteorologists are interested in predicting with relatively high accuracy whether or not it will rain on the day after tomorrow. Climatologists are interested in predicting--with roughly the same degree of accuracy--whether or not average precipitation will have increased in 100 years. The trade-off between immediacy and precision in forecasting the future of chaotic systems is perfectly illustrated in this distinction. 214 kind of parameterization and spin up procedure discussed above can be seen, in this more critical light, as a pernicious practice of curve-fitting: the CGCMs are designed to generate the predictions that they do, as model builders simply adjust them until they give the desired outputs. However, as Oreskes argues, even the basic situation is more complicated than the naive Popperian view implies: in even uncontroversial cases, the relationship between observation and theory is a nuanced (and often idiosyncratic) one. It's often non-trivial to decide whether, in light of some new evidence, we ought to discard or merely refine a given model. Oreskes' discussion cites the problem of the observable parallax for Copernican cosmology and Lord Kelvin's proposed refutation of old-earth gradualism in geology and biology--which was developed in ignorance of radioactivity as a source of heat energy--as leading cases, but we need not reach so far back in history to see the point. The faster-than-light neutrino anomaly of 2011-2012 is a perfect illustration of the difficulty. In 2011, the OPERA lab at CERN in Geneva announced that it had observed a class of subatomic particles called "neutrinos" moving faster than light. If accurate, this observation would have had an enormous impact on what we thought we knew about physics: light's role in defining the upper limit of information transmission is a direct consequence of special relativity, and is a direct consequence of geometric features of spacetime defined by general relativity. However, this experimental result was not taken as evidence falsifying either of those theories: it was greeted with (appropriate) skepticism, and subjected to analysis. In the end, the experimenters found that the result was due to a faulty fiber optic cable, which altered the recorded timings by just enough to give a significantly erroneous result. We might worry even in standard cases, that is, that committed scientists might appropriately 215 take falsifying observations not as evidence that a particular model ought to be abandoned, but just that it ought to be refined. This should be taken not as a criticism of mainstream scientific modeling, but rather as an argument that computational modeling is not (at least in this respect) as distinct from more standardly acceptable cases of scientific modeling DMS might suggest. The legitimacy of CGCMs, from this perspective, stands or falls with the legitimacy of models in the rest of science. Sociological worries about theory-dependence in model design are, while not trivial, at least well-explored in the philosophy of science. There's no sense in holding computational models to a higher standard than other scientific models. Alan Turing's seminar 1950 paper on artificial intelligence made a similar observation when considering popular objections to the notion of thinking machines: it is unreasonable to hold a novel proposal to higher standards than already accepted proposals are held to. We might do better, then, to focus our attention on the respects in which computational models differ from more standard models. Simons and Boschetti (2012) point out that computational models are unusual (in part) in virtue of being irreversible: "Computational models can generally arrive at the same state via many possible sequences of previous states ." 181 Just by knowing the output of a particular computational model, in other words, we can't say for sure what the initial conditions of the model were. This is partially a feature of the predictive horizon discussed in Chapter Five: if model outputs are interpreted in ensemble (and thus seen as "predicting" a range of possible futures), then it's necessarily true that they'll be irreversible--at least in an epistemic sense. That's true in just the same sense that thermodynamic models provide predictions that are "irreversible" to the precise microconditions 181 Simons and Boschetti (2012), p. 810 216 with which they were initialized. However, the worry that Simons and Boschetti raise should be interpreted as going deeper than this. While we generally assume that the world described by CGCMs is deterministic at the scale of interest--one past state of the climate determines one and only one future state of the climate--CGCMs themselves don't seem to work this way. In the dynamics of the models, past states underdetermine future states. We might worry that this indicates that the non-physicality that worried Sen Gupta et. al. runs deeper than flux couplers: there's a fundamental disconnect between the dynamics of computational models and the dynamics of the systems they're purportedly modeling. Should this give comfort to the proponent of DMS? 6.4.3 Tools for Deciding This is a problem only if we interpret computational models in general--and CGCMs in particular--as designed to generate positive and specific predictions about the future of the systems they're modeling. Given what we've seen so far about the place of CGCMs in the broader context of climate science, it may be more reasonable to see them as more than representational approximations of the global climate, or even as simple prediction generating machines. While the purpose of science in general is (as we saw in Chapter One) to generate predictions in how the world will change over time, the contribution of individual models and theories need not be so simple. The sort of skeptical arguments we discussed in Section 6.4.2 can't even get off the ground if we see CGCMs (and similar high-level computational models) not as isolated prediction-generating tools, but rather tools of a different sort: contextually-embedded tools 217 designed to help us figure out what to do. On this view, computational models work as (to modify a turn of phrase from Dennett [2000]) tools for deciding. . Recall the discussions of 182 pragmatic idealization and ensemble modeling earlier in this chapter. I argued that CGCMs are not even intended to either approximately represent the global climate or to produce precise predictions about the future of climate systems. Rather, they're designed to carve out a range of possible paths that the climate might take, given a particular set of constraints and assumptions. We might take this two ways: as either a positive prediction about what the climate will do, or as a negative prediction about what it won't do. This may seem trivial to the point of being tautological, but the two interpretations suggest very different roles for pragmatic idealization generally (and CGCMs in particular) to play in the larger context of climate-relevant socio-political decision making. If we interpret CGCMs as generating information about paths the global climate won't take, we can capitalize on their unique virtues and also avoid skepical criticisms entirely. On this view, one major role for CGCMs' in the context of climate science (and climate science policy) as a whole is to proscribe the field of investigation and focus our attention on proposals worthy of deeper consideration. Knowledge of the avenues we can safely ignore is just as important to our decision making as knowledge of the details of any particular avenue, after all. I should emphasize again that this perspective also explains the tendency, discussed in Chapter Four, of progress in climatology to involve increasing model pluralism rather than convergence on any specific model. I argued there that EMICs are properly seen as specialized 182 This view is not entirely at odds with mainstream contemporary philosophy of science, which has become increasingly comfortable treating models as a species of artifacts. van Fraassen (2009) is perhaps the mainstream flagship of this nascent technological view of models. 218 tools designed to investigate very different phenomena; this argument is an extension of that position to cover CGCMs as well. Rather than seeing CGCMs as the apotheosis of climate modeling--and seeking to improve on them to the exclusion of other models--we should understand them in the context of the broader practice of climatology, and investigate what unique qualities they bring to the table. This is a strong argument in favor of ineliminable pluralism in climatology, as supported by Parker (2006), Lenhard & Winsberg (2010), Rotmans & van Asselt (2001), and many others. I claim that the root of this deep pluralism is the dynamical complexity of the climate system, a feature which necessitates the kind of multifarious exploration that's only possible with the sort of model hierarchy discussed in Chapter Four. Under this scheme, each model is understood as a specialized tool, explicitly designed to investigate the dynamics of a particular system operating under certain constraints. High-level general circulation models are designed to coordinate this focused investigation by concatenating, synthesizing, and constraining the broad spectrum of data collected by those models. Just as in the scientific project as a whole, "fundamentalism" is a mistake: there's room for a spectrum of different mutually-supporting contributions 219 Coda Modeling and Public Policy 1. In 1989, a relatively young software company released their first hit video game, which dealt with the unlikely topic of urban planning. Players of the game-which was called SimCity-took on the role of a semi-omnipotent mayor: sort of a cross between an all-powerful god, a standard city planner, and a kid playing in a sandbox. The player could set tax rates, construct (or demolish) various structures, set up zoning ordinances, and so on, all while trying to keep the city's residents happy (and the budget balanced). Even in its first iteration (the success of the original spawned generations of successor games that continue to be produced today), the simulation was startlingly robust: incorrect tax rates would result in bankruptcy for the city (if they were too low), or stagnation in growth (if they were too high). If you failed to maintain an adequate power grid-both by constructing power plants to generate enough electricity in the first place and by carefully managing the power lines to connect all homes and businesses to the grid-then the city would experience brownouts or blackouts, driving down economic progress (and possibly increasing crime rates, if you didn't also carefully manage the placement and tasking of police forces). Adequate placement (and training) of emergency forces were necessary if your city was to survive the occasional natural disaster-tornados, earthquakes, space-monster attacks , &c.. 183 The game, in short, was a startlingly well thought-out and immersive simulation of city 183 If the player was feeling malicious (or curious), she could spawn these disasters herself and see how well her police and fire departments dealt with a volcanic eruption, a hurricane, Godzilla on a rampage, or all three at the same time. 220 planning and management, though of course it had its limitations. As people played with the game, they discovered that some of those limitations could be exploited by the clever player: putting coal power-plants near the edge of the buildable space, for instance, would cause a significant portion of the pollution to just drift "off the map," with no negative impact on the air quality within the simulation. Some of these issues were fixed in later iterations of the game, but not all were: the game, while a convincing (and highly impressive) model of a real city, was still just that-an imperfect model. However, even imperfect models can be incredibly useful tools for exploring the real world, and SimCity is a shining example of that fact. The outward goal of the game-to construct a thriving city-is really just a disguised exercise in model exploration. Those who excel at the game are those who excel at bringing their mental models of the structure of the game-space into the closest confluence with the actual model the designers encoded into the rules of the game. The programmers behind the Sim-series of games have given a tremendous amount of thought to the nature of their simulations; since the first SimCity, the depth and sophistication of the simulations has continued to grow, necessitating a parallel increase in the sophistication of the mechanics underlying the games. In a 2001 interview, lead designer Will Wright 184 described a number of the design considerations that have gone into constructing the different simulations that have made up the series. His description of how the design team viewed the practice of model building is, for our purposes, perhaps the most interesting aspect of the interview: The types of games we do are simulation based and so there is this really elaborate simulation of some aspect of reality. As a player, a lot of what you're trying to do is reverse engineer the 184 Pearce (2001) 221 simulation. You're trying to solve problems within the system, you're trying to solve traffic in SimCity, or get somebody in The Sims to get married or whatever. The more accurately you can model that simulation in your head, the better your strategies are going to be going forward. So what we're trying to as designers is build up these mental models in the player. The computer is just an incremental step, an intermediate model to the model in the player's head. The player has to be able to bootstrap themselves into understanding that model. You've got this elaborate system with thousands of variables, and you can't just dump it on the user or else they're totally lost. So we usually try to think in terms of, what's a simpler metaphor that somebody can approach this with? This way of looking at models-as metaphors that help us understand and manipulate the behavior of an otherwise intractably complicated system-might be thought of as a technological approach to models. On this view, models are a class of cognitive tools: constructions that work as (to borrow a turn of phrase from Daniel Dennett) tools for thinking . This is not entirely at 185 odds with mainstream contemporary philosophy of science either; van Fraassen, at least, seems to think about model building as an exercise in construction of a particular class of artifacts (where 'artifact' can be construed very broadly) that can be manipulated to help us understand and predict the behavior of some other system . Some models are straightforwardly artifacts 186 (consider a model airplane that might be placed in a wind tunnel to explore the aerodynamic properties of a particular design before enough money is committed to build a full-scale prototype), while others are mathematical constructions that are supposed to capture some interesting behavior of the system in question (consider the logistic equation as a model of population growth). The important point for us is that the purpose of model-building is to create something that can be more easily manipulated and studied than the system of interest itself, with the hope that in seeing how the model behaves, we can learn something interesting about the system the model is supposed to represent. 185 Dennett (2000) 186 See, e.g., Van fraasen (2009) 222 All of this is rather straightforward and uncontroversial (I hope), and noting that simulations like SimCity might work as effective models for actual cities is not terribly interesting-after all, this is precisely the purpose of simulations in general, and observing that the programmers at Maxis have created an effective simulation of the behavior of a real city is just to say that they've done their job well. Far more interesting, though, is a point that Wright makes later in the interview, comparing the considerations that go into the construction of models for simulation games like SimCity and more adversarial strategy games. In particular, Wright likens SimCity to the ancient board game Go, arguing that both are 187 examples of games that consist in externalizing mental models via the rules of the game. In contrast to SimCity, however, Go is a zero-sum game played between two intelligent opponents, a fact that makes it more interesting in some respects. Wright suggests that Go is best understood as a kind of exercise in competitive model construction: the two players have different internal representations of the state of the game, which slowly come into alignment 188 with each other as the game proceeds. Indeed, except at the very highest level of tournament play, games of Go are rarely formally scored: the game is simply over when both players recognize and agree that one side is victorious. It's not unusual for novice players to be beaten 187 Go is played on a grid, similar to a chess board (though of varying size). One player has a supply of white stones, while the other has a supply of black stones. Players take turns placing stones at the vertices of the grid (rather than in the squares themselves, as in chess or checkers), with the aim of capturing more of the board by surrounding areas with stones. If any collections of stones is entirely surrounded by stones of the opposite color, the opponent "captures" the stones on the inside, turning them into the stones of her color. Despite these simple rules (and in contrast to chess, with its complicated rules and differentiated pieces), incredibly complex patterns emerge in games of Go. While the best human chess players can no longer defeat the best chess computers, the best human Go players still defeat their digital opponents by a significant margin. 188 It's important to note that this is not the same as the players having different models of the board. Go (like chess) is a game in which all information is accessible to both players. Players have different functional maps of the board, and their models differ with regard to those functional differences-they might differ with respect to which areas are vulnerable, which formations are stable, which section of an opponent's territory might still be taken back, and so on. 223 by a wide margin without recognizing the game is over-a true beginner's mental model of the state of play might be so far off that he might not understand his defeat until his more skilled opponent shows him the more accurate model that she is using. A large part of becoming proficient at playing Go consists in learning how to manipulate the relevant mental models of the board, and learning how to manipulate the pieces on the board such that your opponent is forced to accept your model. Of course, disagreement about model construction and use has consequences that range far beyond the outcome of strategy games. In the late 1990s, the designers behind the Sim series created a project for the Markle Foundation called "SimHealth." SimHealth worked much like SimCity, but rather than simulation the operation of a city, it simulated the operation of the national healthcare system-hospitals, doctors, nurses, ambulances, &c. Even more interestingly, it exposed the assumptions of the model, and opened those up to tinkering: rather than working with a single fixed model and tinkering with initial/later conditions (as in SimCity), SimHealth's "players" could also change the parameters of the model itself, experimenting with how the simulation's behavior would change if (for example) hospitals could be efficiently run only a dozen doctors, or if normal citizens only visited the emergency room for life-threatening problems. Wright argued that tools of this type made the process of health care policy debate explicit in a way that simple disagreement did not-that is, it exposed the fact that the real nature of the disagreement was one about models. WW: When people disagree over what policy we should be following, the disagreement flows out of a disagreement about their model of the world. The idea was that if people could come to a shared understanding or at least agree toward the model of the world, then they would be much more in agreement about the policy we should take. CP: So in a way, a system like that could be used to externalize mental models and create a 224 collective model....you have an externalized model that everyone agrees to abide by. WW: Yeah, which is exactly the way science works . 189 There's a fantastically deep point here: one that (it seems to me) has been underemphasized by both philosophers of science and political philosophers: to a very great extent, policy disagreement is model disagreement. When we disagree about how to solve some social problem (or even when we disagree about what counts as a social problem to be solved), our disagreement is-at least in large part-a disagreement about what model to apply to some aspect of the world, how to parameterize that model, and how to use it to guide our interventions . Nowhere is this clearer than when public policy purports to be guided by scientific results. 190 Taking the particular values that we do have as given, a sound public policy that aims to make 191 the world a certain way (e.g. to reduce the heavy metal content of a city's drinking water) is best informed by careful scientific study of the world-that is, it is best informed by the creation and examination of a good model of the relevant aspects of the world. One consequence of this is that some of the difficulties of designing good public policy-a practice that we can think of, in this context, as a kind of social engineering-are inherited from difficulties in model building. In our deliberations about which laws to enact, or which policies to reform, we may need to appeal to scientific models to provide some relevant data, either about the way the world is now, or about how it will be after a proposed intervention is enacted. We 189 Ibid., emphasis mine 190 This is not to suggest that policy can be straightforwardly "read off" of scientific models. Understanding relevant science, however, is surely a necessary condition (if not a sufficient one) for crafting relevant public policy. See Kitcher (2011) for a more detailed discussion of this point. For now, we shall simply take it as a given that understanding scientific models play a role (if not the only role) in deciding public policy. 191 I want to avoid becoming mired in debates about the fact/value distinction and related issues. None of what follows rests on any particular theory of value, and the reader is encouraged to substitute his favored theory. Once we've identified what we in fact ought to do (whether by some utilitarian calculus, contemplation of the virtues, application of Kant's maxim, an appeal to evolution, or whatever), then we still have the non-trivial task of figuring out how to do it. Public policy is concerned with at least some of the actual doing. 225 may need to rely on models to allow us to explore the consequences of some proposed intervention before we try out a new policy in socio vivo; that was the intended application of SimHealth, but the model in question need not be so explicit as a computer simulation. If we disagree about which model to use, what the model implies, or how to tune the model parameters, then it may be difficult (or even impossible) to come to a policy agreement. In many cases, the lack of scientific consensus on a single model to be used (or at least on relatively small family of models to be used) when working with a particular system is a sign that more work needs to be done: we may not agree, for instance, about whether or not the Standard Model of particle physics is the one we ought to work with in perpetuity, but this disagreement is widely appreciated to be an artifact of some epistemic shortcoming on our part. As we learn more about the world around us, the scientific community will converge on a single model for the behavior of sub-atomic systems. However, this is not always the case. Suppose we have a pressing public policy decision to make, and that the decision needs to be informed by the best science of the day. Suppose further that we have good reason to think that the sort of singular consensus trajectory that (say) sub-atomic particle models seem to be on is unlikely to appear in this case. Suppose, that is, that we're facing a policy decision that must be informed by science, but that the science seems to be generating a plethora of indispensable (but distinct) models rather than converging on a single one. If we have good reason to think that this trend is one that is unlikely to disappear with time-or, even more strongly, that it is a trend that is an ineliminable part of the science in question-then we will be forced to confront the problem of how to reform the relationship between science and policy in light of this new kind of science. Wright's pronouncement that 226 model convergence is "just how science works" might need to be reexamined, and we ignore that possibility at our peril. As we shall see, policies designed to deal with complex systems buck this trend of convergence on a single model, and thus require a novel approach to policy decision-making. If there is any consensus at all in climate science, it is this: the window for possibly efficacious human intervention is rapidly shrinking, and if we don't make significant (and effective) policy changes within the next few years, anthropogenic influence on the climate system will take us into uncharted waters, where the best case scenario-complete uncertainty about what might happen-is still rather unsettling. Critics of contemporary climate science argue that the uncertainty endemic to our "best" current models suggests that we should adopt a wait-and-see approach-even if the climate is warming, some argue that the fact that our 192 current models are scattered, multifarious, and imperfect mandates further work before we decide on how (or if) we should respond. This position, I think, reflects a mistaken assumption about the trajectory of climate science. The most important practical lesson to be drawn here is this: if we wait for climate scientists to agree on a single model before we try to agree on policy, we are likely to be waiting forever. Climate scientists seem interested in diversifying, not narrowing, the field of available models, and complexity-theoretic considerations show that this approach is conceptually on firm ground. 192 Again, Isdso & Singer (2009) is perhaps a paradigm case here, given the repeated criticism of climate modeling on the grounds that no single model captures all relevant factors. This argument has also been repeated by many free-market-leaning economists. Dr. David Friedman (personal communication), for instance, argued that "even if we were confident that the net effect was more likely to be negative than positive, it doesn't follow that we should act now. It's true that some actions become more difficult the longer we wait. But it's also true that, the longer we wait, the more relevant information we have." Reading this charitably (such that it isn't trivially true), it suggests a tacit belief that climate science will (given enough time) converge on not just more particular information, but a better model, and that the gains in predictive utility in that model will make up for losses in not acting now. 227 Our policy-expectations must shift appropriately. This is not to suggest that we should uncouple our policy decisions from our best current models-quite the opposite. I believe that the central point that Will Wright makes in the quotation from his discussion of SimCity and SimHealth is still sound: disagreement about policy represents disagreement about models. However, the nature of the disagreement here is different from that of the past: in the case of climate science, we have disagreement not about which model to settle on, but about how to sensibly integrate the plurality of models we have. The disagreement, that is, revolves around how to translate a plurality of models into a unified public policy. My suggestion is: don't. Let the lessons learned in attempts to model the climate guide our attempts to shape our influence on the climate. Rather than seeking a single, unified, top-down public policy approach (e.g. the imposition of a carbon tax at one rate or another), our policy interventions should be as diverse and multi-level as our models. Those on both sides of the climate policy debate sometimes present the situation as if it is a choice between mitigation-trying to prevent future damage-and adaptation-accepting that damage is done, and changing the structure of human civilization to respond. It seems to me that the lesson to be drawn here is that all these questions (which strategy is best? Should we mitigate or adapt?) are as misguided as the question "which climate model is best?" We should, rather, take our cue from the practice of climate scientists themselves, encouraging innovation generally across many different levels of spatio-temporal resolution. By way of a single concrete example, consider the general emphasis (at least at the political level) on funding for alternative energy production (e.g. solar, hydrogen fuel cells). It is easy to see why this is a relevant (and important) road to explore-even if the possible threat of climate 228 change turns out to (so to speak) blow over, fossil fuels will not last forever. However, engineering viable replacements to fossil fuel energy is an expensive, long-term investment. While important, we should not allow ourselves to focus on it single-mindedly-just as important are more short-term interventions which, though possibly less dramatic, have the potential to contribute to an effective multi-level response to a possible threat. For instance, directing resources toward increases in efficiency of current energy expenditure might be more effective (at least in the short run) at making an impact. Innovations here can, like EMICs, take the form of highly specialized changes: the current work on piezoelectric pedestrian walkways (which harvest some of the kinetic energy of human foot impacting sidewalk or hallway and store it as electrical energy) is an excellent example . Unfortunately, research programs like 193 this are relatively confined to the sidelines of research, with the vast majority of public attention (and funding) going to things like alternative energy and the possibilities of carbon taxes. A more appropriate response requires us to first accept the permanent pluralism of climate science models, and to then search for a similarly pluralistic set of policy interventions. 2. There's one last point I'd like to make connecting complexity modeling and public policy. In a way, it is the simplest point of the whole dissertation, and it has been lurking in the background of all of the preceding 200-some-odd pages. Indeed, it was perhaps best phrased way back in the first chapter: the world is messy, and science is hard. We've examined a number of senses in which that sentence is true, but there's one sense in particular that's been illuminated in the course of our discussion here. I want to close with a brief discussion of that sense. 193 See, for example, Yi et. al. (2012) 229 The advent of what the loosely related family of concepts, methods, theories, and tools that I've been referring to collectively as "complexity science" or "complexity theory" has changed the face of scientific practice in ways that are only beginning to be appreciated. Just as when quantum theory and relativity overthrew the absolute rule of classical physics in the first part of the 20th century, much of what we previously took ourselves to know about the world (and our place in it) is now being shown to be if not exactly wrong then at least tremendously impoverished. The view that I've associated variously with traditions in reductionism, eliminativism, and mechanism--the view that the world consists in nothing over and above, as Hume put it, "one little thing after another"--is proving increasingly difficult to hold onto in the face of contrary evidence. Novel work in a variety of fields--everything from ecology to network science to immunology to economics to cognitive science--is showing us that many natural systems exhibit behavior that is (to put it charitably) difficult to explain if we focus exclusively on the behavior of constituent parts and ignore more high-level features. We're learning to think scientifically about topics that, until recently, were usually the province of metaphysicians alone, and we're learning to integrate those insights into our model building. While this complexity revolution has changed (and will continue to change) the practice of scientific model building, it must also change the way we talk about science in public, and the way we teach science in schools. The future impact of complexity must be neither confined to esoteric discussions in the philosophy of science, nor even to changes in how we build or scientific models. Rather, it must make an impact on how the general public thinks about the world around them and their place in that world. Moreover, it must make an impact on how the general public evaluates scientific progress, and what they expect out of their scientific theories. 230 I've emphasized a number of times here that many of the criticisms of climate science are, to some extent, founded on a failure to appreciate the unique challenges of modeling such a complex system. The scientists at work building working climate models, of course, by and large appreciate these challenges. The public, however, very clearly does not. The widespread failure to accept the urgency and immediacy of the call to act to avert a climate change disaster is one symptom of this failure to understand. This is not just a matter of clear presentation of the data, or of educating people about what climate models say--though these are certainly very important things. Instead, the disconnect between the scientific consensus and the public opinion about the reliability and effectiveness of climate models is a symptom of science education and science journalism that has been left behind by scientific progress. The demands for more data, better models, further research, a stronger consensus, and so on would be perfectly sensible if we were dealing with predictions about a less complex system. Science is presented to the public--both in primary/secondary education and in most popular journalistic accounts--as aiming at certainty, analytic understanding, and tidy long term predictions: precisely the things that complexity theory often tells us we simply cannot have. Is it any wonder, then, that the general public fails to effectively evaluate the reliability of climate predictions and models? Climatology (like economics, another widely mistrusted complex systems science) does great violence to the public perception of what good science looks like. The predictions and methods of science bear little resemblance to the popular paradigm cases of science: Issac Newton modeling the fall of an apple with a neat set of equations, or Jonas Salk working carefully in a forest of flasks and beakers to isolate a vaccine for polio. 231 If we're to succeed in shifting the public opinion of climate science--and if we're to avoid engaging in a precisely analogous public fight over the reliability of the next complex system science breakthrough--then we need to communicate the basics of complexity-based reasoning, and we need to help the public understand that science is a living enterprise. We need to communicate to the average citizen the truth of the maxim from Chapter One: the world is messy and science is hard. 12/07/2010 8/05/2014 232 Works Cited Aharanov, Y., & Bohm, D. (1959). Significance of electromagnetic potentials in quantum theory. Physial Review , 485-491. Aizawa, K., & Adams, F. (2008). The Bounds of Cognition. 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Realism, Science, and Pragmatism Edited by Kenneth R. Westphal First published 2014 by Routledge 711 Third Avenue, New York, NY 10017 and by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN Routledge is an imprint of the Taylor & Francis Group, an informa business © 2014 Taylor & Francis The right of the editor to be identified as the author of the editorial material, and of the authors for their individual chapters, has been asserted in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Realism, science, and pragmatism / edited by Kenneth R. Westphal. pages cm. - (Routledge studies in contemporary philosophy ; 58) Includes bibliographical references and index. 1. Realism. 2. Science-Philosophy. 3. Pragmatism. I. Westphal, Kenneth R., editor of compilation. B835.R3265 2014 149′.2-dc23 2013041610 ISBN: 978-1-138-01882-2 (hbk) ISBN: 978-1-315-77951-5 (ebk) Typeset in Sabon by Apex CoVantage, LLC 6244-263-0FM.indd 6 1/25/2014 1:18:29 PM Realism, Science, and Pragmatism "This is a first-rate collection of essays on the general issue of realism, on the relation of realism to contemporary philosophy of science and epistemology, and on the challenge that has been made to traditional realism by classical pragmatism and neo-pragmatism. The contributors are among the leading scholars in the field, and their essays advance the debates in ways that will provoke response and further inquiry. Anyone interested in the topic of realism, its history and current controversies, will benefit from paying the close attention that these essays deserve." -John Ryder, American University of Ras al Khaimah, United Arab Emirates This collection of original essays aims to reinvigorate the debate surrounding philosophical realism in relation to philosophy of science, pragmatism, epistemology, and theory of perception. Questions concerning realism are as current and as ancient as philosophy itself; this volume explores relations between different positions designated as 'realism' by examining specific cases in point, drawn from a broad range of systematic problems and historical views, from ancient Greek philosophy through the present. The first section examines the context of the project; contributions systematically engage the historical background of philosophical realism, re-examining key works of Aristotle, Descartes, Quine, and others. The following two sections epitomize the central tension within current debates: scientific realism and pragmatism. These contributions address contemporary questions of scientific realism and the reality of the objects of science, and consider whether, how or the extent to which realism and pragmatism are compatible. With an editorial introduction by Kenneth R. Westphal, these fourteen original essays provide wide-ranging, salient insights into the status of realism today. Kenneth R. Westphal is Professorial Fellow in the School of Philosophy at the University of East Anglia, UK 6244-263-0FM.indd 1 1/24/2014 7:28:48 PM From advance reviews of Realism, Science, and Pragmatism The project is an important one – the reconciliation of realism and pragmatism .... The book is attractive for ... the intellectual strength of the contributions [and its] ... organiziation [which] does justice to the wide range of related issues. The idea of considering the topic of realism from various points of view is original. [It] ... is not easy to find ... a volume ... which ... combines the various discussions. This ... is a strong set of original essays on a number of significant aspects of the ongoing debate concerning philosophical realism. [Most] ... contributors are prominent figures [who] ... bring ... considerable experience and knowledge to bear in these essays to good effect. The editor [has] ... done a first-rate job of selecting and organizing the essays. Most of the authors are from Scandinavian universities, and ... build on the strong philosophical tradition that has built up around Helsinki. The first section provides a general and high altitude overview of the topic, and then several essays that review highlights of the historical background. The rest of the volume is cleverly divided into two ... themes that more than any other capture the tension in the current debates: scientific realism and pragmatism, thus ... assuring ... a comprehensive study of the question in its current incarnations. ... the book ... has the strength of being pluralist in the breadth of its essays. ... These are interesting and provocative essays, and they should generate further discussion and debate. ... Any reader who goes through these essays carefully will have a good command of the topic generally and of the cutting edge discussions and debates. One cannot ask for more ... from a single volume. ... I [anticipate] ... the book being discussed in any Ph.D. program in which contemporary analytic and/or pragmatist philosophy is being studied. The book would be a valuable source in many kinds of courses in philosophy. ... Across Europe, especially in Scandinavia and countries in Central Europe, in Turkey, to some extent in Russia, and well beyond, there is growing interest in philosophy in these styles and dealing with these issues. It will be wise to market the book as much as possible around the world. Introduction 1 KENNETH R. WESTPHAL PART I Realism Contextualized 1 What Is Real(ism)? 13 JAAKKO HINTIKKA 2 Aristotle's Direct Realism and Some Later Developments 21 MIKA PERÄLÄ 3 Late Mediaeval Realisms: Key Arguments Supporting Non-semantic Universality 46 LAURENT CESALLI 4 Descartes on the Formal Reality, Objective Reality, and Material Falsity of Ideas: Realism through Constructivism? 67 DERMOT MORAN 5 Quine's Conception of Objects: Beyond Realism and Anti-realism 93 ANTTI KESKINEN 6 Did Sherlock Holmes Inhale Pipe Smoke through a Hole in His Forehead? 115 PETER SWIRSKI PART II Scientific Realism 7 Realism: Metaphysical, Scientific, and Semantic 139 PANU RAATIKAINEN Contents 6244-263-0FM.indd 7 1/24/2014 7:28:48 PM viii Contents 8 Scientific Realism: Independence, Causation, and Abduction 159 ILKKA NIINILUOTO 9 Cognitive Semantics and Newton's Rule 4 of Experimental Philosophy: Scientific Realism without Empiricism 173 KENNETH R. WESTPHAL 10 Naturalism without Metaphysics 200 JONATHAN KNOWLES PART III Pragmatism and Realism 11 Majesty of Truth and the Moral Sentiment: Emerson's and Peirce's Ethico-Ontological Realism 221 HEIKKI A. KOVALAINEN AND DOUGLAS R. ANDERSON 12 Concepts and the Real in C. I. Lewis' Epistemology 243 LAURI JÄRVILEHTO 13 Pragmatic Realism 251 SAMI PIHLSTRÖM 14 McDowell's Pragmatist Anti-anti-realism 283 EIRIK JULIUS RISBERG Contributors 303 Index 309 6244-263-0FM.indd 8 1/24/2014 7:28:48 PM Issues about realism are as current and as ancient as philosophy itself. Plato in the Theatetus comments on the long-standing battle between philosophical giants and gods about whether, in addition to the physical objects and events we perceive, there are also non-physical, and hence non-perceptible forms or ideas of kinds or characteristics, variously instantiated in physical particulars, but which exist independently both of their instances and of what we may happen to say, think, believe, or know about them. In philosophical usage, the term 'realism' is both basic and polysemic. For example, one can hold realism- in contrast to idealism, irrealism, or agnosticism-if one holds that material objects exist and have various characteristics regardless of what we may say, think, believe, or know about them. One can be a direct realist in the theory of perception by holding that perception is direct awareness of external objects, a moral realist if one believes that there are objective moral values, a scientific realist if one holds that scientific knowledge is about theory-independent phenomena and that such knowledge is possible even about unobservable entities, or a modal realist if one believes that possible worlds are as real as the actual world. In ontology, realism indicates that one grants-in ways which vary from case to case-extra-mental existence to certain kinds of entities, processes, or structures and at least some of their features, such as physical objects, universals, relations, structures, or propositions. Realism about particular objects and about their features or relations became problematic in Twentieth Century philosophy when it became generally recognized that we cannot, as it were, set aside our concepts, theories, beliefs, or, in general, our language to inspect the facts themselves and on that basis assess our beliefs, statements, or theories about them. Realism has remained fraught since. The fourteen original essays presented here explore the relations that different positions designated as 'realism' may have to each other by examining specific cases in point, drawn from a broad range of systematic problems and historical views from Ancient Greek philosophy up to the present day. Individually and taken together, these essays show how much can be gained by examining issues about realism both systematically and historically. The essays form three groups, Part I: Realism Contextualized, Part II: Scientific Realism, and Part III: Pragmatism and Realism. Introduction Kenneth R. Westphal 6244-263-00Intro.indd 1 1/24/2014 7:29:24 PM 2 Kenneth R. Westphal Part I contextualizes issues about realism regarding both physical particulars and universals by critically re-examining several major historical and systematic positions on these topics. A hallmark of pragmatism is that both the understanding and the assessment of current views, and the development of improved views, benefit, often centrally, by re-assessing prior views on the same or related issues, whether these prior views be familiar, neglected, under appreciated, or misunderstood. Accordingly Part I attempts neither a historical nor a systematic review of issues about realism; excellent surveys are available elsewhere. Instead, its six chapters re-investigate key historical and systematic issues where new and unexpected insights can be discovered-and discoveries there are in these chapters, which plumb philosophical depths in Ancient, Mediaeval, Modern, and Contemporary philosophy. The issues are launched, officially yet non-technically, by Jaakko Hintikka in "What Is Real(ism)?" (Chapter 1). Hintikka articulates what is involved in claiming realism about any domain or issue or particular(s), in part by arguing that the use of possible world semantics requires a richer domain of discourse than is provided by any such semantics tailored to any one domain and its attendant possible-worlds model. To use such a logic we must be able to identify individuals across such domains and models, in order to identify any actual individuals within possible worlds, and to identify their merely possible, non-actual existence in some possible worlds, including that possible world which is our actual world. Accordingly, "actuality and existence do not go together," and epistemic logic requires a richer domain of discourse than that of Frege-Russell first-order quantification logic and the possible-worlds semantics built upon it. We need not only the 'is' of identity, the 'is' of predication, and the 'is' of instantiation, but also the 'is' of identifiability, as was implicitly recognized in Aristotle's logic and metaphysics. Epistemic logic requires possible objects as well as actual ones, and epistemic logic is required to use first-order quantification theory and possible-worlds semantics in any actual domain of inquiry. In "Aristotle's Direct Realism and Some Later Developments" (Chapter 2), Mika Perälä elucidates Aristotle's direct theory of perception within Aristotle's general explanatory project. Aristotle's formal cause is specified in terms of the relevant efficient cause, the activity of which occurs in the activity of the patient. This model implies that, in perception, seeing, e.g., a white object is the type of perception it is because it is caused by a white object, where the object's white color exercises its power of being perceived in the sense of vision as someone's seeing that white of that object. That object's white color is the term to which the activity of the perceiver's sense, as the patient of the object's color's activity, is relative. This analysis enables Aristotle to solve problems in the Megarean and Protagorean anti-realist accounts of perception. Aristotle's view thus contrasts in important regards to those of, e.g., Aquinas and Scotus. Perälä examines two important contrasts. First, he argues that Aristotle, unlike Aquinas-and many of his successors even today-did not resort to the concepts of intentional being and 6244-263-00Intro.indd 2 1/24/2014 7:29:24 PM Introduction 3 likeness (and their cognates such as representation) to explain why a perception is about its proper object. Second, he points out that Aristotle, unlike Scotus, did not allow that a mental act could be directed at its object even when the object is not the efficient cause of that act (for example, when God or Devil induces such an act in us). Accordingly, Aristotle provides a cogent direct realist theory of perception. Laurent Cesalli begins his examination of "Key Arguments Supporting Non-semantic Universality" in "Late Mediaeval Realisms" (to invert the sub-title and title of Chapter 3) by noting the highly favorable reception of Scotus's realism about universals by both C. S. Peirce and David Armstrong. Cesalli then critically examines key arguments for the real existence of universals developed by Aquinas, Scotus, Burley, Ockham, Buridan, Richard Brinkley, Nicolas of Autrécourt, Francesco da Prato, John Wyclif, and Dietrich of Freiberg, who argue that (i) semantic universality depends upon the existence of metaphysical universals, or that (ii) scientific knowledge requires metaphysical universals if it is not to be reduced to psychology or linguistics, or that (iii) essences of things exist objectively, no less than do their matter and forms. They analyze relations between universals and particulars in several ways: in terms of a merely formal distinction, mereology, concomitance, partial identity, real identity, or platonic-atomism. Cesalli further elucidates their views by critical comparison with modern forms of realism developed by Bergmann, Armstrong, and Cocchiarella. In "Descartes on the Formal Reality, Objective Reality, and Material Falsity of Ideas: Realism through Constructivism?" (Chapter 4), Dermot Moran revisits Descartes' account of the formal and objective reality of ideas in order to ascertain more exactly Descartes' commitment to realism. For Descartes, as for the Scotist tradition in general, 'real' means something that can be a 'res': to be real is to be possible. The reality of some possible thing is expressed by its essence which is reflected in its 'objective reality'. Objective reality comes in degrees. Some 'real' entities also have actuality as a result of being caused. This is their 'formal reality'. For Descartes certain ideas (e.g., of God) have an objective reality so great that it can be accounted for only by those ideas also having formal reality. Other ideas have objective reality but fail to have a formal cause and may even mislead in presenting the kind of objective reality they possess. These are 'materially false' ideas. The received view is that Descartes' employment of these Scholastic notions is confused and that Antoine Arnauld in his Objections makes a number of valid criticisms of Descartes' account. Moran contends that Descartes' distinctions offer powerful insights into the intentionality of the mind and the manner in which the phenomenological character of our experiences can (or cannot) reliably lead us to grasp the nature of reality in itself. Descartes has a complex conception of intentional content that deserves more attention and credit than it has hitherto received. In "Quine's Conception of Objects: Beyond Realism and Anti-realism" (Chapter 5), Antti Keskinen argues that the apparent tension between 6244-263-00Intro.indd 3 1/24/2014 7:29:24 PM 4 Kenneth R. Westphal Quine's scientific realism and his epistemological conception of objects as theoretical posits is not resolved by appeal to Quine's naturalism, but instead by the genuinely reciprocal containment between science and epistemology. Quine's scientific realism and his epistemological conception of objects as posits are consistent because the notion of reality is itself always part of a theory; otherwise it is meaningless. The appearance of a tension between those two aspects of Quine's view arises only if it is assumed that objects can be real in some sense other than as posits of a theory included within our best current science. Quine disallows any metaphysical realism that has primacy over epistemology; hence his scientific realism is consistent with his epistemological view of objects as theory-dependent posits. Yet this conception of objects does not entail that the objects talked about in our best current science are less than real, in any admissible, theory-external sense of "real." However, Quine's view further implies that, although de re attitude ascriptions have sense, they are rarely true because the conditions necessary for their truthful ascription are rarely satisfied. Though no reductio, this implication of Quine's view is highly counter-intuitive. The answer to Peter Swirski's titular question is no mystery: "Did Sherlock Holmes Inhale Pipe Smoke through a Hole in His Forehead?" (Chapter 6). The question is how we know that Holmes did no such thing, despite his being fictional, and why such knowledge matters. Swirski argues that such knowledge is not based simply upon the text, nor upon possible worlds implicated by the text or the author, not even when guided by a Reality Principle, a Mutual (Shared) Belief Principle, or by the fictional persona of the narrator. None of these proposals properly specify the relevant background beliefs or information required to understand literary texts. Recognizing what is relevant, Swirski argues, requires the reader's reflexive recognition of the real author's successfully executed reflexive intentions relevant to a given fiction, facilitated in part by our recognition of an author's use (or abuse) of genre conventions, and by our quintessentially human, natural capacities to understand one another's intentions and acts of directing joint attention. Our understanding of real intentions is required for comprehending one another, and for comprehending fictional truths. Our reflections on realism began with quantification logic, according to which to be is to be the value of a bound variable, to argue that our understanding of possibility and of possible existence is required to use possible world models of any domain to understand actual features of actual objects or events, thus renewing our appreciation of Aristotle's insights into the 'is' of identification. The systematic and historical trajectory thus launched carries through the essays of Part I to conclude that our understanding of fictional truths requires our understanding of actual intentions and acts of joint attention, without which we could not communicate, and indeed could not be human. Part I thus provides the systematic and historical context of issues about the reality of objects, events, their characteristics, and their relations, within which Parts II and III examine these issues in greater detail. 6244-263-00Intro.indd 4 1/24/2014 7:29:24 PM Introduction 5 Part II considers a specific domain of these issues: scientific realism; Part III considers a distinctive approach to these issues: pragmatism. Part II opens with Panu Raatikainen's "Realism: Metaphysical, Scientific, and Semantic" (Chapter 7). Raatikainen distinguishes and interrelates three influential forms of realism: realism about the external world, construed as a metaphysical doctrine; scientific realism about non-observable entities postulated in science; and semantic realism as defined by Dummett. He first contrasts metaphysical realism about everyday physical objects with idealism and phenomenalism, reviews several potent arguments against these latter views, and argues briefly by induction in support of realism about physical objects. Scientific realism-the idea that natural sciences discover and explain genuine features of natural phenomena-may be commonsense, and may be regarded as the paragon of empirical knowledge, though it has been widely out of philosophical favor, not only since Kuhn's Structure of Scientific Revolutions (1962), but throughout the history of empiricism, from Hume to the Logical Positivists, Logical Empiricists and today's Constructive Empiricists. Raatikainen distinguishes three forms of scientific realism: (i) scientific theories and their existence postulates should be taken literally; (ii) the existence of unobservable entities posited by our most successful scientific theories is justified scientifically; and (iii) our best current scientific theories are at least approximately true. Raatikainen argues that only some form of scientific realism can make proper sense of certain episodes in the history of science. He then considers Dummett's influential formulation of semantic issues about realism. Dummett argued that in some cases, the fundamental issue is not about the existence of entities, but rather about whether statements of some specified class (such as mathematics) have an objective truth value, independently of our means of knowing it. Dummett famously argued against such semantic realism and in favor of anti-realism. Raatikainen examines the relation of semantic realism to the metaphysical construal of realism, presents Dummett's main argument against semantic realism, and focuses on Dummett's key premise, that understanding the meaning of a declarative sentence involves knowing the conditions which would make that sentence true. Raatikainen argues against that key premise by appeal to semantic externalism. Ilkka Niiniluoto's "Scientific Realism: Independence, Causation, and Abduction" (Chapter 8), examines three related criteria of realism within the sciences: mind-independence, causal power, and knowledge by abductive reasoning, by considering what a scientific realist should say about the reality of the past. He argues that realism about the past effectively rules out many anti-realist philosophical positions, such as subjective idealism, phenomenalism, solipsism, positivism, internal realism, social constructivism, and nonor anti-realist varieties of pragmatism. His analysis takes up the theme, announced by Hintikka's opening chapter, of object identification, in connection with Peirce's, Putnam's and with Pihlström's versions of pragmatism. In "Cognitive Semantics and Newton's Rule 4 of Experimental Philosophy: Scientific Realism without Empiricism" (Chapter 9), Kenneth 6244-263-00Intro.indd 5 1/24/2014 7:29:24 PM 6 Kenneth R. Westphal Westphal argues that Evans' analysis in "Identity and Predication" (1975) provides the basis for a powerful semantics of singular, specifically cognitive reference which directly and strongly supports Newton's Rule 4 of (experimental) Philosophy in ways which support Newton's realism about gravitational force. He first examines Newton's Rule 4 and its role in Newton's justification of realism about gravitational force and then summarizes Evans' account of predication and examines its implications for the semantics of singular cognitive reference. Westphal argues that this semantics of singular cognitive reference is embedded in and strongly supports Newton's Rule 4, and that it rules out Cartesian, infallibilist presumptions about empirical justification generally. He then argues that this semantics of singular cognitive reference reveals a key defect in Bas van Fraassen's main argument for his anti-realist "Constructive Empiricism," and also in many common objections to realism, both commonsense and scientific. More generally, Westphal argues, "realism" has appeared problematic to the extent that, in their focus upon the semantics of conceptual content or linguistic meaning, philosophers have neglected the further requirements for specifically cognitive reference. In "Naturalism without Metaphysics" (Chapter 10), Jonathan Knowles argues-against wide-spread consensus to the contrary-that scientific naturalism, the thesis that natural science is our unique source of fundamental knowledge and explanation, does not require metaphysical realism, so that a scientific naturalist can reject metaphysics. Drawing on the work of Huw Price, Knowles argues against a naturalistic form of metaphysical realism that builds on a substantive notion of reference, and also argues (contra Devitt and Searle) that one cannot have a substantive realistic position without such a notion. Further, science itself does not militate for a naturalistic metaphysical realism. The correct alternative to realism, Knowles argues, is not Huw Price's "subject naturalism," which purports to explain scientifically the pluralism exhibited by language, including scientific language, by a global "expressivist" theory of content. Knowles argues that Price's approach leads to another, equally problematic kind of metaphysics, and its semantics lacks the scientific credentials claimed for it. The proper middle ground, Knowles contends, recognizes that semantic minimalism need not reduce truth to justification nor to warranted assertability, and that the systematic search for truth, which grapples with what is yet unknown, remains the prerogative of the sciences. These are the keys to a scientific naturalism without metaphysics. Pragmatist and neo-pragmatist themes are sounded repeatedly in Parts II and III; they are examined in detail in Part III: Pragmatism and Realism. In "Majesty of Truth and the Moral Sentiment: Emerson's and Peirce's Ethico-Ontological Realism" (Chapter 11), Heikki Kovalainen and Douglas Anderson argue that, although they are often logically independent doctrines, realism about physical objects and realism about universals intersect in the religiously influenced interpretation and reception of Plato from the 6244-263-00Intro.indd 6 1/24/2014 7:29:24 PM Introduction 7 early church Fathers up to Romantics such as Samuel Taylor Coleridge, according to whom Christianity embodies universal truths, nonhuman in origin yet knowable to human reason by intellectual intuition-a faculty Kant famously denied. This Platonic, religiously inclined ontological realism enters American philosophy via Ralph Waldo Emerson. Emerson's notion of the moral sentiment is an intellectual faculty of intuiting universal truths-à la Coleridge-and a non-human real force operating in reality- anticipating Peirce. Kovalainen and Anderson argue that both Emerson and Peirce advocate ethico-ontological realism. Understanding Peirce's theory of inquiry and his ontology requires recognizing their moral and theological aspects. More generally, they contend, understanding the issues of realism about particular objects, their features, and their relations requires grappling with the moral and theological dimensions of these issues. In "Concepts and the Real in C. I. Lewis' Epistemology" (Chapter 12), Lauri Järvilehto argues that Lewis developed an aspectual realism that avoids relativism. According to Lewis, concepts guide our attention in what we experience. Concepts combine to form conceptual principles, which function as categorial laws by which we classify whatever we experience. If an experience does not conform to our conceptual expectations, we classify that experience as non-veridical. Consequently, our attributions of reality depend in part on the conceptual principles we employ. This seems to result in a very strong relativism: what is real depends upon the concepts and classifications we devise. This apparent relativism arises from a terminological ambiguity: Lewis uses the term 'real' both to designate that to which we attribute reality within our conceptual scheme, and to designate what there actually is, which we encounter, experience, and classify. The fact that a classification works for our purposes, and thus serves to attribute reality to some kinds of particulars, shows that what we so classify is, albeit aspectually, metaphysically real. Thus Lewis advocates perspectivalist or aspectualist realism rather than relativism. Indeed, the very logic of relativity, Lewis argues, undercuts relativism. In "Pragmatic Realism" (Chapter 13), Sami Pihlström-himself a major exponent of this view-reassesses and further develops his pragmatic realism by re-examining the Kantian roots and character of pragmatic realism and the debates about realism in the classical pragmatism of Peirce, James, and Dewey, and by differentiating and defending his view by critically examining the views of three other contemporary pragmatic realists: Margolis, Westphal, and Vihalemm. Pihlström argues that sustained controversy about and tensions between realism and pragmatism are not a plague, but instead a strength of pragmatism and an important source of its continuing vitality. Pihlström argues that, in both its classical and in its contemporary forms, pragmatic realism is distinct, e.g., from both social constructivist and metaphysical realist accounts of the world and of our knowledge of it, both commonsense and scientific. Recognizing the distinctive virtues of pragmatic realism requires recognizing that whatever we justifiably regard 6244-263-00Intro.indd 7 1/24/2014 7:29:24 PM 8 Kenneth R. Westphal as real must result from inquiry, and those results and the form(s) they take can be neither pre-determined nor presupposed. That is the cardinal mistake of many commonsense, scientific, and metaphysical forms of realism. Pihlström develops a pragmatic realism which combines a (naturalized) transcendental idealism with pragmatic realism and naturalism in a circular (though not viciously circular) structure: We transcendentally constitute the world through engaging in worldly (and entirely natural) practices, including practices of inquiry, which themselves are constituted through this same continuing process; there is no Archimedean fundamentum of our transcendental world-constitution. In "McDowell's Pragmatist Anti-anti-realism" (Chapter 14), Eirik Julius Risberg re-examines the debate between Rorty, Davidson, McDowell, and now joining them, the 'New Pragmatists', about the character and status of objectivity in a philosophy cleansed of the dualism of conceptual scheme and empirical content. Rorty has long argued that any supposed answerability of our words to a world beyond the linguistic community is fundamentally misguided. The New Pragmatists contend instead that the notion of objectivity is not inimical to pragmatism. In particular, Ramberg argues, against Rorty, that Davidson's insistence upon the irreducibility of the intentional marks a post-ontological distinction between the intentional and the non-intentional, which provides for our intentional thought to be answerable to the non-intentional, in a way compatible with pragmatism. Rorty has accepted Ramberg's criticism, but still maintains that McDowell's view of the answerability of thought to the world is metaphysical and beyond the pale of pragmatism. Against Rorty, Risberg argues that Davidson's postontological distinction between the intentional and the non-intentional, and Rorty's accepting that distinction, suffice to show that McDowell's "anti-anti-realism" belongs within the pragmatist fold. Although McDowell and Davidson disagree about the boundary between the intentional "space of reasons" and the non-intentional "space of nature," McDowell's distinction is as post-ontological as Davidson's. Consequently, McDowell's view is tantamount to pragmatist anti-anti-realism. Many of these essays originated from the conference, "Realism in Its Multiple Forms: A Case of Mere Homonymy or Identifiable Common Commitments?" (6–9 June, 2011), hosted by the Helsinki Collegium for Advanced Studies. It was sponsored jointly by the Helsinki Collegium for Advanced Studies; by the research project, "The Ethical Grounds of Metaphysics" (Universities of Helsinki and Jyväskylä), funded by the Academy of Finland; by the Nordic Pragmatism Network, funded by NordForsk; and by the Swiss National Science Foundation. All of the contributors, and I in particular, express our gratitude to these sponsors for their manifest confidence in, and concrete support of, our research. Papers presented there have been substantially revised for the present volume, and several contributions have been specially written for it. 6244-263-00Intro.indd 8 1/24/2014 7:29:24 PM Introduction 9 Originally I had proposed to Sami Pihlström that we co-edit this volume. I have consulted him at every step, yet by happy coincidence I handled the editing: I had the time, whereas Sami was busy directing the Helsinki Collegium for Advanced Studies. More significantly, the contributors have all been wonderful collaborators; hence no problems arose which required extra brain-storming. I wish to thank each contributor for his fine contribution and exemplary cooperation; my special thanks are to Sami for his thoughts, advice, and assistance. I believe all the contributors join me in thanking Sami very warmly for having organized the very successful conference which launched this project, and in thanking the sponsors who made that conference possible, and hence this volume too. Last though not at all least, I wish to thank the Helsinki Collegium for Advanced Studies for its warm hospitality and ideal working conditions, where as a (former) member of its Academic Advisory Board I was allowed to spend the final quarter of 2011, when most of the editorial work on this volume was undertaken. The editorial and production staff at Routledge have been thoroughly professional, the very model of integrity in academic publishing, for which we are all very grateful indeed. 6244-263-00Intro.indd 9 1/24/2014 7:29:24 PM Douglas R. Anderson is Professor of Philosophy at Southern Illinois University at Carbondale. He is author and editor of several books and numerous essays dealing with issues in American philosophy and culture. He is former Editor-in-Chief of the Transactions of the Charles S. Peirce Society and of the Journal of Speculative Philosophy. Presently he edits a book series in American thought for Fordham University Press. His recent work includes Philosophy Americana (2006) and, with Carl Hausman, Conversations on Peirce (2012). Laurent Cesalli, Chargé de rechrerche at the CNRS (UMR 8163, Savoirs, Textes, Langage, Université de Lille 3), was born in 1968 in Vevey, Switzerland. He studied philosophy and musicology in Fribourg (1988–93), survived training as a mountain guide (1994–97), and bicycled from Geneva to Beijing (1995–96). Since 1998 he has worked at the universities of Geneva (PhD in Mediaeval philosophy, supervised by Alain de Libera), Freiburg in Breisgau, Lausanne, Fribourg (CH), and Cornell, NY. His field of research is semantics and ontology in mediaeval as well as in Austro-German philosophy. He is completing a monograph on Anton Marty's philosophy of language. Jaakko Hintikka is Honorary Fellow in the Helsinki Collegium for Advanced Studies, and has received many international honors and awards. He was a Junior Fellow at Harvard (1956–59) and has held Professorships at the University of Helsinki, the Academy of Finland, Florida State University, and Boston University, and was affiliated with Stanford University (1965–82). His interests span philosophy of language, mathematical and philosophical logic, epistemology, philosophy of science, cognitive science, philosophy of mathematics, and history of philosophy, especially Aristotle, Descartes, Kant, Peirce, and Wittgenstein. He is the main architect of game-theoretical semantics and of the interrogative approach to inquiry, and one of the architects of distributive normal forms, possibleworlds semantics, tree methods, infinitely deep logics, and contemporary theory of inductive generalization. He has authored or co-authored over Contributors 6244-263-BM1.indd 303 1/24/2014 7:32:23 PM 304 Contributors 30 books and monographs in nine languages. He has edited or co-edited 17 volumes and authored or co-authored over 300 scholarly papers, including five volumes of Selected Papers (1996–2003). The Philosophy of Jaakko Hintikka appears in the Library of Living Philosophers (2006, vol. 30). Lauri Järvilehto, PhD, is a researcher and trainer at Filosofian Akatemia, Helsinki, and a Post-doctoral Affiliate at the Aalto University Systems Analysis Laboratory. His current research on 'Intuitive Thinking and Decision Making' draws upon recent developments in the philosophy of mind, cognitive psychology and neuroscience, and also the epistemology and semantics of C. I. Lewis, whose conception of a priori knowledge was the topic of Järvilehto's doctoral dissertation. Järvilehto has lectured on theoretical and applied philosophy at many Finnish universities, and at universities in Great Britain, France, Germany, Sweden, Denmark, and Estonia. Antti Keskinen is a Post-doctoral Researcher at the University of Tampere, where he earned his PhD in 2010 with his dissertation, Quine's Critique of Modal Logic and his Conception of Objects. He has interpreted Quine's rejection of de re modalities on the basis of Quine's view of objects as posits. Currently he is researching the role of social cognition (empathy) in Quinean epistemology and philosophy of language, and problems related to the ascription of de re propositional attitudes. Part of the research on his contribution to this volume was done as a Post-doctoral Researcher at the University of Uppsala. Jonathan Knowles is Professor of Philosophy at the Norwegian University of Science and Technology in Trondheim, Norway. He studied at the universities of Oxford, Edinburgh, and London, gaining his PhD in 1995 with a thesis on the philosophy of cognitive science and linguistics. He has written Norms, Naturalism and Epistemology: The Case for Science Without Norms (Palgrave 2003) and edited (with Henrik Rydenfelt) Pragmatism, Science and Naturalism (2011). He has published papers on folk psychology and explanation, knowledge of grammar and linguistic competence, reductionism in psychology, naturalistic theories of norms, the nature of naturalism, and realism and naturalized epistemology. He sits on the board of the Nordic Pragmatism Network. Heikki A. Kovalainen is a Philosophy Researcher at the University of Tampere, Finland. His work centers on the intersections of American and Continental thought, including classical authors (especially Emerson and Nietzsche) as well as contemporary thinkers (especially Cavell and Rorty), focusing on philosophies of life richer than Lebensphilosophie and phenomenology, and encompassing both life's vulgarity and its sublimity. His 6244-263-BM1.indd 304 1/24/2014 7:32:23 PM Contributors 305 publications include the monographs Self as World-The New Emerson (2010), Emerson ja filosofia [Emerson and Philosophy] (2007), and the articles "Emersonian Moral Perfectionism: An Alternative Ethics-But in What Sense?" (2010) and "Emersonian Self-Culture and Individual Growth: The American Appropriation of Bildung" (2013). Dermot Moran holds the Chair of Philosophy (Metaphysics & Logic) at University College Dublin and is a Member of the Royal Irish Academy. A graduate of University College Dublin and Yale University, he has taught at Queen's University Belfast and Maynooth College, and held visiting professorships at Yale University, Connecticut College, Rice University, and Northwestern University. He has published widely on mediaeval philosophy (especially Christian Neoplatonism) and contemporary European philosophy (especially phenomenology). His books include The Philosophy of John Scotus Eriugena (1989, 2004), Introduction to Phenomenology (2000), and Edmund Husserl: Founder of Phenomenology (2005). He is Founding Editor of The International Journal of Philosophical Studies. Ilkka Niiniluoto is, since 1977, Professor of Theoretical Philosophy at the University of Helsinki, and former Chair of the Department of Philosophy (1983–88, 1992, 1995–2000). He is former Rector of the University of Helsinki (2003–2008) and former Chancellor of the University (2008–2013). He has published widely on philosophy of science, and especially scientific realism. His books include Is Science Progressive? (1984), Truthlikeness (1987), and Critical Scientific Realism (1999). He was Associate Editor of Synthese (1973–76), then Editor (1977–79), and a member of its editorial board (1980–2007). He is Editor-in-Chief of Acta Philosphica Fennica (since 1980), and President of the Philosophical Society of Finland. Mika Perälä holds an Academy of Finland Postdoctoral Research Fellowship in the University of Helsinki. Previously he was Postdoctoral Researcher in the Philosophical Psychology, Morality and Politics Research Unit, University of Jyväskylä, Academy of Finland. His doctoral dissertation examined Aristotle's account of perceiving that we see and hear. After obtaining his doctorate in Helsinki (2010), he pursued postdoctoral studies at Oriel College, Oxford, 2010–2011. He is currently writing articles on Aristotle's theory of perception, and a monograph on Aristotle's account of memory. Sami Pihlström is, since 2006, Professor of Practical Philosophy at the University of Jyväskylä and, since 2009, Director of the Helsinki Collegium for Advanced Studies. He has published widely on the problem of realism and related matters, including the pragmatist tradition. His recent books 6244-263-BM1.indd 305 1/24/2014 7:32:23 PM 306 Contributors include Pragmatist Metaphysics: An Essay on the Ethical Grounds of Metaphysics (2009), and Transcendental Guilt: Reflections on Ethical Finitude (2011); he edited the Continuum Companion to Pragmatism (2011) and is an Executive Editors of SATS: North European Journal of Philosophy and is the Book Review Editor of the Transactions of the Charles S. Peirce Society. Panu Raatikainen is Adjunct Professor of Theoretical Philosophy at the University of Helsinki and the University of Tampere. His research embraces logic, philosophy of mathematics, philosophy of language, philosophy of mind, philosophy of science, and the history of analytic philosophy. He has held Research Fellowships in the Academy of Finland and in the Helsinki Collegium for Advanced Studies, and visiting research posts at the University of St. Andrews, the University of London, and the City University of New York. He publishes in such journals as Analysis, Dialectica, Erkenntnis, Synthese, British Journal for the Philosophy of Science, and Journal of Symbolic Logic. Eirik Julius Risberg is a doctoral student in philosophy at the University of Oslo, researching his dissertation on meaning and objectivity, including whether pragmatism can accommodate a notion of objectivity. Risberg holds an MA in philosophy from the University of Oslo and has been a visiting student at the University of California, Berkeley, and at the University of British Columbia, Vancouver. With support from Fullbright Foundation (USA) and the Norwegian Research Council, he spent the academic year 2011–12 as a visiting research student at the University of California, Berkeley. Peter Swirski is Professor of American Literature and Culture at the University of Missouri–St. Louis, Honorary Professor in American Studies at Jinan University (China), Honorary Professor in American Studies at the South China University of Technology, and former Research Fellow in the Helsinki Collegium for Advanced Studies. His research ranges from American Literature and American Studies to interdisciplinary studies in literature, philosophy, and science. He is the foremost critic on the late writer and philosopher, Stanislaw Lem. His books include Between Literature and Science (2000), From Lowbrow to Nobrow (2005), Of Literature and Knowledge (2007), Literature, Analytically Speaking (2010), and was a National Book Award finalist for his Ars Americana, Ars Politica (2010). Kenneth R. Westphal is Professorial Fellow in the School of Philosophy, University of East Anglia, Norwich. He has held Humboldt Research Fellowships at the universities of Heidelberg, Bielefeld, and Göttingen; and been Visiting Professor at Northwestern University and at 6244-263-BM1.indd 306 1/24/2014 7:32:23 PM Contributors 307 the Martin-Luther-Universität Halle-Wittenberg. His research on the character and scope of rational justification in non-formal domains integrates systematic with historical, and analytic with hermeneutical philosophy. His books include Kant's Transcendental Proof of Realism (2004); he edited The Blackwell Guide to Hegel's Phenomenology of Spirit (2009). His recent articles include: "Kant's Critique of Pure Reason and Analytic Philosophy" (2010); "Self-Consciousness, AntiCartesianism & Cognitive Semantics in Hegel's 1807 Phenomenology" (2011); "Norm Acquisition, Rational Judgment & Moral Particularism" (2012); "Hume, Empiricism & the Generality of Thought" (2013); "Natural Law, Social Contract & Moral Objectivity: Rousseau's Natural Law Constructivism" (2013); and "Rational Justification & Mutual Recognition in Substantive Domains" (2013). 6244-263-BM1.indd 307 1/24/2014 7:32:23 PM Index abduction 143, 147, 149, 159, 166–7, 169, 225–7 Absolute, the 222, cf. 231; absolute truth see truth, absolute; absolute vs relative see relative Achinstein, Peter 147 acquaintance, knowledge by, see knowledge, by acquaintance actualism 15, 165 affairs, states of see states of affairs, also see fact(s) affordance (perceptual) 214 Ahab (fictional character) 120 Aladdin 121 Alanen, Lilli 68, 82 Albert the Great (Albertus Magnus) 48, 77, 238n9 Allison, Henry 268, 278, 279 Anderson, Carl 163 Anderson, Douglas 221, 255–6 anti-anti-realism 283–99 anti-realism 47, 93–6, 100, 105, 111, 150, 153–5, 166, 169, 179, 214, 216, 284, 286, 291, 294–5, 298, also see empiricism, Constructive; empiricist 179, also see empiricism, Constructive; --, metaphysical 111; --, semantic 150–5, 169; also see bivalence, Dummett anti-representationalism 214–5, 283–9 Aquinas, Thomas 22–3, 30–4, 39, 48–50, 52 Arabian Nights 116 Aristotle 17–19, 21–42, 49–50, 58, 61n5, 183 arithmetic see mathematics Armstrong, David 47, 48, 58–60, 141, 160, 280n41 Arnauld, Antoine 67–8, 70, 74, 76, 78 ascription of characteristics or properties see predication; --, of propositional attitude 94–5, 106–12, 124, 196n13, 287–8, 297–9; --, de dicto vs. de re 94, 106–7, 110–12; --, notional vs. relational 108–12 assent 97–99, 102, 109, 196n9 also see dissent assertability, warranted, see justification, warranted assertability atomism 59 attention, joint, 129, 131–3 also see intentions attitude, natural 70, 88, 143–4; --, ascription see ascription, of propositional attitude; --, non-egoistic see egoism; --, religious 221 Audi, Robert 257 author's intentions (authorial intentions) see intentions, author's Avicenna 52 Ayer, Alfred Jules 141 Bacon, Roger 77 Baudelaire, Charles Pierre 117 Baudolino 129 belief, fixation of (Peirce) 223, 255, cf. 227 belief system 108, 110, 112, 123–5, 127, 200, cf. 88, 213, 239n17 (Note: close cognate terms are all listed under one main form, without mentioning the latter variants; e.g., 'actualism' also includes 'actualist'.) 6244-263-BM2.indd 309 1/25/2014 8:36:52 PM 310 Index Bequerel, Henri 148 Bergmann, Gustav 56–9 Berkeley, George 85, 141, 142, 169, 174, 229 Bhaskar, Roy 152, 165 biology 209, 214, 215, 275, 288, 272; --, evolutionary 129, 131, 141, 167–9, 212, 213, 234, 266, 275, also see paleontology; primatology 130 bivalence, principle of 96, 150–3, 169; also see intuitionism, mathematical Boghossian, Paul 204 Bohm-Aharonov experiment 20 Bohr, Niels 19 Bolzano, Bernard 73 Bonaparte, Napoleon 167 Borges, Jorge Luis 117, 119 Bouwsma, O. K. 184 Boyd, Richard 145, 147, 160, 167 Brandom, Robert 191, 200, 214–5, 284 Bratman, Michael 132 Brentano, Franz 57, 73, 83, 283 Brinkley, Richard 50 Bruner, Jerome 129 Buddhism 238n1 Buridan, John 48–9 Burley, Walter 47, 48, 50–3, 55–6, 59–60 Canberra plan 204 Carnap, Rudolf 144, 161, 183, 196n11 Cartwright, Nancy 146 category, categories 56, 102, 104, 160–3, 165, 206–7, 209, 213, 230, 243–9, 258–9, 272–3, also see observation categorical, observation terms causal agnosticism 179 causation, causality, cause 22–42, 53, 68–70, 76–7, 79, 81–3, 86–9, 103, 117, 128, 140, 142, 146, 148–9, 159, 160, 163–9, 173–9, 185, 190–3, 196n17, 206, 210–11, 216n6, 258, 268, 271–3, 283, 287, 290–1, 294, 297, 299 ceteris paribus clause 190–1, 196n17 Chisholm, Roderick 142 Chomsky, Noam 208, 212, 215, 217n15 Circularity Accusation, the see justification, circularity Clark, Herbert 129 classical pragmatism see pragmatism, classical classification(s) see category Cocchiarella, Nino 48, 58–9 cognitive science 212–15; also see Theory of Mind (ToM) Coleridge, Samuel Taylor 232 Comte, August 167 Conan Doyle, Arthur 119 conceptual scheme 144, 160, 243–9, 257, 263, 283, 285–7, 295, 298, 299n1; also see categories; belief system confirmation holism see holism, confirmation Constructive Empiricism see empiricism, Constructive constructivism 14, 87, 93, 99–100, 104, 144, 150, 152, 161–2, 168, 196n11, 226, 251–2, 255–6, 258, 260–2, 264–9, 274–7; --, causal vs. conceptual 67, 82, 85, 87–8, 104, 168–9, 247–9, 260–2, 265–9, 274–6; --, social see social constructivism content, of assertion 55, 99, 122, 164, 182, 189, 193, 214–15, 291–2, 295; --, cognitive 54, 107, 184–5; --, conceptual 68, 87, 181–4; also see conceptual scheme; --, empirical 94, 97, 98, 99, 101, 106, 109–12, 169, 216n9, 283, also see conceptual scheme; --, experiential 68, 72, 76, 83, 85, 87–8, 161, 271, 296; --, fictional 71, 88, 115–33, 163; --, of idea see objective reality; --, information 170, also see Dretske; --, intensional 30–1, 59, 67, 69; --, of judgment 55, 68–9, 71–6, 83–7, 120, 128, 167, 182–4, 228, 244–6, 259–60, 271; --, mental 55, 184–5, 279n33, also see ascription, of propositional attitude; --, perceptual 22–42, 164; --, phenomenological 68, 72, 75, 87; --, representational 70, 74–5, 77, 79, 207 also see anti-representationalism; --, semantic 86, 99, 185, 207, 215, also see content, conceptual; 6244-263-BM2.indd 310 1/25/2014 8:36:52 PM Index 311 --, of sentence 99, 109; --, sensory 22–42, 76, 81, 83, 85, also see --, experiential, --, phenomenological, phenomenalism; --, of statement 214–15 also see reference, Thesis of Singular Cognitive; --, of thought 22–3, 54, 55, 70, 74–5, 77, 79, 81, 83, 84, 86, 88–9, 207, 295, also see ascription, of propositional attitude; objective reality Convergent Realism see realism, Convergent Copernicus, Nicholas 187 Coppola, Francis Ford 119, 122, 123 Corleone, Don (fictional character) 119, 122 Corleone, Michael (fictional character) 123 Crawford, Sean 108, 110 critical scientific realism see scientific realism, Critical Crusoe, Robinson (fictional character) 128 culture 104, 115, 128–31, 159, 160, 163, 165, 168, 170, 231, 233–4, 261, 265, 287 Curie, Marie 148 Curie, Pierre 148 Davidson, Donald 195n3, 283–92, 296–9 Davies, David 128 denoting see reference dependence see independence Descartes, René 41, 42, 67–89, 187 description, knowledge by, see knowledge, by description descriptivism 144 Devitt, Michael 96–7, 100, 105, 111, 146, 151, 155, 202, 205–7, 216n13 Dewey, John 164, 169, 222, 230, 251, 253, 254, 256, 259–62, 266, 268, 274, 277n9, 295 Diamond, Cora 234 dictum 61n6, 163, 228, 285 Dietrich of Freiberg 56–7 Ding an sich see things-in-themselves Disch, Thomas M. 124 disjunctivism 41 disputed class (Dummett) 150–1, 153 dissent 48, 97, 99, 102, 109 distinction, de re vs. de dicto 107, also see ascription, of propositional attitude; --, formal 47, 54, 59, 60n3, 75, 78; --, among relatives (relations) 23, 27, 31, 40–2; --, relational vs. notional 113n15 domains, formal vs. non-formal (substantive) 183 Donnellan, Keith 182 dream 41, 163, 235, 243, 247–8 Dretske, Frederick 184 Duhem-Quine Thesis, the, 155 Dummett, Michael 96, 150–5, 166, 169, 286, 292–5 Dupin, Auguste C. (fictional character) 117 Dupré, John 208 dynamic, dynamics 161, 254–5 also see Newton, Isaac Eco, Umberto 115, 121, 129 Edison, Thomas 148 egoism 221, 225, 226, 231, cf. 227 Einstein, Albert 144, 165, 174 Emerson, Ralph Waldo 221–37 empirical adequacy 177, 178, 188–93, 196n15 empirical content, see content, empirical empiricism 41, 100, 103–4, 110, 167, 178–9, 186, 259, 262, 287; --, concept or meaning 161; --, Constructive (van Fraassen) 173, 184, 188–95; --, radical 143, 154, 239n17; --, reductive 144; --, verification 112n10, 183–4 enactivism 214 enuntiabile 61n6 epistemic, epistemology 16–19, 31, 50, 54, 59, 67, 78, 84, 93–8, 99–101, 103–6, 108–12, 118, 143, 152, 154, 159–60, 162–5, 167, 169, 173–4, 181–8, 194–5, 236–7, 243–9, 252–3, 257–9, 264–71, 296, 298; --, naturalistic 58, 93–4, 101, 103, 105, also see naturalism; -- utilities 154 epistemic conception of understanding see understanding, epistemic conception of 6244-263-BM2.indd 311 1/25/2014 8:36:52 PM 312 Index essence 38, 46, 49–54, 56, 59, 67, 69–70, 74, 82–4, 205, 229, 230, 259–60, 262; also see re ethics, ethical (also moral) 140, 159, 160, 162, 213, 262, 267, 275, 289; --, Christian 221–37; --, of inquiry 221–37; --, moral sentiment 221, 224, 231–7 Euclid 18 Evans, Gareth 173, 180–2, 196n9 evidence, empirical or experimental 20, 95–7, 100, 103, 130, 148, 150, 163, 168–9, 174–6, 179, 186–9, 228, 288, 292; see also observation evolution, biological see biology, evolutionary; --, of beliefs or ideas 226, 254, 299n4; --, of reality 254, 255 existence 1–20, 28, 51, 54, 58–9, 69–70, 93–5, 97, 99–103, 105, 108, 118, 140, 145–50, 154–5, 165–6, 174, 176, 179, 183, 200, 206, 209, 210, 223, 247, 252–3, 260, 296; --, vs. being 13, 15, 17, 162–3, 223–4, 247 explanation, Aristotle's model of 26, 28–9; scientific -- 143–4, 148–9, 159–60, 167, 174–9; also see science; boundary conditions 16; initial conditions 16, 177 exportation 108–9 expressivism 210–11 fact(s) 19, 55–60, 104, 117, 152, 162, 164, 185, 192–4, 200–10, 249, 257, 283, 286–99 faith (object of) 48, 221 also see God fallibilism see justification, fallible falsity, formal vs. material 71–89 Feyerabend, Paul 144, 147 fictional truth see truth, fictional Field, Hartry 152, 153 Fisch, Max 266 Flasch, Kurt 56 Flash Stockman (fictional character) 124–5 Fogelin, Robert J. 96–7, 100, 110–11 formalitas 47, 52, 68–9, 77–8 Francesco da Prato 53, 59, 60 Frege, Gottlob 13–17, 73, 152, 162, 206 fundamentum in re 50, 51 Galilei, Galileo 144, 195n7, 227 gavagai 129 General Theory of Relativity 174, 179 generalization by induction see induction generals (Peirce) 223, 229, 236–7, 254–6, 266 Gettier, Edmund 194 Gibson, J. J. 214 Gibson, Roger F. 94, 100–1, 104 Giles of Rome 48 Gilson, Étienne 69, 77–8 Glymour, Clark 179 God, gods 16, 22–3, 39, 40, 49, 69–70, 73, 77–83, 86–7, 89, 141, 142, 169, 221, 223–4, 226–7, 229, 233, 235–7; God's-eye view 162, 257, 265, 269, 273, 276 Gödel, Kurt 15 Godfather, The 119, 122–3 Goodman, Nelson 169, 190, 191, 195n3, 258 Grice, Paul 131 Grubbe, Emil 148 Haack, Susan 95 Hacking, Ian 146, 165 haecceitas 52, 164 Hale, Bob 152 hallucination 21, 38, 41, 42, 140, 246 Hannibal 121, 122 Hanson, Norwood Russell 147 Harper, William 173–86, 186, 187, 188, 192, 193, 196n12 Hausman, Carl 227, 254–6 Hector (fictional character) 120 Hegel, G.W.F. 214, 231, 261, 264–9 Hempel, Carl G. 190, 191, 195n3 history (discipline of historical inquiry) 170, 213; -- and philosophy of science (HPS) 173–80, 183, 187, 192–3; geohistory 168; --, human 168, 234; --, natural 168, cf. 189, 192 also see past; natural -- of the intellect see intellect, natural history of; -- of philosophy 17, 18, 251, 283 also see philosophy (periods); -- of science 145–9, 163, 165–8, 173–80, 187 holism, of attitude ascription 288–90; --, confirmation 99, 155; --, semantic 98, 101, 109; --, social 132 6244-263-BM2.indd 312 1/25/2014 8:36:52 PM Index 313 Hollywood 120 Holmes, Sherlock (fictional character) 115, 117, 119, 120, 127 Homer 18, 19 Horwich, Paul 152 Hume, David 69, 164–5, 174, 195n3, 238n10 Husserl, Edmund 67, 70, 73, 81, 83–5, 88, 162, 165 Hutcheson, Francis 232 idealism 47, 140–4, 150–2, 161, 214, 251–6, 261–3, 275, 276; --, conceptual 162; --, Mediaeval 46, 56–7; --, metaphysical 223, 236, 272; --, objective 162, 163, 169, 222, 223, 255; Refutation of -- (Kant) 271, (Moore) 161; --, subjective 159, 161, 168; --, transcendental 162, 232, 251, 253, 259, 263, 265, 266, 270–5; --, transcendental pragmatic 263–9 ideally justified see justification, ideal identification see 'is', of identification identity 24, 46, 99, 100, 101, 105, 109, 163, 230, also see 'is', of identification; --, conditions (or criteria) 112n10, 163, 180–2, 274; --, partial 53, 59; --, and predication 173, 180–2; --, temporal 141 illusion 25, 38, 41, 228, 243, 260–1 implication see logic incommensurability 144 independence vs. dependence 13, 96, 161, also see relative, vs. absolute; --, of language 57, 144, 200, 252–9, 284, 291; --, logical 14–15; --, of mind or the mental 19, 21, 23, 32, 39, 50, 54, 56, 59–60, 88, 93, 140, 143, 146–8, 150, 152, 155, 159–68, 200–2, 205–7, 213, 216n13, 224, 235, 243, 246–9, 264–7, 269, 272–3, 276, 284, 286, 292; --, of practice 252, 261, 263, 274; --, of theory 96–105, 108, 111, 144, 196n11, 200, 252 indeterminacy of translation see translation, indeterminacy of induction, generalization 149, 166, 173–6, 195n3 inductive argument for realism, see realism, inductive argument for infallibilism see justification, infallible inference to best explanation, see abduction inquiry 276, 284; --, interdisciplinary 274–5; --, phenomenological 214; --, scientific 159–60, 162, 164, 168, 221–37, 252, 253–6, 259–62, 264, 274–5, also see method, scientific instantiation 13–14, 40, 54, 58, 60, 84, 99, 207, 230, 236 instrumentalism 144, 145, 159, 164, 167, 170, 259–62 intellect 32, 34, 39, 40, 50–7, 69, 71–2, 78, 82, 85–8, 222, 227–30, 235; --, intuitive 232; --, understanding (Verstand) see understanding; --, natural history of, 222, 228, 230, cf. 169, 234–5, 264 intentional, irreducibility of 283–99 intentionality (object-directedness) 22, 30–2, 41–2, 48, 56–9, 67, 69, 72–89, 124–7, 129, 182, also see ascription; attention, joint; content;--, 'we', 129, 130–3, also see attention, joint intentions, author's 115, 123, 124, 126–9; --, reflexive 124, 126, 127, 128, 131–2 intuition, forms of (Kant) 165, 232, 271; --, intellectual see intellect, intuitive intuitionism, ethical 232; --, mathematical 151, 169–70 also see bivalence; --, modal 193 'is', of existence 13, 18 also see instantiation; --, of identity 13, 17 also see identity; --, of identification 17–18, 19, cf. 142, 162–3, 168, 180–6, 194, 271–3, 274; --, of instantiation see instantiation; --, of predication 13 also see predication; --, of subsumption 13, also see instantiation, predication Iser, Wolfgang 116 6244-263-BM2.indd 313 1/25/2014 8:36:52 PM 314 Index Jackson, Frank 202, 204, 209 Jacobs, W. W. (William Wymark) 125–6 James, Henry 124 James, William 164, 222, 227, 229, 230, 231, 239n17, 251, 253, 256–9, 261, 262, 266–7, 268, 284–5 Je t'aime, je t'aime 128 joint attention see attention, joint justification, (vicious) circularity charge 142, 149, 154, 279n38; --, cognitive (epistemic) 149, 175, 182, 184–95, 195n3, 210, 212, 283, 285; --, fallible 101, 125, 127, 160, 162, 166, 183, 186–93, 226, 227, 264–6, 273, 278n24, 279n35; --, ideal 152, 154, 164, 169, 224, 228, 237, 257; --, infallible 173, 183, 186–93, 262; --, logical gaps 191–2; --, methodological 187; --, warranted assertability 169, 299n3, cf. 227 Kant, Immanuel 16, 18, 56, 57, 144, 160, 161–2, 165, 230, 232, 251–3, 255, 258–9, 263–72, 274, 277, 283, 284 Kaplan, David 108, 196n11 Kepler, Johannes 175–6, 195n7 knowing who/what 16–19, 54, 59, 108, 115ff, 181–4, 244, 247–9 knowledge, by acquaintance 19; --, by description 181–4; --, scientific 16, 46, 58, 95, 110, 139–215 passim, 223, 235, 237, 253, 259–61, 274; --, of truth conditions see meaning, truthconditional theory of Kruse, F. E. 222, 227 Kuhn, Thomas 144, 147, 154, 179 language, ideal 160; --, natural 14, 46, 97, 132–3, 143, 153, 165–6, 180–1, 190, 196n9, 213, cf. 245; --, ordinary see language, natural; --, philosophy of 184–5, 194–5, cf. 272, 290–1; --, pragmatics 188, 194–5; --, private 143, 293; --, truth-functional 191 language game (Wittgenstein) 160, 273 Latour, Bruno 144, 164 Leibniz, G. W. 15, 179 Lem, Stanislaw 118, 124, 127–9 Leplin, Jarrett 145–6, 167 Lewis, C. I. 141, 183, 191, 243–9 Lewis, David 119–127 Loar, Brian 108 logic, epistemic 13–20, also see knowing who/what; --, erotetic 18–20; Law of Weakening 189–91; material implication, paradox of 191; modal 16–20, 27, 193, also see modality; possible worlds 16, 122; quantificational 14–20, 95–6; --, social theory of 225; strict implication 190 Logical Empiricism 144, 146–7, 195n3 logical gap 42, 183, 187, 191–2 Logical Positivism 102, 141, 144, 159, 168, 195n3, 216n9, 287, cf. 243 Lotze, Hermann 73 Malbranche, Nicholas 142 Margolis, Joseph 251, 255–6, 263–70, 272, 276 mathematics 15, 18, 39, 68, 79, 84, 95, 140, 150–4, 163, 166, 169–70, 183, 193, 223, 233, 259–60, 292, 294; mereology 53, 58, 59; numbers 15, 22, 103, 104, 140, also see intuitionism, mathematical mature sciences see sciences, mature Maxwell, Grover 144, 147, 167 McCluskey (fictional character) 123 McDowell, John 200, 283–99 McGilvray, J. 215 M.D.: A Horror Story, The 124 meaning 13–14, 17–18, 47, 51, 59, 73, 101, 126, 129, 130–3, 151, 153–5, 180–4, 224, 244, 246–7, 258, 272, 286, 290, 292; --, full-blooded vs. modest theory of 300n8; --, truth-conditional theory of 153–5, 292–5, 297–9; also see content, conceptual; pragmatic maxim; reference; semantic(s) meaningful/meaningless(ness) 102, 103, 104, 105, 106, 109, 111, 112n10, 143, 150, 152, 161, 169–70, 200–1, 258, 273, 276 Meinong, Alexis 57, 73 Menzies, Paul 202–4 6244-263-BM2.indd 314 1/25/2014 8:36:53 PM Index 315 mereology see mathematics, mereology Merleau-Ponty, Maurice 214 Mersenne, Fr. Marin 67, 86 metaphor thesis (Dummett) 151 metaphysical anti-realism, see antirealism, metaphysical; --, idealism, see idealism; --, naturalism, see naturalism, metaphysical; --, realism, see realism, metaphysical metaphysics 14–20, 46–50, 57–9, 155, 162, 163, 169, 174, 200–15 passim, 230, 232, 237, 243, 246–9, 254, 264, 266–74 passim, 283, 285–6, 289, 298; --, Aristotelian 17–19, 33; --, Cartesian 69, 71, 73, 84; --, modal see logic, modal; --, Neo-Platonic 48, 54, 58–9, 226, 227, 230, 236; --, Platonic 15, 47, 49, 51, 60, 151–2, 237, 294; --, pragmatist 162, 221–300 passim method, scientific see explanation, scientific Mill, John Stuart 141 Miller, Alexander 151, 153–4 mind and matter 222, 230 mind-independence see independence, of mind mine disease (medical diagnosis) 148 minimal realism see realism, minimal Minimal Scientific Realism see realism, minimal scientific miracle argument 147, 149 Misak, Cheryl 266, 284, 299n4 modality, causal 193–4; --, logical 191, 193 also see logic; --, metaphysical 16, 19, 27, 79–80; also see re "Monkey's Paw" (Jacobs) 125–6 moral sentiment see ethics, moral sentiment Morris, Charles 194 Mother Teresa 115 Mrs. Dalloway 115 Murdoch, Iris 234, 236 Mutual Belief Principle 120–2, 125–6 Mutual Contextual Beliefs 127 natural attitude see attitude, natural; --, history see history, natural; --, kind 205, cf. 217n13, 288; --, language see language, natural; --, phenomena 174–5, 185, 187, 192–3; --, science see science naturalism (philosophical) 93–112, 169, 200–80 passim; --, Canberra plan (Jackson) 204; --, liberal 200, 201, 212; --, metaphysical 200–8, 209–15; --, minimal 200, 275–4; --, object 216n7; --, pragmatic 260–77, 279n38; --, reductive 205; --, scientific 93, 95, 100, 200, 201, 207;--, subject (Price) 209–15; also see epistemology, naturalistic; science, natural nature, space of 285, 297, 298, 299 neo-pragmatism see pragmatism, neoneural stimulation 96–9, 113n13 Newton, Isaac 122, 165, 173–80, 185–8, 191–4 Newtonian mechanics see Newton, Isaac Nicolas of Autrécourt 51, 54, 59–60 Niiniluoto, Ilkka 261, 274 Noë, Alvin 214 nominalism 47–9, 57, 60, 161, 224, 229–30, 235–7, 262–3 non-intentional see intentional, irreducibility of; space of nature normative, norms 212, 221, 224–5, 235–6, 252, 256, 275, 290–1, 295, 297–9, also see ethics numbers see mathematics O'Regan, K. 214 object, abstract 13–14, cf. 32, 88, also see mathematics, numbers; --, mathematical, see mathematics, numbers; --, physical 13–14, 93–106, 140–9, 151, 155, 160, 162, 164–9, 185, 254, 270–1, also see independence, of mind; realism; --, perspectival 19, cf. 161, 244, 247, 249, 257; object, possible 16–20, cf. 41, also see re; object, public 34–5, 97, cf. 106–12, also see attention, joint; observation, sentence; --, of a sign, immediate vs. dynamic 161–2, 254, 255, cf. 180–2 objective reality (representational content of an idea) 42, 50, 54–6, 67–89 6244-263-BM2.indd 315 1/25/2014 8:36:53 PM 316 Index objectivity 51, 164, 200–2, 206, 207, 224, 231, 255, 260–1, 266, 268–9, 283–99, also see independence, of mind; realism; --, ontological vs. postontological 285–92, 297–9 observation, categorical (Quine) 95–101, 110; free vs. focal forms 98, 112n5; --, sentence 95–101; --, terms, vs. theoretical terms see scientific terms, observational vs. theoretical; --, theory-laden see scientific observation, theory laden occasion sentence 97 Ockham, William of, 47–50, 59; Ockham's Razor 192–3 ontogeny 130 ontological commitment 93, 95, 96, 104, 105, 107, 165–6, cf. 192, 273, 291 ordinary language see language, natural Ortcutt, Bernard J. 107–10 "Oval Portrait, The" (Poe) 121–2 paleontology 168, cf. 141, 166, 168–9, 206 parsimony, explanatory see Ockham's Razor particulars see object, physical parts 46–7, 180–1; --, concomitant 53; --, formal 52–3; --, integral 52; --, metaphysical 49, cf. 58; --, proper 53 past 128, 150, 159, 165–70, 292, also see history Peirce, Charles Saunders 47–8, 143, 152, 160–7, 221–39, 251–6, 262–9, 272, 275 perceptual relativity see relativity, perceptual perceptual similarity 97, cf. 245, 270–1 also see observation sentence Pessimistic Meta-Induction 149 phenomenalism 141–3, 150, 152, 153, 156n2, 159, 168 phenomenology, phenomenological 68, 71, 73, 75–6, 81, 162, 165, 214, 225, 230, 231, also see content, phenomenological philosophy [listed are express mentions of these terms, not evident from the Table of Contents or (sub) section headings; also see names of representative figures. –Ed.]; --, analytical 13, 18–19, 116, 243, 261, 268; --, ancient 229, 236; --, mediaeval 31, 77, 78, 160–1, 163, 221; --, modern (17th–18th C.) 38, 41, 209, 226, 296; --, phenomenological 68, 162, 165, 214, 225, 230, 231; --, postmodern 226; --, pragmatic 243 also see pragmatism; --, Scottish 231–2, 233 phlogiston 73, 145–6 physicalism 202, 205, 208, 213, 287, 288, 299n2, also see mind and matter; --, reductive 202, 205, 208, 213, 215n2, 287, 288, 299n2 Planck, Max 144 Plato 15, 18, 27–8, 47, 49, 51, 53, 54, 58–9, 60, 151, 152, 227, 230, 232, 234, 236, 237, 294, 300n7 pluralism, conceptual 160, 162, 243–9, cf. 209, 210; --, pragmatic 243–9 Poe, Edgar Allan 116–22, 227 Popper, Karl 143, 144, 160, 163, 165 posit 93–6, 99, 101, 103–5, 110–1, 269 possible beings (objects) 18–20, 70, 73, 79, 80, 84, 88, 140, 142, 187, cf. 166, 183; -- experience(s) 245, 270–1, 273; -- worlds 15–20, 108, 117–22, 125–6, 128; also see logic, possible worlds practical realism see realism, practical pragmatic maxim (method for clarifying ideas) 258, 268–9, 275–6; -- realism see realism, pragmatic; -- pluralism see pluralism, pragmatic pragmaticism 221, 254 pragmatics (of language, vs. syntax, semantics) 188, 194–5 pragmatism, pragmatist [listed are express occurrences not evident from the Table of Contents or (sub)section headings; also see names of representative figures. –Ed.] 159, 160, 168–9; --, classical 164, 251–63, 267, 272, 276; --, neo200, 6244-263-BM2.indd 316 1/25/2014 8:36:53 PM Index 317 210, 212, 215n2; --, new 284–6, 297–8, 299n4; --, transcendental 161, 251–77 predicate(s) 16, 18, 49, 55–6, 96–101 passim, 109, 181 predication 13, 51–2, 55, 107, 109, 160, 173, 180–1, 184, 196n9, 248; --, as cognitive achievement 181–3, 248; --, of agency 290ff., also see ascription of propositional attitude presentism 165–6, also see actualism, history, past Price, Huw 200, 202–5, 209–11, 214–15 primatology see biology Principle of Bivalence see bivalence, Principle of propositio in re 55 propositio realis 56 propositional attitude see ascription of propositional attitude Protagoras 27–8 psychology 59, 84, 86, 88, 97, 118–19, 122, 128–32, 209, 210, 237, 275, also see ascription of propositional attitude Putnam, Hilary 57, 139–40, 143, 147, 150, 152, 154, 161–4, 167, 169, 196n11, 200, 216n6, 223, 254, 255, 257, 263, 265, 271, 278nn16–17 Puzo, Mario 119, 122 quantification, quantifier (logic) see logic, quantificational quiddity 50, 53–4, 72, 75, also see essence, re Quine, W.V.O. 14, 15, 93–113, 129, 152, 155, 163, 165, 196n9, 215n2, 283, 286–8, 296, 298, 299n2 quotation 107, 295, also see truth, disquotational account radiation, radioactivity 148 Ramberg, Bjørn 209, 284–94, 296, 297–9 Ramsey sentence 203–4, 216n9 re, res 47, 67, 70, 79, 80, also see essence, quiddity real, reality, vs. existence 13–20; reality, formal vs. objective 67–89 realism also see actualism; independence, of mind; logic, quantificational; mathematics; object; --, causal 148–9, 164–9, 174–95; --, common sense 42, 47, 82, 93, 95, 97, 101, 104, 140, 185, 187, 191, 192, 206, 231–4, 257; --, conceptual 47–8, 58–9; --, direct (perceptual) 21, 33–42, 67, 164; --, Emersonian 224, 227–32, 234–7; --, entity vs. theory 146; --, ethico-ontological 221–37; --, external world see independence; --, formal 51–2; --, immanent 47; --, indirect (perceptual, representational) 21, 67; --, inductive argument for 143; --, intensional 48, 58; --, internal 159, 161, 162, 169; --, metaphysical 94, 95, 96–7, 105, 139–40, 146, 161, 200–2, 206, 207; --, mereological 58, 59; --, minimal 265, 275–6, 279n37; --, minimal scientific 145, 147; --, ontological see independence; --, Platonic 51; --, practical (Vihalemm) 274–6; --, pragmatic 221–99 passim; --, propositional, 46, 54–6, 60, also see propositio in re, realis; --, robust or unregenerate 93, 94, 100, 103–5; --, with lower-case 'r' 57; --, with capital 'R' 57; --, scholastic 47, 61n5, 223, 236, 254, 255, 266; --, scientific see scientific realism; --, semantic 150–5, 160, 169–70; --, about universals 46–54, 58–60, also see generals; realism, scholastic reality, actual 14–20 also see actualism, presentism; --, existence, actuality 17–20; --, formal vs. objective 77–89; --, Inaccessibility of Reality Argument 164; --, possible see object, possible; re; --, 'sideways-on' view of, 214; Reality Principle 117, 118, 120–2, 126 6244-263-BM2.indd 317 1/25/2014 8:36:53 PM 318 Index reason 56–7, 224–5, 233, 264 also see mind; --, vs. understanding 232; reasons, space of 285, 297–9 reciprocal containment (of science and epistemology) 100, 111, cf. 279n38 reductive empiricism see empiricism, reductive reference also see content, intensional; logic, quantificational; semantics; reference, reference relation(s), denoting 14, 17, 18, 70–2, 75, 84, 93, 95, 97–104, 130, 132, 142, 145, 148, 150, 161, 162, 167–8, 173–4, 189–95, 210, 247, 252, 258; --, naturalistic theory of 202–7, 288–99; --, Thesis of Singular Cognitive, 180–8; --, to scientific non-observables 20, 95, 100, 143, 145–8, 155, 167, 189, 261–2, cf. 177 reflexive intentions see intentions, reflexive Reid, Thomas 232 reification, reify 93 relative, vs. absolute or non-relational 14, 16, 39–40, 163, 246–9, 274–5, also see independence; mind-independent relativism 144, 159, 160, 243, 246, 249, 251–2, 257, 262, 263 relativity, conceptual 263, also see conceptual scheme; --, perceptual 142 representationalism 21, 23, 32–4, 47, 67, 76, 88, 131, 200–2, 205–7, 210–11, 214, 226, 269, 272–3, 283, 287, also see content, reference, truth Resnais, Alain 128 Rip Van Winkle (fictional character) 123 Röntgen, Wilhelm 148 Rorty, Richard 161, 200, 208, 209, 212, 226, 255, 256, 263, 283–92, 295–9 Rule 4 (Newton), see scientific method(s) Russell, Bertrand 13–19, 73, 169, 181 S-worlds see storyworlds Scheherazade (fictional character) 116, 118–21 Schelling, F.W.J. 222, 223 scheme vs. content see conceptual scheme Schlick, Moritz 152, 153, 195n3 science also see biology, paleontology, Theory of Mind (ToM); cosmology 15; General Theory of Relativity 174, 179; --, natural 57, 58–9, 86–8, 93–7, 100–6, 110–11, 116–19, 139–215 passim, 221–8, 233, 235, 246, 252, 259–62, 287–9; --, mature 146–7; --, social 93, 273, 275 scientia 18, 46, 50, 59–60, 183, 186, 188 scientific, community 237, 255–6; -- explanation see explanation, scientific; -- method(s) 147, 149, 164, 173–9, 185–7, 193, 221, 223, 226, also see abduction; Rule 4 (Newton) 173–5, 179–80, 185–8, 191, 195n7; -- observation, theoryladen 147, cf. 274; -- realism 47–8, 139–215 passim; -- realism, Convergent 146; -- realism, Critical 159–60, 274; -- realism, minimal 145; -- realism, Newton's 173–93; -- realism, Standard 145–6; -- realism, Strong 146; -- terms, observational vs. theoretical 99, 144, 149, also see empiricism, Constructive; observation; observation, sentence; -- theory, well-established 146 Scotus, John Duns 22–3, 39–41, 47, 51–2, 54, 59–60, 77, 163, 166 Searle, John 81, 132, 202, 205–7 Sellars, Wilfrid 143, 144, 152, 160, 164, 261, 284, 285, 286, 297 semantic(s) 46–59 passim, 139–40, 159, 212, 252, also see bivalence; truth; -- content see content, semantic; -- externalism 272; -- minimalism 210, 212; -- paradox of material implication (Brandom) 191; -- postulates 183; -- vs. pragmatics, syntax (Morris) 194–5; -- realism 150–5, 160, 162–4, 203–5, 210, 212, 252, 292, also see representationalism; truth; --, cognitive 173, 180–8; --, Frege-Russell 6244-263-BM2.indd 318 1/25/2014 8:36:53 PM Index 319 17–20; --, scientific 212; --, truth-conditional 151, 153–4, 296, also see truth; --, verificationism 151; -- anti-realism see anti-realism, semantic; -- realism see realism, semantic; --, possible worlds see logic, possible worlds semiotics 161, 194–5, 252, 254 sentence, observation, see observation, sentence; --, occasion, see occasion sentence; also see content, of sentence; content, of statement Seventh Voyage, The 118, 127–9 shared cultural background 129, 131–2, also see Mutual Shared Belief Sheikh Yerbouti 118, 121, 122 simplex apprehensio 82–3 Sinbad (fictional character) 116, 118 Six Walks in the Fictional Woods 121 skepticism 140–1, 145, 149, 159, 160, 170, 183, 184, 193, 202, 203, 223, 252, 261, 262, 270–1 Slater, Michael 19, 256–9 Smart, J.J.C. 144, 147, 167 Smith, Adam 232 Smith, George 177 social constructivism 144, 148, 159, 164, 168, 252, 255–6, cf. 93, 161, 261, 265 Socrates 18, 25, 49, 51–4, 58 solipsism 159, 168 space of nature see nature, space of space of reasons see reasons, space of "Sphinx, The" (poem, Emerson) 222, 231 spontaneity, vs. receptivity 284, 287 Standard Scientific Realism see scientific realism, Standard states of affairs 19, 48, 58, 132, 160, 184, 252, 275; also see fact(s) Stewart, Dugald 232 Stich, Steven 203, 205 storyworlds (S-worlds) 117–19 Stout, G. F. 141, 284 Strawson, Peter 142, 267 Strong Scientific Realism see scientific realism, Strong structure(s) 13, 16, 58–9, 85, 99, 109, 130–2, 139, 147, 151, 160–5, 179, 180, 189, 207, 214–15, 247–8, 251, 255–6, 264, 272–3 Suarez, Francesco 60n3, 68, 74, 77, 82–3 success argument see miracle argument system of beliefs see belief system Tarski, Alfred 160, 164, 168 Terminator, The 128 theory, scientific, see science Theory of Mind (ToM) 129–31 theory-ladenness of observation see observation, theory-laden things-in-themselves (Ding an sich) 144, 160, 161–2, 211, 214, 255, 259, 265, 267–8, 273, 297 Thompson, E. 214 "Thousand-and-Second Tale of Scheherazade, The" (Poe) 116 Tichy, Ion (fictional character) 127–9 Time Machine 128 "Tlön, Uqbar, Orbis Tertius" (Borges) 119 Tomasello, Michael 129–30 Tractatus logico-philosophicus (Wittgenstein) 14, 15, 278n17 Transcendental Idealism (Kant) see idealism, transcendental transcendental pragmatism see pragmatism, transcendental Transcendentalism, New England 222, 264 translation, indeterminacy of 216n6, 286–90 Travis, Charles 184 truth 23, 47, 51, 188–9, 211, 213, 223, 225, 247, 288, 295–99; --, absolute 231, 248, 257, cf. 253, also see God's-eye view; --, aim of inquiry 144, 226, 229, 231, 235, 252–4; --, approximate see truthlike(ness); --, coherence account of 254; -- conditions 153–4, 164, 286, 292–9; --, consensus account of 254; --, correspondence 56, 74, 82, 139, 144, 146, 152, 159–60, 206, 207, 216n13, 252, 254, 256–7, 274, 296–7; --, criteria vs. analysis (or nature of) 258; --, deflationism about 203, 206, 210; --, disquotational account of 145, 297; --, epistemic account of 169, 258, also see justification, warranted assertability; -- evaluability 182, 184, 186, 214, cf. 162; --, fictional 88–9, 115–33; 6244-263-BM2.indd 319 1/25/2014 8:36:53 PM 320 Index truth in fiction see truth, fictional; -- maker(s) 168, 208, 217n17; --, objective 164, also see independence, of mind; realism; --, ontological 82, 232–3; --, pragmatist account(s) of 164, 251, 258, 297–9; --, propositional 55; --, Tarskian account of 160, 164, 168; --, verificationtranscendent 150–2, 163–4, 212, 286, 292, 295; --, verificationist account of 151, 154, 112n10; --, as warranted assertability 169 truthlike(ness) 160, 162, 167, 262, cf. 147, 149, 231, 235 Turn of the Screw 124 Twardowski, Kasimiercz 73–4, 79, 83–4 Ugly Dave (fictional character) 124–5 underdetermination of theory by observation 149, 174, 287–8 understanding, epistemic conception of 153–4, 170, cf. 292–4; --, (Verstand), vs. reason (Vernunft) 165, 232 use vs. mention 107 van Fraassen, Bastian C. 167, 173, 179, 185, 188–95 verification-transcendent truthconditions see truth conditions, verification-transcendent Vienna Circle, the, 152; also see Logical Positivism Vihalemm, Rein 251, 274–6 Walsh, Peter (fictional character) 115 Walton, Kendall 120–1 warranted assertability, see justification, warranted assertability; truth, as warranted assertability Watson, John H. (fictional character) 127 'we' intentionality see attention, joint; intentionality Wee, Cecilia 68, 81–2 Welles, Orson 118 Wells, H. G. (Herbert George) 128 Wells, Norman 68, 78, 82 Westphal, Kenneth R. 251, 266, 270–6, 278n23 Williams, D. C. 143 Wilson, Catherine 67–8 Wilson, C.T.R. 167 Wilson, Margaret 79–82 Wittgenstein, Ludwig 14, 15, 129, 143, 200, 234, 270–3, 276 Wolff, Michael 183 Woolgar, Steven 144, 164 worlds, possible see logic, possible world; object, possible worlds, story (S-worlds) see storyworlds Wright, Crispin 152, 154 Wyclif, John 49–50, 54–6, 59–60 Zappa, Frank 118, 121, 122 6244-263-BM2.indd 320 1/25/2014 8:36:53 PM 24 Philosophical Delusion and its Therapy Outline of a Philosophical Revolution Eugen Fischer 25 Epistemology and the Regress Problem Scott F. Aikin 26 Civil Society in Liberal Democracy Mark Jensen 27 The Politics of Logic Badiou, Wittgenstein, and the Consequences of Formalism Paul M. Livingston 28 Pluralism and Liberal Politics Robert B. 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OR I G I N A L A R T I C L E Slurs and register: A case study in meaning pluralism Justina Diaz-Legaspe | Chang Liu | Robert J. Stainton Department of Philosophy, The University of Western Ontario, London, Ontario, Canada Correspondence Robert J. Stainton, Department of Philosophy, The University of Western Ontario, 1151 Richmond Street, London, ON, Canada N6A 5B8 Email: [email protected] Funding information Research School of Social Sciences of the Australian National University; Social Sciences and Humanities Research Council of Canada Abstract Most theories of slurs fall into two families: Those which understand slurring terms to involve special descriptive/informational content (however conveyed), and those which understand them to encode special emotive/expressive content. Both offer essential insights, but part of what sets slurs apart is use-theoretic content. Slurring words belong at the intersection of categories in a sociolinguistic register taxonomy, one that usually includes [+slang, +vulgar] and always includes [-polite, +derogatory]. What distinguishes "Chinese" from "chink," for example, is neither a peculiar sort of descriptive nor emotional content, but the fact that "chink" is lexically marked as belonging to different registers. Moreover, such facts contribute to slurring being ethically unacceptable. KEYWORD S meaning pluralism, pragmatics, semantics, slurs, sociolinguistic register, use-theoretic meanings Words, though abstract, are human artifacts that in addition to phonological, grammatical, and semantic structure have all sorts of other properties. They have etymology, history, regional or foreign provenance, field restrictions (anatomical, botanical, etc.), currency (obsolescence, rarity, etc.), tone (archaic, humorous, etc.), discourse level (slang, formal, etc.), collocational associations (there are other words whose company they typically keep), proscriptions (word taboos), offensiveness levels, degrees of insultingness, and unsavoury associations. Some of them are no more neutral and inoffensive than a concealed switchblade. That is the key to the power of both slurs and curses (Pullum, 2018, §8). Received: 21 March 2018 Revised: 24 July 2018 Accepted: 18 August 2018 DOI: 10.1111/mila.12236 Mind Lang. 2019;1–27. wileyonlinelibrary.com/journal/mila © 2019 John Wiley & Sons Ltd 1 1 | INTRODUCTION: AIMS AND A "FIRST PASS" We have twin aims. A modest one is to introduce a promising account of the meaning of slurs.1 It is promising both in terms of descriptive adequacy and insight into why slurring is ethically wrong. All hands agree that there is typically something ethically unacceptable about using slurs; we present a novel account of the source. More ambitiously, we aim to illustrate the attractions of pluralism about linguistic meaning. The point is not to offer an exhaustive defense. In particular, we will not undertake a refutation of other approaches to slurs: Insofar as we discuss alternatives, it is to give credit where due, and to contrast our position. Even the "introducing" element carries a caveat: Being pluralistic, our view borrows from, or at least coincides with, aspects of existing positions. So far as we know, however, the resulting synthesis is original. By way of a "first pass," we begin with examples of slurring words: 1. (a) chink (b) dyke (c) kike (d) kraut (e) proddie (f) retard (g) spic (h) wop There are two dominant approaches to such words. One takes them to convey, by whatever means, some linguistically and ethically puzzling message. The other takes them to encode not propositional content, but something emotive.2 Both have something right. There is a place in semantics and philosophy of language generally for meanings understood as descriptive/informational-whether about worldly things and their properties, social events, abstract mathematical facts, or what have you. There is also a place for meanings understood as ways to express and inculcate feelings, pro-attitudes, and other "non-cognitive" mental states. But, both in general and specifically with respect to the meaning of slurs, a third element is required. A far-from-dominant alternative, put very roughly for now, is that (part of) giving the meaning of an expression or construction is (sometimes) to specify the socio-cultural action one performs with it. Linguisticophilosophical ideology aside, such use-theoretic meaning would be recognized in (2a–c) and (3a–c): 2. (a) Congrats! (b) Hello! (c) Welcome! 1 It is an unfortunate feature of our project that we must mention a host of rude and offensive words, including ethnic, racial and religious slurs. Where possible, we have chosen the less offensive ones, but even these are jarring expressions that many reasonable people would prefer, no matter the context, not to encounter. We apologize in advance for forcing readers to do so. 2 The literature on slurs is vast. Painting with broad strokes, a representative sampling of the first family includes Hom (2008, 2010, 2012), Hom and May (2013, 2018), and Williamson (2009); the second includes Jeshion (2013a), Saka (2007), Schlenker (2007) and Whiting (2013). 2 DIAZ-LEGASPE ET AL. 3. (a) Gesundheit! (b) Shush! (c) There, there! We maintain that it also shows up in sociolinguistic registers, illustrated by the differences among the terms in (4): 4. (a) poop (b) shit (c) feces (d) excrement Note also in (5) three variants on the second person singular pronoun in Spanish, running from the most respect-oriented to the most informal3: 5. (a) usted (b) tú (c) vos We will apply register differences of the sort found in (4) and (5) to slurring words. In particular, (1a–h) are usually lexically marked, in the public language, as slang and vulgar, and always marked as impolite and derogatory-so that, for example, what distinguishes "chink" from "Chinese" is this, rather than a difference in reference or sense, or in what emotions a user of "chink" is expressing. 2 | PART ONE: ELEMENTS OF THE VIEW 2.1 | Slurs What exactly do we mean by "slurs"? We are, as per recent usage in the literature, using it to stand for natural language expressions. In common parlance, "He slurred me" is sometimes used in a way comparable to "He slandered me." Our use departs doubly from this colloquial one. For us, a slur will be a word or phrase which typically stands for what is an out-group vis-à-vis the larger, dominant community, which is marginalized and oppressed. That "typically" is justified by seeming exceptions. Exceptions to the marginalized-condition include Spanish "gringo" and "yanqui" for Americans, and English "boche," "fritz," "hun," and "kraut" for Germans. These targets are, if anything, would-be oppressors. (One might think of these as "retaliation slurs," a derivative sub-kind.) An interesting exception to the stands-for condition, drawn to our attention by Michael Martin, are words treated as referring, but where there is no worldly referent. Martin's example is "pleb," which given our 21st-century class system may well 3 Pronouns of this sort are, of course, familiar from numerous languages: French's "tu"/"vous," German's "du'/"Sie," Italian's "tu"/"lei," Russian's "ty"/ "vy." As a by-the-way, register is at least reminiscent of Frege's (1892, p. 61) and (1918, p. 295) notion of coloring/shading/tone, where expressions such as "dog"/"cur" or "horse"/"steed" may fail of intuitive synonymy without ceasing to be referentially equivalent, and even while expressing the same sense. DIAZ-LEGASPE ET AL. 3 lack a determinate referent but is definitely a slur in the United Kingdom; another might be "crone" for witches. What is more, if radical anti-realists about race are correct-that is, racial categories are pernicious myths with no worldly correlates-then "chink" and such are exceptions too. These pejorative epithets have notorious facets. We begin with ethical/social ones. As Hom (2008, p. 427), Anderson and Lepore (2013a, 2013b) and others have stressed, the use of slurring words is taboo. More than "to be avoided in polite company," the use of certain ones is full-on prohibited (Hom, 2010, p. 165). Another hallmark is that the resulting derogation is not merely towards the particular person or persons that the speaker may have in mind. It extends to the entire group which the slur picks out. Thus, if Donnie refers to a woman as "That spic Justina," she is demeaned-but so are Hispanics generally. It is also difficult for the linguistic or worldly context to erase the derogation/offense. In particular, a speaker's lack of offensive intent is not sufficient. This is often called derogatory autonomy.4 Nor is derogation cancelled by embedding: For example, in the antecedent of a conditional or under negation. This related feature has been termed non-displaceability.5 Someone who says, for example, "If there are chinks in Toronto my brother will be upset" or "There aren't any chinks in Toronto" is still slurring the Chinese. The same holds for embedding under a propositional attitude or discourse-reporting verb: "My brother thinks that there are chinks in Toronto" or "My brother asked whether there are chinks in Toronto" are both as offensive as (6): 6. There are chinks in Toronto As a result of being taboo, group-insulting, and hard-to-cancel, slurring words are, as Hornsby (2001, p. 129) insightfully noted very early on, "useless" for those who wish to be enlightened, fairminded, and so forth. Some slurs are even useless when reporting speech directly: Pronouncing them can give offense. "Cunt" as a term for women in English-speaking North America, and "nigger" worldwide, are notorious cases in point. (In light of this, where appropriate we will refer to these hereafter as "the C-word" and "the N-word.") As Hom rightly notes: Uses of [slurring and other offensive] epithets are subject to strict social constraints, if not outright forbidden. There seem to be very rare instances for the appropriate use of epithets (for example, under explicit quotation in the courtroom, in a discussion about language and the media, appropriated uses among members of the target class). Even when a speaker intends a benign use, the result is often an unintended violation of strict speech codes, especially in cases of public speech. For many, the taboo surrounding epithets is not limited to their direct use, but covers their occurrence within quotation, fiction, intensional contexts, questions, negations, conditional antecedents, and even extends to phonologically similar, but semantically distinct, expressions (Hom, 2008, p. 427). All of this granted, and as Hom suggests, the offense can sometimes be mitigated or even erased. Consider a young child or recent immigrant who uses "retard" without having mastered its standing meaning in North American English. Add that the speaker would absolutely have avoided that term 4 This is the term used in Hom (2008, p. 426) and Jeshion (2013a, p. 233). Bolinger (2015, p. 1) calls it "offensive autonomy." 5 Hom (2010, p. 168), Potts (2007, p. 166) and Schlenker (2007, p. 238) use this term, Croom (2011, p. 345) labels the phenomenon "scopelessness," while Bolinger (2015, p. 1) writes of "embedding failure." 4 DIAZ-LEGASPE ET AL. if they had known it was pejorative, and that their ignorance is non-negligent. Though it is not wholly erased, it is clear that the violation is reduced. (Unlike its use by an adult American politician, who even if by some miracle did not know that "retard" was insulting, is expected to.) A second kind of offense-cancellation is much more notorious. When a group collectively chooses to apply the word to themselves, this may fully overcome the "uselessness." Non-derogatory re-appropriated uses of the N-word by members of the black community directed at fellow members are commonplace, for example. Addressing another important case, Mark Richard notes: Slurs can be used without displaying contempt or causing hurt. This happens, for example, when a slur is appropriated by its targets: it is an insult to no one, save perhaps the homophobe, for gay people to call themselves queer (Richard, 2008, p. 12).6 Turning to notorious linguistic facets of slurs, they almost always have non-derogatory correlates which (seemingly) stand for the very same group. As Hornsby (2001) put it, "for each such word, there is, or at least perfectly well could be, another that applies to the same people but whose use does not convey these things-there is, that is, a neutral counterpart" (p. 129; see also Whiting, 2013, p. 364).7 As a result, a slurring word and its neutral counterpart at least seem to contribute the same message-type content to complete sentences: For example, though the point has been contested, it is initially plausible that any possible world at which "There are Chinese in Toronto" holds is ipso facto a world at which (6) holds and vice versa. Unquestionably, it is difficult to paraphrase any alleged difference in terms of description/information: If real at all, it is descriptively ineffable. Thus, as Schlenker (2007, p. 239) points out, speakers are never fully satisfied when one paraphrases "slurring contents" with descriptive terminology. For instance, (7) seems ill-suited to translate (6): 7. There are Chinese in Toronto and I despise the Chinese Nonetheless, someone who took "Chinese" and "chink" to be interchangeable content-wise (e.g., when translating English or writing a children's book) would make a serious mistake. Indeed, what makes the topic of slurs of such great interest to semanticists and philosophers of language is this pre-theoretical difference in meaning. Related to the (seeming) non-informational difference is a puzzle about non-redundancy (Potts, 2007, p. 166, and Schlenker, 2007, p. 239 call it "repeatability.") If (6) were synonymous with (7), then "There are chinks in Toronto and I despise the Chinese" should be semantically anomalous. Its content would include that I despise the Chinese twice over: Its first conjunct would contribute it as an analytic entailment, and then its second conjunct would repeat this overtly. But, and this is the datum, we do not hear "There are chinks in Toronto and I despise the Chinese" as bizarrely 6 For additional discussion see Hom (2008, p. 428), Jeshion (2013a, p. 233) and Kennedy (2002). An important distinction between normalizing and out-group slurs is addressed by Diaz-Legaspe (2018). 7 Consider, by way of contrast, "pedophile,", "torturer," and "traitor." These are words and phrases which we apply to shunned out-groups, and which are likely to be insulting. Still, they are not slurring expressions in our sense because there really is something blameworthy about belonging to such groups-whereas, the prejudices of bigots notwithstanding, there is absolutely nothing despicable about being, say, Chinese, Hispanic, Jewish or Protestant. (See also Jeshion, 2013b, p. 325ff.) Potential exceptions to the neutral-counterpart generalization arise with blameless groups who are nonetheless so widely and deeply reviled that any word for them quickly develops negative connotations. For discussion of this "euphemism treadmill," as he terms it, see Pinker (1994a); it builds on ideas from Pinker (1994b). Both observations will re-emerge below. DIAZ-LEGASPE ET AL. 5 redundant. (The phenomenon arises, of course, whatever one takes the "extra" propositional content of "chink" to be.) A final notorious facet is variation. English, for example, has a surfeit of slurring words for ethnic and racial groups; Spanish has comparatively few of those, but contains numerous ones for what are considered sexually deviant groups; and contemporary Mandarin contains slurs specific to the social classes of Communist China, and for the general category of non-Chinese foreigner. There is also variation in degree of offensiveness: A single tongue can contain numerous slurs for the same targeted group, but which are more or less offensive. As Jeshion (2013a) puts the point, though without endorsing it, slurs: [A]re said to possess derogatory variation, to vary in their degree of offensiveness. The N-word is said to be more offensive than 'chink', as well as 'spook' and 'jigaboo', terms used for the same socially relevant group. 'Kike', 'yid', and 'hymie' are said to differ in their offensive intensity (Jeshion, 2013a, p. 233).8 2.2 | Use-theoretic meanings Continuing to explicate the key elements of our view, we turn to meaning pluralism, understood as including use-theoretic meanings. Pursuing the via negativa, use-theoretic meaning is to be contrasted with meaning understood as wholly a matter of externalizing mental states. Linguistic interaction is not always and essentially a matter of speaker S attempting to voice something psychological, typically aiming to modify inner mental states of some interlocutor. It leaves the error intact to allow that what S expresses may be neither a belief nor a bit of knowledge, but rather a desire, a feeling, and so forth. What is wrongheaded is deeper, according to the use theorist: We need to posit linguistic contents which simply are not in that mentalistic line of work. As Austin (1962) stressed, at least sometimes the speaker S simply performs a socio-cultural action by producing words, often in collaboration with a hearer H. As examples of not-externalizing-but-acting, recall (2) and (3): 2. (a) Congrats! (b) Hello! (c) Welcome! 3. (a) Gesundheit! (b) Shush! (c) There, there! "Hello!" is not a way of making public a belief, nor is "Gesundheit!" Their contents not being truthevaluable, they are not the sort of things which can be believed. Neither do they correspond to some peculiar, highly abstract and addressee-directed update rule. Instead, maintains the use theorist, to 8 Other authors who have discussed such within-a-language variation are Anderson and Lepore (2013a, p. 350), Bolinger (2015, p. 1) and Hom (2008, p. 426). Note a connection with propositional ineffability: The theorist who would treat slurring words in terms of sense must find different ones for "kike," "yid," and "hymie." 6 DIAZ-LEGASPE ET AL. give the meaning of "Hello" is simply to say: It is the English expression used for greeting. And the English (3a), adapted long ago from German, has its meaning captured by: It is the expression to be used when someone else sneezes. (Some tongue-in-cheek evidence: If a non-native speaker asked what "Gesundheit" meant in English, one should respond with this very simple rule, not with a lesson in dynamic logic.) Undoubtedly, an utterance of (2b)/(3a) can be motivated by beliefs, or create new ones in the hearer. The bone of contention is whether such potential belief-involving causes/effects are built into, let alone exhaust, their linguistic meaning. Turning to the emotive, "Congrats!" is not a way of placing one's positive emotions or feelings on display, nor is "There, there!" a way of voicing sympathetic sadness. Rather, these are linguistic tools for performing the actions of congratulating and comforting respectively. Again, undoubtedly there are emotions which cause such speech, and there will frequently be an emotional effect on the addressee: Typically, that is why one congratulates and comforts. Nonetheless, and tongue-in-cheek once more, to give the meaning of (2a)/(3c) does not require a lesson in comparative moral psychology. The connection between content and these mental happenings is far less direct. (Compare: One may say "It's raining yet again" because of feeling glum, but the sentence does not encode such reasons for speaking.) Still pursuing the via negativa, though there is an insight behind the slogan "Meaning is use," we do not endorse a Wittgenstein-style hostility to traditional meaning theories. We are not urging: "Stop studying so-called meaning systematically and instead describe in-context use." Nor do we intend, as per some use-theoretic traditions, to exclude the mental from the theory of meaning. Relatedly, our position is not that linguistic meaning is exhausted by the use-theoretic, at least not in the general case. (Examples (2a–c) and (3a–c) are outliers in this respect.) We embrace complex compositional rules for deriving the content of slur-containing sentences. We equally do not interpret the slogan such that the theorist should provide an exhaustive catalogue of how an expression has in fact been used, and maybe even of all its potential future uses. That is obviously intractable. Besides, actual usage being an interaction effect, and words being deployed in heterodox ways, an exhaustive catalogue would not give proper insight into literal meaning. If giving meaning involves "describing how a linguistic expression is used," does that amount to capturing brute statistical patterns? Again, no. Though there is a crucial evidential relationship between projectible corpus patterns and use-theoretic meaning, there is additionally a teleological, normative element to the latter. Turning to what we do have in mind, what is sought is a description of the function or job of the word, phrase, sentence or grammatical construction-that is, how the term is supposed to be used according to the shared linguistic conventions (Stainton, 2016). A familiar tool metaphor is apt. Just as a hammer is a tool for driving nails into solid surfaces, "Hello!" is a conventional job-specific "vocal implement for greeting." (This is, partially, what underwrites the statistical patterns in actual talk.) This notion of conventional function permits non-literal utterances even in the case of usetheoretic contents. They depart from how the expressions is to be used. A hammer can be used in ways which flout its function: Throwing it at an intruder, framing it and hanging it on a wall as "found art." So too a word. For instance, though the use theorist will cash out the meaning of "Gesundheit" in terms of the action that speakers are supposed to carry out, there is room for convention-flouting instances: For example, as a one-off to frighten a dog, or to poke fun at a colleague's absurd statement by pretending that it was merely a sudden involuntary expulsion of air. We now contrast two sub-varieties of use-theoretic meaning. The more familiar one involves pairing a formative with a type of illocutionary action: Giving a rule regarding (i) which type of action is performed, and (ii) what felicity-type conditions must be in place. Examples we are inclined DIAZ-LEGASPE ET AL. 7 to treat this way include (2a–c), Austin-style explicit performative sentences such as "I now pronounce you husband and wife" and "I divorce you, I divorce you, I divorce you," and the sentential moods (e.g., declarative, imperative, interrogative).9 The other sub-variety requires specifying only felicity-type conditions-there is no illocutionary act in play. Examples we would tentatively place in this category include (3a–c) and also "Bye!," "Go Leafs go!" and "Fuck off!" That little word "hereby" provides a rough-and-ready test for distinguishing. Where there is an illocutionary act, ceteris paribus it may be executed using an explicit performative containing "hereby": One can say, for example, "I hereby congratulate you," "I hereby greet you," and "I hereby welcome you to my home." Where there is no such full-blown speech act, there can of course be no explicit "hereby"sentence for doing "it." Thus, for example, there is no straightforward explicit performative corresponding to "Fuck off!," "Gesundheit!," or "Shush!" Importantly, in both sub-varieties it remains appropriate to speak of "what the expression is conventionally used for": Use-theoretic meaning appears in each. Further dividing the taxonomy, consider two sub-classes within the non-illocutionary family. There are particular ad hoc expressions. Such are the cases we have encountered so far: Specialpurpose terms like "Bye!," "Fuck off!," "Gesundheit!" and "There, there!." But there exist classes of words such that they are only appropriate in conversational situations so-and-so. This takes us to sociolinguistic register. 2.3 | Register We will explain register in three steps: Parallels with metadata; a tentative feature-geometry for registered lexical items; and three clarifications. The term "metadata," made familiar in linguistics by Geoffrey Nunberg and Geoffrey Pullum, merits elucidation. The shortest possible definition is "data about data." This is, of course, a wide category. Our specific interest is metadata regarding content-related aspects of words-not specifying a definition but listing facts like collocation proclivities and origin. Granted, that it co-occurs regularly with "effect" and "in my stomach" is orthogonal to the necessary and sufficient conditions for being a satisfier of "butterfly." Granted, metadata about "emesis" to the effect that it derives from Greek, and that "blackguard" is antiquated and nearly obsolete, is not part of their definitions. And yet, speakers may readily take advantage of such knowledge to achieve meaning-effects: For example, reporting a pompous and pedantic colleague's claim, but mockingly substituting for his flatfooted wording antiquated, low-frequency terms of Greek origin. Though the technical term "register" is equivocal, our usage ties it to metadata of the latter sort.10 In developing our favored notion, we borrow freely from sociolinguistics and Functional Linguistics, especially in the tradition of Michael Halliday. (See especially Gregory, 1967, 2001; Halliday, 1973; and Halliday & Matthiessen, 2004. Our position is also influenced by certain insights in Caldwell, 2018.) The fundamental idea is: To locate a word in a register category is to specify the contexts for which it is socio-culturally appropriate. This involves contrasting it with expressions in a system of 9 For an illocutionary treatment of the meaning of the (matrix) interrogative mood-as opposed to a descriptive/informational treatment in terms of sets of propositions-see Stainton (1999). The key idea is that this mood encodes, in every sentence in the class, to-be-used-for-asking. 10 Mark Jary reminded us of another usage of "register" in sociolinguistics which may latterly prove useful in understanding slurs. "Register," so used, is a word for micro-dialects, also called "varieties": For example, the grammatical constructions and lexis specific to English-speaking police officers, sports broadcasters, video gamers, and so on (cf. Gregory, 1967, 2001.) Jary's promising idea, reminiscent of Nunberg (2018), is that slurring words are marked as belonging to a micro-dialect of (something like) a bigoted community of speakers. 8 DIAZ-LEGASPE ET AL. terms which may co-designate but are suitable for other discourse genres. Because register-features are shared by classes of expressions, we indicate them using a feature geometry. Comparable to a phoneme, the register of a particular word is then a matter of the intersection of its register-features. Recall (4a), "poop." Just as "the p-sound" is marked with the features [+anterior, +coronal, +labial] among others, (4a) is similarly marked with [+child-oriented, -formal, -vulgar]. Such talk of "features" can mislead. It suggests binary polarity. That is insufficient for a descriptively adequate theory of register. For instance, one might wish to contrast "poop" as [+child-oriented] to "crap" as [-child-oriented] versus "shit" as [--child-oriented]. Again, Spanish "usted" is adequate for contexts that require respectful address, while "tú" is to be used when conversing with social equals or those lower in the hierarchy. The latter pronoun would thus be marked as [-formal, -respectful] while "usted" would be [+formal, +respectful]. "Vos," however, is even less formal and deferential-this too is a matter of register. What is more, many terms are unmarked with respect to, say, [polite]. Take "elm" or "Montevideo": Instead of lying on either side of a binary divide, they are neutral, being neither [-polite] nor [+polite]. Despite the affinities between metadata generally and register, two contrasts merit underscoring. Register, we hold, always falls within autonomous knowledge of language. In contrast, that "like" is now a very high-frequency word among young people is metadata not of that sort: That is, not part of "linguistic competence." And register-features, in our sense, mark something normative. They specify how the term is appropriately used-metaphorically, what discursive situations a word is "designed for." As a final step in explicating sociolinguistic register, three clarifications are in order about "appropriateness to a discourse context." Our notion of conversational context differs from a narrow notion familiar from formal semantics. Specifically, Kaplan's (1989) "context of utterance" consists in a limited n-tuple of comparatively objective parameters: Speaker, addressee, place and time of utterance, and so forth. Our notion tracks "context" in a much wider sense. In the broadly Hallidayan tradition, it would include: • Topic area (called "field" in Systemic Functional circles): For example, technical, scientific, medical, nautical, legal, military. • Communicative medium ("mode"): For example, journal article, e-mail, text message, classroom lecture, barroom chat. • Participants' social relations ("tenor"): For example, degree of formality, social hierarchy, age or sophistication of the interlocutor. Regarding "appropriateness," a first point is this: That "usted" is appropriate for respect-worthy contexts does not mean that it must occur there and only there. A speaker can choose to utter "usted" in socially impermissible contexts to achieve some communicative effect: For example, in a heated argument with an old friend who would ordinarily be called "tú." (In this regard, register for linguistic expressions may be compared to dress codes. Socio-cultural convention dictates that it is inappropriate to attend a university lecture in a bathing suit, or sporting jeans at an Embassy dinner party. And yet, a protester could appear dressed in those ways to make a point.) Second, that a term is "appropriate" in the sense at play is somewhat orthogonal to normative ethics. Anticipating, talk of "when a slur should be used" seems puzzling, and rightly so. It may appear that, re-appropriation aside, there is no social context in which using one is appropriate-just as there is no social context in which it is appropriate to be cruel or unjust. We take the point. But, in the first instance, register pertains to what is culturally and discursively apposite. To give an analogy from DIAZ-LEGASPE ET AL. 9 descriptive ethics: An egalitarian who insists that it is morally wrong to build social hierarchy into a language may nonetheless observe, about a local tongue, that some of its terms are appropriately used by those born into the aristocracy, others by those who have merely become rich, and so forth. In the same vein, one may contrast (i) describing, qua empirical sociolinguist, what a group in fact tolerates/permits when it comes to slurring terms with (ii) prescribing, qua normative ethicist, what they ought to. We end this overview of register by mentioning a significant limitation. There is at present no agreed upon definition of "register" in the sociolinguistics literature. Indeed, as hinted above, there will undoubtedly be readers who find our usage unfamiliar. Even among those who understand "register" roughly as we do, there is disagreement about what kind of metadata-type information should be included: For example, do etymology, collocation patterns and frequency-of-use belong? This relates to the most significant theoretical challenge. A question which confronts semanticists of all stripes concerns what belongs within the language proper. Ultimately, register theorists too need to address this vexing issue of the boundary between general-purpose information versus semantics-here as applied to which facts about usagepatterns belong in a lexical entry. To note these limitations is not to encourage despair. For instance, while there is no agreement on the exact and exhaustive list of register-features, certain ones appear repeatedly. They include: [+child-oriented]: "ca ca", "poop", "bunny", "bum bum", "belly button" [+formal]: "excrement", "vomit", "posterior" [+medical/clinical]: "feces", "umbilicus", "emesis", "anus" [+polite]: "please", "pardon me" [+scientific/technical]: "leporidae", "vomitus" [+slang]: "take a dump", "lit", "two-four", "heave", "butt" [+respectful]: "your honour", "sir/madam" [+vulgar]: "shit", "ass", "puke" Regarding placing something within/outside language proper, there are familiar clues: How systematic are the rules-of-use, how do they interact with grammar, and so forth? The Spanish formality system, for instance, clearly is not mere "worldly knowledge." Which of (5a–c) the speaker chooses fixes which second-person conjugation is required: The present tense of "fumar" ("to smoke") is "Usted fuma" in formal, "Tú fumas" in informal and "Vos fumás" in highly informal. And, as Ethan Nowak noted in conversation, Korean verb declension reflects politeness level. To offer a simplified example, the verb "keuda" ("to grow") in present-tensedeclarative varies among: Informal low ("keo"), informal high ("keoyo"), formal low ("keunda") and formal high ("keumnida"). Here, register is patently grammaticalized. In short, the field remains underdeveloped but promising. This limitation matters because a thoroughgoing defense of our register-based approach will eventually require an empirically and theoretically motivated taxonomy. The list above does not come close. We can only express the reasonable hope that once register (again, in the specific sense presented above) becomes of central interest in philosophy of language and semantics/pragmatics, future work will provide the requisite theoretical framework. On the other hand, introducing the overarching shape of our meaning-pluralist approach thankfully requires much less. 10 DIAZ-LEGASPE ET AL. 2.4 | A meaning pluralist, register-based theory of slurs: Beyond the "first pass" Few readers will be familiar with all elements of our synthesis. So, a brief recap is in order. As a "first pass" we said that slurring words are usually lexically marked as slang and vulgar, and are always marked as impolite and derogatory. Drawing (sometimes loosely) on existing research programs, we then spelled out three key notions: Slurs as pejorative terms for an out-group; use-theoretic meanings, and in particular that sub-kind which involves only felicity-type conditions; and sociolinguistic register. Using Table 1, we can summarize by saying that slurs belong in the bottom-right quadrant. In this "second pass," we can employ the feature geometry and simply say: Slurring words are always marked as [-polite] and typically as [+slang, +vulgar]. Indeed, some are [--polite, ++ vulgar]. Expanding upon each, to mark slurring words as [-polite] is to put them in the same use-theoretic class as "No probs!" and "Bye!" as opposed to "Thank you" and "Farewell." To mark them as typically [+slang] is to class them as falling within the popular argot of a culture, along the same lines as "puke" and "to take a dump" as opposed to "vomit" and "to defecate." To say that slurring words are lexically marked with [+vulgar] is to "fit them" to the same sort of situations where "fuck," "shit" and "twat" are discursively appropriate. There is much more to add about these three, but the lack of a definitive theory of register-type content limits what we can say. Besides, though more will be added about them, we must focus on [+derogatory]. It is, after all, the most crucial and most novel. Obviously, words marked [+derogatory] are insulting. But this observation threatens to re-label things. Somewhat more helpful is to say that [+derogatory] terms are always directed at a target (whether real or merely represented as such): They are not pure expletives. And a certain special kind of "put down" is included in their meaning. Our crucial clue about that comes from "derogatory speakings" carried off by words which are not themselves marked [+derogatory]. Consider, for instance, what occurs when a person uses any word in (8) in a derogatory way: 8. (a) bureaucrat (b) capitalist (c) feminist (d) intellectual (e) liberal (f) relativist TABLE 1 Varieties of use-theoretic meaning Illocutionary act Only felicity-type appropriateness conditions Particular words/constructions (i) (2a–c): "Congrats," "Hello," "Welcome" (ii) Explicit performative verbs (i) (3a–c): "Gesundheit," "Shush," "There, there" (ii) "Bye," "Ciao," "Go Leafs Go," "Fuck off" Whole classes of words (i) Sentential moods: Declarative, interrogative, imperative (i) (4a–d): "poop," "shit," "feces," "excrement" (ii) (5a–c): "usted," "tú," "vos" DIAZ-LEGASPE ET AL. 11 In such uses the speaker typically feels a strong negative attitude towards the target and wishes to demean it. In any case, she has no qualms about doing so. In light of this, we infer: Expressions marked as [+derogatory] are appropriate to conversational contexts where the speaker either wishes to insult a target group or is indifferent about doing so. That is the nub of things. We can spell out [+derogatory] further by considering its relationship to the other features of slurring words, thereby also expanding upon their respective meanings. [+derogatory] does not entail [+slang]. Some words began as slang, latterly entered the "standard tongue," yet remain derogatory: "Coolie" and "hillbilly" are examples. Sometimes a word belongs to the standard dialect but becomes derogatory: "Colored" and "negro" are cases in point. "Gypsy," though not slang, is definitely a slur by our lights, because there now exists a neutral counterpart, namely "Roma." It is even more obvious that [+slang] does not entail [+derogatory]. Patently there are [+vulgar] expressions which are not [+derogatory]: For example, "shit" and "Jesus motherfucking Christ." (The latter exemplifies [++vulgar].) There also seem to be [+derogatory] words which are not [+vulgar]. To take an example we owe to a conversation with Glenda Satne, "fairy" might well be used by an elderly person who eschews coarse words but is homophobic.11 Similarly for "papist" as used by "refined" anti-Catholics. The slangier "oreo" and "FOB" are also not [+vulgar], but they are [+derogatory] epithets: Respectively, for certain black people ("black on the outside, white on the inside") and for unacculturated recent immigrants. The feature [-polite] can seem otiose in our account, because [+vulgar] surely entails [-polite]. What is more, how could one be wholly polite while wishing to insult, or being indifferent about doing so? Thus a [+derogatory] word will also, it seems, automatically be [-polite]. Ultimately, whether [-polite] should be included can only be settled in light of a developed theory of register. Regardless, we include it pro tem: As previous examples attest, [-polite] does not entail either [+vulgar] or [+derogatory], so it clearly is a different notion. More importantly, a slurring word being impolite is one thing, it being marked [-polite] in the lexicon is quite another. Our tentative-butpromising conjecture is that the latter obtains. 3 | PART TWO: MEETING THE TWIN AIMS 3.1 | Descriptively and ethically promising We now argue both that our novel synthesis genuinely affords a promising account of slurs, and that it thereby provides a case study in the value of meaning pluralism. We begin with ethical/social facets. That slur use is taboo follows directly from the content of their register-features. Their use will frequently be taboo simply because slurs tend to be [+slang, +vulgar] and are always [-polite]. More importantly, to be marked [+derogatory] is, we inferred from disparaging uses of (8a–f), to be a word appropriate for speakers who care not about insulting a target group. A slurring word will, for the same reason, be not just taboo but full-on prohibited when it exhibits some combination of [++derogatory], [--polite] and [++vulgar]. They offend all, not just the specific persons being spoken about, because the very existence of a task-specific word for speakers willing to insult it is per se offensive to the group. The offensiveness is hard to cancel 11 Kennedy (2002) observes that the N-word was avoided, even by some virulent racists, not because it was derogatory and offensive, but because it was an uncouth word: "Nigger has been a familiar part of the vocabulary of whites high and low. It has often been the calling card of so-called white trash ... Partly to distance themselves from this ilk, some whites of higher standing have aggressively forsworn the use of nigger. Such was the case, for example, with senators Strom Thurmond and Richard Russell, both white supremacists that never used the N-word" (pp. 7–8). 12 DIAZ-LEGASPE ET AL. because it is, for us, built into the term's standing meaning: Whatever the speaker may have in mind, the type "kike" just is [+derogatory]. (A parallel is an L2 speaker who wishes to describe delicately a scene in a romance novel but utters "They fucked": Her intentions notwithstanding, she has used a vulgar word.) As for lack of cancellation when embedded, just as "to take a dump" and "twat" do not cease to be vulgar slang under negation or in the antecedent of a conditional, slurring words do not lose their register-features therein either. As a result, slurring words are "useless" for enlightened and kind-hearted people. To reiterate, the reason is not that a non-bigot would somehow, despite herself, express hatred and contempt for Hispanics by deploying "spic"; nor is it that she would, again despite herself, state or otherwise meanNN that Hispanics are despicable, should be discriminated against, and so forth. Rather, good people will want to avoid bigot-ready words for innocent out-groups. The proper analogies are: "Fuck" is useless for the prude; and the informal-low "keo" in Korean is useless for one who insists on exaggerated respectfulness. Why can mitigation nonetheless occur? We distinguished how a word is to-be-used from the sundry ways it has and can be used in actual talk. Though use-theoretic, the former is normative/teleological, and thus admits of violations. (Recall using "Gesundheit!" to scare off a dog.) Given this, a child or non-native speaker merit correction when they utter an English word with a "nasty function"; their culpability is reduced because they do so in ignorance of how it is to-beused, and without negligence. The treatment of re-appropriation is much more fraught. Our very tentative suggestion is that it involves an unfamiliar sort of non-literal use. As with the L2 speaker's use of "kike," the word which re-appropriators utter genuinely remains [+derogatory] in the wider standard tongue; but, at least initially, in-group speakers are deploying it in a non-derogatory way.12 To introduce another analogy, two friends of longstanding might develop the humorous habit of greeting each other with "Fuck you dude!" The point of their inside joke would be lost if "Fuck you" had ceased to be [+vulgar] in English. So, in some admittedly puzzling sense, the friends are flouting its meaning rather than altering it. Similarly, we conjecture, for the re-appropriation of the N-word, "queer," and so forth. Moving beyond the ethical/social, the notorious linguistic facets of slurs are also captured by our register-based view. There will almost always be a neutral counterpart because slurring words refer to groups. Hence, modulo some potential outlier cases, there can be a term which shares that referent but lacks its morally problematic register-features. This shared referential component will fix the truth-conditions of complete sentences containing slurring words. That is why (6) at least seems to hold true at precisely those possible worlds where (9) does: 6. There are chinks in Toronto 9. There are Chinese in Toronto 12 This is to be contrasted with treating in-group speakers as using a wholly non-derogatory homonym in their "variety of English." That is a tempting way of conceiving of the phenomenon. In particular, it is a plausible account of the varietyspecific "niggah," a term with different register features than "nigger." (The relation between these two is fascinating and important but must be set aside here.) As a general strategy, however, it threatens to treat re-appropriation of slurs as too similar to the use of /'shag/ in Canada as a kind of rug versus its use in the United Kingdom as a synonym for "a fuck." By the way, our "at least initially" is essential because a term may lose its [+derogatory, -polite, +slang, +vulgar] features. This has happened almost entirely with "gay" (also "limey" for the English). And there are in-between stages, when things get muddy indeed. These, however, are rich topics for another day. DIAZ-LEGASPE ET AL. 13 One is never fully satisfied with any paraphrase of "the propositional difference" because there is no such difference. There is a contrast in standing meaning, but it is not of that descriptive/informational sort: To echo Davidson (1978) on metaphor, just as words are the wrong coin to exchange for pictures, messages are the wrong coin to exchange for use-theoretic meanings. (An important piece of evidence that ours is the right approach to descriptive ineffability is that it is not specific to slurs but pertains to register-type contents generally: It is equally difficult to identify contrasting modes of presentation of the rabbit corresponding to "bunny" versus "laporidae.") Non-redundancy is explained similarly: In producing "There are chinks in Toronto and I despise the Chinese," the speaker may "show" through her use of "chink" that she despises Chinese people. Still, this is not part of the truth-conditional content of the first conjunct: It encodes something truth-conditionally thinner. Hence, there is not a restatement when the despising attitude is subsequently "said" in her second conjunct. Our novel approach is also promising with respect to two sorts of socially relevant variation. The descriptive/informational part of our account has it that slurring words typically stand for marginalized out-groups. Now, which groups are the scorned "others" vis-à-vis the dominant majority will vary from one language community to another. Cross-linguistic variation arises thereby. Regarding varying degrees of offensiveness within a language, registers in general are not inevitably + or -. Take [vulgar]: "Vagina" is not vulgar at all; "beaver" is mildly vulgar; "pussy" and "twat"; are quite vulgar; while in Canada and the US the C-word is extremely so. Parallel within-language variation arises for [derogatory] and [polite]: Part of knowing the difference in register-content among slurs is to know, for example, that the N-word is, as a matter of lexicography, derogatory and impolite to the point of obscenity. Before continuing with promising characteristics, a brief methodological excursus is called for. In one respect, we are not semantic minimalists. Qua anti-monists, we posit additional richness in linguistic meaning: Three sub-kinds, in fact. This inclination towards "thickness" in semantics derives from Hallidayan influences and Ordinary Language Philosophy. In another respect, within each subkind, we lean methodologically to "thin" semantics. A case in point are the lexical entries we propose for slurring words. Very crudely, such an entry includes only: That the term is a "naughty word" (compare "fuck" or "cock"); that it stands for group such and such; and that it is "used by people who don't like that group." Much information which other theorists include as part of a slur's encoded meaning is, for us, merely about the world: Which stereotypes are associated with Chinese people, lesbians, Jewish people, and so forth, gets categorized by us as real-world knowledge, that is, nonlinguistic socio-cultural information about referents; the prototypical mental causes and effects of using a slur are equally, by our lights, sundry facts about talk-among-humans. In excising such worldly facts from linguistic meaning proper, our minimalist inspirations are Chomsky/Fodor, Grice (1975), and especially Sperber and Wilson's (1986) Relevance Theory. Finally, we are minimalistically inclined when it comes to the explanatory burdens of semantics within philosophy. A fully successful lexical semantics will leave many issues open for philosophers: When the semanticist pairs a formative with something, she need not delve into the corresponding epistemology, ethics or metaphysics. She may, for instance, content herself with explaining that "persona" in Spanish stands for persons-without feeling compelled to expand upon why persons should not be treated as mere means. She may say that "er" in Mandarin means the number two, while blamelessly failing to explore the epistemology and ontology of numbers. Here, we follow in the footsteps of Borg (2004) and Cappelen and Lepore (2005). Returning to the main theme, we have shown the promise of our view with respect to both the ethical/social aspects of slurs and their widely studied linguistic facets. With these methodological 14 DIAZ-LEGASPE ET AL. remarks in place, we can turn to its potential with respect to the epistemology of slurring, and to a few especially pressing worries. On the one hand, mastering the meaning of (1a–h) is psychologically undemanding. A young child or a novice L2 learner can (unfortunately) be linguistically competent with slurring words. They need not, for example, know the stereotype associated with "kike" to master the word, nor have familiarity with the (purported) grounds for discriminating against Jewish people (Jeshion, 2013b). Similarly, the linguistically competent user need not be an expert in social psychology, recognizing which precise emotions are expressed with "kike," and which it tends to evoke. On the other hand, there definitely are rich interpretive effects among more sophisticated speakers, pertaining to prejudices and emotive potentials. The quasi-minimalist nature of our lexical entries for slurs captures both "hands." Grasping the meanings is psychologically easy because slur-semantics is "thin." Nonetheless, the resulting psychological, social and ethical effects can turn out to be very "thick" because of all the worldly knowledge people have about slurring and its targets. A case in point: There is no denying that the social/psychological effects of (6) versus (9) sharply contrast. In particular, one may learn important things about a speaker who chooses the first over the latter: 6. There are chinks in Toronto 9. There are Chinese in Toronto A critic may suggest that the sentences therefore cannot express the same proposition. Now, our critic's argument is implicitly abductive: The best account of the contrasting informational uptake and other effects is that these sentences encode different truth-conditional contents, owing to the contrasting senses of "chink" and "Chinese." A superior explanation, however, is that a person who selects (6) over (9) merely shows something ethically suspect about herself-she does not state a richer proposition. A hearer can glean this shown-information using general-purpose cleverness and knowledge about people who use words marked [+derogatory]. (Compare Marta who says, "Polysynthetic languages are multifaceted." So speaking will provide her interlocutor with information: Marta can pronounce polysyllabic words, is highly educated, and so forth. Yet no one would explain that information as deriving from the sentence's encoded standing meaning.) Our defensive strategy thus runs: While we emphatically agree with propositionalists and Emotivists that there are vital effects hereabouts, we resist including anything extra in slurs' encoded propositional or emotive content to account for them. Here is another application of our quasi-minimalism, this time about the "thin philosophical burden" of lexical entries. One might complain: Our position is not promising vis-à-vis the social/ethical facets of slurring because we have not said what it is to insult. Still less have we explained what it is to wish to insult, or to be indifferent about doing so. A tempting concessionary response is that this is essential work for later. That, however, is not our stance. We insist that not even a promissory note of the forfuture-research variety is owed. Comparable to "persona" and "er," accounts of the nature of insults, or the moral psychology of the insult-maker, do not fall within the remit of philosophical semantics. We declined to present a full-dress defense of our register-based account. We are content, given present purposes, if the reader acknowledges an approach well worth developing. We have already achieved much towards that end. Still, several worries are so natural and important that we cannot conclude without mentioning them. A social/ethical worry, traceable to Hom and May (2018) and Richard (2008), is that a view along our lines cannot be promising because the slur-user can state something deeply offensive yet DIAZ-LEGASPE ET AL. 15 perfectly true. For instance, (6) is true according to us; and "chinks" has an extension. It allegedly follows that there exist people-more than a billion, including roughly half a million in Toronto- who deserve to be insulted and discriminated against based on being Chinese.13 Now, adopting anything like deserve to be insulted and discriminated against on the basis of being Chinese as the content of "chinks" already begs the question in favor of propositionalism. Beyond pointing to burden of proof violations, and borrowing again from Austin, we would add that sentences can be unhappy in numerous ways. In particular, they can be true without being morally acceptable: For example, when phrased in ethically revolting terminology. Such are "true slurs": For example, (6) is offensive not due to its propositional content, but due to "putting it that way." This is not ad hoc, as a comparison shows: There is nothing offensive about the propositional content Dr. X swabbed Eric's anus for fecal matter; but phrasing this as "Dr. X grubbed some shit from Eric's asshole" would be morally unacceptable given certain audiences. Our response, in sum, is to bite the bullet: Claims made using slurring words really can be both offensive and true. However, the true bit is not offensive; and the offensive bit is not even truth-apt (being use-theoretic). Another worry is that our view is not promising with respect to an epistemological/psychological facet of slurs. Patently, it is only public language words, not mental representations, which exhibit register-features. So, we cannot accommodate slurring thoughts (cf. Copp & Sennet, 2017). Our response requires disentangling two notions. The alleged problem then evaporates. In one sense, we do reject "slurring thoughts": There are no slurring propositions, just slurring language. Pace the objector, this is the right result. To stretch an analogy, "slurring proposition" is akin to "rhyming proposition." In another sense, we are happy to grant that "slurring thoughts" exist: Namely, thoughts couched in inner speech using slurring expressions. Just as we acknowledge rhyming thoughts in this latter sense, we happily grant that some of the sentences which run through our heads contain words marked as [+vulgar], [+slang], and so forth. Worry number three: There must be a difference in reference between slurring words and their socalled counterparts, as is shown by failure of substitution in transparent contexts. Consider, in this light, (10a) versus (10b): 10. (a) Chang isn't Chinese, he is Chinese (b) Chang isn't a chink, he is Chinese Only the second sounds true. Our response will not surprise. We take the folk to interpret (10b) charitably, for pragmatic reasons, as a metalinguistic negation. That is, despite the propositional content encoded by (10b) in English, we hearers easily construe it as in (11): 11. Chang is not a so-called 'chink', he is Chinese Why does charity yield this result? First, the pragmatically modified reading avoids attributing a contradictory assertion to a speaker. More than that, reconstruing (10b) as (11) allows one to hear its speaker as putting forward a morally important truth. 13 To come at the point another way, these philosophers would presumably disagree with us that "pedophile"/"torturer" are offensive and insulting in a different way than "chink"/"kike": For them, all would express an offensive property. They would add that while there are satisfiers of "pedophile" and "torturer," who thereby merit opprobrium, the extensions of "chink" and "kike" must be empty-otherwise, one would make a problematic concession to the bigot. 16 DIAZ-LEGASPE ET AL. Is this appeal to metalinguistic negation ad hoc? It is not, because the same effect shows up where there patently is co-reference. The sentence "Paul isn't gay, he is gay" is manifestly contradictory. Yet if Robert, in responding to a homophobic remark, pronounced only the first "gay" with an exaggerated tone stereotypically associated with flamboyant homosexual men, he could easily be understood as correcting a prior speaker, and stating something non-contradictory. This possible use cannot show that the lexical item "gay" fails to co-refer with "gay." Put otherwise, focusing only on the linguistically encoded propositional content of (10b), there is coreference. As a result, (10b) is not just false but self-contradictory-in just the way that (10a) is. (Compare "That is not a bunny rabbit, that is a leporidae," which typically would not be, but can be, heard as contradictory under special circumstances.) On the charitable re-construal of (10b), one hears it as true; but the upshot at best is that "so-called 'chink"' cannot be substituted salva veritate for "Chinese." The heard contrast between (10a) and (10b) does not establish failure of co-reference between slurring words and their non-slurring correlates. Still, it may seem that substitution in opaque contexts, of a slur for its correlate, changes the truth conditions of the whole sentence. Truth conditions, runs the objection, are fixed by modes of presentation. So, even if slurs and their counterparts corefer, they must have differing senses. For instance, if the sense of "chink" and "Chinese" were the same, then the truth-conditions of (12a)/(12b) should be identical. Similarly for (13a)/(13b)14: 12. (a) Justina thinks that all Chinese are Chinese (b) Justina thinks that all Chinese are chinks 13. (a) Trump argued that Republicans should try to woo Hispanic voters (b) Trump argued that Republicans should try to woo spic voters Again, that is not how we hear them: There are scenarios in which the (a) sentences sound true while the (b) sentences do not. In particular, (12a) is truistic about Justina; but we assure the reader that no one who knows her would accept (12b), because Justina holds no morally worrisome beliefs whatever about the Chinese. It is straightforward to construct a similar scenario for (13). Our reply coincides with Jeshion's (2013b, p. 327ff.) defense of her Emotivism in the face of related examples. The folk do indeed hear the (a)/(b) sentences as having different truth-conditions. As Kripke's puzzling cases teach, however, what follows from this is fraught-because propositional attitude and discourse reporting verbs are in play. It is certainly unsafe to infer contrasting senses between the substituends. At most, we can infer some sort of meaning difference. But register-features already provide one. On these grounds alone, our view plausibly makes the right predictions. More than that, there is independent reason to think that register-features block substitution in various opaque contexts. "Poop" and "shit" surely share not just reference but sense, yet (14a)/(14b) are readily heard as non-equivalent in (literal) propositional content: 14. (a) Overindulgent parents think that all shit is shit (b) Overindulgent parents think that all shit is poop 14 Examples along these lines are owed to Christopher Hom, Robert May, and Adam Sennet. DIAZ-LEGASPE ET AL. 17 Only the second, for example, would be taken to express something informative and judgmental. The pair in (15), which parallels the discourse-reporting (13a)/(13b), is similar: One who accepts (15b) will reasonably take Harvey to be guilty of very poor taste, and probably of workplace harassment. Not so (15a): 15. (a) Harvey argued that wait staff should present their buttocks as buttocks (b) Harvey argued that wait staff should present their buttocks as asses The lesson is that, in the face of this fourth worry, our view about slurs remains promising-because, given independently attested facts about register-features under certain opaque contexts, it automatically predicts what one will hear as the truth-conditions.15 3.2 | Promising qua meaning pluralist It remains to defend the claim that, if correct, our account of slurring words provides a case study in the value of meaning pluralism. One should not feel concern that the view is not pluralist. There is the vital role, for us, of what slurring words stand for. (Here is another illustration: While "asshole" shares all four register-features with slurring words, it is not itself one given its descriptive/informational content.) In addition, ideas from the Emotivist tradition are crucial: We have specified the pre-conditions for apt use in terms of negative mental attitudes. Finally, part of what sets slurring words apart for us has to do with contrasting felicity-type rules of use. The natural worry, rather, is whether register is a kind of meaning. After all, what differentiates a slur is reminiscent of metadata about etymology, co-occurrence proclivities, and so forth. As a preliminary, we must sidestep a terminological issue. Suppose a theorist declares by fiat that "meaning" shall pertain, for her, only to descriptive/informational content. The existence and importance of register in particular, and rules-of-use in general, cannot convince her that meaning includes more. Or again, suppose that by her verbal legislation "meaning" must involve the expression of mental states, whether "cognitive" or "non-cognitive." None of our examples can refute her. In short, we are powerless to falsify stipulations about how to use the vocable /mi:nIŋ/. That, happily, is not the issue. Instead, a first substantive question is whether there is a theoretically and practically important kind in play. A kind not pertaining to "form" (e.g., phonology and morphosyntax). A kind which explains salient phenomena such as our pre-theoretical conceptions of acceptable translations, of (in)correct understanding, of when a L2 learner has mastered a word, and so forth. A kind which has fascinated pragmaticians, philosophers of language and semanticists. Though not something we can establish here-it would require, for example, refuting Quinean indeterminacy-it is plausible enough that such a kind exists. To avoid terminological battles, let us label it the significance-kind. Granting this, a second substantive question is whether register-features belong therein. Here is a "master argument" for a positive answer. There definitely is a difference in usage patterns among coreferring words with contrasting registers. Given the above, such a difference must trace to the form-kind or the significance-kind: No other "linguistic causes" are in the offing. Some such patterns trace to form: Tongue twisters are avoided because of how hard they are to pronounce; multiple center-embeddings are avoided because their syntactic trees are difficult to process online; and so forth. One cannot discount 15 A larger lesson-as a "by the way"- is that postulating an additional sub-kind of meaning, which initially looks antiminimalist, can reduce the "thickness" of the meanings we countenance within each sub-kind: Here, the slurs' senses are kept "thin" by relying on use-theoretic register-type meaning. 18 DIAZ-LEGASPE ET AL. explanations based on linguistic form-kind a priori. However, barring potential future discoveries, the best explanation of patterns in slur-usage does not involve phonetics, phonology or morphosyntax: For example, "chink" does not differ from "Chinese" in relevant ways when it comes to ease of pronunciation, nor with respect to morphological complexity. Assuming significance-kind the best explanation, we should therefore place register-features, including those attaching to slurring words, among the "meaning causes." Adding to this abductive inference, it is agreed by (almost?) all hands that, ceteris paribus, it is meaning which is metaphysically connected to regularities in usage. In sum, stubbornness about labels notwithstanding, our case study suggests that content-monism should be rejected. So should content-dualism. 4 | PART THREE: COMPARE AND CONTRAST 4.1 | Slurring registers are not just descriptive/informational Our goal has been to lay out a positive view: To place another promising option on the table. This means we will disappoint readers seeking a definitive defense. What we can offer instead is a "third pass"-that is, additional clarification-in the shape of a compare-and-contrast. (Be forewarned: This will not take the form "they posit propositional content and we do not," nor "they invoke emotions and desires and we do not." That would not be pluralist.) The first contrast is with the overarching approach which conceives of language as a vehicle for message transfer-so as, on some variations, to thereafter coordinate on action. Grice (1975/1989) introduced an influential taxonomy within this framework. He categorized the various "causes" which can give rise to information exchange and introduced several sub-varieties of "effects." We will organize our comparing/contrasting around that taxonomy. The first sub-division is between very-largely-conventional causes versus causes which go well beyond what conventions afford, invoking in particular a great deal of general-purpose reasoning. ("Very-largely-conventional" rather than "wholly conventional" because Grice acknowledges that conventional linguistic meanings alone seldom determine the message meant, for example, because of disambiguation and context-sensitive items.) Grice also sub-divides the very-largely-conventional in terms of "informational effects." There exist his what is said and his conventional implicatures. Into the latter category, Grice places those contents which would not, to his ears, make an utterance strictly speaking false should the conventionally encoded proposition(s) prove non-factual. He seems to us to operationalize his intuitive contrast in a way contemporary logicians and semanticists might find peculiar. Roughly, his "what is strictly said" can be straightforwardly/isomorphically translated into logical languages familiar during the 1960s. In contrast, the propositional message linguistically encoded in conventional implicatures requires, for example, additional conjuncts in the corresponding logical translation. Taking Grice's most famous example, while there was an item in mid-20th century logics corresponding to "and," there was not one corresponding to "but." So, whereas one could translate "Jane is poor and she is honest" constituent-for-constituent as [POOR (jane) and HONEST (jane)], for "Jane is poor but she is honest" one needed to resort to something like: {[POOR (jane) and HONEST (jane)] and CONTRAST (poverty, honesty)}. (CONTRAST being that relation encoded by "but" in English, however explicated.) Table 2 summarizes. An essential feature of Grice's taxonomy is that, in a broad sense of the phrase "truth-conditions," all three quadrants involve them: The speaker meansNN something in each case, but only judgeable-contents can be meantNN. Relatedly, in all three the contents meantNN will have a translation into a logical DIAZ-LEGASPE ET AL. 19 symbolism. This can be missed because writers, Grice unhelpfully among them, sometimes use "truthconditions" in a narrow way, for something like the strict and literal truth conditions of the speech act. Turning to compare-and-contrast, on a "what is said" approach, a slurring word exhibits a conventional difference-one within the public, synchronic language-in both sense and reference from any (alleged) neutral counterpart. Being unable to canvass variations, we illustrate with Christopher Hom and Robert May's ideas. Adapting from Hom (2008, p. 431), the symbolic logic correlate of (6) would be something like (16), where CHINK is true of x iff (a) x is Chinese and (ii) x ought to be subject to the local anti-Chinese discriminatory practices because of exhibiting the locally-attributed negative properties which Chinese people exhibit qua Chinese: 6. There are chinks in Toronto 9. There are Chinese in Toronto 16. (9x) [CHINK(x) & IN(x)(Toronto)] Hom (2008, 2010, 2012) and Hom and May (2013) would not translate the "neutral" (9) this way. For them, while "Chinese" has an extension, and (9) is true, CHINK lacks an extension and (6)/ (16) are false. We acknowledge that there is a difference in type meaning. Not, however, in sense or reference; only in "fittingness" for discursive situations. For instance, only "chink" is marked in the lexicon as [-polite, +slang, +vulgar] and most importantly [+derogatory]. Put otherwise, we would translate "chink" and "Chinese" into the same predicate in a purely logical meta-language: Both would correspond to CHINESE. Hence both terms have the same (rather large) extension. A comparison with conventional implicature views is challenging because the label is used equivocally.16 Beginning with the easier case where the usage follows Grice (1975/1989), conventional implicatures-like what is said and what is conversationally implicated-are meantNN. One can believe what is conventionally implicated, intend others to judge it true, and so forth. Moreover, this content will be part of standing linguistic meaning. Applied to slurs, the translation of the type meaning of (6) might thus be: 17. (9x) [CHINESE(x) & IN(x)(Toronto)] & [(8x) (CHINESE(x) ! MERIT-DISCRIMINATION(x)] TABLE 2 Grice's taxonomy Very largely conventional Much general-purpose reasoning Straightforward isomorphic translation into formal logic What is said Non-isomorphic translation required Conventional implicatures Conversational implicatures 16 Who embraces the conventional implicature approach to slurs? This is not an unambiguous question, and hence demands exegetical caution. This is not, thankfully, the place to engage the issue. Regarding who self-applies the label, they include McCready (2010), Potts (2005, 2007), Whiting (2013), and Williamson (2009). As presented, none of these seem notational variants on our view-but, as will emerge immediately below, the appearances may mislead (cf. Bach 1999, 2006). 20 DIAZ-LEGASPE ET AL. Now, register (in our sense) is included, as a matter of linguistic convention, in the lexicon. A fortiori, we concur that slurring words like (1a–h) are not synonymous with "Chinese," and so forth. Put psychologically, we agree that knowledge of a word's register-features is knowledge of language properly so-called. We disagree that truth-conditions, even of the broad sort, are involved: In place of (17)'s second conjunct, for example, we introduce something strictly use-theoretic. In short, the logical translation not just of "what is said" by "There are chinks in Toronto" but of its truthconditions tout court is (18): 18. (9x) [CHINESE(x) & IN(x)(Toronto)] Some theorists use Grice's nomenclature such that conventional implicatures do not fit anywhere on Table 2. Specifically, some seemingly eschew a difference in truth conditions even in our "broad" sense when a conventional implicature is added to "what is said." The contrasting meaning is sui generis. Recalling "Jane is poor and/but she is honest," these quasi-Gricean theorists would take their logical translations to be precisely the same: [POOR (jane) and HONEST (jane)]. This latter approach closely approximates ours. Indeed, there may be variations which we would welcome as fellow travelers: for example, if conventional implicatures are cashed out in terms of use-theoretic appropriateness-conditions. Differences might then persist only in the details: Is some cousin of metadata invoked? Are register-features made use of-and the same four that we attribute to slurring words? When characterizing the discursively appropriate situations for slurs, is appeal made to the willingness-to-insult which speakers should feel? And so on. The final sub-variety of descriptive/informational approaches contrasts with the prior two in terms of Gricean "cause." Conversational implicature accounts involve a heavy dose of general-purpose reasoning about rational cooperation. Phrased in cognitive scientific terms, understanding of the special content of slurs is not achieved by drawing upon meaning-information stored in the language faculty; other parts of the mind turn the trick. Yes, there is some conventional difference between a slurring word and its neutral correlate; but, goes the idea, the speaker pragmatically conveys the extra content by relying on the audience's worldly cleverness. Geoffrey Nunberg (2018), a proponent of this kind of view, provides a very telling analogy: The week after the Monica Lewinsky story broke, the New York Times Week in Review section ran its story about it under a picture of the White House at night that was headed Scandale. When I asked an editor at the section why they felt the need to put that final e on the word, he said, "Oh, that's so readers will know it's about sex and not money". Now most Americans would assume, correctly, that French scandale and English scandal are synonyms: when Frenchmen say Quel scandale! they express pretty much the same thought that we would express with 'What a scandal!' The added implications of using the French word in an English context arise from a familiar cultural stereotype of the French' (Nunberg, 2018, p. 267). The headline managed to cleverly convey "broad truth conditions" by using formally and culturally distinct synonyms. The suggestion with respect to slurring words is that something very similar is at work: "Chink" and "Chinese" are complete synonyms as far as knowledge of meaning-in-English DIAZ-LEGASPE ET AL. 21 goes, but, we well-versed speakers know metadata which allows us to creatively meanNN contrasting things with them.17 We share with such a conversational implicature theorist the commitment to the same sense and reference. We equally take on board the very original idea that metadata is centrally involved. We disagree, of course, that the "effect" is generally a message, however conveyed. Such implicating sometimes occurs with slurring speech: a person can intend to induce the belief that it is acceptable to insult Chinese people, and so on in familiar Gricean fashion, by showily selecting the word "chink." But we deny both that this conversational effect is inevitable, or that it captures the heart of slurring words' content. More subtly, we differ on the "cause" of slur-effects. First, unlike "scandal/scandale," we take the cause to be a matter of content. (Register-features belong to the significance-kind, not the form-kind.) Second, to reiterate, unlike cultural stereotypes about the French, we take this content to be linguistically encoded.18 We have contrasted our register-based view with that family of theories which takes slurring content to be exhausted by descriptive/informational messages. We end with the other dominant family. 4.2 | Slurring registers are not just expressive/emotive There is not a doctrine of Emotivism in semantics, nor a doctrine of Emotivism about slurring words. So, we cannot compare our register-based view with "it." Our clarificatory aim can be advanced, nonetheless, by constructing as our foil one sophisticated ur-variant, couched at a high level of abstraction from specific proposals. Thinking of it as a general-purpose approach, the idea would be that sentence (19) and (for some philosophers) sentence (20) do have a sort of propositional content. They are also, however, used to express something psychologically rich and subtle, such as disapprobation: 19. I am appalled by that behavior 20. Torture is morally wrong That is, beyond descriptive/informational content, there is an Emotivist rule-of-use for these. It specifies which "non-cognitive" mental state such a sentence is suited to express. Applying this general approach to slurring words, they too would have both kinds of content. The rule-of-use specific to them might be something like: To be used by people who have a certain conceptually rich negative attitude towards the referent of the slurring word. Thus "chink," for example, 17 Renee Bolinger (2015) seems to have a similar account in mind. She sagely develops it, however, in terms of her (purposely ambiguous) notion of "signaling," to allow for unintentional communication of contents which are not properly propositional. 18 We may reformulate this first half of our compare-and-contrast by forefronting another terminological tangle. The term "semantic" is used in half a dozen cross-cutting ways (Ezcurdia & Stainton, 2013). Of central interest is semanticstype versus semanticstruth. The first pertains to content which conventionally attaches to the expression in the shared language. Put psychologically, semanticstype pertains to meaning-proclivities stored as part of knowledge of language proper. The second pertains to truth-conditions, whether narrow or broad. A use-theorist will not just concede but insist that "Hello!" and "Gesundheit!" lack semanticstruth. But, they will insist, it is a fallacy of equivocation to infer that these lack semanticstype. Given this terminology, the fundamental difference between descriptive/informational views and our own can be stated thus. Those in our Gricean table all hold that slurring involves semanticstruth. We think otherwise. They disagree among themselves about whether slurring words differ from their neutral counterparts in terms of semanticstype, and we align with the what-issaid and conventional-implicature theorists. In brief, there is a difference in semanticstype between "chink" and "Chinese," but there is no difference of any sort in semanticstruth. 22 DIAZ-LEGASPE ET AL. would be a term fitted for use by people who hold the Chinese to be contemptible, unworthy of equal standing, appropriately subject to a discriminatory perspective (Camp, 2013), not fully persons, or what have you.19 Such a refined Emotivism shares many features with our register-based approach. It is agreed that, for example, "chink" does refer, and "There are chinks in Toronto" is truth-evaluable; but that one cannot capture the complete content of slurring words via sense/reference. Put otherwise, a slur's content cannot be wholly captured by translation into even an especially esoteric judgement-centred logical symbolism. Shared, in short, is a meaning dualism. Moreover, there are not two kinds of sentences-one with solely referential and truth-conditional content, one with only emotive content. Rather, as in (19), the two kinds of content interconnect in a single expression. (Contrast a very crude and old-fashioned Emotivism (unfairly) associated with Ayer (1936), Stevenson (1944). "I am in pain," goes the idea, has exclusively emotive/expressive content: It's merely a highly conventionalized way of grunting/groaning. Whereas "The speed of light is constant" has exclusively empirical/factual content.) Equally shared is the postulation of normative rules-of-use for slurring words. Our ur-Emotivist has it that slurring words like (1a–h) are tools for expressing mental attitudes. She thus predicts, correctly, that speakers may utter them in many circumstances without having or externalizing a sophisticated negative attitude, let alone a strong negative feeling like hatred, fear, and so forth. (For a variety of examples of "warm-hearted" uses of slurring words, see Diaz-Legaspe and Stainton, 2018.) Finally, we coincide in holding that appeal to attitudinal states is important in explaining both the meaning and ethics of slurring words. The foregoing Emotivist view thus closely parallels ours. Indeed, it achieves our ambitious aim. If true, it would entail a full-blown pluralism about contents: There are not just propositional and expressive contents, but also rules-of-use for job-bearing linguistic devices. We thus welcome anyone falling under this broad umbrella as, again, fellow travelers. There are, however, contrasts to be drawn. As a minor point, the specific attitudinal state we focus on is different. More significantly, consider the relationship among slurring words, the associated negative attitudes, and offense-potential, as represented in Figure 1: Our sophisticated Emotivist claims a direct meaning-based connection between slurring terms and attitudes. These mental states-whether anger, contempt, disrespect, a discriminatory perspective, and so forth-are offensive to the targeted group. Hence using the slur is derivatively/indirectly morally objectionable. In effect, the wrongness goes via the right side and then the bottom of the Slurring words Offensiveness Negative attitudes FIGURE 1 Elements in slurring 19 Though not exegetical, in formulating this sophisticated ur-Emotivism we have drawn heavily on the incisive work of Robin Jeshion. She writes, for instance: "slurring terms are used to express contempt for members of a socially relevant group on account of their being in that group or having a group-defining property" (2013a, p. 240). DIAZ-LEGASPE ET AL. 23 mutually-agreed-upon triangle. For us, the connections are more intricate. On the one hand, we do recognize a slur-to-attitude connection via the right-hand line, because the situation appropriate to [+derogatory] terms happens to be characterized in terms of willingness to insult. With the sophisticated Emotivist, we here locate a path to explaining the ethical/social features of slurring: Slur ! attitude ! offensiveness. On the other hand, we additionally posit a direct meaning-based connection to offensiveness, along the left-hand line of the triangle. Contra every Emotivism, for a speaker to select a slur, purposely rejecting a neutral correlate, may be immediately wrongful: That is, in opting for a [+derogatory, -polite, +vulgar] word, the speaker may ipso facto merit blame, whatever her mental proclivities. We thus predict, but by dual routes, the pattern which the Emotivist rightly emphasizes, namely that people uttering slurs typically exhibit contempt, hatred, and so forth. The deepest difference between our view and this ur-Emotivism is closely related. For us, not all talk involves making public one's internal mental states, whether "cognitive" or "non-cognitive." Much of it involves directly pulling off, collectively, social actions. Seemingly every Emotivism about slurring words must resist this essential lesson of Functional Linguistics and Ordinary Language Philosophy.20 5 | CONCLUDING REMARKS At the risk of over clarifying, we conclude with a reminder of: The ingredients in our synthesis; their sources; and how they combine to meet our twin aims. We borrow from the two families of accounts which have dominated discussion. From the descriptive/informational one we take on board that, certain philosophically intriguing outliers aside ("pleb," "crone"), slurring words refer to special sorts of groups. Moreover, this family's compositional semantic rules are necessary to capture the whole meaning of sentences in which words like (1a–h) embed. From the expressive/emotive family, we take over the important idea that slurs involve negative attitudes and feelings. We complement these with use-theoretic meanings. They do not pertain to externalizing mental states, but rather link directly to social actions. There are indirect connections to mental states, of course, both because beliefs and desires motivate our speakings, and because felicity conditions sometimes mention the mental (e.g., "It is infelicitous to bet $2 on Argentina's Pride if you have no intention of paying up"). Nonetheless, use-theoretic meanings are fundamentally rules for sociocultural interaction. We particularly highlighted those where there is no illocutionary act-no 20 We are not aiming to establish the superiority of our view over every contender, so we have not attempted to discuss each theory of slurs. In the interest of recognizing their insights, however, we mention two additional candidate fellow travelers (i.e., beyond the non-Gricean conventional implicature theorist and the sophisticated Emotivist). Predelli's (2013) position shares many important commonalities with ours, including specifically a discerning appeal to appropriate contexts of use (which he calls "bias") and even register-features. Simplifying, one difference is that Predelli seems to be pursuing a reductionist tack: He is keen to find a single kind of "meaning" to which a single logical formalism straightforwardly applies. Relatedly, Predelli's overall project seems one of quasi-instrumentalist modelling, so that it is permissible by his lights to sideline various psychological facets of slurring words. We hold, in contrast, that a realist description of the use and understanding of natural language demands rejecting even his kind of monism (cf. Kaplan 1999). Anderson and Lepore's (2013a, 2013b) approach to slurring words as socially prohibited taboo words is deeply resonant with ours, not least in failing to fall into either the propositional or the Emotivist camps. However, rather than treating slur-prohibition as primitive, we want to explain it in terms of problematic register-features. We hope to address both views in more detail (and more justly) in a future paper. 24 DIAZ-LEGASPE ET AL. reasonable "hereby"-paraphrase-but only sociolinguistic appropriateness conditions. (Recall "Gesundheit!" as opposed to, say, "I hereby bet that Argentina's Pride will place first.") We also made use of the Functionalists' idea of words and constructions as purpose-specific tools, that is, a system of devices which a speaker may choose from. In particular, we emphasized lexical and grammatical "tools" differing only in their suitability for contrasting discourse situations. That a speaker may flout an expression's function permits non-literal uses, even in the case of use-theoretic meanings. Methodologically, our view is minimalist in two senses. It endorses generative grammar's distinction between usage-phenomena which trace to properly linguistic causes versus other non-linguistic "performance effects." It also endorses the stance of Borg (2004) and others that semantic content owes relatively few philosophical debts. The result is a fairly "thin" content for slurring words. The final ingredient is lexical metadata. The sort we invoke is similar to familiar sorts of metadata about collocation, frequency, and so forth, but diverges in two key respects: It belongs within knowledge of language proper, that is, within "linguistic competence"; and it is normative. From these half-dozen sources, we constructed an account of the meaning of slurring words. They typically refer, specifically to marginalized out-groups who do not actually merit opprobrium. Appeal to this referential aspect helps distinguish a non-slur like "pedophile" or "asshole" from a slur like "chink." Slurring words are also marked with register-features. These encode non-illocutionary usetheoretic meaning, specifically conditions concerning which contexts (in a wide-ranging sense) they are appropriate for. Importantly, there exist systems of lexical items whose members differ only in their register-features: For example, "usted"/"tú"/"vos" and "excrement"/"poop"/"shit." This register aspect yields the contrast between the otherwise synonymous "chink" and "Chinese." Phrased using our feature geometry, slurring words are usually marked [+slang, +vulgar] and are always marked [-polite], [+derogatory]. This last merits special attention. Its content, as a matter of descriptive sociolinguistics, is tentatively given thus: Terms marked with [+derogatory] are appropriate to discourse situations where the speaker either wishes to insult the designated out-group or is unconcerned about doing so. (Given our second goal, the pluralism herein merits underlining: There is reference, a rule-of-use regarding fittingness to a discourse situation, and that situation is characterized in terms of negative attitudes.) Turning to why using a slurring word is typically ethically unacceptable, it is not because the speaker conveys (in whatever fashion) a "slurring proposition." There are no such things. It is not (solely) because the person experiences a morally repugnant mental attitude while speaking, and externalizes it verbally. Rather, on the one hand, it can sometimes be unacceptable-offensive, rude-to use slurring words simply because they are [-polite, +slang, +vulgar]. Much more importantly, it will standardly be ethically unacceptable for this reason: As a matter of its meaning in the shared language, a given slurring word exists so that it may be chosen over a neutral term by people who are willing to insult an entire out-group. Anyone who opts for such a perfectly nasty word will, absent mitigating factors, be blameworthy. ACKNOWLEDGMENTS Draft versions of this material were presented at: Sociedad Argentina de Análisis Filosófico, Buenos Aires; the Seventh Workshop on Language, Cognition and Context, Valparaiso; the Department of Philosophy, University of Queensland; the Pragmatics Research Group, University College London; the Department of Philosophy, University of Reading; and the Philosophical Society, University of Oxford. We are very grateful to audiences at these venues, and especially to our commentator in DIAZ-LEGASPE ET AL. 25 Chile, Adam Sennet, for extremely helpful suggestions. Thanks are also due to Renee Bolinger, Maren Fichter, Henry Jackman, Jessica Keiser, Eliot Michaelson, Ethan Nowak and Deirdre Wilson for written comments. Our greatest thanks go to University of Western Ontario doctoral students Michael Korngut and Jiangtian Li, fellow members of a reading group on slurs, and co-authors of the conference paper out of which this long article eventually evolved. Financial support was provided by the Social Science and Humanities Research Council of Canada and the Research School of Social Sciences of the Australian National University, through grants to Robert J. Stainton. ORCID Robert J. Stainton https://orcid.org/0000-0001-6661-8266 REFERENCES Anderson, L. & Lepore, E. (2013a). What did you call me? Slurs as prohibited words. Analytic Philosophy, 54(3), 350–363. https://doi.org/10.1111/phib.12023 Anderson, L. & Lepore, E. (2013b). Slurring words. Noûs, 47(1), 25–48. https://doi.org/10.1111/j.1468-0068.2010.00820.x Austin, J. L. (1962). 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What is Radical Recursion? Steven M. Rosen Departments of Psychology and Philosophy (Emeritus) College of Staten Island/City University of New York [email protected] [This paper is published in S.E.E.D. Journal, 2004, 4 (1), 38–57.] ABSTRACT Recursion or self-reference is a key feature of contemporary research and writing in semiotics. The paper commences by focusing on the role of recursion in poststructuralism. It is suggested that much of what passes for recursion in this field is in fact not recursive all the way down. After the paradoxical meaning of radical recursion is adumbrated, topology is employed to provide some examples. The properties of the Moebius strip prove helpful in bringing out the dialectical nature of radical recursion. The Moebius is employed to explore the recursive interplay of terms that are classically regarded as binary opposites: identity and difference, object and subject, continuity and discontinuity, etc. To realize radical recursion in an even more concrete manner, a higher-dimensional counterpart of the Moebius strip is utilized, namely, the Klein bottle. The presentation concludes by enlisting phenomenological philosopher Maurice Merleau-Ponty's concept of depth to interpret the Klein bottle's extra dimension. 1 SEMIOTICS, POSTSTRUCTURALISM, AND RECURSION In classical signification, the stability of the relationship between the signifier and what it signifies is maintained by preserving the anonymity of the former. Attention is fixed solely on the meanings that are signified, not on the act of signification itself. With the advent of semiotics this changes. Semiotics is the discipline that studies the process of signification. Here the sign becomes recursive; instead of focusing exclusively on signified meanings, it comes to focus on itself. The signifier, which had played a predominantly tacit role in classical semiosis, is now itself explicitly signified. Despite this role reversal inherent in the very existence of the discipline of semiotics, structuralist semioticians like Saussure still sought to preserve the invariance of the link between the given signifier and what it signifies. The problem is that, once classical sig2 nification is surpassed by signifying the signifier, the door is opened to an infinite regress. For now, it seems that no signifier is exempted from mutation into that which is signified. A new signifier is presumably needed to signify what had been the signifier, but this new signifier is subject to signification by a still newer signifier, and so on ad infinitum. And each time the tacit operation of the signifier is undermined by being explicitly signified, the functioning of what had been signified by that signifier is also affected. Ultimately then, we have in this "hall of mirrors" neither signifier nor signified in any stable, abidingly meaningful form. Poststructuralist writing exemplifies the recursive "sliding" or "slippage" of the signifier. The approach of psychoanalyst Jacques Lacan is a prime illustration. For Lacan, language "is constituted by a set of signifiers" that involves what "I call the Other" (1966/1970: 193). The "otherness" of language results from the fact that, in its "chain of signifiers" (194), every act of self-reference, rather than affirming the identity of the self or subject that is referred to, always slips away into what is other, into a new and anonymous signifier. As Lacan puts it: All that is language is lent from this otherness and this is why the subject is always a fading thing that runs under the chain of signifiers. For the definition of a signifier is that it represents a subject not for another subject but for another signifier. This is the only definition possible of the signifier as different from the sign. The sign is something that represents something for somebody, but the signifier is something that represents a subject for another signifier. The consequence is that the subject disappears...." (1966/1970: 194) In this way, the sign-which had constituted for earlier semioticians a fixed relationship between a signifier and its signified meaning, with the subject operating stably behind the scenes (the "somebody" to which Lacan alludes)-now dissolves into an evanescent flux of differences wherein the subject loses its substance, becoming a "nobody," a ghost-like quasi-presence. Much the same process of dissolution is reflected in the deconstructionist writings of Jacques Derrida. In the "primary writing" (1976: 7) of which he speaks, "[s]ign will always lead to sign, one substituting the other...as signifier and signified in turn" (Spivak 1976: xix). In Derrida's own words, language must be understood as a field "of freeplay, that is to say, a field of infinite substitutions" (cited by Spivak 1976: xix) in which identity fragments into sheer difference (différance). The specific way this takes place is by the process of self-referential mirroring in which, time and again, the signifier is displaced by being made into what is signified by a newly implicit signifier. 3 It seems clear that the slippage of the signifier results from the indefinite repetition of recursion. This is the nature of semiosis, we are told; it is a "mirror game" in which self-identity is perpetually subverted and we continually slide into otherness and difference. While I certainly agree with the general consensus among semioticians that symbolic operations are inherently recursive, I propose that the infinite regress to which poststructuralism is prone actually derives from its failure fully to achieve recursion. Again, when the signifier "slips," it becomes something that is now itself signified. Yet does this really constitute a concrete instance of self-signification, or does it merely entail the close juxtaposition of two semiotic acts neither of which are recursive in themselves? Initially, X signifies Y. Then there is the reversal of this in which X itself becomes signified by a new signifier, Z. This is obviously not to say that X signifies X. Poststructuralist "self-signification" then does not truly involve a signifier's reference to itself within the same actual occasion, to use Whitehead's (1978) term for a fundamental concrete event; it entails only a switching of roles between signifier and signified from one occasion to another. (No doubt occasions may follow each other in close succession and may be broadly construed as belonging to the "same" occurrence; but, on a more concrete level, the occasions of poststructuralism constitute distinct semiotic acts.) Therefore, poststructuralist signification is not radically recursive, not recursive all the way down into the roots of semiosis, for the signifier of occasion 2 does not signify itself but only that which was the signifier on occasion 1. If poststructuralism fails to meet the challenge of radical self-reference because its signification within the actual occasion is strictly a reference to what is other, does radical recursion involve reference to the self in the sense of simple self-identity (X≡X)? It surely cannot. For without the aspect of the other, of difference, meaning is trivialized and collapses. Radical recursion therefore entails neither external reference nor selfidentity. What it constitutes, I suggest, is the dialectical interplay of these. I propose that, in radical recursion, though the self that is signified is not simply the same self that does the signifying; though the very act of reflecting upon the self turns it into what is other; this other flows right back into the source from which it arises, rather than appearing merely as an other cast before a new self. The semiotic act I am intimating thus would give us neither self nor other, in the categorically opposed sense of these terms. We would realize instead their paradoxical interpenetration. I suggest that this dialectic is what we require to supersede the supremacy of linear signification in a meaningful way. Signifier and signified would be more than reciprocally interdependent in such a self-signification. They would be identical, utterly one. Yet they also would be two. By virtue of the latter aspect, meaningful signification would continue; by virtue of the for4 mer, recursion would go all the way down; it would be realized concretely in the heart of the actual occasion. To be sure, this construal of radical recursion requires further explication. 2 RADICAL RECURSION IN TOPOLOGY Psychoanalyst Jacques Lacan had turned to the science of linguistics in order to clarify the language of the psyche. What we see precisely in the slippage of the signifier through which the subject is "a fading thing" (1966/1970: 194) is the functioning of the unconscious. But Lacan was not content to stop with a merely linguistic clarification of psychic process. In an effort to achieve an even higher level of precision, he appealed to mathematics, and, in particular, to topology, the qualitative sub-discipline that deals with the properties of surfaces. By way of elucidating the signifying activity that constitutes the unconscious discourse of the human subject, Lacan presented a diagram of a Moebius strip: This diagram can be considered the basis of a sort of essential inscription at the origin, in the knot which constitutes the subject. This goes much further than you may think at first, because you can search for the sort of surface able to receive such inscriptions. You can perhaps see that the sphere, that old symbol for totality, is unsuitable. A torus, a Klein bottle, a cross-cut surface, are able to receive such a cut. And this diversity is very important as it explains many things about the structure of mental disease. If one can symbolize the subject by this fundamental cut, in the same way one can show that a cut on a torus corresponds to the neurotic subject, and on a cross-cut surface to another sort of mental disease. (Lacan 1966/1970: 192–193) Comparing the sphere and the Moebius strip, we can say that both are recursive, insofar as they both turn back upon themselves. But unlike "that old symbol for totality," the Moebius possesses a "fundamental cut"; a knot, twist or gap. In Lacan's view, this cut represents the division inherent in the subject that prevents it from realizing the selfidentity symbolized by the sphere. The cut functions like a crack in a mirror, leading the Moebius to signify itself in such a way that it distorts or displaces itself. Or, speaking diachronically, we can say that Moebius recursion is interrupted by the cut and we cut to a new occasion. In moving into the Moebius's twist, the signifier we started with is twisted into that which is signified by a newly implicit signifier that is a "mirror image" of the original yet out of step with it. On Lacan's reading then, the Moebius strip embodies the 5 self-alienating kind of recursion that falls short of what I have called radical recursion. I suggest, however, that there is a different way of reading the Moebius. To see how Moebius recursion can be grasped in the radical sense adumbrated above, let us look more closely at this curious topological structure. We may bring out most effectively the dialectical character of the Moebius strip by comparing it to a non-dialectical structure more similar to it than is the sphere: the cylindrical ring (see Rosen 1994, 2004a). Figure 1. Cylindrical ring (a) and Moebius strip (b) A cylindrical ring (Fig. 1a) is constructed by cutting out a narrow strip of paper and joining the ends. The surface of Moebius (Fig. 1b) is produced by giving one end of such a strip a half twist (through an angle of 180°) before linking it with the other. The cylindrical ring possesses the familiar property of two-sidedness: at any point along its surface, two distinct sides can be identified. Commencing on either side, rotation about the ring traces out a circle of simple self-return like that found on the sphere. The two-sidedness of the cylinder of course precludes continuous passage from one side to the other. Such a transition is inevitably cut short at the surface's edge; the singularity we encounter there tells us that we cannot reach the far side without a break in contact, a cut to a new occasion. We are therefore able to say that, whereas rotation about a single side of the twosided ring signifies the simply continuous affirmation of self-identity, passage between sides expresses a simply discontinuous cut to what is other. Now, in the case of the Moebius strip, it is true that if you place your index finger anywhere on the surface, you will be able to put your thumb on a corresponding point on the opposite side. The Moebius strip does have two sides, like the cylinder. But this only 6 holds for the local cross-section of the strip defined by thumb and forefinger. Taking the full length of the strip into account, we discover that points on opposite sides are intimately connected-they can be thought of as twisting or dissolving into each other continuously, as being bound up internally. Accordingly, mathematicians define such pairs of points as single points, and the two sides of the Moebius strip as but one side. I want to emphasize that the Moebius surface is not one-sided in the homogeneous sense of a single side of the cylindrical ring. It is one-sided in the paradoxical sense, onesided and also two-sided, for the local distinction between sides is not just negated with expansion to the Moebius as a whole. In coming to interpenetrate each other, the sides do not merely lose their distinct identities. And yet, though the sides remain different, they also become one and the same. Thus, if the cylindrical ring embodies the dualism of identity and difference, of continuity and discontinuity, the Moebius strip signifies their dialectical entwinement. We can say as well that while the cylinder dualistically expresses both trivial recursion (through movement on a single side) and non-recursion (through passage to the other side), the Moebius models radical recursion. Let us focus on the unique recursive action of the Moebius. With 360° of rotation about this surface, we appear to return to our point of origin. But this return is in fact also a departure, since, instead of remaining on the same side of the strip as in the case of cylindrical rotation, we are carried to the opposite side. So Moebius recursion incorporates an element of discontinuity not evident in its cylindrical counterpart. It is this distinctive feature that Lacan picked up on in contrasting the Moebius with the sphere. What Lacan apparently missed is that the discontinuity of the Moebius, its twist or cut-unlike the cut required in passing from one side of the cylinder to the other-is also continuous. It is in glossing over the paradox of Moebius recursion that Lacan stopped short of radical recursion. Lacan was apparently unable to recognize that the Moebius signifier does not merely short-circuit its reference to itself by prematurely cutting away from itself to an alterself operative on a new occasion. Rather, the Moebius signification of self as other (and other as self) transpires within the same concrete occasion thereby surpassing the dualism of self and other. The application of Moebius topology has been taken up by thinkers with diverse orientations and disciplinary backgrounds. From a feminist perspective emphasizing embodiment, theorist Elizabeth Grosz expands on Lacan's use of the Moebius by portraying it as expressing "the inflection of mind into body and body into mind" (1994: xii). Anthropologist Peter Harries-Jones (2002) suggests that the paradoxical link between culture and environment as understood by Bateson is best depicted in the form of a Moebius strip. Communications philosopher Brian Massumi (carrying forward Deleuze and Guat7 tari's call for a "topology of multiplicities" [1987: 483]) demonstrates the need to reconceive human transactions via a "strange one-sided topology" that recursively surmounts the old dichotomies by working at a "paradoxically creative edge" (1998). Semiotician Floyd Merrell (1998) uses the paradox of the Moebius to model C. S. Peirce's concept of abduction. And philosopher Yair Neuman-in this issue of SEED-applies the Moebius to the structure of boundary events in semiotic systems. (My own work with the Moebius dates back to the 1970s; see Rosen 1994.) In these writings, sustained emphasis on paradox allows the authors to surmount Lacanian "slippage" and employ Moebius topology to question effectively "the binary oppositions...[of] mind/body, nature/culture, subject/object and interior/exterior" (Grosz 1994: 164). It is all too easy, however, to lose one's paradoxical edge. This is evidenced in Grosz and Massumi when-after using topology to successfully challenge binary opposition on one level of analysis, they appear to fall prey to it on another. Thus, in the case of certain root philosophical oppositions that implicitly structure their thinking-such as the one and the many, identity and difference, being and becoming-they wind up privileging "the fields of difference, the trajectories of becoming" (Grosz 1994: 210). The "onesidedness" of such a reaction to the totalizing propensities of structuralism (and classicomodernism in general) is certainly not of the Moebius kind. Instead of genuinely questioning the categorial purity of the old approach by consistently applying topological paradox to the most basic philosophical dichotomies, there is a slippage into a sort of "reverse purism" (Rosen, 2004b). Pure identity (totality, unity, being, continuity, etc.) is supplanted by a mode of difference every bit as pure: Derridean différance. It is in the process of unambiguously affirming one member of the philosophical binary over the other that the Moebius edge is lost. So what I am proposing is that the application of topological paradox needs to be implemented in a consistent and thoroughgoing manner all the way down. From my own experience, I know how difficult this is to achieve and it would not surprise me to learn that I myself lose my edge in places in this very text. As a dweller in a "glass house," I must be careful then about the "stones" I hurl. We are all challenged to avoid limiting our applications of topological paradox to the surface of our discourse while allowing our deepest assumptions and forms of expression to remain tacitly governed by the system of binary logic that has controlled our thinking for so many centuries. To keep our topological edge all the way down and thus achieve radical recursion, we must consistently exceed mere "logics of presence or position" and employ "qualitative topologics," as Massumi (1998) so well puts it. 8 3 THE KLEIN BOTTLE I must now acknowledge a limitation in the Moebius expression of radical recursion. The Moebius does effectively signify the dialectic of continuity and discontinuity. In traversing the twist, we depart from the circle of self-identity associated with continuous rotation about a single side of the cylindrical ring, and we make the transition to the other side of the surface, which, when enacted on the cylinder, brings simple discontinuity. Yet, in the Moebius case, the departure from cylindrical continuity happens continuously. However, the discontinuous aspect of the Moebius dialectic is in fact something of an abstraction. While the effect of discontinuity is surely created in passing through the twist to the far side, there is never any true cut or break, as occurs when actually crossing an edge. The Moebius therefore signifies the continuity-discontinuity dialectic in a continuous way; the discontinuous element is symbolized but not concretely embodied. I suggest that, for a full-fledged realization of the radically recursive dialectic, we require a topological structure in which continuity and discontinuity are interwoven not merely in effect but in actual fact. There exists a higher-dimensional counterpart of the Moebius surface. By way of introduction, consider an interesting attribute of the Moebius: its asymmetry. Unlike the cylindrical ring, the Moebius has a definite orientation in space; it can be produced either in a leftor right-handed form (depending on the direction in which it is twisted). If both a leftand right-oriented Moebius surface were constructed and then "glued together," superimposed on one another point for point, a Klein bottle would result (Lacan's passing allusion to this topological structure is cited above). Figure 2. The Klein bottle The Klein bottle (Fig. 2) has the same property of asymmetric one-sidedness as the twodimensional Moebius surface but incorporates an added dimension (Rosen 1994). Note, however, that we cannot actually produce a continuous model of this curious container, for leftand right-facing Moebius bands cannot be superimposed on each other in three9 dimensional space without tearing the surfaces. Therefore, while each Moebius enantiomorph is continuous within itself, joining these mirror twins to form a Klein bottle brings discontinuity. The feature of Kleinian discontinuity can be illustrated by means of a different but mathematically equivalent way of making the bottle. Once again a comparison is called for. Figure 3. Construction of torus (upper row) and Klein bottle (lower row) Both rows of Figure 3 depict the progressive closing of a tubular surface that initially is open. In the upper row, the end circles of the tube are joined in the conventional way, brought together through the three-dimensional space outside the body of the tube to produce a doughnut-shaped form technically known as a torus (a higher-order analogue of the cylindrical ring). By contrast, the end circles in the lower row are superimposed from inside the body of the tube, an operation requiring the tube to pass through itself. This results in the formation of the Klein bottle. Indeed, if the structure so produced were cut in half, the halves would be Moebius bands of opposite handedness. But in threedimensional space, no structure can penetrate itself without cutting a hole in its surface. So, from a second standpoint, we see that the continuous construction of a Klein bottle cannot be carried out in the three dimensions available to us. The Klein bottle thus seems to possess the element of concrete discontinuity missing from its lower-dimensional Moebius counterpart. Whereas the twist in the Moebius mediates the transition from one side of the surface to the other in a continuous fashion, the Kleinian passage from inside to outside requires a hole. Of course, a simply discontinuous structure will serve us no better than a simply continuous one if we are seeking to express the dialectic of continuity and discontinuity. What is needed is a structure that embodies the paradoxical inter10 weaving of continuity and discontinuity. And, in fact, the Klein bottle does just that, provided that we approach it in a truly dialectical way. How does modernist mathematics approach the Klein bottle? Mathematicians certainly do not just accept the discontinuity of the Klein bottle. Instead they rely on the idea that a form that penetrates itself in a given number of dimensions can be produced without cutting a hole by invoking an added dimension. The point is nicely illustrated by the mathematician Rudolph Rucker (1977). He asks us to imagine a species of "flatlanders" attempting to assemble a Moebius strip. Rucker shows that, since the space inhabited by these creatures would be limited to two dimensions, when they would try to make an actual model of the Moebius, they would be forced to cut a hole in it. Of course, no such problem arises for us human beings, who have full access to three dimensions. It is the continuous construction of the Klein bottle that seems problematic for us, since this would appear to require a fourth dimension, but, try as we might, we find no fourth dimension in which to execute the operation. For modernist mathematics, however, there is actually no problem. Although dimensions higher than the third may be unavailable to concrete experience, mathematicians feel free to proceed abstractly, calling forth as many extra dimensions as they wish. Added dimensions are summoned into being by extrapolation from the known three-dimensionality of the physical world. This theoretical procedure of dimensional proliferation presupposes that the nature of dimensionality itself is left unchanged. In the case of the Klein bottle, the "fourth dimension" required to complete its formation remains an extensive continuum as is three-dimensional space, though the "higher" space is taken as "imaginary"; the Klein bottle, for its part, is regarded as an "imaginary object" embedded in this space. Enclosed as it is in the hypothesized fourdimensional continuum, the imaginary Klein bottle itself is presumed simply continuous. Like the Moebius strip of three-dimensional space, it is thought to possess nary a hole. Now, in his phenomenological study of topology, the mathematician Stephen Barr advised that we should not be intimidated by the "higher mathematician....We must not be put off because he is interested only in the higher abstractions: we have an equal right to be interested in the tangible" (1964: 20). The tangible fact about the Klein bottle that is glossed over in the higher abstractions of modernist mathematics is its hole. Because the standard approach has always presupposed extensive continuity, it cannot come to terms with the inherent discontinuity of the Klein bottle created by its self-intersection. Therefore, all too quickly, "higher" mathematics circumvents this hole by an act of abstraction in which the Klein bottle is treated as a closed object embedded in a hyper-dimensional continuum. To be sure, an "added dimension" is needed if the Klein bottle is not to be regarded as merely discontinuous. When limited to the three dimensions of ordinary 11 space, the Klein bottle cannot give expression to the dialectic of continuity and discontinuity. But the "added dimension," rather than being a continuum, must itself blend continuity and discontinuity. In a continuum, all interactions occur between fixed terms that are externally related. The point elements of which the continuum is composed are themselves related in this manner. As philosopher Milič Čapek put it in his reflection on classical space, "no matter how minute a spatial interval may be, it must always be an interval separating two points, each of which is external to the other" (1961: 19). In the words of Martin Heidegger, the continuum is essentially constituted by the "'outside-of-one-another' of the multiplicity of points" (1927/1962: 481). Given the fundamental exteriority of classical space, relations among objects and events contained within it must also be external. In the continuum, systems "interact through forces that do not bring about any changes in their essential natures...[they interact] only through some kind of external contact" (Bohm 1980: 173). Generally speaking then, the notion of the continuum implies that all boundaries are external in nature. This includes the point elements that bound space; the boundaries between and among interacting objects, systems, and events in space; and the figure-ground boundary that distinguishes an entity from its spatial context. One other kind of exterior boundary is implicit in the classico-modernist approach: the one that separates the object being observed from the subject that observes or analyzes it. Whereas objects are embedded in the extensive continuum, the subject entails discontinuity. In the language of Descartes, the object is res extensa and the subject res cogitans, thus unextended, not manifested in space. However, this distinction is complicated by the subtlety of the continuum idea. The continuum actually possesses its own aspect of discontinuity. Even though the points composing space are packed densely together, because these points are related to one another externally, the continuum is infinitely divisible; it can be indefinitely partitioned into ever smaller segments (the mathematician Charles Muses was thus prompted to describe the continuum as actually constituting an "infinite discontinuum" [1968: 37]). It naturally follows that the objects embedded in this medium are themselves partible; they can be rendered discontinuous. The discontinuity associated with the subject, on the other hand, signifies its transcendence of the continuum. So, whereas the breach one may produce in an object in fact reflects a property of the continuum, the subject constitutes a break with that continuum. Although it may rightly be said that classico-modernism favors continuity over discontinuity, what we are seeing is that there is indeed a place for discontinuity in the conventional paradigm, albeit a tacit or negative one. At the deepest level, it is the division of continuity and discontinuity that classico-modernism upholds. 12 4 RADICAL RECURSION AND THE DIMENSION OF DEPTH The classical concept of dimension has prevailed from the time of Descartes and Kant to the physics and mathematics of today. In mainstream science and philosophy, the exteriority of relations among objects in space, and between object and subject, has not been questioned in a fundamental way. Yet countercurrents do exist. We find evidence of these in the works of process-oriented thinkers such as Heidegger (1962/1972), Gendlin and Lemke (1983), and Bateson (see Neuman's topological interpretation of Bateson in this issue of SEED, and Harries-Jones's [1995] account of Bateson's recursive vision); in each case, internal dynamics are given precedence over the static externality of the spatial continuum (see also Rosen 1994, 2004a). One of the most explicit formulations of process dimensionality is found in the notion of depth advanced by the phenomenological philosopher Maurice Merleau-Ponty (1964). This idea provides us with an insight into dimension that permits us to surpass the limits of classico-modernism and arrive at a radically recursive understanding of space that is well suited for expressing the Kleinian dialectic of continuity and discontinuity. By way of introducing Merleau-Ponty's depth dimension, let us consider in greater detail the traditional dichotomy between the objects contained in space and their spatial container, or, as Plato put it, between "that which becomes [and] that in which it becomes" (1965: 69). A visible form "becomes," whereas that "in which it becomes" is "invisible and formless" (1965: 70). Whatever changes may transpire in the objects that "become," however they may be transformed, the containing space itself does not change. Indeed, for there to be change, there must be difference, contrast, dialectical opposition of some kind. But the point-elements that make up the classical continuum, rather than entailing opposition, involve mere juxtaposition. Unextended and thus devoid of inner structure, the elements of space possess no gradations of depth; no shading, texture, or nuance; no contrasts or distinctions of any sort. Instead of expressing the dialectical interplay of shadow and light, space itself is all light, as it were. A condition of "total exposure" prevails for the point-elements of the continuum, since these elements, having no interior recesses, must be said to exist solely "on the outside." All that can be said of the relations among such eviscerated beings is what Heidegger said: the points of classical space are "'outside-of-one-another'" (1927/1962: 481). So, rather than actively engaging each other as the beings that are contained in space seem to do, the densely packed elements of the classical container sit inertly side by side, like identical beads on a string. 13 In fact, even though the beings that dwell in such a space can be described as "actively engaged," we have seen that the quality of their interaction is affected by the context in which they are embedded: since the continuum is constituted by sheer externality, the relations among its inhabitants must also be external. Classical dynamics are essentially mechanistic; instead of involving a full-fledged dialectic of opposition and identity wherein beings influence each other from core to core, influence is exerted in a more superficial fashion, "only through some kind of external contact" (Bohm 1980: 173). We may say then that classical space contains dialectical process in such a way that it externalizes it, divesting it of its depth and vitality. It is the classico-modernist view of space that Merleau-Ponty calls into question. What he demonstrates is that the spatial continuum appearing to contain dialectical process actually originates from it. In his essay "Eye and Mind," Merleau-Ponty emphasizes the "absolute positivity" of traditional Cartesian space (1964: 173). For Descartes, space simply is there; possessing no folds or nuances, it is the utterly explicit openness, the sheer positive extension that constitutes the field of strictly external relations wherein unambiguous measurements can be made. Merleau-Ponty speaks of this space without hiding places which in each of its points is only what it is....Space is initself; rather, it is the in-itself par excellence. Its definition is to be in itself. Every point of space is and is thought to be right where it is-one here, another there; space is the evidence of the "where." Orientation, polarity, envelopment are, in space, derived phenomena inextricably bound to my presence [thus "merely subjective"]. Space remains absolutely in itself, everywhere equal to itself, homogeneous; its dimensions, for example, are interchangeable. (1964: 173) Merleau-Ponty concludes that, for Descartes, space is a purely "positive being, outside all points of view, beyond all latency and all depth, having no true thickness" (1964: 174). Challenging the Cartesian view, Merleau-Ponty insists that the dialectical features of perceptual experience ("[o]rientation, polarity, [and] envelopment") are not merely secondary to a space that itself is devoid of such features. He begins his own account of spatiality by exploring the paradoxical interplay of the visible and invisible, of identity and difference, that is characteristic of true depth: The enigma consists in the fact that I see things, each one in its place, precisely because they eclipse one another, and that they are rivals before my sight precisely because each one is in its own place. Their exteriority is known in their envelopment and their mutual dependence 14 in their autonomy. Once depth is understood in this way, we can no longer call it a third dimension. In the first place, if it were a dimension, it would be the first one; there are forms and definite planes only if it is stipulated how far from me their different parts are. But a first dimension that contains all the others is no longer a dimension, at least in the ordinary sense of a certain relationship according to which we make measurements. Depth thus understood is, rather, the experience of the reversibility of dimensions, of a global "locality"- everything in the same place at the same time, a locality from which height, width, and depth [the classical dimensions] are abstracted. (1964: 180) Speaking in the same vein, Merleau-Ponty characterizes depth as "a single dimensionality, a polymorphous Being," from which the Cartesian dimensions of linear extension derive, and "which justifies all [Cartesian dimensions] without being fully expressed by any" (1964: 174). The dimension of depth is "both natal space and matrix of every other existing space" (1964: 176). Merleau-Ponty goes on to observe that primal dimensionality must be understood as self-containing. This is illustrated through a discussion of contemporary art, and, in particular, the work of Paul Cézanne: "Cézanne knows already what cubism will repeat: that the external form, the envelope, is secondary and derived, that it is not that which causes a thing to take form, that this shell of space must be shattered, this fruit bowl broken" (1964: 180). In breaking the "shell," one disrupts the classical representation of objects in space. Merleau-Ponty asks: [W]hat is there to paint, then? Cubes, spheres, and cones...? Pure forms which have the solidity of what could be defined by an internal law of construction...? Cézanne made an experiment of this kind in his middle period. He opted for the solid, for space-and came to find that inside this space, a box or container too large for them, the things began to move, color against color; they began to modulate in instability. Thus we must seek space and its content as together. (1964: 180) The work of Cézanne is Merleau-Ponty's primary example of the exploration of depth as originary dimension. The foregoing passage describes Cézanne's discovery that primal dimensionality is not space taken in abstraction from its content, but is the unbroken flow from container to content. It is in this sense of the internal mediation of container and content that Cézanne's depth dimension is self-containing. Merleau-Ponty also makes it clear that the primal dimension engages embodied subjectivity: the dimension of depth "goes toward things from, as starting point, this body to which I myself am fastened" (1964: 173). In commenting that, "there are forms and defi15 nite planes only if it is stipulated how far from me their different parts are" (180; italics mine), Merleau-Ponty is conveying the same idea. A little later, Merleau-Ponty goes further: The painter's vision is not a view upon the outside, a merely "physical-optical" relation with the world. The world no longer stands before him through representation; rather, it is the painter to whom the things of the world give birth by a sort of concentration or coming-toitself of the visible. Ultimately the painting relates to nothing at all among experienced things unless it is first of all "autofigurative."....The spectacle is first of all a spectacle of itself before it is a spectacle of something outside of it. (1964: 181) In this passage, the painting of which Merleau-Ponty speaks, in drawing upon the originary dimension of depth, recursively draws in upon itself. Painting of this kind is not merely a signification of what is other, but a concrete self-signification that undercuts the external boundary between signifier and signified. In sum, the phenomenological dimension of depth as described by Merleau-Ponty is (1) the "first" dimension, inasmuch as it is the source of the Cartesian dimensions, which are idealizations of it; it is (2) a self-containing dimension, not merely a container for contents that are taken as separate from it; and it is (3) a dimension that blends subject and object concretely, rather than serving as a static staging platform for the objectifications of a detached subject. In realizing depth, we surpass the concept of space as but an inert container and come to understand it as an aspect of an indivisible cycle of action in which container, contained, and "uncontained"-space, object, and subject-are integrally incorporated. The work of Merleau-Ponty provides us with an insight into the nature of the "added dimension" that is required for the Kleinian signification of radical recursion. It would not be enough to say that the Klein bottle makes use of the dimension of depth to realize its dialectic of continuity and discontinuity-not if "makes use" connotes the operation of a model employing a containing medium to signify a meaning external to itself. It is perhaps more accurate to say that the Klein bottle is the depth dimension. For, rather than being a model contained as object-in-space, the Klein bottle-grasped in terms of depth-is the inseparability of object, space, and subject, the unbroken circulation of these intimated by Merleau-Ponty. It is the unique hole in the Klein bottle that plays the pivotal role. This loss in continuity is necessary. One certainly could make a hole in the torus, or in any other object in three-dimensional space, but such discontinuities would not be necessary inasmuch as 16 these objects could be fully assembled in space without rupturing them. It is clear that whether an object like the torus is cut open or left intact, the closure of the space containing that object will not be brought into question; in rendering such an object discontinuous, we do not affect the assumption that the space in which it is embedded is a continuum. Indeed, we have seen that the divisibility of an ordinary object derives from the infinite divisibility of the continuum itself. With the Klein bottle it is different. Its discontinuity does challenge the continuity of three-dimensional space as such, for the necessity of the hole in the bottle indicates that space is unable to contain the bottle the way ordinary objects appear containable. We know that for the Kleinian "object" to be brought to completion, assembled without a hole, an "added dimension" is required, and I am proposing that the dimension to be engaged is that of Merleau-Pontean depth (assuming we do not wish merely to skip over the hole by a continuity-maintaining act of abstraction, as in the standard mathematical stratagem for dealing with the Klein bottle). In the depth dimension, while the hole in the Klein bottle is no mere breach in an object in space, neither is it simply a rupture in space per se that corresponds to the subject. Rather, the Kleinian "hole" is in fact a dialectical (w)hole resulting from an act of selfintersection wherein the purported object does the "impossible": it passes unbrokenly through itself, and, in so doing, flows backward into its own subjective ground (in Merleau-Ponty's terms, it is "autofigurative"). Elsewhere, I noted the resemblance of the Klein bottle to the hermetic vessel of old alchemy (Rosen 1995). The design of the enigmatic vessel is essentially that of the uroboros, the serpent that consumes itself by swallowing its own tail. To contain itself, the serpent must intersect itself, an operation requiring a hole (corresponding to the opening that is its mouth). The hole in the Klein bottle is of this sort. It is neither solely a hole in a container, nor a hole in that which it contains, but the hole produced by the recursive act of self-containment that integrates the container with its contents in this way giving (w)holeness. The Kleinian process of self-containment enacted through the dimension of depth is surely no trivial recursion, no regression of meaning to simple self-identity. The Klein bottle refers to itself, but it also makes reference to what is other and a boundary is crossed. Of course, the boundary in question is not of the exterior sort so familiar to us; instead it is paradoxical, a boundary that is not a boundary (see Neuman, this issue of SEED, and Rosen 1997, 2004a). In passing through Kleinian depths from self to other (subject to object, the discontinuous to the continuum), we cross over the boundary to the "far side," yet at once remain on the "near side." In this way, while the self-other distinction is not just abrogated, the supremacy of this distinction is overcome and we realize a harmony of self and other so intimate that the prior meanings of these terms are trans17 muted. The erstwhile categorical purity of self and other is supplanted by an odd uroboric hybrid, a "hermaphroditic" fusion wherein self and other, though assuredly different, are one and the same. This profoundly paradoxical manner of self-reference is what I mean by radical recursion. 5 SEMIOTIC POSTSCRIPT The subtlety of the notion of radical recursion has not been exhausted by what I have written above. I will note another layer of meaning before I conclude. Consider the words "Klein bottle." Although these signifiers point to the depthdimensional structure that embodies the paradox of radical self-signification, the signifiers themselves-"K-l-e-i-n" and "b-o-t-t-l-e"-are but arbitrarily devised, conventionally agreed upon tokens that refer to their content in a merely external manner. These onedimensional typographic marks appearing on the two-dimensional surface of this page obviously fall short of tangibly delivering the three-dimensional Kleinian depth they signify. To the extent that "unmotivated" conventional marks constitute the primary mode of signification for this text, the old division between signifier and signified will be upheld and the meaning of the Klein bottle will remain an abstraction. In seeking to close the gap between signifier and signified, it might be feasible to place greater emphasis on our two-dimensional images of the Klein bottle (Figs. 2 and 3), or, better still, to work with a full three-dimensional model of this paradoxical structure (a model can be constructed with a flexible length of tubing such as that illustrated in Fig. 3). However, it should be clear by now that for the Klein bottle to be fully dimensioned, our model cannot be limited to an object in three-dimensional space. The Klein bottle must be realized in Merleau-Pontian depth. To this end, rather than regarding the Klein bottle as but an object appearing before us, something "out there" in space that we see and can handle, we must resist this compulsion of naïve realism and take the bottle as something that we read. What I am suggesting is that, while the Klein bottle cannot be actualized in the abstract universal medium of words alone, neither can it be brought to fruition as but a particular concrete thing; to be realized in depth, it must be realized as a hybrid of word and thing (a "general thing"; Merleau-Ponty 1968: 139), as a "tangible word" or iconic text. In the capacity of iconic sign, the Klein bottle can serve to "motivate" the process of signification by raising it from one-dimensional arbitrariness to a fully committed three-dimensional intercourse of signifier and signified. 18 In reading our iconic Kleinian text, we must of course read its hole. Instead of interpreting the hole as a gap in an ordinary object contained in space, we are to read it as an opening to a "higher dimension," and read that dimension "autofiguratively," grasping it as the prereflective source of our very own reading. The hole then becomes a (w)hole that we actualize in depth as we pass unbrokenly from text back to subtext in a uroboric act of radical recursion. Here the Klein bottle signifies radical recursion by signifying itself. Or in Peircian terms, we may say with semiotician Paul Ryan that the Klein bottle is a "sign of itself" (1993: 345–347). REFERENCES Barr, Stephen. 1964. Experiments in Topology. New York: Dover. Bohm, David. 1980. Wholeness and the Implicate Order. London: Routledge and Kegan Paul. Čapek, Milič. 1961. Philosophical Impact of Contemporary Physics. New York: Van Nostrand. Deleuze, Gilles, and Felix Guattari. 1987. A Thousand Plateaus: Capitalism and Schizophrenia. Minneapolis: University of Minnesota Press. 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Donato, eds., The Languages of Criticism and the Sciences of Man: The Structuralist Controversy, pp. 186–200. Baltimore: The Johns Hopkins University Press. Massumi, Brian. 1998. Strange Horizon: Buildings, Biograms, and the Body Topologic. Retrieved 15 February 2003, from the Indiana University Web Site: http://www.indiana.edu/~thinkmat/strange.doc. Merleau-Ponty, Maurice. 1964. Eye and Mind, in J. M. Edie, ed., The Primacy of Perception, pp. 159–190. Evanston, Ill: Northwestern University Press. ---. 1968. The Visible and the Invisible. Evanston, Ill.: Northwestern University Press. Merrell, Floyd. 1998. Simplicity and Complexity. Ann Arbor, Mich.: University of Michigan Press. Muses, Charles. 1968. Hypernumber and Metadimension Theory. Journal of Consciousness Studies, 1: 29–48. Neuman, Yair. 2003. Moebius and Paradox: On the Abstract Structure of Boundary Events in Semiotic Systems. SEED Journal, this issue. Plato. 1965. Timaeus and Critias. Trans. D. Lee. New York: Penguin. Rosen, Steven M. 1994. Science, Paradox, and the Moebius Principle. Albany, N.Y.: State University of New York Press. ---. 1995. Pouring Old Wine into a New Bottle, in M. Stein, ed., The Interactive Field in Analysis, pp. 121–141. Wilmette, Ill.: Chiron. ---. 1997. Wholeness as the Body of Paradox. Journal of Mind and Behavior, 18 (4): 391–424. ---. 2004a. Dimensions of Apeiron: A Topological Phenomenology of Space, Time, and Individuation. Amsterdam and New York: Editions Rodopi, in press. ---. 2004b. Topologies of the Flesh: A Multidimensional Exploration of the Lifeworld. Unpublished manuscript. Rucker, Rudolph. 1977. Geometry, Relativity, and the Fourth Dimension. New York: Dover. Ryan, Paul. 1993. Video Mind/Earth Mind: Art, Communications, and Ecology. New York: Peter Lang. 20 Spivak, Gayatri C. 1976. Translator's Preface, in J. Derrida, Of Grammatology, pp. ixlxxxvii. Baltimore: The Johns Hopkins University Press. Whitehead, Alfred North. 1978. Process and Reality. 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www.crossingdialogues.com/journal.htm ORIGINAL ARTICLE Crossing Dialogues Association Losing control: the hidden role of motor areas in decision-making OWEN P. O'SULLIVAN St. John of God / St. Vincent's University Hospital / University College, Dublin (Ireland) Decision-making has traditionally been viewed as detached from the neural systems of sensory perception and motor function. Consequently, motor areas have played a relatively minor role in discussions surrounding the control processes and neural origins of decision-making. Empiric evidence, catalysed by technological advances in the past two decades, has proven that motor areas have an integral role in decision-making. They are involved in the generation, modulation, maintenance and execution of decisions and actions. They also take part in a complex hierarchical assessment of multi-modal inputs to ensure that the most appropriate action is generated given the context presented. Clinical conditions such as, alien hand syndrome and utilisation behaviour exemplify the importance of these regulatory controls. This review charts the trajectory of our understanding of the hidden role of motor areas in decision-making and refl ects upon the implications of our deepened understanding. The convergence of evidence from multiple modalities underpinning our current knowledge is discussed and the potential applications thereof considered. Keywords: motor cortex, cingulate cortex, decision making, cognitive science, neuroanatomy, alien hand syndrome DIAL PHIL MENT NEURO SCI 2014; 7(2): 45-49 45 INTRODUCTION We like to think that our destiny is sculpted by the conscious decisions we make along the way. However, the simplicity of such a thought is as deceptive as it is alluring: humans tend to overestimate the degree of conscious control they wield over their decisions. Perhaps it is because of this fallacy that we are so taken aback by the blatantly irrational decisions we see ourselves and others make during times of turmoil, low mood or extreme anxiety. Simply put, it is all too easy to forget the degree to which we are governed by our emotions and instincts. We have a tendency to idealise ourselves as rational creatures capable of making clear, logical decisions and ultimately, actions based on the information at hand, the environmental context and our past experiences. Decision-making refers to the cognitive process of evaluating a number of possibilities, and selecting the most appropriate thereof in order to further a specifi c goal, or task. This faculty is a fundamental component of executive function. The debate as to whether, or not we have the capacity to make free choices is neither new, nor settled. However, the cross-cultural experiential phenomenon in humans of a subjective sense of control in volition is well-established (Sarkissian et al., 2010). Traditionally, psychological theories have considered the process of decision-making to be distinct, higher and separate from the neural systems of sensory perception and motor function. Motor areas have thus played a relatively minor role in discussions surrounding the control processes and neural origins of decisionmaking. Research narratives and hypotheses were dominated by the prevailing dogma that decision-making was simply composed of two broad stages with a signifi cant level of interaction between the two. The fi rst stage, occurring in the orbitofrontal cortex (OFC), encompassed predominantly cognitive or executive features of decision-making. The second stage, originating from the limbic regions, concerned itself with the infl uential role of emotion (Bechara et al., 2000). It is only in the past two decades that motor regions have begun to be afforded increased attention. They are, after all, known to be crucial in the planning, initiation and execution of movement. Action has traditionally been portrayed as representing the endpoint of cognition however, this Dialogues in Philosophy, Mental and Neuro Sciences DIAL PHIL MENT NEURO SCI 2014; 7(2): 45-49 O'Sullivan, 2014 assertion is being challenged and consequently the lines between executive and motor functions are fading. Action is increasingly being viewed as a cognitive process in its own right. Researchers from various fi elds within cognitive neuroscience are actively striving to re-defi ne the role of motor regions in the hope of gaining a deeper appreciation of their complexity and the extent of their involvement in the decision process. The nascent fi eld of "decision neuroscience" has presented innumerate fascinating and challenging questions. Are these motor regions purely limited to a "back-seat" role of silent effectors, or do they have a more complex modulatory function? What role do these areas have in decision-generation? How does learning affect this role and what happens when they are damaged? DISCUSSION Classical views of cortical motor area function In 1870, it was fi rst demonstrated experimentally that electrical stimulation applied to the pre-central cortex was capable of eliciting limb movements (Fritsch and Hitzig, 1870). Fulton (1935) fi rst proposed and initiated the cascade of subdivision of the motor cortex into primary and pre-motor regions. Intra-operative electrical stimulation later gifted neuroscience with the somatotopic map of the primary motor cortex, which famously became known as Penfi eld's homonculus (Penfi eld and Boldrey 1937; Penfi eld and Rasmussen, 1952). This was undoubtedly a watershed moment in the trajectory of our knowledge of motor function. The hierarchical and sensorimotor machine models later put forward by the prolifi c Hughlings Jackson (1958) were emphatically challenged by evidence from landmark anatomical studies on non-human primates in the 1990s demonstrating non-primary motor cortical areas with direct projections to the spinal cord (Dum and Strick, 1991). Division of labour a brief overview of the motor areas The primary motor cortex (M1) is responsible for the execution of actions and is somatotopically organised. Each side of the body is under contralateral hemispheric control. The premotor cortex is involved in action preparation. The lateral part of the premotor cortex is associated with linking objects in the environment with a repertoire of specifi c actions. It does so by taking advantage of the structural affordances proffered by the object presented and uses these heuristically to help determine the most appropriate action in a given context. The medial premotor cortex is also known as the supplementary motor area (SMA). The SMA is associated with postural control, locomotion and the performance of well-rehearsed actions: these are movements which require minimal monitoring of the surrounding environment (Graziano and Afl alo, 2007). The anterior cingulate cortex (ACC) has connections with the motor cortex and spinal cord and is also closely connected with the pre-frontal cortex (PFC). A functional overlap is suggested to exist in the ACC and hence, a translative role in converting intentions into actions has been proposed for this region (Paus, 2001). Conceptualising the convergence of cognition and action We are constantly making decisions in our lives. The sheer fl uidity with which it comes to us is astounding. Optimal, and seemingly effortless, human behaviour belies continuous analysis of incoming multimodal sensory information. Contextual factors are then weighted in deciding between competing potential actions and the most appropriate behaviour is eventually selected (Bestmann, 2006). The physical parameters of the ultimate action must also be determined prior to execution. But how to possibly reconcile all of these processes into a feasible, integrated model for decision-making and action behaviour? Cisek (2006) challenged the established notion that decision and action planning were serial processes spatially separate from one another. He proposed a computational model that sees potential decision and action alternatives competing simultaneously across multiple stages of the cortical hierarchy (Bestmann, 2006). The model he proposed was of a unifi ed process, as distinct from previous "separatist" hypotheses. The experimental paradigm employed simulation of anatomically-informed neural networks to demonstrate evidence of the involvement of specifi c premotor areas (dorsal premotor cortex i.e., PMd) in actual decision-making. Spatial 46 www.crossingdialogues.com/journal.htm features, salience and valence information were combined and weighted accordingly to ultimately determine which of the competing representations would come out on top. Decisions took place in an all-or-nothing manner once a certain "quenching threshold" has been met in the PMd (Grossberg, 1973; Bestmann, 2006). The model was supported by existing electrophysiological (Cisek and Kalaska, 2005) and behavioural data (Bock and Eversheim, 2000) based on similar suppositions. Connectional complexity of the cingulate cortex The ACC's extensive (Barbas and Pandya, 1989) motor, spinal cord and PFC connections make it a likely candidate for having a role in the translation or, modulation of decisions into actions. The cingulate sulcus in particular exemplifi es the high degree of connectional complexity observed in this area. It represents a confl uence of inputs from the PFC, M1 and SMA in addition to giving rise to corticospinal projections (Dum and Strick, 1991). Paus' (2001) review article on the role of the ACC discussed the convergence of cognitive and motor processes with the processes involved in maintaining the arousal state. Positron emission tomography (PET) has been employed to discern if the ACC is involved in the willed control of actions (Paus et al., 1993). Paus et al. used a behavioural model to try to identify whether differential ACC engagement was observed based on the output system used to execute the response. In the experiment, simple responses were challenged by the presence of over-learned stimulus-response alternatives. PET fi ndings confi rmed differential activation based on whether the motor output was manual, oculomotor or verbal. Electrophysiological correlates The error-related negativity (ERN) is an event-related potential (ERP) that peaks at approximately 100 ms after movement onset. It is characteristically associated with error detection and confl ict monitoring. The ERN's neural origins have been localised in the area of the ACC (Dehaene et al., 1994). The PFC-ACC connections have been shown to be crucial for ERN generation (Gehring and Knight, 2000). This is highlighted by the observation that patients with unilateral lesions to the lateral PFC display ERN in response to both erroneous and correct items; they are also less likely to demonstrate "corrective behaviour" (Paus, 2001). This evidence suggests a role for the ACC in corrective behaviour and judgement. Automatic motor responses Saccades and visual grasp refl exes are examples of automatic activation of motor responses. In these cases, perceptual processing of a visual stimulus occurs which culminates in the execution of a motor action without the intention of the observer to act (McBride et al., 2012). Visual objects can automatically generate motor responses and they do so via their structural affordances. These affordances refer to facilitative features of the object suggestive of its function and manner of use. When an object is observed, potential motor plans are generated based on data presented. Functional neuroimaging studies have demonstrated the exquisite sensitivity of these visual systems by showing that motor areas are activated by simply looking at an object (Grèzes and Decety, 2002), without necessarily the intention to act upon its presence. We are told from a young age, "we learn by doing, not by seeing" however, this neuroimaging evidence raises intriguing philosophical questions which blur the boundaries of thought and action, hitherto perceived as categorically distinct from one another. Lessons from clinical neurology Automatic activation plays a vital role in everyday life: facilitating desirable responses and inhibiting others is crucial for normal functioning in a dynamic environment rich in diverse visual stimuli. This ability to inhibit inappropriate responses is crucial: otherwise effi cient goaldirected behaviour cannot occur (McBride et al., 2012). Its central role in ensuring day-to-day functionality is laid bare in conditions such as, alien hand syndrome and utilisation behaviour, wherein such behaviour becomes exceedingly diffi cult. These striking conditions demonstrate what happens when the volitional restraints exerted to suppress these automatic actions in response 47 DIAL PHIL MENT NEURO SCI 2014; 7(2): 45-49 O'Sullivan, 2014 to an object are released. Patients with alien hand syndrome reach and grasp for objects in their environment at random. They are aware that the movements are occurring and that they originate from their own body however, they do not experience any subjective feeling of control over these actions whatsoever. (Scepkowski and Cronin-Golumn, 2003). In utilisation behaviour, similar phenomena are observed except in this case the actions are usually purposeful (Boccardi et al., 2002). For example, an individual with this condition may be irresistibly drawn to repeatedly opening and closing the drawers in a desk as they walk past. The brain region affected in these conditions is predominantly the SMA however, damage to the corpus callosum, parietal lobes and basal ganglia can also be implicated in similar conditions (Boccardi et al., 2002). Decision neuroscience: fertile ground at the crossroads of research Decision neuroscience is a relatively new fi eld that has grown steadily over the past two to three decades. As the above examples have illustrated, it has fi rmly shed its reliance on purely psychological paradigms and now transcends disciplines and modalities: encompassing everything from single-neuron recordings to non-invasive functional neuroimaging and everything in-between. It is more intellectually accessible and less abstract than other areas in neuroscience and hence, its popularity and lure is unsurprising. Additionally, the potential for exponential growth in our understanding carries obvious immediate appeal and will certainly bear potential for wider interest outside of cognitive neuroscience. The study of economics, marketing, artifi cial intelligence, investment psychology and risk management are to name but a few of its many plausible applications. The psychology of decision-making has an undeniable philosophical legacy. And this inheritance has proved diffi cult to avoid, if not inevitable. However, rather than being disheartened by this spectre, decision neuroscience promises to provide the empiric evidence needed to aid us in addressing the neural basis of many perplexing questions regarding concepts of free will, conscious and unconscious volition, agency and individual culpability. CONCLUSION Motor areas have an integral role in decisionmaking. They are involved in the generation, modulation, maintenance and execution of decisions and actions. They take part in a complex hierarchical assessment of multimodal inputs with a view toward ensuring that the most appropriate action is generated given the context. The psychopathology demonstrable in patients with frontal lobe syndromes offers insight into the crucial need for executive controls for normal functioning. In order for humans to engage intelligently and effi ciently with their environment, the integrity of these cortical connections must be ensured. Decision-making and action planning can however, no longer be considered as processes, for the most part, limited to the prefrontal cortices. Empiric evidence has proven otherwise and demonstrated that the increase in prominence of the motor areas in this regard is both timely and deserved. Furthering our understanding of cognitive systems such as this in a meaningful, cross-disciplinary and integrative manner remains an ongoing ambition previously precluded by technological limitations. Research into the applications of the neural basis of decision-making will no doubt prove to be a vibrant, highly competitive and potentially extremely lucrative pursuit. 48 REFERENCES Barbas, H, Pandya, DN. (1989) Architecture and intrinsic connections of the prefrontal cortex in the rhesus monkey. J Comp Neurol, 286:353-375. Bechara, A, Damasio, H, Damasio, AR. (2000) Emotion, decision making and the orbitofrontal cortex. Cereb Cortex, 10:295-307. Bestmann, S. (2006) A new unifi ed framework for making and implementing decisions. J Neurosci, 26:13121-13122. Boccardi, E, Della Sala, S, Motto, C, Spinnler, H. (2002) Utilisation behaviour consequent to bilateral SMA softening. Cortex, 38:239-308. Bock, O, Eversheim, U. (2000) The mechanisms of movement preparation: a precuing study. Behav Brain Res, 108:85-90. Cisek, P. (2006) Integrated neural processes for defi ning potential actions and deciding between them: a computational model. J Neurosci, 26:9761-9770. Cisek, P, Kalaska, JF. (2005) Neural correlates of reachwww.crossingdialogues.com/journal.htm 49 Corresponding Author: Dr. Owen P. O'Sullivan St. John of God/St. Vincent's University Hospital/ University College Dublin Basic Specialist Training Scheme in Psychiatry St. John of God Hospital, Stillorgan, Co. Dublin (Ireland) E-mail: [email protected] Copyright © 2014 by Ass. Crossing Dialogues, Italy ing decisions in dorsal premotor cortex: specifi cation of multiple direction choices and fi nal selection of action. Neuron, 45:801814. Deheane, S, Posner, MI, Tucker, DM. (1994) Localization of a neural system for error detection and compensation. Psych Sci, 5:303-305. Dum RP, Strick, PL. (1991) The origin of corticospinal projections from the premotor areas in the frontal lobe. J Neurosci, 11:667-689. Fritsch, G, Hitzig, E. (1870) Uber die electrische Erregbarkeit des Grosshirns. Arch Anat Physiol wiss Med, 37:300-332. Fulton, J. (1935) A note on the defi nition of the "motor" and "premotor" areas. Brain, 58:311-316. Gehring, WJ, Knight, RT. (2000) Prefrontal-cingulate interactions in action monitoring. Nature Neurosci, 3:516520. Graziano, MSA, Afl alo, TN. (2007) Mapping behavioral repertoire onto the cortex. Neuron, 56:239-251. Grèzes, J, Decety, J. (2002) Does visual perception of object afford action? Evidence from a neuroimaging study. Neuropsychologia, 40:212-220. Grossberg, S. (1973) Contour enhancement, short term memory, and constancies in reverberating neural networks. Stud Appl Math, 52:213-257. McBride, J, Boy, F, Husain, M, Sumner, P. (2012) Automatic motor activation in the executive control of action. Front Hum Neurosci, 6:82. Paus, T. (2001) Primate anterior cingulate cortex: where motor control, drive and cognition interface. Nat Rev Neurosci, 2:417-424. Paus, T, Petrides, M, Evans, AC, Meyer, E. (1993) Role of the human anterior cingulate cortex in the control of oculomotor, manual, and speech responses: a positron emission tomography study. J Neurophysiol, 70:453-469. Penfi eld, W, Boldrey, E. (1937) Somatic motor and sensory representation in the cerebral cortex of man as studied by electrical stimulation. Brain, 60:389-443. Penfi eld, W, Rasmussen, T. (1952) The cerebral cortex of man. Macmillan, New York. Sarkissian, H, Chatterjee, A, De Brigard, F, Knobe, J, Nichols, S, Sirker, S. (2010). Is belief in free will a cultural universal? Mind Lang, 25:346-358. Scepkowski, LA, Cronin-Golumb, A. (2003) The alien hand: cases, categorizations and anatomical correlates. Behav Cogn Neurosci Rev, 2:261-277.

Conditionals and Truth Functionality 1 Conditionals and Truth Functionality Rani Lill Anjum Introduction The material interpretation of conditionals is commonly recognized as involving some paradoxical results. I here argue that the truth functional approach to natural language is the reason for the inadequacy of this material interpretation, since the truth or falsity of some pair of statements 'p' and 'q' cannot per se be decisive for the truth or falsity of a conditional relation 'if p then q'. This inadequacy also affects the ability of the overall formal system to establish whether or not arguments involving conditionals are valid. I also demonstrate that the Paradox of Indicative Conditionals does not actually involve a paradox, but instead contains some paralogistic elements that make it appear to be a paradox. The discussion of the paradox in this paper further reveals that the material interpretation of conditionals adversely affects the treatment of disjunctions. Much has been said about these matters in the literature that point in the same direction. However, there seems to be some reluctance against fully complying with the arguments against the truth functional account of conditionals, since many of the alternative accounts rely on the material conditional, or at least on an understanding of the conditional as a function of antecedent and consequent in a similar sense as the material conditional. My argument against truth functionality indicates that it may in general involve similar problems to treat conditionals as such functions, whether one deals with theories of truth, assertability or probability. 1. The inadequacy of the material conditional If natural conditionals 1 'if p then q' have truth conditions that are truth functional, there seems to be no option to the material conditional interpretation of these truth conditions, namely that 'p  q' is defined as true whenever we have one of the following combinations of truth-values for 'p' and 'q': (TT), (FT) or (FF). This means that a conditional would be 1 I will use the term "natural conditionals" for expressions of the following kinds: 'if p then q', 'q if p', 'p unless q', 'p only if q', 'supposing p, q', and so on. That these expressions are "natural" should be taken to mean that they are expressions in our ordinary (natural) language. This should be understood as opposed to the material conditional that is merely defined as a function within a logical system. I thus distinguish between natural language expressions and logically defined terms and functions. Rani Lill Anjum 2 truth functionally defined as false only when the antecedent 'p' is true and the consequent 'q' is false; otherwise true. However, we do not normally consider a natural conditional 'if p then q' as true or false according to these truth functionally defined truth conditions. For instance, the following conditionals (though true within the material interpretation) seem all to be, at best, false: 'If all philosophers learn to swim, then there would be no war'; 'If eggs come from hens, then we can use them (the eggs) to make omelet'; 'If time is money, then Bill Gates can afford to go to the opera every day'; 'If I can't do without Beetle Bailey, then Mort Walker will make his sons continue his cartoon career'. This kind of discrepancy between material and natural conditionals has caused an extensive debate, attracting both advocates and critics of the material conditional. The opening passage in Farrell's paper 'Material Implication, Confirmation, and Counterfactuals' illustrates the confusion that the material interpretation of natural conditionals can cause in the classroom: Students of truth-functional logic frequently regard material implication to be patently absurd. Most of us who teach elementary logic have encountered intelligent students who frustratedly exclaimed something to the effect that: Any logic which pronounces true a sentence such as, "If the moon is a green cheese, John F. Kennedy was the 35th President of the United States," is illogical. A great deal of printer's ink has been spilled in the attempt to rationalize away the paradoxes of material implication... I am at last inclined to throw in the towel and admit the endeavor is fruitless, that the paradoxes and problems generated by material implication are intolerable embarrassments. 2 The main problem with the truth functional approach to conditionals is that we normally use conditionals for asserting or denying a kind of dependency relation between some facts or events, and not to claim a certain combination of truth-values for any pair of statements 'p' and 'q'. In natural conditionals, one obviously cannot infer a conditional dependency relation merely from the truth-values of an arbitrary pair of statements, since natural conditionals are in essence hypothetical, while the material conditional must have assigned truth-values in order to be determined as true or false. It seems to be a general view that the case of counterfactuals represents the main problem for the material interpretation of conditionals. If the antecedent is false, the material 2 Farrell (1979), p. 383. Conditionals and Truth Functionality 3 conditional is defined as true, and this is in fact a problem. In my opinion, though, this problem is no more serious than the fact that a true consequent results in a true material conditional, or the fact that two true or two false statements necessarily give a true material conditional. None of these problems will be solved if one treats counterfactuals as special cases. The conditional 'If I was born in 1711, then I was born the same year as David Hume' is true independently of the truth-values of the antecedent and the consequent. It is true because David Hume was born in 1711. Further, the conditional 'If I was born in 1887, then I was born the same year as David Hume' is false independently of the truthvalues of the antecedent and the consequent. None of these aspects can be explained or accounted for in an extensional, truth functional approach. In his paper 'The Logic of Implication', Balzer stresses the point that conditionals are used to express a dependency relation between the antecedent and the consequent, while the assumption of such a relation is no part of the material conditional: The most puzzling aspect of implication is that of the relation between the antecedent and the consequent. It would seem natural to suppose there must be a connection of some sort between the antecedent and the consequent for a meaningful implication to be made. For example: "If this is water, then it contains hydrogen and oxygen", "If you touch a red-hot poker, then you will be burned" and "If this is lemon, then it will taste sour" would all be regarded as reasonable implications from antecedent to consequent. However, many logicians admit as true implications "If 2 + 2 = 5, then New York is a large city" and "If a horse is a fish, then I can jump over the moon". These are implications without any apparent connection between the antecedent and the consequent. 3 In Aristotelian logic, one did not have a truth functional account of language, where the contentual aspects were disregarded. This means that, within this logic, it is possible to treat the conditionals 'If Socrates is a man, then he is mortal' and 'If Socrates is mortal, then he is a man' as logically different, even though it is true both that Socrates was a man and that he was mortal. Since Aristotelian logic first of all is a system of syllogisms, it was an uncontroversial fact that the truth of a single, existential conditional of the form If a is G, then a is H is dependent on the truth of a corresponding categorical statement All Gs are Hs. The conditional 'If Socrates is a man, then he is mortal' is then true based on the truth of the categorical statement 'All men are mortal', while the conditional 'If Socrates is 3 Balzer (1990), p. 253. Rani Lill Anjum 4 mortal, then he is a man' is false because the categorical statement 'All mortals are men' is false. In Fregean logic, however, not only conditionals that are derived from categorical statements are included, but also conditionals of the form 'if p then q', that are not singular instantiations of a corresponding categorical statement. Hence, other conditions had to be given for the truth of a conditional, namely the truth functional account. However, Garland, a Roman medieval logician, helps us illustrate how a purely truth functional account of conditionals is fruitless, since the same combinations of truth-values can give a true or a false conditional, dependent on the relation expressed: A consequence is true in four ways. 1 One is composed of two true propositions, as in 'If Socrates is a man, he is an animal'... 2 Another is composed of two false propositions, as in 'If Socrates is a stone, he is inanimate'... 3 Another one is composed of a false antecedent and a true consequent, as in 'If Socrates is an ox, he is an animal'... 4 Still another is composed of parts neither of which is true or false, such as you can discern in this example: 'If it were a man, it would be an animal'; for neither of these is true or false... On the other hand, a consequence is false in five ways. 1 One is false with both components being true, as in 'If Socrates is an animal, he is a man'. 2 Another consists of two false components, e.g. 'If Socrates is inanimate, he is a stone'. 3 Still another one is made of false antecedent and true consequent, as in 'If Socrates is a stone, he is a man.' 4 Another one is composed of two parts neither of which is either true or false, e.g. 'If Socrates were an animal, he would be a man.' 5 And still another one is false which has a true antecedent and a false consequent. 4 We note that on Garland's account, if we assign different truth-values to the antecedent and the consequent, we can find more different kinds of circumstances in which a natural conditional can be false than it can be true. This is opposed to the definition of a material conditional, according to which a conditional is true unless the antecedent is true and the consequent false, which is merely one of Garland's five alternatives for false conditionals. As a result, the material conditional is true of more possible circumstances than the 4 Boh (1993), p. 4-5. Conditionals and Truth Functionality 5 corresponding natural conditional, which implies that the material conditional is not an adequate representation of natural conditionals. 2. The harmfulness of the material conditional For reasons like these, it seems to be commonly accepted among most logicians that the material conditional is not an adequate representation of natural conditionals. Many will however still claim that this replacement is harmless. This means that one believes that tests for validity will judge arguments containing natural conditionals as valid if the corresponding arguments containing material conditionals are valid, though not necessarily vice versa. 5 In other words, one claims that formally valid inferences are also contentually valid, that is; true premises guarantee a true conclusion in a formally valid inference. Unfortunately, given the lack of adequacy, it is impossible for the material interpretation of conditionals to be harmless. In his article 'A Confusion About If..Then', Edwards demonstrates that if the material conditional 'p  q' is true of more possible circumstances than the corresponding natural conditional 'if p then q', it will be possible to construct material conditional expressions that are true of more, as well as expressions that are true of fewer, possible circumstances than their corresponding natural expressions: Copi claims that 'If p then q' may assert more than 'p  q'. Suppose that in a given case it does. This means that 'p  q' is true of more possible circumstances than is 'if p then q'. It is assumed by Copi that this means that a premise containing 'p  q' is true of more possible circumstances than the corresponding premise containing 'if p then q'. But this may or may not be the case, as a few examples will show. There is an equal chance that the premise containing 'p  q' will be true of fewer possible circumstances than the corresponding premise containing 'if p then q'. 6 Edwards then gives examples of material conditional expressions that are true of, respectively, more and fewer possible circumstances than their corresponding natural expressions: 7 5 That the material conditional interpretation of natural conditionals is harmless, seems to be the view of for instance: Barker (1997); Copi (1965), p. 17-22; Grice (1989); Jackson (1987) and (1991); Quine (1966), p. 12; and Richards (1969). 6 Edwards (1973/74), p. 85. 7 Edwards (1973/74), p. 86. Rani Lill Anjum 6 True of more possible circumstances True of fewer possible circumstances p  q (p  q)  r [(p  q)  r] (p  q)  r r  (p  q) [r  (p  q)] (p  q) [(p  q)  r] (p  q)  r (p  q)  r r  (p  q) (p  q)  r This means that while 'p  q' is logically weaker than 'if p then q', asserting less than the natural expression, '(p  q)' is too strong in relation to 'not (if p then q)'. The negated material conditional is true when 'p' is true and 'q' is false, while the negation of the natural conditional can also be true under other circumstances. According to Edwards, this has serious implications for the validity of arguments. An argument is contentually valid if there is no possibility of the premises being true and the conclusion false at the same time. The replacement of natural conditionals with expressions containing material conditionals will then affect whether or not it is more, or less, likely that the argument containing them will get the combination of true premises and false conclusion. These matters will again necessarily affect the determination of the formal validity of the argument. For instance, an argument containing '(p  q)' as premise will be formally valid of more possible circumstances than an argument containing 'not (if p then q)' as premise, since the material conditional expression is true of fewer possible circumstances than the corresponding natural expression. This means that the argument containing the material expression will have true premises in fewer possible circumstances than the argument containing the natural expression. Likewise, an argument containing '(p  q)' as conclusion will be formally valid of fewer possible circumstances than an argument containing 'not (if p then q)' as conclusion. This is because one in the argument containing the material conditional expression will have a true conclusion in fewer possible circumstances than the corresponding argument containing the natural expression. When expressions containing material conditionals get more complex than the ones above, it will be difficult, if not impossible, to know beforehand whether they are true of more or fewer possible circumstances than their corresponding natural expressions. Moreover, whether they appear as premises or conclusions also affects the formal validity of the Conditionals and Truth Functionality 7 argument, and makes it even more complicated to control the outcome of the test. One example that I think clearly demonstrates Edwards' point, is the following proof of God's existence: 8 P1 If God does not exist, then it's not the case that if I pray, my prayers are heard P2 I don't pray C God exists Given a plausible formal interpretation where the natural conditional is interpreted as a material conditional, this argument is formally valid; i.e. valid according to the rules of formal logic. 1. p  (q  r) P1 2. q P2 3. q  r 1, 2, T 4. (q  r) 3, T 5. p 1, 4, T 6. p 5, T This particular argument has its main weakness in the first premise. First of all, it contains a negated material conditional in the consequent, which is false of more possible circumstances than its corresponding natural expression. This means that when we introduce 'q' as the second premise, we must negate '(q  r)', and thus also 'p'. The other weakness in this argument is that the first premise contains nested material conditionals, which makes the outcome even more unpredictable. It seems clear, then, that Edwards is justified in his conclusion that the material interpretation of natural conditionals is not and cannot be harmless on the assumption that the former is true of more possible circumstances than the latter. 3. The paradox of indicative conditionals I will now argue that this inadequacy and harmfulness of the material conditional also affects the treatment of other connectives, and in particular disjunctions. To demonstrate this point, I will present and discuss Jackson's Paradox of Indicative Conditionals (PIC), which is supposed to demonstrate that it is impossible to deny the equivalence between 8 The example is found in Edgington (1991), p. 187. Rani Lill Anjum 8 natural and material conditionals. 9 The PIC is constructed with the help of the following principles: 1. The Truth Functionality Principle: The material conditional 'p  q' is equivalent to the disjunction 'not-p or q'. 2. The Uncontested Principle: The indicative conditional 'if p then q' implies the material conditional 'p  q'. 3. The Passage Principle: The disjunction 'p or q' implies the indicative conditional 'if not-p then q'. All these principles seem plausible, but together they allegedly prove that the material conditional is equivalent to the corresponding natural conditional: 1. p  q  not-p or q (the Truth Functionality Principle) 2. p or q  if not-p then q (the Passage Principle) 3. not-p or q  if p then q (from substitution in 2 and the rules for negation) 4. p  q  if p then q (from 1 and 3) 5. if p then q  p  q (the Uncontested Principle) 6. if p then q  p  q (from 4 and 5) This equivalence between material and natural conditionals is hard to accept given the account of adequacy and harmlessness above. However, a paradox is said to occur when we reject the equivalence and introduce a fourth principle: 4. The Principle of the Paradox of Material Implication: 'not-p, therefore, if p then q' and 'q, therefore, if p then q' are invalid forms of inference. If one assumes that the first three principles hold, then the inferences mentioned in the fourth principle must be valid. The fourth principle supports the plausible impression that they do not seem to be valid. It is then impossible for all these four principles to be true together. But which of the principles may be rejected? The source of the problem seems to be the disjunction in the Truth Functionality Principle, which appears to be formulated as a natural disjunction 'not-p or q', but simultaneously gets treated as purely truth functional. While the conditional is presented both with a 9 Jackson (1987), p. 4-6. Conditionals and Truth Functionality 9 natural and a formal form in the PIC, namely 'if p then q' and 'p  q', the disjunction is only given with a natural form 'p or q'. But in fact, the disjunction is treated differently in the Truth Functionality Principle from how it is treated in the Passage Principle. In the truth functional treatment of the disjunction in the Truth Functionality Principle, what is actually stated is the occurrence of certain combinations of truth-values for the antecedent and the consequent of the material conditional. This means that we do not thereby assert anything about the facts or events referred to by 'p' and 'q', but only about their truthvalues. In the Passage Principle, however, the disjunction cannot be treated as a truth functional expression like in the Truth Functionality Principle. In order to imply the indicative conditional, we need to treat the disjunction as stating something about the objects and events that the disjuncts are about, since the indicative conditional certainly is intended to be about these objects and events. Hence, the PIC demands that we identify different level of communication, thus producing a phenomenon that Place calls linguisticism; the "equating of the existence of a situation with the proposition it makes true." 10 Now consider the following formulations of a disjunctive expression: (1) 'not-p or q' (2) 'Either "p" is a false sentence or "q" is a true sentence.' (3) 'Either it is false that p or it is true that q.' In (1) we assert a disjunctive relation between whatever is represented by 'p' and 'q', i.e. we are stating something about the relations between the states of affairs signified by 'p' and 'q'. In (2) and (3), however, we assert a disjunctive relation between the truth-values of an arbitrary pair of statements 'p' and 'q', i.e. we are stating something about the allowed combinations of truth-values associated with arbitrary 'p' and 'q'. We are thus not saying something (directly) about the relations of the states of affairs signified by 'p' and 'q'. Rather, we are defining the truth functional conception of the disjunction. Thus (1) on 10 Place (1996), p. 106-107. Rani Lill Anjum 10 the one hand, and (2) and (3) on the other hand, are by no means the same statements. They may be related in a determinate fashion, but they are actually stating something about (usually) different kinds of objects. To make the distinction explicit between the disjunction in the Truth Functionality Principle and the disjunction in the Passage Principle, then, the disjunction in the Truth Functionality Principle ought to be given a different expression than 'not-p or q', namely one that expresses a relation between truth-values, as in (2) and (3) above. The disjunction in the Passage Principle must, on the other hand, be one that corresponds to (1), where the relation expressed is between events or facts referred to by 'p' and 'q', and not between truth-values. Accordingly, one of the following principles should replace the Truth Functionality Principle: TTP': The material conditional 'p  q' is equivalent to the disjunction 'either "p" is false or "q" is true'. TTP'': The material conditional 'p  q' is equivalent to the disjunction 'either it is false that p or it is true that q'. When this difference between the two disjunctions in the PIC is not explicitly marked, we are led to believe that the disjunction in the Truth Functionality Principle and the disjunction in the Passage Principle are identical, hence the paradox. In the Truth Functionality Principle, the disjunction is treated as truth functional, in the sense that there is no need for a contentual relation between 'not-p' and 'q' to take place in order to make the disjunction true: One is truth functionally allowed to infer 'p or q' from the truth of 'p'. I can for instance on the knowledge that 'I am going straight home after work' infer that 'I'm going straight home after work or I am meeting the King at the pub'. In the Passage Principle, however, the disjunction is treated as non-truth functional. It must then be understood as stating that there is a relation between the state of affairs referred to by 'p' and the state of affairs referred to by 'q', in order to avoid that the disjunction in the Passage Principle is based on the truth of one of the disjuncts. – Or worse, that it follows from the truth of one disjunct and the independent falsity of the other. Otherwise, the passage from 'p or q' to 'if not-p then q' would not be a valid inference. Conditionals and Truth Functionality 11 So even though I am going straight home after work, and not meeting the King at the pub, I can truth functionally infer the disjunction 'I'm going straight home after work or I am meeting the King at the pub'. From this the Passage Principle allows me to infer that 'If I'm not going straight home after work, I am meeting the King at the pub', which is false, even though the disjunction, if it is understood truth functionally, is true, as is the corresponding material conditional. 11 It is therefore necessary, in order to avoid this kind of formally valid – but contentually invalid – inferences, to insist that the natural expression 'p or q' is understood non-truth functionally in the passage principle, in a sense that justifies the inference to a natural conditional. The disjunction in the Passage Principle must then be a natural non-truth functional disjunction, in order for the principle to be valid, while the disjunction in the Truth Functionality Principle is truth functional, and thus not the kind of natural disjunction that would make the passage in the Passage Principle valid. Since the disjunction in the Truth Functionality Principle now differs from the disjunction in the Passage Principle, step 1 in our proof above of the equivalence between 'if p then q' and 'p  q' must be rejected. And since the disjunctions in the first and the third principle now differ, step 4 must also be rejected. The validity of the inference from the material conditional to the natural conditional is hence not established. Place's distinction also elucidates the distinction between natural and material conditionals. While the natural conditional is used to assert (or deny) a kind of dependency relation between two events or facts, the material conditional asserts that 'if 'p' is true, then 'q' is true', provided that we are allowed to use a natural conditional to define or give meaning to the material conditional. So at best we can say that a material conditional expresses a kind of dependency relation between a pair of truth-values. But then we have already interpreted the material conditional according to our natural conditional understanding of sufficient and necessary conditions, and of dependency or causal relations. This means that we can at least not say that a natural conditional can be reduced to or explained by means of the truth functionally defined material conditional. 11 Edgington uses examples of this kind in her discussion of truth-functionality of conditionals in her (1991) paper. According to Grice, such examples demonstrate the distinction between what is false and what is misleading. Rani Lill Anjum 12 4. A triviality result for the material conditional Brandom offers a triviality result of the material conditional in his paper 'Semantic Paradox of Material Implication'. The paradox is based on conditionals with a simplistic structure, that is, sentences of the form 'p  q' for primitive 'p' and 'q'. This is opposed to in the classical paradoxes of the conditional, he says, that often involve the embedding of one conditional in another or the use of some connective other than the conditional. In Jackson's PIC, for instance, we have ' p  (p  q)' and 'q  (p  q)'. Brandom's triviality result is that given the truth functional material interpretation of natural conditionals, we get that any consistent assignment of truth-values to conditionals determines the truth-values of all the primitive sentences 'p', 'q' and so on: "This is absurd, because no set of purely hypothetical facts should determine all of the categorical facts."12 However, as already mentioned, the material conditional cannot represent the hypothetical character of natural conditionals since the material conditional 'p  q' will only have an assigned truth-value if both its components 'p' and 'q' have assigned truthvalues. Brandom's paradox demonstrates the reverse problem, namely that (consistent) assignments of truth-values to a set of material conditionals involve assignments of truthvalues to their components. I will not go into details of his paradox here, but merely notice that these results are not unexpected considering the discussion in this paper. Many philosophers agree that the material conditional is not a perfect (or anyway, exhaustive) interpretation of natural conditionals. It seems however to be a widespread view, in particular after Grice, that the material conditional expresses the truth conditions of natural conditionals, at least of the indicatives, while some conditions of assertability or probability must be recognized in addition. This is for instance the view of Jackson, who finds that the material conditional lacks robustness with respect to the truth of the antecedent, a property that he finds necessary for the assertability of a natural conditional, but not for its truth. Furthermore, Lewis claims that the material conditional holds between all true statements within a possible or actual world, even though it cannot be used to account for counterfactual or subjunctive conditionals. Adams and Edgington, on the other hand, deny that conditionals have truth conditions at all, and maintain that they only have probability conditions, that are found by considering the probability of 'p', 'q' and 'p & q'. 12 Brandom (1981), p. 129. Conditionals and Truth Functionality 13 In this connection it can also be mentioned that Jackson determines robustness in terms of conditional probability. Any account of natural conditionals that in one way or the other relies on the material conditional, is dependent on a valid and sound proof of the equivalence between the material and natural conditionals. Considering all the problems involved in insisting on this equivalence, including Brandom's triviality result, the burden of proof now lies with its advocates. 5. Concluding remarks I have argued here that the material conditional is not an adequate or harmless interpretation of natural conditionals, and that the main problem with this interpretation is that it is truth functional. Truth functionality is, when you come down to it, an approach to natural language that is concerned merely with different combinations of truth-values of statements or sentence elements. My main objection to this is that natural language is not truth functional; it does not have a truth functional structure. The truth functional material conditional is founded on an understanding of natural conditionals, but only with focus on combinations of truth-values. In principle, it is totally correct to say that "in a true natural conditional, if the antecedent is true, then the consequent must be true as well". But this differs from the conditional relations that we claim to hold between facts, events and states of affairs. Within a truth functional logical system, conditionals are represented as "relations" between linguistic entities. A linguisticism occurs when we say that "in a true conditional, the antecedent is a sufficient condition for the consequent and the consequent is a necessary condition for the antecedent". Again we find ourselves talking about relations between linguistic entities, not between situations or events in our world. So it seems that one underlying problem that affects conditionals, is to keep track of when we talk about the world and when we talk about truth of linguistic entities. In examining the logical properties of a conditional, we move without noticing from the first to the second. Even though we use and understand conditionals on the first level, when we try to analyze them within a logical system, we find ourselves operating on the level of linguisticism. This is why we are led to believe that we talk about natural conditionals Rani Lill Anjum 14 when we characterize them as expressing "a relation between the antecedent and the consequent". However, in claiming or expressing a natural conditional, we do not really say anything about its truth-values, or even about the relation between the antecedent and the consequent. Rather, the conditional itself expresses a relation between facts, events or states of affairs in our world. We seem to forget that what we are interested in when we assert, investigate or hear a conditional, is the world in which we find ourselves exploring, understanding, stating, communicating and, even sometimes, trying to find out what is true and what is false. Acknowledgments I want to thank Ernest W. Adams for useful and detailed comments on an earlier version of this paper. I will also express my thanks to Jan H. Alnes, Ray E. Jennings, Laurie Keenan, Johan W. Klüwer, Tarjei M. Larsen, Johan Arnt Myrstad and Mariann Solberg, who have contributed in various ways to the improvement of this paper. I am especially grateful to Myrstad, who has read and commented numerous versions of this paper and in general contributed to my work with expertise and persistent optimism. Thanks also to the University of Tromsø and the Research Council of Norway, NFR, for financial support. References Adams, E.W.: The Logic of Conditionals, Dordrecht, Reidel 1975. Balzer, N.: 'The Logic of Implication', The Journal of Value Inquiry 24, 1990, pp. 253268. Barker, S.F.: 'Material Implication and General Indicative Conditionals', The Philosophical Quarterly 47 (187), 1997, pp. 195-211. Boh, I.: Epistemic Logic In the Later Middle Ages, London, Routledge 1993. Brandom, R., 'Semantic Paradox of Material Implication', Notre Dame Journal of Formal Logic 22, April 1981, pp. 129-132. Copi, I.: Symbolic Logic, New York, Macmillan 1965. Edgington, D.: 'Do Conditionals Have Truth-Conditions?', In Jackson (ed.) 1991, pp. 176201. Edwards, J.S.: 'A Confusion About If..Then', Analysis 34, 1973-4, pp. 84-90. Farrell, R.J.: 'Material Implication, Confirmation, and Counterfactuals', Notre Dame Journal of Formal Logic 20 (2), 1979, pp. 383-394. Conditionals and Truth Functionality 15 Grice, P.: Studies in the Way of Words, Cambridge, Harvard University Press 1989. Jackson, F.: Conditionals, Oxford, Blackwell 1987. Jackson, F. (ed.): Conditionals, Oxford, Oxford University Press 1991. Jackson, F.: 'Assertion and Conditionals', in Jackson (ed.) 1991, pp. 111-135. Lewis, D.: 'Probability of Conditionals and Conditional Probabilities', in Jackson (ed.) 1991, pp. 76-101. Place, U.T.: 'Structural Properties: Categorical, dispositional or both?', Dispositions: A debate, Armstrong, Martin, Place (eds.), London, Routledge 1996, pp. 105-125. Quine, W.: Methods of Logic, London, Routledge, 1966. Richards, T.J.: 'The Harmlessness of Material Implication', Mind 78, 1969, pp. 417-422.

Revista GeoAamazônia http://www.geoamazonia.net/index.php/revista/index eISSN: 2358-1778 Universidade Federal do Pará Programa de Pós-graduação em Geografia Revista GeoAmazônia Belém v. 7, n. 13 p. 53–67 2019 Página 53 DESCRITIVIDADE COMO UM PRINCÍPIO DA GEOGRAFIA AMAZÔNICA: O CHAMADO DE EIDORFE MOREIRA DESCRIPITIVITY AS A PRINCIPLE OF AMAZON GEOGRAPHY: THE CALL OF EIDORFE MOREIRA DESCRIPCION COMO UN PRINCIPIO DE LA GEOGRAFÍA AMAZONICA: EL LLAMADO DE EIDORFE MOREIRA Wallace Wagner Rodrigues Pantoja Secretaria de Estado de Educação do Pará, Belém (PA), Brasil [email protected] RESUMO No ensaio pretendo revalorizar o princípio descritivo na geografia amazônica contemporânea, como nos apresentou o geógrafo paraense Eidorfe Moreira (1960). Lateralmente, chamo atenção de geógrafos/as amazônicos/as para sensibilidade de sua obra, pouco presente na bibliografia dos cursos de formação em Geografia no Pará. A estratégia metodológica é descritivo-interpretativa, de tom fenomenológico. Concluo: a recusa da descrição se instala por um efeito preconceituoso de nossa formação atual em relação aos procedimentos considerados tradicionais; a produção de saber, ansiosa pela explicação/análise, pode produzir violência ética; a generalização apressada de certas ideias sobre a Amazônia, via estudos geográficos, instala um corte artificial entre o simbólico e o emocional no ato de fazer geografia em nossa região. Palavras-chave: Descrição interpretativa; Singularidades amazônicas; Fenomenologia. ABSTRACT In the essay I intend to revalue the descriptive principle in the contemporary Amazonian geography, as presented to us by the geographer from Pará Eidorfe Moreira (1960). Laterally, I call the attention of Amazonian geographers to the sensitivity of his work, which is not present in the bibliography of the training courses in Geography in Pará. The methodological strategy is descriptive-interpretative with a phenomenological tone. I conclude: the refusal of the description is installed by a prejudiced effect of our current formation in relation to the procedures considered traditional; the production of knowledge, eager for explanation/analysis, can produce ethical violence; the abstraction which flirts with abstractionism imposed by the hasty generalization of certain ideas about the Amazon, from geographical studies, installs an artificial cut between the symbolic and the emotional in the act of making geography in our region. Keywords: Interpretative description; Amazonian singularities; Phenomenology. RESUMEN En el ensayo pretendo revalorizar el principio descriptivo en la geografía amazónica contemporánea, como nos presentó el geógrafo paraense Eidorfe Moreira (1960). Quiero llamar la atención de geógrafos amazónicos para la sensibilidad que presenta su obra, que está poco presente en los curso de formación de geografía en Pará. La planificación metodológica es descriptiva – interpretativa con un tono fenomenológico. Concluyo: el rechazo de la descripción se instala por un efecto DESCRITIVIDADE COMO UM PRINCÍPIO DA GEOGRAFIA AMAZÔNICA: O CHAMADO DE EIDORFE MOREIRA Revista GeoAmazônia Belém v. 7, n. 13 p. 53–67 2019 Página 54 discriminativo de nuestra formación actual de procedimientos considerados tradicionales; la producción de saber, ansiosa por la explicación / análisis, puede producir violencia ética; la abstracción -que se coquetea con el abstraccionismoimpuesta por la generalización apresurada de ciertas ideas sobre la Amazonia, a través de estudios geográficos, instala una división artificial entre lo simbólico y lo emocional en el acto de hacer geografía en nuestra región. Palabras clave: Descripción interpretativa; Singularidades amazónicas; La fenomenología. INTRODUÇÃO Descrição, a definição básica do que deveria ser a atividade realizada por geógrafas/os. Porém, hoje, na melhor das situações, "descrever" é momento necessário para apresentar os dados do real que serão explicados/analisados – atividades nobres por excelência que Tradicional que precisamos ultrapassar para atingir um lado ou outro da linha abissal1 da modernidade. São posições hegemônicas, aqui e ali questionadas, mas o questionamento não tem força para recompor os termos do debate. Descrever é momento menor a ser superado pela geografia explicadora e analítica. Não só explicadora e analítica, mas apressada para transformar a realidade, embalada pela afirmação autocontraditória de Marx (1980, original 1845): "Os filósofos têm apenas interpretado o mundo de maneiras diferentes; a questão, porém, é transformá-lo"2. Afirmação que, assimilada pelas ciências humanas, eriça nossa vontade de potência. Sua autocontradição está na separabilidade implícita entre interpretar e transformar, bem como secundarização do ato de interpretar, sem falar que sua sentença em favor da ação já embute, nela mesma, uma interpretação3. A referência ao pensamento marxiano se justifica pela nossa formação calcada na bibliografia da Geografia Crítica (ao menos no campo da Geografia Humana) que, apesar dos evidentes avanços, parece contribuir para um efeito difuso da recusa a priori de procedimentos ou métodos colados ao pensamento tradicional – percebidos como não 1 Faço referência ao debate levantado por B. S. Santos – com foco na decolonialidade – sobre o corte que as epistemologias modernas teriam feito entre saberes científicos e outros saberes. Porém, na medida em que a decolonialidade marca sua posição frente a outros modos de pensar modernos, quase como uma grife que etiqueta o que coloniza e o que não coloniza o pensamento, pode produzir deslocamentos, mas não rupturas crítica das linhas abissais, colocando em risco a própria prescrição de B. Santos (2007) sobre a "ecologia de saberes", apesar de seu alerta agostiniano para eterna autorreflexividade de quem pensa de maneira pós-abissal. 2 Teses sobre Feuerbach. Disponível em: https://www.marxists.org/portugues/marx/1845/tesfeuer.htm. 3 Na crítica ao Programa de Gotha, Marx sentencia que: "cada passo do movimento real é mais importante do que uma dúzia de programas" (MARX, 2012, p. 20. Originalmente 1875, publicado em 1891). E é chamativo – não ativa, um video curto de Heidegger criticando esta Tese de Marx pode ser encontrado em https://www.youtube.com/watch?v=96xeh_6vYU0. na realidade até meio óbvio – que a frase famosa ilustre a capa da edição da Boitempo com um balão dito pela caricatura de Marx. Em tempo, ao polarizar interpretação e ação para fazer frente a uma filosofia pretensamente fundamentariam a ação no plano socioespacial – e, na pior, um estorvo herdado da Geografia DESCRITIVIDADE COMO UM PRINCÍPIO DA GEOGRAFIA AMAZÔNICA: O CHAMADO DE EIDORFE MOREIRA Revista GeoAmazônia Belém v. 7, n. 13 p. 53–67 2019 Página 55 participativos ou engajados – contra o qual marxistas, incluindo geógrafos, performam. Se o interpretativo pode ser desvalorizado, o descritivo não só pode como deve ser! Ainda que nenhuma interpretação ou explicação esteja dissociada da descrição (MOREIRA, 2012, p. 46. Original de 1960). Já escutei em banca de doutorado – e não só nesta situação – que a argumentação estava "descritiva demais"! É verdade que a fala pode ser direcionada à qualidade inferior da descrição, mas o fato é que o descritivo tem reduzido valor no conjunto do trabalho sobre o fenômeno geográfico, mais ainda, seu valor parece ser fraqueza, demérito (MOREIRA, 2012). [...] descrever significa para o geógrafo moderno uma função destituída de mérito científico, função consequentemente negativa, tornando-se como tal um fator de descrédito para os estudos geográficos. Dir-se-ia que a descrição constitui uma sorte de "pecado original" da Geografia, pecado de que ela se deve remir e quanto antes, sob pena de invalidar sua formação científica (MOREIRA, 2012 [originalmente 1961], p. 45). situação se complica ainda mais quando estamos falando de Geografia da e sobre a Amazônia, onde o desconhecimento geográfico exigiria trabalhos efetivamente descritivos, preteridos em pouco contextualizada na realidade vivida nos diferentes lugares da "região". As aspas se justificam na medida em que a região – como escala-signo – se impõe de maneira apriorística sobre outras conexões escalares e escalas geográficas na maioria dos debates sobre o amazônico Quais as vantagens da descritividade, defendida por Eidorfe Moreira e, ao fazer geográfico da Amazônia atual? Este ensaio se propõe responder esta questão. A proposta não caminha na simples aceitação do que nos disse Moreira em 1960, mas diálogo tensível – demarco a recusa da descrição de maneira geral, posteriormente busco articular o sentido de descritividade ao fazer-saber geográfico em termos metodológicos, éticos e estéticos, por fim, considero, a partir do exposto, consequências importantes ao trabalho geográfico na Amazônia. 4 Totalização que se torna certo tipo de profecia autorrealizada de maneira retórica e não argumentativa. Há uma totalidade, por si, dariam conta de erigir uma totalidade apreensível – ao lermos os trabalhos, salvo honrosas exceções, temos a sensação frustrante de que tal totalização anunciada não se realizou. 4 tendência estranha na produção da geografia amazônica para supor que certos termos, tais como totalização e Eidorfe Moreira, geógrafo paraense (nasceu na Paraíba, veio com menos de 2 anos ao Pará) chamava atenção à desvalorização do ato de descrever por parte dos geógrafos: Esta tendência não arrefeceu nas últimas décadas, ao contrário, generalizou-se. A relação à análises e explicações algo apressadas, desejosas de totalização (PANTOJA, 2018). tensão jamais apartada de abertura sensível ao Outro (PANTOJA, 2018) –. Inicialmente, DESCRITIVIDADE COMO UM PRINCÍPIO DA GEOGRAFIA AMAZÔNICA: O CHAMADO DE EIDORFE MOREIRA Revista GeoAmazônia Belém v. 7, n. 13 p. 53–67 2019 Página 56 ANTIDESCRIÇÃO NA FORMAÇÃO ATUAL DE GEÓGRAFOS/AS AMAZÔNICOS/AS Moreira (2012), discutindo a importância do que chamou descritividade5, atenta ao fato desta ser subestimada pela nova metodologia em voga – referência implícita à Geografia Teorético-quantitativa que ganhou força no Brasil nos anos de 1960 e 1970. Eu acrescento a Geografia Crítica com sua totalização explicativa em oposição à fragilidade/parcialidade "modeladora" da Teorética e "descritiva" da Geografia Tradicional (SANTOS, 2002; MORAES, 2005) ou de um de seus polos epistemológicos (GOMES, 2005). A descrição, a enumeração, a classificação dos fatos referentes ao espaço são momentos de sua apreensão, mas a Geografia Tradicional se limitou a eles; como se eles cumprissem toda a tarefa de um trabalho científico. [...] sempre concluindo com elaborações do tipo formais, a-históricos e, enquanto tais, abstratos (sem correspondência com os fatos concretos) (MORAES, 2005, p. 40). É absolutamente questionável que este guarda-chuva chamado "Geografia Tradicional" tenha se limitado à descrição (ou enumeração ou classificação), além disso, mesmo reconhecendo na descrição um momento da apreensão dos fatos, é preterido por outros momentos (interpretação, análise, explicação), não dando conta do trabalho científico, produzindo resultados a-históricos e "abstratos". E, sem fulanizar, é interessante como nossos objetivos de pesquisa fogem da descrição até mesmo quando deveria figurar enquanto objetivo secundário. Minha formação em geografia e, penso, a formação de muitos geógrafos e geógrafas atuais, teve como aspecto central a depreciação do descritivo e sua rotulação como procedimento de geografia ultrapassada, não histórica e desconectada dos fatos concretos. Há razões sedimentadas6 no imaginário da ciência geográfica amazônica para crítica ao descritivo – e sua recusa como procedimento central – por parte de um seguimento numeroso de geógrafos: a descrição seria nivelamento asséptico das coisas e relações, negação do participativo em privilégio da observação "de fora", morte da dialética porque destruiria a tensão existencial em favor da ausência de uma concepção (ou método) que viabilize "[...] a compreensão organizada dos móveis essenciais da vida" (LUKÁCS, 1965, p. 80). 5 Contido no livro "Ideias para uma concepção geográfica da vida", originalmente publicado em 1960, com edição comemorativa organizada por Maria Stella Faciola Pessoa Guimarães, publicada pela Secretaria Municipal de Educação de Belém (SEMEC) em 2012. 6 Faço referência ao processo de Sedimentação de conceitos da qual nos fala Husserl (CF. DERRIDA, J. Edmund Husserl s Origin of Geometry. An Introduction). A sedimentação como um longo processo de agregação conceitual que no presente aparece como pronto, mas que pode ser reativado na direção dos vários horizontes históricos que formaram o referido conceito. DESCRITIVIDADE COMO UM PRINCÍPIO DA GEOGRAFIA AMAZÔNICA: O CHAMADO DE EIDORFE MOREIRA Revista GeoAmazônia Belém v. 7, n. 13 p. 53–67 2019 Página 57 Entretanto, um equívoco dessa crítica ou, ao menos, sua traficância para outros campos – e o herdamos em nossa formação geográfica contemporânea – é a incompreensão de que toda descrição é, necessariamente, interpretativa. Não há descrição sem intencionalidade (visada posicional que tensiona o mundo), inevitável entrelaço entre o pesquisador e o que se quer expor em processo de [...] fixação de nossa experiência sensorial com respeito às coisas: uma sistematização das impressões sugeridas pelas suas qualidades, relações e modos de ser, implicando como tal certa presunção de certeza relativamente ao que se estuda ou descreve – e isto contraria o estado de volubilidade que caracteriza o pensamento moderno (MOREIRA, 2012, p. 47). Moreira encontra na tendência "volúvel" da sua contemporaneidade a aversão ao descritivo – porque ato de fixação das coisas – o que revela algo sobre a geografia produzida em seu tempo e como ele se colocava diante dela. Uma virada estava em curso contra o qual pretendia erguer trincheira e acabaria perdendo, tanto que sua presença em nossa formação é quase ausente. Não invalido a crítica à "volubilidade" que, de certa maneira, apreende o tom da modernidade líquida (BAUMAN, 2001), o coroamento dos afobados7 (HAN, 2015, p. 37) ou a tendência ao achatamento constitutivo da modernidade tardia, onde tudo parece vazamentos de superfícies desorientadoras8 em uma "nova falta de profundidade" (JAMESON, 1984). Embora volubilidade não deva ser confundida com dinamicidade ou transformação abrupta, se vivemos/criamos/reproduzimos espaços movediços, será preciso inventar ou retomar métodos suficientemente reconfiguráveis para interpretá-los e descrevê-los. Obviamente, há um tipo de descrição formalista que responde aos protocolos de publicação que pouco contribui à descritividade de que nos fala Moreira. Porém, não é reduzindo a descrição à sua caricatura de inventário de recursos – caricatura feita por não poucos geógrafos críticos, tais como Moraes (2005), Santos (2002) já citados – que podemos operar mudança efetiva nos modos de reapropriação desta atitude no fazer geografia. Reabilitar o descritivo como princípio para Geografia da Amazônia não significa marcar oposição a priori entre a descrição e outras pretensões fundamentais à investigação; nem defender um uso da descrição que foi (e não foi só) inventariante, colonizador e 7 Han cita diretamente Nietzsche: "Por falta de repouso, nossa civilização caminha para uma nova barbárie. Em nenhuma caráter da humanidade fortalecer em grande medida o elemento contemplativo". 8 A conexão entre volubilidade e modernidade tardia e sua falta de profundeza, inclusive teórica, é tanto mais interessante na medida em que Moreira exerce a descrição enquanto modo próprio de conhecer científico que não tem profundeza de sentido (MOREIRA, 2012, p. 48, recorrendo a HUSSERL), explicitando outro sentido de "superfície de expressão" que não é "falta de profundidade". outra época os ativos isto é, os inquietos, valeram tanto. Assim, pertence as correções necessárias a serem tomadas quanto ao DESCRITIVIDADE COMO UM PRINCÍPIO DA GEOGRAFIA AMAZÔNICA: O CHAMADO DE EIDORFE MOREIRA Revista GeoAmazônia Belém v. 7, n. 13 p. 53–67 2019 Página 58 classificatório9. Significa assumir a descritividade como: a) procedimento metodológico central ao enfrentamento de questões amazônicas simplesmente desconhecidas, muitas vezes precariamente descritas para já passar, apressadamente, à explicação ou análise; b) criação do conteúdo de expressão das relações humanas e intermundanas, sua impressionabilidade (MOREIRA, 2012, p. 48) que a descrição em termos estéticos/cênicos pode e deve evocar10, exercitando a "[...] inexaurível capacidade de significação" (MOREIRA, 2012, p. 49), que não se confundiria com mero subjetivismo; c) reconhecimento do sentido ético de-limitante para com a criação da ciência geográfica, na medida em que descrição está na ordem da aparência, da superfície existencial que, pelo dobramento entre-nós11, será incomensurável para além de qualquer profundeza pretendida e, por isso, exigindo sempre renovados esforços para atingir este superfície de contato. DESCRITIVIDADE – EXPERIÊNCIAS DO FAZER ENTRE-NÓS A recusa e a má-fé12 para com a descritividade revela a vergonha pela origem da Geografia como ciência de lugares e não de homens13 (LA BLACHE, 1913), diante de certo modo de compreensão desta origem que se generalizou, apesar de Dardel (2011) – e, em seus termos, do próprio Moreira (NUNES, 1989, p. 27)14 – acreditar que Geografia é antes relação essencial com a Terra do que ciência, uma situação-limite enquanto experiência que demanda outros caminhos, por exemplo, para geohistória da "região" que, mesmo nas abordagens 9 Seria um exercício esclarecedor rastrear – como gostava de dizer Moraes – como a crítica da Geografia Tradicional se tornou recusa de qualquer método tradicional de fazer geografia, em uma posição menos crítica e mais preconceituosa e desconhecedora do que se admitiria nos cursos de formação em Geografia, particularmente no Pará. 10 Assim não oponho o descritivo ao narrativo, como faz Luckács (1965), mas também não diluo simplesmente um no outro, já que embora sejam – ao menos em termos formais – de ordem diferentes, só se realizam em uma contribuição efetiva a partir da situação existencial em que são demandados. 11 Merleau-Ponty (2012), nos fala da dobra como o ponto em que o verso toca o anverso e aí que a negatividade verdadeiramente existe, de onde se pode ver coisas. 12 Próximo ao sentido sartreano de esconder a angústia que não se pode esconder de si mesmo, apesar de querer disfarçar para si e para outros – a angústia aparece. 13 La Bache provoca esta afirmação tão evocada pelos críticos para apedrejar – não sem propriedade – a dita Geografia Tradicional. No mesmo texto, La Blache (1913, p. 297) dedica um subtópico ao método descritivo: "La géographie se dislingue comme science essentiellement descriptive Non pas assurément elle renonce explication étude des rapports des phénomènes de leur enchaî nement et de leur évolution sont autant de chemins qui mènent Mais cet objet même oblige plus que toute autre science suivre minutieusement la méthode descriptive". 14 Benedito Nunes (1989) faz a apresentação das Obras Reunidas de Eidorfe Moreira, chama atenção para como Moreira enraíza sua visão de mundo na geografia, que é mais do que um saber, é relação existencial reveladora, como as situações-limite de que nos fala Jaspers (1958), tais como a morte, onde não é possível divisar o que vem após ou mesmo vencer, de modo que é preciso um esforço interpretativo e compreensível sempre renovado um salto. Guimarães (2012; 2015) expõe esta posição de Nunes em relação a obra de Moreira. Já Pereira (2014), sugere uma possível leitura de Dardel por Moreira, já que este era leitor de geógrafos franceses do período e usa o termo "geograficidade" cunhado por Dardel. O mais provável é que atmosfera fenomenológico existencial marcou a ambos, Dardel e Moreira. para esclarecer (e nos esclarecer na) a situação-limite e poder vivê-la como fracasso incontornável que demanda DESCRITIVIDADE COMO UM PRINCÍPIO DA GEOGRAFIA AMAZÔNICA: O CHAMADO DE EIDORFE MOREIRA Revista GeoAmazônia Belém v. 7, n. 13 p. 53–67 2019 Página 59 críticas, segue o modelo rígido que se inicia na ocupação europeia (passando pelo período pombalino e economia gomífera, pontuada por eventos singulares como a Cabanagem), com vaga menção às formações espaciais originárias, salvo raras exceções generalistas e com frágil conexão entre geografia e antropologia arqueológica e linguística amazônicas. Outras ciências, tão modernas quanto a Geografia, parecem não sofrer tanto pelo fantasma descritivo. Não é sem razão que Geertz (2008, p. 13) abre A Interpretação das Culturas, clássico da antropologia, reunindo descrição densa e interpretação15. Todorov (2014), ao anunciar seu projeto de estudo do campo simbólico afirma que se manterá descritivo, porque pretende dar conta da abrangência das teorias e investigar as condições de emergência das "vontades de interpretar". Ele sabe que a estratégia de descrição fundamenta sua interpretação; esta, sem àquela, seria virtualmente impossível. Ainda que possamos levantar diferenças entre descrição densa, método descritivo na filosofia da linguagem e descritividade geográfica, há um fundo comum: qualquer inteligibilidade abrangente exige este contato com a extensão superficial na qual nos enredemos, cuja variabilidade de significação incontornável demanda trabalho infindável de fixação (ainda que, fatalmente, parcial), podendo trazer importantes resultados que integrem o escopo científico, filosófico e, alguns acreditam, até artístico. Este fundamento renegado deveria ser encarado de maneira educativa, enquanto exercício do fazer geográfico, mesmo por iniciantes – estudantes, jovens pesquisadores ou pessoas que, intuitivamente, fazem da Geografia uma cosmovisão16 (MOREIRA, 2012). Nenhuma geografia é possível sem descrição – que não é só gênero literário, nem simples enumeração (MOREIRA, 2012) – "[...] forma específica de fixar e relacionar os fatos em função do espaço, o que importa em dizer um conhecimento da vida em termos extensivos, com isso a sua unidade e expressão de conjunto" (MOREIRA, 2012, p. 50). É possível vislumbrar aqui a pretensão totalizadora e niveladora17 da descrição – assim como algo de naturalismo "em função do espaço" – mas também a extensividade vivida que se expressa numa relação entre geógrafo e paisagem. Esta tensão é notada por Nunes (1989) e Pereira (2014), este prefere falar de tensão articulada, embora eu pense que a tensão é já articulação. Descrever, como exercício, é tensionamento entre os limites de um geógrafo em 15 O título do capítulo é: "Uma descrição densa: por uma teoria interpretativa da Cultura". 16 Pereira (2014) interpreta a concepção do trabalho de Eidorfe Moreira em termos de método que permite "desdobrar" sua obra, chama atenção para como o mesmo entende a geografia não só como ciência, mas como cosmovisão e mundivivência. Benedito Nunes (1989, apud GUIMARÃES, 2012) como já mencionado na nota anterior, vai mais além, entende que para Moreira, a geografia é "situação-limite da existência", tal como propôs Jaspers. 17 É como Lukács se refere à descrição nos romances criados por expoentes de seu tempo, como Zola e Flaubert: "A narração distingue e ordena. A descrição nivela todas as coisas" (LUKÁCS, 1965, p. 62). DESCRITIVIDADE COMO UM PRINCÍPIO DA GEOGRAFIA AMAZÔNICA: O CHAMADO DE EIDORFE MOREIRA Revista GeoAmazônia Belém v. 7, n. 13 p. 53–67 2019 Página 60 apreender o mundo e a extensão espacial existente não docilmente disponível à apreensão. Mesmo que na ordem da aparência, se dá entre uma vontade de comunicação para si e para os outros, entre objetividade e subjetividade, entre o fixar e o relacionar, revela a tensão do descritivo no narrativo (RICOEUR, 1994) mais do que sua polaridade dialeticamente amarrada em tese como faz Lukács (1965). E embora o narrativo esteja ligado ao tempo (RICOEUR, 1994; CASEY, 2001) assim como o descritivo através do "processo evolutivo" da paisagem (MOREIRA, 2012, p. 50), há incontornável fundamento espacial que configura o ato de narrar (ENTRIKIN, 1991; MALPAS, 2001) e, acrescento, o ato de descrever em Geografia; porque posiciona18 lugares, sujeitos e o sujeito que narra/descreve o/no mundo que não está já pronto, mas constitui-se na descrição colaborativa e em choque do campo científico que, mais do que nos fazer saber certos aspectos, viabiliza um quadro de interpretabilidade que vai se transformando justamente pelas tensões não só descritivas ou narrativas, mas também explicativas, etc. que fissuram e provocam o conhecimento. O que não significa atribuir o conhecer ao subjetivismo, como parece à primeira vista19, mas afirmar que nosso conhecer o mundo depende dos quadros de interpretabilidade que se constituem no e constituem o real pela tensão descritiva, narrativa, explicativa e vivida da ciência geográfica em uma ressonância entre nós, o Outro20 e o mundo, fazendo-se21. Penso que este "fazer" tem como um dos pilares o exercício descritivo por jovens geógrafas/os, tendo em vista a prática da auto-educação do olhar geográfico com "boas ou más aplicações" (MOREIRA, 2012, p. 51) daí resultantes. Assim, promover a co-educação metodológica, que não se constitui mera repetição "do jeito certo e já pronto de fazer ciência"22, mas experimentação mesmo da descritividade como contribuinte do aprofundamento deste exercício ao tematizar a realidade amazônica por conhecer, em grande medida, capturada nas grandes linhas explicativas que enviesam os olhares e; ao mesmo 18 Posicionar pode ser uma vontade de controle que perseguirá sempre os geógrafos, evocando um aspecto ético do nosso trabalho que abordo mais adiante. 19 Embora não seja, nem de longe, algo novo em ciência, desde pelo menos Heisenberg, passando por Feyerabend e Prigogine. O subjetivo é incontornável porque exprime a conexão simbólico-emocional pessoal e social (GONZÀLEZ-REY; MARTINÊS, 2017) que configura o fazer-saber, inclusive os mais objetivos. 20 Estou aqui pensando na Fenomenologia do Rosto de Lévinas. 21 Esta ideia é diretamente inspirada na noção de construção do mundo e seu conhecimento sistemático de que da imagem de simples formigas tateando o ambiente fixo que vai se descobrindo na medida em que se tateia. 22 Uma crítica que Feyerabend (2010) faz ao modelo de ciência que temos é uma suposição de que as mesmas tiveram ao longo de sua afirmação no campo em que são trabalhadas. De certo modo ele se aproxima da noção de sedimentação conceitual de que fala Husserl, expresso na nota 6. nos fala Feyerabend (2010), quando exemplifica o ato de conhecer o mundo muito mais como um ambiente de ressonâncias que nos transforam e transformam o mundo a medida que avançamos no seu conhecimento ao invés metodologias são empregadas de maneiras prontas, se apagando a história de erros, teimosias, recuos que as DESCRITIVIDADE COMO UM PRINCÍPIO DA GEOGRAFIA AMAZÔNICA: O CHAMADO DE EIDORFE MOREIRA Revista GeoAmazônia Belém v. 7, n. 13 p. 53–67 2019 Página 61 tempo, pode ser vislumbre de realidades amazônicas cujo contato superficial – incontornavelmente superficial em não poucos casos – exigiria um respeito descritivo junto aos trabalhos explicativos/analíticos sobre as mesmas realidades. Este empenho diante do processo de construção do conhecimento dos Outros Espaços23 e, sobretudo, dos Lugares Vividos por Outros, não se dá apenas pela possibilidade de abertura das experimentações de saber, embora fundamentais. A descritividade – enquanto atividade "menor" – também evoca o senso de aproximação por contato dos Lugares, inscrevendo uma dimensão ao mesmo tempo ética e emocional do saber-fazer. Ética porque se relatar a si mesmo24 já é atividade temerária tendo em vista o que entendemos por verdade ou, ao menos, a realidade proximal que auxilia numa relação compreensiva; relatar o Outro e seu(s) Lugar(es) nos exige comprometimento que vai além do esclarecimento, na direção do encontro efetivo que interrogue os lugares amazônicos de dentro dos mesmos e nos interrogue nesta relação, de maneira que a descrição possa assumirse co-participante, tornando-se aprofundamento existencial neste contato superficial entre nós. Assim, ainda que Moreira me desperte para descritividade na Geografia, não posso concordar com muito de sua argumentação que superlativa o papel da Geografia como: "[...] a mais alta aplicação do método descritivo no campo científico, uma vez que mostra os fatos na sua mais ampla escala de grandeza, quer dizer, na sua plena configuração no espaço" (MOREIRA, 2012, p. 50). Sem falar que deixa termos, tais como "alta aplicação", "plena configuração", "escala", entre outros, bastante nebulosos e até mesmo retóricos, talvez pelo tom ensaístico claramente assumido do texto. O próprio uso do método descritivo não é questionado em seus princípios de ordem, a não ser em seus bons ou maus resultados25, o que torna aceitável a crítica de Almeida (2008) ao conceito de Amazônia cunhado por Moreira e encampado pela SPVEA (Superintendência de Plano de Valorização Econômica da Amazônia), que lança mão justamente de rubricas descritivas para se estabelecer, ao denunciar que "Os modelos para produzir tais critérios [de 23 A referência óbvia é o texto de Foucault "De Outros Espaços", onde prefere falar dos "espaços de fora" já que os fenomenólogos, segundo ele, se ocuparam muito dos "espaços de dentro". E embora tenha um incontestável apelo à imaginação, sem falar que é bem gostoso de ler, não deixa de assumir um triunfalismo colonizador ao colocar os navios e as civilizações desbravadoras como as verdadeiramente sonhadoras e tudo fora delas apenas espionagem, policiamento e desilusão. 24 Judith Butler (2015) tem um texto com este título e versa sobre este argumento em um sentido mais abstrato e Entretanto, há óbvios paralelos entre relatar e descrever ou, o que me parece mais enfático, a descritividade está na base de performances como o relato. 25 Moreira (2012, p. 51) enfatiza que: "Como qualquer método, a descrição pode ter boas ou más aplicações, pode ser utilizada de modo apropriado ou não, e é nisso que reside o ponto crucial em questão". Porém, precisemos questionar os fundamentos ontológicos de um método e não apenas se ele, já existente, terá bons ou maus empregos, como quer Moreira. amplo. Vale ainda explicitar que enquanto ela fala de relato eu estou dialogando em torno do descritivo. DESCRITIVIDADE COMO UM PRINCÍPIO DA GEOGRAFIA AMAZÔNICA: O CHAMADO DE EIDORFE MOREIRA Revista GeoAmazônia Belém v. 7, n. 13 p. 53–67 2019 Página 62 conceituação e delimitação da região, presente no livro de Eidorfe Moreira: Amazônia – o conceito e a paisagem, de 1958], considerados 'objetivos' e 'racionais', são de inspiração naturalista, amarrados em conceitos biológicos, que permeavam inclusive os argumentos demografistas e as categorias censitárias do IBGE" (ALMEIDA, 2008, p. 29. Inserções minhas). Portanto, não advogo uma aceitação pura e simples da descritividade tal qual idealizada por Moreira, mas recolocar – em diálogo com ele – a importância de descritividade enquanto exercício metodológico, ético e inseparável de um componente emocional para o enfrentamento de questões geográficas na Amazônia na contemporaneidade, não só em termos de resultados, mas de processos de experimentação formativa de geógrafas/os. O que me leva a um último acento que aparece neste texto de Moreira de maneira não tão explícita, mas que deve emergir: a cisão artificial entre o simbólico e o emocional na feitura das descrições geográficas. Em um pequeno texto, bastante acessível e com insights convidativos – inicialmente pensado como conferência de temas filosóficos para crianças/jovens – Didi-Huberman costura que A partir de Nietzsche [...] é toda a vida sensível que é questionada – como na poesia e na literatura [...]. A vida sensível será descrita em sua energia, inclusive passional, e não somente prescrita em seus deveres de razão e ação [...] Maurice Merleau-Ponty dirá que o evento afetivo da emoção é uma abertura efetiva [...] um tipo de conhecimento sensível e de transformação ativa de nosso mundo (DIDI-HUBERMAN, 2016, pp. 24-26. Grifos no original). Eu lia ao mesmo tempo e por razões diferentes o texto de Didi-Huberman e de Eidorfe Moreira26, anotei a passagem acima no meu exemplar de Ideias para uma concepção geográfica da vida. Por sinal, imediatamente após o texto que me inspirou a escrever, Moreira abre outro: "Geografia e Poesia", onde ensaia uma ontologia artística da geografia na medida em que: Toda alma poética é uma alma povoada e alimentada de paisagens, como se o sentimento poético estivesse em função de um certo grau de 'geograficidade'27, se assim podemos dizer, das nossas representações [...] de 26 Este ensaio surgiu destes cruzamentos textuais e do convite para uma Roda de Conversa sobre Geografia Cultural e Humanista na Amazônia (ocorrida na Univesidade Estadual do Pará em 23/11/2018), na qual me propus relacionar elementos fenomenológicos presentes na Geografia de Eidorfe Moreira e Amélia Nogueira, um geógrafo e uma geógrafa amazônicos que inscreveram modos de conexão entre Geografia, Paisagem e Espaço Vivido em perspectiva regional. 27 Já comentei em nota anterior (cf. nota 14), as possíveis conexões entre o pensamento de Dardel e Moreira, PEREIRA, 2014) e sua forte afinidade de estudo pela literatura de geografia e história francesa da primeira metade do século XX (COELHO, 2012; PEREIRA, 2014). O que me parece mais importante é o fato da força criativa e claramente envolvida na atmosfera fenomenológico-existencial de Moreira passar despercebida, DESCRITIVIDADE COMO UM PRINCÍPIO DA GEOGRAFIA AMAZÔNICA: O CHAMADO DE EIDORFE MOREIRA Revista GeoAmazônia Belém v. 7, n. 13 p. 53–67 2019 Página 63 um dado meio – [...] esse 'meio' visto menos como 'fisiografia' do que como 'atmosfera' (MOREIRA, 2012, p. 54). Este apelo à geograficidade, esta importância do sensível que pode ser acessado, inicialmente, pelo exercício da descrição geográfica, exige repensar o próprio procedimento em conexão com a dimensão emocional do pesquisador em ato de pesquisa, bem como dos sujeitos envolvidos/participantes em sua vida sensível, tendo em vista o comprometimento com a atmosfera geográfica partilhada. Assim, a própria descrição pode ser abertura ao conhecer sensível, constitui ação – e o é em termos etnometodológicos (GARKINKEL, 1984) – transformação ativa como acredita Merleau-Ponty. O que afirmo acima não está presente no texto de Moreira – que não ignora o aspecto subjetivo, mas seguramente não o encara como estou encarando a emoção na criação da ciência, muito embora suas descrições sejam surpreendentes em termos emocionais –, me parece que é para onde é preciso avançar se a descritividade merece ser revalorizada para fins geográficos. A descrição não pode e não deve ser um capítulo enfadonho tendo em vista encadear sumariamente os envolvidos nos processos pesquisados, a descrição deve ser uma feitura da imersão nos lugares, ou o quanto seja possível esta feitura em termos de representação que evoque o emocional, abrindo assim uma conexão entre ciência e subjetividade (GONZÁLEZREY; MARTINÊS, 2017). Afinal, mesmo com algum reconhecimento da subjetividade, sobretudo por geógrafos da fenomenologia e pós-estruturalistas, normalmente, somos dicotômica do que deveriam ser o objetivo e o subjetivo. E mesmo quando reconhecido, a subjetividade é um termo amplamente utilizado em textos da geografia humanista e até decolonial, mas sem um enfrentamento sério do seu significado e sentido em termos geográficos, especialmente em termos amazônicos contemporâneos. Há consequências – inclusive políticas, impossíveis de aprofundar neste ensaio – da dicotomia entre objetivo-subjetivo que pode contribuir para desvalorização da subjetividade não só do/a pesquisador/a, mas dos sujeitos amazônicos em sua relação fundamental com os lugares, o que abriria uma geografia aqui e ali já tematizada, mas ainda subestimada na completamente, por geógrafos reunidos em torno de grupos humanistas no Brasil, preocupados demais em olhar para o próprio umbigo e construir suas próprias "geografias heroicas" em torno de si, sem se questionar – ou escavar, para usar um termo husserliano – os fundamentos da emergência de experimentações geográficas humanistas fora do eixo centro-sulista. Esta, por sinal, é uma crítica explicita no artigo de Pereira, comprometido com a decolonialidade geográfica. constrangidos a separar emoção da pesquisa e seus resultados "objetivos", numa compreensão DESCRITIVIDADE COMO UM PRINCÍPIO DA GEOGRAFIA AMAZÔNICA: O CHAMADO DE EIDORFE MOREIRA Revista GeoAmazônia Belém v. 7, n. 13 p. 53–67 2019 Página 64 "região", uma "geografia pavulagem"28 como pode bem ser taxada, porque entendida como pedante ou presunçosa. E justamente aí o esforço de descritividade destas relações entre o simbólico, o emocional e os lugares em termos expressivos e escalarmente abrangentes queremos nos enxergar – unificando/reduzindo a variabilidade pulsante do real que nossos relatos coerentemente analíticos apressam em delimitar, gerindo o que é pensável e interpretável nos horizontes sedimentados da Amazônia. Descrever, geograficamente, a realidade dos lugares amazônicos nos exige ir além da comunicação de um saber, na direção da experiência de sentir. Ainda que sejamos cobrados e constrangidos a prescrever (prescrições que compõem o agir científico, mas não podem ser afobadas), não podemos subestimar a necessidade de descrição como instituição de ambiência amazônica. E Moreira dá algumas pistas da instituição dessa cosmovisão com vários exercícios da paisagem geográfica em sua obra – nos atirando no intricado contexto29 regional, em situação (JASPERS, 1958) para daí intencionar um conhecimento geográfico, uma experiência via descrição como método e o não apartamento do simbólico (discurso e representação) do emocional no ato de fazer ciência e imaginar, hoje, outro conceito de Amazônia. CONSIDERAÇÕES FINAIS A constatação do abandono e depreciação do método descritivo pela geografia dos anos de 1960, denunciada por Eidorfe Moreira, se generalizou em nosso presente e se tornou a postura exigida e reproduzida na formação geográfica amazônica. Obviamente, o descritivo está presente nas várias publicações de pesquisas que fazem avançar a ciência geográfica. Não só porque a Geografia é uma ciência essencialmente descritiva, como categorizava Moreira, mas porque o descritivo é um modo de realização comunicativa inerente ao fazer científico e, não raro, é possível ler termos "mais nobres" quando, na realidade, o que temos é efetiva descrição em não poucos trabalhos. Por outro lado, esta recusa explícita do descritivo impede que tomemos para nós as potencialidades do método – sem que isto implique submeter-se apenas a ele – o que poderia levar ao desenvolvimento do próprio método na contemporaneidade, ao mesmo tempo em que 28 Pavulagem é um termo paraense que significa "algo ou alguém pretensioso, metido à besta, pedante". 29 Lukács (op. cit.) reserva ao narrativo o poder de nos atirar no contexto, como participantes, no plano literário. Penso que o método descritivo ou a descritividade em Geografia, pode resguardar um valor participativo também. (MOREIRA, 2012) seria um passo coerente. Esforço real, senão antes, junto às meta-narrativa inevitáveis ou chaves explicativas que nos atirem à mudança geopolítica – que é como sempre DESCRITIVIDADE COMO UM PRINCÍPIO DA GEOGRAFIA AMAZÔNICA: O CHAMADO DE EIDORFE MOREIRA Revista GeoAmazônia Belém v. 7, n. 13 p. 53–67 2019 Página 65 sua experimentação pode abrir uma impressionalidade e uma expressividade necessárias para a compreensão da Amazônia atual. Sendo da ordem da aparência do contato superficial, a descritividade vai na direção do extensivo e do não sintético, uma abrangência aos tateios, ao contato da pele e aí mora suas possibilidades infindas, nos atirando dentro dos movimentos, ao encontro com os sujeitos e os lugares sem uma ordenação prévia que já emoldure nosso pensar, sentir ou emoldure os Outros, caso assumamos como exercício ético e estético. O que implica jamais perder de vista nossas ambiguidades/nuances e a dos outros, apesar de nossa tara (necessária) por coerência hierárquica e explicação generalizante, de modo que o exercício da descritividade pode se refundar em um trabalho de auto-educação e co-educação do olhar geográfico e do geográfico do olhar em uma perspectiva subjetivamente corporificada e politicamente comprometida com as geograficidades amazônicas por conhecer e por construírem/participarem do conhecer socializado. O chamado de Eidorfe Moreira, ainda que não comporte todos esses aspectos, estabelece o senso crítico para com uma geografia de corte – entre o tradicional e o moderno, entre o explicar/analisar e o descrever, entre a generalização que alimenta a ação política e a cosmovisão cuidadosa com a abrangência escalar do mundo – em favor de uma essência fundante do ato de geografar tão necessário na contemporaneidade amazônica, para que nós não só lembremos do que é valorizável e expressivo, mas dos meios que temos na instituição de valor potencial, já que, nas palavras de Moreira (2012, p. 49): " Ainda que nos revele infinitos aspectos da Natureza, a ciência jamais a tornará completamente inteligível [...] há sempre a possibilidade de um além". REFERÊNCIAS BIBLIOGRÁFICAS ALMEIDA, A. W. B. Antropologia dos Archivos da Amazônia. Rio de Janeiro: Casa 8/Fundação Universidade do Amazonas, 2008. Disponível em: http://191.98.188.189/Fulltext/11524.pdf, acesso em 06.12.2018. BAUMAN, Z. Modernidade Líquida. Rio de Janeiro: Jorge Zahar Editor, 2001. BUTLER, J. Relatar a si mesmo. Crítica da violência ética. Belo Horizonte: Autêntica, 2015. CASEY, E. J.E. Malpas's Place and Experience: A Philosophical Topography (Cambridge University Press, 1999) Converging and diverging in/on place. Philosophy & Geography, v. 4, n. 2, 2002, pp. 225-230, Disponível em: https://www.tandfonline.com/doi/abs/10.1080/10903770123141?journalCode=cpag20, acesso em 10.12.2018. DESCRITIVIDADE COMO UM PRINCÍPIO DA GEOGRAFIA AMAZÔNICA: O CHAMADO DE EIDORFE MOREIRA Revista GeoAmazônia Belém v. 7, n. 13 p. 53–67 2019 Página 66 COELHO, G. M. Eidorfe Moreira e o conhecimento transdisciplinar. Novos Cadernos NAEA, v. 15, n. 2, 2013. Disponível em: https://periodicos.ufpa.br/index.php/ncn/article/view/1079/1529, acesso em 02.12.2018. DARDEL, E. O homem e a Terra. São Paulo: Perspectiva, 2011. DERRIDA, J. Edmund Husserl s Origin of Geometry. An Introduction. Lincoln and London: University of Nebraska Press, 1989. DIDI-HUBERMAN, G. Que emoção! Que emoção? São Paulo: Editora 34, 2016. ENTRIKIN, N. The Betweenness of Place. Towards a Geography of Modernity. London: Palgrave Macmillan, 1991. GARFINKEL, H. Studies in Ethnomethodology. Cambridge: Polity Press, 1984. GEERTZ, C. A interpretação das Culturas. Rio de Janeiro: LTC, 2008. GUIMARÃES, M. S. F. P. Caminhos para ler Eidorfe Moreira. In: MOREIRA, E. Ideias para uma concepção geográfica da vida. Apêndices, pp. 213-276. Belém: SEMEC, 2012. GUIMARÃES, M. S. F. P. O olhar de Benedito Nunes sobre a obra de Eidorfe Moreira. Bol. Mus. Para. Emílio Goeldi. Cienc. Hum., Belém, v. 10, n. 3, p. 605-625, 2015. Disponível em: http://editora.museu-goeldi.br/bh/artigos/chv10n3_2015/olhar(guimaraes).pdf, acesso em 03.12.2018. GONZÁLEZ-REY, F.; MARTINÊS, A. M. Subjetividade: teoria, epistemologia e método. Versão e-book. São Paulo: Alínea, 2017. GOMES, P. C. C. Geografia e Modernidade. 5a ed. Rio de Janeiro: Bertrand Brasil, 2005. FEYERABEND, P. Adeus à Razão. São Paulo: Ed. Unesp, 2010. JASPERS, K. Filosofía. Madrid: Universidade de Puerto Rico, 1958. JAMESON, F. Postmodernism, or, the cultural logic of late capitalism. Durham: Duke University Press, 1991. HAN, B. C. A sociedade do cansaço. Petrópolis, RJ: Vozes, 2015. LUKÁCS, G. Ensaios sobre Literatura. Rio de Janeiro: Civilização Brasileira, 1965. MALPAS, J. E. Comparing topographies: Across paths/around place: A reply to Casey. Philosophy & Geography, v. 4, n. 2, p. 231-238. Disponível em: https://www.tandfonline.com/doi/abs/10.1080/10903770123850, acesso em 05.12.2018. MARX, K. Crítica ao Programa de Gotha. São Paulo: Boitempo, 2012. MORAES, A. C. R. Pequena História Crítica. 20a ed. São Paulo: Annablume, 2005. DESCRITIVIDADE COMO UM PRINCÍPIO DA GEOGRAFIA AMAZÔNICA: O CHAMADO DE EIDORFE MOREIRA Revista GeoAmazônia Belém v. 7, n. 13 p. 53–67 2019 Página 67 NUNES, B. Notas críticas. In: MOREIRA, E. Obras reunidas. v. 1. Belém: CEJUP, 1989. MOREIRA, E. Ideias para uma concepção geográfica da vida. Belém: SEMEC, 2012. PEREIRA, E. A. D.Uma leitura da concepção geográfica de Eidorfe Moreira. , v. 16 n. 31 (Doutorado em Geografia), Universidade de Brasília, Brasília, 2018, 449 f. Disponível em: http://repositorio.unb.br/handle/10482/32855, acesso em 02.12.2018. GEOgraphia Transamazônica: geocartografia da inexistência entrelugares 24-50, 2014. Disponível em: PANTOJA, W. W. R. . Tese http://periodicos.uff.br/geographia/article/view/13671, acesso em 06.12.2018. RICOEUR, P. Tempo e Narrativa. Tomo I. São Paulo: Papirus, 1994. SANTOS, B. S. Para além do pensamento abissal. Novos Estudos, n. 79, 2007, pp. 71-94. Disponível em: http://www.scielo.br/pdf/nec/n79/04.pdf, acesso em 10.12.2018. SANTOS, M. Por uma geografia nova. São Paulo: EdUSP, 2002. TODOROV, T. Simbolismo e interpretação. São Paulo: Ed. Unesp, 2014.

ARRAN GARE ARRAN GARE is Reader in Philosophy and Cultural Inquiry and Director of the Joseph Needham Centre for Complex Processes Research, Swinburne University, Melbourne, Australia. The focus of his research is on transforming culture to create an environmentally sustainable civilization. He is the author of a number of books, including Postmodernism and the Environmental Crisis (London: Routledge, 1995) and Nihilism Inc.: Environmental Destruction and the Metaphysics of Sustainability (Sydney: Eco-Logical Press, 1996). INTRODUCTION: ARCHITECTURE AND THE ENVIRONMENT FROM A HEIDEGGERIAN PERSPECTIVE The German sociologist Ulrich Beck has well characterized the present predicament: The transformation of the unseen side-effects of industrial production into global ecological trouble spots is ... not at all a problem of the world surrounding us-not a so-called "environmental problem"-but a far-reaching institutional crisis of industrial society itself.... What previously appeared "functional" and "rational" now becomes and appears to be a threat to life, and therefore produces and legitimates dysfunctionality and irrationality.... Just as earlier generations lived in the age of the stagecoach, so we now and in future are living in the hazardous age of creeping catastrophe. When generations before us discovered despite resistance, and had to shout out loud at the world, we have come to take for granted: the impending "suicide of the species" ...1 In the face of this predicament, Beck argued, ethics can be compared to "a bicycle brake on an intercontinental jet."2 It is virtually impossible to imagine the discourse of ethics being able to have significant impact on the forces leading to the destruction of the global environment. The emerging conventional view is that the only hope for the future lies in developing more energy-efficient and less polluting technologies, discovering new forms of energy, increasing the efficiency of agriculture, better comprehending and managing ecosystems, and better understanding markets to harness the entrepreneurial talents of capitalists to implement the discoveries of science.3 If ethics is regarded as irrelevant in an age of creeping catastrophe, what role could art (even when broadly conceived to include architecture) play? Looked at in this light, it would appear that art is virtually irrelevant to addressing environmental problems, although architecture might play some role in developing more energy-efficient buildings. Art is even more ineffectual than ethics. However, Martin Heidegger advanced the provocative thesis that science is fundamentally limited and that it is to art that we should look for salvation. Science, Heidegger argued, enframes the world to reveal it only as standing reserve to be dominated. It is the very quest for control, which is problematic.4 Nature and even people are now evaluated only as instruments. It is this way of thinking that has created a global ecological crisis, and the crisis will not be solved through such thinking. Heidegger called for an inversion of the standing granted to science, ethics, and art. "Ethics" emerged in Ancient Greece with Plato and Aristotle when thinking was becoming a science, and to the present day is contaminated by the same domineering orientation. What is really needed is not the abstract thinking of science or ethics but a different "ethos"; that is, a different way of dwelling on the earth, a way of dwelling such that Being is revealed.5 How can this be achieved? Heidegger invoked art. Art reveals (that is, unveils) the Being of beings to reveal their truth as presencing. Truth (conceived of as unconcealedness of beings) and beauty are indissociable. As Heidegger put it: "Truth is the truth of being. Beauty does not occur alongside and apart from this truth. When truth sets itself into the work, it shines forth. The shining forth- as this being [here to be understood actively and transitively] of truth in the work and as work-is beauty. Thus the beautiful belongs in the selfhappening of truth."6 Artworks are events of truth to which belongs the beautiful, and these events open a path to dwelling in the light of Being. Buildings and architecture as forms of art are crucial to this. They are happenings of truth, sites or places in which geometrical space is subordinated to the more primordial possibility of an event in excess of the merely physical or of the merely human, an event in which the human finds its proper place: its home. Buildings open the possibility of genuine dwelling, not just amid things, but as the event of the coming into presence of things out of a non-thingly horizon. As Heidegger wrote of a Greek temple: ARCHITECTURE AND THE GLOBAL ECOLOGICAL CRISIS: FROM HEIDEGGER TO CHRISTOPHER ALEXANDER Published in The Structurist, 'Toward an Ecologicl Ethos in Art and Archictecture, No. 43/44, 2003/2004, pp.30-37. It is the temple-work that first fits together and at the same time gathers around itself the unity of those paths and relations in which birth and death, disaster and blessing, victory and disgrace, endurance and decline acquire the shape of destiny for human being.... The temple-work, standing there, opens up a world and at the same time sets this world back again on earth, which itself only thus emerges as native ground.7 Through art, and architecture in particular, man dwells poetically on this earth.8 The challenge for architecture is to make possible such dwelling, and architecture is beautiful when it does so. It is suggested, without ever being claimed, that dwelling in this way is what is required to properly respond to the ecological crisis. EVALUATING HEIDEGGER How can we evaluate such provocative suggestions? Heidegger's oracular style and language have made it difficult to assess his work. Most of his disciples devote themselves to exegesis, as though to have shown how Heidegger developed his ideas and what he really meant is to attain the truth. Even from this perspective, Heidegger promises little to those concerned with addressing the global ecological crisis. After supporting Nazism, Heidegger adopted a very passive orientation to the problems he identified. In the face of the "extreme danger" that "the frenziedness of technology may entrench itself everywhere," Heidegger concluded that "The closer we come to the danger, the more brightly do the ways into the saving power begin to shine and the more questioning we become. For questioning is the piety of thought."9 At best we can hope to recover what had been revealed to the Ancient Greeks, who were quite destructive of their natural environment. Taking a more critical stance towards Heidegger, Fredric Jameson concludes that: "Heidegger's 'field path' is, after all, irredeemably and irrevocably destroyed by late capital, by the green revolution, by neocolonialism and the megalopolis, which runs its superhighways over the older f ields and vacant lots and turns Heidegger's 'house of being' into condominiums, if not the most miserable unheated, rat-infested tenement buildings."10 This is not simply a gratuitous display of disrespect for Heidegger. It suggests that Heidegger's diagnosis of the problem is incomplete-it is a claim that we can no longer dwell in the world as suggested by Heidegger because the global market no longer permits it. Heidegger's call for authentic dwelling is irrelevant before the destructive power of global capitalism. In his study of Heidegger's critique of modernity and its relevance for the environment, Michael Zimmerman is even more critical.11 Zimmerman finally comes to the conclusion that Heidegger's philosophy was fundamentally flawed, undermining its own critical force, something Heidegger himself came to appreciate. As Zimmerman puts it: Heidegger could read modernity as the most constricted mode of disclosure only by viewing Western history as decline and fall from a nobler origin. Eventually abandoning this view, he could say only that technological modernity excluded the ancient Greek disclosure of being, but ancient Greece excluded the technological disclosure. I would add that ancient Greece also excluded modernity's egalitarian commitments.12 This would seem to provide even more reason to dismiss Heidegger. However, Heidegger's work belongs to the tradition which began with the early Romantics of the late eighteenth and early nineteenth centuries.13 Influenced by Kant, Herder, Goethe, Schiller, and Fichte, it was a reaction against the prevailing ideas of the Enlightenment: the mechanistic view of nature, the atomistic view of society, and a highly abstract notion of rationality. The most outstanding representative of this tradition, and the figure who defined its goals, was Schelling. Heidegger was far more indebted to Schelling than he acknowledged, as Sonya Sikka has shown.14 It was Schelling who argued that art is superior to science as a means to comprehend ultimate reality and who opposed the reduction of nature to a mere instrument for human purposes. It was also Schelling who argued in opposition to Hegelian idealism that there is an "unprethinkable being" preceding all thought, particularly reflective thought. Schelling not only developed a profound critique of the approach to nature of the sciences very similar to Heidegger's and influenced the tradition of hermeneutics with which Heidegger aligned himself, following Herder and Goethe; he also attempted to lay the foundations for a new form of post-mechanistic science in which nature is seen as first and foremost "productivity" or process, and only derivatively as "products" or things. This has developed as the 2 tradition of process thought, a tradition that includes Peirce, Bergson, and Whitehead and the scientists such as Ilya Prigogine, David Bohm, and Brian Goodwin influenced by them.15 If we evaluate Heidegger's work as the contribution of a highly original thinker to the developing tradition of thought begun by the early German Romantics, not as a finished body of ideas but as a possible creative advance within this tradition, then we do not have to either embrace Heidegger's work as a whole or dismiss it because of its obvious failings. We can assess his analysis of modernity and claims for art, assimilate his insights, and go beyond him. When viewed in this way, Heidegger's claim that his own work is fundamentally different and much deeper than science or speculative metaphysics is no longer tenable; but this makes his insights more defensible. While Heidegger himself might have concluded that he had no basis for judging the superiority of one way of revealing the world over another, when seen as part of the broader tradition of process thought and the struggle to overcome the mechanistic world view, Heidegger's work can be seen as a contribution to this struggle for a more adequate comprehension of nature and our place within it. From this perspective, it is not enough to note that the space and time of science are incompatible with space and time as it is lived; it is necessary to revise the notions of space and time within science to make intelligible the possibility of lived space and time. Such ideas can be evaluated according to how they overcome the blind spots and aporias of rival ways of understanding the world. Advancing process thought, Heidegger has also advanced our understanding of what it is to understand the world. In particular, Heidegger has helped overcome the bias in favor of contemplative thinking and visual analogies. Process metaphysics should no longer be understood as a "world view" but as a mode of being in the world whereby it discloses itself and ourselves more adequately. What matters most is not how we think about the world contemplatively but how we orient ourselves while practically engaged within it; that is, how we dwell within it. For speculative metaphysicians and scientists, what is important is not so much the ability to manipulate abstractions in order to make correct predictions, but (in Michael Polanyi's terminology) to make sense of the world by using these abstractions to more adequately "indwell" in what we are trying to understand.16 BEYOND HEIDEGGER Going beyond Heidegger, we can see that the modern way of enframing the world as standing reserve is not merely a matter of the forgetting of Being. It is an orientation reinforced and extended by the dynamics of the global market. This is being promoted and manipulated by powerful states and the wealthy elites who control them, emancipated from democratic control in any meaningful sense, as a means to accelerate economic growth, exploit their working classes, and exploit the resources of poorer nations more effectively while providing the military means to maintain and extend the market and control access to resources. Treating the world as a world of things to be exploited is legitimated by mainstream economic theory, management theory, social Darwinism, Darwinist evolutionary theory, and mainstream reductionist science generally, all of which cohere as the integrated world view of scientific (or mechanistic) materialism.17 The global market, techno-science, and the scientific materialist cosmology which supports these are associated with massive concentrations of power that have neutralized almost all opposition to its present dynamics. The commodification of even the most creative art by the culture industry is an aspect of this neutralization. The situation is far worse than that portrayed by Heidegger. Miguel de Beistegui, in his study of Heidegger, states that The question, with respect to our contemporary situation, is to know the extent to which the global economy-an economy beyond the general economy of the polis, for no longer tied to the polis of the nation-state, an economy, in other words, which has transformed our very being, and set itself almost entirely free from the "place" in which it was traditionally anchored, reconfiguring also the private space itself as no longer private, but as entirely traversed by this essentially fluid and plastic force: capitalism- that is sweeping us away is not simply a state of homelessness depriving us of any sense of place, deterritorializing the nation and the homeland.... Is there not, in such a context, an erring and a lack of place far more threatening and colossal than anything hitherto experienced, a sort of perpetual banishment fed and kept alive by the economic machine?18 Any place with the slightest trace of authentic dwelling is immediately packaged to attract the tourist dollar (with tourism now accounting for twenty per cent of all greenhouse gas emissions). With virtually every aspect of nature and social life 3 commodified and subject to the logic of the global market, almost all effective resistance to its destructive imperatives has dissolved. Throughout the world people have been seduced into giving up their political power (that is, democracy) for the promise of higher levels of consumption that a globally free market will supposedly deliver. Since the 1970s when the environment was first put on the agenda, this logic has accelerated the rate of virtually every form of environmental destruction, despite the emergence of a global environment movement.19 What could stop these destructive dynamics? We know from the failure of communism that a centrally planned economy is not the solution. One plausible program is to revive democracy, the power of the people to shape their destiny for the common good, wresting power from existing power elites and subordinating the market to the common good of the whole of humanity including future generations. There are enormous obstacles standing in the way of achieving this, not least the power of the power elites, but the most problematic is the dominance of the prevailing scientific materialist cosmology which construes life as a struggle for survival and domination, values above all the unlimited power to satisfy appetites, and denies even the possibility of self-determination, a construal of life continuously reconfirmed by the behaviour of people within a market economy. While Heidegger had virtually nothing to say about freedom and democracy and the scientific materialist cosmology, these were central concerns of the broader tradition to which he belonged. Fichte argued that it is only through being recognized by others as free that we become free agents, and freedom consists in limiting ourselves in accordance with the appreciation of others as free. That is, humans become free, self-determining agents only through their relations to other people whereby the freedom of others is acknowledged. Schelling developed his alternative conception of nature to justify this conception of humans, which were then seen as emergent phenomena within a dynamic, creative nature. This is the conception of humans and their place within nature required to support the struggle for genuine democracy able to appreciate the creativity of both people and natural processes and act accordingly. Much work has been done elaborating this cosmology and developing it as an alternative foundation for the sciences. However, there are major difficulties in developing science on these foundations. When setting up experiments the aim is to set up initial conditions, controlling the environment and constituents of entities so as to be able to predict outcomes. This approach does not give a place to immanent causation of processes and our own participation in the dynamics of what is being investigated. There is also the problem of giving a place to mathematics (which implies determinism) in the world while reconciling this with a conception of nature as creative process. While efforts are being made to deal with these problems, the work is at such an abstract level and is so difficult to comprehend that it is diff icult to persuade on this basis more than a small minority of people that they need to fundamentally alter their modes of being in the world. However, such problems have almost invariably led exponents of this alternative tradition to appreciate art as complementary to science in comprehending the world. To effect a fundamental transformation of our modes of being in the world, art, with its cognitive claims properly defended, could be more important than post-mechanistic science. And it is in this context that Heidegger's work can be appreciated, although appreciating its full significance requires that it be reinterpreted. Heidegger can be interpreted as forging a new language (usually by privileging verbs over nouns) to enable people to understand themselves as creative participants within a creative nature; that is, in a world of processes. By developing this to illuminate the mode of being in the world of the Ancient Greeks, he was able to expose the takenfor-granted assumptions of modernity which have hidden from people both the creative activity of processes within their world and also their indebtedness to this creative activity in their own becoming and in their productive activities. Most importantly, those assumptions have hidden from people the meaning of processes in nature, in social l ife, and in their own individual l ives. Heidegger's new language not only enables people to appreciate these processes but also shows how works of ar t and their buildings disclose people's worlds and thereby the meanings of their lives. Of the arts, no art form has greater impact on the way people experience and orient themselves within the world than architecture. That is, the original tentative suggestion, repudiated by Heidegger himself, that 4 art generally and architecture in particular might play a fundamental role in engendering a superior ethos or way of dwelling in the world, is justified when Heidegger is reinterpreted as part of the tradit ion of process thought deriving from Schelling. Rather than lamenting a lost mode of revealing Being, architecture can be appreciated as central to the struggle to develop a more adequate way of dwelling which discloses the world as interrelated processes rather than as merely a collection of things. REVOLUTIONIZING ARCHITECTURE: THE WORK OF CHRISTOPHER ALEXANDER While Heidegger has had a major impact on architectural theory through the influence of his notion of lived space,20 it is Christopher Alexander, an architectural theorist who has not been influenced by Heidegger, who has made the most vigorous effort to overcome the form of architecture and town planning that discloses the world and people only as standing reserve. Alexander did not conceive his work in such terms and has shown little interest in ecological problems. His concern has been to provide the means for people to create beautiful buildings, buildings that enable people to feel at home. This, Alexander believes, is the condition for basic improvements in the institutions of society required to put things right socially. To this end, he has attempted to develop an approach to architecture that focuses on living processes, and was led to examine not only the order underlying all that we build, including cities in all ages, but all that grows throughout nature. Opposing the underlying mechanistic assumptions which, he contends, pervade modernist architecture, he has aligned himself with the tradition of process thought and with post-mechanistic science. He claims that architecture can now play a leading role in developing this new understanding of the world. That is, Alexander belongs to the same tradition of thought going back to Goethe and Schelling to which Heidegger, process metaphysicians, and post-mechanistic scientists belong. My contention is that concern with processes leads to a different mode of disclosing the world. People are led to experience themselves as participants in an active world rather than as subjects trying to control it. Alexander's life and work exemplify this, leading to the struggle within architecture for a more adequate way of dwelling in the world in accordance with the process tradition of thought. It is this reorientation which could fulfill Heidegger's intimation that architecture could change the way people dwell within the world and change what they aspire to, a change that could be central to overcoming the destructive forces of modernity and to creating environmentally sustainable forms of life. Alexander was preoccupied from his student days with the question of what makes things, especially buildings, beautiful. He believed that most people agree on what is beautiful; it is a quality of reality that most people have no difficulty appreciating. He also believed that most people could see that it is mostly old buildings and cities that are beautiful; if people are honest, they can see that modernist architecture is not beautiful, it is oppressive and ugly. He believed this to be a fact about reality, not a subjective view. The evolution of Alexander's ideas was largely a matter of clarifying the nature of beauty with a view to enabling people to create beautiful buildings. He first set out to discover what is actually going on in good design. Originally, he argued that the object of design is form, and that the problem of design is to fit the form (over which we have control) to its context, which puts demands on this form. He argued that "A well-designed house not only fits its context well but also illuminates the problem of just what the context is."21 Alexander attempted to show a deep underlying correspondence between the pattern of a problem and the process of designing a physical form answering that problem. The structure, if successful, will clarify the life it accommodates. This led to a concern to explicitly map the problem's structure and to provide a language to do so, enabling end users to participate in the design process. Alexander's focus was on the design process by which people make beautiful buildings rather than on the buildings as end products. As his ideas evolved, process came to be emphasized more and more. Initially, Alexander portrayed problems and their solutions as hierarchical structures, the solutions being an exact counterpart to the functional hierarchy established during the analysis of the problems. However, he soon came to see that the free functioning of the system depended not so much on meeting a set of external requirements but on the system's coming to terms with itself, being in balance with the forces generated internally by the system. After comparing planned cities and 5 unplanned cities, he came to see that it was the treelike structure of plans with all spaces being allotted to particular functions that deadened social life and alienated people. Noting that self-respecting children far prefer to play in abandoned construction sites than in playgrounds, which are cut off from the rest of society and where everything they can do has been planned, Alexander realized that planned cities have fragmented social life and have prescribed what is to be done in each functional space, eliminating any room for creativity. In his famous paper "A City is Not a Tree," he argues that the richness of pattern of unplanned cities, a richness manifest in their vibrancy, is the result of an overlapping structure that interrelates social activities and leaves open the possibility for ever new uses of space.22 They [Author: do you mean cities, structures or spaces? Please clarify.] are semi-lattices. Reflections on these issues led Alexander to focus on the processes which actually produce the structures of environments, that is, on "generativity." Pursuing this line of inquiry led Alexander to see that the environment consists of relations or patterns rather than things, and that these patterns are generated by language-like systems of rules. Patterns are recurring problems along with the solutions to these (such as reconciling a range of needs) which can be used a million times over without ever doing so in exactly the same way. "Things" are merely convenient labels we give to patterns or bundles of patterns. Alexander was concerned with those patterns which solve some sort of architectural or social problem, which are embodied in the structures we build. Patterns work by co-existing, competing, and co-operating in some dynamic balance to build up complex wholes, which in turn form higher-level patterns, generating a complex fabric from relatively simple generative rules. On this assumption Alexander attempted to develop a pattern language based on identifying rules operating at all levels of the environment to analyze and facilitate the interaction of human needs in space as a generative process comparable to the form-generating processes in nature. This language was designed to enable lay people to participate in the design process. He argues that "towns and buildings will not be able to become alive, unless they are made by all the people in society, and unless these people share a common pattern language, within which to make these buildings, and unless this common pattern language is alive itself." 23 Increasingly, Alexander was seeing the generation of form in buildings as merely a special case (with "needs" being a particular kind of force) of the generation of form throughout nature, and that such form generation is the essence of life. Beauty was now equated with life. As Alexander put it, "The beauty of a thing is not purely in how it looks. It has to do with how it is. Now how it 'is' essentially involves a relationship between the various events that are going on there.... So it is ultimately the inner life which is the thing that matters."24 Concerned with living both in relation to the design and building activities and in relation to people affected by the resulting buildings, Alexander concluded that this must lead to a new understanding of nature: I have come to believe that the problem of physical order-the kind of order which creates quality in architecture ... this problem is of so great a stature, that we shall have to modify our picture of the whole physical universe in order to see it clearly.... I believe we are on the threshold of a new era, when this relationship between architecture and the physical sciences may be reversed-when the proper understanding of the deep questions of space, as they are embodied in architecture ... will play a revolutionary role in the way we see the world.25 It was this conviction that led Alexander to embark upon his most important work, a study of the nature of order. At the time he embarked on this, Alexander thought he would write one book. It is now a fourvolume work, The Nature of Order: An Essay on the Art of Building and the Nature of the Universe, consisting of Book One, The Phenomenon of Life, Book Two, The Process of Creating Life, Book Three, A Vision of a Living World, and Book Four, The Luminous Ground. In these works, Alexander characterizes the discipline of architecture as "a mass psychosis of unprecedented dimension, in which the people of earth ... have created a form of architecture which is against life, insane, image-ridden, hollow."26 He attributes this to the domination of the mechanistrationalist picture of the world. To combat this, Alexander elaborates a new view of order in terms of which "statements about relative degree of harmony, or life, or wholeness-basic aspects of order-are understood as potentially true or false."27 Alexander's work on order is similar to, although not identical with, complexity theory. His ideas are particularly close to the speculations on order and wholeness of the physicist David Bohm. However, drawing on his 6 work in his own architecture and on the design process, Alexander has made an original and significant contribution to the tradition of process thought. Providing a sharable perspective based on a holistic view of the goal of life, Alexander aspires to enable people to work together, reconciling the many factors and needs influencing the environment, to make buildings with a profound, living order; that is, to create beautiful buildings. Architecture would again be aligned with life. THE RELEVANCE OF ALEXANDER TO THE GLOBAL ECOLOGICAL CRISIS Alexander has gone well beyond the speculative thinking of architects influenced by Heidegger; he is engaged in a struggle to transform architectural practice. While he has not addressed himself to the global ecological crisis, the development of his ideas has led him towards a new way of disclosing nature and people as other than standing reserve, as life with its own internal dynamics. In developing these ideas there has been an astonishing convergence with Heidegger in granting centrality to process over things, and in allowing appreciation of significance within the world. Most importantly, despite appearances, the way Alexander understands the beautiful is strikingly similar to Heidegger's. The beautiful is life, understood as the harmonious ordering of diverse forces and events, disclosed as such in all its relationships (being in the light of Being) and augmented with this disclosing. Like Heidegger, Alexander has been concerned with our sense of place in the world and is concerned to recreate the feeling of belonging, which at the same time is to clarify (or reveal) what this belonging is. Through architecture, he has exposed and offered alternatives to not only the assumptions underlying modernist architecture, but also the institutions of modernity which have generated the global ecological crisis. His work is a major contribution to developing a cosmology that does justice to life and our experience of it as beautiful. Alexander has addressed more fully than Heidegger the social dimension of our relation to the world and the effect of the market on society. He has forged a real, practical alternative to instrumentalist/functionalist thinking and has struggled to provide the means by which the public can participate in designing and building beautiful environments. It is a challenge to the notion of architects selling a special expertise to the public which takes out of the public's hands any participation in the building of its environments. Perhaps more importantly, Alexander is concerned to overcome the division between the public and the private and to recreate or revive those spaces and institutions that in the past mediated between individuals and the State. This is associated with efforts to revive institutions and corporations as self-organizing processes. That is, through architecture, Alexander is addressing the atomization of society and the corrosion of public life, institutions, and organizations that have undermined real democracy. The implications of Alexander's work for the global ecological crisis can be clearly seen when we consider what his work implies for individuals. It has been widely noted how the atomizing functionalism dominating the modern world has dehumanized people. Having lost the unity of the social and the individual, people have lost their individuality, their power as citizens, and, with the globalization of the economy, their economic security. They have been swindled into an endless pursuit of money to compensate for what they have lost. They consume more than ever before because, as Baudrillard notes, they now are consuming symbols rather than what is useful.28 They buy commodities to define their identities, but once bought, these commodities almost immediately lose their symbolic significance. Consequently, consumers can never be satisfied. And to escape the emptiness and ugliness of their own environments they become tourists, but take this emptiness with them and debase everywhere they visit with their presence. Alexander promises to regenerate life by engaging such people in designing and building beautiful environments, reviving their social spaces and their institutions to enrich their lives and their appreciation of beauty. Involving people in this way requires of them that they come to understand in a practical way a cosmology that validates their experiences of beauty and life. If this project is successful, people will dwell within the world in a different way and again be aligned with life, and with revived communities, they will also be empowered to further the interests of life. At this stage it is difficult to finally assess Alexander's work and its potential. The successful appropriation of his ideas by computer scientists gives some indication of the power and generality of his analysis of order. This project is a major advance in the 7 tradition of process thought, a tradition developed in opposition to mechanistic thinking and the social order founded upon it. By calling for a transformation of architecture on this basis to enable people to appreciate and participate in the creation of beauty, Alexander is promoting a mode of dwelling on earth that will reveal and augment life. While architecture by itself is unlikely to overcome the destructive power of the global market and the political institutions and power elites which are now supporting it,29 once the problems he is grappling with and the relationship between these and the broader tradition of process thought are understood, it is difficult to imagine the global ecological crisis being addressed successfully without the kind of revolution in architecture that Alexander is striving to bring about. ■ NOTES 1. Ulrich Beck, "Risk Society and the Provident State," in Risk, Environment & Modernity, ed. Scott Lash, Bronislaw Szerszynski, and Brian Wynne (London: Sage, 1996), 32, 34, 40. 2. Ulrich Beck, "From Industrial Society to Risk Society," in Cultural Theory and Cultural Change, ed. Mike Featherstone (London: Sage, 1992), 106. 3. See for example Paul Hawken, Amory Lovins, and Hunter Lovins, Natural Capitalism: Creating the Next Industrial Revolution (Boston: Little, Brown and Co., 1999). 4. Martin Heidegger, The Question Concerning Technology, tr. William Lovitt (New York: Harper, 1977), 5. 5. Martin Heidegger, "Letter on Humanism" in Basic Writings, ed. David Farrell Krell (London: Routledge & Kegan Paul, year?), 232. 6. Martin Heidegger, Holzwege (Frankfurt am Main: Vittorio Kloserman, 1950/1980), 68, translated and cited in Miguel de Beistegui, Thinking with Heidegger: Displacements (Bloomington: Indiana University Press, 2003), 153. 7. Martin Heidegger, "The Origin of the Work of Art" in Poetry, Language, Thought, tr. Albert Hofstadter (city: Harper & Row, 1971), 42. 8. Martin Heidegger, "...Poetically Man Dwells..." in Poetry, Language, Thought, 225. 9. Heidegger, The Question Concerning Technology, 35. 10. Fredric Jameson, Postmodernism or, The Cultural Logic of Late Capitalism (Durham: Duke University Press, 1991), 34. 11. Michael E. Zimmerman, Heidegger's Confrontation with Modernity: Technology, Politics, Art (Bloomington: Indiana University Press, 1990). 12. Michael E. Zimmerman, Contesting Earth's Future: Radical Ecology and Postmodernity (Berkeley: University of California Press, 1994), page #?. 13. See Andrew Bowie, Aesthetics and Subjectivity: From Kant to Nietzsche, 2nd ed. (Manchester: Manchester University Press, 2003), esp. 8. 14. Sonya Sikka, "Heidegger's Appropriation of Schelling," in Southern Journal of Philosophy Nol. 31, No. 4 (1994): 421–428. 15. See Arran Gare, "The Roots of Postmodernism: Schelling, Process Philosophy and Poststructuralism" in Process and Difference, ed. Catherine Keller and Anne Daniell (New York: S.U.N.Y. Press, 2002), 31–53. 16. Michael Polanyi, Knowing and Being, ed. Marjorie Grene (Chicago: Chicago University Press, 1969), 148. 17. See Arran Gare, Nihilism Inc.: Environmental Destruction and the Metaphysics of Sustainability (Sydney: Eco-Logical Press, 1996), 112–188. 18. Miguel de Beistegui, Thinking with Heidegger: Displacements (Bloomington: Indiana University Press, 2003), 163. 19. See Joel Kovel, The Enemy of Nature, The End of Capitalism or the End of the World? (London: Zed Books, 2002). 20. See for example Christian Norberg-Schulz, Existence, Space & Architecture (New York: Praeger, 1971) and other works by Norberg-Schulz. 21. Stephen Grabow, Christopher Alexander: The Search for a New Paradigm in Architecture (Stocksfield: Oriel Press, 1983), 36. 22. Christopher Alexander, "A City is Not a Tree" in Human Identity in the Urban Environment, ed. Gwen Bell and Jacqueline Tyrwhitt (Harmondsworth: Penguin, 1972), 401–428. 23. Christopher Alexander et.al. A Pattern Language: Towns, Building, Construction (New York: Oxford University Press, 1977), x. 24. Ibid., 56. 25. Ibid., x. 26. Christopher Alexander, The Nature of Order, Book One: The Phenomenon of Life (Berkeley: Centre for Environmental Structure, 2002), 6. 27. Ibid., 22. 28. Jean Baudril lard, "The System of Objects" in Jean Baudrillard: Selected Writings, ed. Mark Poster (Cambridge: Polity Press, 1988), 10–22. 29. As James M. Mayo argues in "Marxism, Architectural Aesthetics, and Practical Ethics," The Structurist, No, 41/42, 2001/2002: 74–82.

See discussions, stats, and author profiles for this publication at: http://www.researchgate.net/publication/259095281 Psychopathic personality and utilitarian moral judgment in college students ARTICLE in JOURNAL OF CRIMINAL JUSTICE * SEPTEMBER 2013 Impact Factor: 1.24 * DOI: 10.1016/j.jcrimjus.2013.06.012 CITATIONS 8 READS 40 2 AUTHORS, INCLUDING: Yu Gao City University of New York Brooklyn College 29 PUBLICATIONS 469 CITATIONS SEE PROFILE All in-text references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. Available from: Yu Gao Retrieved on: 03 November 2015 Author's personal copy Psychopathic personality and utilitarian moral judgment in college students Yu Gao a,⁎, Simone Tang b a Department of Psychology, Brooklyn College and the Graduate Center of the City University of New York, Brooklyn, NY, USA b Fuqua School of Business, Duke University, Durham, NC, USA a b s t r a c ta r t i c l e i n f o Available online 19 July 2013 Purpose: Although psychopathy is characterized by amoral behavior, literature on the association between psychopathy and moral judgment pattern is mixed. Recent evidence suggests that this may be due to the moderation effect of anxiety (Koenigs, Kruepke, Zeier, & Newman, 2011). The current study aims to examine the psychopathy-utilitarian judgment association in college students. Method: In this study, a group of 302 college students completed amoral judgment test involving hypothetical dilemmas. Their psychopathic traits were assessed by the Psychopathic Personality Inventory – Short Form (PPI-SF) (Lilienfeld & Andrews, 1996). Results: Individuals with higher psychopathic traits were more likely to make utilitarian responses to moral dilemmas. Furthermore, the association between utilitarian responses and psychopathy was more salient for the behavioral factor of psychopathy (PPI-II), and this association was mediated by self-reported aggression. However, the moderating effect of anxiety was not found. Conclusions: These results build upon work on utilitarian moral judgment in psychopathic individuals in a non-incarcerated, non-institutionalized sample, and have important implications for the behavioral correction system. © 2013 Elsevier Ltd. All rights reserved. Introduction Psychopathy is a constellation of personality traits characterized by callous and impulsive antisocial behavior. Emotional deficits, in particular low fear reactivity, and behavioral dysregulation have been argued to be the core processes in the development of psychopathy (Fowles & Dindo, 2009; Hare, 2003; Patrick, Bradley, & Lang, 1993). Unsurprisingly, their emotional and behavioral traits have devastating effects on society. According to Kiehl and Buckholtz (2010), psychopaths cost the nation roughly $250 to $400 billion a year due to trials, prison stays and inflicted damage. Bernard Madoff, the psychopathic operator of one of the largest financial frauds in the U.S., stole about $65 billion from his clients. And psychopaths are not just isolated to the few "bad, corrupt apples" at the top. Psychopaths exploit the trust of coworkers and their managers, andmanipulate friends and strangers (Babiak & Hare, 2007; Coid, Yang, Ullrich, Roberts, & Hare, 2009). Yet although psychopaths have such a devastating effect on society, there is only limited experimental data on how they form moral judgments. One moral decision-making task that has been used to explore the psychological and neurobiological processes underlying moral judgment in psychopathy is the scenario task. In its general form, participants read a hypothetical scenario, and are asked to decide whether they would commit some harm in order to achieve a certain goal. Based on Greene's dual process theory, there are two competing processes, supported by neurological systems, which are part of making moral judgments. On one hand, we may strongly feel that an action is inherently wrong. On the other hand, we may engage in cost-benefit analysis and decide that the action can serve the greater good. Greene uses this theory to explain why people report being unwilling to push a man down a footbridge to stop a trolley from running over five people, and yet are willing to flip a switch so that the trolley changes course to hit one man rather than five. In the first, "personal" moral scenario, a dominant, negative emotional response is associated with the action, which elicits moral disapproval. In contrast, the latter, "impersonal" moral scenario does not have any associated dominant response, and thus produces a rational utilitarian decision (Greene, 2007; Greene, Nystrom, Engell, Darley, & Cohen, 2004; Greene, Sommerville, Nystrom, Darley, & Cohen, 2001). This theory has been supported by brain imaging data which have identified that during moral decision-making tasks the "executive brain center" prefrontal cortex is associated with rational processing (Forbes & Grafman, 2010; Young & Koenigs, 2007) whereas emotional regions including anterior temporal lobes and anterior cingulate gyrus are involved in emotional processing (Robertson et al., 2007). In addition, patients with fronto-temporal lobe dementia or ventromedial prefrontal cortex damage who have noted emotional deficits, have also shown more utilitarian responses (Ciaramelli, Muccioli, Ladavas, & di Journal of Criminal Justice 41 (2013) 342–349 ⁎ Corresponding author at: Department of Psychology, Brooklyn College, 2900 Bedford Avenue, 5401 James Hall, Brooklyn, NY 11210. Tel.: +1 718 951 5000x6033; fax: +1 718 951 4814. E-mail address: [email protected] (Y. Gao). 0047-2352/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jcrimjus.2013.06.012 Contents lists available at ScienceDirect Journal of Criminal Justice Author's personal copy Pellegrino, 2007; Koenigs et al., 2007; Moretto, Ladavas, Mattioli, & di Pellegrino, 2010). Taken together, these findings suggest that brain regions including the anterior cingulate cortex, the ventromedial prefrontal cortex, the amygdala, and hippocampus are critically involved in moral reasoning. Given the critical role of emotion in moral judgment and striking social/emotional deficits observed in psychopaths, one might expect to findmore utilitarian judgment in individuals with high psychopathic traits. However, the evidence has been mixed. Using Greene's moral task, some empirical studies have supported this hypothesis (Bartels & Pizarro, 2011; Landon & Delmas, 2012), whereas others have failed to find the association between psychopathy and utilitarian preferences (Cima, Tonnaer, & Houser, 2010; Glenn, Raine, & Schug, 2009; Pujol et al., 2011). Despite the mixed findings at the behavioral level, fMRI data have indicated that reduced activity in regions including ventromedial prefrontal cortex and amygdala during the evaluation of moral dilemmas is associated with psychopathy scores (Glenn et al., 2009). In addition, a baseline network alteration that leads to a functional disconnection between emotional and cognitive elements in moral judgment has also been found in criminal psychopaths (Pujol et al., 2011). Koenigs and colleagues (2011) have hypothesized that failure to find the association between psychopathy and utilitarian moral judgment may be due to the heterogeneity of the group. In a group of incarceratedmales, they found that although compared to non-psychopathic controls, both lowandhigh-anxious psychopaths aremore likely to endorse the impersonal moral actions, only the low-anxious psychopath exhibit abnormally utilitarian personal moral judgment (Koenigs et al., 2011), suggesting that anxiety may moderate the relationship between psychopathy and moral judgment. However, it is unknown if these associations between psychopathy, anxiety, and utilitarian responses can be generalized in non-institutionalized populations. In the current study, the State and Trait Anxiety Inventory (STAI) – Trait scale (Spielberger, Gorsuch, Lushene, Vagg, & Jacobs, 1983) was used to assess anxiety level in college students. This 20-item scale has been used in several studies to examine the associations between anxiety and psychopathic traits (Uzieblo, Verschuere, & Crombez, 2007) in college samples. The goal of the current study was to examine the association between psychopathic personality traits and moral judgment in a non-clinical, non-forensic sample. Although Bartels and Pizarro (2011) found that psychopathic personality traits, as measured by the Self-Report Psychopathy form (Paulhus, Hemphill, & Hare, 2012), were positively associated with utilitarian responses to moral dilemmas, it was unclear which of the factor(s) of psychopathy would be associated with utilitarian moral responses. This issue is important as this has implications for gaining a more comprehensive picture of the processes behind psychopathy and moral decision-making. Thus, one of the goals of this study was to explore the factor(s) associated with utilitarian moral judgments. In the current study, we assessed psychopathic traits using the Psychopathic Personality Inventory – Short Form (PPI-SF; Lilienfeld & Andrews, 1996). The Psychopathic Personality Inventory (PPI; Lilienfeld & Andrews, 1996) is one of themost widely used instruments to assess psychopathic characteristics in undergraduate samples, and the PPI-short form (PPI-SF) was developed from the original PPI and includes 56 items. In a sample of 758 college students a three-factor model (PPI-I, PPI-II, and Coldheartedness) has been derived with the PPI-SF (Smith, Edens, & Vaughn, 2011). Studies have suggested that PPI-I is strongly positively associated with interpersonal dominance, and negatively related to internalizing psychopathology, including anxiety and depression (Benning, Patrick, & Iacono, 2005; Edens & McDermott, 2010; Lilienfeld & Widows, 2005). In contrast, PPI-II tends to be positively correlatedwith internalizingmeasures and is positively correlated with externalizing measures including impulsivity, hostility, and aggression (Edens &McDermott, 2010; Lilienfeld &Widows, 2005). Behavioral genetic studies have indicated that PPI-I was associated with a reduced genetic risk for internalizing disorders, and PPI-II was associated with an increased genetic risk for externalizing disorders (Blonigen, Hicks, Krueger, Patrick, & Iacono, 2005). More interestingly, Smith and colleagues (2011) have found that PPI-II is positively associated with Machiavellianism and deception and negatively correlated with extraversion. Since greater endorsement of utilitarian solutions have been found to be positively related to Machiavellianism (Bartels & Pizarro, 2011), we hypothesize that utilitarian judgment will be more closely related to PPI-II than PPI-I. Another issue concerns the possible mediating effect of aggression on the relationship between psychopathy andmoral judgment. Amediation relationship is said to occurwhen thepredictor variable influences the outcome indirectly through its relationship with a mediating variable (Baron & Kenny, 1986). Significant associations have been found between psychopathy and aggression (Porter & Woodworth, 2006), but no study has yet examined the link between aggression and moral judgment. It is possible that people high in psychopathic personality are simply more aggressive and thus willing to endorse killing actions. It is therefore also important to investigate if high aggression could partly account for the association between psychopathy and utilitarian judgment. Given that PPI-II but not PPI-I is associated with aggressive behavior, we hypothesize that aggression may mediate the association between PPI-II and utilitarian judgments. The present study We examine the associations between utilitarian judgment, psychopathic personality traits, anxiety, and aggression in a group of male and female college students. Based on Koenigs et al. (2011)'s finding, we hypothesize that: 1) Individuals with high psychopathic personality traits would be more likely to make utilitarian judgments for impersonal and personal moral dilemmas. 2) The association between psychopathy and utilitarian response to personal moral dilemmas would be moderated by anxiety: only individuals with high psychopathy AND low anxiety would show significantlymore utilitarian responses to personal moral dilemmas. 3) Utilitarian moral judgment would be more closely related to PPI-II but not PPI-I, and this associationwould bemediated by self-report aggression. Method Participants Three hundred and two undergraduate students (73% females, mean age = 21.97, SD = 6.19) from an urban college participated for course credit. They were from diverse racial and ethnic backgrounds: 39.6% Caucasian, 25.2% Asian, 14.4% African American, 11.7% Hispanic, and 9.1% from other racial backgrounds. They were tested in a small group setting (typically one to five participants per session). Participants responded to 15 dilemmas and a series of self-report questionnaires measuring their psychopathic personality traits, trait anxiety, aggression, and demographic information. All procedures were approved by the university IRB, and informed consent was obtained from all participants. Measures Moral dilemmas A total of 15 dilemmas (4 non-moral, 4 impersonal moral, and 7 personal moral scenarios, see Appendix A) selected from a previously published set (Greene et al., 2001; Koenigs et al., 2007) were presented in random order. To be consistent with the question format used in previous studies (Cima et al., 2010; Koenigs et al., 2011), 343Y. Gao, S. Tang / Journal of Criminal Justice 41 (2013) 342–349 Author's personal copy participants were asked to answer "Yes" or "No" to the hypothetical questions relating to the dilemmas, in the form of, "Would you ... in order to ...?" In non-moral dilemmas, participants were asked to make decisions that involved no harm. For example, they had to decide whether to substitute preferred macadamia nuts for walnuts in a batch of brownies, even though the recipe called for walnuts. Non-moral dilemmas served as a control and baseline condition. In impersonal moral dilemmas, participants were asked to make decisions that involved harm, but did not require them to directly commit harm to another person. For example, they were asked to decide whether they would flip a switch so that noxious fumes from a room with three people would flow into a room with one person. In personal moral dilemmas, participants were asked to make decisions that involved harm, but this time required them to directly commit harm to another person. For example, they had to decide whether to transplant the organs of a relatively healthy patient to five dying patients. "Yes" responses corresponded to "utilitarian" decisions that maximized utility or the greater good (Greene, 2003). There was no time limit for reading the scenario description or responding to the question. Following themethod of Koenigs et al. (2011), the proportion of "yes" responses for each type of scenario was calculated for each individual. Psychopathic personality inventory – short form (PPI-SF Lilienfeld & Andrews, 1996) Psychopathic personality was assessed using the PPI-SF and participants answered on a 4-point Likert scale (1 = false, 2 = mostly false, 3 = mostly true, 4 = true). The PPI-SF consists of eight subscales with internal consistencies, as measured by Cronbach's alpha for the undergraduate sample, ranging from .63 to .80 (Smith et al., 2011), and has good test retest reliability (Lilienfeld & Andrews, 1996; Poythress, Edens, & Lilienfeld, 1998). More recent factor analysis by Benning et al. (2003) and Smith et al. (2011) on these eight subscales revealed evidence for two largely orthogonal factors. Factor 1(PPI-I) is the sum of the standardized scores on Stress Immunity, Social Potency, and Fearless subscales, while Factor 2 (PPI-II) is the sum of the standardized scores on Blame Externalization, Machiavellian Egocentricity, Carefree Nonplanfulness, and Impulsive Nonconformity. The eighth subscale, Coldheartedness, failed to load substantially onto either factor (Benning, Patrick, Hicks, Blonigen, & Krueger, 2003; Smith et al., 2011). Internal consistency in the current sample for the subscales ranged from .50 (Coldheartedness) to .81 (Blame Externalization). In the following analyses, PPI-I, PPI-II, Coldheartedness, and PPI total score (sum of the eight subscales) were used. Reactive-Proactive Aggression Questionnaire (RPQ Raine et al., 2006) RPQ was used to indicate predatory (proactive) and impulsive (reactive) dimensions. The RPQ consists of 23 items of which 12 items make up the proactive subscale (e.g. "How often have you got others to gang up on someone else?") and 11 make up the reactive subscale (for example "How often have you gotten angry or mad or hit others when teased?"). Earlier studies have shown good internal reliabilities for total RPQ, with reactive and proactive subscale scores all above .81 (Raine et al., 2006).The RPQ has demonstrated good validity and the two factors of reactive and proactive aggression have been substantiated with confirmatory factor analysis (Raine et al., 2006). Internal consistency in the current sample was .82 for the total RPQ. For the reactive subscale internal consistency was .76, for the proactive subscale it was .69. The State-Trait Anxiety Inventory –Trait version (STAI–T, Spielberger, Gorsuch, Lushene, Vagg, & Jacobs, 1983) The STAI-T is a questionnaire comprising of 20 items designed to measure trait anxiety (e.g. "I worry too much over something that really doesn't matter"). Participants answer according to how they generally feel on a 4-point Likert scale (1 = almost never, 2 = somewhat, 3 = moderately, and 4 = very much). It assesses trait anxiety, but also relates to measures of depression and negative affect (Bados, Gomez-Benito, & Balaguer, 2010). We used the STAI-T because of its wide usage, history of good internal reliability, good test-retest reliability, and good convergent validity (Antony, Orsillo, & Roemer, 2001; Spielberger, 1989; Spielberger et al., 1983). In addition, we used the trait version rather than the state version because it is more theoretically consistent with our hypotheses – we are interested not in how anxiety at a particular moment would affect the relationship between psychopathy and moral judgments. Rather, we are interested in how a general feeling of anxiety is likely to moderate the relationship. Internal consistency in the current sample was .90. Statistical analyses Paired sample t-tests were conducted to compare participants' responses to different types of dilemmas. Effect sizes were reported using Cohen's d (Cohen, 1988). Multiple regressions were used to examine the relationships between psychopathic traits and utilitarian responses. To test for potential moderating effects of anxiety, hierarchical multiple regression analyses were conducted for the proportion of "yes" responses to overall moral dilemmas, personal moral dilemmas, and impersonal moral dilemmas separately. For these analyses, anxiety and PPI-total score were entered by subtracting the sample mean from each participant's score. In step 1, the proportion of "yes" responseswas regressed onto gender and PPI-total score. In step 2, the main effect of anxiety and a multiplicative interaction term was entered into the equation to test for the interaction between anxiety and psychopathy. To examine the potential mediating effect of aggression, we first followed Baron and Kenny (Baron & Kenny, 1986)'s criteria to assess if true mediation was present. The paths between independent, dependent and mediator variables were assessed by ordinary least squares (regression). The Sobel test (Sobel, 1982) was used to test whether themediator significantly attenuated the influence of the independent variable on the dependent variable. Finally, a 2-step regression analysis was conducted to examine if any remaining relationship between psychopathy and utilitarian responses was still significant after controlling for the aggression mediator. Results Moral (impersonal vs. personal) vs. non-moral dilemmas We first compared participants' responses between moral and non-moral scenarios. The proportion of "yes" responses to moral dilemmas was significantly lower than that to non-moral dilemmas (t (301) = 29.16, p b .001; for moral dilemmas, M = .40, SD = .19; for non-moral dilemmas, M = .85, SD = .19, d = 2.37). Participants also gave fewer utilitarian responses (i.e. smaller proportion of "yes" responses) to personal compared to impersonal moral dilemmas (t (301) = 4.15, p b .001; personal moral dilemmas, M = .38, SD = .25; impersonal moral dilemmas, M = .44, SD = .22, d = 0.24) (Fig. 1). Psychopathic personality and utilitarian responses Table 1 shows inter-correlations among main study variables. Participants who had higher PPI total scores (overall psychopathy) indicated a greater preference for "yes" responses to the moral dilemmas. This was true for the overall analysis, when responses across all 11 moral dilemmas were collapsed (r = .16, p b .01), as well as for the personal moral dilemmas in particular (r = .14, p = .02). The relationship between PPI total scores and utilitarian responses to impersonal moral dilemmas approached significance (r = .11, p = .05). Consistent with our third hypothesis, these significant relations with 344 Y. Gao, S. Tang / Journal of Criminal Justice 41 (2013) 342–349 Author's personal copy utilitarian responses were mainly driven by PPI-II (rs = .18 to .24, p b .05). PPI-I was not correlated with any of the decision variables. Compared to females, males scored significantly higher on PPI total (male M = 129.39, SD = 13.99, female M = 119.49, SD = 13.72, d = 0.71) and factor scores (PPI-I, male M = 0.28, SD = .77, female M = -.10, SD = 0.66, d = 0.53; PPI-II, male M = 0.20, SD = 0.61, female M = -0.07, SD = 0.62, d = 0.44; and Coldheartedness, male M = 0.21, SD = 1.06, femaleM = -0.08, SD = 0.97, d = 0.29). Males also scored lower on anxiety (male M = 40.88, SD = 9.85, female M = 45.22, SD = 10.61, d = -0.42) and were more likely to make utilitarian responses to personal moral dilemmas (male M = 0.47, SD = 0.26, female M = 0.34, SD = 0.23, d = 0.53). However, males did not differ from females in their responses to impersonal moral, non-moral dilemmas or reported aggression. To control for the observed effects of gender on the responses to moral dilemmas, we conducted multiple regressions for the utilitarian responses using PPI-II as a predictor while controlling for gender. As Table 2 shows, the relationship between utilitarian preferences and PPI-II was robust even when gender was taken into account.1 Moderating effect of anxiety Anxiety was not correlated with any of the decision measures. Hierarchical multiple regression analyses were conducted to test for potential moderating effects of anxiety. As shown in Table 3, there was no significant interaction between anxiety and psychopathy for the utilitarian responses to overall moral dilemmas or personal moral dilemmas. For impersonal moral dilemmas, the addition of an interaction term and anxiety added 3% of the variance to the prediction of the utilitarian responses, and the interaction between total psychopathy and anxiety approached significance (p = .06). Similar analyses were conducted with PPI-II replacing PPI-total in the equations. There was no significant interaction between anxiety and PPI-II in predicting for utilitarian responses to any type of the moral dilemmas. Thus, our second hypothesis was not supported. Mediating effect of aggression As can be seen in Table 1, participants who scored higher on aggressionmeasuresweremore likely tomake utilitarian responses for overall moral dilemmas, in particular the impersonal moral dilemmas (r = .14 to .33, p b .05). None of the aggression variables were significantly correlated to responses to personal moral or non-moral dilemmas. Since proactive and reactive aggression showed similar relationship to all other variables (see Table 1), the following analyses were conducted using total aggression score. After controlling for total aggression, the amount of variance in utilitarian responses to overall moral dilemmas explained by PPI-II was reduced from 5.6% (F = 17.88, p b .001) to 1.7% (F = 3.77, p = .05). The Sobel test showed that this reduction was statistically significant (z = 3.36, p b .001). Similarly, the amount of variance in utilitarian responses to impersonal moral dilemmas explained by PPI-II was reduced from 3% (F = 6.75, p = .01) to 0.1% after controlling for aggression. The Sobel test showed that this was a statistically significant reduction (z = 4.90, p b .01). Regression analyses indicated that the residual variance in utilitarian responses explained by PPI-II after controlling for aggression was no longer significant (F = 0.04, p = .84), suggesting that aggression completely mediated the psychopathyutilitarian responses relationship for impersonal moral scenarios.2 Fig. 1. Proportion of "yes" responses to overall moral, non-moral, personal, and impersonal dilemmas in college students. Bars refer to 1 standard error of the mean. Table 1 Intercorrelations between main study variables, together with means and SDs PPI-total PPI-I PPI-II Coldheartedness Anxiety Aggression Proactive aggression Reactive aggression Moral Personal moral Impersonal moral Non-moral Gender (1 = M, 2 = F) PPI-total 1 .73*** .69*** .32*** -.09 .33*** .39*** .21** .16** .14* .11a -.03 -.30*** PPI-I 1 .06 .19** -.41*** .05 .13a -.02 -.01 -.01 -.01 .09 -.24*** PPI-II 1 -.02 .39*** .50*** .50*** .41*** .24*** .21*** .18** -.09 -.19** Coldheartedness 1 -.31*** -.18** -.09 -.22** .03 .04 -.01 -.14* -.13* Anxiety 1 .21** .12 .24** .11 .08 .09 -.06 .18* Aggression 1 .86*** .91*** .21** .09 .33*** -.08 -.07 Proactive Aggression 1 .58*** .14* .03 .29*** -.06 -.11 Reactive Aggression 1 .23** .12a .30*** -.09 -.03 Moral 1 .92*** .63*** .03 -.16** Personal Moral 1 .27*** .02 -.22*** Impersonal Moral 1 .04 .03 Non-moral 1 -.02 Gender 1 Mean 122.13 0.00 0.00 0.00 44.16 12.51 2.97 9.25 .40 .38 .44 .85 SD 14.47 .71 .63 1.00 10.60 5.92 3.02 3.79 .19 .25 .22 .19 Note. * p b .05, ** p b .01, ***p b .001, a p b .10. Table 2 Relationships between PPI-II and utilitarian responses, controlling for gender – standardized betas PPI-II Gender (1 = Male, 2 = Female) Moral .21*** -.12* Personal Moral .17** -.19** Impersonal Moral .19** .06 345Y. Gao, S. Tang / Journal of Criminal Justice 41 (2013) 342–349 Author's personal copy Discussion In this study, we investigated the association between psychopathic traits and moral judgments. Results supported our first and third hypotheses. As expected, individuals with high psychopathic traits were generally more willing than those with low traits to endorse impersonal or personal harms or rule violations in order to achieve certain beneficial outcomes. More interestingly, utilitarian responses to personal and impersonal moral dilemmas were positively correlated with the second factor of PPI (PPI-II, including Blame Externalization, Machiavellian Egocentricity, Carefree Nonplanfulness, and Impulsive Nonconformity) but not the first factor (PPI-I, including Stress Immunity, Social Potency, and Fearlessness) or Coldheartedness, and the association between psychopathy and utilitarian moral judgments was mediated by aggressive behavior. Our second hypothesis, however, was not supported: anxiety did not moderate the association between psychopathy and utilitarian response to personal moral dilemmas. Our finding suggests that the association between utilitarian judgment and psychopathic traits is mainly driven by these externalizing characteristics rather than the affective/interpersonal factor of psychopathy. As mentioned in the introduction, PPI-I and PPI-II largely represent a global indicator of (reduced) internalizing and (increased) externalizing psychopathology, respectively. These differential relationships further demonstrate that it is critical to disaggregate potentially heterogeneous psychopathological and personality constructs into narrower, homogeneous components (Fowles & Dindo, 2009; Lynam & Widiger, 2007; Smith et al., 2011) to fully understand the mechanism and processes underlying moral judgments in psychopathic individuals. In contrast to our prediction, we failed to find moderating effect of anxiety, suggesting that the behavioral/emotional deficit that gives rise to utilitarian judgments characterizes all those who score high on psychopathic traits, regardless of anxiety level. One possible reason for the discrepant finding from Koenigs et al. (2011) is that Koenigs et al. examined the association in an incarcerated sample whereas we focused on non-institutionalized college sample. Different findings seem to suggest that differences between criminal and noncriminal psychopathy are both quantitative and qualitative in nature. We also found that males gave significantlymore utilitarian answers to personal moral dilemmas when compared to females, although there were no gender differences in utilitarian responses to non-moral dilemmas or impersonal moral dilemmas. This was consistent with prior literature (Fumagalli et al., 2010; Youssef et al., 2012), suggesting that the cognitive-emotional processes and the underlying neurobiological mechanisms involved in evaluating personal moral dilemmas differ in males and in females (Fumagalli et al., 2010). Although Coldheartedness was negatively correlated with the proportion of "yes" responses to non-moral dilemmas, it was not significantly correlated with the utilitarian responses to personal or impersonal moral dilemmas in our study. This may be due to the relatively low internal reliability of the subscale (alpha = .50), and a Type II error is possible. Nevertheless, we found Coldheartedness to be significantly correlated with PPI-I but not PPI-II, which are consistent with prior literature (Smith et al., 2011). One of the limitations of the study is that self-report scales were used throughout this research. Despite its practical and effective quality of assessing traits of interests, this method may be prone to response bias. Although psychopathic traits, aggression, and moral judgment have been found to be significantly related to socially desirable responding (Warren, 2009), the same study has found that socially desirable responding did not have a significant impact on the associations between psychopathy, aggression, and empathy. Nonetheless, future studies could use other measures, such as utilizing role-play in the laboratory or other behavioral paradigms to replicate and build upon this research. The second issue is representativeness. There were significantly more females than males in our sample, although in the general population psychopaths are predominantly males. This was mainly due to the gender imbalance inherent within the psychology course make-up at the college, with approximately 70-80% of psychology students being female. Although we controlled for the effect of gender in the analyses, future studies with a large sample or an equal number of males and females are required to investigate the effect of gender imbalance on the associations of interest. Additionally, despite our participants being ethnically and socioeconomically diverse, we used a convenience sample of college students. They may differ qualitatively from those who have been convicted or incarcerated, and hence the relationship between the dimensions of psychopathic traits may relate differently to moral judgments. This, in turn, would imply different antisocial behavior prevention strategies for the two sets of populations. Conclusions In addition to building upon prior research on moral decisionmaking of psychopaths, our research has contributed to the understanding of the specific factors of psychopathy that are associated with utilitarian decision-making. Our results suggest that externalizing traits contribute more strongly to utilitarian responses, and Table 3 Hierarchical regression analyses testing for the potential moderating role of anxiety Moral Personal Moral Impersonal Moral β R2 R2 change β R2 R2 change β R2 R2 change PPI-total Gender -.20** -.27*** .04 PPI-T .06 .03 .07 0.05 .08 .01 Gender -.24** -.30*** -.01 PPI-T .06 .04 .08 Anxiety .15* .13a .10 Anxiety × PPI-T .07 .01 .14a 0.08 0.03a .10 .02 .03 .03a PPI-II Gender -.19* -.26** .04 PPI-II .19* .17* .11 .08 .11 .01 Gender -.21** -.27*** .02 PPI-II .16a .14a .10 Anxiety .07 .07 .03 Anxiety × PPI-II .04 -.01 .10 .09 .01 .11 .01 .02 .01 Note. * p b .05, ** p b .01, ***p b .001, a p b .10. 346 Y. Gao, S. Tang / Journal of Criminal Justice 41 (2013) 342–349 Author's personal copy this externalizing – utilitarianism association becomes more salient in the impersonal dilemmas when the prepotent negative emotional responses that oppose acts of harmareweaker or nonexistent. Psychopathy incurs an enormous toll on society, and understanding how psychopathic individuals make moral decisions is important. If their tendency to maximize outcome is largely associated with certain characteristics such as blame externalization rather than fearlessness, then correction programs should focus on exercising responsibility rather than trying to intimidate psychopathic offenders with the negative consequences of offending. Thus, for example, the California Department of Corrections and Rehabilitation Honor Program, which emphasizes personal responsibility and commitment to personal growth (Hartman, 2007), may be more successful than a program in which senior inmates recount stories of how their crimes destroyed their futures. Appendix A Non-Moral Scenarios: 1. You are bringing home a number of plants from a store that is about two miles from your home. The trunk of your car, which you've lined with plastic to catch the mud from the plants, will hold most of the plants you've bought. You could bring all the plants home in one trip, but this would require putting some of the plants in the back seat as well as in the trunk. By putting some of the plants in the back seat you will ruin your fine leather upholstery which would cost thousands of dollars to replace. Would you make two trips home in order to avoid ruining the upholstery of your car? 2. You have decided to make a batch of brownies for yourself. You open your recipe book and find a recipe for brownies. The recipe calls for a cup of chopped walnuts. You don't like walnuts, but you do like macadamia nuts. As it happens you have both kinds of nuts available to you. Would you substitutemacadamia nuts for walnuts in order to avoid eating walnuts? 3. You need to travel from New York to Boston in order to attend a meeting that starts at 2PM. You can take either the train or the bus. The train will get you there just in time for your meeting no matter what. The bus is scheduled to arrive an hour before your meeting, but the bus is occasionally several hours late because of traffic. It would be nice to have an extra hour before the meeting, but you cannot afford to be late. Do you take the train or the bus? 4. An old friend has invited you to spend the weekend with him at his summer home some ways up the coast from where you are. You intend to travel there by car, and there are two routes that you can take: the highway and the coastal road. The highway will get you to your friend's house in about three hours, but the scenery along the highway is very boring. The coastal route will get you to your friend's house in about three hours and fifteen minutes, and the scenery along the coastal road is breathtakingly beautiful. Would you take the coastal route (scenic but slower) or the highway (boring but faster)? Impersonal Moral Scenarios: 5. You are the late-night watchman in a hospital. Due to an accident in the building next door, there are deadly fumes rising up through the hospital's ventilation system. In a certain room of the hospital are three patients. In another room there is a single patient. If you do nothing the fumes will rise up into the room containing the three patients and cause their deaths. The only way to avoid the deaths of these patients is to hit a certain switch, which will cause the fumes to bypass the room containing the three patients. As a result of doing this the fumes will enter the room containing the single patient, causing his death. Would you hit the switch in order to avoid the deaths of the three patients? 6. You work for the Bureau of Health, a government agency. You are deciding whether or not your agency should encourage the use of a certain recently developed vaccine. The vast majority of people who take the vaccine develop an immunity to a certain deadly disease, but a very small number of people who take the vaccine will actually get the disease that the vaccine is designed to prevent. All the available evidence, which is very strong, suggests that the chances of getting the disease due to lack of vaccination are much higher than the chances of getting the disease by taking the vaccine. Would you direct your agency to encourage the use of this vaccine? 7. You are walking down the street with a friend, and he drops his wallet as he is getting into his car to leave. You open the wallet and find that it contains several hundred dollars and some credit cards. Your friend is wealthy. You, however, have been hit by hard times recently and could really use some extra cash. You consider sending the wallet back to your friend without the cash, keeping the cash for yourself. Would you keep the money you found in the wallet? 8. You have been trying to find a job lately without much success. You figure that you would be more likely to get hired if you had a more impressive resume. You could put some false information on your resume in order to make it more impressive. By doing this you might ultimately manage to get hired, beating out several candidates who are actually more qualified than you are. Would you put false information on your resume? Personal Moral Scenarios: 9. You are a doctor. You have five patients, each of whom is about to die due to failing organs. You have another patient who is healthy. The only way you can save the lives of the first five patients is to transplant five of this young man's organs (against his will) into the bodies of the other five patients. If you do this, the young man will die, but the other five patients will live. Would you perform this transplant in order to save five of your patients? 10. You are on a cruise ship when there is a fire on board, and the ship has to be abandoned. The lifeboats are carrying many more people than they were designed to carry. The lifeboat you're in is sitting dangerously low in the water – a few inches lower and it will sink. The seas start to get rough, and the boat begins to fill withwater. If nothing is done it will sink before the rescue boats arrive and everyone on board will die. However, there is an injured person who will not survive in any case. If you throw that person overboard, the boat will stay afloat and the remaining passengers will be saved. Would you throw this person overboard in order to save the lives of the remaining passengers? 11. You are the leader of a small group of soldiers. You are on your way back from a completed mission deep in enemy territory when one of your men has stepped in a trap that has been set by the enemy and is badly injured. The trap is connected to a radio device that by now has alerted the enemy to your presence. They will soon be on their way. If the enemy finds your injured man they will torture him and kill him. He begs you not to leave him behind, but if you try to take him with you your entire group will be captured. The only way to prevent this injured soldier from being tortured is to shoot him yourself. Would you shoot this soldier in order to prevent him from being tortured by the enemy? 12. You are the captain of a military submarine travelling 347Y. Gao, S. Tang / Journal of Criminal Justice 41 (2013) 342–349 Author's personal copy underneath a large iceberg. An onboard explosion has caused you to lose most of your oxygen supply and has injured one of your crew who is quickly losing blood. The injured crew member is going to die from his wounds no matter what happens. The remaining oxygen is not sufficient for the entire crew to make it to the surface. The only way to save the other crew members is to shoot dead the injured crew member so that there will be just enough oxygen for the rest of the crew to survive. Would you kill the fatally injured crew member in order to save the lives of the remaining crew members? 13. It is wartime and you and your two children, ages eight and five, are living in a territory that has been occupied by the enemy. At the enemy's headquarters is a doctor who performs painful experiments on humans that inevitably lead to death. He intends to perform experiments on one of your children, but he will allow you to choose which of your children will be experimented upon. If you refuse to bring on of your children to his laboratory, he will find both of them and experiment on both of them. Would you bring one of your children to the laboratory in order to avoid having them both die? 14. A viral epidemic has spread across the globe killing millions of people. You have developed two substances in your home laboratory. You know that one of them is a vaccine, but you don't know which one. You also know that the other one is deadly. Once you figure out which substance is the vaccine you can use it to save millions of lives. You have with you two people who are under your care, and the only way to identify the vaccine is to inject each of these people with one of the two substances. One person will live, the other will die, and you will be able to start saving lives with your vaccine. Would you kill one of these people with a deadly injection in order to identify a vaccine that will save millions of lives? 15. You are in a hospital waiting to visit a sick friend. A young man sitting next to you explains that his father is very ill. The doctors believe that he has a week to live at most. He explains that his father has a substantial life insurance policy that expires at midnight. 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Modal disagreements Justin Khoo [email protected] Forthcoming in Inquiry Abstract It is often assumed that when one party felicitously rejects an assertion made by another party, the first party thinks that the proposition asserted by the second is false. This assumption underlies various disagreement arguments used to challenge contextualism about some class of expressions. As such, many contextualists have resisted these arguments on the grounds that the disagreements in question may not be over the proposition literally asserted. The result appears to be a dialectical stalemate, with no independent method of determining whether any particular instance of disagreement is over the proposition literally asserted. In this paper, I propose an independent method for assessing whether a disagreement is about what's literally asserted. Focusing on epistemic modals throughout, I argue that this method provides evidence that some epistemic modal disagreements are in fact not over the proposition literally asserted by the utterance of the epistemic modal sentence. This method provides a way to break the stalemate, and reveals a new data point for theories of epistemic modals to predict-that is, how there can be such modal disagreements. In the rest of the paper, I motivate a general theory of how to predict these kinds of disagreements, and then offer some brief remarks about how contextualist, relativist, and expressivist theories of epistemic modals might accommodate this new data point. Keywords: epistemic modals, disagreement, contextualism, relativism, expressivism. Consider two types of disagreement. Believer asserts that God exists. Atheist disagrees, asserting that God does not exist. Agnostic also disagrees-but unlike Atheist, he doesn't think that what Believer says is false (and he also doesn't think that what Atheist says is false). Exactly what Believer and Agnostic disagree about is not obvious-perhaps they disagree about whether to believe (or whether one ought to believe) that God exists. No matter: there is a clear difference between the two types of disagreement. Let's say Atheist Type-1 disagrees with Believer's assertion, and that to Type-1 disagree with an assertion is to believe (or claim) that what is asserted (or said) by it is false. By contrast, let's say 1 that Agnostic Type-2 disagrees with Believer's assertion, and that to Type-2 disagree with an assertion is dispute it in some way but not believe (or claim) that what is asserted by it is false. This is vague, since we've only been told about Type-2 disagreement by learning what it is not, but still hopefully enough to get a sense of it-disagreement centered on an assertion but not over the truth or falsity of what's asserted. If sometimes in rejecting an assertion (by saying "No" in response to it, e.g.), one thereby Type-2 disagrees with it, this raises trouble for various arguments from disagreement that are often marshaled against contextualist theories about a range of expressions. The trouble is that such arguments depend on the assumption that the disagreements in question are of Type-1. As a case study, we'll look at the recent debate about the semantics and pragmatics of epistemic modals-words like might and must, which allow us to talk about possibilities of a distinctively epistemic flavor (think: the kind of possibilities we aim to distinguish between in inquiry, draw on evidence to rule out, and so on)-focusing entirely on sentences containing a single unembedded epistemic modal (e.g., 'The keys might be in the drawer'). Consider the following story:1 Mobster. Fat Tony is a mobster who has faked his own death in order to evade the police. He secretly plants highly compelling evidence of his murder at the docks. The evidence is discovered by the authorities, and word gets out about his apparent death. Several forensic experts have examined the evidence and concluded that Fat Tony is dead. One of the experts who has examined the evidence, Smith, offers a more cautious assessment. Smith says, (1) Fat Tony might be dead. However, Beth, another investigator on the case, has already learned that Fat Tony faked his death and is currently hiding out in his warehouse. Overhearing Smith's conversation, Beth jumps in to say, (2) No, Fat Tony is alive. He faked his death! Here, in rejecting Smith's assertion, Beth expresses her disagreement with it. Let's call Smith and Beth's disagreement a modal disagreement (generally, a modal disagreement is one in which one party assertively utters an epistemic modal sentence and the other 1Modified slightly from Knobe & Yalcin, 'Context-Sensitivity of Epistemic Modals'. 2 expresses disagreement by rejecting that assertion). The standard disagreement argument against contextualism requires an additional assumption-that in rejecting Smith's assertion, Beth Type-1 disagrees with it (hence that Beth thinks that what Smith asserted is false). The trouble for contextualism is that there doesn't seem to be any single proposition about some body of evidence that Smith is warranted in asserting and Beth thinks is false. For instance, suppose that the proposition Smith asserts, ♦p, is the proposition that it is compatible with the evidence examined by Smith that Fat Tony is dead. This is something Smith would be warranted in asserting, but not something Beth would think is false! Suppose alternatively that ♦p is the proposition that it is compatible with the best evidence that Fat Tony is dead. Although this is something Beth would think is false (since she knows Fat Tony is alive), it's not something Smith would be warranted in asserting, since, presumably, he has no idea what the best evidence says about whether Fat Tony is dead (only what his evidence says). In a slogan: the first strategy fails to predict that Beth thinks that what Smith asserts is false, while the second fails to predict that Smith is warranted in what he says. This disagreement argument has been used to motivate relativism about epistemic modals (the view that the truth value of the proposition expressed by an epistemic modal claim is relative to an assessor) over orthodox contextualism about epistemic modals (the view that the proposition expressed by an assertive utterance of an epistemic modal sentence depends on the context in which it is uttered).2,3 However, the existence of rejections expressing Type-2 disagreements (such as perhaps between Believer and Agnostic) suggests that the crucial assumption of the argument-that in rejecting Smith's assertion, 2Relativism can predict a unique proposition ♦p that is asserted by Smith and believed to be false by Beth because it abstracts the evidence from the proposition expressed by an utterance of (1) and makes it a parameter that the truth of that proposition is relative to. According to relativism, ♦p is true as assessed by j just in case j's evidence is compatible with p. Given this rough gloss on the relativist theory, ♦p will be true as assessed by Smith since the evidence he examined is compatible with p. This ensures that Smith's assertion of ♦p is warranted. However, ♦p will be false as assessed by Beth since her evidence is not compatible with p (cf. Egan et al., 'Epistemic Modals in Context'; Stephenson, 'Judge Dependence'; Macfarlane, 'Epistemic Modals', Assessment Sensitivity). 3A view that tries to split the difference between relativism and contextualism is the pluralist theory of von Fintel & Gillies, 'Might Made Right'. On that theory, an utterance of an epistemic modal sentence 'puts in play' a plurality of propositions which are available for assessment of truth and falsity. Thus Smith's utterance may put in play both the proposition that it is compatible with Smith's evidence that Fat Tony is dead and the proposition that it is compatible with Beth and Smith's evidence that that Fat Tony is dead (perhaps among others). Smith's assertion is warranted if he knows (or reasonably believes) at least one of these propositions-in this case, the former-while the assessment that what Smith said is false is also correct since the latter proposition is also available for assessment and is false. I'll set aside this view in what follows. 3 Beth Type-1 disagrees with it-is controversial. Indeed, in the most recent iteration of the dialectic over disagreement arguments, many contextualists argue that we should reject this assumption rather than accept the anti-contextualist conclusion of the argument.4 However, although proponents of disagreement arguments sometimes recognize that not all disagreements are of Type-1, they continue to push disagreement arguments against their opponents.5 Thus, such theorists must think that, despite the possibility of Type-2 disagreements, the ones under consideration are Type-1. Since this assumption is exactly what (at least some of) their contextualist opponents reject, the intuitions of the various theorists involved about such disagreements are clearly not going to settle the matter. One reaction to this result is to conclude a stalemate and look for other data or theoretical considerations with which we might distinguish the theories.6 However, I think this pessimistic reaction is not forced on us. In this paper, I will propose an independent method for distinguishing which rejections express Type-1 and Type-2 disagreements. My strategy will be to look at experimental evidence that allows us to tease apart intuitions about rejecting a claim from intuitions about whether the claim is false. I argue that this 4Björnsson & Finlay, "Metaethical Contextualism Defended'; Montminy, 'Epistemic Modals and Indirect Weak Suggestives'; Plunkett & Sundell, 'Disagreement and the Semantics of Normative Terms'; Sundell, 'Disagreements About Taste'; Huvenes, 'Epistemic Modals and Credal Disagreement'. Of these theorists, only Montminy and Huvenes engage with epistemic modals. Since both differ from my own account of Type2 modal disagreements (see §3–4), I pause to register a concern about each. One trouble with Montminy's proposal that utterances of epistemic possibility claims make two speech acts-one primary assertion of some proposition and the other a 'weak suggestive'-is that we are not told enough about what a weak suggestive is and why utterances of these sentences express them to evaluate the proposal. For instance, Montminy says that to weakly suggest that p is to 'merely express low credence in the proposition that p' (Montminy, 'Weak Suggestives', 17). However, what is it to express a low credence in a proposition? On one understanding (cf. Yalcin, 'Epistemic Modals', 'Nonfactualism'), it is to do something that has a certain characteristic effect on the common ground of the conversation. Let's suppose that's right. Then, still, Montiminy's account is incomplete, since he never explains why utterances of epistemic possibility sentences are (sometimes) used to make weak suggestives (only that they are 'regularly used to indirectly perform weak suggestives' (Montminy, 'Weak Suggestives', 20)). For a possible answer to this question (which I'm not sure Montminy would agree with), see my discussion of contextualism in §4. Huvenes argues for the possibility that some epistemic modal disagreements are disagreements in credence (that is, arising from the two parties having different credences in some proposition). I worry that, given that disagreeing with someone's claim is generally sufficient to felicitously reject it (by saying 'no'), Huvenes' theory incorrectly predicts that the following dialogue should be felicitous: (i) A: I think it's likely to rain. Self-report of a high credence in rain B: #No, I think it's not likely to rain. Self-report of a low credence in rain 5Lasersohn, 'Disagreement and Predicates of Taste'; von Fintel & Gillies, 'Might Made Right'; Macfarlane, Assessment Sensitivity. 6As in Björnsson & Finlay, "Metaethical Contextualism Defended' and Plunkett & Sundell, 'Disagreement and the Semantics of Normative Terms'. 4 method provides evidence that in rejecting Smith's assertion in Mobster, Beth Type-2 disagrees with it. Indeed, I am optimistic that many other modal disagreements are of Type-2-though implementing this methodology to other scenarios is beyond the scope of this paper. If correct, my conclusion about Smith and Beth's disagreement changes the landscape on drawing semantic conclusions from data about disagreement, at least with respect to epistemic modals. First, it confirms the above contextualist response that we cannot immediately infer from every modal disagreement that the rejector thereby thinks that what is asserted is false. Second, it reveals a new data point for theories of epistemic modals to predict: that is, how and when Type-2 modal disagreements arise.7 The rest of this paper is structured as follows. In §1, I argue that we should be open to the possibility that, in rejecting Smith's assertion in Mobster, Beth Type-2 disagrees with it. I do so by arguing against a widely held view of what it is to reject a claim that entails that every felicitous rejection expresses a Type-1 disagreement. Then, in §2, I appeal to new empirical evidence that, in rejecting Smith's assertion in Mobster, Beth Type-2 disagrees with it. This reveals a new independent method for distinguishing Type1 and Type-2 disagreements. It also leaves us with a puzzle-what, if not what Smith asserts, could Smith and Beth be disagreeing about? In §3, I show how a well-known theory of conversation due to Robert Stalnaker may be naturally extended to allow for the possibility of rejections expressing Type-2 disagreements. Then in §4, I sketch how we can understand Type-2 modal disagreements within such a theory. The basic idea draws on the observation that generally, when someone asserts an epistemic possibility claim like (1), they propose that the negation of its prejacent (in this case, that Fat Tony is dead) not be taken for granted in their conversation. Someone may thus reject such an assertion to express that they refuse to comply with this proposal. In so doing, the two parties disagree, and their disagreement seems to be over whether to take some proposition (in this case, that Fat Tony is dead) for granted in the conversation. I show how how some of the existing semantic/pragmatic theories of epistemic modals might be able to predict such Type-2 modal disagreements in just this way. 7I pause to point out a limitation in the scope of my discussion-I will focus on disagreements expressed by way of the rejection of some assertion, and hence will not discuss disagreements that may arise merely in the thoughts of two parties who are not in conversation with one another. I hope to extend the thoughts in this paper to such other cases in future work. 5 1 Rejection and disagreement We usually express disagreement with someone using words like no, false, and wrong. To fix some rough terminology, say that when someone utters a sentence of the form No, . . . / That's false, . . . / Wrong, . . . targeting some assertion, they express that they reject that assertion. Now, one might think that there's not much difference to the choice of words we use to reject an assertion-each amounts to the same thing: asserting that what the other person said is false. Thus, consider the following thesis about rejection:8 rejecting is contradicting: to reject an assertion just is to claim that what is asserted by it is false. It follows from rejecting is contradicting that there can't be any Type-2 modal disagreements. In this section, I'll argue that rejecting is contradicting is false- there is an important difference between saying no in response to someone's assertion (such as what Beth does in Mobster) and calling what they say false (I'll set aside wrong for now). In particular, my claim in this section is that in some cases, by rejecting an assertion, one thereby Type-2 disagrees with it, and thus that it's an open possibility that in rejecting Smith's assertion, Beth Type-2 disagrees with it. A prima facie motivation for rejecting is contradicting is the fact that it accounts for the contrast between minimal pairs like the following:9 (3) a. Sue: I'm a doctor. b. Tim: #No, I'm not a doctor. c. Tom: No, you're not a doctor. The explanation goes as follows: both Tim and Tom express that they reject Sue's assertion; however, only Tom in fact rejects Sue's assertion since only he claims that what Sue asserts is false. Thus, according to rejecting is contradicting, the reason Tim's utterance is infelicitous is because he attempts to reject Sue's assertion, but in fact he fails to do so, since he does not claim that what she said is false. Tom's utterance is felicitous because it succeeds in its rejection aim, since he claims that what Sue said is false. It naturally follows that in general a necessary condition for a rejection of some assertion to be felicitous is for 8It's unclear whether anyone holds rejecting is contradicting in full generality. Nonetheless, it will be illustrative to rehearse some counterexamples to it to make room for my positive claim in §2. 9This minimal pair comes from Lasersohn, 'Disagreement and Predicates of Taste', 647; Moore, Philosophical Studies, 332-334. 6 the utterer to thereby claim that what is asserted is false. But then it follows, given that individuals believe what they say (the normal case), that in cases of felicitous rejections targeting some assertion, the rejector thinks that what is asserted is false. Thus, it follows that every felicitous rejection will express a Type-1 disagreement. And thus it follows that since Beth's rejection (in Mobster) is felicitous, she must therefore Type-1 disagree with Smith's assertion. To make room for the possibility that Smith and Beth's modal disagreement is of Type-2, therefore, we must find some reason to reject rejecting is contradicting. As it turns out, the thesis faces several counterexamples-cases of felicitous rejections in which the utterer intuitively rejects the targeted assertion but does not not claim that what is asserted is false. Consider the following example: (4) A and B are wondering whether the bank is open (it's a Saturday). A has just called a friend who told A that the bank was open last Saturday. A: The bank is open today. B: No, the bank might be open today. Banks are never open on national holidays and we still don't know whether today is a national holiday. In the example, B's rejection of A's assertion is felicitous, but B clearly does not claim that the bank is not open. To see this, notice the contrast between B's rejection in (4) and in (5): (5) A: The bank is open today. B: #That's false, the bank might be open today. Banks are never open on national holidays and we still don't know whether today is a national holiday. Unlike in (4), in (5), B claims that the bank is not open today, which clashes (Mooreparadoxically) with her subsequent claim that the bank might be open today. Nothing like this clash is felt in B's utterance in (4). But since B's utterance here isn't odd like Tim's in (3), the example shows that there are felicitous rejections in which the rejector intuitively rejects the targeted assertion but does not not claim that what is asserted is false. Here's another example:10: (6) A: Jim ate some of the cookies from last night. 10Horn, 'Metalinguistic Negation', A Natural History of Negation; Geurts, 'The Mechanisms of Denial'; van der Sandt & Maier, 'Denials in Discourse'. 7 B: No, he ate all of the cookies from last night. Again, B's rejection of A's assertion is felicitous even though B doesn't thereby claim that what A says is false-of course B believes that Jim ate some of the cookies, since she believes that he ate all of them.11 Thus, we have two prima facie counterexamples to rejecting is contradicting. In response, the defender of rejecting is contradicting might hold that our examples are not genuine cases of rejections. However, in making this move, the theorist threatens to make her claim uninteresting. What defines the class of 'genuine rejections' her theory is supposed to apply to-are they just those in which the rejector thereby claims that what the assertor says is false? If so, the rejecting is contradicting will be vacuously true. However, even if the defender of rejecting is contradicting could find a non-circular way to delimit the scope of her principle, it's not clear it would be of much interest. After all, we've just seen intuitive cases of rejections that are sensible but in which the rejector does not claim that the targeted assertion is false. This calls out for explanation-why are these rejections felicitous, unlike Tim's in (3)? Unfortunately, rejecting is contradicting can offer no account here. My conclusion is that these counterexamples to rejecting is contradicting are genuine-they demand an alternative theory of rejection that can explain why they are felicitous (I'll sketch such a theory in §3). Although we have several counterexamples to rejecting is contradicting, we do not yet have an instance of a Type-2 modal disagreement (remember, a modal disagreement is one in which what's rejected is an assertion made by assertively uttering an epistemic modal sentence). Thus, it doesn't follow from the falsity of rejecting is contradicting that the kinds of disagreement cases motivating relativist and pluralist theories (such as Mobster, where it is the epistemic possibility claim which is being rejected) involve Type11Here's one more, just for good measure, from Stalnaker, 'On the Representation of Context'. Suppose A met two women at the party, one of whom was a philosopher and the other of whom was the Secretary of Health and Human Services. However, as a practical joke, the two women A met were introduced to A as each other, so that A thought the first was the politician and the second the philosopher. We know all this. A then tells us: (i) I met an interesting woman last night who was a member of Obama's cabinet. We could then sensibly utter: (ii) No, that was the other woman! However, strictly speaking, we all believe that what A asserts when she utters (i) is true: after all, she did meet a female member of Obama's cabinet last night. 8 2 disagreements. Furthermore, some theorists even express a direct intuition that such modal disagreements must be of Type-1.12 However, the contextualist response that such disagreements may be of Type-2 should give us pause if we think mere intuition can settle whether any particular disagreement is Type-1 or Type-2. If there's nothing else to appeal to in deciding whether some disagreement is of Type-1 or Type-2, then perhaps a stalemate on this point is forced. However, in the next section, I will sketch a new method for distinguishing Type-1 and Type-2 disagreements, and show that it provides evidence that Smith and Beth's disagreement is of Type-2. 2 Type 2 modal disagreements My choice to begin with Knobe and Yalcin's 'Fat Tony' story was no accident. In a series of experiments surveying ordinary speaker intuitions about various versions of the Fat Tony story (all of which made it clear that Fat Tony is still alive), Knobe and Yalcin found that ordinary speakers overwhelmingly disagree with the statement that what the person claiming that Fat Tony might be dead (Smith, in my version) said was false. Notice that, if ordinary speakers tended to reject Smith's assertion when presented with the same vignette, we would have evidence that in rejecting Smith's assertion they Type-2 disagree with it. Furthermore, since these speakers are in an epistemic situation exactly analogous to Beth's in the Mobster vignette, this would be evidence that in rejecting Smith's assertion Beth Type-2 disagrees with it (that is, disputes it in some way, without claiming that what Smith says is false). Let's turn now to the experiment. 2.1 Methods 60 participants completed a questionnaire via Amazon Mechanical Turk (AMT). Participants were randomly assigned to either the False condition or the Rejection Condition. All participants received the following control vignette: Control. Two of your coworkers, Alex and Beth, are talking about their plan to meet a mutual friend Jim after work. Alex asks Beth: 'Where is Jim now?' 12For instance, Macfarlane says (about an analogous modal disagreement) that, 'It is crucial to such disputes that the participants take themselves to be contradicting each other when one says "it might be that p" and the other says "no, it can't be that p" ' (Macfarlane, 'Epistemic Modals', 148). 9 Moments ago, Beth received a call from Jim that she thought came from his house in the suburbs. Beth says, 'Jim is at home right now.' As a matter of fact, Jim is still working in his office down the hall, although neither Alex nor Beth knows this. Participants were told that they know all of the above. Those in the False condition were then asked whether they agreed with the claim that what Beth said is false. Those in the Rejection condition were then asked whether they would respond to Beth by saying something along the lines of 'No, Jim is actually working in his office right now'. Next, each participant received the experimental vignette, the following truncated version of Mobster: Modal. Fat Tony is a mobster who has faked his own death in order to evade the police. He secretly plants highly compelling evidence of his murder at the docks. The evidence is discovered by the authorities, and word gets out about his apparent death. Several forensic experts have examined the evidence and concluded that Fat Tony is dead. One of the experts who has examined the evidence, Smith, offers a more cautious assessment. 'Fat Tony might be dead,' Smith says. Participants were told that they were also on the case, and know all of the above. Those in the False condition were then asked whether they agreed with the claim that what Smith said is false. Participants in the Rejection condition were then asked whether they would respond to Smith by saying something along the lines of, 'No, Fat Tony is alive. He faked his death.' For all questions, participants responded on a 7 point Likert scale, with 1 being 'completely disagree', 4 being 'in between', and 7 being 'completely agree'. Before I report the results of the study, consider your own intuitions regarding these cases. Do you find a difference between Control (False) and Control (Rejection)? What about between Modal (False) and Modal (Rejection)? 2.2 Results The results of the experiment are displayed in Figure 1: 10 Figure 1 (displaying the means, with error bars displaying the standard error of the means) The crucial observation is that the mean ratings in Modal (False) were significantly lower than the mean ratings in Modal (Rejection), as revealed graphically by the gulf between those respective means on the graph.13 This reveals the following observation: The Difference Observation: when presented with Modal, ordinary speakers are strongly inclined to reject Smith's assertion but are also strongly inclined to disagree with the statement that what Smith said is false. This is direct evidence that ordinary speakers rejecting Smith's assertion thereby Type-2 disagree with it. Ordinary speakers tended to agree with rejecting Smith's assertion, which reflects (unlike Tim's rejection of Sue's assertion) their inclination to express some kind of disagreement with it. But ordinary speakers tended to disagree with the statement that what Smith said is false when presented with the same scenario. This suggests that they Type-2 disagree with Smith's assertion. Furthermore, since the speakers presented with the vignette Modal were in an analogous epistemic situation to Beth's in the Mobster vignette, The Difference Observation is evidence that in rejecting Smith's assertion, Beth Type-2 disagrees with it. 13The difference between Modal (Rejection) (M = 5.03, SD = 1.77) and Modal (False) (M = 2.42, SD = 1.61) was highly significant, t(59) = −6.04, p < .001. The difference between Control (Rejection) (M = 5.60, SD = 1.13) and Control (False) (M = 6.10, SD = 1.35) was not significant, t(59) = 1.55, p = .13. Finally, the difference between Control (False) (M = 6.10, SD = 1.35) and Modal (False) (M = 2.42, SD = 1.61) was significant, t(30) = 8.74, p < .001. This last piece of data confirms the results of Knobe & Yalcin, 'Context-Sensitivity'. 11 2.3 Discussion Recall that standard relativist theories like those defended by Egan et al. and Macfarlane are designed to predict a unique proposition expressed by an assertion of an epistemic modal claim whose truth is relative to an assessor.14 Thus, relativist theories are designed to make the right predictions about Smith and Beth's disagreement in Mobster, assuming that it is of Type-1; furthermore, such theories claim support from the fact that rival theories (for instance, contextualist theories) have trouble predicting a single proposition that Smith asserts and reasonably thinks is true and Beth reasonably thinks is false. However, this support is undermined if Smith and Beth's disagreement is of Type-2, for in that case the fact that they disagree provides no pressure to think that they disagree about whether what Smith says is true or false. On the contrary, the data provide a challenge to relativist theories, which is to explain how Beth, in rejecting Smith's claim, Type-2 disagrees with it (rather than Type-1 disagrees with it). Of course, this is a challenge faced by all theories of epistemic modals-something I'll return to discuss in §3-so the dialectical situation is not obviously worse for the relativist. However, importantly, unless the relativist can find a clear case of a Type-1 modal disagreement, she is no better off dialectically than her contextualist (or expressivist) peers.15 The only systematic study of the various disagreement cases proffered by relativists is Knobe & Yalcin's, and the results of that study are not promising for relativism.16 The method sketched above provides a way of testing for whether the rejector of some assertion Type-1 or Type-2 disagrees with it. If speakers tend to reject a particular assertion but not assess what was said by it as false, then we have evidence that in rejecting the assertion, they Type-2 disagree with it. If instead speakers tend to reject that assertion and also tend to assess what was said by it as false, then we have evidence that in rejecting the 14Egan et al., 'Epistemic Modals in Context'; Macfarlane, 'Epistemic Modals'. 15Of course, disagreement arguments are not the only arguments relativists have used to motivate their view. For instance, MacFarlane 2011 presents two other arguments for relativism: one which proceeds from direct truth value intuitions in eavesdropping cases like Mobster, and one which proceeds from intuitions about retraction in similar examples (see Macfarlane, 'Epistemic Modals', 146-148. See Knobe & Yalcin, 'Context-Sensitivity' for some challenges to both kinds of arguments. Finally, relativists have also appealed to data from cross-contextual indirect speech reports involving epistemic modals (as in Egan et al., 'Epistemic Modals in Context'). However, see Cappelen & Hawthorne, Relativism and Monadic Truth for a contextualist response to these challenges. 16In Knobe & Yalcin, 'Context-Sensitivity', four cases are explored. The first is analogous to the scenario discussed here, the second adds an extra-contextual assessor character, and the third and fourth compare retraction to assessments of falsity. In all of their cases they found that ordinary judgments go against those predicted by standard relativist theories. 12 assertion, they Type-1 disagree with it. The latter case is what we find in Control. In that vignette, we found that speakers are inclined to reject Beth's claim, and similarly inclined to judge that what she said is false. Thus, in that case, the most obvious conclusion to draw is that the reason they reject her claim is because they think that it is false-and this is just what it is to Type-1 disagree with it. Let's recap. So far, we have shown that there are possible cases in which someone rejecting an assertion thereby Type-2 disagrees with it (§1). We have also seen evidence that Smith and Beth's disagreement in Mobster is Type-2 (§2). However, we do not yet have a sense of what Smith and Beth could be disagreeing over, if not what Smith says. In the next section, I'll sketch a general framework of conversation that allows for the possibility of rejections expressing Type-2 disagreements. Then, in §4, I'll offer an account of what Smith and Beth could be disagreeing about, and supply some brief remarks about how some of the main existing theories of epistemic modals might be able to predict this. 3 Toward a theory of Type-2 disagreement Pre-theoretically, to reject something is to refuse to accept it. But what is someone who rejects an assertion thereby refusing to accept? A natural place to look for an answer to this question is the influential work of Robert Stalnaker.17 Stalnaker develops a theory of assertion on which to assert that p is to propose that p be mutually presupposed (or taken for granted) among the participants in the conversation.18 Call the set of propositions that are mutually presupposed in the conversation the common ground of that conversation. Thus, according to Stalnaker's theory of assertion, to assert a proposition p is to propose that p be common ground in one's conversation. We might extend this idea to other speech acts in a minimal way by holding that to inquire as to whether p is to propose making it common ground that whether p is a topic of discussion, and to command p to be the case 17Stalnaker, 'Assertion', 'On the Representation of Context', 'Common Ground'. In what follows, I'll make crucial use of Stalnaker's theory of conversation. However, other theories of rejection might be amenable to a related treatment in a different theory of conversation-see for instance Bach & Harnish, Linguistic Communication and Speech Acts; Brandom, 'Asserting', Making it Explicit. 18Mutually presupposed in the iterative sense, so that p is mutually presupposed among A and B iff A and B both presuppose p, and each presupposes that both presuppose p, and presupposes that both presuppose that both presuppose p, and so on. I follow Stalnaker in holding that presupposition is the attitude of acceptance for conversational purposes (which may be belief, or some attitude less committed than belief). See Stalnaker, 'Common Ground', Context. 13 is to propose making it common ground that p is demanded of someone, and so on.19 Remember that our aim is a plausible theory of conversation that allows for the possibility of rejections expressing Type-2 disagreements. In particular, we want to make sense of cases where A asserts that p and B rejects A's assertion, even though B doesn't think that p is false. To do so, we need to extend the basic Stalnakerian framework in two ways. First, we must say what accepting and rejecting an assertion is. Second, we must extend this notion to allow for rejections that (in some sense) target something other than the proposition asserted that is communicated by that assertion. I discuss both extensions in the next two sections. 3.1 Accepting and rejecting assertions Stalnaker does not take a stand on what accepting or rejecting an assertion amounts to. However, we may extend his theory of assertion in a natural way as follows: acceptance: to accept an assertion that p in some conversation is to reflexively presuppose p, conditional on everyone else in that conversation doing so.20 In effect, what accepting an assertion that p amounts to is doing your part to make p common ground on the condition that everyone else also does so. Thus, suppose that A,B, and C are in a conversation and A asserts that p. Suppose that both B and C believe p, and are thus prepared to reflexively presuppose it, conditional on each other doing so. Then, if it becomes common ground that both B and C accept A's assertion, p will be common ground in their conversation, and A's assertion will have succeeded. Suppose instead that B does not believe p, and is not prepared to presuppose p, regardless of what others in the conversation do. In that case, B refuses to accept A's assertion. If it becomes common ground that B refuses to accept A's assertion, then it will be common ground that p is not common ground in their conversation, and A's assertion will have failed. 19This extension to other speech acts is strictly optional. Officially, I want to remain neutral as to whether we should model all speech acts as proposals to update the common ground, or if some are best modeled as updating some other parameter of context (as in Lewis, 'Scorekeeping in a Language Game'). For instance, Craige Roberts argues that questions directly update a distinct contextual parameter which represents the questions under discussion in the context (see Roberts, 'Information Structure in Discourse'), and Paul Portner argues that imperatives directly update the To-Do List of their addressees (where this is a distinct contextual parameter which assigns a set of properties to each individual in the context; see Portner, 'Imperatives Within a Theory of Clause Types', 'Imperatives and Modals'). 20To reflexively presuppose p in some group G is to presuppose p, and presuppose that everyone in G presupposes p, and that everyone in G presupposes that everyone in G presupposes p, and so on. p is mutually presupposed in G iff everyone in G reflexively presupposes p. 14 Combining acceptance with hypotheses about how one can make it common ground that one accepts or refuses to accept an assertion will generate predictions about various conversational exchanges.21 This is where rejection comes into the picture. I propose that we combine this extension of the Stalnakerian theory with the following hypothesis about what it is to say no in response to an assertion: the rejection hypothesis: (a) To reject a proposal is to assert that one refuses to accept it. (b) In English, the answer-word no is conventionally used to reject proposals. Part (a) of the hypothesis connects rejecting a proposal with making it common ground that one refuses to accept it. Thus, by rejecting an assertion that p, you propose that it be common ground that you refuse to accept that assertion, and thus that it's common ground that p is not common ground. Evidence for part (b) of the hypothesis comes from the fact that we use no to reject different kinds of proposals, both linguistic and nonlinguistic.22 However, note that it doesn't follow from the rejection hypothesis that every utterance of no in response to an assertion results in rejecting it, or that uttering no in response to an assertion is the only way to reject it.23 The resulting theory (acceptance 21I won't go into acceptance here, but one thing to note is that it seems to be the default. Thus, if one does nothing in response to an assertion, presuming that it's common ground that the assertion was made and that one has had enough time to respond, this will be sufficient to make it common ground that one accepts that assertion. 22In addition to rejecting assertions, we find no used to reject commands, and even non-linguistic proposals such as romantic gestures: (i) A: Answer the phone! B: No, you answer the phone! (ii) 〈A leans in to kiss B〉 B: No, please don't. It's also not surprising, given the rejection hypothesis, that no would come to be used to express negative emotions (or even a kind of irrational denial) towards non-proposals. For instance, in observing an outcome of a competitive sports match, one might utter no! to express dismay at said outcome. 23For instance, if asked a yes-no question, no has both uses: (i) A: Did Bill quit smoking? B: No. C: No, your question is out of place-he never smoked in the first place! Also, one may sometimes use yes to reject an assertion, as in: 15 + the rejection hypothesis) entails that one way to reject an assertion that p is to say no in response to it, and that you ought to do so if you want to make it common ground that p not be common ground in your conversation. Before we get to the final refinement of the theory, I want to show how, in its current form, it already predicts some of the cases we've seen in which someone rejecting an assertion Type-2 disagrees with it. Recall the contrast between B's response in (4) and (5): (4) A and B are wondering whether the bank is open (it's a Saturday). A has just called a friend who told A that the bank was open last Saturday. A: The bank is open today. B: No, the bank might be open today. Banks are never open on national holidays and we still don't know whether today is a national holiday. (5) A: The bank is open today. B: #That's false, the bank might be open today. Banks are never open on national holidays and we still don't know whether today is a national holiday. the rejection hypothesis allows us to predict this contrast, since according to it, to reject an assertion is simply to make it common ground that you refuse to accept it. This might be because you think what is asserted is false, or because, as with B in (4), you are agnostic as to whether what is asserted is true. This allows us to predict that, in rejecting A's assertion in (4), B is merely refusing to reflexively presuppose what A asserts, not claiming that what A says is false (hence allowing us to predict the contrast in B's responses in (4)/(5)). On my extension of Stalnaker's theory, A and B's disagreement in (4) is merely over whether to presuppose that the bank is open today.24 Thus, adopting acceptance and the rejection hypothesis allows us to predict rejections that express Type-2 disagreements. (ii) A: It's raining. B: Yes, it might be. But it also might not be. Thanks to Vann McGee (p.c.) for pointing this out to me. 24I assume that the notion of disagreeing over whether to φ (for some action φ) is intuitively clear enough to take as a given for now. As such, I won't take a stand on what it amounts to or is grounded in. For instance, perhaps X and Y 's disagreement over whether to φ is grounded in their having conflicting (in the sense that both cannot be satisfied) desires about whether to φ. Or perhaps it is grounded in their having contradictory beliefs about whether each ought to φ. 16 3.2 Rejections beyond the asserted proposition We need to generalize the theory to handle cases like (6), where one rejects an assertion not because one does not want the proposition thereby asserted to be common ground, but because one wants to avoid something else the assertor thereby communicates in making that assertion becoming common ground (in this case, something she implicates, assuming the Gricean theory of scalar implicatures is broadly correct): (6) A: Jim ate some of the cookies from last night. B: No, he ate all of the cookies from last night. There is an interesting literature about uses of negation that seem to target not what's asserted but other, backgrounded, information also communicated by that assertion.25 My extension of Stalnaker's theory to handle cases where one rejects an assertion for such reasons is similar to a helpful proposal of van der Sandt's discussed by Geurts. I propose that we generalize the Stalnakerian theory to the level of what I will call the communicative impact of a speech act. To begin, recall that the common ground of a conversation may be represented by a set of propositions C. Let's call the communicative impact of an assertion that property of sets of propositions that the assertor thereby proposes the common ground of her conversation have.26 In the simplest case where someone asserts that p and doesn't thereby communicate anything else, the communicative impact of this assertion is the property of having p as a member (thus the theory makes the same predictions as the simple Stalnakerian theory in such cases). We can generalize from this case by adding other propositions to the communicative impact which are thereby communicated by the making of the assertion-they will include every proposition implicated and pragmatically presupposed by making that assertion (basically, any proposition that assertor intends to communicate by asserting what she does).27 We can also generalize to cases in which the communicative impact of an assertion is the property of not having certain propositions as a member (this will be characteristic of epistemic possibility claims, as we'll see in §4).28 25Horn, 'Metalinguistic Negation', A Natural History of Negation; Geurts, 'The Mechanisms of Denial'; van der Sandt & Maier, 'Denials in Discourse'. 26Standardly, a property of sets of propositions may be modeled as a set of sets of propositions. Thus, you can think of the communicative impact of an assertion as such a set of sets of propositions, each member of which represents a way the common ground may be compatible with how that impact aims for it to be. 27Importantly, P will not include information that will become common ground for other reasons-for instance, that the assertor, since she uses an English sentence to make her assertion, speaks English. 28Those familiar with dynamic semantic treatments of epistemic modals will recognize this kind of approach to thinking about how assertions can update what's common ground (cf. Veltman, 'Defaults in 17 In example (6) above, the communicative impact of A's assertion is the property of having as a member the proposition that Jim ate some of the cookies and that Jim didn't eat all of the cookies. Thus, in asserting that Jim ate some of the cookies, A thereby proposes that both of these propositions be common ground. With this notion of the communicative impact of an assertion in hand, we generalize acceptance as follows:29 acceptance*: to accept an assertion with communicative impact P in some conversation is to do your part to ensure that the common ground of that conversation has P , conditional on everyone else in that conversation doing the same. The only difference between acceptance and acceptance* is that in the latter we exchange a proposition p (the one asserted) for a property of sets of propositions P (the communicative impact of the assertion). On our generalized theory, then, accepting an assertion involves doing more than reflexively presupposing the proposition asserted (conditional on everyone else in the conversation doing so)-crucially, it involves doing your part to ensure that the common ground of your conversation has the property P (conditional on everyone else in the conversation doing so). This move allows our theory to predict the possibility of rejecting an assertion without thinking that what is asserted is false. Let's see this how our theory predicts this in the case of (6). As we saw above, the communicative impact of A's assertion is the property of having the proposition that Jim ate some but not all of the cookies as a member. Thus, in asserting that Jim ate some of the cookies, A thereby proposes that both of these propositions be common ground. Therefore, by acceptance*, were B to publicly accept A's assertion, it would thereby become common ground that Jim ate some but not all of the cookies. However, B believes that Jim ate all of the cookies, and thus wants this to be common ground. Therefore, B rejects A's assertion, thereby making it common ground that she refuses to accept A's assertion, thereby making it common ground that it is not common ground that Jim ate some but not all of the cookies. However, B cannot stop there, if she wants it to be Update Semantics'; Gillies, 'A New Solution to Moore's Paradox'; von Fintel & Gillies, 'An Opinionated Guide to Epistemic Modality'; Willer, 'New Dynamics for Epistemic Modality'). However, I don't take a stand here on the distinguishing claim of such theories, which is that we should think of the meaning of a sentence entirely in terms of its communicative impact. 29This is strictly speaking an extension of Stalnaker's theory of context, but one which keeps with the general spirit of that approach to conversations. Stalnaker all along recognized that what's pragmatically presupposed by an utterance will often be accommodated, and thus become common ground in similar fashion to what's asserted. My presentation here aims to incorporate what's implicated in the same vein. 18 common ground that Jim ate all of the cookies. Therefore, B follows up her rejection of A's assertion by counter-asserting that Jim ate all of the cookies. Notice that given acceptance* + the rejection hypothesis, we predict that in rejecting A's assertion, B Type-2 disagrees with it, since A and B agree that what A says is true.30 We will return to modal disagreements in the next section. However, before doing so, I want to highlight one final upshot of the theory of disagreement sketched here, which is that it can account for the minimal pair that initially motivated rejecting is contradiction: (3) a. Sue: I'm a doctor. b. Tim: #No, I'm not a doctor. c. Tom: No, you're not a doctor. Tim and Tom's utterances differ on their proposed reasons for rejecting Sue's assertion. Tim rejects Sue's claim because he thinks that he is not a doctor, while Tom does so because he thinks that Sue is not a doctor. Since the fact that Tim believes that he is not a doctor isn't (in any obvious way) a reason for him to refuse to accept Sue's assertion that she is a doctor, and given that Tim has no other reason to reject Sue's assertion,31 it follows that Tim has no reason to reject Sue's assertion. But then his rejecting her assertion makes no sense, and this accounts for why we perceive his rejection of her assertion to be infelicitous. Tom, on the other hand, believes that Sue is not a doctor, and this is a reason for him to reject Sue's assertion; he thinks the proposition she asserts is false, and thus wants it to be common ground between them that it is not common ground-therefore, he rejects it. Hence, we predict that Tom's rejection is felicitous. Thus, our generalized version of Stalnaker's theory of conversation predicts the range of felicitous and infelicitous rejections we seem to find, thus confirming the theory. 30I pause to respond to a criticism raised by Geurts towards the van der Sandt theory of denial which also applies to my generalized Stalnakerian theory of rejection (see Geurts, 'The Mechanisms of Denial', 286). The problem is that on my theory, rejection seems to target the entire communicative impact of the assertion, preventing any of it from becoming common ground. However, often when we reject an assertion, we object to some but not all of its communicative impact. We see this clearly in (6), where B objects to something A implicates, but not what A asserts. However, I don't think this is a serious problem for my theory. According to it, agents can still explicitly add back in any information that's part of the conversational impact of the rejected assertion in their own assertion. This is what B does when she explicitly counter-asserts that Jim at all of the cookies, and it's what we find generally with rejections: after rejecting an assertion, one often makes explicit ones grounds for so doing, which clarify what it was about the assertion that one found objectionable. 31This follows from what Tim says plus the assumption that he is following the Maxim of Quantity (be informative!). 19 We've just seen evidence in favor of acceptance* and the rejection hypothesis. In particular, we've seen that this combination of theses can account for the data rejecting is contradicting could (the contrast in (3)) and more besides (the examples of rejection that express Type-2 disagreements). However, we have not yet seen how the theory predicts Type-2 modal disagreements where it's an assertion made by uttering an epistemic modal sentence that is rejected (as in Smith and Beth's conversation in Mobster). In the next section, I outline how we might account for such modal disagreements within the theory of conversation motivated here. 4 Predicting Type-2 modal disagreements We begin with the key observation that at least one characteristic effect of accepting an assertion of epistemic possibility claim like (1) is that it will not be common ground that Fat Tony is alive-in other words, to ensure that there are some live possibilities compatible with what's mutually presupposed in the conversation in which Fat Tony is dead.32 (1) Fat Tony might be dead. Where the prejacent of (1) is the proposition that Fat Tony is dead (intuitively, the proposition in the scope of the modal operator denoted by might), we can put the observation as follows (note that this observation is in principle neutral between competing semantic theories-as we'll see below, different theories will be able to predict The Update Observation in different ways):33 The Update Observation: generally, assertively uttering an epistemic possibility sentence involves proposing that it not be common ground that its prejacent is false. (Thus, generally, the communicative impact of assertively uttering an epistemic possibility sentence will involve the property of not having as a member the negation of its prejacent.) 32Stalnaker, 'Pragmatics', Context; Swanson, 'How Not to Theorize'; Yablo, 'A Problem about Permission and Possibility'; Yalcin, 'Epistemic Modals', 'Nonfactualism', 'Bayesian Expressivism'; Willer, 'New Dynamics'. 33I do not want to commit to anything more than this being a general and plausible observation. In fact, I think there are instances in which it fails (for instance in Egan et al.'s 'surprise party' scenario, or von Fintel & Gillies' 'mastermind' case; see Egan et al., 'Epistemic Modals in Context', 13-14; von Fintel & Gillies, 'CIA Leaks', 83-84), but that these are exceptional cases. I won't be able to go into this point further here for the sake of space. Also, presumably, there will be similar update observations for epistemic necessity sentences, and graded (or probabilistic) modal sentences. 20 Let's look at our example from Mobster. There, Smith utters (1), and Beth knows that Fat Tony is alive. Given The Update Observation (and that this context is one in which it applies), Smith proposes that it not be common ground that Fat Tony is alive. Since Beth knows Fat Tony is alive and (presumably) wants to communicate this information, the aim of Smith's proposal is something she wants to avoid-instead, she wants it to be common ground that Fat Tony is alive. Therefore, Beth has a reason to reject Smith's assertion. Hence, Beth's rejection of Smith's assertion is felicitous. If this is the reason for Beth's rejection, then it follows that Beth disagrees with Smith about whether to presuppose that Fat Tony is alive-Smith aims for it not to be common ground in their conversation that Fat Tony is alive, while Beth aims for it to be common ground in their conversation that Fat Tony is alive. Furthermore, notice that in characterizing Smith and Beth's disagreement, we didn't need to say anything about whether Beth believes what Smith asserts. Thus, we have an account of how Beth could Type-2 disagree with Smith's assertion. Nonetheless, it would be nice to round out this merely prima facie explanation with a more concrete theory, which both predicts The Update Observation, as well as allows for cases like Mobster, where Beth rejects Smith's assertion without thinking that what he asserts is false. There are various ways to do this, depending on what kind of semantics we want to give for epistemic modals. In the rest of this section, I will show how we can combine this kind of explanation with three existing classes of theories. First, consider contextualism about epistemic modals. On a contextualist theory, epistemic modal sentences express propositions about some contextually salient body of information.34 Different contextualist theories embrace different metasemantic views about what body of information is contextually salient, but one of the most plausible versions holds that the salient body of information will be the best available evidence. On that view, by assertively uttering (1), Smith asserts the proposition that the best available evidence is compatible with Fat Tony being dead. Interestingly, although no contextualist theory has attempted to predict the dynamic update effects of uttering epistemic modal sentences, this one has a shot at predicting The Update Observation. The derivation goes roughly as follows: (i) it is generally common ground that everyone will aim to presuppose just what's entailed by the best available evidence, so (ii) learning that the best available evidence is compatible with p (the effect of assertively uttering might p) should result in (iii) it being 34Kratzer, 'Modality', Collected Papers, DeRose, 'Epistemic Possibilities'; Schaffer, 'Contextualism for Taste Claims and Epistemic Modals'; Dowell, 'Flexible Contextualism about Epistemic Modals'; Yanovich, 'Standard Contextualism Strikes Back'. 21 common ground that no one presupposes ¬p, which should result in it not being common ground that ¬p-and this just is the update effect described by The Update Observation. But now consider a context in which X, Y and Z are talking, and X assertively utters might p. Further, suppose that X, Y and Z all believe that the best available evidence is compatible with p. Nonetheless, Z knows that p is false-this is possible even though Z believes the best available evidence is compatible with p as long as Z thinks that her evidence that p is false is not available to X and Y.35 In that case, if Z accepts X's assertion, then by the above derivation it will not be common ground that ¬p. Nonetheless, Z has a reason to reject X's assertion since she (presumably) wants it to be common ground that ¬p. Therefore, this version of contextualism can predict cases like this in which Z doesn't think that what X said is false even though she may sensibly reject his claim (and hence in which Z's rejection expresses a Type-2 modal disagreement). Although the strategy remains to be fleshed out and independently motivated, it should be clear that at least some contextualist theories of epistemic modals are in a position to predict Type-2 modal disagreements. What about expressivism? Roughly, on such a theory, epistemic modal sentences do not express propositions but rather conditions on information states (like belief states, or context sets). According to Yalcin's expressivist theory, what it is to assertively utter such a sentence is to express a property of one's state of mind.36 This is importantly different from saying that one is in that state of mind-the two have different aims, for instance. To express a property of one's state of mind is to aim to get others to share it, while to say that one is in some state of mind is to aim to get others to believe that one is in that state of mind. According to the expressivist, when Smith assertively utters (1), he expresses a property of his belief state, namely, that it is compatible with Fat Tony being dead. In so doing, he aims to get others' belief states to share this property of his belief state, and hence come to not believe that Fat Tony is alive. Thus, expressivist theories of epistemic modals are well-suited to predict The Update Observation. However, can expressivism also predict that Beth, despite sensibly rejecting Smith's assertion, also not think that what 35An anonymous reviewer challenges this claim on the grounds that X and Y could easily get the information by asking Z. In response I hold that it depends on the notion of "availability" one has in mind. If availability is a matter of being able to make use of the evidence, then it is not available to X or Y. If availability is a matter of being able to easily get the evidence, then it may be available to X and Y. In other work, I propose drawing on this flexibility to predict the variation in judgments about whether what was said by some epistemic modal claim is false (see Papafragou, 'Epistemic Modality and Truth Conditions'; Dowell, 'Flexible Contextualism about Epistemic Modals', 'Flexible Contextualism about Deontic Modals'). 36Yalcin, 'Epistemic Modals', 'Nonfactualism'. 22 Smith says is false? Yalcin holds that, since according to expressivism, what Smith asserts is not a proposition (but rather a property), it is neither true nor false.37 Hence, when presented with the question of whether what Smith said is false, ordinary speakers are thereby forced to try to figure out what the questioner is after, and he suggests they may consider the following surrogates: • Rational: Would someone equipped with Smith's evidence be responding appropriately to it by accepting what Smith said? • Advisable: Would someone with full information be responding appropriately to it by accepting what Smith said? Yalcin contends that the answer to Rational is yes and Advisable is no. Hence, someone who thought it rational for Smith to have said what he did but that it would not be advisable to accept what Smith said might reject Smith's claim but then, interpreting the question about whether it is false as the Rational question, think that Smith's claim wasn't false.38 Finally, consider relativist theories of epistemic modals. Although we've undermined one standard argument for such theories, there is still life yet for relativism. According to a standard version of relativism, epistemic modal sentences express propositions whose truth value depends on a body of information that is salient in the context in which they are assessed.39 Relativists thus have room to maneuver out of predicting that Beth believes what Smith said is false in Mobster as long as they go flexible and don't predict that the information salient in Beth's context is automatically the evidence she has.40 When it comes to predicting The Update Observation, relativists have a few options. One is to try to derive it in a manner similar to the contextualist. Another is to appeal to the resources from de se assertion to derive the update effects.41 However, although it remains unclear at this time whether going flexible will make trouble for predicting The Update Observation, it nonetheless seems that relativist theories of epistemic modals are in a position to predict Type-2 modal disagreements. 37Yalcin, 'Nonfactualism', 311-312. 38Dynamic theories of epistemic modals may be able to say something similar to predict Type-2 modal disagreements. 39Egan et al., 'Epistemic Modals in Context'; Egan, 'Epistemic Modals, Relativism, and Assertion', 'Relativism and Epistemic Modals'; Stephenson, 'Judge Dependence, Epistemic Modals, and Predicates of Taste'; Macfarlane, 'Epistemic Modals'. 40Macfarlane, 'Epistemic Modals', 174-177. 41See Stephenson, 'Judge Dependence', 508-511; Egan, 'Epistemic Modals, Relativism, and Assertion'. 23 5 Final thoughts The main upshot of this paper is a new diagnostic for when a disagreement is of Type-1 or Type-2, and new evidence that some modal disagreements are of Type-2. The result is a new twist on what semantic/pragmatic conclusions we should draw from data about disagreement: not only can we not assume in a given disagreement that it is of Type-1, but the best theory of the semantics and pragmatics of epistemic modals should predict that certain modal disagreements (such as Smith and Beth's in Mobster) are of Type-2. The rest of the paper aimed to motivate a background theory of conversation which would allow for Type-2 disagreements (§3) and then outline how several existing theories of epistemic modals might be able to predict Type-2 modal disagreements (§4).42 Supplemental data The underlying research materials for this article can be accessed at http://semanticsarchive.net/Archive/Tc0NmIzY/ References Bach, Kent, & Harnish, Robert M. 1979. Linguistic Communication and Speech Acts. Cambridge: MIT Press. Björnsson, Gunnar, & Finlay, Stephen. 2010. Metaethical Contextualism Defended. Ethics, 121, 7–36. Brandom, Robert. 1983. Asserting. 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Rastko Jovanov Das Leben als Dokument. Die Genealogie des registrierten Lebens als biopolitische Institution Einleitung: Das Dokument als vitam instituere In der gegenwärtigen Philosophie ist die Bemühung um eine Begri¤ichkeit zu beobachten, die sich als der Versuch zur Erfassung und Auslegung des menschlichen Lebens innerhalb der politischen Bio-Regimenen als des dokumentierten Lebens bemerkbar macht. Dabei wird behauptet, dass das nichtdokumentierte Leben eigentlich überhaupt nicht existieren könne. (vgl. Douzinas 2007; Ferraris 2007) Das Zusammenleben der Menschen – ob in den archaischen Gesellschaften auf den anerkannten sittlichen Normen und Regeln gegründet oder in rechtliche Formulierungen und gesetzliche Kodifikationen eingeschrieben – könne nicht ohne die Institutionalisierung des dokumentierten Lebens gestiftet werden. Das ist die sozialontologische These, auf der dieser Text basiert. Jeder Einzelne existiert, insofern er instituiert ist. Wenn ich vom dokumentierten Leben spreche, dann wird der Terminus "Institution" nicht primär als Bedingung einer sozialen Interaktion oder der menschlichen Gesellschaft im allgemeinen verwendet, sondern der Begri– der Institution wird mit dem Begri– des menschlichen Lebens in Verbindung gebracht und (auf den 10 Rastko Jovanov Spuren von Pierre Legendre (Legendre 1985)) behauptet, dass die Institution des dokumentierten Lebens nur ein besonderer Teil des grundelegenden Modus des menschlichen Seins ist – des Modus des institutionalisierten Lebens. Als eine solche Institution – um gleich am Anfang eine mögliche und diesem Text nicht entsprechende Lesart abzuwehren – kann auch die in den archaischen Jägergesellschaften vorkommende Versammlung einer Gruppe um die Beute herum und ihre Teilung gemäss der stillschweigend anerkannten Hierarchie verstanden werden. In diesem Sinne repräsentiert die Institution die stillschweigend akzeptierten oder eingeschriebenen Normen und Regeln des Verhaltens des menschlichen Lebens in Rahmen einer Gruppe (sei es, dass es sich dabei um Stamm, Gesellschaft, Staat oder Korporation etc. handelt) Die Institution des registrierten1 Lebens stellt nur eine Art dieser grundlegenden Institutionalisierung des Lebens dar und zwar eine späte Errungenschaft innerhalb der Entwicklung der modernen bürgerlichen Gesellschaft. Das Problem liegt darin, dass wir mittlerweile Zeugen davon sind, dass diese Errungenschaft sich dahingehend entwickelt hat, dass sie heute am Anfang des 21. Jahrhunderts das menschliche Leben wesentlich durchdringt und es auch in ein Computerdokument sublimieren kann, das falsifiziert werden und mit dem man politisch und sozial manipulieren kann. Bis zum Ende des 19. Jahrhunderts zeugten üblicherweise zwei Daten vom Leben eines Menschen in den aufbewahrten kirchlichen und staatlichen Dokumenten: das Geburtsund Todesdatum. Die Nekrologe in den Tageszeitungen beinhalteten ja die zusätzlichen Angaben, die das erloschene Leben einer Person in Detail schildern: der Professor an der Universität, der Weltreisende, der Entdecker, der Räuber oder im Falle Kants die Tatsache, dass er nur einmal und sehr kurz seine Heimatstadt 1 Zwischen den Termini „registriertes" und „dokumentiertes" Leben mache ich keinen Unterschied: der Unterschied besteht nicht auf der sozialontologischen Ebene, sondern nur wenn man das Problem des institutionalisierten Lebens bloss sozial-politisch anschaut. 11 Politiken des Lebens verlassen hat. Das waren aber nur zusätzliche Beleuchtungen eines abgeschlossenen Lebens. Für die innere Politik eines souveränen Staates blieben solche Tatsachen ohne Bedeutung. Dieser Zugang zum menschlichen Leben war vorherrschend auch in der Kriminologie und im Strafrecht bis zum Ende des 19. Jahrhunderts. Die wesentliche Wandlung in der Beziehung staatlichen Institutionen zum menschlichen Leben, die in dieser Zeit erst aufkam – bzw. die Vermischung von Kriminologie und Psychologie (Psychiatrie) und ihr Einfluss auf das Strafrecht – hat Foucault eingehend erforscht und in seinen einflussreichen und bahnbrechenden Vorlesungen über das Geburt der „Biomacht" dargestellt. Für die Zwecke dieser Arbeit, die die Genealogie der biopolitischen Institutionalisierung des registrierten und dokumentierten Lebens darstellen möchte, ist es nötig, die Aufmerksamkeit gleich auf die Foucaultsche Betrachtung der Gestalt des Verbrechers und die Wandlung des Zugangs zu der Tat des Verbrechens seitens der rechtlichen Institutionen zu lenken, was ich mit einer kurzen historischen Einleitung dokumentieren möchte. Neben der Vorlesung, die Foucault unter dem Titel About the Concept of the Dangerous Individual in 19th Century Legal Psychiatry2 1978 auf dem Symposium „Law and Psychiatry" an der York Universität in Toronto (Foucault 1978) gehalten hat, benutze ich auch seine Archivforschungen aus dem Buch Überwachen und Strafen. Die Geburt des Gefängnisses (Foucault 1976). Foucault zufolge wurde das in der Mitte des 19. Jahrhunderts erfolgte Verlassen des bestehenden rechtlichen Systems durch die Lehre der italienischen Biokriminologie iniitiert, in der behauptet wurde, dass das Verbrechen und seine Ursache im Wesentlichen das Vorhandensein des gefährlichen Elements signalisieren sollen. Dieser Bruch mit der rechtlichen Tradition ist mit dem Zumessen der grösseren Wichtigkeit dem Begri– der psychologischen Symptomatologie der Gefahr und 2 International Journal of Law and Psychiatry 7 (1978), 1–18. 12 Rastko Jovanov nicht mehr mit dem bislang grundlegenden Begri– der Kriminologie, nämlich jenem der objektiven Verantwortung verbunden. Die objektiven Umstände, die der verbrecherischen Tat vorausgingen, machten den gesetzlichen Boden des bislang geltenden Strafrechts aus. Das erwähnte Verlassen des Rechtssystems ereignete sich durch die Interventionen der Medizin, der Psychologie und ihres Derivats der Psychiatrie und zwar unter der Einbeziehung der Begri–e „Gefahr" und „gefährliches Individuum" in die Grundlagen der strafrechtlichen Gesetzbücher. Wenn also die Umstände des Verbrechens als solchen nicht die einzigen sind, die den Anspruch auf Rechtsgültigkeit erheben, sondern lediglich die naturhaften und angeborenen Voraussetzungen des Verbrechers für das verübte Vergehen darstellen, dann wird eine der entscheidenden Folgen sein, dass man anfängt, das politische Vergehen wie das alltägliche bürgerliche Verbrechen zu betrachten. Demnach wären die Feinde der Gesellschaft in gleicher Weise ein Revolutionär oder ein Protestteilnehmer wie ein Totschläger. Das Strafrecht erhebt sich dadurch zu einem Mechanismus für die Verteidigung der Gesellschaft, während der rechtliche Begri– der Verantwortung aufhört, als Sache der richterlichen Entscheidung zu gelten und wird in den Verantwortungsbereich des Expertengerichts verschoben, d. h. zu einer Angelegenheit des technischen Urteils der Experten der Psychiatrie, Psychologie, Medizin gemacht. Ich zitiere Foucault, der seinerseits einen Vertreter der italienischen Schule der Biokriminologie namens Pugliese zitiert: "The commission of medical experts to whom the judgment ought to be referred should [...] render a real decision." (Foucault 1978: 14) In dieser Arbeit beabsichtige ich, die Genealogie und die gegenwärtigen Auswirkungen solcher Einmischung in die Wahlfreiheit und die Biographie der Individuen zu verfolgen, 13 Politiken des Lebens die zu der gegenwärtigen politischen Tatsache3 der biopolitischen Institutionalisierung des „registrierten Lebens" geführt hat. Die heutige Institution des registrierten Lebens verdankt sich der Bemühung während der Phase der Erstarkung der modernen bürgerlichen Gesellschaft, die bestehende Kriminalität in der Gesellschaft zu verhindern und zu verringen4, welche man als „Krankheit", die die Existenz der Gemeinschaft bedroht, au–asste. (vgl. noch Jhering 1877: I, 481)5 Im Laufe seiner historischen Entwicklung wird sich zeigen, dass das Phänomen des registrierten Lebens vielmehr ein Nusprodukt der bürgerlichen Gesellschaft ist und Indikator des unaufhebbaren Bestands der Dialektik der Gewalt und des ursprüngliches Agons ist, den keine gesellschaftliche Verfassung bis dato aufheben konnte. Man kann daher sagen, dass sich jede gesellschaftliche Verfassung als sich gegenüber gewalttätig gezeigt hat.6 3 Der Akzent liegt auf die „Tatsache", denn das registrierten Leben ist kein Begri–, keine Idee, kein Konzept, schon gar nicht eine Fiktion. 4 Man kann sagen, dass eine solche Tendenz jedem Zeitgeist inhärent ist. Vgl. Platon, Staat, IV, 444d. 5 „...Verbrechen ist die von seiten der Gesetzgebung constatirte Gefährdung der Lebensbedingungen der Gesellschaft." 6 Schon bei Hegel wird die Souveränität des Staates vorzugsweise im Zusammenhang der negativen Begri–e wie Aufopferung, Krieg, Vergehen betrachtet, die als Bedingung der innerstaatlichen konkreten Verwirklichung des Rechts fungieren. Wenn der Staat jedoch die Beziehungen zu anderen Staaten eingeht und versucht, seine 'Souveränität nach Aussen' zu bestätigen, hebt sich das im Inneren des Staates verwirklichte Recht auf, durch die souveränen Entscheidung über Krieg und Frieden, die ihr Topos nur in der Individualisierung des Staates in der Gestalt des Souveräns hat. Bereitschaft für die Opferung alles 'Endlichen' – des Eigentums, Lebens, Genusses im Materiellen – gewinnt seine eigene Existenz und ist laut Hegel die wahre Voraussetzung der Erhaltung jeder politischen Ordnung. Ich bin der Meinung, dass man, ohne dabei die Systematik der Hegelschen Rechtsphilosophie zu stören, mit Recht sagen kann, dass im Inneren des Staates das Recht herrscht, während die Politik an seinen Grenzen erscheint. Vgl. Hegel 1977: §§320-343. Biopolitik als modernes Paradigma, die gleichzeitig mit der Hegelschen 14 Rastko Jovanov Diese anfänglichen Strategien der Verminderung der in einer rechtlichen Ordnung vorhandenen Kriminalität gehen von der folgenden Frage aus: Wie sollten der Verbrecher und der Gesetzesübertreter klassifiziert und qualitativ zu bestimmt werden? Für eine derart gestellte Frage wurden klar definierte Kriterien darüber benötigt, was das sei, was ein Individuum als einen Verbrecher bestimmt bzw. was einen Menschen biologisch zum Verbrechen „antreibt". Die Frage lautet nicht: Welche Umstände, die sich objektiv konstatieren lassen, zum Verbrechen führen? Welche objektiven Tatsachen haben die Verbrechertat verursacht und den Verbrecher dazu geführt, sie auszuüben. Diese Frage, die in die wesentlichen metodologischen Grundlagen des Strafrechts gehört, wird von nun an ganz anders gestellt: Welcher Typus des Menschen hat das Verbrechen begangen? Lässt sich die Veranlassung des Verbrechens im Charakter der menschlichen Natur des Verbrechers finden, die von der gesellschatlich anerkannten Norm (Normalität) der Lebensführung abweicht? Die Tendenzen zum registrierten Leben kamen nicht wie etwa die Vorschriften und Verordnungen einer zentralisierten Regierung von einer übergeordneten Instanz, sondern wurden durch die Verschränkung der Auslegung der rechtlichen Normen mit den Einwirkungen des Darwinismus auf die Naturwissenschaften initiiert. Und als solche wurden sie im Laufe der Zeit institutionalisiert, weil die Regierungen die Vorteile solEntdeckung der modernen bürgerlichen Gesellschaft entsteht, weist wesentlich auf die innere Umgestaltung des Rechts hin. Es geht dabei nicht mehr um die Revolution und ihre Gewalt bzw. die äussere Tat des 'grossen Verbrechers' und seine Begründung einer neuer Rechtsordnung. Biopolitik will auf die Umwandlung der Beziehung zwischen Recht und (Bio-) Politik hinweisen. Die traditionelle Schranke zwischen der Souveränität nach Innen und nach Aussen, die für Hegel noch gilt, wird dadurch aufgelöst. Deswegen hat auch Foucault Recht, wenn er behauptet, dass der klassischen Souveränitätsbegri– nur noch vergangene, nicht mehr geltende Gestalt ist: die bürgerliche Gesellschaft fängt an, in die Souveränität der Entscheidung einzugreifen. 15 Politiken des Lebens cher Methoden der Registrierung der individuellen Biographien erkannt haben. Diese Vorteile wären nicht wahrgenommen und angewandt worden, wenn der Sicherheit der Gesellschaft nicht der Vorrang über das Recht des Souveräns und das Recht der souveränen Macht, über Leben und Tod zu entscheiden bzw. „sterben zu machen und leben zu lassen" (Foucault 1983: 162), eingeräumt worden wäre. Da es um die Sicherheit der Gesellschaft geht, lässt es sich unschwer begreifen, dass die Registrierungstendenzen einerseits an Qualität gewinnen, andererseits aber auch ihrer eigenen Verfälschungen ausgesetzt sind. Das betri–t vor allem den Ausnahmezustand, d. h. den Zustand des realen und flüchtigen Kriegs nach Aussen und des „unsichtbaren", dauerhaften und allgemein-durchdringenden Kriegs innerhalb einer Gesellschaft und zwar eines solchen zwischen den gegnerischen Gruppen bzw. der politischen Parteien. Foucault zufolge sind wir pazifiziert, in die militärischen Techniken der Bewährung des inneren Friedens und des Nicht-Versinkens in den Zustand der permanenten stasis eingeschrieben. Im Laufe der Zeit haben wir die permanente Gewalt um uns herum weitgehend vergessen, die uns mittels der Techniken der Kontrolle und der disziplinierenden Institutionen umgestalten und „normalisieren".7 Eines der Hauptziele, die Foucault in Überwachen und Strafen verfolgt, ist zu zeigen, wie der traditionelle, äusserliche Krieg und die souveräne Entscheidung, ihn zu führen, in den inneren Krieg transformiert wurden, der durch die Produktion der auf die Kontrolle der modernen bürgerlichen Gesellschaft ausgerichteten Wissensformen durchgeführt wird. Auf diese Art wurde das Wissen besonders in den sozialen Wissenschaften untrennbar an die Machtbeziehungen gekoppelt, die immer mehr auf die Kontrolle und Überwachung des Körpers und der Handlung von Subjekten ausgerichtet wurden. Es handelt 7 Für eine frühe Betrachtung der Begri–e „Norm" und „Normalität" vgl. Windelband 1906. 16 Rastko Jovanov sich um die grundlegende Einwirkung des Krieges und der Heeresorganisation auf die Ausbildung der sozialen Institutionen. Die scheinbare Einrichtung des inneren Friedens und die Vertreibung des Krieges über die Grenzen des eigenen Territoriums durch den Staat, bedeutet nicht, dass die Politik an sich nicht eine Form des Krieges ist, d. h. die Fortsetzung des Krieges mit anderen Mitteln. Laut Foucaults berühmten Formulierung ist die Politik „die Fortsetzung [...] des militärischen Modells [...] als grundlegendes Mittel zur Verhütung der bürgerlichen Unordnung [bzw. stasis]" (Foucault 1976: 217) zu begreifen und zwar durch die Anwendung der militärischen Mechanismen zum Zweck der Erzielung der disziplinierten und folgsamen Truppen. Wenn man aber die einfache jedoch wesentliche Frage stellt, was es bedeute, das „Leben" in den Mittelpunkt der politischen Anfrage zu stellen, dann könnte diese Arbeit die folgende Antwort anbieten: Nur als ein registriertes, dokumentiertes, nur als das soziale Faktum besitzt das Leben im 21. Jahrhundert politische Geltung. Um für diese Behauptung eine brauchbare Argumentation anbieten zu können, werde ich die folgenden Thesen aufstellen und versuchen, sie im weiteren Text zu rechtfertigen. In der ersten These dieser Arbeit wird behauptet, dass das „registrierte Leben" eine, jedoch die einflussreichste von den vielen Folgen ist, welche einer ab dem Ende des 19. Jahrhunderts immer enger werdenden Verbundenheit zwischen Recht und Psychiatrie entspringen. Das ist ein Leben, das sich heutzutage im Rahmen der biopolitischen Herrschaftssysteme in den Computerdateien und Computerordnern, in den Registern der Regierungen gegenwärtiger Staaten und multinationaler Korporationen niedergeschlagen hat. Es handelt sich also um das biopolitische Leben par excellence und zwar in dem Sinne, dass die heutige Politik sich mehr auf die Körperlichkeit des Individuums richtet, deren Bewegungen und Handlungen nur als ein Abbild der vorausgehenden Determiniertheit des Men17 Politiken des Lebens schenwillens durch die naturhaften neuronalen Prozesse anzusehen sind. Man kann mit Recht behaupten, dass sich dieser Zustand auf die informatisch-technologische Revolution vom Ende des 20. Jahrhunderts zurückführen lässt. Da das „Projekt" des registrierten Lebens innerhalb einer Vielfalt der miteinander verflochtenen Informationen, der Ansammlungen unterschiedlichen (richtigen oder falschen) Erkenntnissen, diversen Voraussetzungen über die gefährlichen Individuen oder die gefährlichen Klassen abläuft, aber auch weil eine solche Evidenzierung systematisch und planmässig durchgeführt wird und auch auf das alltägliche Leben des Menschen einwirkt, indem er im voraus und potentiell als das „gefährliche Element" bezeichnet wird, zeigt uns, dass es sich hier um eine neue Institution handelt, die sich ganz auf die klassische Weise gestiftet wird: ohne einer vorherigen Konsensabsicht im Rahmen einer bestimmten Gruppe von Individuen, sondern so, als wäre sie in den Zeitgeist oder, wie es Heidegger nennt, ins Gestell der Zeit bereits eingewoben. In der zweiten These wird behauptet, dass sich drei wesentliche Verschiebungen in der modernen und gegenwärtigen Geschichte identifizieren lassen, die am Ende zu dem, das wir heutzutage als die „Institution des registrierten Lebens" nennen könnten, geführt haben. Alle drei Verschiebungen zeigen ferner stets eine Konvergenz zum Krieg oder genauer ausgedrückt zu einigen militärischen Strategien und ihren Konsequenzen für das Leben des Individuums innerhalb der Gesellschaft. Nicht nur dass die Kriegsführung zu einer tieferen Grundlegung dieser neuen biopolitischen Institution beigetragt hat, sondern sie wurde wesentlich auch an die Entscheidung zur Kriegsführung gekoppelt. Für eine solche Entscheidung braucht man Gründe. Wir werden weiter im Text sehen, dass sich solche Entscheidungsgründe immer auf der vorausgesetzten Existenz eines „gefährlichen Elements" beruhen, das die positiven Werte und das Dasein einer im Staat organisierten Gesellschaft bedroht oder sie in der nahen Zukunft bedrohen 18 Rastko Jovanov wird. Die erste Verschiebung lässt sich zwar von einer unmittelbaren Beziehung zur Kriegsführung trennen, was aber nicht bedeuten soll, dass diese Verschiebung nicht im Einvernehmen und in Harmonie mit jenem Zeitgeist stand, der durch die Völkerpsychologie Wilhelm Wundts und die Naturalisierung des hegelschen Volksgeistes in grossen Schritten zu den Kriegsapologien der deutschen Philosophen im ersten Jahrzehnt des 20. Jahrhunderts führte. Wenn diese Genealogie des „registrierten Lebens" richtig ist, würde ich mich dann am Ende der Arbeit mit der Frage beschäftigen, ob und welches Leben heutzutage möglich ist, ohne registriert, überwacht oder durch die Massenmedien geleitet zu sein? Welcher Widerstand und welche Aktion können die bestehenden biopolitischen Regime zerstören und die Institution des dokumentierten Lebens vom Grund auf umgestalten, ohne dabei das Leben als solches zu beschädigen, welches, wie ich schon betont habe, notwendigerweise ein institutionalisiertes Leben ist? Vorausblickende Bemerkung Ich fange zunächst mit einer kurzen Überlegung bezüglich der Institution des registrierten Lebens im Werk von Cesare Lombroso und innerhalb der italienischen kriminologischen Anthropologie an. Dann werde ich mich der zweiten Verschiebung zu den Körperpolitiken in den nationalsozialistischen Programmen der Rassenhygiene sowie bei den NS-Rechtstheoretikern, die auf die NS-Justiz und NS-Richter im Hinblick auf die Motive und Gründe für ihre Entscheidungen bei den Rechtssprüchen eingewirkt haben. Die dritte jedoch entscheidende Verschiebung lässt sich in den Folgen jenes politischen Ereignisses aufspüren, dem umgehend das Akronym 9/11 (nineeleven) beigefügt wurde, als wäre die Zeit gekommen, einem Ereignis, das weitaus o–ener und ö–entlicher als alles zuvor die Institution des registrierten Lebens gestalten wird, einen e–ektiven, leicht zu merkenden und zu erkennenden Namen 19 Politiken des Lebens zu geben, jedoch nur um auf diese Weise die Absichten und Konsequenzen der Existenz einer solchen Institution zu verbergen und den Eingang in eine Serie kriegerischer Aktionen und eine Reihe der später sowohl in den USA als auch in der UN und der EU vollzogenen Gesetzerlassungen leichter zu finden. Es wurde versucht, die Legitimität solcher Verfahren durch den sog. „war on terror" und den Hinweis auf die Gefahr, die der radikale Islamismus mit sich bringt, zu sichern. Im Hintergrund und im Schatten solcher ö–entlichen Spektakel der Sicherheitserhöhung und der Abwehr der gefährlichen Elemente stand die biopolitische Institution der neoliberalen Disziplinierungen durch die Einsicht in die Biographie-Register der gesamten Bevölkerung. Denn heute ist jeder ein potentieller Feind und kann die vorhandenen Werte der Wohlfahrt der westlichen Gesellschaft gefährden. Die Institutionalisierung der Register der individuellen Biographien und der Lebensgeschichten verdankt sich auch der Verschiebung, die von dem Monopol der Drucktechnologie auf das Computer-Code vollzogen wurde. Das Archiv wurde durch die Datenbank ersetzt, welche im Gegensatz zum Archiv durch die Möglichkeit definiert ist, die durch die verschiedenen Überwachungsapparaturen gesammelte Informationen innerhalb einer Millisekunde in die Computer-Dateien organisieren zu können. Gerade diese Möglichkeit der unmittelbaren (also ohne jegliche menschliche Arbeit vollziehbaren) technologischen Organisation der Informationen über eine Person verbirgt im Hintergrund die Entstehung einer neuen bioinformatischen Epistemologie von Wissen/Macht. Das Subjekt wird letztendlich zu einer völlig berechenbaren, organisierten Entität. Das Subjekt wird produziert, konstituiert und institutionalisiert durch das System des „supervision-writing", durch das „uninterrupted supervision, continual writing" (Foucault 2006: 55-56).8 8 Die deutsche Übersetzung dieser Vorlesungen Foucaults, die der Macht der Psychiatrie gewidmet sind, wurde erst Anfang dieses Jahres bei 20 Rastko Jovanov Das Subjekt hört auf, eine selbstproduzierte Entität zu sein, die durch die Einheit des Selbstbewusstseins und der Selbsterhaltung charakterisiert wird. (vgl. Henrich 1973) Wir können heute auf jeden Fall noch nicht mal erahnen und schon gar nicht wissen, welche Auswirkungen das auf die Philosophie bzw. auf das, was man „human sciences" nennt, haben wird. Biokriminologie Der anfängliche Schritt zu einer umfangreicheren Beeinflussung der Bevölkerungsbiographien durch die Staatspolitik wurde im Feld des Theoretischen vollzogen und zwar im Rahmen der in der zweiten Hälfte des 19. Jahrhunderts zu verzeichnenden Bemühung der Kriminologie, die für diese frühe Phase des Kapitalismus und der postrevolutionären Kodifikationen der strafrechtlichen Gesetzbücher charakteristische massive Verbrechensraten zu verringern. In den Jahrzehnten zwischen der ersten (1789) und der zweiten (1848) Revolution wuchs erheblich die Bedeutung der politischen Dimension des Verbrechens in den breiten Bevölkerungsschichten: das waren die Jahrzehnte der sozialen Auseinandersetzungen, der Kämpfe gegen die politische Ordnung, des Widerstandes gegen die steigende Industrialisierung und am Ende dieser Periode das Einmünden der Arbeiterkämpfe (Streiks, verschiedene bürgerliche und korporative Vereinigungen) in die politischen Revolution von 1848. (vgl. Foucault 1976: 351-352) Wie Foucault richtig bemerkt, sollen die Ursachen der Verbrechenssteigerung in den „neuen Rechtsformen [gesucht werden], die strengen Reglementierungen, die Anforderungen des Staates, der Grundbesitzer und Unternehmer sowie die stra–eren Überwachungstechniken die Gelegenheiten zu Delikten vermehrten und viele Individuen zu Rechtsbrechern machten, die unter anderen Umständen nicht kriminell geworden wären" (Foucault 1976: 353-354; hervorgehoben R. J.) Suhrkamp verö–entlicht. 21 Politiken des Lebens Dieselbe Zeit kennt auch die umfassende Entwicklung der Polizeikontrolle und die wachsende Überwachung der Bevölkerung ist, wie Foucaults in seinen Archivarbeiten bemerkt, schon damals anwesend. Die damalige Überwachungspraxis ist „eine stumme, geheimnisvolle, unbemerkte Wachsamkeit [...] sie ist das ununterbrochen geö–nete und unterschiedslos über alle Bürger wachende Auge der Regierung, das sie gleichwohl keiner einzigen Zwangsmassnahme unterwirft [...] Sie muss nicht einmal im Gesetz niedergeschrieben sein."9 (hervorgehoben R. J.) Diese frühen Formen des polizeilichen Überwachens setzten doch „die Organisation eines Dokumentationssystems" (Foucault 1976: 362) voraus, welches dazu dient, die dokumentierte Personenbeschreibung mit den Haftbefehlen zu koppeln. Schon „ab 1833 wird nach dem Vorbild der 'Naturforscher, der Bibliothekare, der Händler, der Geschäftsleute'10 ein Karteisystem mit Einzelblättern eingeführt, mit dem sich neue Daten und zu jedem gesuchten Individuum gehörige Informationen leicht einbauen lassen". (Foucault 1976: 362-363; hervorgehoben R. J.) Infolge des riesigen Fortschritts in den Naturwissenschaften und einer zunehmenden Naturalisierung und Psychologisierung der philosophischen Begri–e kommt es zu einer folgenreichen Berührung des Strafrechts und der Biokriminologie, die in einem grösseren Mass die zukünftige Beziehung zwischen Individuum und gesellschaftlicher Ordnung bestimmen wird. Man ist nun dabei, den Verbrecher und sein Verbrechen vom Standpunkt der Biologie, der Medizin und der Psychiatrie zu erforschen. Die determinierenden Faktoren eines Verbrechens werden nun in der Naturhaftigkeit des Menschen gesucht.11 9 A. Bonneville, Des institutions complementaires du systeme penitencier, 1847, S. 397–399. Zitiert nach Foucault 1976: 362. 10 A. Bonneville, De Ia recidive, 1844, S. 91 f. Zitiert nach Foucault 1976: 362. 11 Der Begri– „Verbrecher" ist doppeldeutig. Einerseits bezeichnet er den bürgerlichen Verbrecher, also jemanden, der ein Verbrechen verübt hat und damit zum Gegenstand des Strafrechts geworden ist. Wenn man 22 Rastko Jovanov Dass die Tendenz zu der Vermischung von Recht und Biologie als der wahrhaftige Fortschritt im Feld der Kriminologie und des Strafrechts aufgefasst wurde, zeigt auch die Internationalisierung dieser pseudowissenschaftlichen Disziplin sowie die auf den diversen internationalen Kongressen der Kriminalanthropologie dokumentierte vorherrschende Haltung zu diesem Thema. Es wurden insgesamt sieben Kongresse in sechs europäischen Staaten abgehalten: Rom (1885), Paris (1889), Brüssel (1892), Genf (1896), Amsterdam (1901), Turin (1906), Köln (1911). Als Begründer der Kriminalanthropologie gilt Cesare Lombroso, der die Grundlagen der Disziplin in seinem Buch L'uomo delinquente (1876) verö–entlichte.12 Darin spricht Lombroso u. A. über die Sicherheit der Gesellschaft; das Strafrechts wird für einen Abschnitt der Psychiatrie gehalten; er verwirft die These von der Willensfreiheit und an ihrer Stelle setzt er die Gefahr, die die geborene Delinquenz des Verbrechers mit sich bringt und welche sich an den Charakteristiken seines Körpers ablesen lässt. Von nun an tritt die Natur an die Stelle des Geistes, Körper an die Stelle des Begri–s und Gefährlichkeit an die Stelle der Verantwortung ein. Für uns ist an dieser Stelle lediglich die Tatsache von Bedeutung, dass es sich dabei um einen Versuch gehandelt hat, mit Hilfe eines anthropologisch-medizinischen Zugangs zum Begri– des Verbrechers den theoretischen Beitrag über den Verbrecher in diesem Sinne spricht, bleibt man dann noch innerhalb ein und derselben Ordnung; die Beziehungen zu ihr sind intakt und ein neues Recht wird nicht gescha–en. Andererseits war das Wesen des Verbrechens immer auch ein Teil der souveränen Macht. Das Recht des Souveräns und seine Position, insbesondere im Hinblick auf den Krieg bzw. auf das Völkerrecht (oder altes Kriegsund Friedensrecht, oder wie man diese Sphäre menschlicher Realität nennen mag), dieses Recht oder Privilegium also hat ihm ermöglicht, jene Art der Macht, die durch das Vergehen das Recht ausübt, in die Hände zu bekommen. Zum Begri– des „souveränen Vergehens", siehe ausführlicher in: Jovanov 2014: 104–110. 12 Ich benutze hier die deutsche Übersetzung aus dem Jahr 1894: Der Verbrecher (Homo Delinquent) in antropologischer, ärztlicher und juristischer Beziehung (2 Bände, Hamburg, 1894). 23 Politiken des Lebens zur (Straf-)Rechtstheorie zu leisten. Die Forschung orinentiert sich hier nicht am Verbrechen, sondern allein am Verbrecher. Der deutsche Autor, der 1894 die Vorrede zur deutschen Ausgabe des Buches von Lombroso schreibt, spricht von der „Embryologie des Verbrechens" (Lombroso 1894b: VII), weil sich Lombroso für die Anatomie und Anthropometrie des Verbrechers, für seine Physiognomie, Haarwuchs, seine Schmerzempfindlichkeit etc. interessiert. Zu dieser biologischen Komponente kommt auch eine psychologische hinzu: die kranken Triebe des homo delinquentus, seine Handschrift, die seine geborene Zuneigung zum Verbrechen verrät, etc. Im Schlussabschnitt des Buches wird eine Therapie bzw. die therapeutische Anleitung angeboten, die einer militärischen Taktik der Disziplinierung der unfolgsamen Abteilungen ähnelt und die lehrt, was man mit dem Verbrecher tun und wie man mit ihm umgehen soll. Ungeachtet dessen sollen wir für die Zwecke dieser Untersuchung festhalten, dass als das Kriterium für die Schuldzuschreibung für das erfolgte Verbrechen die genetisch und biologisch bedingte potentielle Gefährlichkeit des Verbrechers für die Gesellschaft betrachtet wird. Der Vorrang vor dem rein formaljuristisch definierten Schuldbegri– wird im Rechtsverfahren der Gestalt des Verbrechers und dem Begri– der eingeborenen Deliquenz eingeräumt. Der Verbrecher muss im Hinblick auf die Sicherheit der Gesellschaft und angesichts der bestehenden Normen neutralisiert werden. Diese Verschränkung des modernen Fortschrittglaubens, der biologistisch naturalisierten Kriminologie und der Bereitschaft für die institutionalisierte Gewalt gegen jede Normabweichung war jedoch bis zum Ersten Weltkrieg den Werten und den Institutionen des bürgerlichen Rechtstaates untergeordnet. (vgl. Stolleis 2003: 209)13 Doch das „Augusterlebnis" und 13 Vgl. dazu auch die Diskusion im Rahmen der Verhandlungen des Ersten Deutschen Soziologentages vom 19.-22. Oktober 1910 und auch die kritische Bemerkungen Max Webers zum biologischen Rassenbegri– (o. A. 1911: 111166). 24 Rastko Jovanov der „Geist von 1914" werden zu einer weiteren Entwicklung der institutionalisierten Lebensregistrierung beitragen, die ihren Abschluss sowohl in der Pervertierung der Begri–e „Moral" und „Recht" als auch in einer intensiven Biologisierung des gesellschaftlichen Lebens in der nationalsozialistischen (Bio-)Politik haben werden. NS-Biopolitiken: Wille statt Tat Der nächste Schritt in Richtung Expansion der registrierten Lebensgeschichten zum Zweck der Gesellschaftssicherung und der Abwehr der gefährlichen „Elemente", die den Fortschritt und die (in der Positivität der Tatsachlichkeit hervorgehobene partikularen) bestehenden Werte bedrohen, wurde im nationalsozialistischen Regime des Dritten Reiches vollzogen. Die Verfolgung der Biographien und die Evidenzierung des Lebens wird im zweifachen Sinne durchgeführt: im negativen, antisemitischen Sinne als die Nichtzugehörigkeit zu der niederen „Rasse", und im positiven, nationalsozialistischen Sinne als die Zugehörigkeit zum deutschen Volke, das als eine ursprüngliche Gemeinschaft verstanden wurde.14 Die Kennzeichnung des gefährlichen Elements und seine Einführung in den Registern konnte ohne Hindernisse vollzogen und aufgrund des institutionalisierten und durch Gesetze durchgeführten antisemistischen Grundeinstellung der ֖entlichkeit als eine legitime Handlungsweise auferlegt werden. Aus heutiger Perspektive betrachtet war zu erwarten, dass die bio-logistische und die rassistische Linie sich vereinen werden, wie Stolleis mit Recht bemerkt: „Kein Zweifel, dass alle diese Studien über „Minderwertige", „Schwachsinnige", „Verbrecher aus Anlage", „Asoziale", „Gemeinschaftsunfähige", „Zigeuner" und andere Gruppen den Weg bereitet haben, den 14 Über die philosophische Volksau–assung als "ursprüngliche Gemeinschaft", vgl. Heidegger 1999: 8, 72, und meine Kritik an seiner Au–assung in: Jovanov 2015a. 25 Politiken des Lebens diese Menschen dann in die Euthanasieanstalten und Vernichtungslager gehen mussten." (Stolleis 2003: 209) Noch Lombroso hat in seiner Schrift Antisemitismus und Juden die Meinung vertreten, dass die Ursachen für die Judenverfolgungen in den "physischen Krankheiten und ihren Gesetzen" zu suchen sind und dass gerade diese „Fehler" für den wachsenden Antisemitismus verantwortlich sind. (Lombroso 1894a: 5-11 –.) Den grössten Einfluss auf die NS-Juristen im Hinblick auf die juristische Legitimierung der Artgleichheit und die zwingende theoretische Konstruktion des Rassenbegri–s hat möglicherweise das im Jahr 1930 verö–entlichte Buch Rassenkunde des deutschen Volkes von Hans Günther (Günther 1930) ausgeübt, der die Rasse – 60 Jahre später nach Lombroso – als eine „Menschengruppe, die sich durch die ihr eigene Vereinigung körperlicher Merkmale und seelischer Eigenschaften von jeder anderen Menschengruppe unterscheidet und immer wieder nur ihresgleichen Zeugt" (Günther 1930: 14), definiert. Die Anfänge der deutschen kriminologischen Psychologie lassen sich nicht vom Einfluss Lombrosos trennen, was bereits 1880 im Buch Die Abscha™ung des Strafmasses: Ein Vorschlag zur Reform der heutigen Strafrechtspflege von Emil Kraepelin (Kraepelin 1880) bemerkbar ist. Im letzten Jahrzehnt des 19. Jahrhunderts ensteht in Deutschland ein ausgeprägtes Interesse an der Frage nach dem geborenen Verbrecher (vgl. Kurella 1893; Koch 1894). Hans Kurella war derjenige, der besonders darauf bestand, dass jedes kriminelle Verhalten biologisch determiniert ist, und der zugleich alle soziologischen Erklärungen des Verbrechens abgelehnt hat. Für die weitere Entwicklung des Strafrechts in Deutschland war jedoch eine engere und unmittelbarere Zusammenfügung der biologischen und der sozialen Verbrechenserklärung charakteristisch, aber auch die theoretische Aufwertung des Begri–s „Kriminalpsychologie" im Unterschied zum Begri– der „Kriminalanthropologie" (vgl. Wetzell 2000: 61). Diese Tendenz lässt sich unmissverständlich am Titel des bis in die 30-er Jahre einflussreichsten kriminologischen 26 Rastko Jovanov Werkes in Deutschland, aber auch am Aufstieg des NS-Regimes merken. Es handelt sich nämlich um das Buch Das Verbrechen und seine Bekämpfung: Kriminalpsychologie für Mediziner, Juristen und Soziologen, ein Beitrag zur Reform der Strafgesetzgebung von Gustav Ascha–enburg (Ascha–enburg 1903).15 Ascha–enburg zufolge ist es „am schwersten zu begreifen und zu verstehen, die Individualität des Verbrechers" (Ascha–enburg 1903: 203), gerade deswegen, weil die Strafe „die Gesellschaft vor den verbrecherischen Angri–en einzelner Individuen schützen" (Ascha–enburg 1903: 210) soll. Die Abwägung der Vorteile und der Mängel verschiedener Strafmitteln – von Todesstrafe, Deportation, Disziplinarstrafe, die Aberkennung der bürgerlichen Ehrenrechte, bis zu Geldstrafen schliesst Ascha–enburg mit der Behauptung ab, dass das wichtigste Strafmittel „die Entziehung der Freiheit" ist (Ascha–enburg 1903: 219). Dabei lässt sich eine klarer Schritt in Richtung auf die begri¤ichen Grundlagen der Institutionalisierung des registrierten Lebens bemerken, weil die Begri–e „Verbrecher" und „Strafe" wesentlich eine bürgerliche Herkunft vorweisen können,16 sodass die „Entziehung der Freiheit" anfängt, ein negatives Verhältnis zu den Fundamenten der bürgerlicher Gesellschaft selbst einzunehmen bzw. sich negativ zu der subjektiven Freiheit des Einzelnen zu verhalten, die – wie noch Hegel als einer der ersten angenommen hat – eine Grenze zwischen der alten und der neuen Welt ausmacht.17 Es ist wichtig zu bemerken, dass Ascha–enburg diesen Freiheitsentzug unmittelbar durch die Methode der „Beaufsichti15 Das Buch wurde bereits 1913 ins Englische übersetzt: G. Ascha–enburg, Crime And Its Repression, Little, Brown, and Company, Boston. 16 Vgl. den einleitenden Satz des Fragments von Horkheimer und Adorno Aus einer Theorie des Verbrechers: „Wie der Verbrecher so war die Freiheitsstrafe bürgerlich." (Horkheimer und Adorno 1947: 269.) 17 Hegel 1977: §124 Anm.: „Das Recht der Besonderheit des Subjekts, sich befriedigt zu finden, oder, was dasselbe ist, das Recht der subjektiven Freiheit macht den Wendeund Mittelpunkt in dem Unterschiede des Altertums und der modernen Zeit." 27 Politiken des Lebens gung der Beschäftigung und Lebensweise [...] in Festungen oder in anderen dazu bestimmten Räumen" unter die Lupe nehmen will (Ascha–enburg 1903: 217). Die Rede über die Räume des Strafrechts begann also bereits damals. Diese Räume werden sich – wir sind die Zeugen davon – im 21. Jahrhundert in die rechtsfreien Räume des nackten Lebens umwandeln. Eine solche durch die Vermischung des Rechts, der Biologie und der Psychiatrie entstandene Tendenz wurde bereits damals vermutet und letztendlich identifiziert. In Sommers Kriminalpsychologie liest man: Die Lehre vom geborenen Verbrecher in der Hand von dogmatischen Vertretern der staatlichen Ordnung kann zu einer furchtbaren Wa–e gegen die persönliche Freiheit der Individuen werden. Nicht in der Richtung der Psychiatrisierung, sondern in der eines Zwangsstaates mit Detention ad libitum liegt die wahre Gefahr dieser wissenschaftlich unvermeidlichen Lehre bei ihrer eventuellen missbräuchlichen Anwendung. Der Wohlfahrtsausschuss der französischen Revolution mit unbeschränkter Macht über die dem augenblicklichen Staate gefährlichen Elemente ist diejenige Form staatlicher Ordnung, für welche die Lehre vom geborenen Verbrecher am besten geeignet ist. (Sommer 1904: 309-310) Auf eine durchdringende Dokumentierung des Lebens wirkten die Schriften von Radbruchs Lehrer Franz von Liszt und seine Strafrechtsreform ein, die wiederum durch die italienische Kriminalanthropologie angeregt wurde, welche mit jugendlicher Kraft und Begeisterung den Kampf gegen die klassische Kriminalistik aufgenommen [hat]; sie bestreitet dem Strafrecht den Charakter einer juristischen Disziplin und verwandelt es in einen Zweig der Gesellschaftswissenschaft; sie misstraut den Wirkungen der Strafe und will diese auf einem grossen Gebiete ihrer bisherigen Herrschaft ersetzen durch Präventivmassregeln („Strafsurrogate"); sie nimmt dem Strafprozesse seine juristische Gestaltung und verwandelt ihn 28 Rastko Jovanov in eine fachmännische psychiatrisch-anthropologische Untersuchung des Verbrechers; sie erblickt ihre Hauptaufgabe in der Erforschung der Ursachen des Verbrechens und ihre medizinischen wie juristischen Anhänger wetteifern in statistischen und anthropometrischen Untersuchungen. (Liszt 1905: 131) Liszt selbst behauptet jedoch, dass seine Strafrechtsreform auf einer anderen Synthese des Rechts mit der Psychologie und der Biologie basiert.18 Der Zweck der Strafe besteht nicht in der Retribution oder überhaupt in einer Art genereller Prävention, sondern in der Behinderung des Verbrechers, ein Verbrechen wieder zu begehen. Auf diese Weise wird die Strafe nicht nur von einer verübten Tat, sondern auch von der potentiellen künftigen Gefährlichkeit eines Individuums abhängig gemacht. Die Strafe wird daher individualisiert und präventiv und „wendet sich gegen den Willen des Verbrechers" (Liszt 1905: 163). Denn Eine[...] konkrete[...] Tat [...] ist untrennbar von der Person des Täters. Mag sie eine Episode in seinem Charakterleben, mag sie der Ausdruck seines innersten Wesens sein: es gibt kein Verbrechen, das nicht der Verbrecher begangen hätte. Tat und Täter sind keine Gegensätze, wie der verhängnisvolle juristische Irrtum annimmt; sondern die Tat ist des Täters. (Liszt 1905: 175) 18 Vgl. Liszt 1905: 178: „Der Erforschung des Verbrechens als sozialethischer Erscheinung, der Strafe als gesellschaftlicher Funktion, muss innerhalb unserer Wissenschaft die ihr gebührende Beachtung werden. Dass es eine Kriminalanthropologie, eine Kriminalpsychologie, eine Kriminalstatistik als besondere, der Wissenschaft des Strafrechtes mehr oder weniger fernstehende Disziplinen gibt, ist der Beweis des schweren Verschuldens, welches die wissenschaftlichen Vertreter des Strafrechtes tri–t; es ist aber auch der Grund für die bisherige Unfruchtbarkeit jener Disziplinen. Nur in dem Zusammenwirken der genannten Disziplinen mit der Wissenschaft des Strafrechtes ist die Möglichkeit eines erfolgreichen Kampfes gegen das Verbrechertum gegeben. Unserer Wissenschaft gebührt die Führung in diesem Kampfe." 29 Politiken des Lebens Die Registrierung individueller Biographien richtet sich nun nicht nur auf die Juden, Serben und andere slawische Völker, sondern erlebte einen allgemeinen Aufschwung im Hinblick auf die Akzeptanz oder Nichtakzeptanz der nationalsozialistischen Werte und zwar ungeachtet dessen, ob sie gesetzlich verordnet oder mit dem Milieu und der Sittlichkeit des alltäglichen Lebens bereits verwoben wurden. Als Bespiel nehmen wir die Dossiers deutscher Philosophen, die der „Sicherheitsdienst für das Reichsministerium für Wissenschaft, Erziehung und Volksbildung" Anfang 1943 zusammengestellt hat.19 Ein tre–endes Beispiel dafür stellen die Informationen dar, welche der Philosoph Kurt Schilling20 dem Sicherheitdienst mitteilte: Betri–t: Bericht über die Erkundungsreise vom 8.11.-10.11/.39 Ich war infolgenden Orten: Freiburg i.B., Frankfurta.M., Marburg (zwei Mal), Köln, Göttingen, Hamburg, Regensburg, Berlin. Zehn Kameraden wurden besucht und zur Teilnahme an der geplanten Abteilung für Philosophie aufgefordert. Ausserdem habe ich noch mit den Professoren Heyse, Heimsoeth und Bäumler ohne etwas von dem Plan zu erwähnen allgemein über die Lage der Philosophie in Deutschland gesprochen. Von den besuchten Kameraden haben sechs ihre Bereitschaft zur Teilnahme erklärt (Schlechta, Matzat, Bröcker, Lipps, Luschka, Mollowitz); zwei habe ich nicht angetro–en (Bollnow, Ritter); zwei haben abgesagt (Gerhard Krüger und Gadamer), und zwar weil sie fürchten, bei ihrer 19 Siehe auch die verschiedenen in den Germanisten-Dossiers notierte Informationen (Simon o. J.). 20 Auch sein Dossier beinhaltet etliche Informationen: „Schilling ist aufrichtig bemüht, das Ideengut des Nationalsozialismus aufzunehmen und seine wissenschaftlichen Erkenntnisse danach auszurichten. Schilling war einige Semester in Prag zur Vertretung, wurde aber ziemlich angefeindet. Im Wintersemester 1941/42 ist er mit der Vertretung des philosophischen Ordinariats (Grunsky) in München beauftragt." (Leaman und Simon o. J.: 40) 30 Rastko Jovanov Arbeitsüberlastung an der Universität nicht mit der erforderlichen Kraft sich für die neue Aufgabe einsetzen zu können. Ich charakterisiere im folgenden kurz die sechs Kameraden, die zur Teilnahme bereit sind. 1. Karl SCHLECHTA Dr.phil.habil., Dozent an der Universität Frankfurt, Kulturreferent der Stadt Frankfurt, Pg., Leiter des Nietzschearchivs und Herausgeber der vom Führer geldlich unterstützten neuen grossen Kritischen Nietzscheausgabe. Geb. 1904. Ganz besonders tüchtiger, wenn auch durch seine Ämter naturgemäss etwas überlasteter Mann. 2. MATZAT. Dr. phil. München Conrad v. Bergstr., Pg., Assistent von Grunsky, Lektor im Amt Rosenberg, früher an leitender Stelle in der Studentenschaft Freiburg tätig. Jüngerer sehr tüchtiger Mann mit guter Schule und besonders einsatzfreudig. 3. Walter Bröcker Freiburg i. B. Schwimmbadstr. 13, Dozent an der Universität, Assistent am philosophischen Seminar. Geb. 1902. Bisher keine politische Betätigung. Auskunft seines Dozentenführers Berger: sehr gut. Kamerad Berger betreibt seine Aufnahme in die Partei. Fachlich sehr tüchtig; etwas langsamer, aber sehr sorgfältiger Arbeiter. 4. Hans Lipps. Dr. phil. et med., Ordinarius für Philosophie an der Universität Frankfurt a.M. geb. 1889. 1914-18 als Bataillonsarzt im Feld, einmal verwundet, SS-Mann (San.Sta–. II/2,Nr.312511). Hauptforschungsrichtung Sprachphilosophie, Logik, philosophische Psychologie. 5. Werner Hubert LUSCHKA Dr. phil. Studienrat in Regensburg, Schüler Heideggers und Vosslers; geb. 1900; Kriegsfreiwilliger, Pg. seit 1917, Preisträger einer Preisaufgabe der philosophischen Fakultät der Universität München. Musste aus wirtschaftlichen Gründen beim Tod seines Vaters sein philosophisches Studium abbrechen und Lehrer werden. Etwas langsamer aber sehr gründlicher Arbeiter. Käme vor allem für französische Philosophie in Betracht. ... (Leaman und Simon 1992: 278) Aus folgenden Beispielen lässt sich klar erkennen, was alles als dokumentarisch „wertvoll" gegolten hat: 31 Politiken des Lebens Becker, Oskar, Bonn ordentlicher Professor, geboren 1889. [...] Kein Parteigenosse, aber loyal zum Nationalsozialismus. 1934 aus der Kirche ausgetreten. [...] Becker wird von den Studenten nicht anerkannt, weil der keine Lebensphilosophie vorträgt. Zudem ist er kein guter Redner. [...] Beurteilung durch den Dozentenbund lautet: Sehr geschätzt. Arbeitet daran, die Existenzphilosophie durch eine Philosophie der Gemeinschaft zu ersetzen. Will den Menschen als einen in der Gemeinschaft stehenden zeigen, wobei Boden, Geburt, Blut, Rasse und Volk als die eigentlichen Lebensmächte gelten. Im Grunde genommen ist Becker Skeptiker. Sieht die Fragwürdigkeit aller Dinge. Die Folge ist, dass er sich nicht vorbehaltlos einzusetzen vermag, obwohl das Verlangen nach solchem Einsatz vorzuliegen scheint. (Leaman und Simon o. J.: 14) Bröcker, Walter, Rostock [...] Sehr scharf sehend, auch selbständig, aber ohne weltanschauliche Ausrichtung. (Leaman und Simon o. J.: 17) Meyer, Hans, Würzburg – ordentlicher Professor seit 1922, katholisch – Politisch: Katholisch stärkstens gebunden (Konkordatslehrstuhl) – Freund von Held (Bayern) – Bayerische Volkspartei – Gilt als „untragbar" in politischer und weltanschaulicher Hinsicht. (Leaman und Simon o. J.: 35) Sartorius von Walthershausen, Bodo, Köln – Wissenschaftlich: sehr begabt und fleissig, aber reiner Theoretiker, insbesondere auf historisch-enzyklopädischem Gebiet – Politisch: Früher national, ohne Aktivität. – Kein Kämpfer. Weiche Natur. – Weltanschaulich: aufgeschlossen und ungebunden. – Charakterlich: Grundanständig, kameradschaftlich, strebsam. – Neigt etwas zur Abkapselung. (Leaman und Simon o. J.: 46) 32 Rastko Jovanov Solche registrierte Tatsachen über die individuellen Leben aus dem Sicherheitsdienstarchiv und anderen Verwaltungszentren der nationalsozialistischen Regierung haben wesentlich auch den Gerichtsprozess selbst bestimmt und eine grundlegende Einwirkung auf die Umwandlung des Strafrechts nach 1933 gehabt.21 Aufgrund dieser Registrierung wurden bestimmte Individuen als „gefährlich" eingestuft, weil ihre dokumentierte Biographien nicht die neuen Werte des Nationalsozialismus – „Treue", „sittliche Pflicht", „Ehre" – gespiegelt haben, die vollständig den traditionellen und formell-rechtlichen Charakter der Schuld, des Verbrechens und des Strafrechts verändert haben (vgl. Pauer-Studer & Fink 2014: 94). Auf diese Weise wurde die Institution der bürgerlichen Grundrechte gegenüber dem Staat in ihren Grundlagen erschüttert. Als neue Rechtsquelle wurden das Führerund Volksgemeinschaftsprinzip aufgestellt: „An die Stelle eines an formalen Verfahren orientierten Rechtsbegri– soll ein materiales, durch Weltanschauung und politisch geprägte Wertvorstellungen angereichertes Verständnis des Rechts treten." (Pauer-Studer & Fink 2014: 19) Mit dem neuen Strafrechtskonzept wird die Entwicklung in Richtung der Institutionalisierung des registrierten Lebens auf die Lehre Lombrosos über den schuldhaften Willen des Verbrechers zurückgeführt. Die Verteidigung neuer gesellschaftlichen Werte reduziert die Schuld des „gefährlichen Individuums" auf sein potentielles Verbrechen und auf die Charakteristiken seines Willens, welche unmittelbar aus seiner registrierten Biographie herausgelesen werden können. Die rechtlichen Normen werden umgestaltet, insofern sie „sich zur Aufgabe stellen, die nationale Lebensordnung des 21 Zum Thema der nationalsozialistischen Strafrechtskonzeption siehe insbesondere die Einleitung von Herlinde Pauer-Studer zum Sammelband der Originaltexte deutscher Juristen und Rechtsphilosophen (Pauer-Studer & Fink 2014: 15–141). In folgenden Paragraphen stütze ich mich auf die Texte aus diesem Sammelband. Vgl. dazu auch Michael Stolleis, Recht im Unrecht. Studien zur Rechtsgeschichte des Nationalsozialismus (Stolleis 2006). 33 Politiken des Lebens Volkes in ihrem Bestande zu schützen und zu entwickeln". (Koellreutter 1938: 11) Insoweit lasse die Lebensregistrierung „keine Neutralität von einzelnen Lebensbereichen" zu (Huber 1934: 35), sondern jede Handlung eines Individuums, jede seine Bewegung und jeder Gestus, jedes sei es voreilig oder mit Bedacht ausgesprochene Wort bringen mit sich die potentielle Schuld, denn der Gegenstand des Strafrechts ist der gefährliche Wille des Täters und nicht etwa seine Tat. Pauer-Studer behauptet, dass sich im Hinblick auf diese Frage bei den NS-Strafrechtstheoretikern bereits ab der Mitte der 1930er Jahre ein Konsensus eingestellt hat (vgl. Pauer-Studer & Fink 2014: 80). Von ausserordentlicher Bedeutung für die Institutionalisierung des registrierten Lebens sind auch die Ideen der Generalund Spezialpräventionen, weil sich die Definition des Willens im NS-Strafrecht, obwohl nicht ausreichend präzise bestimmt, vorwiegend auf die „Intentionen, eine Handlung oder einen Plan auszuführen", bezieht (Pauer-Studer & Fink 2014: 87).22 Deshalb sind auch die Dossiers und die Archive individueller Lebensgeschichten ausserordentlich bedeutsam geworden, weil sich aus ihnen die Intentionen eines gefährlichen Willens herauslesen lassen. Demnach wird die Lebensregistrierung sozusagen zur Reinigungsapparatur der Gesellschaft. Eine Gesellschaft verteidigen, heisst nun, sie von den Tätertypen wie Volksverräter, Volksschädling, Gewohnheitsverbrecher oder Korruptionsverbrecher zu reinigen (vgl. Pauer-Studer & Fink 2014: 89). Die Tat und der Täter stellen daher eine untrennbare Einheit dar. Möglicherweise hat das am klarsten Edmund Mezger definiert: Schuld ist Tat-Schuld, aber auch Lebensführungs-Schuld, und deshalb richtet sich die Strafe nicht nur nach der Einzel-Tat, sondern auch nach der Persönlichkeit des Täters, soweit aus ihr gegen den Täter ein Vorwurf erhoben werden kann. Das 22 Zu den Begri–en der Generalund Spezialpräventionen vgl. Ascha–enburg 1903: 209 f. 34 Rastko Jovanov bedeutet noch nicht eine Ausrichtung der Strafe nach der Persönlichkeit des Täters und seiner Gefährlichkeit schlechtweg. (zitiert nach Pauer-Studer & Fink 2014: 90; hervorgehoben R.J.)23 Eine solche „Lebensführungsschuld" wurde schon nach der Reichstagsbrandverordnung ermöglicht, welche der Regierung die Eingri–e in Postund Briefwechsel erlaubte und welche mit dem „Gesetz über die Geheime Staatspolizei" vom 10. Februar 1936 völlig institutionalisert wurde. In diesem kann man Folgendes lesen: Alle staatsgefährlichen Bestrebungen im gesamten Staatsgebiet zu erforschen und zu bekämpfen, das Ergebnis der Erhebungen zu sammeln und auszuwerten, die Staatsregierung zu unterrichten und die übrigen Behörden über für sie wichtige Feststellungen auf dem Laufenden zu halten und mit Anregungen zu versehen. (zitiert nach Pauer-Studer und Fink 2014: 103; hervorgehoben R. J.) Die Institution des registrierten Lebens und die (geheime) Polizei werden zum Hütter der Gesellschaft. Diese Aufgabe wird nicht mehr der Rechtsnormen anvertraut, die die Gleichheit aller vor dem Gesetz sichern. Das ist nun eine Gleichheit vor der Treuepflicht und zwar in einer Form der zum Totalitarismus verkommenen Solidarität. Auf dieser Weise haben sich Recht und Gesetz getrennt, weil man der überpositiven, ungeschriebenen und supralegalen Rechtsquelle Vorrang eingeräumt hat. Die neue Rechtsquelle befreit einerseits den Richter im Gericht von dem Zwang der formalen Norm. Anderseits kommt es deswegen zur völligen Verzerrung der Beziehung zwischen Recht und Sittlichkeit, d. i. zur Aufhebung ihres Unterschieds. Ein klares Zeugnis davon geben die Schriften der NS-Rechtsphilosophen Julius Binders und Karl Larenzs. 23 Edmund Mezger, Deutsches Strafrecht. Ein Grundriss, Junker & Dünnhaupt, Berlin 1938, S. 135. Zitiert nach Pauer-Studer & Fink 2014: 90. 35 Politiken des Lebens Der rechtskonservative Neuhegelianismus interessierte sich für eine Sittlichkeit, die durch die Gegnerschaft zu den anderen Völkern gestaltet und verkörpert wurde. Das bedeutet zugleich, dass die Vertreter dieser theoretischen Bewegung die vorherigen Bedingungen völlig ausser Acht gelassen haben, welche Hegel als notwendig für die Sittlichkeitsausbildung in einem Staate betrachtete. Die Sittlichkeit ist Hegel zufolge die zweite Natur des Menschen und sie gründet niemals auf der ersten Natur, auf den natürlichen Gegebenheiten eines Volkes. Die Hegelsche Sittlichkeit lässt sich als die institutionalisierte Bildung der europäischen postrevolutionären Bürgerschaft über die wahren Bedingungen des gerechten Zusammenlebens au–assen, d. h. als etwas ganz anderes als was man sie im Nationalsozialismus aufgefasst hat. Der Stellenwert der Begründung des Rechts des neues deutschen Staates im Rahmen des Ausnahmezustands hat auf der theorethischen Ebene ermöglicht, dass man die bestehenden sittlichen Institutionen (die sich in erster Linie auf die Werte des Einzelnen als Person und als Bürger beziehen) des Individualismus und Atomismus beschuldigen und zugleich die Sittlichkeit auf die erste Natur des Menschen, auf die natürlichen Grundlagen des menschlichen Seins, d. h. auf die rassistischen „Blut und Boden"-Au–assungen zurückführen konnte. Die Sittlichkeit definiert Binder auf folgender Weise: das „Gemeinschaftsleben ist ihre Gesittung und ihr Wissen von dem Rechte dieses Gemeinschaftslebens, dessen Daseinsform der Staat ist, ist ihre Sittlichkeit. Sittlich in diesem Sinne ist das unbewusste, unreflektierte Leben [...] und darin liegt zugleich, dass kein Gegensatz bestehen kann zwischen Sitte, Sittlichkeit und Recht; dass das Recht und die sog. Moral nicht verschieden, sondern wesenseins sind." (Binder 1934: 20–21) Das unbewusste, unreflektierte Leben hat seinen Grund aber in einem natürlichen Zusammenhang, „der Zusammenhang zwischen den Generationen und Familien, der durch die Ehe, Zeugung und Geburt vermittelt wird, ein auf der natürlichen Grundlage von 36 Rastko Jovanov Blut und Verwandtschaft [...] als Einheit gefühlter und begriffener Zusammenhang von Menschen einer Rasse, eines Stammes, eines Blutes, der in einem bestimmten, von ihm in Besitz genommenen und erhaltenen Gebiete lebt und wirkt." (Binder 1934: 19) Das Volk wird also zur Grundlage des Rechts und des Staats und nicht mehr der Freiheit, als dies noch bei Hegel der Fall war. Ich vertrete die Meinung, dass man mit gutem Recht behaupten kann, dass, obwohl der Freiheitsbegri– einer der in diesem Zusammenhang am meisten gebrauchten Termini ist, die Gleichheit und nicht die Freiheit der vorherrschende Begri– in diesen Theorien ist. Warum aber Gleichheit? Weil es sich hier um ein revolutionäres Zeitalter handelt, die keine neutrale Position zulässt. Erinnern wir uns an Saint Juiste und Französische Revolution. In einem solchen Zeitalter werden Werte zu Prinzipien. Beispielsweise das Solidaritätsgefühl, das Hegel zufolge – und in diesem Punkt pflichte ich ihm bei – im Volk nur im Falle der Bedrohung durch Naturkatastrophen oder beim Verteidigungskrieg zum Vorschein kommt, wird in NS-Deutschland zu einem allgemeingültigen Prinzip, das zwingend dem Anderen aufzuerlegen ist oder das der Andere mimetisch übernehmen soll. Als Person hat der Mensch keine Rechte, sondern nur als „Bauer, Soldat, Geistesarbeiter, Ehegatte, Familienmitglied, Staatsdiener" (Larenz 1934: 40), also nur in seiner Funktion innerhalb der Volksgemeinschaft. Der Wille des Führers wird dann zur einzigen Quelle alles Rechts. Die Sittlichkeit, die in der Einheit von Recht und Pflicht besteht, wird dadurch auf einen dem Hegelschen ganz fremden philosophischen Gedanken zurückgeführt. Wenn der Wille des Subjekts aber nicht das Gebot der Treuepflicht erfüllen vermag, dann ist er nach der nationalsozialistischen Rechtsau–assung schuldig und verbrecherisch. Der Staatswille allein könne niemals verbrecherisch sein, weil er den Volkswillen zur eigenen Sicherung und Erhaltung re37 Politiken des Lebens präsentiert: „Recht und Staat sind wesentlich Wille; so können wir den Staat auch definieren als die Form, die sich die Nation gibt, um in der Aussenwelt wirken zu können, oder kurz als den Willen der Nation zu Dasein und Wirksamkeit." (Binder 1934: 21) Der Einzelne kann daher den Staat nicht zur Verantwortung ziehen. Fühlt man sich nicht an den Fall Snowden erinnert? Der Bürger als potentieller Verbrecher Bedingt durch das Phänomen des global agierenden Terrorismus, das von einer enormen Zunahme der biopolitischen Forschungen begleitet wurde, vollzieht sich die dritte Verschiebung nun aber auf einem globalen Niveau. Das Leben und die Bewegungen der einzelnen Personen werden massiv und auf verschiedenste Weise registriert. Das Leben wird zu einer Computer-Datei: „In a biopolitical world, life is registered life, while undocumented life does not exist", behauptet Douzinas (Douzinas 2013: 151). Die Durchführung und die Verwaltung registrierter Biographien wird sowohl im Inneren des Staates als auch innerhalb der internationalen und der supranationalen Sphäre institutionalisiert, die durch die Regierungen der neoliberalen Ordnung mit Hilfe der grossen internationalen Korporationen geleitet werden. Es handelt sich nun nicht mehr nur um die Verteidigung und die Sicherheit einer Gesellschaft, sondern es werden neue normative Werte ins Spiel gebracht: Die (liberale) Demokratie als ein universaler Wert, den die westliche Gesellschaft unter grosser Aufopferung im Zweiten Weltkrieg erkämpft hat. Indem dafür plädiert wird, dass sich mit den präventiven Kriegen die ganze Welt sicherer für die Demokratieentwicklung und rule of law machen lässt, unterziehen die westlichen Staaten ihre Bürger der wachsenden Überwachung und Kontrolle. Im Kontext des „war on terror" werden die neuen Technologien der Gouvernementalität institutionalisiert, um Profile zu erstellen und um die Bedrohungen für die neoliberale Ordnung und ihre Werte zu registrieren und zu dokumentieren. Laut einer 38 Rastko Jovanov Foucaultschen Aussage aus einer Vortragsreihe aus den Jahren 1972/73 brauchen diese neue Technologien „an organ of generalized and constant oversight; everything must be observed, seen, transmitted: organization of a police force; instituting of a system of records (with individual files)" (Foucault 1997: 35). „War on terror" erweckt und generalisiert die Angst, diese gewaltige Emotion, die nun als Medium benutzt wird, um für die biopolitischen Regime sichern zu können, dass ihre eigene Bevölkerung sie bei einer Kriegsführung (die hauptsächlich den Interessen der multinationalen Korporationen dient) unterstüzen wird. Das Überwachen und die Registrierung der Handlungen von Individuen – vom Kauf der Flugtickets bis zur Aufnahme von Hypotheken in der Nähe von „gefährlichen" Gebieten, wie Flughäfen und Bahnhöfen – dienen der sozialen Sortierung und der Katalogisierung, genauso wie die NS-Klassifikationen der Tätertypen für die Bestimmung der „gefährlichen Individuen", die sich dem inneren und äusseren Kurs der neoliberalen Herrschaft widersetzen oder schlichtweg aufgrund ihres Erscheinungsbildes nicht mit den Vorgaben der Selbstdarstellung des neoliberalen Subjekts übereinstimmen. Die Institution des registrierten Lebens wird in ihrem Wesen nicht mehr politisch bestimmt, sondern sie wird zu einem wichtigen sozialen Faktor, welcher imstande ist, auf die Schicksale der Menschenleben einzuwirken. Das Überwachen und Dokumentieren der Handlung des Individuums „has become systematic, embedded in a system that classifies according to certain pre-set criteria, and sorts into categories of risk and opportunity [...] Such classification is very important for people's life-chances and choices. Surveillance is becoming a means of placing people in new, flexible, social classes" (Lyon 2007: 371) und das gerade in solchen Klassen, die man als (nicht) „gefährlich" definiert. Die Institution der Registrierung der Lebensgeschichten ist Douzinas zufolge der Preis, der gezahlt werden muss, nachdem dem Individuum in der Moderne „freedom of choice" (Douzinas 2007: 115) eingeräumt wurde. 39 Politiken des Lebens Die erste Sichtbarkeit der Bewegung in Richtung der Praxis der umfangreichen und durchdringenden Lebensregistrierung nach 9/11 wurde zunächst im Verhältnis zu den ausländischen Bürgern, vor allem jenen aus dem Nahosten bemerkbar. Das ist aus dem „Military Order of November 13, 2001" ersichtlich, das unmittelbar nach dem Angfri– auf das World Trade Center von dem präsidialen Kabinett der USA verabschiedet wurde: The term 'individual subject to this order' shall mean any individual who is not a United States citizen" und "has engaged in, aided or abetted, or conspired to commit, acts of international terrorism, or acts in preparation therefor, that have caused, threaten to cause, or have as their aim to cause, injury to or adverse e–ects on the United States, its citizens, national security, foreign policy, or economy. (o. A. 2001) Dieser Akt beinhaltete auch die diskret eingefügte Absicht, die Bürger des eigenen Staates in die umfangreiche Überwachensund Registrierungspraktiken einzubeziehen. Denn Verbrecher ist nun auch jeder, der mein/unser Reichtum und mein/unser Wohlstand bedroht und nichts ausser einer Unsicherheitsund Gefahrenquelle darstellt. Mit dieser Bedrohung meines/unseres Reichtums wird dieser andere zum Anderen, ganz von einer strategischen Begri¤ichkeit erfasst und zwanghaft in die Netze begri¤icher Zusammenhänge verstrickt, welche ihn ausserhalb der Grenzen der Gesellschaft stellen. Der Andere ist kein Bürger mehr, sonder nur der erkrankte Körperteil, dem man sich entledigen soll. Mit einer solcher Argumentation hören wir dann auf, überhaupt darauf zu achten, wann der Zwang legitim ist und wann nicht. In diesem Sinne kommt die grösste Gefahr einer Gesellschaft nicht mehr von Aussen, sondern ist im Gesellschaftskörper selbst zu finden. Die letzten Terroranschläge der radikalen Islamisten in Paris scheinen dieses Argument nur zu bestätigen. Die Existenz der Institution des registrierten Lebens weist jedenfalls darauf hin, dass die Verbindung zwischen der 40 Rastko Jovanov gesetzlichen Ordnung (nomos) und der Krankheit (nosos)24 bzw. den „kranken" oder den „gefährlichen" Individuen, einen latenten Druck auf die herrschende Klasse25 auszuüben vermag, der als Anlass dazu dienen kann, vom Gesetz ausgenommene Zonen zu scha–en. Darüber hinaus gibt er der herrschenden Klasse die Gelegenheit dazu, sich auch weiterhin an der Macht zu behaupten, ungeachtet des Preises, den in einer globalisierten Welt alle bezahlen müssen. Die Konsequenzen lassen sich auch an der diesbezüglichen Rechtgestalt ablesen: As law is disseminated throughout society, its form becomes detailed and full of discretion, its sources multiple and diffused, its aims unclear, unknown or contradictory, its e–ects unpredictable, variable and uneven. All the key themes of legality are weakened. Rule and normativity are replaced by normalisation, value by discretion and the legal subject by administratively assigned roles and competencies. (Douzinas 2007: 125) Das Recht bedarf zweifelsohne eines Akts der Gewalt, um zur Anwendung zu kommen. Die Vielschichtigkeit der Beziehung zwischen Recht und Kraft (Gewalt) ist der Forschung nicht verborgen geblieben, wovon etliche prominente Untersuchungen wie etwa jene von Benjamin (Benjamin 1999), Foucault (Foucault 2008: 254)26 und Derrida (Derrida 1991) zeugen. 24 Eugene Thacker zeigt, dass die zweite Häfte des VIII. Teils und das ganze IX. Teil von Platons Staat auf einem einzigen Argument zurückgeführt werden kann: "... the greatest threat to the body politic comes from within. [...] That is, of central concern for Plato is the relation between the order of law (nomos) and the various elements that would threaten law with disorder or ‚disease' (nosos)." (Thacker 2015) 25 Die Betonung liegt gerade auf der „Klasse", denn die gegenwärtigen neoliberalen Ordnungen und Projekte sind eng mit einer einzigen Klasse verbuden. 26 "[...] since formulation of the law implies a parliament, discussion, and decisions taken. It is in fact a reality, but it is not only this reality. So then, on the other hand, there is the set of instruments by which this prohibition 41 Politiken des Lebens Die Neuigkeit aber, die das 21. Jahrhundert mit sich brachte, betri–t das gewandelte Verhältnis zwischen der Legitimität und der Wirksamkeit des Rechts, das vergleichbar der aus dem Nationalsozialismus bekannten Praxis durch den Begri– der Gerechtigkeit moralisiert wurde27 Aus diesem Grund soll auch die neoliberale Insistierung auf dem Begri– „human rights" auf eine ähnliche Weise behandelt werden.28 Die Menschenrechte können zwar ein Individuum vor ungerechten Umständen, in denen es sich befinden kann, beschützen, diese Institution der Menschenrechte fungiert heutzutage aber auch als das Instrument einer Macht, die die „Feinde" der Werte der westlichen „Demokratie" disziplinieren, ausschliessen und kriminalisieren soll. Die Figur des Feindes erscheint heute wieder in ihrer vollen gewaltätigen und grandiosen Gestalt. Deswegen lässt sich die Schlüsselfrage des jetzigen Augenblicks folgendermassen formulieren: Auf welche Weise lassen sich die neuen kohäsiven Gestalten der selbständigen und autonomen Gebiete des Rechts, der Politik und der Freiheit des Subjekts in einer Welt begründen, die unter dem steten Blick der institutionellen Formen des registrierten Lebens stehen? Der gegenwärtige Zustand Bisher habe ich versucht, die wesentlichen Ausprägungen einer Genealogie der Institutionalisierung des registrierwill be given a real 'force'. This idea of a force of law is expressed in the frequently encountered word, enforcement, which is often translated in French by 'reinforcement (renforcement)' of the law. It is not reinforcement. Law enforcement is more than the application of the law, since it involves a whole series of real instruments which have to be employed in order to apply the law." 27 Die Auslegungsarten der Gerechtigkeit bei den NS-Rechtstheoretikern sind noch nicht genug erforscht. Diese Aufgabe steht noch aus. 28 Bojanić verweist mit Recht auf einen inflationären Gebrauch des Begri–es oder richtiger des „Projekts" der „human rights" (Bojanić 2015). Vgl. auch (Douzinas 2007). 42 Rastko Jovanov ten Lebens zu schildern. Nach den anfänglichen Anregungen seitens pseudowissenschaftlicher Ansätze und der Theorie der italienischen kriminologischen Anthropologie und danach unter der Mitwirkung der Technik und durch die Zunahme der Vernetzung aller Gebiete des menschlichen Lebens hat die biopolitische Perspektive auf das Lebens als Dokument die Grundfunktionen des souveränen Staates übernommen oder diese zumindest gründlich umgestaltet. Die elementare politische Frage, die Frage nach der gerechten gesellschaftlichen Verfassung, soll heute erneut gründlich durchdacht werden. Denn das Politische stellt weder die einzige noch die entscheidende Instanz mehr für diese grundlegende Frage des Zusammenlebens dar. Neue wissenschaftliche Disziplinen, nämlich biound neurotechnologischen Wissenschaften einerseits und Informationswissenschaften anderseits „entziehen" sich dem Horizont der politischen Entscheidungen und man kann sogar sagen, dass gerade sie für die politischen Entscheidungen bestimmend sind. Das eigentümliche Verbrechen und das Eindringen des Staates in die Sphäre des Privaten und Biographischen eines menschlischen Lebens haben längst die Schwelle überschritten, hinter die man nicht mehr zurückgehen kann. Eine der wichtigsten Konsequenzen davon ist, dass wir alle heute potentielle Verbrecher sind, weil die Register und die Dokumente gerade die Potentialität des Verbrechens steigern, die sich aus ihnen herauslesen lässt. In der Moderne wurde das Verbrechen durch die Tat und nicht die potentielle Absicht bestimmt. Heutzutage in einer „post-post modernen" Welt, in der Welt nämlich, die wir noch nicht richtig benennen können (denn wir wissen nicht was derzeit überhaupt geschieht), ist die Handlung kein actus, sondern potentia. Deswegen muss die Gefahr eines Verbrechens nicht real und objektiv dokumentiert sein, sondern es ist ausreichend, wenn sie bloss möglich ist. Mit anderen Worten wird potentia zwangsweise einem (künftigen) Akt zugeschrieben, der eigentlich nicht notwendig eintreten muss. Demnach kann auch jemand ein „gefährliches Indivi43 Politiken des Lebens duum" sein, der gegebenenfalls eine grosse Geldsumme von seinem Bankkonto abhebt, weil die geschilderte Potentialität ermöglicht, den möglichen Zweck dieses Aktes im Voraus zu kriminalisieren, indem es im Register dieses Individuums als potentieller verbrecherischer Akt bezeichnet wird. Gewiss verliert das Recht durch das Eindringen in die private, subjektive Sphäre weitgehend an seiner Legitimität, die es nur gewaltsam durch die Legalität der Verordnungen bzw. die Dokumentalität ersetzen kann. Die Legitimität wird somit aus dem Wesen des Rechts verdrängt und durch die EÂzienz ersetzt. Man ist mit aller Kraft bemüht zu behaupten, dass die Legitimität nur in der EÂzienz der Verteidigung demokratischer Werte liegt. Eine weitere von der biopolitischen Institutionalisierung des registrierten Lebens zu erwartende Konsequenz wird sich, wenn sie nicht schon eingetreten ist, wahrscheinlich auf dem Feld der Neurobiologie zeigen. Das dokumentierte, klassifizierte und registrierte „Leben" hat – wir erinnern uns – seine Wurzel in denkriminologisch-anthropologischen Theorien über die angeborenen Anomalien des Verbrechers, die unabwendbarlso naturhaft auf den Willen des Verbrechers wirken und somit auch die verbrecherische Tat massgeblich bedingen. Die weitere Geschichte dieser Institutionalisierung lief über die NS-Rechtstheoretiker, die die moderne Konzeption des Strafrechts auf der Grundlage des Begri–s des Willens umwandelt und zugleich eine Tätertypologie erstellt haben, bis zu den Folgen des „war on terror", eines Phänomens, das aufgrund der versammelten Dokumentation individueller Biographien erfolgte Vorrangstellung der potentia gegenüber dem actus des Verbrechens entscheidend mitbedingt hat. Mit Hilfe der Neurowissenschaften wurde die Schuld letztendlich an den Körper und das Hirn und nicht mehr an den Geist und seinen Willen gekoppelt. Der Körper ist schuldig, weil er dazu vorbestimmt ist, das Böse zu wählen! Die Freiheit und der Wille des Subjekts werden dadurch in die vordeterminierte neuro-logischen Folgen transformiert. 44 Rastko Jovanov Einen Einblick in den möglichen Einfluss der Neurobiologie auf das Recht und somit auch auf die voranschreitende Minderung der bürgerlichen Freiheiten gewähren auch die 2007 im Rahmen der Erö–nungsveranstaltung des Instituts für Kriminalwissenschaften in Göttingen einer Stadt, die zumindest bis Ende des 18. Jahrhunderts als die Stadt des „Praeceptores Germaniae" galt abgehaltene Vorträge und Diskusionen. (vgl. Harrendorf 2008) Die neueren Erkenntnisse der Hirnforschung versuchen zu zeigen, dass „auf neuronaler Ebene bereits Handlungsimpulse nachweisbar sind, bevor der Mensch die bewusste Entscheidung zu einer Handlung tri–t, sie stellen also die Willensfreiheit des Menschen in Frage" und behaupten "dass dem Schuldstrafrecht damit die Grundlage entzogen sei". (Harrendorf 2008: 41) Damit wird der innere Grund des Strafrechts entwertet, den die deutschen Entscheidungen des Bundesgerichtshofs in Strafsachen hervorheben: „Der innere Grund des Schuldvorwurfs liegt darin, dass der Mensch auf freie, verantwortliche, sittliche Selbstbestimmung angelegt und deshalb befähigt ist, sich für das Recht und gegen das Unrecht zu entscheiden." (BGHSt 2, 194; zitiert nach Harrendorf 2008) Und weil die Thematik des Verbrechens niemals aus dem Horizont des Zeitgeistes herausfällt, folgert Harrendorf mit Recht: Dennoch befinde sich das Schuldstrafrecht derzeit in einer Krise. Diese Krise sei jedoch eher einem tiefen Misstrauen der Politik in die Leistungsfähigkeit der Gerichte und Gutachter unter der Ägide des Schuldprinzips zuzuschreiben. So wolle die Politik immer mehr abrücken von der konkreten Beurteilung der Tat. Der Fokus verschiebe sich auf die Person des Täters und dessen Gefährlichkeit, weg vom Schuldprinzip hin zum Prinzip polizeilicher Prävention. Die Einführung der nachträglichen Sicherungsverwahrung belege dies eindrucksvoll. Auch psychologischen und psychiatrischen Gutachtern begegne die Politik mit wachsendem Misstrauen. Das gehe bis zu dem Vorschlag, 'schwarze Listen' zu führen 45 Politiken des Lebens mit Gutachtern, deren Prognosen sich als unzutre–end end erwiesen hätten. (Harrendorf 2008: 44) Wenn man Strafe, Schuld und Verantwortung an das neurobiologische Prinzip koppelt, dann werden die in einem Gericht vorgelegte Beweise immer mehr auf der Institution des registrierten Lebens basieren, d. i. auf der Dokumentation der Bewegung, Handlung, aber auch auf den in dem globalen informatischen Netz geäusserten politischen Positionen. Der Gerichtsbeschluss wird somit praktisch im Voraus gefällt. Alles spricht dafür, dass das Prinzip der aufgezwungenen und in die demokratischen Grundsätze verpackten Gleichheit der Werte, welche also immer mehr totalitäre Umrisse bekommt, wenn man sie nicht als Wert, sondern als Prinzip auffasst (ähnlich wie bei der Empfindung und dem Wert der Solidarität, die im Nationalsozialismus als Prinzip der Handlungen aufgefasst wurden), den Vorrang vor der Freiheit bekommen wird, sei es, dass es sich um die subjektive, private Freiheit des Individuums oder die objektive, intersubjektive Freiheit des gerechten Zusammenlebens innerhalb einer (Welt-)Gemeinschaft geht. Wo wächst das Rettende? Eine Form des Widerstandes gegen die durchdringende Institutionalisierung des registrierten Lebens und ihrer politisch-sozialen Folgen für das Leben des Individuums besteht im Versuch, die Antwort auf die von mir bereits gestellte Frage zu geben: Wie lassen sich die neuen Gestalten des Zusammenhaltes der autonomen Gebiete des Rechts, der Politik und der Freiheit des Subjekts in einer Welt, die unter dem Blitzlicht von Reflektoren (die zugleich die Sehkraft, d. i. die Theorie verbirgt) der institutionalisierten Formen des registrierten Lebens steht, normativ begründen und rechtfertigen? Diese Widerstandsform wird doch mit dem Einwand konfrontiert, dass sie eigentlich die Rückkehr zur klassischen 46 Rastko Jovanov Politik darstellt, welche das ֖entliche vom Privaten unterscheidet und in welcher die politische Gewalt nicht unmittelbar mit dem Körper verbunden ist, sondern auf dem Niveau der Prinzipien (gesellschaftlicher Vertrag bei Hobbes, Anerkennungstheorie bei Hegel) argumentativ durchgeführt wird. Das erschwert zugleich selbst die Formulierung eines solchen Widerstandes und belässt ihn, so würde ich sagen, auf der Ebene der reinen theoretischen Reflexion, die nicht die Konkretheit des gegenwärtigen politischen Lebens erreichen kann. Dieser Widerstand stellt gewissermassen das dar, was Hegel die schlechte Abstraktheit genannt hat, da er lediglich eine leere Versprechung einer gerechten Welt anzubieten vermag. Ich neige zu der Au–assung, dass die zweite Widerstandsform mehr verspricht, wenn man das Problem der Durchführung des Widerstandes gegen die biopolitischen Regime von „Überwachen-Schreiben" bzw. gegen die Institution des registrierten Lebens in Betracht zieht. Diese zweite Widerstandsform finde ich in Anlehnung an die grundlegende Umwandlung des metaphysischen Weltbildes. Unter „Metaphysik" verstehe ich die Stütze, subjectus, der in der Grundlage einer Epoche gelegt ist und als der Grundstein eines Hauses seiner Zerstörung leibhaft entgegenwirkt und sie aufhält. Demgemäss nenne ich metaphysisch heutzutage die eingefahrenen Formen politischer Aktionen und ökonomischer Vorherrschaft internationaler und suprainternationaler Transaktionen im Rahmen der neoliberalen Sachordnung. Als metaphysisch nenne ich also die herrschende, auf den menschlichen Körper gerichtete Ideologie neoliberaler Politik, die primär durch die bioinformatischen Systeme geleitet wird, die imstande sind, die politische Handlung zu regulieren und die sozialen Politiken zu gestalten. Das, was ich die Institution des registrierten oder dokumentierten Lebens nenne, stellt die Stütze der gegenwärtigen Politik dar. Wenn es richtig ist, dass die Biopolitik das Subjekt konstruiert und die individuellen Biographien manipuliert, dann 47 Politiken des Lebens soll dahinter eine grundlegendere Instanz gefunden werden, welche eigentlich die gegenwärtigen biopolitischen Regime der Gewalt, die Suspendierung des Rechts und das Entstehen der rechtsfreien Räume ermöglicht. Diese Instanz ist nicht anderes als das neoliberale metaphysische bzw. ideologische Weltbild. Deswegen soll der Widerstand gegen die biopolitische Gewalt und die Institution dessen, was ich „ÜberwachenSchreiben" nenne, anfangs nicht in der Form des Widerstandes gegen die neoliberale (Bio-)Politik, sondern eher als Widerstand gegen die neoliberalistische „Metaphysik" erfolgen, in welcher das menschliche Leben auf die Form einer Computerdatei – derer versammelte Informationen im Voraus jede Möglichkeit einer selbstständigen Aktion und eines Widerstandes unwahrscheinlicher machen – zurückgeführt wird. Deshalb soll sich die zweite Form des Wiederstandes auf den Begri– der Aktion stützen. Diese Aktion und dieser Widerstand können jedoch nicht eine Tat der Masse oder einer politisch engagierten Gruppe mit einem klar definierten Ziel (in diesem Fall mit dem Ziel des Widerstandes gegen den neoliberalen Kapitalismus und die biopolitische Macht) sein.29 Wie uns die Geschichte lehrt, endet die politische Aktion der Masse in unkontrollierten Gewaltausbrüchen und mit den Ergebnissen, die keineswegs jenen zuvor intendierten entsprechen. Andererseits sind die politische Aktionen der engagierten sozialen Gruppen (wie etwa Occupy) immer gegen andere Gruppen (in diesem Fall gegen die neoliberale Eliten) gerichtet und durch einen blossen Agonismus geleitet, der, obwohl er durchaus einen klar definierten und aufrichtigen Zweck intendieren kann, auch im Fall des positiven Ausgangs bzw. des siegreichen Erreichens des intendierten Zwecks lediglich in einer Neubesetzung der Plätze an der Spitze der gesellschaftlichen Hierarchie endet (vgl. Jovanov 2015: 125). 29 Zu den ontologischen Kategorien der Masse und der Gruppe, vgl. Jovanov 2015b: 122–129. 48 Rastko Jovanov In den letzten Jahren hat m. E. Costas Douzinas am zutre–endsten die Formen des Widerstandes inklusive der Möglichkeiten des Widerstandes gegen den biopolitischen neoliberalen Kapitalismus erforscht. Ich werde hier eine seiner Thesen untersuchen, in welcher seine Behauptung von der Möglichkeit eines Widerstandes und einer politischen Aktion zusammengefasst sind: Collective resistance becomes political and may succeed in radically changing the balance of forces when it condenses di–erent causes, a multiplicity of struggles and local and regional complaints, bringing them together into a common place and concurrent time. (Douzinas 2014: 96) Da geht Douzinas gerade von dem aus, was ich oben kritisiert habe: von einer sich selbst spontan regulierenden Masse: „Individual disobedience and isolated acts of defiance converge and become collective resistance." (Douzinas 2014: 97) Der kollektive Widerstand in der Form der massenhaften Versammlung ist doch kurzatmig, während der Staat (und Douzinas bemerkt das richtig) immer die Möglichkeit hat, sich entgegen den Interessen der Protestierenden anzupassen, sobald diese Interessen irgendeine Partikularität in ihrer Anforderung zeigen. Daher wird in der Perspektive des biopolitischen Neoliberalismus als wahrhaft gefährlich nur jenen Widerstand betrachtet, der „a force that can transform the relations of law and present itself as having a 'right to law'" (Douzinas 2014: 97) beinhaltet. Meiner Meinung nach findet man in diesem „having a right to law" die rettende Aktion, die das „Leben" de-institutionalisiert, aber nur in dem Masse, in dem sie ihm belässt, autonom Besitz über eine sakrale Sphäre zu ergreifen, d. i. die Sphäre der persönlicher 'mytischen' Gesinnung, die dem Einzelnen notwendigerweise für die Orientierung in der ihm gegebenen Welt erforderlich ist. Dieser Anteil des Mythos, welcher niemals aus der menschlichen Lebenswelt verschwindet, lässt sich ge49 Politiken des Lebens rade in der auf das menschliche Denken einwirkende Macht der Sprache finden. Daher ist jede Umänderung des metaphysischen Weltbildes symbolisch und mytisch. Die Sprache unterliegt heutzutage völlig der Kolonisierung durch die disziplinierende Macht und man soll sie aus dieser Lage herausreissen. In dieser Situation hat sich das zeremonielle Ritual der Bestrafung durch die souveräne Macht letztendlich auf alle Sphären des menschlichen Lebens ausgebreitet. Die neoliberalen Regime ritualisieren jedes Moment in einem Tag des Menschen: in Schulen, Fabriken, Kirchen etc. Überall lässt sich die Sprache der Disziplinierung finden, die zugleich eine Sprache der Moralisierung ist. Also die Sprache, die in die innere persönliche Sphäre greift. Z. B. die ökonomische Seite des Terminus 'Schuld' ist durch die disziplinierende Macht kolonisiert, die jene bestraft, welche sich über die 'positiv' geltenden Werte hinwegsetzen und in das Tabufeld eingreifen. Deswegen ist „a right to law" als die Weise des selbstgerechten Gefühls für die Gerechtigkeit im Menschen notwendig dem kodifizierten Gesetz entgegengesetzt. Das ist der Akt des Verbrechens und der Übertretung, von dem also, was Foucault zusammen mit Bataille Transgression nennt. Als Akt ist das Verbrechen ein Ereignis – das Geschehen in der Welt. Man soll das Verbrechen nicht als etwas au–assen, das rein rechtzerstörerisch wirkt, denn es scha–t das Recht zugleich mit seiner Zerstörung. Eines bleibt immer fraglich (und das ist beim Recht das Wesentliche): wir wissen nicht, was das Recht von uns verlangt. Hier liegt die Quelle seiner Macht. Foucault bemerkt daher mit Recht: If it were self-evident and in the heart, the law would no longer be the law, but the sweet interiority of consciousness. If, on the other hand, it were present in a text, if it were possible to decipher it between the lines of a book, if it were in a register that could be consulted, then it would have the solidity of external things: it would be possible to follow or disobey it. Where then would its power reside, by what force or prestige 50 Rastko Jovanov would it command respect? In fact, the presence of law is its concealment. (Foucault 1987: 33) Wir sind nicht nur dem Recht unterworfen, sondern sind auch seine Subjekte. Aus dieser Sicht erö–net das Verbrechen ein Raum für die Freiheit selbst. Deshalb muss das Eingreifen der rettenden Macht zu den Ursprüngen der Biopolitik und der Institution des registrierten Lebens zurückkehren, um auf diese Weise in der politischen Domäne kriminell und in der informatischen piratisch zu werden. Insofern die biopolitischen Regime heutzutage überhaupt von der Suspendierung des Rechts und von dem Ausnahmezustand leben, wie von Agamben behauptet, dann könnte sich die allgemeine Suspendierung des Rechts lediglich durch das selbstgerechte Gefühl für die Gerechtigkeit legitimieren. Dieses Gefühl, welches in dem Masse vorhanden ist, dass es den metaphysischen Hintergrund der gegenwärtigen Welt verändern könnte und als Erschütterung, Verletzung, Skandal in Bezug auf 'business as usual' erscheint, kann jedoch nicht unmittelbar, d. h. durch die regulierten weltweiten Massenproteste und –aktionen eine neue Ordnung hervorbringen. Dazu sind zwei Dinge vonnöten: (a) eine kriminelle Aktion erfordert eine klare symbolische Struktur, um ihre Aktionen, im Moment, in dem sie das positive Recht bricht, legitimieren zu können. Sie ist notwendig an das neue (im Augenblick der Aktion abwesende) metaphysische Weltbild gebunden, sodass sie gezwungen ist, die bestehenden gesellschaftlichen Rituale der disziplinierenden Macht mit der Forderung zur Umwandlung des Rechts zu verbinden. Diese symbolische Struktur ist eine narrative, weil sie von der Macht der Sprache über das Denken ausgeht, ist aber auch immer eine messianische, weil sie sich auf das Kommende beruft. Man darf nicht vergessen, dass jede solche Aktion in einem bestimmten Mass das Risiko mit sich bringt, auch jenes Verbrechen zu legitimieren, welches die Form totaler Dehumanisie51 Politiken des Lebens rung annimmt. Das Beispiel, das insbesondere in Europa immer mehr vergessen wird, ist das NS-Regime.30 (b) die kriminelle Aktion bedarf das, was Benjamin der grosse Verbrecher nennt. Sie verlangt also nach einer Repräsentationsbehörde, die die institutionelle Kontrolle übernehmen kann, denn – wie ich am Anfang dieses Textes behauptet habe auch das biopolitische Leben wird seine eigene Institutionalisierung (vitam instituere) nicht vermeiden können. Wenn man unter dieser Aktion dasjenige versteht, was Benjamin in dem gleichen Schriftstück die „wahren Kriege" (Benjamin 1999: 203) genannt hat, dann sieht es danach aus, dass die Veränderung jedes metaphysischen Weltbildes nur durch den Krieg herbeigeführt werden kann. Ob sich eine solche Notwendigkeit des Krieges vermeiden lässt, hängt von jener Schlüsselfrage ab, ob eine mächtige multinationale Korporation die Rolle des grossen Verbrechers auf sich nehmen kann, da die Stiftung einer neuen Ordnung in der gegenwärtigen globalisierten Welt nur von innen kommen kann, also aus dem System selbst. 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Leaman, Georg; Simon, Gerd (o. J.): „SD über Philosophie-Professoren". Abgerufen am 17.11.2015 von https://homepages.uni-tuebingen.de/ gerd.simon/philosophendossiers.pdf. Legendre, Pierre (1985): L'Inestimable Objet de la transmission. Étude sur le principe généalogique en Occident. Paris: Fayard. 54 Rastko Jovanov Liszt, von, Franz (1905): „Der Zweckgedanke im Strafrecht (1882)". In: Strafrechtliche Aufsätze und Vorträge. Erster Band (1875-1891). Berlin: Guttentag S. 126–179. Lombroso, Cesare (1894a): Der Antisemitismus und die Juden im Lichte der modernen Wissenschaft. Leipzig: Wigand. Lombroso, Cesare (1894b): Der Verbrecher (Homo Delinquent) in antropologischer, ärztlicher und juristischer Beziehung. Hamburg: Verlagsanstalt und Druckerei A.-G. Lyon, David (2007): „Everyday Surveillance: Personal Data and Social Classifications". In: Hier, Sean; Greenberg, Joshua (Hrsg.) The Surveillance Studies Reader. New York: Open University Press. (2001): Military Order of November 13, 2001. 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A History of German Criminology, 1880-1945. Chapel Hill & London: The University of North Carolina Press. Windelband, Wilhelm (1906): „Über Norm und Normalität". In: Sonderabdruck aus der Monatsschrift für Kriminalpsychologie und Strafrechtsreform. Ascha–enburg, Gustav (Hrsg.). Heidelberg.

© Contrastes. Revista Internacional de Filosofía, vol. XVIII (2013), pp. 49-68. ISSN: 1136-4076 Licenciatura de Filosofía, Universidad de Málaga, Facultad de Filosofía y Letras Campus de Teatinos, E-29071 Málaga (España) Teoría y práctica en Musonio Rufo: Un análisis crítico de las Disertaciones 5 y 6 Theory and practice in Musonio Rufo: A critical analysis of Lectures 5 and 6 RODRIGO SEBASTIÁN BRAICOVICH Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) (Argentina) Recibido: 3-8-2012 Aprobado definitivamente: 10-12-2012 RESUMEN Los objetivos específicos son los siguientes: (i) reconstruir en forma sistemática la relación entre λόγος y ἔθος/ἄσκησις desarrollada por Musonio Rufo en las Disertaciones 5 y 6; (ii) postular las reflexiones de Aristóteles sobre el problema de la habituación como un marco conceptual relevante para encuadrar el análisis de ambas disertaciones; (iii) analizar las posibles tensiones lógicas entre la concepción de Musonio de ἔθος/ἄσκησις y la concepción intelectualista de la acción humana defendida por la ortodoxia estoica. Sugeriré asimismo que el recurso a las Dissertationes de Epicteto puede ofrecer una respuesta tentativa a interrogantes no resueltos en las Disertaciones 5 y 6. PALABRAS CLAVE EJERCICIOS ESPIRITUALES, ESTOICISMO, INTELECTUALISMO, PSICOLOGÍA DE LA ACCIÓN ABSTRACT The specific goals are the following: (i) to put together in a systematic manner the relationship between λόγος and ἔθος/ἄσκησις presented by Musonius Rufus in Lectures 5 and 6; (ii) 50 RODRIGO SEBASTIÁN BRAICOVICH Contrastes vol. XVIII (2013) to propose Aristotle's reflections on the problem of habituation as a relevant framework to make sense of both lectures; (iii) to analyze the possible logical conflicts between Musonius' conception of ἔθος/ἄσκησις and the intellectualist conception of human agency defended by Stoic orthodoxy. I will further suggest that Epictetus' Discourses may offer a tentative answer to questions that are left unanswered in Lectures 5 and 6. KEYWORDS SPIRITUAL EXERCISES, STOICISM, INTELLECTUALISM, PSYCHOLOGY OF ACTION I. IntroduccIón SI bIen laS dISertacIoneS que Se han preServado de Musonio Rufo a partir del registro de Lucio, uno de sus alumnos,1 constituyen el capítulo más inexplorado del período romano de la escuela estoica, uno de los dos elementos por los que dichas disertaciones suelen ser citadas está representado por la defensa que Musonio realiza de la necesidad de concebir la filosofía en términos eminentemente prácticos.2 El tipo de abordaje que se realiza, a este respecto, cubre desde la alusión ocasional a la primacía que el autor habría otorgado a la práctica por sobre la teoría, hasta la inclusión explícita de Musonio en una corriente de pensadores que intentan distanciarse de la concepción esencialmente contemplativa que el aristotelismo propone de la filosofía para construir una alternativa primordialmente práctica, centrada en ciertos «ejercicios espirituales» que pasarían a ocupar el rol central en el diseño de la praxis pedagógica de la filosofía.3 El objetivo general de estas páginas consistirá en analizar las Disertaciones 5 y 6 de Musonio [D5 y D6, de ahora en más],4 concentrándome en los siguientes 1 G. Roskam 2005:99-100 ofrece un abordaje sintético de la problemática de la conservación de las disertaciones de Musonio. 2 El otro elemento por el que se suele recurrir a Musonio está representado por su posición respecto de los roles y derechos de la mujer en la sociedad. 3 Tal es el caso de la lectura propuesta por John Sellars (vid. J. Sellars 2003 y 2007), inspirada fundamentalmente en las interpretaciones previas de Pierre Hadot y Michel Foucault. 4 Utilizaré, tanto para las citas en griego como para las referencias, la edición de Cora Lutz (C. Lutz 1947). Las traducciones al español son propias. 51Teoría y práctica en Musonio Rufo Contrastes vol. XVIII (2013) objetivos específicos: i) reconstruir el concepto de habituación/ejercitación (ἔθος/ ἄσκησις) que encontramos en ambas disertaciones a la luz de la oposición de ambas nociones al λόγος, entendido como comprensión teórica de principios filosóficos; ii) traer a la luz el posible trasfondo aristotélico (y no únicamente estoico) de la distinción entre ἔθος y λόγος; iii) analizar los posibles conflictos que la relación desarrollada por Musonio entre ἔθος/ἄσκησις y ἀρετή puede generar respecto de la concepción intelectualista de la acción defendida por el estoicismo antiguo. Como estrategia hermenéutica, por último, sugeriré que ciertas lagunas dejadas por argumentación de Musonio en D5-6 pueden ser tentativamente abordadas desde los desarrollos de la problemática de la ἄσκησις que encontramos en las Dissertationes de Epicteto. II. La oposIcIón ἔθος / λογος en la Disertación 5 El objetivo de la Disertación 5 conservada por Lucio/Estobeo se encuentra establecido en forma sencilla al inicio de la misma: Αὖθις ἐνέπεσεν ἡμῖν ζήτησις πότερον ἀνυσιμώτερον πρὸς κτῆσιν ἀρετῆς ἔθος ἢ λόγος, εἰ ὁ μὲν λόγος διδάσκοι ὀρθῶς τί εἴη ποιητέον, τὸ δὲ ἔθος γίνοιτο κατὰ τοιοῦτον λόγον πράττειν ἐθιζομένων. τῷ δὲ Μουσωνίῳ τὸ ἔθος ἐδόκει εἶναι ἀνυσιμώτερον. (5.1-2 [1-4])5 En otra ocasión cierto surgió entre nosotros la pregunta acerca de qué era más efectivo para alcanzar la virtud, el hábito o la teoría, dado que la teoría enseña qué es correcto hacer, mientras que el hábito surge en aquellos que están acostumbrados a actuar de acuerdo con dicha teoría. Musonio opinaba que el hábito era más efectivo. Los pasajes siguientes (5.3-10) representan un contrapunto entre dos tipos de individuos mediante el cual Musonio pretende ofrecer razones en favor de la afirmación volcada en 5.2: la primera alternativa consiste en un médico/navegante/músico que es capaz de hablar versadamente (ἱκανοῦ λέγειν) sobre los principios teóricos de su arte (medicina/ navegación/música), pero que carece de experiencia en llevar dichos principios a la práctica; la segunda alternativa consiste en un médico/ navegante/músico que no posee la capacidad de hablar versadamente6 5 En lo sucesivo, las referencias a las disertaciones de Musonio siguen el siguiente formato: (número de disertación.sección de Lutz [líneas en la edición de Lutz]) 6 En el caso del navegante, Musonio presenta la alternativa de un individuo que es capaz de hablar sobre (los principios de) la navegación sólo de manera precaria e insegura (ένδεῶς καὶ παντάπασιν ἀσθενῶς); en el caso del músico, de un individuo que es inferior (ἥττονος) en su conocimiento de la teoría de la música que su contraparte. 52 RODRIGO SEBASTIÁN BRAICOVICH Contrastes vol. XVIII (2013) sobre dichos principios teóricos pero que posee experiencia en llevarlos a la práctica. Luego de presentar ambas alternativas, Musonio interroga a su interlocutor acerca de cuál de las dos elegiría en cada caso, ante lo cual éste responde sistemáticamente en favor de la segunda opción. A partir de lo establecido hasta aquí, y luego de extender la analogía al caso de la templanza (σωφροσύνης) y el autocontrol (ἐγκρατείας) (5.11), Musonio deduce una serie de conclusiones pretendidamente equivalentes o, al menos, mutuamente implicadas: τὸ μὲν ἔθος πρὸς τὸ δύνασθαι πράττειν ἄγει, τὸ δ' ἐπίστασθαι λόγον τοῦ πράγματος πρὸς τὸ δύνασθαι λέγειν. (5.12 [32-33]) El hábito nos conduce a la capacidad para actuar, pero conocer la razón detrás de los actos nos permite hablar sobre esos actos. συνεργεῖ μὲν γὰρ καὶ τῇ πράξει ὁ λόγος διδάσκων ὅπως πρακτέον καὶ ἔστι τῇ τάξει πρότερος τοῦ ἔθους· οὐ γὰρ ἐθισθῆναί τι καλὸν οἷόν τε μὴ κατὰ λόγον ἐθιζόμενον. (5.13 [34-36]) La teoría que nos enseña cómo actuar opera junto con la acción y la precede, dado que no es posible acostumbrarse a hacer algo bueno a menos que nos acostumbremos a hacerlo de acuerdo con la teoría. δυνάμει μέντοι τὸ ἔθος προτερεῖ τοῦ λόγου, ὅτι ἐστὶ κυριώτερον ἐπὶ τὰς πράξεις ἄγειν τὸν ἄνθρωπον ἤπερ ὁ λόγος. (5.14 [36-38]) Sin embargo, lo cierto es que el hábito es superior a la teoría, en la medida en que es más efectiva para conducir a los hombres a actuar que la teoría. Al considerar la totalidad de la disertación, se abren dos posibles líneas de interpretación acerca de cuál es su objetivo último: si nos concentramos fundamentalmente en las analogías provistas por Musonio, el motivo central de esta disertación parece ser la crítica (frecuente asimismo en Epicteto) a aquellos que pronuncian discursos públicamente y hacen gala de sus profundos conocimientos de la teoría de una escuela y, sin embargo, no actúan siguiendo esos principios, una idea que aparecía en forma explícita ya en D3: «ὥσπερ γὰρ ἰατρικοῦ λόγου ὄφελος οὐδέν, ἐὰν μὴ πρὸς ὑγίειαν φέρῃ σώματος ἀνθρωπίνου, οὕτως οὐδ' εἴ τινα φιλόσοφος ἔχει ἢ διδάσκει λόγον, οὐδὲν 53Teoría y práctica en Musonio Rufo Contrastes vol. XVIII (2013) ὄφελος αὐτοῦ, ἐὰν μὴ φέρῃ πρὸς ἀρετὴν ψυχῆς ἀνθρωπίνης»7 (D3.22 [63-65]). Si nos atenemos, por el contrario, a la discusión planteada en D5.1-2 y 5.12-14, parece evidente que ese no es el motor real de la disertación (no al menos el motor central), y que la contraposición entre la mera capacidad de hablar (λέγειν) sobre una teoría y la capacidad de poner en obras dicha teoría es, a fin de cuentas, secundaria respecto de la distinción entre conocer la teoría (independientemente de que podamos pronunciar discursos acerca de la misma) y actuar de acuerdo con ella. Ambos objetivos no son, sin embargo, inconciliables, y podemos suponer que D5 está atravesada en su totalidad por dos ideas complementarias: i) el mero conocimiento de una teoría es suficiente para hablar versadamente sobre ella, pero no para actuar de acuerdo con la misma; ii) para actuar de acuerdo con una teoría el conocimiento de la misma debe ser complementado con el hábito, la práctica reiterada de cierto tipo de acciones. Esta interpretación ha sido efectivamente adoptada por buena parte de los comentaristas de Musonio, y ha sido señalada en ciertas ocasiones como el momento inicial en una nueva forma de concebir la filosofía: una concepción que, distanciándose radicalmente de la matriz contemplativa de la filosofía aristotélica, concibe la dimensión práctica de la tarea filosófica como igual, o incluso como más importante, que la incorporación de la teoría.8 A pesar de ello, la aparente transparencia hermenéutica de D5 depende fundamentalmente de traducciones ciertamente cuestionables, y un análisis de las fuentes griegas revela que la superficie clara de D5 esconde en realidad dificultades lógicas sustantivas. Que esto es así se hace evidente si atendemos al esquema argumental de la disertación: 5.1: Planteo de la pregunta: ¿qué es más efectivo para adquirir la virtud, la teoría o el hábito? [p1] 5.2: Respuesta a p1: el hábito es más efectivo que la teoría para adquirir la virtud [r1]; 5.3-10: Razones en favor de r1; 5.11: A partir de la evidencia de 5.3-10, Musonio concluye que es mejor ser temperado que poder hablar sobre la templanza [r1']; 5.12: A partir de lo dicho en 5.1-11, Musonio concluye que conocer la teoría de algo nos permite hablar sobre eso, pero es el hábito el que nos permite actuar de acuerdo a la teoría [r2], y que 7 «Así como la teoría médica no tiene ninguna utilidad si no conduce a la salud del cuerpo del hombre, que un filósofo enseñe su teoría no tiene ninguna utilidad si no conduce a la virtud del alma del hombre». 8 Cf. paradigmáticamente J. Sellars 2003. Cf. Asimismo G. Reydams-Schils 2005:156; G. Roskam 2005:100; M. Morford 2002:195, quienes aceptan (quizás tomando a la letra la traducción de Lutz), la idea de que la práctica es más importante que la teoría. 54 RODRIGO SEBASTIÁN BRAICOVICH Contrastes vol. XVIII (2013) 5.13: a pesar de que la teoría es anterior al hábito en tanto no es posible actuar correctamente si no conocemos la teoría [r3], 5.14: el hábito es anterior a la teoría en tanto es más efectivo para conducir a los hombres a la acción [r4]. A pesar de que las dificultades aquí presentes son múltiples, me interesa detenerme únicamente sobre tres elementos: en primer lugar, y como se desprende de los ejemplos proporcionados por Musonio en D5.3-10, la disyuntiva allí planteada es falsa, en la medida en que, como se indica en D5.13 (y en otros pasajes que veremos más adelante), el hábito no es posible sin la teoría. De lo que se trata, en otras palabras, no es de elegir entre la teoría y el hábito, sino de elegir entre solamente la teoría o, por el contrario, la teoría y el hábito (de actuar de acuerdo con la misma9). Como se deduce de D8.710 y como se verá en D6, el hábito no puede darse sin la teoría, y si esto es así, la pregunta de Musonio acerca de si el hábito es más efectivo que la teoría (p1) carece por completo de sentido, en tanto el hábito es hábito de actuar de acuerdo con una teoría, es decir: la puesta en acción de ciertos principios teóricos. La pregunta que Musonio debería formular –y que de hecho va a formular explícitamente en D6– no es si la teoría es más efectiva que el hábito, sino si aquella es suficiente, sin el hábito, para alcanzar la virtud. Una segunda dificultad se refiere exclusivamente a D5.14, y consiste en el hecho de que r4 no es una respuesta a p1, y la diferencia entre r1 y r4 es sustantiva: afirmar que el hábito es más efectivo para conducir a los hombres a la acción (r4) no es lo mismo que afirmar que es más efectivo para alcanzar la virtud (r1). Aun si concediéramos –caritativamente– a Musonio que a través de D5.3-10 ha demostrado r4, eso no significa que haya demostrado r1, como pretende haber hecho. En rigor, a diferencia de r1 (que, como veremos, adquiere una dimensión plena cuando la interpretamos 9 Se pone en cuestión, con todo esto, la adecuación de las analogías proporcionadas por Musonio, en la medida en que, contra lo afirmado en D5.13, al menos en el caso de la música y la navegación es efectivamente posible ser capaz y experimentado en ambas artes sin ser conocedor de la teoría. No es esta posibilidad la que tiene en mente Musonio: tanto en el caso de la música como de la navegación, el contrapunto no se produce entre el conocedor de la teoría y que puede hablar versadamente sobre ella y el que no puede hablar en absoluto acerca de la teoría, sino entre aquél y el que es inferior en la teoría musical o el que puede hablar de manera precaria sobre la teoría de la navegación; en ninguno de los dos casos, en suma, se trata de posesión versus carencia de cierto conocimiento, sino más bien grados de conocimiento. 10 δίκαιος δὲ πῶς ἂν εἴη τις μὴ ἐπιστάμενος δικαιοσύνην, ὁποῖόν τί ἐστιν; πάλιν οὖν καὶ ταύτῃ φιλοσοφητέον τῷ βασιλεῖ, ὅτι δικαιοσύνην καὶ τὸ δίκαιον οὐκ ἂν ἄλλως φαίνοιτο γνούς, εἰ μὴ φιλοσοφήσειεν. «¿Cómo podría alguien ser justo si no comprendiera qué es la justicia? Es por ello que el rey debe estudiar filosofía, dado que, si no lo hiciera, no estaría claro si sabe qué es la justicia y lo justo» (D8.7 [22-25]). 55Teoría y práctica en Musonio Rufo Contrastes vol. XVIII (2013) en el contexto conceptual apropiado) r4, sin especificaciones adicionales, carece por completo de sentido. Ninguna de estas dos dificultades, sin embargo, resulta decisiva, en la medida en que ambas pueden ser explicadas apelando al contexto esencialmente pedagógico en el que se inscriben las disertaciones recogidas por Lucio:11 desde esa perspectiva, tanto la consideración de ἔθος como un aspecto más importante que λόγος como el pasaje de r1 a r4 no pasan de ser meras estrategias retóricas, cuya intención consiste en transmitir en forma provocativa la idea de que la teoría, aun cuando sea necesaria, no es suficiente para alcanzar la virtud.12 Existe, no obstante, un tercer problema: aun si interpretamos r4 como una variante retórica de r1, ambos exigen una explicación, en la medida en que plantean una dificultad que no puede ser desarticulada -como en los otros casosrecurriendo al contexto pedagógico y nos exige indagar más allá. La dificultad en cuestión se reduce a la siguiente pregunta: ¿cómo es posible que estar habituado a hacer x sea condición de posibilidad de hacer x? Vg.: ¿en qué sentido estar habituado a actuar templadamente puede ser condición de posibilidad de actuar templadamente?13 Para comprender lo que se halla en juego aquí, no obstante, debemos antes abordar la discusión complementaria que Musonio desarrolla en D6. 11 Cf., a este respecto, G. Roskam 2005:100-101; Ch. Pomeroy Parker 1896:126-127. 12 Si bien estas dos dificultades pueden parecer triviales en vistas de la solución propuesta, la confusión que ellas suscitan es el punto de partida de la lectura ofrecida por Sellars respecto de la esencialidad de la práctica en Musonio (centralidad que Sellars proyecta a Epicteto y Marco Aurelio). D8 ofrece evidencia adicional respecto de esta necesidad de una interpretación enmarcada en las intenciones pedagógicas que guían cada disertación, dado que allí encontramos la idea de que es la filosofía, en tanto ἐπιστήμη, la que produce las virtudes: πῶς δ' ἂν ἢ αὐτὸς σωφρονήσειέ τις μὴ μελετήσας κρατεῖν τῶν ἐπιθυμιῶν, ἢ ἀκόλαστος ὢν ἄλλους ποιήσειε σώφρονας; τίς μέντοι ἐπιστήμη πρὸς σωφροσύνην ἄγει πλὴν φιλοσοφίας, οὐκ ἔστιν εἰπεῖν· αὕτη γὰρ διδάσκει μὲν ἐπάνω ἡδονῆς εἶναι, διδάσκει δ' ἐπάνω πλεονεξίας, διδάσκει δὲ ἀγαπᾶν εὐτέλειαν, διδάσκει δὲ φεύγειν πολυτέλειαν, ἐθίζει δ' αἰδῶ ἔχειν, ἐθίζει δὲ γλώττης κρατεῖν, τάξιν δὲ καὶ κόσμον καὶ εὐσχημοσύνην περιποιεῖ καὶ ὅλως τὸ ἐν κινήσει καὶ σχέσει πρέπον. ταῦτα δὲ ἀνθρώπῳ προσόντα παρέχεται σεμνὸν καὶ σώφρονα αὐτόν (D8.15-17 [42-50]). Lo decisivo aquí es que, a diferencia de D5-6, donde uno de los objetivos centrales consiste en convencer al interlocutor acerca de la necesidad de no limitarse al estudio de la filosofía, el objetivo de D8 consiste en convencer a un rey respecto de la necesidad de (al menos) iniciarse en el estudio de la misma si desea volverse un gobernante justo. 13 Optar por traducir ἔθος por «práctica», como hacen Lutz y King, no representa ninguna solución, en tanto afirmar que solo podremos hacer x si antes hemos practicado x carece igualmente de sentido. 56 RODRIGO SEBASTIÁN BRAICOVICH Contrastes vol. XVIII (2013) III. La oposIcIón ἄςκηςις / λογος en La Disertación 6 El inicio de la Disertación 6 da cuenta de la continuidad temática y argumental respecto de D5: Παρώρμα δὲ πρὸς ἄσκησιν τοὺς συνόντας ἐντεταμένως ἀεὶ τοιοῖσδέ τισι λόγοις χρώμενος. Ἡ ἀρετή, ἔφη, ἐπιστήμη ἐστὶν οὐ θεωρητικὴ μόνον, ἀλλὰ καὶ πρακτικὴ καθάπερ ἥ τε ἰατρικὴ καὶ ἡ μουσική. (D6.1 [1-4])14 Siempre incitaba con vehemencia a los que lo rodeaban a que se ejercitasen usando discursos como éste: la virtud, decía, no es solamente conocimiento teórico sino también práctica, tal como lo son la medicina y la música. A pesar de que los cinco parágrafos siguientes evidencian un giro terminológico importante, en la medida en que desplazan el eje de la discusión desde la relación entre ἔθος y λόγος a la contraposición entre una serie de términos vinculados a la comprensión o el aprendizaje de principios teóricos, por un lado, y la ejercitación15 de dichos principios, por otro, el uso reiterado de ἐθίζω como equivalente de esta segunda serie de términos justifica que asumamos que ἔθος y ἄσκησις cumplen la misma función en el argumento en contra de la suficiencia del λόγος para alcanzar la virtud.16 Como se puede ver en el siguiente esquema, la idea fundamental que atraviesa D6 sigue siendo la misma que encontrábamos en D5.1-2 y 11-14: 6.2: Para ser bueno es necesario no sólo aprender en profundidad los principios (μαθήματα) que conducen a la virtud sino también entrenarse (γυμνάζεσθαι) de acuerdo con los mismos con celo y rigor. 6.3: Para ser templado es necesario no sólo saber (εἰδείη) que uno no debe ser derrotado por los placeres sino también estar entrenado en oponerse a los mismos. 14 Cf. asimismo 14.29 [86-88]: οὐ γὰρ δὴ φιλοσοφεῖν ἔτερόν τι φαίνεται ὂν ἢ τὸ ἃ πρέπει καὶ ἃ προσήκει λόγῳ μὲν ἀναζητεῖν, ἔργῳ δὲ πράττειν. («Es evidente que filosofar no es otra cosa que investigar mediante la razón qué es apropiado y correcto y transformarlo en obras»). 15 Los tres conceptos fundamentales empleados por Musonio son: ἄσκησις/ἀσκέω; μελέτη/μελετάω; γυμνάζω. 16 Esto no hace menos cuestionable, no obstante, la decisión de Cora Lutz y Cynthia King de traducir ἔθος por «practice» en D5 (estableciendo una absoluta homogeneidad conceptual entre ambas disertaciones), en la medida en que tal decisión contribuye a ocultar uno de los problemas centrales que señalé anteriormente, a saber: ¿cómo es posible que Musonio suponga que estar habituado a hacer X es condición de posibilidad de hacer X? 57Teoría y práctica en Musonio Rufo Contrastes vol. XVIII (2013) 6.4: Para ser justo es necesario no solo aprender (μεμαθηκὼς) que uno debe amar la igualdad sino también estar adiestrado (μεμελετηκὼς) en evitar los excesos. 6.7: Por lo tanto, el aprendizaje (μαθήσει) de los principios de cada virtud debe ser seguido del ejercicio (ἄσκησιν) de los mismos. Ahora bien: ¿Cuál es la diferencia entre hacer x y entrenarse para hacer x? ¿Qué características debe poseer un acto que realizo para que sea considerado como un acto en sí mismo y no un ejercicio para el mismo? En otras palabras: ¿por qué el entrenamiento no es ya la acción? La referencia a Aristóteles, llegados a este punto, se vuelve ineludible: τὰς δ ἀρετὰς λαμβάνομεν ἐνεργήσαντες πρότερον, ὥσπερ καὶ ἐπὶ τῶν ἄλλων τεχνῶν: ἃ γὰρ δεῖ μαθόντας ποιεῖν, ταῦτα ποιοῦντες μανθάνομεν, οἷον οἰκοδομοῦντες οἰκοδόμοι γίνονται καὶ κιθαρίζοντες κιθαρισταί: οὕτω δὴ καὶ τὰ μὲν δίκαια πράττοντες δίκαιοι γινόμεθα, τὰ δὲ σώφρονα σώφρονες, τὰ δ ἀνδρεῖα ἀνδρεῖοι. μαρτυρεῖ δὲ καὶ τὸ γινόμενον ἐν ταῖς πόλεσιν: οἱ γὰρ νομοθέται τοὺς πολίτας ἐθίζοντες ποιοῦσιν ἀγαθούς. (Aristóteles, EN 1103a32-1103b3) Adquirimos las virtudes después de ejercerlas primero, como es el caso también en las demás artes, pues las aprendemos haciendo lo mismo que se debe hacer después de haberlas aprendido; por ejemplo, se llega a ser constructor de casas construyendo casas, y citarista, tocando la cítara. De ese modo, pues, también llegamos a ser justos realizando actos justos, moderados realizando actos de moderación, y valientes realizando actos de valentía. [Lo] atestigua también lo que sucede en las ciudades, esto es, que los legisladores hacen buenos a los ciudadanos acostumbrándolos [a serlo].17 Las dificultades presentes en este célebre pasaje de EN son parcialmente análogas a las señaladas respecto de D5: en primer lugar, la elección de ejemplos por parte de Aristóteles adolece del mismo problema que señalábamos anteriormente respecto del pasaje de Musonio de r1 a r4, a saber, que la idea de que (a) hacer x (vg.: construir casas) en repetidas ocasiones hace posible hacer x ; no es lo mismo que la idea de que: (b) hacer x (vg.: construir casas) en repetidas ocasiones hace posible hacer x virtuosamente. 17 Cito de acuerdo con I. Bywater 1962; la traducción es de E. Sinnott (Aristóteles 2007). 58 RODRIGO SEBASTIÁN BRAICOVICH Contrastes vol. XVIII (2013) Sin especificaciones adicionales, (a) carece por completo de sentido. En segundo lugar, la idea de que adquirimos una virtud después de ejercerla representa, prima facie, un contrasentido tan evidente como el que indicábamos en el caso de D5.14. Aristóteles, sin embargo, y a diferencia de Musonio, se anticipa inmediatamente a dicha objeción: ἀπορήσειε δ ἄν τις πῶς λέγομεν ὅτι δεῖ τὰ μὲν δίκαια πράττοντας δικαίους γίνεσθαι, τὰ δὲ σώφρονα σώφρονας: εἰ γὰρ πράττουσι τὰ δίκαια καὶ σώφρονα, ἤδη εἰσὶ δίκαιοι καὶ σώφρονες [...] τὰ δὲ κατὰ τὰς ἀρετὰς γινόμενα οὐκ ἐὰν αὐτά πως ἔχῃ, δικαίως ἢ σωφρόνως πράττεται, ἀλλὰ καὶ ἐὰν ὁ πράττων πῶς ἔχων πράττῃ, πρῶτον μὲν ἐὰν εἰδώς, ἔπειτ ἐὰν προαιρούμενος, καὶ προαιρούμενος δἰ αὐτά, τὸ δὲ τρίτον ἐὰν καὶ βεβαίως καὶ ἀμετακινήτως ἔχων πράττῃ. [...] εὖ οὖν λέγεται ὅτι ἐκ τοῦ τὰ δίκαια πράττειν ὁ δίκαιος γίνεται καὶ ἐκ τοῦ τὰ σώφρονα ὁ σώφρων: ἐκ δὲ τοῦ μὴ πράττειν ταῦτα οὐδεὶς ἂν οὐδὲ μελλήσειε γίνεσθαι ἀγαθός. (Aristóteles, EN 1105a17-1105b13) Se podría plantear la dificultad de cómo decimos que se debe llegar a ser justo realizando actos justos, y moderados realizando actos moderados, pues si se realizan actos justos y moderados, ya se es justo y moderado. [...] Las acciones originadas de acuerdo con las virtudes no se realizan de manera justa o moderada solo si son de un cierto modo, sino también si el que actúa se halla [dispuesto] de un cierto modo cuando actúa: primero, si [actúa] a sabiendas; después, [si actúa] eligiendo, y eligiendo [el acto] de por sí; y, tercero, si actúa de manera firme e inconmovible. [...] Es, pues, acertado decir que el justo llega a ser tal por realizar actos justos, y el moderado, por [realizar] actos moderados; y si no realiza esos actos, nadie tiene siquiera la posibilidad de llegar a ser bueno. Con todo, el común de los hombres no practica esas acciones, sino que, refugiándose en la teoría, creen que filosofan, y que así llegarán a ser virtuosos.18 18 ¿Leyó Musonio EN? Varios elementos en los dos pasajes citados parecerían sugerir una lectura directa de Aristóteles por parte de Musonio: (i) la analogía con el citaredo, (ii) la presencia de σωφροσύνη y δικαιοσύνη como ejemplos de virtudes, (iii) el uso de ἐθίζω, (iv) la idea de que nos volvemos virtuosos mediante la práctica de las virtudes, y (v) la crítica a los que creen que el conocimiento de la teoría es suficiente para alcanzar la virtud. Si bien la coincidencia entre Aristóteles y Musonio respecto de cada uno de estos elementos no indica nada si los consideramos por separado, la coexistencia de todos estos puntos parecería representar evidencia suficiente respecto de una influencia directa. Todo ello, sin embargo, es meramente circunstancial: la cítara como objeto de referencia en las analogías entre la filosofía y otras artes no contribuye en absoluto un patrimonio exclusivo de Aristóteles, sino que es frecuente en la filosofía antigua; la convivencia de (iii) y (iv), por su parte, bien puede proceder de cualquier tratado peripatético y no requiere suponer necesariamente la lectura directa de EN 1103a3259Teoría y práctica en Musonio Rufo Contrastes vol. XVIII (2013) Los elementos fundamentales en este pasaje son dos: en primer lugar, la distinción entre (a) la realización de un acto justo (moderado, valiente, etc.) y (b) la realización de ese mismo acto tal como lo realizaría un hombre virtuoso (lo cual incluye la disposición del agente); en segundo lugar, la idea de que la realización repetida de (a) puede conducir a (b). Si bien la primera idea es esencialmente incontestable (siempre y cuando supongamos la existencia de algo así como la virtud), es la segunda idea la que ha generado importantes polémicas entre los comentaristas de Aristóteles a lo largo de la historia de Occidente, fundamentalmente en lo que respecta a las características específicas del proceso de habituación. La pregunta central ha sido, una y otra vez, la siguiente: ¿cómo es posible que la mera repetición de ciertos actos (o el mero hábito de realizarlos) sea causante de un eventual cambio en la disposición psíquica del agente?19 Si bien existe un consenso relativamente extendido entre los comentaristas en cuanto a que el proceso de habituación al que hace referencia Aristóteles no constituye un proceso de repetición puramente mecánico, a partir del cual el cambio de disposición en el alma del agente sobrevendría de un modo cuasi mágico,20 no existe un consenso claro acerca de lo que sí se encuentra implicado en dicho proceso.21 El análisis de este problema excede, sin embargo, los objetivos del presente artículo, y no representa un obstáculo para la argumentación general, en la medida en que la referencia a Aristóteles solo tenía por objetivo presentar un contexto posible en el cual enmarcar la discusión que articula D5 y D6. Se podría argumentar, no obstante, que la referencia a Aristóteles no era en absoluto necesaria, en la medida en que los desarrollos teóricos del estoicismo antiguo ofrecen una alternativa conceptual (parcialmente) análoga a la distinción aristotélica entre el acto en sí mismo y el acto tal como lo realizaría el hombre virtuoso, a saber, la distinción entre los conceptos de καθήκοντα y κατόρθωμα.22 Podemos, por ejemplo, componer a partir de los testimonios 1105b13 por parte de Musonio. La elección de σωφροσύνη y δικαιοσύνη como ejemplos de virtud, por último, tampoco requiere de la lectura de tal pasaje, en la medida en que las mismas, además de ser dos de las cuatros virtudes cardinales estoicas, aparecen en forma recurrente a lo largo de las disertaciones de Musonio en conjunción con otras virtudes. 19 Un tratamiento reciente de esta problemática específica se encuentra en N. Bowditch 2008; cf. asimismo S. Broadie 1991:103-110. 20 Cf., entre otros, P. Gottlieb 2009:186; R. Sorabji 1973:124-127; R. Hursthouse 1998:210-212; N. Bowditch 2008:315. 21 Para una interpretación esencialmente cognitivista del problema, vid. R. Sorabji 1973. 22 Cf., fundamentalmente, Stob. 2.85,13-86,4 [LS 59B; SVF 3.494]; Cic. Fin. 3.58-9 [LS 59F] y 3.24-5 [LS 64H; SVF 3.11]; Philo Cher. 14-15 [LS 59H; SVF 3.513]; Stob. 2.93,14-18 [LS 59K; SVF 3.500]. Para una discusión general de este problema, cf. B. Inwood 1985: 205-215. 60 RODRIGO SEBASTIÁN BRAICOVICH Contrastes vol. XVIII (2013) legados por Cicerón y Estobeo una imagen mínima de lo que los estoicos antiguos pudieron haber entendido por tal distinción: si atendemos a Cicerón, sería lícito denominar κατόρθωμα (recte; τὸ δέον, en el griego de Filón23) a una acción que ha sido realizada 'de la forma correcta' (iuste24) y contiene todas la medidas de la virtud (omnes numeros virtutis continent;25 πάντας ἀπέχον τοὺς ἀριθμούς, según la versión griega de Estobeo26). No obstante, la distinción estoica entre καθήκοντα y κατόρθωμα, carece –al menos hasta donde sabemos– de un elemento que resulta central para D5 y D6, a saber, la idea de la habituación o ejercitación, y es precisamente ese elemento el que me interesa resaltar. IV. La Idea de habItuacIón/ejercIcIo en eL horIzonte InteLectuaLIsta estoIco Fundamentalmente a partir de los estudios de Pierre Hadot en las décadas de 1960 y 1970, un aspecto que ha sido reiteradamente señalado por los comentaristas como uno de los aportes decisivos del período romano de la escuela estoica (fundamentalmente tomando en consideración a Séneca, Musonio, Epicteto y Marco Aurelio) consiste en la dimensión esencialmente práctica del abordaje que tales autores ofrecen de la filosofía. Trascendiendo el mero 'giro hacia la ética' que se verifica en dicho período, las obras que se han conservado de los estoicos romanos representan un repertorio compuesto por una serie de «ejercicios espirituales» destinados a poner al proficiente en camino hacia la virtud: desde el aprendizaje memorístico hasta la repetición incesante de ciertos principios teóricos, pasando por ejercicios específicos como abstenerse de beber agua o de ingerir comida, el estoicismo romano ofrecería –de acuerdo con dicha línea de lectura– una visión más realista y cotidiana de la labor filosófica que la alternativa ofrecida por los desarrollos lógicos y físicos de Crisipo o, yendo más atrás, por la concepción esencialmente teorética de la filosofía desarrollada por Aristóteles.27 Ahora bien, el énfasis –muchas veces descontextualizado28– sobre este tipo de ejercicios ha impedido percibir claramente hasta qué punto ese tipo de estrategias puede constituir una amenaza directa al núcleo intelectualista defendido por el estoicismo (con la posible excepción de Posidonio) a lo largo de 23 Cf. Philo Cher. 14-15 [LS 59H; SVF 3.513]. 24 Si iuste depositum reddere in recte factis sit, in officiis ponatur depositum reddere; illo enim addito «iuste» fit recte factum, per se autem hoc ipsum reddere in officio ponitur. (Cic. Fin. 3.58-9 [LS 59F]). 25 Cf. Cic. Fin. 3.24-5 [LS 64H; SVF 3.11]. 26 Stob. 2.93,14-18 [LS 59K; SVF 3.500]. 27 Respecto de este último sentido en particular, cf. J. Sellars 2003. 28 Cf. a modo de ejemplo, las interpretaciones recientes de D. Robertson 2010 y E. Buzaré 2011. 61Teoría y práctica en Musonio Rufo Contrastes vol. XVIII (2013) toda su historia: desde una perspectiva monista, en efecto, la suposición de que ciertas estrategias no intelectuales (tales como la repetición, la memorización, etc.) son necesarias para alcanzar la virtud se vuelve sumamente peligrosa, en la medida en que tales estrategias parecen evidenciar necesariamente la existencia de procesos no reductibles al discurso racional.29 Las razones de que ello sea así son claras: una vez desterrada la existencia de partes irracionales del alma y habiendo traducido todo acontecimiento psíquico a un acontecimiento cognitivo, la única vía posible de modificación de la disposición del agente consiste en la demostración racional de la verdad de un determinando principio teórico. Como lo afirmará el discípulo más célebre del propio Musonio: πέφυκεν δὲ πᾶσα ψυχὴ ὥσπερ τῷ ἀληθεῖ ἐπινεύειν, πρὸς τὸ ψεῦδος ἀνανεύειν, πρὸς τὸ ἄδηλον ἐπέχειν, οὕτως πρὸς μὲν τὸ ἀγαθὸν ὀρεκτικῶς κινεῖσθαι, πρὸς δὲ τὸ κακὸν ἐκκλιτικῶς, πρὸς δὲ τὸ μήτε κακὸν μήτ' ἀγαθὸν οὐδετέρως [...] τὸ ἀγαθὸν φανὲν εὐθὺς ἐκίνησεν ἐφ' αὑτό, τὸ κακὸν ἀφ' αὑτοῦ. οὐδέποτε δ' ἀγαθοῦ φαντασίαν ἐναργῆ ἀποδοκιμάσει ψυχή. (Epicteto, Dissertationes 3.3.2-4).30 Toda alma, por naturaleza, igual que asiente a lo verdadero, niega lo falso y ante lo incierto se abstiene, así también ante el bien reacciona con deseo; ante el mal, con rechazo; ante lo que no es ni bueno ni malo, de ninguna de las dos maneras. [...] Cando se presenta el bien, inmediatamente se mueve hacia ello; cuando el mal, se aleja de ello. El alma nunca rechazará una impresión clara del bien.31 ¿Descansan, entonces, D5 y D6, como parece temer Houser,32 sobre una concepción posidoniana de la habituación,33 una concepción que depende de la aceptación –contra la ortodoxia estoica– de la existencia de una parte irracional del alma (i.e., del sacrificio del monismo anímico en pos de una concepción de tipo platónico o aristotélico)? ¿En qué consiste, en última instancia, la ejercitación anímica que Musonio sugiere como un posible camino hacia la virtud? 29 Como se hace evidente, esta dificultad no afecta a Aristóteles, dado que el proceso de habituación en cuestión se refiere a las virtudes éticas, las cuales se vinculan explícitamente con la parte irracional del alma. 30 Cf. asimismo Epicteto, Dissertationes 2.26.1-7. 31 Cito de acuerdo con W.A. Oldfather 1961; la traducción es de Ortíz Garcia (Epicteto 1993). 32 Cf. A. Houser 1997:35-37. 33 Respecto de Posidonio, cf. paradigmáticamente F169g: χρὴ γὰρ καὶ τοῦτο μὲν ἐπιστήμην λαβεῖν τῶν ἀληθῶν καὶ τὰς κατὰ πάθος δὲ κινήσεις ἀμβλυνθῆναι χρηστοῖς ἐπιτηδεύμασιν ἐθισθείσας, εἴ τις μέλλοι βελτίονα τὸ ἦθος ἀποδείξειν τὸν ἄνθρωπον. Cf. asimismo F31; F164; F165 (cito según la numeración de Kidd) y R. Sorabji 2000: passim. 62 RODRIGO SEBASTIÁN BRAICOVICH Contrastes vol. XVIII (2013) A juzgar por al menos dos pasajes similares que veremos a continuación, el proceso de habituación/ejercitación propuesto por Musonio parecería apoyarse en mecanismos no intelectuales. En efecto, si retomamos D6, la alusión a una continuidad entre el cuerpo y el alma en cuanto a los efectos que determinadas acciones pueden tener sobre ellos alude a un proceso donde la comprensión de principios teóricos no parece cumplir ningún papel: τῆς οὖν ἀσκήσεως ἡ μέν τις ἰδία τῆς ψυχῆς μόνης γίνοιτ' ἂν ὀρθῶς, ἡ δέ τις κοινὴ ταύτης τε καὶ τοῦ σώματος. κοινὴ μὲν οὖν ἄσκησις ἀμφοῖν γενήσεται, συνεθιζομένων ἡμῶν ῥίγει, θάλπει, δίψει, λιμῶ, τροφῆς λιτότητι, κοίτης σκληρότητι, ἀποχῇ τῶν ἡδέων, ὑπομονῇ τῶν ἐπιπόνων. διὰ γὰρ τούτων καὶ τῶν τοιούτων ῥώννυται μὲν τὸ σῶμα καὶ γίνεται δυσπαθές τε καὶ στερεὸν καὶ χρήσιμον πρὸς ἅπαν ἔργον, ῥώννυται δὲ ἡ ψυχὴ γυμναζομένη διὰ μὲν τῆς ὑπομονῆς τῶν ἐπιπόνων πρὸς ἀνδρείαν, διὰ δὲ τῆς ἀποχῆς τῶν ἡδέων πρὸς σωφροσύνην (6.12-14 [36-44]) Un tipo de ejercitación es apropiado únicamente para el alma; otro tipo lo es para el cuerpo; otro es común para el alma y el cuerpo. Realizamos el ejercicio común a ambos [i.e., ejercitamos ambos] cuando nos habituamos a soportar el frío, el calor, el hambre, la sed, la escasez de comida, un lecho duro, a abstenernos de los placeres y a sobrellevar los sufrimientos. A través de estos métodos y otros parecidos, el cuerpo se fortalece, se vuelve capaz de tolerar los sufrimientos y apto para soportar cualquier tarea. El alma también se ve fortalecida, dado que se entrena en la valentía al sobrellevar los sufrimientos y se entrena en la moderación al abstenerse de los placeres.34 Si nos limitamos a este pasaje, σωφροσύνη y ἀνδρεία constituirían virtudes cuya adquisición no requiere ningún proceso específico de comprensión intelectual, sino un simple proceso de habituación mecánica y repetitiva: tal como el estómago se termina habituando eventualmente y en forma automática a la escasez de alimentos, así el alma se vuelve moderada habituándose a abstenerse de los placeres.35 Si sumamos a este pasaje unas líneas de la Disertación 4 34 Diógenes Laercio atribuye a Diógenes el Cínico una argumentación similar en DL 6.70-71. No obstante, aunque dicho pasaje coincide con D6 en distinguir, en primera instancia, entre los ejercicios corporales y los ejercicios anímicos para luego vincular a cierta clase de gimnasia corporal con la adquisición de la virtud, la similitud entre ambos pasajes es meramente superficial, en la medida en que el argumento de Diógenes consiste en señalar que para alcanzar la ἀρετή relativa a una cierta τέχνη, es necesario realizar un esfuerzo (πὸνος) incesante, todo lo cual es, en el mejor de los casos, secundario para el argumento general de Musonio. 35 Hadot alude, analizando este pasaje, al hecho de que el gymnasion era el lugar en el que, además de los ejercicios físicos, se impartían con frecuencia las lecciones de filosofía (cf. 63Teoría y práctica en Musonio Rufo Contrastes vol. XVIII (2013) que refuerzan esta idea, el temor expresado por Houser respecto de que estemos ante una recaída platónico/posidoniana parece sumamente atendible: μὴν τὸν παιδευόμενον ὀρθῶς, ὅστις ἂν ᾖ, εἴτε ἄρρην εἴτε θήλεια, ἐθιστέον μὲν ἀνέχεσθαι πόνου, ἐθιστέον δὲ μὴ φοβεῖσθαι θάνατον, ἐθιστέον δὲ μὴ ταπεινοῦσθαι πρὸς συμφορὰν μηδεμίαν· δι' ὅσων ἄν τις εἴη ἀνδρεῖος. (4.26 [78-82]) Un niño que es educado correctamente, ya sea varón o niña, debe ser habituado a tolerar las tribulaciones, a no temer a la muerte y a no ser derrotado por los infortunios; de esa forma será valiente. V. Los ejercIcIos InteLectuaLes en La Disertación 6 y epIcteto El panorama que se ha ido construyendo hasta este punto (i.e., en la concatenación de D5 y D6.1-7) parece confirmar la idea compartida por buena parte de la crítica especializada respecto del carácter decisivo que asume la práctica en Musonio y respecto de la necesidad de concebir dicha práctica (o habituación o ejercitación) como un proceso que trasciende la mera comprensión teórica. No obstante, cuando llega el momento de ofrecer precisiones respecto de qué debemos entender por dicha ἄσκησις, habituación o ejercitación, el panorama se modifica en forma decisiva: 6.8-9: [El proceso de ejercitación referido en 6.1-7] es más importante en la filosofía que en otras artes (τέχνης) porque el hombre se inicia en la filosofía habiendo aprendido (μεμαθηκότες) lo contrario de lo que debe ahora aprender y porque su alma está dañada a causa de la corrupción (διαφθορᾷ) y el mal de su entorno.36 6.15: El ejercicio que es específico del alma37 consiste en mantener a la mano (προχείρους) las demostraciones (ἀποδείξεις) referidas a los bienes aparentes y en habituarse a reconocer (γνωρίζειν) los bienes reales, a distinguirlos (διακρίνειν) de los que no lo son, 6.16: y en preocuparse (μελετᾶν) en no huir de los falsos males y en no perseguir los falsos bienes. P. Hadot 1998:208). 36 Un análisis de esta problemática de la «corrupción» del ambiente en el que el individuo es educado se encuentra en R. Valantasis 1999. 37 Musonio distingue en D6.12-14 entre una ejercitación que es propia del alma, otra que es propia del cuerpo y otra que es común a los dos. 64 RODRIGO SEBASTIÁN BRAICOVICH Contrastes vol. XVIII (2013) Si tomamos en cuenta estas especificaciones, este ejercicio espiritual, como lo denomina Sellars, se vuelve un ejercicio esencialmente intelectual: la práctica en cuestión no es otra cosa, a fin de cuentas, que el acto de evaluar cada situación (problemática) a la luz del criterio ofrecido por la axiología estoica, esto es, la tripartición bueno/malo/indiferente, estando siempre atentos al hecho de que aquello que aparece prima facie como un bien o un mal puede no serlo.38 La causa del vicio es, en el fondo, puramente intelectual: a pesar de que afirmamos estar de acuerdo con que nada que no sea el vicio debe ser temido y nada que no sea la virtud debe ser perseguido, «cuando sobrevienen los infortunios, pensamos (ἡγούμεθα) que nos ha ocurrido algo malo; cuando viene el placer, pensamos (ἡγούμεθα) que nos ha sucedido algo bueno» (6.20 [62-64]). El remedio (i.e., el camino hacia la virtud) no puede ser, en consecuencia, otra cosa que una modificación de nuestros hábitos intelectuales: enfrentar cada situación armados con los principios apropiados para evaluarla correctamente (6.21 [69]: ἀκολούθως ταῖς ὀρθαῖς ὑπολήψεσι τοῖς πράγμασι χρώμεθα), en lugar de permitir que sean los principios heredados de nuestro entorno los que decidan sobre el valor de cada una de las alternativas que se abren ante nosotros. Esto resuelve el interrogante acerca de si la insistencia de Musonio sobre los conceptos de ἔθος/ἄσκησις es compatible con la concepción intelectualista de la acción humana defendida por el estoicismo, y disipa parcialmente los temores respecto de que D5-6 representen una recaída posidoniana. Queda sin resolver, a pesar de ello, una pregunta importante: ¿Por qué insiste Musonio en la idea de que debemos habituarnos a evaluar cada situación a través de los principios teóricos correctos, en lugar de establecer que debemos evaluar cada situación a través de dichos principios? ¿Qué es lo que aporta dicha habituación? En otras palabras: ¿en qué sentido la habituación a evaluar las situaciones armados con los principios teóricos apropiados puede producir que, eventualmente, alcancemos la virtud? La respuesta a esta pregunta, desafortunadamente, no la podemos obtener directamente a partir de los escasos registros que se han conservado de las disertaciones de Musonio. No obstante, las Dissertationes de Epicteto pueden ser utilizadas, según propongo, para ensayar una respuesta provisoria. En ambas obras, en efecto, encontramos en forma recurrente una discusión análoga a la que hemos estado analizando en D5-6, presentada en términos virtualmente idénticos a los de Musonio39 y reiterando, en ciertos casos, estructuras argumentativas 38 Así como lo será en Epicteto la distinción entre lo que depende y lo que no depende de nostros, esta última advertencia representa el principio central sobre el que parecen haberse articulado las estrategias retórico-pedagógicas de Musonio. 39 Sobre la problemática de la ἄσκησις en general en Epicteto, vid. B.L. Hijmans 1959, A.A. Long 2002 y, fundamentalmente, J. Cooper 2007. 65Teoría y práctica en Musonio Rufo Contrastes vol. XVIII (2013) análogas. Tales paralelos, no obstante, coexisten con un aporte específico que Epicteto realiza al problema de la ἄσκησις y que surge a la luz cuando, a partir de la estructura aparentemente caótica de las lecciones recogidas por Arriano, reconstruimos una serie de estrategias sumamente coherentes y precisas que articulan en forma silenciosa la totalidad de las reflexiones de Epicteto referidas al progreso moral (i.e., al camino que recorre el proficiente en su trayecto hacia la virtud). Dichas estrategias se construyen fundamentalmente sobre dos ejes: en primer lugar, la distinción entre la mera comprensión abstracta de los principios teóricos y la aplicación de los mismos a los casos particulares; en segundo lugar, el énfasis por parte del autor en la necesidad de una comprensión analítica y sistemática de los conceptos que se hallan en juego en cada evaluación que realizamos de las situaciones a las que nos vemos enfrentados cotidianamente.40 Si incorporamos estos dos elementos,41 sugiero, podemos no sólo ofrecer una respuesta concreta al interrogante que señalábamos hace un momento como irresuelto en Musonio (i.e., en qué sentido la habituación a evaluar las situaciones que enfrentamos cotidianamente a la luz de los principios correctos puede contribuir a que eventualmente alcancemos la virtud), sino también contribuir a responder a otra pregunta, íntimamente vinculada a aquella, que todavía no hemos abordado, a saber: ¿en qué se distingue la acción del proficiente de la acción del individuo virtuoso? La respuesta a ambos interrogantes puede ser presentada en forma unificada: a pesar de que una determinada acción realizada por un individuo virtuoso y uno que todavía no lo es pueden aparecer externamente como idénticas, la diferencia entre ambas acciones consiste en que la acción del individuo virtuoso, en primer lugar, no se funda sobre una comprensión puramente abstracta de ciertos principios teóricos, sino sobre la capacidad de aplicarlos correctamente 40 Cf., a este respecto, Braicovich 2012 y 2013. Dissertationes 3.20 y 4.1.63-85 ofrecen dos ejemplos claros de este proceso complejo mediante el cual, partiendo de una idea de carácter general a la que el interlocutor asiente sin comprender todavía sus consecuencias lógicas, Epicteto despliega dialécticamente lo que significa adquirir una comprensión profunda de un principio teórico. Una aplicación sumamente interesante, desde el punto de vista pedagógico, de esta estrategia se encuentra en Dissertationes 2.19.20-34, donde Epicteto despliega las verdaderas consecuencias y el verdadero precio a pagar por ser un «filósofo estoico». Cf. asimismo Dissertationes 3.16.7-13 y 1.22.9ss. 41 Esta estrategia tentativa de complementar las lagunas presentes en las disertaciones de Musonio con los desarrollos ofrecidos por Epicteto ciertamente supone que asumamos una continuidad doctrinal sustantiva entre ambos pensadores. Lo cierto, en este sentido, es que nada en las Dissertationes hace suponer que Epicteto se haya desviado en lo más mínimo respecto de las enseñanzas de su maestro, y las únicas divergencias evidentes son fundamentalmente temáticas, y son atribuibles a una diversidad de intereses entre ambos, y no a una diferencia en cuanto al enfoque doctrinal de cada uno. 66 RODRIGO SEBASTIÁN BRAICOVICH Contrastes vol. XVIII (2013) a la situación a la que se ve enfrentado; y, en segundo lugar, en que su acción se ve acompañada por una captación profunda y sistemática de las consecuencias lógicas que se derivan de dicha acción desde la perspectiva de la consistencia interna de su alma (i.e. de sus opiniones). La acción del individuo que todavía no ha alcanzado la virtud, por el contrario, carecerá de tales características (he ahí la respuesta a la segunda pregunta), pero la realización sostenida y sistemática del tipo correcto de acciones contribuirá a que el individuo adquiera gradualmente una comprensión cada vez más profunda de las múltiples dimensiones éticas que se derivan de sus acciones hasta que, eventualmente, las acciones que realice constituirán acciones correctas (καθήκοντα) realizadas por los motivos correctos (κατόρθωμα), o, puesto en términos aristotélicos, acciones realizadas a partir de una disposición virtuosa. vI. concluSIón Como señalé en la introducción, los objetivos específicos que me proponía en estas páginas eran los siguientes: en primer lugar, reconstruir en forma sistemática la relación entre λόγος y ἔθος/ἄσκησις tal como la misma es presentada por Musonio en D5-6; en segundo lugar, postular las reflexiones de Aristóteles sobre el problema de la habituación como un marco conceptual relevante para encuadrar el análisis de D5-6; por último, analizar las posibles tensiones lógicas entre la concepción de Musonio de ἔθος/ἄσκησις y la concepción intelectualista de la acción humana defendida por la ortodoxia estoica. Sugerí, de modo complementario, la posibilidad de recurrir a las Dissertationes de Epicteto en busca de desarrollos parciales que permitiesen ofrecer una respuesta tentativa a los interrogantes que no podemos responder a partir de la sola evidencia de D5-6. La conclusión fundamental que creo que es posible extraer a partir de este desarrollo consiste en el hecho de que la «práctica» frecuentemente señalada por los comentaristas como central para la concepción de la filosofía defendida por Musonio, puede legítimamente ser concebida como un proceso puramente intelectual, un proceso de captación cada vez más profunda y concreta de la teoría. Esto no agota, ciertamente, las tareas a realizar en la búsqueda de una comprensión más acabada de la concepción específica que los estoicos romanos desarrollaron de la actividad filosófica (de sus tareas, sus fundamentos y sus estrategias), pero permite al menos establecer una continuidad clara entre las reflexiones de Séneca, Musonio Rufo y Epicteto en cuanto al hecho de que en ninguno de los tres autores el énfasis en procesos activos y prácticos de reconstrucción de la subjetividad supone un recurso velado a una concepción dualista del alma ni un conflicto con la concepción intelectualista de la acción humana defendida por la ortodoxia estoica. 67Teoría y práctica en Musonio Rufo Contrastes vol. XVIII (2013) referencIaS bIblIográfIcaS ARISTÓTELES. 2007: Ética nicomaquea, tr. Eduardo Sinnott. Buenos Aires: Ediciones Colihue. BOWDITCH, N. 2008: «Aristotle on habituation: The key to unlocking the Nicomachean Ethics», Ethical Perspectives, 15.3, pp. 309-342. BRAICOVICH, R.S. 2012: «Critical assent, intellectualism and repetition in Epictetus», Apeiron, 45, pp. 314-337. _____, 2013: «Ejercicios espirituales e intelectualismo en Epicteto», Classica, 23.2. [En prensa] BROADIE, S. 1991: Ethics with Aristotle. Oxford: Oxford University Press. BUZARÉ, E. 2011: Stoic Spiritual Exercises. North Carolina: Lulu. BYWATER, I. 1962: Aristotelis. Ethica Nicomachea. Oxford: Clarendon Press. COOPER, J.M. 2007: «The Relevance of Moral Theory to Moral Improvement in Epictetus», en A.S. Mason y Th. Scaltsas (eds.), The Philosophy of Epictetus. Oxford: Oxford University Press, pp. 9-19. EPICTETO. 1993: Disertaciones, tr. Paloma Ortiz García. Barcelona: Gredos. FOUCAULT, M. 1994: Histoire de la sexualité, tome 3: Le souci de soi. París: Gallimard. GILL, C. 2006: The Structured Self in Hellenistic and Roman Thought. Oxford: Oxford University Press. GOTTLIEB, P. 2009: The Virtue of Aristotle's Ethics. Cambridge: Cambridge University Press. HADOT, P. 1993: Exercices spirituels et philosophie antique. Paris: Albin Michel. _____, 1998: ¿Qué es la filosofía antigua?. México: Fondo de Cultura Económica. HIJMANS, B.L. 1959: Askesis. Notes on Epictetus' Educational System. Assen. HOUSER, J.S. 1997: The Philosophy of Musonius Rufus. A Study of Applied Ethics in Late Stoa. Tesis Doctoral. Brown University. HURSTHOUSE, R. 1988: «Moral Habituation. A review of Troels Engberg-Pedersen, Aristotle's Theory of Moral Insight», Oxford Studies in Ancient Philosophy, 6, pp. 201-19. INWOOD, B. 1985: Ethics and Action in Early Stoicism. Oxford: Clarendon Press. _____, 2004: Reseña de The Art of Living: the Stoics on the Nature and Function of Philosophy, de John Sellars. Notre Dame Philosophical Reviews, 04 de Abril, http://ndpr.nd.edu/news/23760-the-art-of-living-the-stoics-on-the-nature-andfunction-of-philosophy. KING, C. 2011: Musonius Rufus: Lectures and Sayings. CreateSpace. LONG, A.A. 2002: Epictetus. A Stoic and Socratic Guide to Life. Oxford: Oxford University Press. LUTZ, C. 1947: «Musonius Rufus: The Roman Socrates», Yale Classical Studies, 10, pp. 3-147. MASON, A.S.; Scaltsas, T. (eds.). 2007: The Philosophy of Epictetus. Oxford: Oxford University Press. 68 RODRIGO SEBASTIÁN BRAICOVICH Contrastes vol. XVIII (2013) MORFORD, M.. 2002: The Roman Philosophers. From the Time of Cato the Censor to the Death of Marcus Aurelius. Londres: Routledge. NUSSBAUM, M. 1996: The Therapy of Desire. Theory and Praxis in Hellenistic Ethics. Princeton: Princeton University Press. OLDFATHER, W.A. 1961: Epictetus: The Discourses as reported by Arrian. The Manual and Fragments. Londres: Heinemann. POMEROY PARKER, Ch. 1896: «Musonius the Etruscan», Harvard Studies in Classical Philology, 7, pp. 123-137. REYDAMS-SCHILS, G. 2005: The Roman Stoics. Self, Responsibility, and Affection. Chicago: University of Chicago Press. ROBERTSON, D. 2010: The philosophy of cognitive-behavioural therapy (CBT) stoic philosophy as rational and cognitive psychotherapy. Londres: Karnac Books. ROSKAM, G. 2005: On the Path to Virtue. The Stoic Doctrine of Moral Progress and its Reception in (Middle-)Platonism. Leuven: Leuven University Press. SELLARS, J. 2003: The Art of Living: the Stoics on the Nature and Function of Philosophy. Ashgate. _____, 2007: «Stoic practical philosophy in the imperial period», Greek and Roman Philosophy, 100 BC-200 AD, Bulletin of the Institute of Classical Studies, Supl. 94.1, pp. 115-140 SORABJI, R. 1973: «Aristotle on the role of intellect in virtue», Proceedings of the Aristotelian Society, 74, pp. 107-129. SORABJI, R. 2000: Emotion and Peace Of Mind. From Stoic Agitation to Christian Temptation. The Gifford Lectures. Cambridge: Cambridge University Press. VALANTASIS, R. 1999: «Musonius Rufus and Greco-Roman ascetical theory», Greek, Roman and Byzantine Studies, 40.3, pp. 207-231. rodrIgo SebaStIán braIcovIch pertenece al Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina Líneas de investigación: La concepción intelectualista de la acción humana en el estoicismo romano y su vinculación con las estrategias terapéuticas desarrolladas por Séneca, Musonio Rufo, Epicteto y Marco Aurelio. Publicaciones recientes: (2010) "Freedom and epistemic determinism in Epictetus' Discourses", The Classical Quarterly (Oxford), 60.1, pp. 202-220. ISSN: 0009-8388. (2012) "Critical assent, intellectualism and repetition in Epictetus", Apeiron: Journal for Ancient Philosophy and Science (De Gruyter). [Aceptado para publicación]. ISSN: 0003-6390. Dirección electrónica: [email protected]

Article Public justification and the reactive attitudes Anthony Taylor University of Oxford, UK Abstract A distinctive position in contemporary political philosophy is occupied by those who defend the principle of public justification. This principle states that the moral or political rules that govern our common life must be in some sense justifiable to all reasonable citizens. In this article, I evaluate Gerald Gaus's defence of this principle, which holds that it is presupposed by our moral reactive attitudes of resentment and indignation. He argues, echoing P.F. Strawson in 'Freedom and Resentment', that these attitudes are so deep a part of us that we are unable to rationally reject them. I examine and reject this defence of the principle. Considering the nature of our commitment to the moral reactive attitudes, I argue that those attitudes need not be grounded in a commitment to public justification. The availability of alternative grounds for these attitudes shows, contra Gaus, that we can rationally reject the principle of public justification while maintaining a wholehearted commitment to the reactive attitudes. Keywords public justification, public reason, moral responsibility, reactive attitudes, liberal legitimacy Introduction According to the principle of public justification, the moral or political rules that govern our common life must be in some sense justifiable to all reasonable citizens. Versions this principle have been endorsed by numerous political philosophers, and perhaps most notably by John Rawls (1993). While Rawls's account is undoubtedly the most Corresponding author: Anthony Taylor, Nuffield College, University of Oxford, New Road, Oxford, Oxford OX1 1NF, UK. Email: [email protected] Politics, Philosophy & Economics 2018, Vol. 17(1) 97–113 a The Author(s) 2017 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1470594X17695070 journals.sagepub.com/home/ppe discussed, this article sets it aside to consider a sophisticated and interesting defence of the principle developed by Gerald Gaus. Specifically, Gaus defends the Basic Principle of Public Justification: A moral imperative 'j'! in context C, based on rule L, is an authoritative requirement of social morality only if each normal moral agent has sufficient reasons to (a) internalize rule L, (b) hold that L requires j–type acts in circumstances C.1 (2011: 263) In order to know when this principle is satisfied, we need to know when we have sufficient reasons to internalize rules and hold that they require particular acts.2 Here, Gaus defends Having a Reason: An agent has a sufficient reason R to hold that  is the thing to believe, or j is the thing to do, if, and only if, (i) they have arrived at R by following the norms of good reasoning and (ii) if they engaged in a 'respectable amount' of reasoning, they would not (or did not) discover defeaters for R. (2011: 249–250) Taking these claims together, we have an account of the justification of social–moral rules that takes seriously the evaluative standards of each individual. Social–moral rules are those 'that require or prohibit action, and so ground moral imperatives that we direct to each other to engage in, or refrain from, certain lines of conduct' (2011: 2). In order for such a rule to be one others' can legitimately demand my compliance with, it must cohere with my perspective in the right way – it must be one I have sufficient reason to endorse. Why should we accept the principle of public justification? The details of Gaus's answer to this question will occupy us for the remainder of the article, but in brief, he argues as follows. As moral persons we are liable to feel the reactive attitudes of resentment and indignation toward those who we perceive to violate moral rules, and these attitudes have appropriateness conditions. Suppose that I believe you have intentionally charged into me causing me to spill my cup of coffee, and I feel resentment on account of your lack of regard for me. If I were to learn that you were in fact caught in a strong gust of wind, my resentment would no longer be appropriate. The object of my attitude, your lack of regard for me, is no longer present, and therefore, it can no longer be rationally sustained. Gaus argues that the reactive attitudes are appropriate toward ostensible rule violators only if they had sufficient reason to internalize the rule in question (2011: 218–224). He then defends the claim, following P.F. Strawson, that our propensity to the reactive attitudes is so deep a part of us that it sits beyond the reach of rational justification. Taking these claims together, he reaches the conclusion that there is a principle of public justification embedded in our social practice of issuing imperatives and following social–moral rules (192). In spite of the foundational importance of this argument for Gaus's defence of public reason liberalism, it has yet to receive sustained critical attention in the literature. In this article, I evaluate and ultimately reject this, as I shall call it, reactive attitudes argument for the principle of public justification. I contend that – even granting some generous 98 Politics, Philosophy & Economics 17(1) assumptions to his strategy – this argument fails. While, for all I argue here, there may be more to be said for this principle, I do not believe it can enjoy the kind of justification Gaus claims for it. It does not follow from our commitment to rationally grounded reactive attitudes. I proceed as follows. In the first section, I set out the reactive attitudes argument, paying particular attention to the nature of the commitment to public justification it aims to establish. Next, I take a closer look at Gaus's account of Having a Reason. The next two sections examine the premises of the argument: the appropriateness conditions that Gaus takes to govern the reactive attitudes and the Strawsonian strategy that seeks to establish the commitment in question. I then set out a potential alternative account of the appropriateness conditions for the reactive attitudes that, I argue, shows the scope of the Strawsonian strategy to be circumscribed. Finally, I reconsider our commitment to public justification in light of this, arguing that the strategy can no longer do the work that the reactive attitudes argument requires of it. The reactive attitudes argument The phrase 'reactive attitudes' is a term of art introduced by P.F. Strawson. These attitudes, Strawson tells us, are those that are responsive to our perception of the qualities of will that others express toward us as manifested in their actions or omissions (2008 [1962]: 5–7). In addition, these attitudes have 'vicarious analogues', namely, the attitudes we adopt in response to our perception of the quality of another's will toward others (15). Resentment is a paradigmatic instance of a reactive attitude. When I believe you have intentionally charged into to me causing me to spill my cup of coffee, my resentment is a response to my perception that your action expresses ill will toward me. When I learn that you were actually caught in a strong gust of wind, given that gusts of wind are not willed by anyone, I learn that my resentment is no longer appropriate, as my belief that you expressed ill will toward me is false. Perhaps some other attitude such as frustration would be appropriate, but not the distinctly reactive attitude of resentment. There is some disagreement both about the range of attitudes that are reactive in Strawson's sense and about what hangs them together as a class.3 However, settling this issue is not essential for the reactive attitudes argument, which can be reconstructed as follows. 1. The reactive attitudes of resentment and indignation are appropriate only toward rule violators who are (i) capable of caring for moral rules and (ii) have sufficient reason to internalize the applicable moral rule. 2. Authoritative moral imperatives are appropriately addressed only to those to whom the reactive attitudes would be appropriate. 3. Therefore, authoritative moral imperatives are appropriately addressed only to those who (i) are capable of caring about moral rules and (ii) have sufficient reason to internalize the applicable moral rule. (Gaus, 2011: 205–258) Taylor 99 The first premise of the argument provides an account of when resentment and indignation are appropriate. As we saw in the spilt coffee example, certain beliefs are necessary to rationally sustain these attitudes. Discerning when resentment and indignation are appropriate is a matter of working out what the propositional content of these attitudes is. That is, answering the question: what must we believe about their object in order to sustain the attitude? Premise (1) expresses Gaus's answer to this question. I will discuss (1) in more detail shortly. Setting it aside for now, the controversial premise here is (2). Why should we believe that the fact that our reactive attitudes presuppose certain claims is sufficient to establish that those claims are correct? After all, an objector will be inclined to think, we can reflect on our attitudes and modify them – or perhaps even reject them completely – if we come to conclude they are unjustified. However, Gaus argues, following a strategy originally developed by Strawson, that rational justification of these attitudes is beside the point. The following passage most clearly expresses that strategy. We are embedded in certain sorts of practices, with certain beliefs and emotions. They form part of the reasons from which we must judge, criticize, and propose changes. A practice such as social morality is deeply embedded in our view of the world; it affects our understanding of interpersonal relations, including love and friendship, and so of what sort of life is worth living. If the presuppositions of our moral practices are so deep a part of the way we see the world, then to renounce the practice would be to renounce most of what we care for and value. But how can we have reason to do that? How could we survey all that matters to us and come to the conclusion that our reasons lead us to give it up, by renouncing the view of the world on which our reasons depends? Where would that reason come from? It is, I think, as difficult to argue a moral person out of her moral practices as it would be to argue the psychopath into them; given who they are, they do not have reasons to change their view of the world. (2011: 192) The inevitability of resentment and indignation for those bound up in the practice of experiencing them leaves them, in a sense, outside the reach of rational justification. If that is true, and we agree with Gaus about what the appropriateness conditions for these attitudes are, then we reach the conclusion. What is important here is the kind of conclusion we reach, and thus, the kind of justification Gaus can say the principle of public justification enjoys. It is not 'an exogenous (external) demand on an acceptable social morality based on some foundational moral intuition but a deep presupposition of our social morality with rational reactive attitudes' (2011: 223). On introducing the conclusion that moral prescriptions are appropriately addressed only toward those who are capable of caring about moral rules and have sufficient reasons to endorse them, Gaus refers to it as The Principle of Moral Autonomy (2001: 211). Its relevance for public justification should be relatively clear. If imperatives that demand we follow particular social moral rules should only be addressed toward those who have sufficient reason to internalize the rule in question, then we will want to know what rules (within the relevant jurisdiction) everyone has sufficient reason to internalize. This is what the principle of public justification tells us. 100 Politics, Philosophy & Economics 17(1) What does it mean to say that public justification is a deep presupposition of our social morality with rational reactive attitudes? Clearly, Gaus does not think the principle of public justification is a conclusion we reach after extensive moral reflection, considering its implications and its coherence with our considered moral judgements about particular cases. Rather, it seems, he thinks it has normative force in the following way. We are, in virtue of the kind of beings we are – moral persons – committed to a certain kind of view of the world, including a certain framework of attitudes. To give up this framework is, to import some language from Strawson, practically inconceivable for us. Since, as premise (1) tells us, a presupposition of this framework of attitudes is The Principle of Moral Autonomy, we are committed to the idea of a publicly justified social morality in virtue of our commitment to this framework. It is the fact that our commitment to this framework is not something we can rationally reject that gives the conclusion its normative force. There is evidence that confirms this understanding in the following passage from the preface to Gaus's earlier Justificatory Liberalism (1996). We understand ourselves and others as capable of putting aside personal valuings, and of acting on norms that can be justified to all. To rid ourselves of this conception [ . . . ] would undermine our understandings of ourselves, and others and social life, leaving us without rational grounding for most of what we hold dear. Given who we are, [ . . . ] we are committed to the idea of a publicly justified morality (vii).4 This completes my account of the reactive attitudes argument. It is undoubtedly an interesting argument for a principle of public justification, and considering whether it is successful will require taking a closer look at what Gaus and Strawson have to say about this strategy for defending premise (2), which I do shortly. First, however, I will briefly consider the idea of Having a Reason and what Gaus says in support of premise (1). Having reasons Part of what makes Gaus's conception of public justification distinctive is his account of what it means to have a reason. Gaus presents us with an internalist, as opposed to an externalist, account of that relation. An externalist account would say that you have a reason R if and only if R is an external reason that applies to you (2011: 233). Gaus rejects such views in the following passage. I believe the Externalist View of Having a Reason to be implausible: it misconstrues the relation between having a reason and being a rational agent. First consider the Externalist View of Having a Reason applied to theoretical reasoning. It implies that Aristotle, when writing on physics, had – possessed – a reason to embrace particle physics, because particle physics is true. But surely he did not have any such reason; to see R as a reason is to see it as justificatory, but Aristotle simply could not employ his rationality in a way that could lead him to see the facts supporting particle physics as justificatory (he could not even understand these facts). Only by not following the conclusions of his rational deliberation – being Taylor 101 irrational – could Aristotle endorse such a 'reason', and he could never see it as justifying a belief. (233–234) What does it mean to have a reason, then? We might suppose we stand in this havingrelation to a reason R when it is a reason that applies to us, and we can be said to have or possess it.5 However, consider a case of false belief, such as Bernard Williams's Gin and tonic: Alice orders a gin and tonic, but unbeknownst to her the bartender hands her a glass of petrol (1981: 102). Does Alice have a reason to take a sip in this case? If the glass had contained gin and tonic, we would naturally say that there is a reason for Alice to take a sip. She wants to drink a gin and tonic and it is a gin and tonic, so there is a reason for her to drink it. Further, she might sensibly be said to have this reason, as she quite justifiably believes it is a gin and tonic, given that that is what she ordered. In the case where the glass contains petrol, however, Alice may still have a warranted belief that it is a gin and tonic, but because this belief is false, we would no longer say that there is a reason to drink it. Therefore, if we think we stand in this having-relation to a reason when it is both a reason that applies to us and one that we can be said to have or possess, we will say Alice does not have a reason to take a sip, as the reason in question does not exist. On Gaus's view, however, whether Alice has a reason to take a sip does not depend on whether there is such a reason. In deciding whether we have a reason R, only our warranted beliefs about whether R is a reason that applies to us are relevant, not whether there is in fact a reason R (2011: 247). In considering Having a Reason, then, it is important to be clear that what we are talking about is simply our warranted beliefs about the reasons that apply to us. With this in mind, nothing related to public justification follows directly from this having-relation. That an agent has a warranted belief that they should j does not tell us whether we may permissibly stop them or issue a morally authoritative imperative that they not take a sip. We need an explanation of why our warranted beliefs about our reasons have normative import for such questions. This is what the conclusion of the reactive attitudes argument aims to provide, so let us now take a look at that argument. The appropriateness of resentment and indignation As I noted above, we discern when resentment and indignation are appropriate by working out what the propositional content of those attitudes is.6 We also saw that Strawson claimed these reactive attitudes are a response to the quality of other's wills toward us, but this suggestion alone will not take us far. The pertinent question is: how should we specify the conditions that need to be met for an agent to count as expressing a relevantly malevolent will? Gaus answers this question by defending two conditions, which I will now set out in turn. First, resentment is inappropriate toward those who are incapable of caring about moral rules when they do not promote their wants, ends or goals (2011: 211). They may care about rules when they correlate with their ends, such as when they know they can 102 Politics, Philosophy & Economics 17(1) avoid an unpleasant punishment by conforming with the rule, but they lack the ability to understand that a moral demand can be authoritative, that it can require them to set aside what they care about and act in line with it. What they are lacking is in part an affective capacity. Not simply the ability to understand moral rules or reasons and their importance in cooperative social life but the capacity to care about such rules. This answer explains part of our everyday practice of holding people morally responsible. One thing we tend to take into account when deciding whether someone is responsible, and thus appropriately the object of resentment or indignation, is whether they possess certain capacities. It is not implausible, prima facie, to describe the relevant capacity as the capacity to care about and internalize moral rules. Young children, psychopaths, and gusts of wind do not have this capacity, but the kind of agents we generally take to be morally responsible do. I will refer to this first condition as the capacity condition. This is not the only condition Gaus claims governs the appropriateness of resentment and indignation. He asks us imagine we are issuing an imperative to someone unable to grasp its authority, in spite of the fact that they meet the capacity condition. I know that she does not see that it is the moral thing to do, and suppose I think her lack of appreciation is quite genuine. I demand 'j!' and she does not see why she is obligated to j. She is puzzled that anyone would think j is obligatory. If I think this, then again I cannot reasonably feel resentment or indignation that she fails to j, any more than I can feel indignation at a four-year-old who is unable to detach himself from what he most wants to do and so steals some favourite candy. She just cannot see how 'j!' has any internal authority over her (2011: 219). He goes on to argue that if we offered this person a reason to adopt the rule that requires she js, and she were unable to grasp the force of that reason, then resentment would still be inappropriate. She cannot grasp that j-ing is the thing to do, because she cannot grasp my reason for holding that j-ing is the thing to do. Therefore, I cannot rationally resent her failure to act (p. 220). I will refer to this second condition as the sufficient reason condition. On this view, the capacity condition is what we can call a global condition. Where it is not met, it tells us that due to some fact about the agent it is not appropriate to hold them responsible in general, and therefore that it would be inappropriate to resent them. The sufficient reason condition, by contrast, is what we can call a local condition. It tells us that due to some fact about the relationship between the agent and this particular event, it is not appropriate to hold them responsible for it, and therefore that it would be inappropriate to resent them. If you do not meet the sufficient reason condition with regard to some particular rule, you are still considered to be a responsible moral agent in general, who might be appropriately resented in other instances. If you fail to meet the capacity condition, however, you are not considered to be a responsible moral agent at all. Taken together, these two conditions have a certain unity. The explanation of why some people can be the object of resentment and indignation is continuous with the explanation of why some specific actions or omissions can be appropriately resented. The explanatory factor in both the global case and the local case is the idea of having Taylor 103 sufficient reason to internalize and comply with social–moral rules. If we accept this picture, the propositional content of these attitudes – what we resent, or feel indignant about, when we do so rationally – is the attitude of the violator's failure to follow a rule that they had sufficient reason to. When we feel indignant because we believe someone has violated a rule, global conditions such as insanity, or being a young child, and local conditions, such as being caught in a strong gust of wind, function to tell us that this attitude was not in fact present, and so our indignation cannot be rationally maintained. However, it is far from obvious that resentment or indignation are never appropriate responses to those who do not meet the sufficient reason condition with respect to some rule. Intuitively, conscientious SS officers often acted wrongly and were thus appropriately resented, even if they would believe after sufficient amount of good reasoning that any rules prohibiting their acts were ones they did not have sufficient reason to follow.7 We might think, therefore, that we should accept an alternative view about when resentment and indignation are appropriate that does not have this implication. This is simply to doubt the normative significance of Gaus's notion of Having a Reason for the question of when the reactive attitudes are appropriate. One way to respond to this doubt would be to appeal to a deeper explanation of why it is inappropriate to resent those who do not meet the sufficient reason condition. However, as we shall see, Gaus's Strawsonian strategy precludes his providing such an explanation. That strategy is to claim that renunciation of the framework of reactive attitudes is practically inconceivable, and even if it were conceivable, it would be irrational. It tells that these attitudes lie beyond the scope of rational justification – so a deeper explanation of the conditions force is neither necessary nor possible. The Strawsonian strategy The preceding section set out what Gaus says in support of premise (1) – his account of when resentment and indignation are appropriate. In spite of the doubt I have just raised, for the sake of argument I will assume for the time being that, as a description of our current practice of holding one another responsible with its attendant liability to the reactive attitudes, this account is correct. That is, I will grant Gaus premise (1) of the reactive attitudes argument. Attending to the second premise of that argument, recall that it states that authoritative moral imperatives are appropriately addressed only to those to whom the reactive attitudes would be appropriate. The Strawsonian strategy that supports this premise is a much-discussed argument in debates surrounding the compatibility of moral responsibility and determinism. The stated aim of Strawson's 'Freedom and Resentment', the paper that introduced this strategy, is reconciliation between the positions of the 'optimist' and the 'pessimist' (2008 [1962]: 1–2). The pessimist holds that the truth of determinism would undermine our everyday practice of holding people morally responsible. Here is an example of how that practice might be undermined. One potential explanation of why we let people off the moral responsibility hook, and thus do not feel resentment on account of their actions or omissions that would otherwise be resentment-worthy is that they could not have acted otherwise. On this explanation, known as the principle of alternate 104 Politics, Philosophy & Economics 17(1) possibilities, the fact that you were caught in a strong gust of wind gets you off the moral responsibility hook because it tells me you could not have acted other than to make me spill my coffee (Frankfurt, 1969: 829). However, if this were the best explanation of why you ought to be let off the hook in this instance, the truth of determinism would render the practice of holding people morally responsible unjustified across the board. That determinism is true tells me that everyone is relevantly like you, unable to do otherwise.8 The optimist, by contrast, holds that our practice of holding people morally responsible is justified by its effectiveness at regulating social behaviour in ways that produce good consequences. According to this optimist then, the truth of determinism does not show our practice to be unjustified, since it does not tell us that our practice has negative consequences. Strawson aims to reconcile these positions by showing they are both mistaken about the grounds of the practice of moral responsibility. On his view, while the grounds of the practice are not undermined by the truth of determinism as the pessimist thinks, the practice also is not justified by its good consequences, as the optimist thinks. His argument rests on an account of the reactive attitudes, and the conditions under which we consider it appropriate to suspend them. One instance where we deem it appropriate to suspend the reactive attitudes, Strawson tells us, is where we learn that we were mistaken about the quality of will in question (2008 [1962]: 7–8). Again, the spilt coffee case is an example of this. When I learn that you were caught in a strong gust of wind, I learn that I was mistaken to hold that your spilling of my coffee expressed ill will toward me – it expressed no such will at all. This is what I called a local condition above. A second instance where we deem it appropriate to suspend the reactive attitudes is where we learn that it would be a mistake to resent this person due to a more general fact about them – their insanity, for example, or their being a young child (8–9). This is what I called a global condition above. Finally, beyond these two kinds of case, Strawson notes a further kind. We can, it seems, suspend the reactive attitudes not because, for one of the two preceding reasons, we believe it would be inappropriate to cling on to them, but rather because we have the ability to suspend them, which we can employ 'as a refuge, say, from the strains of involvement; or as an aid to policy; or simply out of intellectual curiosity' (10). Presented with this account of our practice of moral responsibility, we will want to know what role it can play in answering the normative question of when we should suspend the reactive attitudes. After all, Strawson's pessimist presumably holds that regardless of when we in fact suspend the reactive attitudes, we ought to do so permanently, across the board, as the truth of determinism renders them unjustified. To see Strawson's answer to this question we should first note that he holds that the alternative to the reactive attitudes is what he calls objectivity of attitude. To adopt the objective attitude to another human being is to see him, perhaps, as an object of social policy; as a subject for what, in a wide range of sense, might be called treatment; as something certainly to be taken account, perhaps precautionary account, of; to be managed or handled or cured or trained; perhaps simply to be avoided, though this gerundive is not peculiar to cases of objectivity of attitude. (2008 [1962]: 9) Taylor 105 Importantly, this objectivity of attitude is fundamentally opposed to human interpersonal relations. To be involved in such relations precisely is, Strawson tells us, to be liable to the reactive attitudes (10). The problem, then, is that that the pessimistic view just described would recommend we take up the objective stance permanently, as it tells us that our practice of holding people morally responsible is always unjustified. In taking up this stance permanently we would thus be sacrificing all interpersonal relations, and it is this that is for us as we are, practically inconceivable. The human commitment to participation in ordinary inter-personal relationships is, I think, too thoroughgoing and deeply rooted for us to take seriously the thought that a general theoretical conviction might so change our world that, in it, there were no longer any such things as inter-personal relationships as we normally understand them; and being involved in inter-personal relationships as we normally understand them precisely is being exposed to the range of reactive attitudes and feelings that is in question. (12) Moreover, Strawson argues, even if we could conceive of having a choice in this matter, it would be irrational to choose to permanently adopt the objective stance. In making such a choice, 'we could choose rationally only in the light of an assessment of the gains or losses to human life; its enrichment or impoverishment; and the truth or falsity of determinism would not bear on the rationality of this choice' (14). If he is right that the decision must be made on the basis of these considerations, then, taking into account the claim that the objective stance is fundamentally opposed to interpersonal relations, it is only a short step to the conclusion that permanent adoption of the objective stance is irrational. So Strawson's answer to the question of what bearing an account of when we in fact suspend the reactive attitudes has on the normative question of when we ought to suspend those attitudes turns on what I will call the Commitment Claim: Given our deep commitment to the reactive attitudes, it is practically inconceivable that we could abandon them; and further, if we could abandon them, it would be irrational to do so. To be sure, this claim raises a number of further questions that would need to be answered in order to establish that it can provide a satisfactory answer to the normative question. As I noted above, this argument is now the subject of a large literature in debates about moral responsibility. I will not be able to offer anything close to an adequate analysis of it with respect to those debates here. What I will do instead is grant, as a further assumption, that the commitment claim is true. Moreover, I want to grant that it provides a satisfactory answer to the pessimist's normative question. We ought not renounce the reactive attitudes in light of the truth of determinism, because our doing so is practically inconceivable in light of our commitment to interpersonal relationships. This is a controversial assumption. However, I will argue in the remainder of the article that even with it in place Gaus's argument for the principle of public justification is unsuccessful. Those who find this Strawsonian approach wanting more generally will simply have a further reason to reject that argument. 106 Politics, Philosophy & Economics 17(1) However, in granting the commitment claim as an assumption, I want to keep in the foreground why Strawson thinks this claim is true. It is because he holds that being involved in interpersonal relationships precisely is being liable to the reactive attitudes that he holds it to be practically inconceivable for us to give them up. An alternative With this thought in mind, I will set out an alternative view of when the reactive attitudes are appropriate. Consideration of this alternative will, I believe, show that the scope of the commitment claim is limited. As we have seen from the discussion so far, there are numerous cases where someone meets the capacity condition and ostensibly violates a moral rule, and yet we are inclined to think that it is not appropriate to feel resentment or indignation toward them, as in the spilt coffee example. As I noted above, if we think that the underlying principle that tells us when resentment or indignation are appropriate is the principle of alternate possibilities, then the compatibility of moral responsibility and determinism will be threatened. Since the truth of determinism will tell us that none of us could ever have acted otherwise, the principle will tell us that no one is ever the appropriate target for resentment or indignation. An alternative principle, expounded by R.J. Wallace, is that of 'no blameworthiness without fault' (1994: 135). On his view, our resentment or indignation can be rationally sustained where we believe a person has violated a moral obligation that we accept and hold them to. At first glance, this may seem to have a limited ability to account for local conditions, because that I accept and hold you to a moral obligation seems like it would be unaffected by facts about your relationship to a particular act or omission. That is, it seems that if you are under a moral obligation to refrain from violating my bodily integrity by knocking into me, for example, the fact that you were caught in a strong gust of wind will be irrelevant to whether or not you have violated this obligation. However, Wallace argues that because holding someone to an obligation we accept involves a commitment to the existence of reasons that support that obligation, moral obligations must be focused on states of affairs that are directly susceptible to the influence of reasons. Therefore, an act that was not susceptible to the influence of reasons cannot constitute the violation of a moral obligation, and thus cannot make a person the object of rationally appropriate resentment (118–147). This insight shows that the view can account for local conditions. When I learn you were caught in the gust of wind, what I learn is that your bodily movement was not the kind of thing that is susceptible to the influence of reasons, and therefore that you did not violate a moral obligation at all. Further, Wallace also supports a condition much like Gaus's capacity condition. It is not appropriate to hold someone responsible in general if they do not have the power of reflective self-control. That is, the power to grasp and apply moral reasons and regulate their behaviour in light of them (154–155). The underlying normative principle here, then, is that it is unfair to hold those who lack such powers responsible. Cases where we tend to let those with such powers off the hook are all cases where the agent turns out not to have performed that action intentionally at all. Importantly for Wallace the compatibility of moral responsibility with determinism is not threatened on Taylor 107 this view, as even in a deterministic world we would not have reason to suppose that no one ever has the power of reflective self-control, or so he argues. I will call Wallace's alternative condition governing when we ought to let the generally responsible off the hook the susceptibility to reason condition. It should be clear that it is quite different from the sufficient reason condition. Provided Alf meets the capacity condition, he can act on his warranted beliefs about the reasons he has and still violate a moral obligation Betty holds him to, making her resentment appropriate. That is, even though he believes after a sufficient amount of good reasoning that he has a moral permission to j, if he j's, where j is both an act susceptible to the influence of reasons, and an act Betty holds him to an obligation not to perform, her resentment is rationally appropriate. On this view, the propositional content of resentment is the belief that someone has violated an obligation we hold them to, where that requires, necessarily, that their act or omission was susceptible to the influence of reasons. I have assumed that Gaus's account of when the reactive attitudes are appropriate in our current practice of moral responsibility is correct. I have granted, then, that within this practice local conditions are governed by the sufficient reason condition, not the susceptibility to reason condition. But supposing that is right, and we are currently participants in a practice governed in part by the sufficient reason condition, might we abandon that condition in favour of the susceptibility to reason condition? That would involve ceasing to suspend our resentment or indignation in the cases in which Gaus thinks we should, those in which we learn that the person in question did not have sufficient reason to internalize the rule we believed they had violated. Instead, we would suspend our resentment or indignation only in cases where we learn that either the person in question does not meet the capacity condition or their act or omission did not meet the susceptibility to reason condition. In answering this question about the possibility of modifying our practice it seems that Gaus would want to appeal to the commitment claim. The sufficient reason condition is a presupposition of the reactive attitudes as we find them when we examine our current practice. Given that the reactive attitudes are constitutive of interpersonal relationships, and we are committed to such relationships, we are committed to the reactive attitudes as we find them. However, this answer to our question is unsatisfactory. While it follows from our commitment to interpersonal relationships that wholesale repudiation of the reactive attitudes is practically inconceivable, it does not clearly follow that revising the conditions under which we deem it appropriate to suspend them is practically inconceivable. To defend that claim we would need to argue that this revision would amount to sacrificing all interpersonal relations, as Strawson thinks would be the case with permanent adoption of the objective stance. Is there any reason to suppose that revising the conditions under which we deem it appropriate to suspend the reactive attitudes would have such severe consequences? Gaus appears to think this is the case. As I pointed out above, he tells us that to reject the understanding of ourselves and others as capable of acting on rules that can be publicly justified would be to 'undermine our understandings of ourselves, and others and social life, leaving us without rational grounding for most of what we hold dear' 108 Politics, Philosophy & Economics 17(1) (1996: vii). Further, he tells us: 'If the presuppositions of our moral practices are so deep a part of the way we see the world, then to renounce the practice would be to renounce most of what we care for and value' (2011: 192). These claims echo Strawson's point about changing our social world so that there are no longer such things as the reactive attitudes and therefore no longer such things as interpersonal relationships. However, it is difficult to see how, in Gaus's case, these claims could be true. When the conditions under which we deem it appropriate to suspend the reactive attitudes are governed by the capacity condition and the sufficient reason condition, we have an expectation of others that they follow the rules that they have sufficient reason to internalize and comply with. When we feel indignant on account of their failure to do so, the object of our indignation is a fact about the relationship between their capacities and their act or omission. The object of our indignation is: 'This person has the ability to care about and internalize social–moral rules, and they have sufficient reason to internalize and comply with this rule in particular – but they have violated it nonetheless!' By contrast, when the conditions under which we deem it appropriate to suspend the reactive attitudes are governed by the capacity condition and the susceptibility to reason condition, we have an expectation of others that they not violate moral obligations we accept and hold them to. The object our indignation, when we feel indignant on account of their failure to do so, is again a fact about the relationship between their capacities and the act or omission. It is: 'This person's action, which violated a moral obligation, was susceptible to the influence of reasons and so can properly be seen to express a judgement on behalf of their object. They have chosen to j, and given that j violates an obligation I hold them to, I resent their choice to j'. Both of these accounts involve demands and expectations. Both have an object that reflects a fact about the relationship between the person's capacities and the act or omission we might take to be worthy of say resentment, indignation or gratitude. There might be numerous theoretical reasons to prefer one account to the other (or indeed to reject both) either as an account of what our practice is or ought to be. However, what seems clearly false is that Gaus can respond to the question of whether we ought to revise our practice from one to the other by appealing to the commitment claim to say that we should not, because that change would amount to renouncing all we care for and value. With this in mind we can see that, even granting the commitment claim, its scope is limited – it does not tell us that renouncing the sufficient reason condition in favour of an alternative is practically inconceivable for us. Before moving on to reconsider the reactive attitudes argument in light of this, it is worth pausing to consider one way in which Gaus might reply to this argument. The susceptibility to reason condition seems to fit more intuitively with an externalist account of reasons and rationality. What makes that condition distinct from Gaus's sufficient reason condition is that it does not take the question of whether resentment is appropriate to hang on whether or not their target had sufficient reason, in his internalist sense, to endorse the rule in question. It takes the fact that there is a moral obligation not to j, alongside the fact that the agent possesses the relevant capacity, as sufficient to ground rational resentment. This only makes sense, we might think, if we take external reasons to be normatively significant, such that an agent is guilty of a rational flaw for failing to respond appropriately to a reason that applies to them, even if they would not Taylor 109 see it as a reason after a sufficient amount of good reasoning. Otherwise it would imply, somewhat counterintuitively, that we can rationally resent someone who is guilty of no rational failing at all. Since, as we have seen, Gaus rejects the externalist account of having a reason, we might think that he could thus reject any conditions that draw on externalism on the same grounds. However, while Gaus rejects externalism as an account of the reasons that we have, he explicitly does not deny externalism about the reasons that there are (2011: 232–233). We can happily accept that he is right to think that his internalism is the right account of the having-relation. The important question is whether or not that relation is of normative significance. As I noted above, Gaus thinks this relation is significant because of the reactive attitudes argument. It is because we only resent those who have sufficient internal reasons to endorse the rules they have violated – and we are committed to doing so in the sense implied by the commitment claim – that the reasons we have are significant for the question of when resentment is appropriate. Since in presenting the susceptibility to reason condition my aim has been precisely to call into question the nature of this commitment, Gaus cannot respond by simply asserting it. Reconsidering our commitment to public justification What are the implications of this argument about the scope of the commitment claim? It shows that the claim does not pick out the attitude of resentment with the sufficient reason condition as one of its appropriateness conditions in particular. Rather, it picks out a set of potential appropriateness conditions for resentment.9 However, the reactive attitudes argument cannot support the principle of public justification unless the commitment claim picks out resentment with the sufficient reason condition as one of its appropriateness conditions in particular. To show this I will now reconstruct that argument taking the proper scope of the claim into account. Premise (1) stated that resentment and indignation are only appropriate toward ruleviolators who are (i) capable of caring for moral rules and (ii) have sufficient reason to internalize the applicable moral rule. Our confidence in this premise will surely drop when we realize that the commitment claim does not provide support for it. We might therefore be inclined to think that (1) is not an accurate description of our practice of holding people responsible. If (1) is false, then Gaus's principle of public justification will not follow from the argument. However, I will now show that the principle does not follow even if we continue to grant this premise as an assumption. We can begin by noting that premise (2) must be modified. Originally, it stated that authoritative moral imperatives are appropriately addressed only toward those to whom the reactive attitudes would be appropriate. But when we ask why this is true Gaus's response is to appeal to the commitment claim. As I have shown, this claim is only plausible with regard to a set of presuppositions of resentment, and at least one member of this set does not have the sufficient reason condition as a presupposition. The argument for this is simply the argument of the previous section: the susceptibility to reason condition is capable of playing the role of rationally grounding valuable interpersonal relationships. The availability of this alternative shows that the commitment claim does not establish that the sufficient reason condition is beyond the reach of rational 110 Politics, Philosophy & Economics 17(1) justification. Rather, it shows that it would be practically inconceivable, and irrational if conceivable, to not endorse a member of the set of appropriateness conditions that is capable of furnishing rational grounds for human interpersonal relationships. Moreover, the commitment claim has nothing to say about how we choose from within this set – it only tells us that we cannot rationally choose to not to endorse any member of it. We must therefore modify (2) to state: authoritative moral imperatives are appropriately addressed only toward those to whom the reactive attitudes would be appropriate, according to an account of when these attitudes are appropriate. Where 'an account' can refer to any conception that is capable of furnishing rational grounds for these attitudes. The scope of appropriate moral authority, according to this premise, is dependent on the scope of the set of appropriateness conditions for the reactive attitudes that we cannot rationally reject. Once the premise is modified in this way, the conclusion no longer follows. The conclusion, recall, was that authoritative moral imperatives are appropriately addressed only toward those who are (i) capable of caring for moral rules and (ii) have sufficient reason to internalize the applicable moral rule. This conclusion shows, Gaus claims, that public justification is a 'deep presumption of our social morality with rational reactive attitudes' (223). However, the way in which (2) gives this conclusion its normative force in the original argument is, we saw, via the commitment claim. Since that claim does not tell us that we cannot modify the practice of social morality such that the sufficient reason condition is abandoned in favour of the susceptibility to reason condition, it does not give the conclusion its normative force in the modified argument. For all that has been said, we could adopt the susceptibility to reason condition, modify our practice as it recommends and still enjoy rationally grounded interpersonal relationships without a commitment to a public justified social morality. Note that I need not even argue that we ought to do this. The mere availability of an alternative that is not ruled out by the claim strips the public justification principle of its normative force. On this reconstructed version of the argument, we are not committed to public justification in virtue of being the kind of beings that we are. Rather we are committed to rationally grounded interpersonal relationships. Public justification is one way, among others, that we can furnish rational grounds for these relationships. Therefore, public justification is not, as Gaus claims, a deep presupposition of our social morality with rationally grounded reactive attitudes, something that we cannot rationally reject given the kinds of beings that we are. Conclusion I will now summarize the argument I have developed over the course of this article. Gaus's argument for the principle of public justification appeals to the presuppositions of our framework of reactive attitudes. That argument aims to show us that we are committed to public justification by virtue of our commitment to this social–moral framework of attitudes. Some may find Gaus's description of our social–moral practice uncompelling and still more may find his Strawsonian strategy wanting. However, I have argued that even assuming that Gaus is right on both of these points, the reactive attitudes argument fails. When we properly delineate the scope of the commitment claim Taylor 111 we see that our commitment is not to public justification but to interpersonal relationships, for which we can furnish alternative rational grounds. Perhaps other arguments can be marshalled in defence of the principle of public justification, but it is hard to see how they could establish that the principle enjoys the kind of justification Gaus claims for it. Author's note The author is a DPhil candidate in political philosophy at the University of Oxford. Acknowledgements For their comments on previous versions of this article, I am very grateful to Paul Billingham, Simon Caney, Will Dahlgreen, Steve Hood, Joe Horton, Sam Kiss, David Miller, Asbjørn Schmidt, Tom Sinclair, two anonymous reviewers and the Editor of this journal. Declaration of Conflicting Interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Work on this article was supported by a doctoral studentship from the Arts and Humanities Research Council. Notes 1. This formulation omits: '(c) moral agents generally conform to L', which raises issues that I cannot consider here. 2. Internalize is used here as a technical term, defined by Allan Gibbard as 'having a motivational tendency of a particular kind to act' on the pattern of behaviour prescribed by the norm, where the kind of tendency in question is related to 'a purpose of coordination' (1993: 68–71). 3. See, for example, O'Neill (2008: 77–83) and Wallace (1998: Ch. 2) 4. This quote summarizes an earlier formulation of the reactive attitudes argument from Gaus's Value and Justification (1990). 5. See, for example Schroeder (2008). This is the factoring account of having a reason that Schroeder rejects. 6. Note that saying that these attitudes have propositional content is not to assume moral cognitivism. Rather, it is simply to say that these emotions are about something: that they have an object. The claim that resentment and indignation have a form of implicit demand as their object is common in discussions of the moral reactive attitudes. See, for example, Darwall (2006). However, the claim that there is a strong link between the object of resentment and an action being wrong is controversial. 7. I adapt this example from Parfit (2011: 158). 8. Note that I do not say what the thesis of determinism says here, as Strawson's argument proceeds without a precise definition. He instead proceeds on the basis that 'if there is a 112 Politics, Philosophy & Economics 17(1) coherent thesis of determinism, then there must be a sense of 'determined' such that, if that thesis is true, then all behaviour whatever is determined in that sense' (2008 [1962]: 11). 9. Some may prefer to refer to this alternative as an alternative attitude to resentment, rather than an alternative appropriateness condition for resentment. Nothing in the following argument hangs on this choice, but I refer to it as an alternative appropriateness condition for simplicity. References Darwall S (2006) The Second-person Standpoint: Morality, Respect, and Accountability. Cambridge: Harvard University Press. Frankfurt H (1969) Alternate possibilities and moral responsibility. The Journal of Philosophy 66: 829–839. Gaus G (1996) Justificatory Liberalism: An Essay on Epistemology and Political Theory. New York: Oxford University Press. Gaus G (1990) Value and Justification: The Foundations of Liberal Theory. New York: Cambridge University Press. Gaus G (2011) The Order of Public Reason. New York: Cambridge University Press. Gibbard A (1993) Wise Choices, Apt Feelings: A Theory of Normative Judgement. Oxford: Clarendon Press. O'Neill M (2008) Freedom, fairness and responsibility. PhD Dissertation, Harvard University, MA, USA. Parfit D (2011) On What Matters: Volume I. New York: Oxford University Press. Rawls J (1993) Political Liberalism. New York: Columbia University Press. Strawson PF (2008) Freedom and Resentment and Other Essays. Oxford: Routledge. Schroeder M (2008) Having Reasons. Philosophical Studies 139: 57–71. Wallace RJ (1998) Responsibility and the Moral Sentiments. Cambridge: Harvard University Press. Williams B (1981) Internal and External Reasons. Moral Luck: Philosophical Papers 1973-1980. Cambridge: Cambridge University Press. Author biography Anthony Taylor is a DPhil candidate in political philosophy at the University of Oxford. Taylor

Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. What Is Time? Marcello Oreste Fiocco University of California Irvine Department of Philosophy Irvine, California U.S.A. [email protected] _________________________________________________________________ Article info CDD: 115 Received: 01.03.2017; Accepted: 14.03.2017 DOI: http://dx.doi.org/10.1590/0100-6045.2017.V40N1.MF ____________________________________________________________________________________________________ Keywords Time Ontology Change ____________________________________________________________________________________________________ ABSTRACT In this paper, I answer the question of what time is. First, however, I consider why one might ask this question and what exactly it is asking. The latter consideration reveals that in order to answer the question, one must first engage in a more basic investigation of what a thing, anything at all, is. Such radical investigation requires a special methodology. After briefly characterizing this methodology, I show how it can be employed to answer the titular question. This answer is significant not merely because it illuminates something of perennial interest, but because it is essential to a comprehensive and fully satisfactory metaphysics of time and, hence, to a view of the full structure in reality. ___________________________________________________________________________________________________ Despite the salience of time in one's life and its centrality to the work of scientists and philosophers, the titular question is rarely (if ever) posed. Of course, in the midst of cosmological ruminations, Augustine famously asks "What, then, is time?"1. But this is more of a rhetorical cri de coeur than the initiation of an ontological inquiry. What Augustine goes on to say-"If no one asks me, I know: if I wish to explain it to one that asketh, 1 See his Confessions, Book XI, Chapter XIV. 44 Marcello Oreste Fiocco Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. I know not."-is surely misleading. The problem is not just one of articulation. One could hardly have firm beliefs regarding such an abstruse matter as what time is prior to a good deal of metaphysical reflection. Even appreciating this, though, it is not obvious how such reflection should be directed. In this paper, I answer the question of what time is. First, however, I consider why one might ask this question and what exactly it is asking. The latter consideration reveals that in order to answer the question, one must first engage in a more basic investigation of what a thing, anything at all, is. Such radical investigation requires a special methodology. After briefly characterizing this methodology, I show how it can be employed to answer the titular question. This answer is significant not merely because it illuminates something of perennial interest, but because it is essential to a comprehensive and fully satisfactory metaphysics of time and, hence, to a view of the full structure in reality. I. Why might one ask this question? One need have no clear sense of time to wonder what it is-if one is just taking it to be closely related to change. Change occurs when a thing is one way, then another, incompatible way. All the changes one experiences or undergoes, taken together, are one's life. So, if change and time are so closely related that the former requires the latter, without time, there would be no lives. Change is the stimulus for some of the most poignant emotions associated with life, bringing about nostalgia and prompting expectation. Moreover, if it is experiencing a preponderance of agreeable changes that provides satisfaction and if striving to effect certain changes is what gives one purpose, then both the quality and the very meaning of life are bound up with change and so with time. It is natural, then, for any contemplative person to consider what this is that is so important to life, especially when one suspects these changes and, thus, this life to be finite. For these reasons, laypersons are intrigued by time. Others with more specialized interests might be as well. Chemists and biologists who study processes of various sorts and physicists who presume that the fundamental structure of the universe is to be characterized in terms of What is time? 45 Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. space-time take time for granted. They might, in a reflective moment, wonder what it is. There are also different areas of philosophy that rely on time in obvious or subtle ways. Consider the philosophy of history, which examines how what was contributed to what is and what this reveals about what will or might be. Consider action theory, in particular, questions regarding free will that examine whether one's choices now can contribute to what will be, or whether what is to come was settled long ago. There are metaphysical questions, pertaining to causation and laws of nature, that examine how things develop and meta-ethical questions about whether the rightness of an action is to be accounted for in terms of the goodness that subsequently comes from it. Surely the work of all those engaged in such inquiry could benefit from an account of what time is and how it contributes to the world.2 While it is plausible that the work of many would benefit from asking the question of what time is, this question seems crucial to the work of others. There is, after all, an area of inquiry ostensibly directed at time- the philosophy of time-in which philosophers purport to investigate, among other issues, the structure or extent of time; its topology; whether it has a direction; how things change and, hence, exist in or through time; what properties things bear just in virtue of being in time (or whether they merely stand in distinctively temporal relations); how one experiences time; whether one can travel in it; whether it in some sense "passes". It seems clear that none of these issues could be resolved without first having an account of what time is. For example, one cannot say what the structure of time is unless one has an account of what it is (and, hence, knows that time is the kind of thing that is structured). It is surprising, then, that the aforementioned issues, concerning putative features of time or matters associated with it, dominate the philosophy of time. Indeed, an examination of the voluminous literature on time shows that the primary question of what time is is no part of the disputes that exercise philosophers of time. Nonetheless, if a comprehensive metaphysics of time is to be attained, with all the insight 2 In Fiocco 2013, I argue that considering the world in time presents formidable challenges for the meta-ethical position of consequentialism. 46 Marcello Oreste Fiocco Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. into the many issues that depend on time that it would provide, it must begin with this question. II. What is this question asking? Say one agrees that the question of what time is is worth asking. Still, it might not be clear what it is asking. If the only handle one has on time is its close relation to change, and change is such a pervasive part of one's life, it might be hard to discern the focus of the question in order to begin answering it. In some cases in which one asks what something is, to be told the kind of thing that thing is suffices to answer one's question, if that kind is at the appropriate level of generality and one has a clear enough sense of what it is to be of that kind. In other cases, though, a thing might be of no specific kind, it might not be an instance of any kind more limited than the summum genus of thing, (i.e., being, entity, existent). In other words, it might be sui generis, the only thing like it, in any robust sense, in the world. Were this so, to answer the question of what that thing is would require an illuminating account of that thing itself, an account that articulates the definitive features of that thing. So the titular question might be asking for the statement of some kind or for an account of time per se, depending on whether time is an instance of some kind or is sui generis. Before determining whether it is the former or the latter, however, one must determine whether time is anything at all. It might not be obvious that it is. There is undoubtedly change, but one might be doubtful that in addition to all the conspicuous changes in the world there is something-literally a thing, to wit, time-underlying or otherwise related to these changes. Therefore, prior to the question of what time is, is the question of whether time is a thing. Of course, this question can only be answered if one has some account of what a thing (i.e., an entity, an existent, a being)-in the most general and inclusive sense-is. All inquiry is directed at something or other, more precisely, at the relations between kinds of thing or particular ones. Inquiry takes things for granted. Although knowing what a thing is might be unnecessary for successful inquiry in most domains, if one's goal is What is time? 47 Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. understanding-knowledge of what exactly it is one is investigating, whether it must be as it is and why-then one needs such an account of thing. This account would reveal what each thing ultimately is and how things in general do (and could) relate. Understanding the world or the fundamental structure in reality is presumably the goal of metaphysics in the grand sense, and so an account of thing is indispensable to it. Since any sort of familiar inquiry takes things for granted, if one is to attain an account of what a thing is, one must engage in a special sort of inquiry. One needs a unique methodology. I discuss elsewhere what seems to me to be the requisite methodology, which I dub original inquiry.3 Here it suffices merely to sketch this methodology and present its upshot. If the goal of original inquiry is an account of what a thing is, one cannot presume at the outset anything about any thing, not even that things exist. With such a severe constraint, it might seem that there is no way to proceed for it might seem that one has forfeited all means of inquiry. However, even if one eschews everything and with it any assumption that might prejudice or otherwise taint this special sort of inquiry into each thing, one is not without a focus with which to start. One still has the world: all this, the heterogeneous array encompassing one. Were one to avail oneself of the assumptions, and the concepts arising from them, that one forgoes in original inquiry, one could characterize this array as a mélange of shapes and colors and odors and textures and sounds and cognitive, conative and affective feels. But here, presuming nothing about the world-not even that "it" is a thing-the world is to be regarded merely as the impetus to inquiry. The world as the impetus to inquiry is no hypothesis, it is an indubitable prerequisite of any inquiry at all. In confronting the impetus to inquiry, one makes no assumption about what, if anything, exists. Nevertheless, this, the world, is a certain way. There must be some explanation for how the world is as it is. This is just the claim that there is some apt characterization, in terms of what exists, of this; hence, the claim should not be deemed contentious. Such a characterization would illuminate how this is as it is. Were there no such explanation, there could be no successful inquiry. Inquiry is directed either at the world at large or 3 See my "What Is a Thing?" (under review). 48 Marcello Oreste Fiocco Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. at some localized phenomenon. Successful inquiry into the latter, which would provide an account of how that phenomenon is, would depend on some account of the former, since any localized phenomenon is subsumed by the world. Therefore, any explanation of any phenomenon at all rests on some explanation of the world being as it is; the very possibility of successful inquiry requires such an explanation. If there are reasons for being dubious about the possibility of successful inquiry, they arise in light of concerns about the elusiveness of the world, the inaccessibility of things or the fallibility of minds, but such concerns have no force here, where nothing is being assumed about the world (or things or minds). In original inquiry, there is no breach between so-called appearance and reality and so skepticism has not even a toehold. An explanation for how the world is as it is must be based on something. Indeed, any explanation must have a basis: one could not have a genuine explanation that is vacuously representational, one that would explain in terms of nothing at all. There must be something or other- the explanans-that in virtue of its relation to what is to be explained- the explanandum-illuminates the latter. Such a view of explanation is hardly controversial. Thus, although confronting the world brings with it no ontological commitment, recognizing that this, the impetus to inquiry, must have some explanation does. The first ontological principle in inquiry is that there is something that underlies an account of how the world is as it is. This raises the question of what a thing must be in order to serve as the ontological basis of at least a partial explanation for how the world is as it is. As the basis of such an explanation, a thing bears some relation to the world; in this sense, it is in the world, making some contribution to all this. If it did not make the contribution it does, it would be nothing at all. Moreover, its contribution is unique. Were some other thing to make its contribution, it itself would not be making one. Consequently, there is no thing that makes another thing be or be what it is. Were there such a thing, the very existence of the latter-and, hence, the contribution it makes to the world-would be attributable to the former. In other words, were a thing made to be, "it" would be ontologically idle and, thus, nothing. The world would not be as it is, even in part, because of "it", but because of whatever is supposed to make "it" be. I have argued, on the What is time? 49 Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. basis of such considerations4, that each thing is fundamental, in that no other thing can explain the existence of a thing or what it is (since its being what it is is attendant upon its very existence). Things provide the means of explanation yet also its limits; things explain but are themselves inexplicable, that is, they are not even susceptible to explanation. A thing contributes to the world as the basis of at least a partial explanation for how the world is as it is. Thus, each thing must be some way(s) or other. Were a thing no way at all, there would be nothing to it, in which case, it could not serve as the ontological basis of any explanation. What a thing is determines, to a significant extent, how it is; how it is are the ways it is. Since nothing explains what a thing is and, hence, the ways that it is, each thing just is what it is. A thing cannot exist without being what it is; its very existence, then, requires that it be at least some of the ways it is. Therefore, each thing is constrained in its being, it must be certain ways just in virtue of existing. In this sense, each thing is natured. If there be familiar concrete objects, universal properties (i.e., attributes), particular properties (i.e., modes or tropes), relations, universal kinds, numbers, propositions, processes, facts, states of affairs, sets, holes, boundaries, privations-what have you-each is natured. It is misleading, however, to say that each thing has a nature, for this suggests that a nature is a thing (something to be had) in virtue of which a thing is or is what it is. But, as argued above, there can be no such thing that explains or grounds another. For similar reasons, it is misleading to talk of the essence of a thing. There are no essences, though each thing is, nonetheless, many ways essentially. Although everything must be certain ways (depending on what it is), many things can also come to be and cease to be other ways they do not have to be. These other ways, though, must be consistent with those ways a thing must be just in virtue of existing at all. A thing, therefore, is a natured entity. Elsewhere5, I consider whether the obvious circularity of this account of thing is problematic, and conclude 4 See my "Each Thing Is Fundamental: Against Hylomorphism and Hierarchical Structure" (under review). 5 See my "What Is a Thing?" (under review). 50 Marcello Oreste Fiocco Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. that it is not. The account is informative in light of the world, the ineluctable impetus to inquiry. III. How should one go about answering it? As noted in the previous section, prior to the question of what time is, is the question of whether time is a thing at all. I then presented an account of what a thing is. If there is any insightful account of time itself, time must be a natured entity. One must determine, then, whether this is the case. Presumably any grounds for maintaining that time is indeed a thing would also inform the account of what it is. To look for such grounds, one might begin by examining how time is regarded, what it is taken to be. Most who regard time do not go beyond the sense, acknowledged above, that it has something to do with change. This sense alone, obviously, is too vague to reveal much of anything about time per se. If pressed, a layperson might offer an observation like time is what makes one grow old or what clocks measure. But what makes one grow old are the features of one's organs or cells; it is their limited durability, that is, the incapacity of these things to preserve their integrity as they interact with other things and undergo their characteristic changes. It is not time itself that makes one age, it is the changes one undergoes given what one's body, organs, and cells are. If time is relevant here at all, it is because it has something to do with change, and so this homely suggestion about time and growing old provides no more insight into time than the vague sense with which one began. Clocks are just machines with cycles coordinated with cycles in other things, usually naturally occurring ones. If clocks measure anything they measure these other cycles. A cycle is just a reoccurring series of changes in some thing (or things). So if a clock is simply a means of marking such changes, then, again, one has a suggestion about time that provides no more insight then one had to begin with. The claim that a clock does not measure only cyclical changes, but also what those changes take place in presents an helpful spatial metaphor. Until one has grounds for maintaining that time is a thing-and a literal account of what it is What is time? 51 Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. based upon such grounds-no metaphor can really be illuminating. Such grounds do not seem to be forthcoming from casual reflection on time. One might think, then, that the cogitations of experts would be more promising. However, as mentioned above, the question of what time is is not central to the philosophy of time. This claim might be perplexing, and not merely because this question is the most obvious one to try to answer if one is interested in the philosophy of time. The claim might be perplexing also because discussions of the metaphysics of time are almost always presented in a context in which there are supposed to be competing theories of time. Every issue is approached in light of the A-theory versus the B-theory or the tensed theory versus the tenseless theory or the block theory versus the passage theory or presentism versus eternalism, etc. Indeed, when considering the philosophy of time at large, one is confounded by what appears to be a host of theories of time. Although there are these many so-called theories of time, those who propound them never make explicit what it is exactly they are theories of. Still, even if they are there only tacitly, one might hope, by examining such theories, to uncover grounds that indicate whether time is a thing and some insight into what this thing is. The first point to note when considering these theories is that none of the terms used to name them- 'A-theory', 'the tenseless theory', 'presentism', etc.-is univocal. Each has been used for distinct theories (some of which are incompatible). Suppose though that the (or an) A-theory is a view according to which there is an infinite series of moments, each one of which bears an absolute monadic temporal property being past, being present, or being future, in addition to standing in binary temporal relations, to wit, earlier than and later than, to other moments. This theory can be contrasted with one, the B-theory, on which there is this infinite series of moments, but the moments only stand in binary temporal relations. Both are theories of what (temporal) properties or relations moments, in an infinite series, bear. Suppose the (or a) tenseless theory is a view on which things do not come into being, but merely exist permanently at some temporal location; whereas the (or a) tensed theory is a view on which things do, in some sense, come into being. Both of these are theories of how things exist, or how they come to exist. Not one of these four theories is about time per se and so they 52 Marcello Oreste Fiocco Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. are of little help in determining whether time is a thing. The same holds for other putative theories of time. Nevertheless, given the theories just considered, one might surmise that what time is is a plurality of related moments, either an infinite series of them (each of which might or might not bear an absolute monadic temporal property) or the totality of temporal locations comprising some lesser array (in which things exist with or without becoming).6 Setting aside that it is by no means clear what a moment is, if this is what time were, time would not be a thing. A plurality, which is many, is not any one, natured entity. (This is not to say that a natured entity cannot be complex; a complex thing is one yet with many parts.) There is nothing that makes a thing be or be what it is; each natured entity just is and is as it is essentially because it is. However, a plurality is made to be (and made to be what it is) by the many things that compose it, and a plurality is as it is because these things are as they are. A plurality makes no contribution to the world beyond those made by the things it includes. If there were some way a seeming plurality were that made such a contribution, this would be some indication that that "plurality" were in fact a thing. Such a way would have to be more than some contrived feature, like being multitudinous or being multi-located, based on the presumption that the plurality is indeed a unified thing. If time were a plurality of moments, there is no obvious feature of the world that could be explained by it rather than those moments. It is, however, not tenable to maintain that time is just a plurality of moments and so nothing in itself. A significant position in the metaphysics of time is presentism. Although there are a great variety of presentist views, the most promising ones deny that there is any array of moments. Yet proponents of such views do not reject time itself. Any such presentist accepts that time exists just as much as any Aor B-theorist or tensed or tenseless theorist, etc. What all these metaphysician of time agree upon-and it might be the only thing-is that time is real. What 6 Some make a claim like this one explicit. See, for example, Tallant 2013: 372 for the assertion that, on the B-theory, the "reality of time consists in nothing more than various objects standing in the 'fixed and permanent' relations earlier than and later than." What is time? 53 Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. they disagree about are the things associated with time or the relations, the structure, among these things. Thus, all these purported theories of time are actually theories of what can be construed more broadly as temporal reality, each of which takes for granted the existence of time. The questions remain, though, of what it is each is taking for granted and what motivates all these controversial theories of temporal reality. There is a long-standing debate in the metaphysics of time, one predating the modern proliferation of theories of temporal reality, that seems to pertain to the key question of whether time is a thing. This debate, concerning relationism versus substantivalism, is supposed to be about whether time is a substance and, hence, a thing. According to relationists, time is nothing distinct from all the occurrences that take place in the world and the relations among them. According to substantivalists, time is a thing in addition to all these occurrences; in particular, it is the thing in which these occurrences take place and exists independently of them. Relationism cannot offer any insight into time per se. On this view, time is, at most, a plurality: all occurrences, that is, all changing things and the relations among them. As such, time is nothing in itself. Substantivalism does not provide any more insight. Although on this view, time is supposed to be a thing, the account of what it is-the "container" or "arena" in which all occurrences take place-seems to be just (a spatial) metaphor. A literal account of what this "container" is, if indeed there is one, would presumably characterize it as an array of moments. Setting aside, again, that it is not at all clear what a moment is, an array of anything (moments or what have you) is merely a plurality, not anything in itself. On neither side of the debate, then, is time really a natured entity. This debate between relationists and substantivalists is often motivated with the question of whether time could exist in the absence of any occurrences or changes. If the answer to this question is negative, then time is supposed just to be any occurrences (or changes) and their relations. (But this does not follow-even if time could not exist without occurrences, it need not be those occurrences.) If the answer is affirmative, time is supposed to be something, whatever it is, distinct from all those occurrences. Regardless of one's answer, however, I want to point out the futility of raising this question prior to having an account of what time is or being able to say whether it is a thing. Without such an account (or 54 Marcello Oreste Fiocco Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. argument), one is no position to determine whether time must exist with changes or could exist in their absence. In general, without some clear sense of what a thing is, one cannot say what that thing must exist with or can exist without. As observed above, one's untutored sense of time is that it has something to do with change, but this sense is surely not sufficient to determine whether it is something that must exist with changes or is really anything at all. Hence, this debate between relationists and substantivalists reveals nothing about time, neither what it is nor even its status as a thing. One does not get an answer to the question of whether time is a thing by considering what laypersons might say about it, nor even by considering the voluminous literature produced by philosophers of time. Without any grounds for maintaining that time is a thing, one can hardly answer the question of what time is. One needs, therefore, a different strategy to get purchase on time, one that goes beyond casual reflection-consulting one's "intuitions"-or examining the work of experts. If time is real and is as vital to the world, and one's experience of it, as it seems to be, one should expect to find some indication of this thing in original inquiry, that is, when confronting the world without presumption. Any such indication should provide the means to illuminate the sense that time has something to do with change. In confronting the world simply as the impetus to inquiry, one is indeed impressed by a world that can be as it is only if it includes a thing plausibly taken to be time. To answer the question of what time is, therefore, one must begin here, with original inquiry. IV. The Answer When one undertakes original inquiry, one encounters a heterogeneous array: the world that is thus. If one undertakes original inquiry again, one encounters a distinct heterogeneous array: the world as so. The world is thus and (then) so. There are, then, two modes of differentiation in the world. The heterogeneity apparent when it was just thus and the heterogeneity apparent in its being thus and (then) so. The first mode of differentiation can be accounted for simply in terms of the existence of What is time? 55 Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. distinct of things. This second mode is no less incontrovertible than the first, so it, too, must have an explanation and, hence, an ontological basis. Call the natured entity underlying the explanation for this second mode of differentiation time. This is, then, a preliminary answer to the titular question. Time is that thing that accounts for the world being differentiated in the distinct ways that it obviously is. This initial characterization is fitting because if change genuinely occurs in the world, it does so via this second mode of differentiation; this second mode is needed for there to be change. Thus, in a way to be elaborated, time is the thing that makes change possible. The sense that change has something to do with time is thereby corroborated, for there is this very close relationship between time and change. The better one illuminates change and how it is made possible, the more insight one will have into time per se. So consider again change. Above, I characterize it as the phenomenon that occurs when a thing is one way, then another, incompatible way. Although, there have been those in the history of philosophy who have denied that there is change, I maintain that it must be real. One experiences change confronting the world in original inquiry, that is, when making no assumptions about anything. Distinct modes of differentiation are presented. Either the world itself changes (going from thus to so) or, if the world is not a thing (and I do not think it is: the world is a plurality, namely, all things), there is change in the inquirer, who encounters the world as thus and (then) as so. To deny change requires some argument ulterior to original inquiry. Any such argument or, for that matter, any argument at all itself requires change in some inquirer. Premises must be presented sequentially, then considered (sequentially) and then some inference drawn. Given that there is compelling reason to accept change-one experiences it in original inquiry-and that no argument can undermine it, I maintain change indeed occurs. Since change is real, there must be some explanation for it and this explanation must have an ontological basis among the things in the world (for there is, of course, nothing else). Change requires incompatible ways. No one thing is incompatible in itself. Change, therefore, is no one thing. Rather, change is what can be called a structural phenomenon, some plurality of things in relation. If something changes, that thing is incompatible 56 Marcello Oreste Fiocco Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. ways, that is, bears inconsistent properties. Since, again, nothing is incompatible in itself-a thing that were both p and not-p would be inherently contradictory and so is impossible-and yet there is change, there must be some thing distinct from the thing that undergoes change that accounts for how the former is incompatible ways. Call whatever it is that accounts for how one thing is incompatible ways a moment. Thus, when change occurs, something is one way at one moment and that very thing is an incompatible way at a distinct moment. This is an elucidation of the plausible account of change adopted above, specifying the basis of the phenomenon in the world of things. It reveals that change is several things in relation: at least one mutable thing, two properties and two moments. Time is the thing that makes change possible; change requires moments. This raises the question of how time is related to moments and what it is that can both underlie the explanation for the second mode of differentiation in the world and also be so related to moments. Answering this question would illuminate what exactly this thing time is. I maintain that there must be some explanation for the second mode of differentiation evident in the world and, furthermore, that time is the ontological basis of this explanation. Yet I also maintain that there is change and, hence, that there must be distinct moments. If the existence of more than one moment underlies an account of change and change is just a manifestation of this second mode of differentiation-call it temporal differentiation-it might seem that moments suffice as the ontological basis of an explanation for temporal differentiation. Distinct moments are not a thing, they are merely a plurality of things, and so one might claim that it is misguided to posit something-time itself-as the basis for such an explanation. Thus, given the existence of moments, time might seem superfluous. This is a reoccurrence of the idea, to wit, that time really is not anything at all, just an array of moments, floated and rejected above. One might see in the foregoing considerations new motivation for this idea; however, these considerations also provide the means to argue against it more conclusively. More is needed to account for the structural phenomenon of temporal differentiation and change than merely distinct moments. These phenomena also require that moments be related in a certain way. For, say, change to occur, a thing must be one way at one moment and an What is time? 57 Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. incompatible way at a distinct moment that succeeds the first (or, conversely, a thing must be one way at a moment that precedes another moment at which it is an incompatible way). This structure-this plurality of things in constraining relations-comprising moments requires explanation. Just adding the relation later than (or earlier than) to the complex of mutable thing, incompatible properties and moments will not do, for something is needed to account for how this relation is related to these other things in the appropriate ways (a Bradleyian regress threatens here). Time is supposed to be the natured entity that underlies the explanation for the second mode of differentiation in the world, and so time is this needed thing. It is time that accounts for how the world encountered in original inquiry is thus and then so. In other words, it is in virtue of time that one moment is appropriately related to another by later than (or earlier than). Given change, there is no denying that moments are related in a distinctive way, one that need not inhere in them as instances of that kind of thing that enables a natured entity to be incompatible ways. Time, therefore, is the ontological basis of an explanation for what is naturally called the temporal order of moments. Note that the explanation for the order of moments, this distinctively temporal structure, cannot be based on any moment itself. A moment is simply a natured entity that enables something to be incompatible ways. If one moment were to exist at another, and a thing can be incompatible ways at distinct moments, then it would be possible for something to be incompatible ways at the same moment. This is impossible, so distinct moments are mutually incongruous; each moment excludes every other. Nothing more seems to follow regarding the relations among moments merely considering a moment per se. Yet if there are distinct moments, as there must be given temporal differentiation and change, these moments must be ordered. Time is the thing that accounts for this order. Considering moments, there is no reason to think any particular moment must exist, and much reason for thinking each moment ceases to be.7 Yet, given that there is change, there must be some moment(s) or other(s). The necessary presence in the world of some moment(s), though any moment can fail to exist, is a structural phenomenon that requires an 7 See Fiocco 2014a. 58 Marcello Oreste Fiocco Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. explanation. What is needed here is a sort of natural-cum-causal explanation, an account of how one thing comes to be given the existence of another. It is not necessary to explain how a particular moment is. There could be no such explanation, for there is no explaining a thing or its existence in this sense. Nevertheless, how there are instances of a certain kind at all is amenable to explanation, and such an explanation is somewhat pressing if there is reason to think that there must be some instance of that kind though each one ceases to be. The source of any moment could naturally determine the order of moments. Therefore, since time is the thing that accounts for this order, it is plausible that time is also the source of each moment, the thing that underlies an explanation for the presence in the world of any such thing. This conclusion confirms the claim that time is more than merely a plurality of moments. The world comprises things standing in constraining relations and time is one of these things. Given this, time makes a contribution to the world that no other natured entity does. It is the ontological basis of an (at least partial) explanation for the second mode of differentiation in the impetus to inquiry, accounting for how the world is thus and then so. As such, time is the thing that enables change; it is in virtue of time that anything whatsoever changes. Change requires moments ordered in a particular way. So time enables change by being the source of any moment, as well as the thing that orders them. This, then, is the answer to the question of what time is: It is the natured entity that is the source of any moment and what orders any moment(s). V. What hangs on this answer? I have answered the question of what time is. Since it seems clear from the foregoing discussion that time is sui generis, rather than an instance of some kind, I have answered this question by articulating the definitive features of time. This answer has some value. First of all, it makes clear what those who muse upon time or worry about it are musing upon or worrying about, namely, the thing that makes change possible, that thing that permits growth and decay, gain and loss. The answer also illuminates what is or, at least, should be the focus of the philosophy of time. Time What is time? 59 Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. is the thing that underlies any temporal phenomenon, and so all accounts of these phenomena must ultimately be based on the thing that gives rise to and orders moments. More importantly, however, I maintain that it is only by recognizing that time is a thing itself-and, more precisely, the one that makes change possible-that one can develop a comprehensive and fully satisfactory metaphysics of time. Investigations of the metaphysics of time are universally taken to reveal an irreconcilable tension. This tension is characterized in different ways. Thus: [There are] two fundamentally different ways in which we conceive of and talk about time. On the one hand, we conceive of time in a dynamic or tensed way, as being the very quintessence of flux and transiency...[On the other hand,] is the static or tenseless way of conceiving time, in which the history of the world is viewed in a God-like manner, all events being given at once in a nunc stans.8 And Distinctions and transitions of tense, between what has been, is and will be past, present and future, divide philosophers into two fundamentally opposed camps. The one, 'tensed', camp takes these distinctions to reflect real nonrelational differences between past, present and future things (events, facts, etc.)...that is what [those] in the 'tenseless' camp deny.9 More recently and succinctly: Contemporary analytic metaphysics of time is shaped by the debate between A-theorists and Btheorists.10 And 8 Gale 1967: 65-66. 9 Mellor 1981: 4. 10 Deng 2013: 713. 60 Marcello Oreste Fiocco Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. Metaphysical theories of time divide into A-theories and B-theories.11 These characterizations of the tension do not make clear what its source is and, hence, it is not obvious what the disputes arising from this tension are really about. Although the characterizations suggest-or in some cases explicitly state-that the tension pertains to time, or different theories thereof, the lack, in these investigations, of any account of what time itself is belies this. Perhaps the only claim that can be taken for granted in contemporary discussions of the metaphysics of time is that time exists.12 There is, then, some common ground among those investigating the metaphysics of time. If any progress is to be made from this common ground, an uncontroversial account of what time is is required. The account I propound is not based on any contentious assumptions. Indeed, it is not based on any assumption at all. It arises through original inquiry, by encountering the world as the impetus to inquiry, noting that it is differentiated in distinct modes and recognizing that this phenomenon must be explicable in terms of what exists. Because it arises in this way, and is so minimal, it is hard to see how anyone could deny this account of time, or why anyone would want to. So here is the required account. If this is what time is, however, the tension that is supposed to be central to investigations of the metaphysics of time vanishes. Time per se is no series, so a fortiori it is neither an A-series nor a B-series; it is itself neither tensed nor tenseless, transient nor static. Harmony is achieved. Yet such quick resolution of these hoary disputes is hardly satisfying. What this dissatisfaction indicates is not that there is some problem with this ecumenical account of time; rather, it shows that these disputes ostensibly about time (or theories thereof) actually have a different focus. If one accepts that there is time and at least one moment, one can see that these disputes are more about these moments of time (and the things that exist at them); they concern the properties of moments, their relations, the extent of them. In other words, the disputes are about the 11 Deasy 2016. 12 McTaggart, of course, denied this. See McTaggart 1908. What is time? 61 Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. relations among distinctively temporal entities (moments, temporal properties, temporal relations, etc.), i.e., the structure in temporal reality. Thus, there is, among others, a dispute concerning whether moments have absolute monadic temporal properties and stand in temporal relations or just stand in temporal relations; a dispute concerning whether moments (and the things that exist at them) come into being or exist permanently at some temporal location; a dispute concerning whether there is more than one moment with the same ontological standing; a dispute concerning whether the properties of moments make them dynamic in some sense. There are several disputes here, not one of which pertains to time per se, and not one of which is primary in the sense that one must accept one of the disputed positions and one's choice there determines one's position on every other dispute. None of these disputes, therefore, reveals the central tension in investigations of the metaphysics of time. In fact, these disputes might seem puzzlingly trivial. One might wonder why anyone would care to argue about, say, the properties of moments or how many there are. What is missing is a view of the real bone of contention in discussions of the metaphysics of time, what it is that motivates any of these disputes in the first place. By examining the disputes together, rather than the details of any one of them, one can see that each arises in connection with a phenomenon that all those who investigate the metaphysics of time acknowledge. Regardless of how one identifies (A-theorist, B-theorist, tensed theorist, tenseless theorist, presentist, block theorist, etc.) one concedes that there is a compelling impression of novelty arising from one's experience of being in a world with time. This impression has traditionally been characterized as a sense of passage or flow and is really nothing more than an awareness of the second mode of differentiation in original inquiry-that awareness of the world being thus and then so-accounted for in terms of time itself. Accompanying this impression of novelty, however, is one of permanence that is no less evident. Given that the world is now thus, it seems that this never changes, that it must always be the case that the world is thus at this moment. Yet, obviously, things do change for the world is now so. This last observation need not be incompatible with it always being true that things and, hence, the world were a different way a moment ago. Thus, the real bone of contention in investigations of the metaphysics of 62 Marcello Oreste Fiocco Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. time, the motivation for all the disputes one encounters therein, is how to reconcile the undeniable impression of novelty in the world with the commensurately compelling impression of permanence. Some metaphysicians of time take more seriously the permanence, and try to account for the impression of novelty by the means they employ to explain this permanence. Others take more seriously the novelty, and try to account for the impression of permanence in terms of what underlies this novelty. There is a divide, then, between those who take the world to be fundamentally permanent and those who take it to be fundamentally transient. This conflict between permanence and transience is supposed to be unavoidable when investigating the metaphysics of time and, as illustrated by the quotations above, usually provides the framework for these investigations. Acquiescing to this framework has led to impasse. A comprehensive metaphysics of time provides the means to account for every temporal phenomena. Insofar as both transience and permanence are irrefragable phenomena, a fully satisfactory metaphysics of time accounts for both without neglecting either. Thus, a comprehensive and fully satisfactory metaphysics of time would resolve the tension between the transience and permanence in the world. Since no thing in itself is both transient and permanent, if the central disputes regarding the metaphysics of time concerned a single thing-time or the world, were it a thing13- the tension giving rise to these disputes would indeed be irresolvable. Transience would have to be accounted for haphazardly in terms of things best-suited to explain permanence or vice versa; there could be no fully satisfactory metaphysics of time. This is why it is so important to recognize what these disputes are really about. They are not about time alone, but about time and the structure it renders. Realizing this presents the prospect of finding a place for both transience and permanence in this structure and, hence, in the world more generally. The key to a fully satisfactory metaphysics of time is distinguishing between the world in time and the world outside of time. This spatial locution is merely suggestive; the operative notion of inclusion has not to do with space, but with ontological dependence, the necessary relations between things 13 As mentioned above, I do not think the world is a natured entity. Rather, it is a plurality of things, to wit, all things. What is time? 63 Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. given what those things are. As a thing, time is in the world. There are some things that are ontologically dependent on time, in that, given what they are, they cannot exist in the absence of time. These things together, this plurality, is what above I dub temporal reality, the world in time. There are, however, other things that are ontologically independent of time, in that they do not have to exist with time merely given what they are. The plurality of these things is atemporal reality, the world outside of time. It is here that having an account of what time is is imperative. The account provides the insight required to elaborate the distinction between temporal and atemporal reality. Time is the thing that makes change possible by yielding and ordering moments. Any moment, then, ontologically depends on time. The mark of existing in time is existing at a moment thereby poised to change. Any mutable thing must exist at a moment and so any such thing is ontologically dependent on time. Transience involves change in a way consistent with the compelling impression of novelty. Therefore, one can expect the transience in the world to be accounted for in terms of those things that ontologically depend on time. Those things that do not ontologically depend on time do not, cannot, exist at a moment (otherwise they would depend on time) and so these things do not (cannot) change. One can expect, then, the permanence in the world to be accounted for in terms of these atemporal things. If the real bone of contention in investigations of the metaphysics of time is how to reconcile the indisputable impressions of novelty and permanence, then by recognizing the world in time and the world outside of time one can discern the proper domain of transience and permanence, and so do justice to both, thereby allaying any tension between them. Crucial to recognizing both the world in time and without it-the full structure in reality-is having the proposed account of what time is and appreciating the distinction between time itself and temporal reality. An account of what time is is essential to a comprehensive metaphysics of time, for it provides its foundation; furthermore, as just observed, such an account is of the utmost importance in articulating a fully satisfactory metaphysics of time. Yet there is still much to be done to attain this metaphysics. One must give an account of the structure in temporal reality consistent not only with change, but with the impression of novelty. One must give an account of atemporal reality, what exists outside of time, in 64 Marcello Oreste Fiocco Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. terms that illuminate the permanence of the world. I have started these projects elsewhere, providing a presentist account of the world in time14 and an account of the simple facts that structure the world without time and, thus, timelessly ground one's true claims about all the things that change and cease to be15. The purpose of this paper, though, is to demonstrate that one cannot have a genuine sense of the basis of or the motivation for modern discussions of the metaphysics of time without first having an account of what time is. Without such an account, there is no way to move beyond the familiar disputes-and the impasse-in these discussions. However, with such an account, there is the promise of a comprehensive and fully satisfactory metaphysics of time. References AUGUSTINE. Confessions, London: Penguin Books, 1961. DEASY, D. Philosophical Arguments Against the A-Theory. Pacific Philosophical Quarterly, 2016. DOI: 10.1111/papq.12151. DENG, N. Our Experience of Passage on the B-Theory, Erkenntnis, 78: 713-726, 2013. FIOCCO, M. ORESTE. Becoming: Temporal, Absolute and Atemporal, in Debates in the Metaphysics of Time, edited by L. Nathan Oaklander (Bloomsbury Press, 2014), pages 87-107. ______ On Simple Facts, Res Philosophica 91: 287-313, 2014b. ______Consequentialism and the World in Time, Ratio 26: 212-224, 2013. ______A Defense of Transient Presentism, American Philosophical Quarterly 44: 191-212, 2007. 14 See Fiocco 2007. 15 See Fiocco 2014b. What is time? 65 Manuscrito – Rev. Int. Fil. Campinas, v. 40, n. 1, pp. 43-65, jan.-mar. 2017. GALE, R. (ed.) The Philosophy of Time (New York: Doubleday and Company), 1967. MCTAGGART, J.M.E. The Unreality of Time, Mind, 18: 457-484, 1908. MELLOR, D.H. Real Time (Cambridge: Cambridge University Press), 1981. TALLANT, J. Recent Work: Time, Analysis 73: 369-379, 2013.

This is a post-peer-review, pre-copyedit version of an article to be published in Journal of Archaeological Method and Theory. The final authenticated version will be available online at: http://dx.doi.org/10.1007/s10816-018-9378-y. Materiality and Human Cognition Karenleigh A. Overmann*1and Thomas Wynn Center for Cognitive Archaeology, Department of Anthropology, University of Colorado, Colorado Springs In this paper, we examine the role of materiality in human cognition. We address issues such as the ways in which brain functions may change in response to interactions with material forms, the attributes of material forms that may cause change in brain functions, and the spans of time required for brain functions to reorganize when interacting with material forms. We then contrast thinking through materiality with thinking about it. We discuss these in terms of their evolutionary significance and history as attested by stone tools and writing, material forms whose interaction endowed our lineage with conceptual thought and meta-awareness of conceptual domains. Keywords: materiality, writing, stone tools, cognitive evolution, Material Engagement Theory In a recent science-fiction movie (Villeneuve, 2016; also see Chiang, 2002), humans learn to communicate with an extraterrestrial species. The plot draws upon ideas from neuroscience and linguistics to suggest that immersion in the alien language would change how humans perceive time: acquiring a second language involves neural change (Abutalebi, 2008); language influences or determines thought (Sapir, 1929; Whorf, 1940); language affects how time is conceived (Whorf, 1950). In emphasizing the consequences of mastering an alien language, however, an important point is conspicuously missed: The characters also interact with an alien material culture (i.e., its writing). While the time-travel effects that result are the stuff of fiction, the idea that brains can be changed by interacting with material forms is not. Rather, it is both something we do every day and implicit to our evolutionary history. For example, learning to read and write is an interaction with a material form that changes functionality in the fusiform gyrus (the part of the temporal lobe that recognizes objects), Broca's and Wernicke's areas (the main centers for producing and comprehending language), and Exner's area (the part of the brain active in handwriting) (Overmann, 2016a). The Neolithic peoples who first realized literacy from the behavior of writing adapted a material form that would eventually yield unprecedented access to and meta-awareness of human conceptual domains (Olson, 1994; Olson & Cole, 2006; Watson & Horowitz, 2011). And species who were our remote ancestors interacted with stone tools in ways that may have produced conceptual thought in the first place (Coolidge & Wynn, 2018). Materiality's influence on human cognition far exceeds its acknowledged role in offloading and storing mental content (d'Errico, 1998). This is not often recognized, for reasons that include the incremental pace and long temporalities involved in co-influential change between brains and material forms. Here we examine what changes in the brain when it interacts with material forms like writing and stone tools, what it is about such material forms that can cause the brain to change, * Correspondence concerning this article should be addressed to Karenleigh A. Overmann, Center for Cognitive Archaeology, Department of Anthropology, University of Colorado, Colorado Springs, 1420 Austin Bluffs Pkwy, Colorado Springs, CO 80918 USA; e-mail [email protected]. and how long it takes brains to reorganize when they interact with these forms. We consider the kind of theoretical framework needed to analyze co-influence between brains and material forms. We discuss thinking through materiality (i.e., incorporating it into our cognition; adapting it through long-term use to become increasingly efficient at eliciting specific psychological, neurological, and behavioral responses; and using it to recreate those changes in newly indoctrinated individuals; Overmann, 2017) and why we often fail to notice its role in cognition, effects of embodied skill and behavioral automaticity that free up attentional resources for other purposes. We contrast thinking through materiality with thinking about it (i.e., forming and manipulating concepts) and explain why thinking about materiality is both wonderful and strange from an evolutionary perspective. We end by reviewing aspects of the archaeological record that suggest the emergence of these abilities: stone tools and writing, the two material forms that have arguably had the greatest influence on the development of the human capacity for conceptual thought. Investigating co-influence between brains and materiality requires a theoretical framework that puts them together as a system, rather than treating them separately. This perspective is provided by Material Engagement Theory (Malafouris, 2013), a theoretical framework in cognitive archaeology in which cognition is viewed as influenced by being in a body (embodied; Lakoff & Johnson, 1999; Prinz, 2009) and situated in an environment (embedded; Smith, 1999); as comprising a system that includes the body and materiality as constitutive components (extended; Clark, 2008; Clark & Chalmers, 1998); as consisting of the dynamic, transformative interactivity among the components (enactive; Hutto, 2013); and as possessing an evolutionary history that continues to unfold (evolving; Malafouris, 2015). For its part, materiality is envisioned to influence human behavior and psychological processing (i.e., materiality has agency); however, it is also acknowledged to have different capacities, potentials, and mechanisms for influencing brain and body and changing in response to their influence than the other components. Materiality is also seen as having and acquiring meaning in virtue of what it is and what humans do with it (what Malafouris calls enactive signification). Applying Material Engagement Theory starts by viewing human cognition as a dynamically interactive system that includes, in addition to brains, bodies and material forms. A systemic view of visual perception, for example, makes it a cognitive state that emerges from the interaction of material stimuli, neural reactions, and physical movements (Gibson, 1977). Humans alter the system by adjusting its components, typically through behaviors with material forms. We are a species that manipulates material forms to produce specific behavioral and psychological responses. An example of this is music. Players of musical instruments produce sounds that elicit physical and emotional responses in those hearing them. Finally, consider the material forms themselves: They are the result of generations and sometimes centuries or even millennia of cooperative effort that has designed and refine them toward producing specific responses, effort often expended without any guiding idea of the behavioral, psychological, or material changes that might ultimately result. They embody and make available accumulated knowledge that functions to decrease the cognitive effort of future generations (Hutchins, 1995), who need merely learn how to use the object (i.e., not reinvent it from scratch), and perhaps extend its application and refine its design. Material forms and the body are not just causally linked but constitutive of cognition (Malafouris, 2013). Reading is a good example of this, as it is a cognitive state that requires a material form, writing, and the behaviors and neural reactions occasioned by its engagement. Indeed, beyond the neural activity occurring in the brain, without words on the page and the eyes' movements over them, a person cannot be said to be reading. Similarly, in stone knapping, the "decision about where to place the next blow, and how much force to use, is not taken by the knapper in isolation; it is not even processed internally. The flaking intention is constituted, at least partially, by the stone itself ... [which], like the knapper's body, is an integral and complementary part of the intention to knap" (Malafouris, 2010a, p. 17). As reading and knapping cannot be reduced to neural activity, nor writing and stone to perceptual stimuli, it is through the active engagement of materiality that such cognitive states are brought forth and the agency of bodies and material forms revealed. However, we grant there are differences in the degree and kind of cognitive contributions made by bodies and material forms: the pen one writes with, and the chair one sits in to write, contribute differently than the written characters produced with them in matters like the amount of sustained attention they receive, the degree to which they engage bodily movement and require embodied skill, and so on. When a cognitive state at one time (C1 at t1) is compared to another at a different time (C2 at t2), any differences between the two states imply that the psychological, behavioral, and material components have changed through their interaction. Specific ontogenetic changes in behaviors and brains are associated with literacy (Dehaene et al., 2010; Nakamura et al., 2012). Less apparent on the ontogenetic timescale is change in the material form, something for which multiple generations may be required. For example, over some 1500 years between the mid-4th millennium and 2000 BCE, Mesopotamian writing changed from signs that conveyed semantic meanings through their resemblance to objects into signs that conveyed both meanings and sounds but no longer resembled the objects they once depicted; this change in form was enabled by change in behaviors and brains, and in turn it influenced further change in both-for example, by intensifying the need for formalized instruction and effectively selecting practitioners into specialized communities with distinct identities (Overmann, 2016a). It is in this temporally laden sense that we use the term "coinfluence" to describe the ability of material forms to change behaviors and brains. We then use archaeologically attested change in material forms to infer related change in behaviors and psychological processes. The ongoing change and transformative capacity of neurons, behaviors, and material forms is "metaplasticity" (Malafouris, 2010b). Here we highlight two aspects of this key idea. First, bodily movement is implicit to everything from moment-to-moment sense-making to cognitive change over time. In perception, movement mitigates the fact that unchanging or overly similar stimuli yield desensitization and habituation. In cognitive change, movement affords the continual engagement of and adjustment to the material forms that comprise our cognitive ecologies (Malafouris, 2013). Second, the interactivity of the neural, behavioral, and material aspects of our cognition extends their inherent plasticity beyond the range endowed through mechanisms like genetics or physics. Thus, humans do not create and use material forms because the species has special brains; rather, humans are a species whose cognition has reached its present state by engaging material forms, past and present. Brain Change, Material Change, and Temporality Literacy nicely illustrates the kinds of things that can change in the human brain when it interacts with a material form, as well as its potential to function in ways that evolution did not specifically equip it to do. Today when someone learns to read and write, the fusiform gyrus, which has an evolved function for recognizing objects through combinations of their local and global features, becomes trained to recognize written characters through their features (Cohen & Dehaene, 2004; Vogel, Petersen, & Schlaggar, 2014). It also becomes trained to interact with Wernicke's and Broca's Areas for comprehending and producing speech and Exner's Area for controlling handwriting (respectively, gyri in the superior temporal, inferior frontal, and middle frontal lobes; Pegado, Nakamura, & Hannagan, 2014). The behavioral component, handwriting, improves hand– eye coordination, fine motor control, the ability to recognize written signs, tolerance for ambiguity in their formation, and recall of the written material (James & Engelhardt, 2012; Longcamp, Zerbato-Poudou, & Velay, 2005; Mueller & Oppenheimer, 2014; Sülzenbrück, Hegele, Rinkenauer, & Heuer, 2011), all of which imply neurological change. Reading and writing are often considered as a mode of language rather than an interaction between brains, bodies, and a material technology. However, the behavioral (production) and material (product) aspects of reading and writing are critical to understanding their effects on the human brain. The material aspect is particularly critical, given the inseparability of looking at written material to understand its meaning and the written form itself. That is, as a cognitive activity, reading does not exist without writing-the letters, syllabograms, and logograms made accessible to vision and touch by material forms like clay, papyrus, paper, computer screen, sign language, and Braille. Someone can recall information gained through reading, but this differs from what occurs when signs on a page are read. Simply, reading is the interaction of psychological processes like vision and language with the material form that is writing through the behaviors like handwriting that interface them. Today, reading involves a material form that has become highly adept at eliciting specific behavioral and psychological responses in both novice and fluent readers. At writing's origins, however, neither material form nor psychological processing supported literacy in the way we understand it. Neither could there have been any idea of what would ultimately be realized once people began handwriting simple characters with conventionalized meanings. This behavior was transformative, however, as it occasioned change in both the brain processes involved in reading (described above) and the material form instantiating writing (described below). As behavior, writing represented an interaction between psychological processes, the body, and material forms. As a material form, it instantiated sequences and patterns of lines and curves that cumulatively resembled and thus denoted physical objects. These sequences and patterns were visually perceivable objects whose associated cultural and linguistic meanings were intelligible and thus communicable between individuals. In conjunction with material attributes like durability, the communicative value of signs also intensified the behavior, opening up multiple cascading opportunities for further change in brains and the material form. Adapting early writing into a form capable of communicating language fluently and influencing psychological and behavioral changes efficiently required the participation of many individuals over multiple generations. In Mesopotamia, one of the earliest known independent inventions of writing, it required the participation of enough scribes to administer a state-level bureaucracy and about 1500 years (Overmann, 2016a).12Characters drawn by hand (as opposed to being carved, stamped, or impressed) appear in the Ancient Near Eastern archaeological record in 12A similar analysis of the other writing systems thought to represent independent invention-those of Egypt, China, and Mesoamerica-has not been performed. Thus, they cannot presently be quantified in terms of the length of time needed to realize literacy. However, all four independent origins (the three mentioned plus Mesopotamia) were similar in being bureaucratic states, implying a similar production demand i.e., repetition of conventionalized, simple, nonnumerical signs by hand at a volume and over a duration of time sufficient to train the fusiform gyrus, garner handwriting effects, adapt the material form, and realize literacy; Overmann, 2016a). the late 4th millennium BC. Many were pictographs, conveying their meaning through iconic resemblance (as a picture of a jar meant a jar), and some were ideographs, meaningful in virtue of social agreement as to what they signified (e.g., a circle divided in fourths by crossed lines meant sheep or other ruminants). Within centuries, under the production demands of a state-level bureaucracy, written characters started to become less recognizable as the objects they depicted or signified (Fig. 1). The loss of iconic resemblance is reasonably attributed to mechanisms that reflect increased skill in handwriting (e.g., biomechanical effectiveness and motor habituation) and training effects in the fusiform gyrus (e.g., recognition of characters through combinations of their local and global features; adjustment to enhance visual discriminability) (Overmann, 2016a). The loss of overt iconicity meant that the features identifying characters and differentiating them from each other were much subtler. Training and practice became necessary to read and produce them; this excluded non-initiates, created communities of specialized practitioners like scribes, and intensified change in the material form of writing and the brains interacting with it. Figure 1. Chronology of cuneiform signs (redrawn from Nissen, Damerow, & Englund, 1993). The early pictures (left) lost their depictiveness somewhere between 3000 and 2400 BC. This change entailed that reading and writing required increasingly formalized training. To highlight similarities, the early signs have been rotated one quarterturn to the left to give them the same orientation as the later signs (right), which were rotated to the orientation shown between the mid-third and late second millennium, possibly to facilitate biomechanical aspects of writing or visuospatial considerations of reading. Other changes (not shown above) occurred before the script would achieve the requisite fidelity to language needed to support literacy, whose possible onset occurred by 2000 BC (see Fig. 9 in Overmann, 2016a). Other types of changes (Overmann, 2016a) were required to adapt the initial picto– /ideographs with this-means-that associations between form and meaning into an abstract script with sufficient expressiveness to support a cognitive state analogous to modern literacy. The salient point is that adapting the material form required massive, distributed participation and a cultural span of time. This is significant for two reasons, one trivial and the other not. The trivial reason is that an extraterrestrial material culture would be less likely to interact optimally or immediately with human cognition (i.e., in presumably lacking the requisite participation over sufficient time to become efficient at influencing human change). The important reason is that because it takes massive, distributed participation and cultural spans of time to develop and refine a technology like writing, change becomes invisible, whether the change is material, behavioral, or neural. Each generation merely uses its material culture, and often fails to notice as both it and they change in the process. Material change can represent increased efficiency in changing brains and behaviors. It also represents the accretion of social knowledge, which reduces cognitive effort by distributing past and present effort to the current and succeeding generations (Hutchins, 1995). Co-influential change between brains, bodies, and materiality occurs on differing but coexisting temporalities, and the longer they are, the less tractable they become to both experience and neuroscience. One temporality is experiential: for example, reading the words on this page. Chances are, most readers do not think of this activity in terms of dynamic, transformative interaction of their psychological processes, behaviors, and a material form. The materiality- words on a page; the page itself; the book containing the pages-seems unchanging. The associated psychological processes and bodily behaviors are mostly unconscious to experience. Another temporality is ontogenetic. Children require several years and specific training to become proficient in reading and writing. The material form of writing as presented to the novice and fluent reader instantiates a spectrum from simple to complex. Change in brain function and form associated with the acquisition of the abilities to read and write can be measured longitudinally, explained theoretically, and appreciated experientially in terms of increased proficiency. Longer still are generational or cultural temporalities. In long temporalities, materiality can change rapidly and profoundly, while accumulating and helping reproduce the incremental changes in behaviors and brains that yield cognitive states like literacy. Over the longest spans of time, which are evolutionary, interaction with materiality has the potential to yield new brain structures (e.g., the regions of the intraparietal sulcus specialized for representing aspects of visual stimuli, proposed to be advantageous in making and using complex tools; Orban et al., 2006). Experientially, the long temporalities are beyond our reach, and neuroscience has at present little theory or methodology to measure or explain them. Thinking Through and Thinking About Materiality There are other reasons why we might not think of our cognition in terms of incorporating materiality as an integral or constitutive component. Materiality's semiotic value is acquired through enculturation and language, mechanisms that may predominate its acquisition through enactivity. That is, an artifact like a hammer is a hammer not only because its use involves behaviors and linguistic labels that can be learned and reproduced, but also because it is an object that is graspable, movable, and durable enough to be used to beat, drive, or shape other objects (e.g., this enables a fist-size nodule of flint to be a hammer, but not an iPhone). We are consciously aware of very little of our cognitive activity as we move our bodies and interact with material forms. We do not deliberately perceive objects or form memories or think through the moment-tomoment details of how we will move or speak; rather, we perceive, learn, move, and speak without much conscious awareness of the details of the implicit cognitive planning and execution (Kihlstrom, 1987, 1989). Additionally, behaviors that we may once have been consciously aware to some degree can become highly automated, freeing attentional resources to focus elsewhere. A familiar example of such automaticity is the degree of conscious awareness given to specific movements when learning to drive, compared to the relative lack of conscious awareness given to the same movements once driving proficiency has been acquired (Charlton & Starkey, 2011). In fact, the behavior can become so highly automatized and the use of the material form so unconscious that it is possible to drive to a destination that is familiar but unintended, something perhaps discovered only upon arrival. Few material devices become a persistent part of the body. Those that do may alter perception and movement: Glasses improve vision; artificial limbs alter mobility, posture, and proprioceptive awareness of where the body is and what it is doing; pacemakers affect the interoceptive awareness of how the body feels, health-wise; and all of them can influence the sense of what the body is, even as they become incorporated to the extent that they receive little conscious attention (de Preester & Tsakiris, 2009). Those material devices that do not become persistent parts of the body-most of the stuff of the environments we traverse and inhabit- nonetheless function to extend the body while they are engaged (de Preester & Tsakiris, 2009).23 The distinction between prosthetics and body-incorporation and tools and body-extension does not preclude the latter from affecting perception and movement. Certainly, neurons controlling finger movements, for example, react to tools as if they were part of the hand, allowing them to function as extended fingers (Maravita & Iriki, 2004). A stick extends tactile perception along its length to its tip, a phenomenon used by visually impaired people when they navigate by cane (Malafouris, 2008). In a sighted person, visual space is also remapped, so that things within the extended reach of the tool seem nearer to them (Maravita, Husain, Clarke, & Driver, 2001; Maravita, Spence, Kennett, & Driver, 2002). Tools are also subject to effects like automaticity and attentional refocus. In reading and writing, psychological processing, behavioral movements, and the material form become seamlessly integrated, so that the decision to move the eyes over the page cannot easily be separated from the comprehension of what is written on it, or the reading whose feedback facilitates the alteration of both production and written content. Focus on reading content can preclude awareness of the book, especially for proficient readers; when someone is aware of the book, it is probably not being read, as it is difficult to keep conscious attention focused on both book and content simultaneously. Even a material form that can become an integral part of our cognitive system, as a book does in reading, is not experienced as such when it is not so integrated. Materiality becomes merely the tools and objects we pick up, manipulate, and discard to accomplish our goals. Even if it seems strange to think through materiality, let alone do so unconsciously, this in fact may actually be typical of how most species engage material forms in general. That is, organisms may simply perform an action with an object without necessarily thinking about the object as something separate from its process of use. This appears to be how non-human primates think with tools: Their focus is on a goal, and tools are a means to that goal but not a separate and 23Beyond common sense, there are few criteria for determining when an object functions as part of the cognitive system. For example, if reading does not exist as a cognitive activity without interacting with the material form of writing, it implies that a book is part of the cognitive system whenever someone is reading it. Its cognitive status while unused but recalled is less certain, its cognitive status unpurchased at a distant bookstore or unfinished by its author even more so. All these connections (and more) can conceivably mean that books are part of the cognitive system, and as concepts, they are certainly anchored by our experience of interacting with books. distinct goal themselves. But the human ability to think about objects-to form and manipulate concepts of them, independent of the processes in which they are used (Coolidge & Wynn, 2018)- is remarkable. It too is arguably part of the ability to recruit and incorporate materiality into the human cognitive system. For example, materiality not only opportunizes the realization of concepts through mechanisms like enactive signification (i.e., things acquire meaning in virtue of what they are as physical substances and how we use them), it also anchors and stabilizes them (Hutchins, 2005), providing the brain with durable, manipulable stimuli. These in turn provide opportunities to realize and recognize new patterns as they are used, organized, and reorganized (Overmann, 2016b). But while even the most purely mental activity may depend on concepts being anchored and stabilized by material structures, such activity can be conducted in the absence of the material structures themselves. And evolutionarily speaking, it is this ability to think about materiality that is wonderful and unique to humans. The role of materiality in human conceptual life may have deep evolutionary roots: Roughly two million years ago, rather than abandoning a tool after use (ad hoc tool use), early members of the genus Homo retained and reused their stone tools, demonstrating a new relationship with tools and possibly the beginnings of a concept of one (Coolidge & Wynn, 2018). This does not entail that they had language. Currently, the available evidence has not yielded certainty on when language may have originated. Some estimations have placed it as early as 1.8 million years ago, either in conjunction with the appearance of the Acheulean handaxe, or with early Homo (e.g., Holloway, Sherwood, Hof, & Rilling, 2009). The latter has long been associated with KNM-ER 1470, a 1.8-million-year-old Homo rudolfensis specimen interpreted as having asymmetry and Broca's cap (Falk, 1987; Holloway, 1983; Tobias, 1981): Asymmetry suggests the neurofunctional lateralization associated with language and handedness, Broca's cap, language. Such features, admittedly, "cannot prove that this or that hominid had language" (Holloway et al., 2009, p. 1330). Parsimonious interpretation is warranted, for several reasons. First, endocasts provide limited insight into neuroanatomical landmarks and within-species, inter-individual variability. Second, while Broca's area (Brodmann's area 44/45) is expanded in humans, something that is reasonably related to language (Schenker et al., 2009), great apes possess "an anatomical and functional homologue of Brodmann's area 44" (Sherwood, Broadfield, Holloway, Gannon, & Hof, 2003, p. 277). Great ape brains are also asymmetric, at least in captivity, where they may be artificially exposed to greater tool use (and even then, again to a lesser degree than is characteristic of human brains) (Hopkins et al., 2017). Further, Broca's area has been implicated in both language and tool use (Binkofski & Buccino, 2004; Higuchi, Chaminade, Imamizu, & Kawato, 2009). Accordingly, even if KNM-ER 1470 is correctly interpreted as having Broca's area, it remains unclear that the feature would necessarily indicate language in addition to the tool production and use archaeologically attested. Further, a recent experimental study suggests that producing an Acheulean handaxe may be more a matter of fracture mechanics than linguistic instruction or intentionality (Moore & Perston, 2016). Others have not found a strong role for verbal instruction in lithic reduction techniques (Putt, Woods, & Franciscus, 2014), at least until those techniques become more complex than those associated with producing handaxes-for example, prepared core strategies like Levallois (Lycett, 2018). Thus, neither the paleontological or archaeological evidence necessarily demonstrates the availability of language at 1.8 million years ago in conjunction with early Homo. Further, many extant non-linguistic species use ad hoc tools: chimpanzees modify sticks to fish for termites, sea otters crack open shellfish using rocks as anvils, crows use twigs and other materials as probes, and octopuses use coconut shell halves for defense (Finn, Tregenza, & Norman, 2009; Hall & Schaller, 1964; Hunt, 1996; Sanz, Call, & Morgan, 2009). Like these species, early Homo too was presumably alinguistic, since they lacked the requisite physiological changes associated with language (e.g., significant altriciality, decoupled respiration, and descended larynx) that would variously appear between 1.8 million to 200,000 years ago with Homo erectus and Homo sapiens, though these features too may not be dispositive regarding language or its absence (Fitch, 2000, 2009, 2017). The possible appearance of a tool concept prior to language would also be consistent with the mosaic evolution that has characterized evolution in the hominid lineage more generally (e.g., bipedalism occurred long before brain size increased; Lovejoy, 1988). Even modern humans may form and mentally manipulate concepts in ways that are independent of language but related to motor activity. In reading, activity in Exner's area, a part of the brain located above Broca's area and anterior to the primary motor control area that has been implicated in the production of handwriting (Pegado et al., 2014), is thought to provide "a core recognition of the gesture in the written word" (Konnikova, 2014). Numbers and mathematics provide another potential example, as modern brains performing mathematical tasks recruit neurological circuits involved in planning and executing motor movements (Andres, Seron, & Olivier, 2007; Heimann, Umilta, & Gallese, 2013; Penner-Wilger et al., 2007; Tschentscher, Hauk, Fischer, & Pulvermüller, 2012). This is especially true of the motor control of the fingers, as attested by both lesion studies and performative skills. Damage to the angular gyrus, which has been implicated in finger control, is associated with finger agnosia and acalculia, the inabilities to know the fingers and perform calculations, respectively (Roux, Boetto, Sacko, Chollet, & Trémoulet, 2003). The mental abacus, an imaginary device used to perform complex mathematical calculations with accuracy and speed, demonstrates the importance finger movements (Brooks, Barner, Frank, & Goldin-Meadow, 2014; Frank & Barner, 2012). Interestingly, such motor planning functions take place whether or not the movements are actually carried out-a kind of internal simulation-and this may be the quality that enables individuals with impaired mobility to participate in human conceptual life. The use of "neural muscles"34to manipulate both objects and concepts is also suggested by mirror neurons and cerebellar activity. Mirror neurons, which become active both when an individual performs a motor action and when an individual sees a conspecific perform a motor action, may provide a gestural underpinning for communication with implications for the evolution of language (Gentilucci & Corballis, 2006). However, their presence in non-human species suggests that mirror neurons provide a largely alinguistic basis for understanding conspecific 34Embodied engagement with material objects involves neural activity (e.g., motor planning and execution), and motor planning in the absence of motor execution has been found in mentally manipulating concepts like numbers. It may be implicit to literacy as well, since Exner's area for controlling the movements of handwriting is active in recognizing characters (i.e., reading), as distinct from their manual production (handwriting). We have proposed the term "neural muscle" for this phenomenon. Specific neural activity continues to be elicited by interactions with specific material forms, even as the original motor movements become obsolete and are discarded (e.g., as typing on computer keyboards obviates writing by hand). This suggests that neural activity associated with higher-level cognitive functions may relate to productive behavior with past cultural forms and behaviors (e.g., prehistoric stone tools and their production and use), with descendent interactions with cultural forms and behaviors perpetuating the associated neural responses. Elsewhere, we have proposed the term "neural fossil" for the persistence of "neural muscles" beyond the material forms that occasioned them (though "fossil" has an inapt connotation of formerly and hence no longer living). We propose that humans have developed a generalized neurological response to material culture that is perpetuated by interacting with descendent material forms and behaviors. None of this discussion should be understood as proposing that the neural activity in question is necessarily representational in nature. motor actions and intent, and these are important in human tool teaching and learning (specifically, the understanding and imitation of behaviors in the absence of verbal instruction). The cerebellum, traditionally ascribed a role in motor learning, fine motor control, and motor movement sequencing, may play an important role in creating and manipulating abstract concepts as well, along with higher-order decisionand rule-making for multiple forms of information (Balsters, Whelan, Robertson, & Ramnani, 2013; Koziol, Budding, & Chidekel, 2010; Vandervert, 2009; Vandervert, Schimpf, & Liu, 2007). This is not to argue that language is unimportant to concepts-far from it. It is, however, to recognize that the early Homo (pre-Homo erectus) cannot be excluded from having had the requisite ability to manipulate conceptual objects as if they were physical forms on the basis of not having language. It is also perhaps a reason why the inclusion of multiple material forms, especially novel and unfamiliar ones, can spark creativity (Kirsh, 2014): Not only do novel material forms opportunize the recognition of new patterns, they may also engage distinct neural muscles (i.e., ones that differ from any previously engaged). And it is to the remote past and long temporalities involved in the evolution of our lineage that we must turn to answer the question of when and why these abilities emerged. The First Stable Category of Thinking About Materiality: The Biface Well beyond the unconventionality of considering the material form of writing as something tractable to archaeological investigation is the problem of discerning the evolutionary shift from thinking through materiality to thinking about it from the archaeological record. One potential criterion is behavioral change, such as when early Homo began to retain and carry flakes and cores from one location to another (Braun et al., 2008). But did this behavioral change entail that early Homo also had a concept of tool? How could archaeologists possibly decide, one way or the other? Another potential criterion is artifact type, the idea that tools can be categorized by form, with the archaeological recurrence of a particular form suggesting both intent and a concept of form on the part of those who recreated it. The idea of artifact type, however, is something with which archaeologists have long struggled. That is, there are almost no ways to confirm how well and in what sense our categories of archaeological analysis correspond to categories recognized by prehistoric humans, especially for the very deep past. Of course, there is no requirement that there be any such correspondence, for archaeologists often employ analytical types to help them investigate the past, without any need for analytical categories to reflect ancestral ones (assuming early hominins even had the ability to categorize to begin with) (Dunnell, 1971). But if archaeologists could identify categories used by early hominins, it might reveal something interesting about how and when a tool concept first emerged. Archaeologists no longer believe that the earliest stone tools were organized into natural categories or types (Toth, 1985; Wynn, 1981; Wynn, Hernandez-Aguilar, Marchant, & McGrew, 2011). Instead, the earliest lithic assemblages are thought to be the result of hominins focusing on task completion and producing a series of temporary products along the way. Many archaeologists suspect that the first imposed artifact categories appeared about 1.8 million years ago in the guise of tools known as bifaces, often also referred to as "large cutting tools" (LCT). With these tools, hominins for the first time manufactured material objects that seem to clump into categories. Glynn Isaac once used the metaphor of a spatial surface to describe the variability of early Paleolithic stone tools, with high points on the surface representing distinct design targets (Isaac, 1969, 1976). The implication of Isaac's analysis was that the hominins themselves would have recognized these peaks in morphological space as differentiating distinct categories of tools. If these hominins were in fact thinking about their tools, then they very well could have thought of them in terms of these morphological categories. But how did these hominins generate these categories in the first place? Fig. 2a is an image of a handaxe excavated at the 1.79 million-year-old Kenyan site of Kokiselei (Lepre et al., 2011). Archaeologists now use the typological term biface or large cutting tool for the general category, and handaxe for the narrower category that encompasses bifaces whose sides converge on a pointed tip, such as the example in Figs 2a and 2b. These different terms reflect a century and a half of uncertainty about what such artifacts may have represented as cultural products, and indeed, whether they could rightfully be described as cultural products in the first place. The best recent description of a biface is that of John Gowlett (2006; Wynn & Gowlett, 2018). He began his definition of a biface with a basic core chopper, a variety of temporary core tool that hominins began using about 3.3 million years ago (Harmand et al., 2015). He then asked what hominins added to the basic chopper to produce a large cutting tool such as a handaxe. Gowlett proposed six essential characteristics, which he termed "design imperatives": 1) Glob-butts: For a tool to be an effective hand tool, it needed a center of gravity that fit in the hominin hand. The solution was to retain a mass of stone, often unmodified, that allowed the tool to fit comfortably in the hand; this feature also provided sufficient weight to enable the tool to function. 2) Forward extension: When making handaxe, the tool maker's primary goal was to produce a cutting edge that was longer than those on a core chopper, as well as to acquire greater leverage. Knappers accomplished this by extending the length of the tool as measured away from the palm and the center of gravity. 3) Edge support: The primary functional feature of a handaxe is a cutting edge. The sturdiest edge is a bifacial edge. Hominins produced this by removing trimming flakes from the edge onto both faces of the tool. This produces an edge that has an effective cutting angle that stands up to repeated use. 4) Lateral extension: A long, narrow tool with a glob-butt and long cutting edge will tend to twist in the hand. To counter this tendency, hominins retained as much breadth as possible, especially at the glob-butt end of the tool. 5) Thickness control: Lighter tools are easier to wield, and cause less fatigue. The hominins strove to reduce the thickness of their handaxes in order to reduce weight. With the constraints of forward and lateral extension, the only avenue for weight reduction was via thickness, especially toward the working end of the tool. 6) Skewness: When the center of gravity was slightly off-center, the result was a tool that was better balanced for single-hand use. If a hominin tool maker deployed these considerations when making a large cutting tool, the result would be what we see in Fig. 2b. All these considerations are ergonomic: They instantiate the basic physics and perceived muscle and skeletal resistances of a hand-held tool. These are embodied resources, and thus the advent of biface technology arguably occurred through developments in embodied cognition. But the question at hand is the development of categorical thinking, the ability to think about materiality. In what sense did these design imperatives constitute an ontological category of tool? (a) (b) Figure 2. Handaxes. (a) Left: Handaxe from Kokiselei, Kenya, ca. 1.76 million years ago (Lepre et al., 2011). (b) Right: Handaxe from FLK West, Olduvai Gorge, ca. 1.69 Ma (Diez-Martín et al., 2015). Though only 70,000 years apart, the FLK West handaxe (right) differs from the Kokiselei example (left) in its size, symmetry, and manufacture technique, suggesting its production involved greater attention to features of the tool. Cognitive science has long been interested in categorical thinking. Historically, two models have dominated. Advocates of one argue that the mind defines categories through a list of required features; an exemplar qualifies for membership in a category if it presents all of the required traits, or in some versions of the model, a preponderance of required traits (Barrett, 2017; Carey, 2009). For the second model, the prototype, an exemplar warrants inclusion in a category based on its degree of resemblance to a prototype, something presumably held in long-term memory. After decades of debate and experiment, cognitive science has resolved the debate in favor of the prototype model: "The existence of prototypicality structure and its importance in the process of categorization are absolutely beyond doubt" (Carey, 2009, pp. 496–497). However, there remains uncertainty about how the mind generates prototypes and how an individual learns them. In some situations, it appears that individuals rely on memory of specific exemplars as prototypes, while in others individuals generate a kind of average "family resemblance" from the metaphorical range of variation of exemplars (Palmeri, 2014; Smith, 2014). In the kinds of natural settings that primates encounter daily, the "family resemblance" solution appears more efficacious than reliance on specific exemplars (Smith, Zakrzewski, Johnson, & Valleau, 2016). From the perspective of grounded cognition (Barsalou, 2008) and Material Engagement Theory (Malafouris, 2013), categories emerge when bundles of co-occurring embodied and extended traits coalesce into a prototype, a variety of "family resemblance" based in neuromuscular, ergonomic, and visual experience. This requires two cognitive processes: attention and association. Categories emerge "when attention is focused repeatedly on the same kind of thing in the world, by utilizing associative mechanisms among modalities, which, in turn, might permit re-enactment and simulation" (Pezzulo et al., 2011, p. 6). This is how a child learns categories: repeated association of salient features in attention, followed by simulation and internal execution of the associated bundle. Note that such categories are not abstract in the usual sense of the word, and need not exist as mental templates or visual images, though visual features can certainly be features of prototypes. At the evolutionary scale of change, the co-activation of perceptual features of tools and the motor component of tool use engage the appropriate neural resources (neuronal recycling; Dehaene & Cohen, 2007) and initiate neural reorganization via Baldwinian natural selection (Wynn, Overmann, Coolidge, & Janulis, 2017), which holds that "under some conditions, learned behaviors can affect the direction and rate of evolutionary change by natural selection" (Depew, 2003, p. 3). There is an interesting irony here for archaeologists interested in stone tool typology. Archaeologists almost always define types via attribute lists, sometimes prescribed, sometimes polythetic. But the mind does not construct categories this way. The handaxe itself is an excellent example. Many Paleolithic specialists have constructed their personal category of "handaxe" through repeated exposure, not through a set of attribute prescriptions. It is a prototype based on exemplars. When pressed to define the "type," they struggle to compile a list of attributes. For example, Corbey and colleagues (Corbey, Jagich, Vaesen, & Collard, 2016) provide an attribute list: "Acheulean handaxes were produced by the bifacial reduction of a block or large flake blank around a single long axis. They have a cutting edge in the secant plane, and range in shape from lanceolate through ovate to orbiculate" (p. 6). The problem is that this definition misses something essential about the Acheulean handaxe, and in fact the definition is so broad that it applies to artifacts from all over the world from almost all time periods, many of which specialists would not consider to be true handaxes (Wynn & Gowlett, 2018). Isaac (1976) actually came closer to describing the nature of artifact categories when he emphasized a metaphorical design space. Gowlett's design imperatives are similarly an excellent example of prototype, but one whose design space was primarily ergonomic. All of the design imperatives consist of bundles of ergonomic and visuospatial features. 1) The glob-butt consists of muscular tensions and resistances tied to heft (perceived weight) and grip security. It is a tactile motor package. 2) Forward extension is also an ergonomic bundle based in the musculoskeletal feel of leverage, and the duration and resistance of a cutting stroke. Here there is also a set of neural visuospatial correlates associating the tactile elements with the length dimension of the artifact. 3) Edge support combines musculoskeletal resistance with assessment of task efficiency and experience of breakage. Here, too, there are visuospatial features, including edge angle. 4) Lateral support is primarily a musculoskeletal bundle linked to grip resistance and security, with visuospatial correlates. 5) Thickness control acts as a counter to heft and fatigue. Heft correlates with size, but because length and breadth have other ergonomic constraints, the only size dimension free to reduce heft is thickness. Thickness control presumes that forward and lateral extension are primary ergonomic concerns. 6) The musculoskeletal aspects of heft and grip are asymmetrical in relation to artifact form, and the optimal biomechanical solution is for heft to be biased toward the location of grip. The result is skewness. Each of these ergonomic bundles assembled via the cognitive mechanisms of attention and association. Hominin butchers, for example, noticed (attention) the feel of a core tool with a longer cutting edge, associating its heft and stroke length with visuospatial features of the tool (forward and lateral extension). Eventually all six of these ergonomic bundles coalesced into an artefactual prototype-a biface-and, more significantly, hominins came to internally execute and simulate the bundle. The first true tool type had emerged. It was in a very real sense a visuospatial, ergonomic category, an embodied and extended concept. The hominins came to an awareness of this prototype as an ontological thing that they could think about. We know this because they soon added a non-ergonomic feature to the mix: visual pleasure. Consider now the artifact in Fig. 2b from the site of FLK West in Olduvai Gorge (DiezMartín et al., 2015). This handaxe differs in several respects from the slightly older Kokiselei example (Fig. 2a): First, the blank is a large flake, not a cobble; second, at over 30 cm in length, it is very large for a biface, or indeed any hand-held tool; and third, it is bilaterally symmetrical. Each of these three features suggests that the maker had thought about the tool itself, not just about a task to complete. The resulting handaxe is arguably too large to have been a hand tool. It also dwarfs the other FLK West handaxes (Diez-Martín et al., 2015). There is no obvious mechanical reason for this tool; a smaller handaxe, like others from the site, would have been much easier to wield. It may have had a role in social display of some sort (Cole, 2014; Shipton, 2010), perhaps as simple as showing off. However we come to understand it, the exceptional size of this handaxe suggests that the tool itself was the goal. Gigantism became a kind of recurring motif for Acheulean knappers, with giant examples occurring in most areas with large enough clasts. There good examples from throughout the African Acheulean (Berlant & Wynn, 2018), but also from Europe, where large clasts are rarer (e.g., Cuxton [Wenban-Smith, 2004]; also see Fig. 3). The spatio-temporal organization of biface technology corroborates the knappers' reliance on a tool concept. The Large Flake Acheulean (Sharon, 2008, 2009) provides the best documented examples. Large flake manufacture is the first step in a two-step procedure to produce a biface(Sharon, 2009). By convention and as the term is used here, large means over ten centimeters in maximum dimension. Producing a flake of this size required that the tool maker use a large core that was probably too heavy to be carried very far. Their immediate goal at the source must not have been task performance, but tool production, or at least blank production. The flake for the FLK West handaxe was over 30 centimeters in length, a very large flake that required an over-the-head, twohanded hammer strike on a boulder-sized core (Jones, 1981). Knappers initially produced flake Figure 3. Giant ficron handaxe from Cuxton, England; stratigraphic age estimated as late as MIS 8 (ca. 300,000 years ago) (Wenban-Smith, 2004). Photograph by Thomas Wynn. blanks at the raw material source, carried the blanks to a second location where they trimmed them into finished artifacts, and then sometimes carried them again to another location. Gallotti and Mussi (2017) refer to such technical sequences as being "fragmented" and have documented their presence as early 1.0 Ma in Ethiopia. Archaeologists have described similar fragmented technical sequences for many Acheulean sites and localities (Barkai, Gopher, & LaPorta, 2006; Goren-Inbar & Sharon, 2006; Hallos, 2005; Paddayya, Jhaldiyal, & Petraglia, 2006; Roberts & Parfitt, 1999; Sampson, 2006). At Gesher Benot Ya'aqov 780,000 years ago, knappers used different blank production procedures at the quarry for handaxes and cleavers, indicating that they anticipated completing particular varieties at a subsequent time and place (Herzlinger, Wynn, & Goren-Inbar, 2017). They initiated two different fragmented sequences that had different final artifact forms as goals. They clearly thought about the tools themselves, not just about a specific task to complete. Their immediate goal at the source must not have been task performance, but tool production, or at least blank production. The bilateral symmetry of this handaxe is an overdetermined quality. That is, bilateral symmetry added nothing to the functional potential of the large flake, yet its maker invested the effort to impose it (there is a long history of attempts to demonstrate an advantage to particular handaxe shapes, but none has been successful; Key & Lycett, 2017; Machin, Hosfield, & Mithen, 2007). The simplest way to account for this overdetermination is through the pleasure the maker experienced in producing a symmetrical shape, an effect referred to as visual resonance (Hodgson, 2000, 2009, 2011, 2015). Cells in the primate visual cortex evolved to be sensitive to bilateral symmetry (symmetrical things are almost always living organisms), and arousal of these cell groups also elicits arousal of opioid releasing cells in the pleasure centers of the brain. Simply, symmetrical patterns are more pleasing to the eye than non-symmetrical patterns. Crucially, there is no need to posit a mental image: Visual resonance would draw a tool maker to produce a symmetrical shape if possible, but only if the knapper attended to features of the tool itself. Attending to material features is essential to literacy as well, suggesting an inherent continuity between stone tools (whose interaction actualized the ability to form and manipulate concepts mentally, as if they were physical objects) and writing (whose interaction actualized the ability to form and manipulate concepts physically, as if they were mental objects). The large size and bilateral symmetry of the FLK West handaxe were qualities that enhanced the basic ergonomic imperatives of biface manufacture instantiated in the handaxe from Kokiselei. Bilateral symmetry is not itself an ergonomic feature; indeed, the design imperative of skewness would bias a tool maker toward a slight asymmetry. If, as seems plausible, large size played a role in display of some sort, then size, too, had become a non-ergonomic feature linked to visual impact. The tool maker must have been thinking about the visual appearance of the tool itself, not just a task to be completed. These extra-ergonomic features are interesting in their own right, but for the purpose of this essay, they serve to confirm that a coherent set of features had coalesced into a stable tool concept available for thought. Hominins had started to think about the materiality they used, and in doing so likely recruited the same kinds of neural muscles seen today as activity in mirror neurons and the cerebellum, during handwriting and mathematics, etc. Neural Muscles: Into the Future The seemingly simple technological/cognitive development of the biface would have immense significance for hominin evolution. It situated hominin technology well beyond anything known for apes and monkeys, and was the seed for the progressive, conscious manipulation of technology that both characterizes and differentiates the human species, though fruition of this trend would be another long time coming. However, our ancestors did not start thinking about materiality one day as a miraculous discovery (nor, for that matter, can it be implausibly ascribed to alien contact). Instead, thinking about materiality was the consequence of several million years of anthropoids making and using tools, and at least a million years of patient stone knapping by hominins. For example, when hominins began to carry cores and flakes from place to place, one cognitive consequence was a temporal extension of the hominins' contact with this materiality, a likely prerequisite for a permanent, stable, tool concept. In these ancestral species, thinking through the materiality of stone tools ultimately gave us the ability to think about them as concepts. In more recent peoples, thinking through the materiality of making marks on things like clay, papyrus, and bone opened up a new way of thinking about concepts, through a material form that allows ideas to be subjected to analysis, reflection, revision, and refinement, and transmitted through space and time to preserve, educate, and provoke (Olson, 1994). In developing neural muscles that interact with material forms, our ancestors became the species that can both think about and think through materiality, and both continue to change us. Over the course of our evolutionary history, these muscles (to continue the metaphor) have become stronger and change more quickly in response to interactions with material forms. Certainly, when biface manufacture is contrasted with writing, what particularly stands out is that the neurological changes involved in the latter required a much shorter span of time than those of the former. What might this suggest for an evolutionary history that continues to unfold? For example, as we increase our use of collaborative media like the Internet and Twitter and tools like smart phones and computers, and decrease behaviors like handwriting, our brains are changing (Sülzenbrück et al., 2011). We already see change in conceptions of privacy (though the implications and future consequences of this are far from clear), and we can speculate about effects on memory. However, our brains will also change in ways we cannot foresee, since we may have little idea of what might emerge next. Further, we are unlikely to see the changes taking place, since the temporalities over which they emerge are multigenerational and longer. We may simply remain content with noticing at some point that the brain has started to do something new, as the Mesopotamians once did when writing began to speak to them, a phenomenon so astonishing that many cultures have ascribed it to divine intervention (Senner, 1989). But as the species that both thinks about and thinks through materiality, we might also consider asking questions along the lines of these: Is the trend toward faster change continuing, or has it slowed or reached some sort of plateau? Can we become aware of cultural/evolutionary changes in brain function as they are taking place? Can we gain any sense of their direction or even control over the process, either deciding whether it is a direction we want to follow or changing its speed of progression? Would we be able to master the process whereby material forms become more effective and efficient in shaping our behaviors and brains? 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Consequentialism and Collective Action Brian Hedden Abstract Many consequentialists argue that you ought to do your part in collective action problems like climate change mitigation and ending factory farming because (i) all such problems are triggering cases, in which there is a threshold number of people such that the outcome will be worse if at least that many people act in a given way than if fewer do, and (ii) doing your part in a triggering case maximizes expected value. I show that both (i) and (ii) are false: Some triggering cases cannot be solved by appeal to expected value, since they involve infinities, and some collective action problems are not triggering cases, since they involve parity. However, I argue that consequentialism can still generally prohibit failure to do your part in those collective action problems where we believe that so acting would be impermissible. Keywords: Consequentialism, collective action, parity, incommensurability, climate change 1 Collective Action Problems Collective action problems take many forms. I will focus on a kind of collective action problem in which a much better outcome would result if all or most people performed a given action than if few or none did so, and yet it is tempting to think that no particular individual could make any difference by acting one way or the other. Examples: A much better outcome would result if everyone reduced her carbon footprint than if no one did. But it is tempting to think that no single individual could make a 1 difference by reducing her carbon footprint. Average global temperatures would be the same regardless of whether one drove a hybrid or a gas-guzzler. If everyone were to refrain from buying factory farmed meat, things would be much better than if no one did so. But it is tempting to think that no single purchase would affect how many animals are raised and slaughtered in factory farms. The supply chain just isn't that sensitive to individual purchases. If everyone were to vote for the better candidate, then that candidate would win. If no one were to do so, then either the worse candidate would win, or else democracy might collapse (if no one voted at all). But it is tempting to think that no single individual would make a difference to the outcome. Only tiny races for dog-catcher are ever decided by a single vote. To fix terminology, say that an exceptional act is one such that things would be better if few or no people did it than if all or most did, but where arguably no single such act would make things worse. The term is supposed to evoke the idea that in performing such an act, one seems to be making an exception for oneself. Not voting, buying factory-farmed meat, and not reducing one's carbon footprint are examples of exceptional acts. My focus in this paper is whether and to what extent consequentialism permits performing exceptional acts. Prima facie, it might seem that consequentialism will often permit such exceptional acts. For consequentialism is concerned with bringing about the best outcome possible, and if any particular exceptional act would make no difference to how good the outcome is, it seems that that act would be permitted by consequentialism. This implication might constitute a serious objection to consequentialism. But my concern is not with whether consequentialism's treatment of collective action problems should make us more, or less, confident in the truth of consequentialism. Rather, I am concerned simply with whether and to what extent consequentialism permits exceptional acts. I leave it to others to decide whether consequentialism's verdicts, and reasons for those verdicts, are adequate. Many consequentialists have argued that, in fact, consequentialism does not permit ex2 ceptional acts (Kagan 2011; see also Singer 1980; Parfit 1984; Norcross 1997, 2004). Following Kagan, say that in a triggering case there is some threshold number of people k such that the outcome would be worse if at least k people were to perform a given exceptional act than if fewer were to (there may be several such thresholds). These consequentialists then make two claims: First, in triggering cases, the exceptional act is impermissible because it has sub-maximal expected value.1 It has a high probability of making things slightly worse, or a low probability of making things much worse, or something in between, but in any case its expected value will be lower than that of not performing the exceptional act. Second, all collective action problems are triggering cases. This standard treatment is attractive and elegant. But it has also been met with skepticism (Nefsky 2011; Budolsfon 2018). I argue that both of its component claims are in fact false, but not for the reasons emphasized by critics, and, more importantly, not for reasons that significantly threaten the overall verdict. In each case, a slight tweak saves the verdict that consequentialism generally prohibits exceptional acts. Here is the plan: In §2, I show that expected value theory provably cannot prohibit exceptional acts in some infinitary triggering cases but argue (contra Budolfson) that it is likely to do so in more realistic cases. In §3, I turn to the second claim. While conceding the flaws in previous attempts to show that all collective action problems are triggering cases, I argue that a better strategy is available, based on the claim that all relations of the form is exactly as F as are equivalence relations. Hence there cannot be a sequence of outcomes (ordered by the number of people who perform the exceptional act) with the first better than the last but each exactly as good as its predecessor. But as I discuss in §4, this is not sufficient to show that all collective action problems are triggering cases. Parity lurks. While there cannot be a sequence of outcomes with the first better than the last and each 1The expected value of an act is the result of adding up, for each possible outcome, the product of the value of that outcome and the probability that the act will result in that outcome. In symbols: EV(A) = ∑ i P (Oi | A)V (Oi). The consequentialist theory then says that an act is permissible just in case no alternative act has higher expected value. 3 outcome exactly as good as its predecessor, the possibility of parity means that there can be a sequence of outcomes with the first better than the last and each outcome not worse than its predecessor. But I show that one attractive theory of decision-making under parity allows consequentialism to prohibit exceptional acts even in parity-laden, non-triggering collective action problems. If this decision theory is correct, then even though not all collective action problems are triggering cases (because of parity) and expected value theory can't prohibit exceptional acts in all triggering cases (because of infinities), consequentialism still prohibits exceptional acts in most of the kinds of collective action problems with which we are most concerned, and where we tend to judge that exceptional acts would be impermissible. 2 The Appeal to Expected Value Let's start with the bad news. The standard consequentialist treatment of collective action runs aground in infinitary cases. Suppose there are (countably) infinitely many people, each facing a switch. If only finitely many people flip their switches, then everyone spends eternity in heaven. But if infinitely many people do so, then everyone spends eternity in hell. This is a triggering case, where the threshold number is א0, the cardinality of any countably infinite set. The outcome would be much worse if at least א0 people flipped their switches than if fewer did so. But here the appeal to expected value is impotent. For no single switch can mean the difference between infinitely many switches being flipped and only finitely many being flipped. Thus each person has probability 0 of making the outcome worse by flipping her switch, and flipping the switch does not have lower expected value than not flipping it.2 2See also Arntzenius, Elga, and Hawthorne (2004) for a related but importantly different case. In a group version of their Satan's Apple, there are infinitely many people, each with a slice of apple. If infinitely many eat their slices, everyone goes to hell, while if finitely many do so, everyone goes to heaven. It is good for each person to eat her slice, but the goodness of the gustatory pleasure is far outweighed by the badness of hell. In this case, for each person, the outcome would be better if she were to eat her slice than if she were not to, holding fixed what others do. Hence she would be required to eat on consequentialist grounds. This yields the puzzling result that every combination of acts is worse than some alternative combination. In my case, for each person, the outcome would be exactly as good whether or not she were to flip her switch. 4 Infinities are weird, and in this case I think we have no considered judgment that the exceptional act is indeed impermissible. Consequentialists may therefore be willing to concede that they cannot prohibit exceptional acts in all triggering cases, hoping instead to do so in those finite cases where we tend to judge that the exceptional act is impermissible.3 Budolfson (2018) argues that even this more modest goal is unattainable. Focusing on the case of factory farming, Kagan (2011) begins by conceding that it is unlikely that each chicken purchased causes one more chicken to be raised and slaughtered.4 But there must still be some threshold, such that if that many chickens are purchased in a given period, approximately the same number of additional chickens will be raised and slaughtered. Thus we know that there is some triggering number T (more or less), such that every T th purchase (more or less) triggers the order of another T chickens (more or less). I don't have any idea what that number is, but I do know that whatever it is, I have a 1 in T chance of triggering the suffering of another T chickens (more or less). And so in terms of chicken suffering, my act of purchasing a chicken still has an expected disutility equivalent to one chicken's suffering. And since, by hypothesis, this is greater than the pleasure I will get from eating the chicken, the net expected utility of my purchase remains negative. As I walk to the butcher counter, then, not only don't I know whether my act will have bad results, I don't even know what the chances are that my act is a triggering act. But I do know, for all that, that the net expected results of my act are bad. So I should not buy a chicken. (2011, 124). As Budolfson explains, this reasoning relies on the assumption that the expected effects of the relevant act are approximately equal to the average effects of that sort of act. Given modest assumptions about the efficiency of the marketplace, each act of purchasing a chicken has, on average, the effect of one chicken being raised and slaughtered. And, Kagan suggests, Hence consequentialist considerations seem to permit, but not require, each person to flip. And here, there are many optimal combinations of acts, namely all those where only finitely many people flip. 3This concession does mean denying that it is part of the nature of morality that it satisfies the 'Principle of Moral Harmony,' which states that 'when all the members of a social group do what they morally ought to do, the group as a whole does benefit more than it would have from the performance of any worse alternative set of actions' (Feldman 1980, 167). Feldman gives independent reasons to doubt this principle, however. But see Portmore (2018) for a defense of a modified principle of moral harmony. 4See Broome (2018) for discussion of thresholds in the case of climate change. 5 if you are ignorant about the location of the thresholds and about how others will act, the expected number of extra dead chickens resulting from an individual chicken purchase will be approximately equal to that average effect. (As a side note, it is worth mentioning that consequentialism will still prohibit buying a chicken even if that act has an expected increase in the number of chickens produced of less than one, provided that the suffering of each chicken far outweighs the difference in the pleasure one gets from eating a chicken and the pleasure one would get from alternative vegetarian meals. If the suffering of a chicken in a factory farm still outweighs, say, the aggregate net pleasure of eating 100 chickens, as it plausibly does, then buying a chicken would be prohibited even if its expected increase in the number of chickens produced were only around 0.01.) Budolfson (2018) argues that Kagan is overly optimistic. This is because we are in a position to know about the presence of 'buffers' in the supply chain which reduce both the probability of an individual act making a difference as well as the size of the difference it will make, if it makes any difference at all (see also Nefsky 2011). A buffer is anything that makes production less sensitive to individual consumer acts. As one example, a chicken wholesaler might have the option of selling unsold meat at cost to a dog food manufacturer or rendering plant, with the result that a small reduction in demand from ordinary consumers will not yield a concomitant reduction in the number of chickens the wholesaler purchases from producers. Budolfson argues that these buffers make the expected effect of an individual act significantly lower than the average effect of acts of that type, with the result that the relevant exceptional act does not have sub-maximal expected value. Does our knowledge of these buffers scuttle the appeal to expected value? I do not think so. It is important to emphasise that buffers in a supply chain do not eliminate the existence of threshold numbers. Instead they help determine what those threshold numbers are, in particular by placing the thresholds farther apart than they would otherwise have been. But Kagan has already conceded that the thresholds may be very few and far between, and that 6 the probability of an act making any difference at all may therefore be very low. So why should the existence of buffers pose a threat to the appeal to expected value? Expected value calculations take into account all probabilities, no matter how small, meaning that there is no positive number such that the probability of an act making a difference must be above that number in order for it to be prohibited on expected value grounds. What matters is the relation between the probability of making a given difference and the size of that potential difference, where an increase in one can compensate for a decrease in the other. And Kagan makes the plausible assumption that, in general, buffers which reduce the probability of an individual act making a difference will yield a compensating increase in the size of the effect that an individual act will have, in the unlikely event that it does make a difference. If this is right, then chicken-purchasing will be prohibited on expected value grounds, regardless of the size and effectiveness of the buffers in question. Budolfson argues that this assumption is mistaken and that buffers can reduce the probability of an act making a difference without yielding a compensating increase in the size of the effect that the act will have, if it does happen to make a difference. He writes (p. 8): even in the very unlikely event that, say, an individual purchase of meat really did succeed in making the price of animals at one point at a production end of the supply chain $0.01 higher than it otherwise would have been, that would not make the dramatic difference to the number of animals that are brought into existence that it would have to make in order for the possibility of such a threshold effect to drive the expected effect toward the average effect, in part because the number of animals that are brought into existence is suprisingly insensitive to very small changes in price at that location for a variety of reasons. In a footnote, Budfolson supports this contention by observing, for the case of cattle raising, that 'insofar as ranchers judge that capital should be invested in raising cattle rather than other investments, they will tend to raise as many cattle as they can afford to breed and feed within that budget, letting the ultimate extent of their profits fall where it may at the feedlot' and that many ranchers 'use the nutritional well-being of their herd as a buffer 7 to absorb changes in market conditions, feeding their cattle less and less to whatever point maximizes the new expectation of profits as adverse conditions develop' (ibid). However, this observation at most shows that individual acts are unlikely to have any large effects in the short term. But consequentialists care about an act's long-term effects.5 Perhaps, over the course of the next year, ranchers will raise and slaughter the same number of cattle regardless of any (relatively small) price changes. But profits one year will affect how things go the year after, and the year after that. At the margins, lower profits discourage new entries into the industry and may lead some existing ranchers to abandon cattle production altogether or to diversify their investments, for instance by shifting toward raising sheep for wool. This makes evident that quite a bit of work is being done by Budolfson's caveat about 'insofar as ranchers judge that capital should be invested in raising cattle.' We cannot hold fixed people's judgments about where to invest capital, since these may themselves be affected by consumer acts. Indeed, it is by influencing investment decisions that individual acts may be most likely to have large effects. Lower profits also mean that ranchers who persist in cattle raising will have less capital the next year. They might still 'raise as many cattle as they can afford to breed and feed within that budget,' but the budget will be lower, possibly resulting in fewer cattle bred and fed. Of course, individual acts are unlikely to have such dramatic effects, but that has already been conceded by consequentialists like Kagan. The point is that there is still a small probability of their having such large long-term effects. I am therefore unconvinced by Budolfson's contention that the effects of individual acts will be either null or too small to have any hope of being prohibited on expected value grounds. Let me now turn to an example Budolfson gives to illustrate the importance of buffers: Richard makes paper T-shirts in his basement that say 'HOORAY FOR CONSEQUENTIALISM!', which he then sells online. The T-shirts are incredibly cheap to produce and very profitable to sell and Richard doesn't care about waste per se, and so he produces far more T-shirts than he is likely to need each month, and 5This focus on the long term raises Lenman's (2000) famous 'cluelessness' problem, however. 8 then sells the excess at a nearly break-even amount at the end of each month to his hippie neighbor, who burns them in his wood-burning stove. For many years Richard has always sold between 14,000 and 16,000 T-shirts each month, and he's always printed 20,000 T-shirts at the beginning of each month. Nonetheless, there is a conceivable increase in sales that would cause him to produce more T-shirts-in particular, if he sells over 18,000 this month, he'll produce 25,000 T-shirts at the beginning of next month; otherwise he'll produce 20,000 like he always does. So, the system is genuinely sensitive to a precise tipping point-in particular, the difference between 18,000 purchases and the 'magic number' of 18,001. (2018, 6) Budolfson argues that, given the facts about buffers in the T-shirt supply chain (the option of selling excess merchandise at cost) and about the historical trends in consumer purchasing decisions, the expected effect on T-shirt production of a single act of purchasing a T-shirt is 'essentially zero' because 'there is virtually no chance that exactly 18,001 people are going to buy Richard's T-shirts this month and trigger a dramatic threshold effect' (ibid, 6). Thus the expected effect of buying a T-shirt is much lower than the average effect of consumers' acts of buying T-shirts. He concludes that the problem with Kagan-style reasoning is 'that it overlooks the fact that we can know enough about the supply chains...to know that threshold effects are not sufficiently likely and are not of sufficient magnitude to drive the expected effect of consumption anywhere close to the average effect' (ibid, 7). Now, we must concede that there can be no decisive, a priori argument that the expected value of purchasing a chicken (or other exceptional acts) will be sub-maximal, because as Budolfson rightly notes, 'the knowledge available about the mechanisms at play in such situations matters greatly' (ibid, 11). After all, it is rational subjective probabilities that matter in calculating expected values, and rational subjective probabilities depend on the agent's evidence. We can even imagine an evil demon planting misleading evidence to suggest to each person that they are nowhere near any thresholds, in which case expected value theory will not prohibit the exceptional act.6 6Note also that in cases where we know we are nowhere near any thresholds, we often do not judge that the exceptional act would be impermissible. For instance, deciding not to engage in any food production is 9 But it is important to be clear about what is going on in Budolfson's example. As noted above, buffers in the supply chain do not eliminate the existence of thresholds, but instead help determine where they are. This means that, when you possess detailed information about the exact workings of buffers, this could in principle provide evidence about what the threshold numbers are. And when you also possess information about historical trends in consumer purchasing decisions, this provides evidence about how many others will perform the relevant act. Now, it is not surprising that if you have evidence about what the threshold numbers are and about how many others will perform the relevant act, the expected effects of your act will probably be lower than if you didn't possess all that evidence. Think of it this way: In cases where threshold numbers are few and far between, we already know that it is very likely that your act will make no difference. Hence it is very likely that, in the limiting case where you are fully informed about all aspects of the situation, and, in particular, about the exact (post-buffer) threshold numbers and about how everyone else will act, the expected effects of your act will be null. More generally, as you gain more and more evidence about what the threshold numbers are (e.g., by learning more and more about the buffers) and how others will act (e.g., by learning more and more about historical demand), the expected effects of your act will probably become lower and lower, the exception being the rare case in which you are in fact right at the threshold number, in which case the expected effects of your act will actually increase as you become more informed. But in order for the expected effect of buying a chicken on the number of chickens raised and slaughtered to be 'essentially zero,' it is not enough to know that there are certain buffers in the supply chain and that there are historical trends in consumer decisions. This is because mere knowledge that there are certain buffers provides little or no evidence about where the new threshold numbers are (or about the possible magnitudes of an individual act's effects, assuming the previous argument is correct), and because mere knowledge that an exceptional act, as things would be worse if everyone did this than if no one did. But it is permissible for me not to produce food, since I know I am nowhere near a threshold where doing so will make things worse. 10 there are historical trends in consumer decisions provides little or no evidence about how many others will perform the relevant act. In order for the the appeal to expected value to fail, you would also need detailed evidence about what the trends in consumer decisions in fact are, as well as detailed evidence about the exact workings of these buffers, so as to be able to better locate where you sit in relation to any thresholds. These facts were simply given to us in Budolfson's T-shirt case. But in real-life we have no such knowledge. I have no idea even approximately how many chickens are consumed worldwide each year. And while I believe that consumption is increasing, with the effects of a growing population and people rising out of poverty outweighing increased vegetarianism, I am ignorant about its rate of increase. Now, some evidence about consumer trends is available online. But more to the point, while I have some idea about the nature of buffers in the global chicken supply chain (e.g., that some excess is sold to rendering plants), I have no idea even approximately what the new threshold numbers are that result from the operation of these buffers. And given the complexity of global economic forces, not even industry experts could determine even roughly what the threshold numbers are that result from these buffers, especially given that those thresholds concern the number of chickens raised and slaughtered over the long run. In real-life cases, then, we are in roughly the situation that Kagan and others suppose. Namely, we are very ignorant about where the thresholds are and about how many others will perform the relevant exceptional act, and so we are also very ignorant about how close or far we may be from hitting the threshold.7 Along with my previous argument that buffers do 7Budolfson might concede that you should have a roughly uniform probability distribution over hypotheses about exactly how many others will buy chickens, as well as a roughly uniform probability distribution over hypotheses abvout what all the threshold numbers are. However, he could rightly point out that this is not enough to vindicate the appeal to expected value. For you might have a non-uniform probability distribution about what the threshold numbers are, conditional on any given hypothesis about how many chickens will be purchased. That is, even if you have no idea what demand will be or where the new, post-buffer thresholds are, you might nonetheless think that the two are correlated, such that the thresholds and anticipated demand, whatever they are, are likely to be far apart. In the T-shirt case, this would be the case if you thought that Richard sets up his supply chain with the aim of ensuring that demand will not approach the new thresholds. But I see little reason to think that demand and the new, post-buffer thresholds will be correlated in this 11 not prevent individual acts from having large long-term effects, this ignorance about where we sit in relation to any thresholds largely vindicates the appeal to expected value.8 In conclusion, I think Budolfson is greatly overstating things when he writes that 'in the real world we generally have access to additional evidence that makes it empirically indefensible to equate the expected marginal effect and average effect in such a way, and that makes it similarly indefensible to assign a probability to making a difference that would be sufficiently high to vindicate the conclusions of the [expected value response to triggering cases]' (ibid, 10-11). Budolfson is correct that the expected value of purchasing a chicken (or any other exceptional act) will not necessarily be sub-maximal regardless of what one's evidence might be, but wrong in thinking that the expected value will not be sub-maximal given our actual evidence.9 I conclude that expected value considerations will still prohibit exceptional acts in most of the triggering cases with which we are most concerned. way in real-world cases like factory farming, given that these cases involve a large and fluctuating number of producers, operating independently, none of whom has the power to unilaterally determine global production and none of whom is likely to care much about exact global demand. 8In this respect, the cases of factory farming and climate change are importantly different from that of voting, contra Budolfson (2018, 8). Polling data and knowledge of the voting rule allow citizens to locate approximately where they sit in relation to the relevant thresholds. In cases where the race isn't close, this will justify a tiny probability of one's vote making a difference. And this probability must be further discounted by one's confidence that one has accurately identified the candidate who is in fact better, as Lomasky and Brennan (2000) note. So consequentialism will not always require voting. But when the race is close, the stakes are huge, and one candidate is clearly better, as in the case of the last several US presidential elections, say, consequentialism may require voting despite the still tiny probability of making a difference. See also Barnett (ms) for an argument that, given two modest assumptions which are often met in real-life, voting will be required on consequentialist, difference-making grounds. 9As a reviewer noted, Budolfson might be interpreted not as making a claim about expected effects, given a typical consumer's actual evidence, but rather as providing us with additional evidence such that, in light of that evidence, we see that the expected effects of the exceptional act are too small for it to be prohibited on expected value grounds. But as noted, the complexity of economic forces means that even industry experts will be ignorant of where the thresholds are and how large the long-term effects of a given consumer act might be. So, even relative to such experts' evidence, the expected effects of purchasing a chicken will be approximately equal to one additional chicken produced. And it would not help to interpret Budolfson as making a claim about expected effects relative to the objective chance function. For the global economy is arguably not a physically chancy system, meaning that the objective chance of an individual act making a difference is either 0 or 1, depending on whether we are in fact right at a threshold, but that we don't know which it is. And if it is the latter, then the expected effect of purchasing a chicken, relative to the objective chance function, will be much greater than the average effect of chicken-purchasing acts. 12 3 Imperceptible Harms Turn now to the second component of the standard consequentialist treatment of collective action cases. This is the claim that all collective action problems are triggering cases: there is always some threshold number of people k such that the outcome would be worse if at least k people performed a given exceptional act than if fewer did so. (As noted, there may be multiple thresholds, and indeed it could be that every additional exceptional act makes the outcome worse.) This claim is intuitively compelling in the cases of voting and factory farming. But it is less obvious in other cases, like that of climate change. Let us consider a famous case which puts pressure on this claim, namely Parfit's (1984, 80) case of the harmless torturers. There is a patient hooked up to a torture machine. Other than the patient, there are n people, including you, each of whom has a switch in front of her. Flipping that switch will slightly increase the voltage going into the patient. If no one flips her switch, the patient will receive no voltage and experience no pain. If everyone flips her switch, the patient will receive a very high voltage and experience great pain. But for all j, the patient cannot tell the difference between the pain involved in the outcome Oj in which exactly j people flip their switches and the pain involved in the outcome Oj+1 in which exactly j+1 people flip their switches. Hence it seems like each possible outcome Oj+1 is just as painful-and therefore just as good-as its predecessor Oj. Thus, the harmless torturers case seems like a non-triggering collective action problem. I begin by giving my own response before explaining how it avoids the problems facing previous consequentialist-friendly responses. I claim that for all F , the relation is exactly as F as is an equivalence relation.10 (Indeed, I think this fact is a conceptual truth, though I 10This seems obvious to me, but oddly, it has been scarcely defended in the literature. Broome (2004, 151-2) does claim that equally as good as is transitive. This follows from his defintion, on which A is equally as good as B just in case (i) A is neither better nor worse than B, and (ii) for any C, C is better (worse) than A if and only if C is better (worse) than B (ibid, 20). He also notes that it would follow from analysis on A is equally as good as B just in case the degree of A's goodness is identical to the degree of B's goodness, given the transitivity of identity. Broome would presumably thinks that the same holds if we substitute any other predicate for 'good.' I do not, however, commit myself to either of Broome's proposed analyses. 13 do not need this stronger claim here.) Being an equivalence relation, it is transitive. Hence there cannot be a sequence of outcomes such that the last is F -er than the first and yet each outcome is exactly as F as its predecessor. In the harmless torturers case, the relevant F is painful. From my general claim, it follows that the relation is exactly as painful as is an equivalence relation, and hence transitive. Thus, there cannot be a sequence of states such that the last is more painful (for the patient) than the first and yet each state is exactly as painful its predecessor. Given that On is indeed more painful than O0, it follows that there must be at least one state that is not exactly as painful as its predecessor.11 This does not mean that the patient can tell the difference between any two adjacent states. The two states may be indiscriminable in the sense that the patient is not in a position to know whether they are exactly as painful as each other (Williamson 2013 (1990)). Indiscriminability is non-transitive. This should not be surprising. Given our limited powers of discrimination, it should not be assumed that one is always in a position to tell whether two states are exactly, as opposed to merely almost exactly, the same as each other, even with respect to some phenomenal property. This is especially true in cases like this one, where the states cannot be experienced simultaneously (meaning that the comparisons rely on memory), not to mention that extreme pain interferes with one's cognitive capacities.12 11See also Barnett (2018) for a different innovative and, in my view, compelling argument that cases like the harmless torturers must be triggering cases. 12Compare Graff Fara (2001) and Mills (2002), who defend the fallibilist claim that we are not always in a position to know whether two things look the same to us. Graff Fara (2001) rebuts an argument that limited powers of discrimination mean that looking the same, understood as sameness of visual phenomenology, is non-transitive. She considers two possible ways of cashing out the claim that our powers of discrimination are limited. First way: 'For some sufficiently slight amount of change (in colour, sound, position, etc.), when we perceive an object for the entirety of an interval during which it changes by less than that amount, we perceive it as not having changed at all during that interval' (917). But this claim is false, for it entails that we never misperceive an object as having changed in the relevant respect when it has in fact not changed at all. Second way: 'For some sufficiently slight amount of change (in colour, sound, position, etc.), we cannot perceive an object as having changed by less than that amount unless we perceive it as not having changed at all (as having changed by a zero amount)' (917). But this claim does not entail that phenomenal sameness is non-transitive. For ruling out the possibility of an interval during which the object appears to change by some amount below the threshold leaves open the possibility that the object will appear to change discretely at some point in an interval where it in fact changes continuously. 14 And defining what it is for one state to be exactly as painful as another in terms of the impossibility of discriminating between them in a pairwise comparison smacks of the crude operationalism that has long since fallen into disrepute in the philosophy of science; we would not, for instance, define what it is for one thing to be exactly as hot as another in terms of the impossibility of some thermometer's giving different readings for them. (It may also be that for each state, it is indeterminate, and not merely unknowable, whether it is exactly as painful as its predecessor, even though it is determinately true that not every state is exactly as painful as its predecessor. Then, it would be determinately the case the harmless torturers case is a triggering case, but indeterminate where the thresholds are. This indeterminacy-laden case can be treated along consequentialist lines by appeal to a decision theory for indeterminacy which is analogous to the decision theory for parity that I explore in the next section. See also footnote 26.) What about the claim that phenomenal properties are response-dependent in the sense that judging that the property applies in a given case makes it the case that it so applies? If the patient judges that each state is exactly as painful as its predecessor, might that make it the case that they are exactly as painful as each other? I do not need to deny that the monadic property of being painful is response-dependent (though I am skeptical of this claim). It may be that whenever a subject judges, of the state she is currently in, that it is painful, then it is painful. But such response-dependence is implausible for relations of comparative painfulness, unless further constraints are imposed. For we can imagine a subject who judges that state S1 is more painful than S2, and also judges that S2 is more painful than S1; an unconstrained response-dependence thesis for more painful than would then entail, falsely, that it is non-asymmetric. Worse, we can imagine a subject who judges that S1 is more painful than S1; unconstrained response-dependence would then entail, again falsely, that the relation is non-irreflexive. Thus, any response-dependence thesis for the relation more painful than must impose constraints that ensure that it satisfies various structural constraints such 15 as irreflexivity, asymmetry, and transitivity. Once these constraints are imposed, it is unclear why the response-dependence theorist would reject constraints which ensure the reflexivity, symmetry, and transitivity of exactly as painful as. Let me consider three objections.13 The first is that I am simply dismissing the Sorites. Nefsky (2011, 383-9) levels this charge at Kagan, whose argument we will briefly consider below. The standard Sorites, applied to the case at hand, involves the three (classically) jointly inconsistent claims that O0 is not painful, that On is painful, and that for all j, if Oj is not painful, then neither is Oj+1. This is a genuine paradox, and I offer no solution here. Nor do I need to, for what matters is not (or at least not only) whether a given state is painful, but rather (or in addition) how painful it is.14 And even if no single switch flipped can change the outcome from not painful to painful (a difference with respect to a vague, monadic property), this does not mean that no single switch flipped can affect the morally significant underlying dimension of how painful it is. Now consider a different Sorites-like paradox, involving the four jointly inconsistent claims that O0 is not painful, that On is painful, that if one state is exactly as painful as another then one is painful just in case the other is, and that for all j, Oj is exactly as painful as Oj+1. Here, the last claim is not intuitively compelling, once we distinguish between states being exactly as painful as each other, and their being merely almost exactly as painful as each other. Of course, one can make a theoretical argument in favor of this last claim, but it does not have the same intuitive pull as the third claim of the standard Sorites. Thus, I am 13A brief comment on a fourth objection: I am not reifying 'feels,' in the way that Dennett's (1978, xix-xx) imaginary society reifies 'fatigues.' That is, I do not focus on the relation feels the same as, analyze it as has the same feel as, and appeal to the fact that identity (and hence identity of feels) is an equivalence relation (see Williamson (1994, 179) for discussion). 14Bacon (2018) argues for the stronger conclusion that it is irrational to care intrinsically about the vague. For example, it is irrational to care about whether one is bald, over and above all the underlying facts relevant to baldness, such as how many hairs one has, how they are distributed, how people react to you, and so on. I am sympathetic to Bacon's claim, and to the analogous view that vague properties are not intrinsically morally significant. But for my purposes, I need only the weaker claim that it is irrational to care exclusively about the vague, and that facts involving vague properties do not exhaust what is morally significant, to the exclusion of the underlying more precise properties and relations. 16 not dismissing the Sorites, but only this latter psuedo-Sorites, which is no paradox at all. The second objection is inspired by Temkin (2012, 164), who considers and rejects the claim that it is a conceptual truth that for all F , is F-er than is a transitive relation:15 Consider the following example. Let us define the relation "larger than" as follows: for any two people a and b, a is larger than b if a is heavier than b or if a is taller than b. Clearly, so defined, a might be larger than b, because heavier, and b might be larger than c, because taller, yet a might not be larger than c, as c might be both heavier and taller than a. So it appears than one could have a "...er than" relation that is not transitive. If Temkin is right, that would cast doubt on my claim that for any F , the relation is exactly as F as is an equivalence relation. For we could imagine defining is exactly as large as thus: for any two objects a and b, a is exactly as large as b if and only if either a is exactly as heavy as b or a is exactly as tall as b. Defined thus, we might have a, b, and c such that a is exactly as large as b, b is exactly as large as c, and yet a is not exactly as large as c. My response is flatfooted: neither the comparative 'larger,' nor the relation it expresses, work in the way Temkin is imagining. And it is even clearer that neither 'is exactly as large as,' nor the relation it expresses, work in the way I just sketched. Admittedly, this dispute is difficult to settle. We are close to bedrock. But standard linguistic treatments of comparatives (Kennedy 2007; Schwarzschild 2008; see also Kamp 1975, 145) agree that 'is exactly as F as' and 'is F -er than' always express transitive relations. Indeed, it is hard to see how to devise a plausible compositional semantics for comparatives, including for the morpheme '-er,' the modifier 'exactly' (and the contrasting modifiers 'approximately' and 'roughly'), and especially for the positive form 'is F ,' which rejects these claims (see Nebel 2018).16 While I do not take this to decisively settle the matter, I think that it remains 15See also Temkin (1996) and Rachels (1998). 16A note on compositionality and 'exactly.' The expression 'is exactly as painful as' is not an idiom; it is not like 'kick the bucket,' where we understand the expression by learning it as a whole rather than by understanding the meanings of the component words and how they are put together. Therefore, our semantics should have it that 'exactly' means the same thing in expressions like 'is exactly as painful as' as 17 overwhelmingly plausible that 'is F-er than' and, more importantly for my purposes, 'is exactly as F as' always express transitive relations. The third objection is that discriminability by humans may be necessary in order for two states to differ in a morally significant way. Differences in painfulness that cannot be detected by humans are not morally significant. On this view, while is exactly as painful as may be transitive, is exactly as painful in the morally relevant sense as is non-transitive, since indiscriminability is non-transitive.17 There are two points to make in response to this objection. The first is that this view entails that is exactly as good as is also non-transitive, which conflicts with the claim that all relations of the form is exactly as F as are equivalence relations, regardless of whether the relevant F is painful, good, or anything else. The second is that there is good reason to doubt that discriminability is necessary for the difference between two states to be morally significant. It is important to distinguish between cognitive (or belief-like) judgments about painfulness and the underlying painfulness itself. As defined above, indiscriminability is understood in terms of cognitive judgments: two states are indiscriminable (for an agent) with respect to painfulness just in case the agent is not in a position to know (or, perhaps, to reliably judge) that they differ in their painfulness. This is the sense in which indiscriminability may be non-transitive. But why privilege these cognitive judgments about phenomenology over the underlying phenomenology itself? If two states differ in how painful they are, why should the agent's inability to have knowledge of their differing painfulness mean that this in expressions like 'has exactly two children' and 'arrived at exactly noon.' It also means that expressions of the form 'is exactly as F as' should work the same way regardless of whether 'F ' expresses a phenomenal property like painful, a non-phenomenal, descriptive property like tall, or a normative property like good. And they should work the same way regardless of whether 'F ' expresses a unidimensional property like tall or a multidimensional property like large. This suggests, for example, that 'is exactly as painful as' cannot mean the same as 'is indiscriminable with respect to pain from' (even setting aside the non-transitivity of indiscriminability). For an expression like 'is exactly as tall as' does not mean the same as 'is indiscriminable with respect to height from.' Perhaps it is impossible tell the difference between two things differing in height by a Planck length and their not differing in height at all (and we can even imagine this to be a nomological impossibility and not merely a practical one); hence two things could indiscriminable with respect to height even if their heights differ by a Planck length, but they would then not be exactly as tall as each other. 17Thanks to an anonymous reviewer for pressing me on this objection. 18 difference is morally insignificant? To press the point further, consider a creature with less capacity for fine-grained introspective knowledge than humans. Certain animals might well fit the bill. Perhaps this creature has no capacity for cognitive judgments whatsover. In this case, all states count as indiscriminable for that creature. But, assuming that the creature can feel pain, and different levels of pain, it is implausible to say that none of the creature's possible pain states differ in their moral significance. Alternatively, we can imagine that the creature has the capacity for introspective knowledge, but only of a very limited and coarse-grained sort. While there are many different levels of pain that the creature can feel, it is only ever in a position to know that two pain states differ when one is very slight and the other very intense. Again, it is implausible that this epistemic limitation means that the difference between slight and moderate pain, or between moderate and very intense pain, is morally insignificant. The lesson is clear: differences in painfulness must sometimes be morally significant even when the subject is not in a position to have knowledge of their differing painfulness.18 Now I want to argue that my approach is superior to existing consequentialist treatments of harmless torturers-style cases. The first reason is that my approach does not rely on contentious claims about verbal reports and their relation to phenomenal states. Kagan (2011) notes that if asked in O0 whether she is in pain, the patient will answer 'no,' while 18The objector might respond that the sense in which the tiny differences between adjacent states of the machine are undetectable is that they feel the same to the agent, and that if two states feel the same, they must be equally morally valuable. I have avoided talking in terms of the locution 'feels the same as' and instead focused on the relation is exactly as painful as, since the former locution is unhelpfully ambiguous. As Keefe (2011) notes, it has at least two distinct readings. First, there is a purely phenomenal reading, on which 'S1 feels the same as S2' can be glossed as 'The feel of S1 is the same as the feel of S2.' On this reading, 'feels the same as' expresses a transitive relation, since identity is transitive. However, one might doubt the legitimacy of reifying feels in this way, as noted in footnote 13. Second, there is a reading of 'feels the same as' on which it means 'feels as though they are the same.' This is a cognitive reading, on which two states feel the same roughly when the agent judges (or is inclined to judge), on the basis of introspection, that they are the same in the relevant respect. So understood, 'feels the same as' expresses a non-transitive relation. But the holding of this relation does not suffice for two states to be equally morally valuable, since as argued above, it is implausible to privilege cognitive judgments about, or based on, phenomenology to the exclusion of the underlying phenomenology itself. Absent some other proposed reading of 'feels the same,' I conclude that it never expresses a relation that is both non-transitive and such that its holding between two states suffices for them to be equally morally valuable. 19 if asked in On whether she is in pain, she will answer 'yes.' Hence there must be adjacent outcomes which differ with respect to the patient's answer to the question whether she is in pain. Kagan says that the two outcomes must therefore feel different. McCarthy and Arntzenius (1997) previously gave a more sophisticated version of this argument. They imagine allowing the patient infinite time to play around with the machine, trying out each of the states multiple (even infinitely many) times, and each time recording her best description of how painful it felt, using whatever language she likes. Allowing the patient to try out each state multiple times mitigates worries about the possible instability of her responses and gives an accurate record of her overall dispositions with respect to how to describe her experience. But the basic argument is the same as Kagan's. The patient's verbal response disposition for O0 clearly differs from her verbal response disposition for On. Thus, there must be two adjacent outcomes that differ, if only slightly, with respect to the verbal response dispositions they yield. And this means that those two adjacent outcomes must not feel the same to the patient. This strategy has two significant limitations. First, verbal reports, and even long-run verbal report dispositions, need not accurately reflect underlying phenomenal states (Nefsky 2011). It might be that the patient is more disposed to report painfulness in Oj than in Oj+1 even though they feel exactly the same. This is because one's verbal reports could be influenced not only by the underlying phenomenal states, but also by non-phenomenal states like tissue damage. The fact that two adjacent states yield differing verbal dispositions does mean that the agent is sensitive to some differences between the two outcome states, but not that those differences show up in her phenomenology.19 19Kagan (2011, 136) is alert to this problem and replies that 'it is important to bear in mind that these [reports] are indeed immediate and spontaneous reports concerning the qualitative aspects of the victim's experiences. The victim is simply reporting how the state feels to him, with regard to whether it involves pain, or whether the amount of pain differs from that involved in other states.' But it is not at all clear that even if we try, we can make our verbal reports sensitive only to our introspective phenomenology. Of course, Kagan can just stipulate that the patient is responding only to her phenomenology, but then the argument could be simpler, and indeed more similar to my own. He could leave out the verbal reports and simply point out that the painfulness of O0 and the painfulness of On differ and that feels the same as is an equivalence 20 Second, even if we assume that differing verbal reports mean a difference in how the states feel, this would only show that there must be two adjacent states that don't feel the same. It would not show that there must be two adjacent states that fail to be exactly as painful as each other. For it is at least logically possible for one state to be exactly as painful as another state even though the two don't feel the same. Similarly, it is logically possible for two visibly different paintings to be exactly as beautiful as each other, or for two things to be (or look) exactly as red as each other without looking the same, for instance if one is a bit greenish and the other a bit blueish, but they are equally far from pure red.20 My strategy also has the advantage of not being tied exclusively to phenomenal properties. Nefsky (2011, 374) considers a consequentialist view on which fairness is morally relevant: Now, imagine that there is a large supply of clean water that two impoverished communities, A and B, have equal claim to and that will be distributed by an international committee. The fair outcome would be for the water to be divided approximately evenly between the two communities. Approximately evenly because-I think we can say-fairness is not, in this case, an extremely precise matter. A few drops of water more or less on one side does not make the distribution unfair (or even any less fair) in any morally relevant sense of the term. But is exactly as F as must always be an equivalence relation, regardless of whether F is a phenomenal property, a normative property, or anything else. Hence is exactly as fair as is an equivalence relation as well. More generally, is exactly as good as is an equivalence relation, and so no matter what is morally valuable, there cannot be a sequence of states with the first better than the last but each exactly as good as its predecessor. relation, and hence there must be adjacent states that don't feel the same. 20Even functionalists about pain will grant that pain involves not only verbal dispositions, but other dispositions as well, such as a disposition to grimace. We can then imagine two states S1 and S2, such that one is slightly more disposed to report being in pain in S1 than in S2, but one is slightly more disposed to grimace in S2 than in S1. These differing dispositions may suffice for the two states to feel different, but if they exactly balance each other, the states will nonetheless count as exactly as painful as each other. One might object that in the cases I've discussed, the two things cannot be exactly as beautiful, as red, or as painful as each other if they look or feel different. Instead they can only be 'on a par' with respect to beauty or redness. I discuss parity in the next section. 21 4 Parity But wait! I have argued that is exactly as F as is always an equivalence relation. But the fact that is exactly as good as is an equivalence relation does not suffice to show that all collective action problems are triggering cases. All that this fact shows is that there cannot be a sequence of states with the first better than the last and each state exactly as good as its predecessor. But a triggering case, as defined above, is one involving a sequence of states with the first better than the last and where some state is worse than its predecessor. Arguably, one thing can be neither better, nor worse, nor exactly as good as another. Some philosophy job P might be neither better nor worse than some journalism job J , which is also neither better nor worse than the philosophy job with an extra $100 (P+). Given the transitivity of is exactly as good as, P and P+ cannot each be exactly as good as J , since P+ is better than P . Instead, at least one of them must be on a par with J (Chang 2002). Hence, it is compatible with the view I defended in the previous section that there be a sequence of states such that the first is better than the last and yet each state is neither better nor worse nor exactly as good as its predecessor in the sequence.21 The harmless torturers case is not such a case. Parity with respect to F -ness can arise only when there are multiple factors relevant to how F a thing is, but no precise way of assigning weights to those factors so as to enable precise trade-offs. For instance, if how large something is depends on both how heavy and how tall it is, but there are no precise weights assigned to the two dimensions of heaviness and tallness, there can be cases where one thing is neither larger nor smaller nor exactly as large as another. But in the harmless torturers case, the kind of pain involved in each state is the same, and the only thing that varies is its intensity. This means there will be no parity between any of the states. But climate change may be a case in which there is a sequence of possible outcomes 21Compare Nefsky's (2011, 382) complaint that Kagan shows at most that some state must feel different than predecessor, and not that it feels worse. She does not discuss parity, however. 22 where the first is better than the last, but none is worse than its predecessor. Things would be better if everyone reduced emissions than if no one did so. But it may be that no tiny increment in emissions would make the outcome worse; instead, it would leave it exactly as good as, or on a par with, how it would otherwise have been.22 After all, even where a small increment in emissions causes a morally relevant difference, the difference needn't be all bad: some people will feel less comfortable, others more so; some animals will have less food, others more; and so on. And there may be no way of assigning precise weights to these various harms and benefits. Indeed, in many collective action problems, our acts may affect the number and identities of the people who exist, and the kinds of harms and benefits that accrue to different people; these are the sorts of differences that may yield parity between outcomes. We can also modify the harmless torturers case to introduce parity: Harmless Torturers (Parity Version) There is a patient hooked up to a torture machine and n other people (including you), each facing a switch. If no one flips, the patient feels almost no pain. If everyone flips, the patient feels excrutiating pain. But there are two kinds of pain: burning pain and throbbing pain. And as more switches are flipped, the pain intensity increases, but alternates between burning pain and throbbing pain. So, O0 means throbbing pain of intensity 1, O1 means burning pain of intensity 2, O2 means throbbing pain of intensity 3, and so on. In addition, there is parity among types of pain, such that burning pain of intensity x is neither better nor worse nor exactly as bad as throbbing pain of intensity x± 1. Here, the outcome will be much better if no one flips their switch than if everyone does, but no outcome is worse than its predecessor. Thus, you know that, regardless of how many others flip, your flipping will not bring about a worse outcome than not flipping. Does that mean that it is permissible for you to flip your switch? That depends on the correct theory of decision making under uncertainty and parity. Schoenfield (2014, 267) endorses a principle for decision-making under parity which says that it is permissible for you 22cf. Andreou (2006), who likens cases of pollution to Quinn's (1990) case of the self-torturer with intransitive preferences, though she does not claim that betterness itself is non-transitive (Andreou 2018). 23 to flip. For her LINK principle says, in part, that 'If you are rationally certain that neither of the two options [A and B] will bring about greater value than the other, it's not required that you choose A, and it's not required that you choose B.' But a different decision theory, Prospectism (Hare 2010; see also Weirich 2004), prohibits flipping, at least given complete uncertainty about how many other people will flip. With parity, the betterness ordering can be incomplete and hence not representable by a value function that assigns one outcome a greater real number than another outcome if and only if the former is better than the latter. For it may be that O1 is not better than O2, O2 is not better than O3, and yet O1 is better than O3. But there are no real numbers such that x ≤ y, y ≤ z, and yet x > y. We can, however, consider value functions representing coherent completions of the betterness ordering, where V represents a coherent completion of that betterness ordering just in case, for all Oi and Oj, V (Oi) > V (Oj) if (but not only if) Oi is better than Oj. (Think of a coherent completion of a betterness ordering as one that respects the original ordering's better than relations but also eliminates parity by taking each instance of some Oi being on a par with Oj and replacing it with Oi's being either better than, worse than, or equally good as Oj.) Prospectism then says: Prospectism: It is permissible to perform an action if and only if, for some value function V that represents a coherent completion of the betterness ordering, no alternative action has higher expected value relative to V . My aim is not to provide a defense of Prospectism, but simply to show that it gives consequentialism a way of prohibiting exceptional acts in parity-laden collective action problems.23 And let me first flag that my argument will rely only on the left-to-right (or 'only if') 23The debate over decision-making under parity and uncertainty has focused on a different kind of case (Hare 2010; Schoenfield 2014). Suppose A and B are on a par. As such, mildly sweetening one of them (say, by adding $5 to it), converting it to A+ or B+, would not make it better than the other. Now suppose there are two opaque boxes. One contains A and the other contains B, with the arrangement determined by a fair coin. And you can see left-hand box has a $5 note on top of it. Question: Are you required to take the (sweetened) left-hand box? Prospectism says 'yes.' For taking the left-hand box is associated with the prospect {0.5 chance of A+, 0.5 chance of B+}, while taking the right-hand box is associated with the prospect 24 direction of Prospectism, meaning that any decision theory which agrees with its necessary condition for permissibility will allow consequentialism to get the same desired result. Now, suppose you are completely uncertain about how many other people will flip their switches, such that you have a uniform probability distribution over the states S0, ..., Sn−1 (where Sj is the state where exactly j-many other people flip). In this case, Prospectism prohibits flipping. This is because each 'intermediate' outcome O1, ..., On−1 has the same ( 1 n ) probability of resulting from you flipping as from you not flipping and hence can be ignored; they cannot make a difference to the relative expected values of flipping and not flipping. (To illustrate, if n > 17, outcome O17 would result from your flipping if S16 is actual, and would result from your not flipping if S17 is actual, but by our setup S16 and S17 are equiprobable.) As for the non-intermediate outcomes O0 and On, flipping has probability 0 of yielding O0 and probability 1 n of yielding On, while not flipping has probability 1 n of yielding O0 and probability 0 of yielding On. But since O0 is better than On, each value function representing a coherent completion of the betterness ordering assigns the former a higher number than the latter. Therefore, relative to each value function representing a coherent completion of the betterness ordering, flipping has lower expected value than not flipping.24 Another way to put this is that flipping is stochastically dominated by not flipping, where A stochastically dominates B just in case for each outcome O, the probability of yielding an outcome at least as good as O is at least as great for A as for B, and for some outcome O∗, {0.5 chance of A, 0.5 chance of B}, and relative to any value function representing a coherent completion of the betterness ordering, the former prospect has higher expected value than the latter. Schoenfield's LINK principle says 'no,' since you know that, no matter how the coin landed, the contents of the left-hand box are not better than the contents of the right-hand one (since A+ is not better than B, nor is B+ better than A). Without going fully into the arguments for and against each response to the two opaque boxes case, let me add that it is a messy affair designing a full decision theory that entails LINK. The only one I am aware of is Hare's Deferentialism, which is considerably more complex than Prospectism. If parsimony is a virtue in normative theorizing as well as in empirical inquiry, this may be one reason to favor Prospectism. 24Put in terms of symbols, the point is that for all Vi: EVi(¬flip) = 1n × Vi(O0) + 1 n × Vi(O1) + ... + 1 n × Vi(On−1) + 0× Vi(On) EVi(flip) = 0× Vi(O0) + 1n × Vi(O1) + ... + 1 n × Vi(On−1) + 1 n × Vi(On) Hence, EVi(¬flip) > EVi(flip) iff Vi(O0) > Vi(On) But, since O0 is better than On, Vi(O0) > Vi(On), and so for all Vi, EVi(¬flip) > EVi(flip) 25 the probability of yielding an outcome at least as good as O∗ is strictly greater for A than for B. And Prospectism prohibits stochastically dominated actions (Bader 2018). (Note that flipping is also stochastically dominated in the original harmless torturers case, provided you have a uniform probability distribution over hypotheses about how many others will flip. Therefore, even if you are unconvinced by my treatment of that case, consequentialism will still prohibit flipping, given any decision theory that prohibits stochastically dominated acts.) Thus, if Prospectism is correct, consequentialism can prohibit the exceptional act of flipping even in this parity-laden, non-triggering case. This result holds provided you are uncertain how others will act. Now, if you knew exactly how many others would flip, Prospectism would permit flipping. As for intermediate cases, where you are neither completely uncertain nor completely knowledgable about how others will act, the devil is in the details; whether Prospectism prohibits flipping will depend on exactly how far your probability distribution deviates from uniformity and 'how much' parity there is in the betterness ordering.25 But 25If you deviate from a uniform probability distribution over the states S0,...Sn−1, flipping is no longer stochastically dominated and hence may be permitted by Prospectism. For there will be some intermediate outcomes that are more probable if you flip than if you don't. If a value function assigns sufficiently high value to those outcomes, flipping will then have highest expected value. But whether such a value function is admissible will depend on the details of both the betterness ordering and your probability function. The point is best seen with a modification of the opaque boxes case in footnote 23. In this new version, suppose that there is probability n 6= 0.5 that A was placed in the (sweetened) left-hand box. Taking the left-hand box (L) is associated with the prospect {n chance of A+, 1 − n chance of B+}, while taking the right-hand box (R) is associated with the prospect {1 − n chance of A, n chance of B}. So, for any value function Vi, EVi(L) = n× Vi(A+) + (1− n)× Vi(B+), while EVi(R) = (1− n)× Vi(A) + n× Vi(B). Without loss of generality, let n > 0.5. Then, EVi(R) ≥ EVi(L) only if both (a) Vi(B) > Vi(A+) and (b) Vi(B)−Vi(A+)Vi(B+)−Vi(A) ≥ 1−n n . Such an admissible value function will exist if the only facts about the betterness ordering are that A+ is better than A and that B+ is better than B. But in more realistic cases, there will be additional facts about the betterness ordering will constrain admissible value functions. When (a) obtains, Vi(B)−Vi(A+) Vi(B+)−Vi(A) increases (with an asymptotic upper bound of 1) as Vi(A+)−Vi(A) and Vi(B+)−Vi(B) decrease and as Vi(B)− Vi(A+) increases. But additional facts about the betterness ordering may put a lower bound on how far A+ and B+ must be ranked above A and B, respectively, and an upper bound on how far B can be ranked above A+. How much by way of additional constraints are needed to render R impermissible depends on the value of n, with more constraints needed as n approaches 1. This is because 1−nn decreases as n increases, thereby making it 'easier' to find an admissible value function that jointly satisfies (a) and (b). The details are messy, but the moral is simple: In both the opaque boxes case and the parity version of the harmless torturers case, Prospectism prohibits taking the right-hand box or flipping your switch if your probability distribution over the relevant states is uniform. Otherwise, it may permit such acts, though this will depend on the details of the probability distribution and the betterness ordering involved. 26 this limitation may not be terribly serious, for we have already seen that consequentialism will not prohibit exceptional acts in all collective action problems. At most, it will do so in realistic cases in which you're very uncertain how others will act. I have shown that parity need not threaten this more modest claim.26 Is Prospectism compatible with consequentialism? There is one way of characterizing consequentialism on which the two are not compatible. On this gloss, consequentialism says that an act is permissible just in case there is no alternative act which would yield a better outcome. But this gloss is incompatible with any decision theoretic version of consequentialism, for it makes no reference to the agent's subjective probabilities. Fortunately, there is another standard gloss that leaves room for a decision theoretic version of consequentialism. On this gloss, consequentialism says that the good is prior to the right, such that an act's permissibility depends only on the values of the possible outcomes of the available acts, as well as the agent's (rational) subjective probabilities. This gloss is neutral with respect to which decision theory is correct and is therefore compatible with Prospectism.27 The issue, then, is not whether Prospectism is compatible with consequentialism, but simply whether it is correct. It is certainly somewhat counterintuitive that Prospectism sometimes prohibits an act which it is known will not yield a worse outcome than its alternatives. But while this constitutes probably the most significant objection to Prospectism, it does not mean that Prospectism is false, for there are also significant objections to non-Prospectist 26Broome (1997) regards incommensurability not as parity, but as vagueness or indeterminacy. Even if he is wrong, it may also be indeterminate what the likely outcome of some action would be, for instance if it is indeterminate what the threshold numbers are in a triggering case, or if it is indeterminate how precisely you would perform the act in question. To deal with indeterminacy, consequentialists would do well to adopt some Prospectism-like theory on which an act is prohibited if if it has sub-maximal expected value on to every admissible way of resolving any indeterminacy. (For epistemicists (Williamson 1994), indeterminacy is a kind of unknowability and hence rational uncertainty, and as such may not require any modification of expected value theory.) See Hare (2011), Moss (2015), and Williams (2017) for further discussion of decision-making given indeterminacy. 27Still, (act) consequentialism may be incompatible with certain ways of motivating decision theories. In particular, it would be in tension with defending a decision theory on the basis of its tending to yield better overall results in a sequence of choice situations (whether faced by a single agent at different times or by different agents). Without going into details, however, let me just say that all of the defenses of Prospectism cited below are based on grounds other than the results it is likely to yield when followed repeatedly. 27 theories that avoid this oddity (Hare, 2010; Bader, 2018; Doody, 2019; Rabinowicz, ms). Settling the matter requires evaluating all the arguments for and against Prospectism and its competitors, which is beyond the scope of this paper. Suffice it to say that Prospectism is a live possibility but that my conclusion is a conditional one: If Prospectism (or at least its necessary condition for permissibility) is correct, then consequentialism can prohibit exceptional acts even in parity-laden, non-triggering collective action problems. 5 Conclusion Consequentialists have responded to a standard sort of collective action problem by arguing (i) that all such cases are triggering cases, and (ii) that exceptional acts are prohibited in triggering cases by virtue of having have sub-maximal expected value. Unfortunately, both claims are false. Expected value theory won't prohibit exceptional acts in some triggering cases, since they involve infinities. And some collective action problems are not triggering cases, since they involve parity. Nonetheless, I conclude that many intuitively impermissible exceptional acts can still be prohibited on consequentialist grounds. First, while consequentialism cannot prohibit exceptional acts in some infinitary triggering cases, it likely does so in more realistic cases where we have strong judgments of impermissibility. In such triggering cases, most of us are sufficiently ignorant-both about the mechanisms involved and about how others will act-that consequentialism will prohibit the exceptional act on expected value grounds. Second, while I have argued that there cannot be a sequence of outcomes with the first better than the last and each exactly as good as its predecessor, parity means there can be a sequence of outcomes with the first better than the last and each not worse than its predecessor. This means that not all collective action problems are triggering cases. Nonetheless, consequentialism can still prohibit exceptional acts in such cases, provided that 28 Prospectism (or, again, at least its necessary condition for permissibility) is correct. I conclude that consequentialism can prohibit exceptional acts in many, if not all, of the sorts of collective action problems where we tend to judge that the exceptional act is indeed impermissible. While consequentialists will likely welcome this conclusion, it does not entail that the consequentialist treatment of collective action problems (let alone consequentialism more generally) is correct. Whether consequentialism gives the correct verdict about permissibility in all collective action problems, and whether it gives the correct explanation of these verdicts, are topics for another paper.28 28For helpful feedback, I would like to thank John Broome, Stephanie Collins, Mark Colyvan, Kevin Dorst, Luke Elson, Daniel Greco, Alan Hàjek, Caspar Hare, Shelly Kagan, Daniel Muñoz, Miriam Schoenfield, Roger Schwarzschild, Sam Shpall, Nicholas JJ Smith, Jack Spencer, and Daniel Wodak, as well as audiences at the University of Colorado-Boulder, MIT, Yale, the Australian National University, the Australian Catholic University, the University of Adelaide, and the University of Sydney. 29 References Andreou, Chrisoula. 2006. 'Environmental Damage and the Puzzle of the Self-Torturer.' Philosophy and Public Affairs 34 (1): 95–108. Andreou, Chrisoula. 2018. 'Better Than.' Philosophical Studies. Early view online. Arntzenius, Frank, Elga, Adam, and Hawthorne, John. 2004. 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Advanced Modalizing Problems Mark Jago Forthcoming in Mind. Draft of December 2014. Abstract: I present an internal problem for David Lewis's genuine modal realism. My aim is to show that his analysis of modality is inconsistent with his metaphysics. I consider several ways of modifying the Lewisian analysis of modality, but argue that none are successful. I argue that the problem also affects theories related to genuine modal realism, including the stage theory of persistence and modal ctionalism. Keywords: Modality, modal realism, advanced modalizing, David Lewis 1 Introduction David Lewis's modal realism (1968; 1971; 1986) has been hugely inuential in the philosophy of modality. It is known to suffer from the advanced modalizing problem (Divers 1999), wherein commitments of Lewis's metaphysics of possible worlds conict with the content assigned to certain statements of possibility. (I outline the problem in section 2.) A revision to Lewis's analysis of possibility seems to be required, if one is to maintain his modal metaphysics. Here, I consider a range of alternative analyses, and argue that they all fail. I conclude that Lewisian metaphysics is unable to provide an adequate analysis of possibility. The conclusion is signicant because one of the main justications given for Lewisian modal realism is precisely that it provides the best analysis of possibility (Lewis 1986, Sect. 1.2, Sect. 3). But if it is unable to provide any such analysis, Lewisian modal realism is seriously undermined. The paper is organized as follows. I outline the commitments of Lewisian modal realism and introduce the advanced modalizing problem in section 2. In sections 3–7, I discuss variant analyses of modality which attempt to overcome the problem without surrendering Lewisian modal realism. I argue that they all fail. In section 8 I show how the problem also affects stage theory (Sider 1996) and modal ctionalism (Rosen 1990). Section 9 is a brief conclusion. 2 Lewisian modal realism and advanced modalizing By 'Lewisian modal realist', I mean someone who holds the following theses: (1) There is a plurality of worlds, all of the same ontological kind (2) Worlds are maximally spatiotemporally connected, and so distinct worlds do not have any parts in common (3) Ordinary individuals are parts of just one world and have their properties simpliciter, not merely relative to this world or that 1 I shall argue that no analysis of modality is available to someone who holds these theses. My target of course includes Lewis (1968; 1971; 1983b; 1986), but is wider: one might disagree with Lewis on what causation is or on what properties or dispositions are, for instance, and yet still count as a Lewisian modal realist. Lewisian modal realists, having accepted that individuals are worldbound, are committed to some version of counterpart theory. The standard account analyses a de dicto statement of possibility such as 'possibly, there are Fs' as saying that there is a possible world at which there are Fs, i.e., a world which has Fs as parts (Lewis 1983b). It analyses a de re possibility statement 'possibly, a is F' as saying that there is a world w and a counterpart a′ of a in w such that, in w, a′ is F. More generally, if 'A(a1, . . . , an)' contains constants (or other unbound terms) 'a1', . . . , 'an', then the standard Lewisian account will analyse 'possibly, A(a1, . . . , an)' as follows: (4) There is a possible world w and counterparts a′1, . . . , a ′ n of a1, . . . , an (respectively) in w such that, at w, A(a′1, . . . , a ′ n) To complete the account, we need to know how the phrase 'at w' works. Lewis (1968) provides an account in terms of a translation from quantied modal logic to the extensional language of counterpart theory. On that translation, 'at w' has the effect of recursively restricting all quantiers in its scope to w's domain. But Lewis (1986) abandons any such attempt at systematic translation from QML. For Lewis (1986), 'at w' works 'mainly by restricting the domains of quantiers in its scope, in much the same way that the restricting modier 'in Australia' does' (Lewis 1986, p. 5). But it need not restrict all such quantiers, just as 'in Australia' does not restrict to Australia the 'any' in 'in Australia, there's some philosopher cleverer than any other' (Lewis 1986, p. 5). Since I want my target to include both Lewis 1968 and Lewis 1986, I won't assume that 'at w' always restricts all quantiers within its scope. I'll call the analysis above the Lewisian analysis. This analysis runs into the advanced modalizing problem (Divers 1999), as follows. The Lewisian modal realist holds that: (5) There are many possible worlds She also accepts the rule of possibility introduction, from 'A' infer 'possibly, A'. For, given the inter-denability of 'possibility' and 'necessarily', possibly introduction is equivalent to the factivity of necessity, from 'necessarily, A' infer 'A', which she accepts. (Divers (1999, p. 218) holds that possibility introduction 'approaches the status of analyticity'.) So, from (5), the Lewisian modal realist is committed to: (6) Possibly, there are many possible worlds She will analyse this as saying (7) There is a possible world at which there are many possible worlds that is, there is a possible world which contains as parts many possible worlds. But she cannot accept (7), for she holds that worlds are spatiotemporally maximal, 2 and hence that each possible world has just one possible world, itself, as a part. So a Lewisian modal realist cannot accept the analysis of modality from above. (Divers (1999) discusses and rejects some attempts to head off this argument; see also Divers 2014. I agree with him that the problem is genuine and requires modication to the analysis of modality.) In what follows, I will investigate alternative (but still broadly Lewisian) counterpart-theoretic analysis of modality which the Lewisian can accept and which overcome the advanced modalizing problem problem from above. I'll argue that none can be sustained. 3 The redundancy analysis Divers (1999) presents a solution to the advanced modalizing problem, as follows. He claims that, in general, the content of an utterance may be restricted to a single world, or unrestricted, applying to all of modal space. So, we may understand: (8) There are no ying hippos in a restricted way, as saying (truthfully): (8r) In our world, there are no ying hippos or we might understand it unrestrictedly, as saying (falsely, by Lewisian modal realist lights): (8u) There are no ying hippos anywhere in modal space For Divers, 'the semantic function of a possibility operator on a non-modal quanticational sentence is always that of quantifying in, by way of a variable that is already reserved for worlds' (Divers 1999, p. 229). So prexing a content of the form: in w, A with 'possibly' has the effect of changing 'in w' to 'there is some world w such that, in w'. So, prexing the restricted content (8r) with 'possibly' will give us the content: (9) For some world w, in w, there are ying hippos which is true by Lewisian modal realist lights. But Divers notes that, in an unrestricted content such as (8u), there is no phrase 'at w' for 'possibly' to alter. So prexing that content with 'possibly' will have no effect. As a consequence, when 'A' is read unrestrictedly, 'possibly, A' will have the same (unrestricted) content as 'A' itself. The same goes for 'necessarily': on unrestricted contents 'A', 'necessarily' has no semantic effect, so that 'necessarily A' and 'A' have the same (unrestricted) content. Using this analysis, Divers can avoid the problem from above. The Lewisian modal realist of course intends her utterance of 'there are many worlds' to be unrestricted (since it is false when restricted to any world). But then prexing 3 'possibly' to that content has no semantic effect, and so (6) has the content: there are many worlds, and not the content assigned by (7). Divers' account has much to recommend it: it is semantically non-ad hoc (Divers 1999, p. 230) and it ties in well with Lewis's own remarks about the semantic effect of 'possibly' (Lewis 1986, p. 5). But it cannot be sustained, for their are (by Divers' lights) unrestricted contents which (by anyone's lights) change their truth-value when prexed by 'necessarily'. Here is one example. Consider Anna, who is taller than Bill. As Anna and Bill are perfectly normal humans, it is a contingent matter that Anna is taller than Bill. Now suppose further that Anna and Bill are not worldmates of one another, but are nevertheless both regular human beings very much like you and me. Then (10) Anna is taller than Bill should be true, but contingently so. (10) has either a restricted content: (10r) In the actual world @: Anna is taller than Bill or it has an unrestricted content: (10u) (In all modal space:) Anna is taller than Bill The former is clearly false, for Anna and Bill are not worldmates and hence are not both parts of any one world. By contrast, (10u) is true, given the genuine modal realist's metaphysics. So, appealing to charity, we should disambiguate (10) to the unrestricted reading (10u). When the genuine modal realist utters (10), we understand her as asserting (10u). But prexing (10u) with either 'possibly' or 'necessarily' is redundant, says Divers. So we can thus infer from the unrestricted content of (10) to: (11) It is necessary that Anna is taller than Bill As a consequence, what the genuine modal realist asserts when she utters (10), namely (10u), is not contingent. But it clearly is a contingent matter that Anna is taller than Bill. Divers' analysis gives us the wrong results on this score, and that is reason enough to reject it. Divers may claim that (11) is harmless, since (by his lights) it says no more than (10). But it is not sufcient for Divers merely to nd an acceptable reading of (11). The data to be explained is the deep intuition that (10) is contingent. Anna and Bill are regular human beings (albeit spatiotemporally separated ones) with regular physical human bodies, embedded within physical environments, and subject to physical laws very much like our own. Then it is certainly true – perhaps it is even a conceptual truth – that it is a contingent matter who out of Anna and Bill is taller. But Divers' redundancy analysis cannot say this, and so must be rejected. We must consider some other analysis of modality. 4 4 The disjunctive analysis In this section, I consider a broadly Lewisian analysis of modality which takes its cue from Divers' redundancy analysis. The idea is to take 'possibly, A' to be true iff it is true either on the Lewisian analysis or on a redundancy reading of 'possibly'. (My thanks to an anonymous referee for suggesting this analysis.) The content assigned to 'possibly, A(a1, . . . , an)', on this theory, is the disjunction: (12) Either there is a world w and counterparts a′1, . . . , a ′ n of a1, . . . , an respectively in w such that, in w, A(a′1, . . . , a ′ n); or A(a1, . . . , an) Call this the disjunctive analysis. One may worry that the approach is semantically ad hoc (on what basis does a sentential operator result in a disjunctive content?), but let us set that worry to one side. The main advantage of this approach is that it overcomes the problems from sections 2 and 3. If we begin with some consistent 'A' and infer 'possibly, A', this cannot lead to contradiction on the disjunctive analysis. A disjunction is inconsistent iff all its disjuncts are; but one disjunct of the resulting analysis of 'possibly, A' is 'A' itself, which we assumed to be consistent. A problem is not far away, however. We generate the problem by forcing the analysis of 'possibly, A' to ignore the redundancy disjunct, 'A'. Take the hunk of desk-shaped matter on which my laptop is currently sitting; call it 'Hunk'. Hunk isn't a world but (by ordinary modal standards) it could have been: things could have been such that Hunk exists unaccompanied by anything wholly distinct from it. Moreover, had Hunk existed unaccompanied, the Sydney Harbour Bridge would not have been a part of Hunk. It certainly need not have been. So: (13) Hunk (h) isn't a world but it could have been a world lacking the Harbour Bridge (b) as a part Applying the disjunctive analysis and using 'W ' for 'is a world', 'P' for 'is a part of', and 'C' for 'is a counterpart of', (13) has the content: (14) ¬Wh& (∃w∃x∃y(Ww& Pxw&Cxh& Pyw&Cyb&Wx& ¬Pyx)∨ (Wh & ¬Pbh)) As 'Wh' in the second conjunct's second disjunct contradicts the rst conjunct '¬Wh', (14) entails (15) ¬Wh & ∃w∃x∃y(Ww & Pxw & Cxh & Pyw & Cyb &Wx & ¬Pyx) But, given Lewisian modal realist metaphysics, this is unsatisable. Supposing that there are entities w, x, and y such that Ww & Pxw & Cxh & Pyw & Cyb & Wx & ¬Pyx we infer w = x (since w and x are worlds and x is a part of w) and hence Pyx & ¬Pyx: contradiction. So (15) as a whole is unsatisable. We have inferred a contradiction from a truth to which the Lewisian modal realist is committed, and so the disjunctive analysis is no better off than the standard Lewisian analysis or the redundancy analysis. Again, we must consider some other analysis of modality. 5 5 The world-free analysis In this section, I consider a 'world-free' analysis of modal language. The idea is to drop the requirement that the counterparts invoked by an occurrence of 'possibly' must be parts of the same world as one another, by dropping quantication over worlds altogether. (This was suggested to me by Harold Noonan.) 'Possibly' then has the semantic effect of existentially quantifying over counterparts; but it no longer existentially quanties over possible worlds (and so does not restrict embedded quantiers to any particular world). So 'possibly, A(c1⋯ cn)' is analysed as: (16) There are counterparts c′1, . . . , c ′ n of c1, . . . , cn, respectively, such that A(c′1, . . . , c ′ n) Call this theworld-free analysis. This approach avoids the advanced modalizing problem from section 2. As (6), 'possibly, there are many possible worlds' is de dicto, it will be assigned the unrestricted content there are many worlds, just as on Divers' redundancy analysis. More generally, if the Lewisian modal realist accepts 'A(c1, . . . , cn)', then inferring 'possibly, A(c1, . . . , cn)' cannot be problematic as a result of the content it is assigned by the world-free analysis. For then, since each individual is a counterpart of itself, c1, . . . , cn themselves will witness the existential commitments of that content. On this analysis, what's possible is xed entirely by the counterpart relation. This brings with it a number of advantages over the standard Lewisian analysis. It allows us to make sense of possibilities involving transworld individuals, abstract entities, and island universes, for example. It allows that (in an appropriate context) I could have had spatiotemporally disconnected parts, for example. This seems a welcome addition of exibility to the analysis of modality, especially given that there will remain contexts in which 'I could have been a transworld individual' is false. In analysing whether the world-free analysis is adequate as a theory of possibility, I want to consider the effect it has on de dicto modality. By ordinary standards, one can truly assert: (17) There could have been no penguins But this is false on the world-free analysis, which treats (17) as having the false content: (18) (Unrestrictedly:) there are no penguins This is rather bad news for the world-free analysis, for there is clearly some sense in which there might have been no penguins. The defender of the world-free analysis might counter by analysing (17) as a de re statement about the actual world, i.e.: (19) The actual world could have been such that, at it, there are no penguins 6 This is true, on the world-free analysis, for the actual world (at least in some contexts) has counterparts which contain no penguins. The general strategy for the world-free analysis, therefore, is to treat each de dicto modal claim as a de re modal claim about the actual world @, with quantiers restricted to @. This way of treating de dicto claims is problematic. There exists something that is not part of our world. So by possibility introduction, the Lewisian modal realist is committed to: (20) It is possible that there exists something that is not part of our world But treating this as a de re claim about the actual world, as above, this is analysed as: (21) There exists a counterpart w of the actual world such that, at w, something is not part of w But this is false. Although we allow (with Lewis 1986) that in general 'at w' need not restrict to w all quantiers within its scope, it must so restrict some of them, else 'possibly' it is not functioning as a modal operator at all (Noonan 1994). Hence in (21), 'something' is restricted to w and so (21) cannot be satised. The world-free analysis faces an additional problem, which I shall introduce in section 7 (since it is a problem for the analyses discussed in sections 6–7 as well). For now, given the issue just raised, I do not think the world-feee analysis is tenable. 6 The many-worlds analysis In this section, I consider yet another variant on the Lewisian analysis. One might locate the problem with the original analysis not in its use of worlds tout court, but in its restriction to single worlds in the analysandum. Since cases of advanced modalizing typically focus on situations involving multiple worlds, it is natural to suspect that these problems can be avoided by relaxing this restriction in the analysis of possibility statements. We can do this for a statement 'possibly, A(a1, . . . , an)' by allowing the counterparts of a1, . . . , an to be parts of distinct worlds. (This move was suggested by an anonymous referee for this journal.) On this approach, 'possibly, A(a1, . . . , an)' is treated as: (22) There are worlds w1, . . . ,wn and counterparts a ′ 1, . . . a ′ n of a1, . . . , an, respectively, such that each ai is a part of wi and, at the plurality w1, . . . ,wn: A(a′1, . . . , a ′ n) Whatever is possible according to Lewis's original analysis is possible according to the many-worlds analysis (just take the case in which w1 = ⋯ = wn) but not vice versa. In particular, the many-worlds analysis provides a consistent reading of (6); but unlike the world-free approach, it handles such de dicto possibility statements in a standard way. In this respect, it is an improvement on the views considered hitherto. 7 The problem with the many-worlds analysis is as follows. Consider Anna (a) and Bill (b) from section 2, who are not spatiotemporally related to one another. The Lewisian modal realist accepts that there is a mereological sum s of (just) Anna and Bill: s = a ⊔ b. So, using 'S' for 'are spatiotemporally related to one another', we have: (23) ¬Sab & s = a ⊔ b So by possibility introduction, we infer: (24) Possibly, ¬Sab & s = a ⊔ b The many worlds analysis assigns this the content ∃w1∃w2∃w3∃x1∃x2∃x3(Ww1 & Px1w1 & Cx1a & Ww2 & Py2w2 & Cx2b & Ww3 & Px3w3 & Cx3s & at w1,w2,w3 ∶ ¬Sx1x2 & x3 = x1 ⊔ x2) (25) But this content is incompatible with Lewisian modal realist metaphysics. Since '¬Sx1x2 & x3 = x1 ⊔ x2' contains no quantiers (or quantier-like terms), the phrase 'at w1,w2,w3' is redundant. Since x3 is a part of w3 and x1 and x2 are both parts of x3, it follows from the transitivity of parthood that both x1 and x2 are parts of w3. Hence (given the denition of 'world') x1 and x2 are spatiotemporally connected, contradicting '¬Sx1x2'. The many-worlds analysis fails. But there is a closely-related approach which avoids this problem, which I'll discuss in the next section. 7 The plurality of worlds analysis The many-worlds analysis treats a de re possibility statement about a1, . . . , an by requiring that, for each ai, there is a world wi and a counterpart a′i of a in wi. This rules out any of the ais having counterparts bigger than any world: hence the problem from section 6. But we can relax the requirement, saying instead that, for some plurality of worlds, a1, . . . , an have counterparts somewhere in that plurality. Thus 'possibly, A(a1, . . . , an)' is analysed as: (26) There are worlds w1, . . . ,wn and counterparts a ′ 1, . . . a ′ n of a1, . . . , an, respectively, such that each ai is a part of the plurality w1, . . . ,wn and, at the plurality w1, . . . ,wn: A(a ′ 1, . . . , a ′ n) This approach seems immune to advanced modalizing problems of the kind used in section 2. If 'A(a1, . . . , an)' is true of some plurality of worlds, then that plurality of worlds plus a1, . . . , an themselves will witness (26), and so no contradiction will be derivable. This is, I think, the best analysis of possibility statements available to the Lewisian modal realist. It overcomes all the problems faced by the analyses discussed so far; it retains the spirit of Lewis's original 8 proposal; and it allows as possibilities states of affairs which, given Lewisian metaphysics, it is very natural to think should be possible. The problem with the plurality of worlds analysis concerns not possibility but truth simpliciter. Utterances are true, not merely relative to some world or other, but true simpliciter. These need not be unrestricted utterances such as (5); everyday (restricted) utterances such as 'I exist' and 'there are no unicorns' are true simpliciter. On Lewis's analysis, 'A' is true simpliciter iff it is true relative to its world of utterance. 'I exist' and 'there are no unicorns', as uttered by me, are true simpliciter because they are true relative to my world, the actual world. On the plurality-of-world analysis (and also the many worlds analysis), however, we are interested in analysing contents relative to some plurality of worlds. So should we continue to analyse truth simpliciter as truth relative to a given world, or as truth relative to a given plurality of worlds? Neither option is very happy. Suppose we take the conservative option and continue to say that, by denition, an utterance of 'A' is true simpliciter iff it is true relative to the world of utterance. Then it is analytic that, for restricted contents, truth simpliciter requires truth relative to some world. So it is also analytic (given how Lewisian metaphysics denes 'world') that truth simpliciter requires truth relative to some spatiotemporally connected entity. But, given the plurality-of-worlds analysis, some possible truths are not like this. 'There are exactly two penguins, and they are not worldmates' is false but possibly true, on the plurality of worlds analysis. The problem is that, on the present approach, it is analytic that it is false simpliciter, and an analytically false statement cannot possibly be true. So we must reject this rst option. We avoid the problem if we allow that 'A' is true simpliciter iff it is true relative to a plurality of worlds, including the world of utterance. But there are many such pluralities. If we require an utterance 'A' to be true relative to all such pluralities in order for it to be true simpliciter, then very little will be true simpliciter. We won't capture the intuitive truth (simpliciter) of 'there are no unicorns' (under its restricted reading), because there are pluralities of worlds which include both ours and a world of unicorns. If on the other hand we require an utterance 'A' to be true relative only to some such pluralities, in order for it to be true simpliciter, then we arrive at a contradiction: 'there are no unicorns' will come out both true (simpliciter) and not true (simpliciter). So we must reject this option, too. Note that the problem applies equally to the world-free and many-worlds analyses. Bricker's (2001) solution to the problem is to adopt an absolute notion of actuality. On this view, some world or plurality of worlds is uniquely actual; all the other worlds are somehow not ontologically on a par with these. They exist but are, in a deep metaphysical sense, non-actual. Bricker's approach resolves the issue because then truth simpliciter can then be treated as truth relative to the unique actual world or plurality of worlds. There are a host of problems with for Bricker's view. But we can ignore them for present purposes, for the approach is not compatible with Lewisian modal realism. It is precisely the Lewisian insistence that each possible world is ontologically on a par which generates the problem; hence Lewisian modal realists cannot accept the plurality of worlds analysis. 9 8 The scope of the problem So far, I have argued that no analysis of modality is compatible with Lewisian modal realism. I have done that by considering and rejecting a number of proposals. Of course, there may be an analysis of modality, compatible with Lewisian modal realism, which I have not considered here. But there is strong reason to believe that, even if there is some further analysis to be given, it cannot differ too greatly from the analyses discussed above. Lewisian modal realism, in accepting non-overlap of worlds, commits one to a counterpart-theoretic analysis of modality. Given the need to analyse restricted de dicto contents such as (17), worlds must play a role at some point of the analysis, where all of those worlds are ontologically on a par. Given those constraints, it is highly likely that any Lewisian modal realist analysis of modality will suffer from the problems discussed above. Before concluding, I want briey to consider whether these arguments also affect metaphysical theories which are parallel to or parasitic upon Lewisian modal realism. I'll consider the stage theory of persistence (Sider 1996) and modal ctionalism (Rosen 1990). Stage theory (Sider 1996) is four-dimensional in its ontology: past, present, and future entities exist. It differs from the Lewisian 'worm view' of persistence (Lewis 1976), on which an ordinary material object is a four-dimensional fusion of object-stages, in that stage theory identies the ordinary objects with the temporal stages. According to the stage view, a person is a particular temporal stage, rather than a fusion of suitably-related person-stages. Stage theory deals with temporal statements in much the same way that the Lewisian analysis deals with modal statements. Lewis's unity relation I (Lewis 1976) and world-stages are for the stage theorist what the counterpart relation C and possible worlds are for the Lewisian modal realist. According to stage theory, 'a was taller than b' says that there is a past world-stage with parts x and y such that x is I-related to a, y is I-related to b and x is taller than y. (It does not say that a past stage I-related to a is taller than a past stage I-related to b, else 'I was taller than my father' would be true, even though I've always been shorter than him.) Given this parallel, an argument can be run against stage theory in just the way it is run against the Lewisian analysis of possibility, with the 'at some time' operator for 'possibly' and the 'is a world-stage' in place of 'is a world'. The various amendments to the Lewisian approach discussed in sections 3–7 are all prima facie available to the stage theorist, but all suffer from analogues of the problems discusses in those sections. (For brevity, I won't discuss those arguments here.) So stage theory appears to be as bad off as Lewisian modal realism. Modal ctionalism (Rosen 1990) does not accept that there is a plurality of worlds; but it nevertheless uses a ction of modal realism (roughly, The Plurality of Worlds) to analyse modal statements. It treats 'A' as true iff 'A@', the Lewisian analysis of 'A' relativised to the actual world @, is true according to the ction of genuine modal realism, suitably supplemented by an encyclopaedia of all actual, non-modal truths. Thus, the ctionalist will not assert things like 'there are ying hippos at other worlds', but will assert that, according to the ction, there are ying hippos at other worlds, and hence that (literally) there could have been ying hippos. 10 The problem with this approach is that the advanced modalizing problem shows that the ction of modal realism is inconsistent with that analysis of possibility statements. The modications to the analysis of possibility discussed in sections 3–7 carry over to the modal ctionalist analysis: 'A' will be true iff the relevant analysis of 'A' is true according to the ction of Lewisian modal realism. But the arguments from sections 3–6 show that the rst four of those analyses are problematic: in each case, both the analysis of 'A' and the negation of the analysis are true, according to the ction. But if the ction is inconsistent, then everything is true according to it; and so truth simpliciter, without the ction, trivialises. I want to consider, briey, whether the modal ctionalist fares better if she adopts the plurality-of-worlds analysis (section 7) as her way of mapping modal statements onto her ction. The worry considered in section 7 concerned truth simpliciter. We cannot take truth simpliciter to be truth relative to the world of utterance, since some truths concern more than a single world. But neither can we take truth simpliciter to be truth relative to some plurality of worlds, for there are many such pluralities and not all will agree on the truth of some 'A'. Modal ctionalism seems to offer a way out of this worry. A unique ctional world or plurality of worlds has a special status, namely, being the world or plurality of worlds which represents concrete actuality. So there is a distinguished ctional world or plurality of worlds (just as there is on Bricker's (2001) solution), even though the ction is Lewisian. Thus, 'A' is true simpliciter iff its plurality-of-worlds analysis is true in the ction, relative to whichever ctional world or plurality of worlds corresponds to concrete reality. (For simplicity, call that unique ctional world or plurality of worlds actual.) So prima facie, the modal ctionalist can avoid the worry from section 7. A problem remains, however. Truth (simpliciter) is governed by the T-scheme: A iff 'A' is true (simpliciter). So given the suggested analysis of truth (simpliciter), (T) A iff the analysis of 'A' is true in the ction, relative to the actual world or plurality of worlds will be an analytic truth. But what modal status does (T) have? Suppose we evaluate the right-hand-side of (T) relative to some world of the ction. To do so, we need to know whether 'the ction' and 'the actual world' behave rigidly. Suppose they do not, so that the referent of 'the ction' varies across worlds. Then, we will have a ctional world w whose ction disagrees with w on what is the case. Then some instance of (T) will be false relative to w and so the ctionalist must take (T) to be possibly false. But (T) is analytic, and so cannot be possibly false. Suppose instead that 'the ction' and 'the actual world' behave rigidly. Then whatever is true relative to the actual world or plurality of worlds is necessarily true relative to the actual world or plurality of worlds. So for some contingent 'A', the right-hand-side of (T) will be true relative to any world of the ction. But since that 'A' is contingent, for some world w of the ction, 'A' will be false relative to w, and so (T) as a whole will be false relative to w. Again, the ctionalist must take (T) to be possibly false. But (T) is analytic, and so cannot be possibly false. 11 So the modal ctionalist cannot make use of the approach from section 7 after all. In sum, modal ctionalism (if based on Lewisian modal realism) must be rejected. 9 Conclusion I have argued that Lewisian modal realism cannot give a satisfactory analysis of modality. If so, the justication for Lewisian modal realism is seriously undermined. I also argued that parallel arguments equally affect stage theory and modal ctionalism (as based on Lewisian modal realism). I dened Lewisian modal realism by its central commitment to (1)–(3). Surrendering (1) in favour of a plurality of ersatz worlds would avoid the advanced modalizing problem with which I began (section 2), as well as the problem from section 5. Surrendering (1) by holding an absolute notion of actuality, as Bricker (2001) does, would avoid the argument from section 7. Surrendering (2) would allow a world to contain other worlds, thus avoiding the problems from sections 2 and 5. Rejecting (3), as McDaniel (2004) and Yagisawa (2010) do, may avoid commitment to counterpart theory at all (since individuals may then be parts of multiple worlds). But it is not clear that this will resolve the issue, given that the problems from sections 2 and 5 concern de dicto possibilities. These are precisely the options that Lewis (1986) argued so forcefully against. So there are options for modal realists, but none the Lewisian would accept. References Adams, Robert 1974: 'Theories of Actuality'. Nous, 8, pp. 211–31. -- 1981: 'Actualism and Thisness'. Synthese, 49, pp. 3–41. Bricker, Philip 2001: 'Island Universes and the Analysis of Modality'. In Preyer and Siebelt 2001, pp. 27–55. Divers, John 1999: 'A Genuine Realist Theory of Advanced Modalizing'. Mind, 108, pp. 217–39. -- 2002: Possible Worlds. London: Routledge. -- 2014: 'The Modal Status of the Lewisian Analysis of Modality'. Mind, 123, pp. 861–72. -- and Melia, Joseph 2006: 'Genuine Modal Realism: Still Limited'. Mind, 115, pp. 731–40. Lewis, David 1968: 'Counterpart Theory and Quantied Modal Logic. The Journal of Philosophy, 65, pp. 113–26. -- 1971: 'Counterparts of Persons and their Bodies'. The Journal of Philosophy, 68, pp. 203–11. -- 1976: 'Survival and Identity'. In Rorty 1976, pp. 17–40. 12 -- 1983a: Philosophical Papers Vol I. Oxford: Oxford University Press. -- 1983b: 'Postscripts to 'Counterpart Theory and Quantied Modal Logic''. In Lewis 1983a, 39–46. -- 1986: On the Plurality of Worlds. Oxford: Blackwell. -- 1996: 'Elusive Knowledge'. Australasian Journal of Philosophy, 74, pp. 549–67. McDaniel, Kris 2004: 'Modal Realism with Overlap'. Australasian Journal of Philosophy, 82, 137–52. Melia, Joseph 2001: 'Reducing Possibilities to Language'. Analysis, 61, pp. 19–29. Noonan, Harold 1994: 'In Defence of the Letter of Fictionalism'. Analysis, 54, 133–9. Paseau, Alexander 2006: 'Genuine Modal Realism and Completeness'. Mind, 115, 721– 30. Preyer, Gerhard and Frank Siebelt (eds) 2001: Reality and Humean Supervenience: Essays on the Philosophy of David Lewis, Lanham: Rowman and Littleeld. Rorty, Amélie (ed.) 1976: The Identities of Persons. Berkeley: University of California Press. Rosen, Gideon 1990: 'Modal Fictionalism'. Mind, 99, 327–54. Sider, Theodore 1996: 'All the World's a Stage'. Australasian Journal of Philosophy, 74, 433–53. -- 2002: 'The Ersatz Pluriverse'. The Journal of Philosophy, 99, 279–315. Yagisawa, Takashi 1988: 'Beyond Possible Worlds'. Philosophical Studies, 53, 175–204. -- 2010:Worlds and Individuals, Possible and Otherwise. New York: Oxford University Press.

Ciência e Determinismo Daniel Durante Pereira Alves 10 de agosto de 2007 A mesa de meu escritório tem 80 centímetros de altura. Isto é um fato real. Vamos agora inventar uma estória usando este fato real. Imaginemos que ao esticar a mão para apanhar uma borracha, eu esbarro no grampeador, empurrando-o para fora da mesa. Como estamos imaginando, nossa estória pode continuar de muitas e interessantes maneiras: o grampeador pode transformar-se em um pára-quedas e cair suavemente, ou pode ficar flutuando à minha frente, ou ainda pode enfurecer-se com meu descuido e atacar-me com seus grampos afiados. Ele pode até cair no chão, demorando 0,4 segundos no percurso. Se você estivesse em um show de perguntas e respostas da TV e fosse perguntado sobre qual destas alternativas ocorreria na realidade, qual você escolheria? Eu sei, antes da ocorrência do fato, que tanto o grampeador, quanto uma moeda ou qualquer coisa não muito leve que cair de minha mesa de 80cm de altura levará 0,4 segundos para chegar ao solo. Nada cairá em menos tempo e apenas folhas de papel ou coisas grandes e leves cairão em mais tempo. Uma velha equação da cinemática nos ajuda a separar a ficção da realidade, indicando qual é a única continuação verossímil para nossa estória. Qualquer coisa não muito leve que caia de uma altura h não muito grande em metros vai demorar √︁ h 5 segundos para chegar ao solo. As outras alternativas de nossa estória podem até render boas bilheterias no cinema, mas não correspondem à realidade. Não há como mudarmos isso. O tempo de queda dos objetos não depende de nossa vontade. Se por acaso eu não puder mais conviver com a idéia de que tudo que cai de minha mesa demora 0,4 segundos para atingir o solo, tenho apenas duas opções: ou mudo a altura de minha mesa, ou instalo meu escritório na Lua, onde as coisas demoram √ 1,2h segundos para cairem de uma altura de h metros. Os objetos levariam 0,98 segundos para caírem de minha mesa, com a monótona desvantagem de que as folhas de papel e demais coisas grandes e leves também levariam os mesmos 0,98 segundos para a cair. Há muitos outros fatos que podemos conhecer antecipadamente com esta segurança e precisão. Sei, por exemplo, que no dia 1∘ de agosto de 2008 haverá um eclipse total do Sol, mas não será visível do Brasil. O próximo eclipse total em nossas terras só ocorrerá em 2045. Este tipo de conhecimento seguro, preciso, que nos torna capazes de separar a ficção da realidade e prever o futuro é, de acordo com o filósofo Karl Popper, o grande legado da ciência natural e a principal característica que define e distingue o conhecimento científico. Mas se a ciência nos torna capazes de prever o futuro, então o futuro já está determinado hoje, e tal determinação nos retira o poder de atuar, interferir e decidir sobre os fatos. Se sei hoje que haverá um eclipse total do Sol em 2045, então este evento futuro *Artigo publicado no volume 89 da revista Comciência (ISSN 1519-7654) eidtada pela SBPC/UNICAMP em agosto de 2007 (http://www.comciencia.br) 1 já está determinado hoje e não há nada que possamos fazer quanto a isso. Se a previsão de uma certa cartomante de que eu jamais me tornarei milionário for verdadeira, então eu não tenho poder algum para mudar esta situação. Faça o que eu fizer. O determinismo, segundo esta abordagem, seria um subproduto inevitável da ciência, e a nossa liberdade de atuação estaria cada vez mais limitada, quanto mais a ciência avançasse. A descrição científica de um fenômeno representaria uma sentença de sua inevitabilidade e de nossa impossibilidade de modificar a situação descrita. Mas um momento! Há dois problemas com esta linha de raciocínio que vincula a ciência ao determinismo paralisante. O primeiro deles, mais simples, já tem sido bastante discutido, desde que o filósofo Francis Bacon, no início do século XVII, ressaltou que o conhecimento das leis gerais e universais que regem os fenômenos naturais, ao invés de retirar-nos poder de atuação, ao contrário, constitui-se em instrumento de poder, controle e dominação sobre a natureza. A maneira mais eficiente de eu evitar que os objetos que por ventura caiam de minha mesa demorem estes irritantes 0,4 segundos para atingir o solo é conhecer a equação t = √︁ h 5 e assegurar-me de que a altura da mesa não seja de 80 centímetros. Para controlarmos e dominarmos a natureza, não devemos desafiar suas leis, mas segui-las com inteligência. A espantosa capacidade de ingerência nos fenômenos naturais que a tecnologia dos nossos dias nos proporciona é uma prova irrefutável do sucesso desta estratégia. O telefone celular, a bomba atômica, o coquetel anti-AIDS, a soja transgênica, o avião e todos os produtos tecnológicos com os quais atuamos, modificamos e controlamos diversos aspectos da natureza não desafiam suas leis. Ao contrário, as seguem. E sua construção só foi possível porque ampliamos muito o conhecimento das inexoráveis leis da natureza. A informação de uma cartomante nos paralisa, porque ela nos dá acesso apenas aos resultados e não às suas regras de previsão. A cartomante ou não nos diz o quê determina nosso futuro ou, se diz, atribui esta determinação a elementos completamente fora de nosso alcance, tais como uma certa carta de baralho, uma predisposição divina, ou uma relação mística entre configurações astronômicas e padrões de comportamento. As previsões da ciência, ao contrário, não são paralisantes porque elas explicitam em detalhes as regras de determinação do futuro e estas regras envolvem fatores naturais manipuláveis. Se queremos modificar o tempo de queda de determinado objeto, mudemos a sua altura. É claro que nem todos os elementos presentes nas leis de previsão científicas são manipuláveis. Não há muito o que podemos fazer para evitar o eclipse total do Sol em 2045. Mas os elementos manipuláveis presentes nas informações científicas são suficientes para nos dar esta espantosa capacidade de controlar e modificar a natureza, distanciando-nos bastante da aparente paralisia determinista. O outro problema ligado à vinculação da ciência ao determinismo é um pouco mais sutil. O aspecto sistemático e organizado do conhecimento científico costuma exigir das teorias mais do que meras regras de descrições dos fenômenos. A equação cinemática t = √︁ h 5 descreve a queda livre, apontando como, sob determinadas circunstâncias, o tempo de queda dos objetos varia com relação a altura da queda. Mas muitas vezes os cientistas vão além da descrição sobre o "como" dos fenômenos e perguntam: por que? A teoria descritiva do movimento dada pela cinemática não é capaz de explicar este porque. Quem explica porque, na queda livre, o tempo varia com relação a altura como descrito na equação acima é a Mecânica que, para explicar a cinemática, postula duas novas entidades: massa e força. A mecânica não se limita a descrever de modo preciso os fenômenos que observa2 mos. Ela atribui causas a estes fenômenos e trata de coisas que não podemos observar. Você já "viu" uma força? Eu nunca "vi". Todas as informações empíricas que temos sobre as forças são indiretas. São, de fato, conseqüência das forças e não as forças propriamente ditas. Sei que quando pulo, volto ao chão. Sinto o movimento, a aceleração inicial, me sinto quase parado quando a subida termina e a queda se inicia, sinto o impacto da queda com o solo,... mas a força de gravidade eu não sinto nem vejo. A força de gravidade é algo que explica estes fenômenos, mas estes fenômenos perceptivos não são a força de gravidade. Talvez um exemplo da astronomia ajude a esclarecer. Disse antes que em 2045 haverá um eclipse do Sol. Pois bem, esta previsão é baseada em uma concepção astronômica que descreve os movimentos da Terra do Sol e da Lua de acordo com a teoria heliocêntrica de Copérnico. O Sol é o centro do sistema solar, é orbitado pela Terra que por sua vez é orbitada pela Lua. Quando a Lua fica diretamente no caminho entre o Sol e a Terra, esta faz uma sombra na Terra, durante o dia, e esta sombra representa o eclipse total do Sol. Conseguimos prever o eclipse do Sol de 2045 porque Kepler e outros astrônomos aperfeiçoaram as teorias de Copérnico e forneceram dados precisos sobre posições, velocidades e percurso das órbitas da Lua, da Terra e demais planetas. Mas você já deve ter ouvido falar na antiga astronomia geocêntrica de Ptolomeu, que descrevia o cosmos de uma maneira radicalmente diferente. A Terra era estacionária e o centro do universo. Sol, Lua, demais planetas e estrelas orbitavam a Terra. Para Ptolomeu havia várias esferas celestes, uma para a Lua, outra para o Sol, uma para as estrelas fixas e para cada planeta. Estas esferas ligavam-se à Terra por finas engrenagens. Todos os movimentos eram descritos como complicadas composições de movimentos circulares. Pois bem, acontece que com esta cosmologia quase mitológica para os nossos padrões contemporâneos, também conseguimos prever o eclipse do Sol que ocorrerá em 2045! Ptolomeu, em seu famoso livro Almagesto, apresenta dados e descrições detalhadas deste sistema antigo de astronomia. E o sistema funciona para calcularmos a posição aparente dos planetas, da Lua, prever eclipses da Lua e do Sol e para praticamente todas as informações astronômicas que conseguimos coletar com instrumentos simples. Ora, como é possível que uma teoria tão equivocada, segundo os nossos parâmetros contemporâneos tenha tamanho sucesso preditivo? Como podem duas teorias tão radicalmente diferentes (a ptolomaica e a copernicana) concordarem na descrição aparente de tantos fenômenos? Para responder a estas perguntas precisamos perceber que as teorias astronômicas, tanto a de Ptolomeu, quanto a Copérnico, não são meramente descritivas. São também explicativas. Ambas vão além dos fenômemos que podemos observar e perceber e incluem elementos que explicam estes fenômenos. Se nos concentrarmos apenas no que podemos observar e perceber, poderíamos fazer um mapa celeste e nele descrevermos os movimentos da Lua, Sol, planetas e estrelas neste mapa. Se esquecermos os significados dos termos tanto dos usados nos cálculos geocêntricos quanto dos heliocêntricos, e nos concentrarmos unicamente na posição dos astros no mapa celeste dadas pelas duas teorias astronômicas, veremos que a única divergência entre elas é que, em cada uma delas, usamos contas diferentes para chegarmos aos mesmos resultados. Elas concordam com a posição dos planetas e estrelas no mapa e prevêem com a mesma acuidade suas posições futuras. Reparem que a única informação empírica que temos dos fenômenos astronômicos é o posicionamento e contínuo deslocamento dos astros no mapa celeste. Ptolomeu acrescentou a estes fenômenos engrenagens, esferas celestes e epicíclos que não vemos, mas 3 que explicam e dão estrutura ao que vemos. Copérnico, por seu turno, também acrescentou estruturas não perceptivas aos fenômenos celestes. Afinal de contas algum de vocês já experimentou a sensação do movimento de rotação da Terra? Eu me sinto bem parado agora, enquanto estou sentado em casa escrevendo este texto. Não temos acesso a nenhuma informação empírica ou experiência fenomênica perceptível que comprove o movimento de rotação da Terra. É claro que não estou defendendo o geocentrismo. Temos hoje um caminhão de razões para preferirmos a astronomia heliocêntrica à geocêntrica, mas não é inconcebível que possamos ter uma astronomia geocêntrica que seja compatível com as demais ciências físicas. O único problema é que uma mudança tão profunda e central em nossas estruturas de entendimento exigiria tantas outras mudanças de concepções, tantas alterações em outras teorias já aceitas, que simplesmente não vale a pena. Melhor concentrar esforços em descobrir novos aspectos da realidade do que gastar tanto trabalho e energia numa mudança deste tipo. Estas mudanças radicais só costumam ocorrer em poucos momentos da história da ciência e constituem o que o filósofo Thomas Khun chamou de Revoluções Científicas. Vale lembrar que a revolução copernicana (a substituição contrária, do geocentrismo pelo heliocentrismo) foi um processo lento e muitas vezes violento, que levou mais de 150 anos para se consolidar. De qualquer forma, nenhuma revolução científica, por mais radical que seja, nos fará esquecer as coisas que já sabemos, ou fará parar de funcionar as coisas que já funcionam. Da mesma forma que não perdemos a capacidade de prever o eclipse total do Sol que ocorrerá em 2045 quando trocamos o geocentrismo pelo heliocentrismo, nenhuma revolução científica fará os aviões caírem, ou os celulares pararem de funcionar, ou o coquetel anti-AIDS perder seus efeitos, ou o meu grampeador levar menos de 0,4 segundos para cair de minha escrivaninha ao chão. As revoluções científicas não interferem na característica mais fundamental das teorias científicas: a sua capacidade preditiva. A ciência parece então possuir dois tipos de conteúdo, um que é cumulativo e imune às revoluções científicas, que vou chamar de descritivo, mas que também poderia ser nomeado preditivo, ou empírico ou, como prefere meu aluno Arthur V. Lopes, a quem atribuo esta classificação, fenomênico. Em Física, a Cinemática e a Termodinâmica são exemplos de teorias cujo conteúdo é quase que exclusivamente descritivo. O outro tipo de conteúdo, a que chamarei de explicativo, é aquele que explica e dá estrutura ao conteúdo descritivo da ciência. É este o tipo de conteúdo que é substituído nas revoluções científicas, que é incerto e contingente, é teórico ou, como prefere Arthur, hipotético. Ele corresponde às idéias mais abstratas das quais não temos experiência fenomênica e que organizam racionalmente as descrições científicas. As teorias científicas definidoras de paradigmas, na nomenclatura de Thomas Khun, são exemplos claros de teorias que apresentam alto grau de conteúdo explicativo, tais como a Mecância de Newton, as Astronomias de Ptolomeu e Copérnico, ou a Teoria da Evolução de Darwin. Estes dois tipos de conteúdo são partes essenciais do conhecimento científico. Sempre que em alguma área ou disciplina um dos tipos prevalece sobre o outro, os cientistas sentem um incômodo, ou porque precisam explicar melhor os fenômenos que descrevem, quando falta conteúdo explicativo e sobra descritivo, ou porque precisam verificar se suas idéias correspondem aos fatos, quando falta conteúdo descritivo e sobra explicativo. Bem, e o determinismo? Por que defendo que esta distinção pode ajudar a nos prevenir de uma postura determinista com relação a ciência? Se por um lado a capacidade preditiva da ciência representa uma aproximação com o determinismo, embora não o 4 paralisante, conforme já vimos, por outro lado, devemos lembrar que o termo determinismo costuma ser relacionado ao princípio segundo o qual um conhecimento suficientemente detalhado do presente, nos daria capacidade para prever qualquer acontecimento do futuro e reconstituir qualquer acontecimento passado. Ora, conforme dissemos, o conteúdo explicativo é essencial à atividade científica. Mas o conteúdo explicativo é contingente, corrigível, não verificável, passível de substituição, como nas revoluções científicas. Mais ainda, mesmo que nenhuma revolução científica seja capaz de destruir o que já sabemos, deixando intacto o conteúdo descritivo da ciência, as revoluções científicas podem (e em geral o fazem) propiciar um grande aumento no conteúdo descritivo das teorias científicas. Por exemplo, já havia descrições cinemáticas bastante precisas antes de Newton propor sua Mecânica, mas não havia nenhuma cinemática lunar, simplesmente porque não havia nenhum movimento por lá para ser descrito. Nossa equação de queda livre lunar t = √ 1,2h só se tornou possível como conseqüência do conteúdo explicativo proporcionado pela Mecânica de Newton. A Revolução Copernicana, a Teoria da Evolução, a Relatividade, a Mecânica Quântica, todas as revoluções científicas trouxeram possibilidades novas e ampliaram muito o conteúdo explicativo da ciência. Como não temos razão nenhuma para crer que nossas teorias exprimem conteúdos explicativos definitivos, uma vez que este conteúdos não são nem testáveis; como uma mudança paradigmática é sempre possível, não temos jamais razão para crer que possamos vir a ter este conhecimento suficientemente detalhado do presente que nos daria acesso a todo o futuro e todo o passado. A ciência possui sim um aspecto cumulativo, sem dúvida prevê o futuro e nos ajuda a distinguir a realidade da ficção, mas ela jamais nos levará à paralisia do determinismo.

Circulation Unbound: Hegel, Heidegger, and the State David Kolb, Bates College Modernity means freedom, we say, and circulation let loose: commodities, technology, choices, the autonomous individual. In contrast to our free exchange, we imagine old traditional societies as regulated exchange along a network of posts defined by fixed roles. In those societies identities and roles were experienced as naturally given. They were not experienced as constituted (and questioned) by the circulation among them, nor as exchangeable or substitutable one for another.1 The extreme opposite of that picture of traditional society would be a Baudrillardian flow of identities and signifiers none of which have any solidity. Or the Heideggerian Gestell where all identities are available for use and consumption, exchange and substitution in a depthless circulation of beings made indifferently available. In such a world our individual freedom can be trivialized by the circulation that we thought guaranteed it. We picture our modern or postmodern selves as unbound from traditional social roles taken as fixed by nature. Have we then entered a realm of total exchange, a realm in which all is malleable, open for use and substitution? Is the circulation that surrounds us domesticated or monstrous? In this essay I examine how Hegel and Heidegger envision the role of the State in binding up the unlimited flows of modernity. I begin with an overall summary, then look at several issues in more detail. For Hegel, the state is the civic totality, encompassing and architectonically allotting space to differentiated subordinate spheres of life. The state provides legitimation and security of function and meaning. It limits and binds together the circulation of goods and mutual recognition. The state is an end (but not a total end). For Heidegger, the state is a kind of beginning (but not a pure beginning). In his Introduction to Metaphysic the polis provides a place [topos] for life; it grounds and preserves all communal activities. The polis works in these and lets them work in its space.2 For Hegel, the state is a whole that is a process with a form. We do not judge a state according to how it was founded; we judge according to what form its constitution enacts. If we ask what is or has been the historical origin of the state in general, still more if we ask about the origin of any particular state . . . these questions are no concern of the Idea of the state. . . . So far as the authority of any existing state has anything to do with reason, these reasons are culled from the forms of the law authoritative within it. (PR 258) 2 For Heidegger, the state is a expression of a destiny, an event, and a task to which no constitutional or social form could be adequate. The beginning "stands before us....it awaits us as a distant command bidding us to catch up with its greatness."3 Heidegger is fascinated by the "violent" act that founds a people or state in responding to that call. In "The Origin of the Work of Art" he speaks of the "state--‐ founding Act" [staatgründende Tat] and its possible relation to the "essential sacrifice" [wesentliche Öpfer] that act can demand.4 Even after his adventures in the thirties and forties, even when he plays down such heroic masculine rhetoric, Heidegger still talks of a beginning and a destiny to be retrieved. The Event bestows a meaning that is never captured by any particular institutional form. While our current social forms are part of our response to the call of our regime of presence and possibility, the call does not legitimate those particular forms. For both Hegel and Heidegger there is a sense in which the state has the function of binding and arresting a circulation that threatens to get out of hand. This is explicit in Hegel's discussion of the relations of civil society and state. The rational state legitimates limits. Circulation seems unbound in civil society, which gets its form from the inter--‐replaceability of commodity circulation, but even there the universal lurks. It finally expresses itself in the state that binds circulation into the rational process. We can see in Heidegger's political engagements a hope that a genuine response to the destiny of the folk might break the inauthentic spiral of technological das Man. Perhaps the political and social pessimism of his later writings reflects a conclusion that there is no vocation in the political that can limit that circulation. The step back that is thinking does not directly accomplish anything social, and it certainly does not legitimize particular limits or forms of social interaction. There is a deeper point. In the end, Hegel's system binds exchange and circulation by showing that, despite its seeming universality the circulations of goods and services and recognition within civil society and the state are only limited operations. They themselves exist within an ontologically fuller motion and circulation (of Spirit) whose "infinitude" includes self--‐defined limits. Heidegger cannot accept this implied metaphysics of fullness. But his own notions about the conditions of the possibility of das Gestell insert that circulation within a "deeper" process (temporalization and our receptive relation to the Event) that has its own finitude and epochal limits. Later thinkers revise the relations of these levels. In Baudrillard and others the deeper process is itself redefined as an unlimited circulation, so that the conditions of the possibility of ordinary events now open things up even further to exchange and substitution. 3 I turn now to examine several of these points in more detail. Although Hegel's analysis of civil society anticipates much of what has been discussed in more recent theories, there are crucial differences. In Hegel's system, Spirit's self--‐ recognition circulates around and comes to itself, but signs and status do not circulate as freely. The "infinite" movement of Spirit puts limits on all other circulations. Hegel analyzes commodities in terms of their use--‐value. Mutual need fuels civil society. But while this or that need can be satisfied, needs in general multiply. Particularity by itself, given free rein in every direction to satisfy its needs, accidental caprices, and subjective desires, destroys itself and its substantive concept in this process of gratification. At the same time, the satisfaction of need, necessary and accidental alike, is accidental because it breeds new desires without end, is in thoroughgoing dependence on caprice and external accident. (PR 185) This multiplication of needs goes on ad infinitum (PR 191) with no qualitative limits (PR 195). In this circulation our needs become both more particularized and more abstract, as do our relations with other people: while particular life--‐styles and occupations are multiplied, I interact with you only in your role as a provider of food (PR 192). Yet the circulation of particulars is not the whole story. "To particularity [the Idea] gives the right to develop and launch forth in all directions; and to universality the right to prove itself not only the ground and necessary form of particularity, but also the authority standing over it and its final end" (PR 184). Heidegger describes a similar multiplied circulation of goods and meanings. He offers two basic descriptions of what he sees as modern total circulation: universal planning and universal use. Universal planning involves a guarantee of the stability of a constant form of using things up . . . it develops the completely equipped plan and certainty of all plans whatsoever . . . the encompassment of areas, the particular realms of human equipment necessarily become 'sectors'. . . . the planing calculation of the guarantee of the whole of beings.5 This picture of a differentiated whole arrayed around a controlling center is not yet the full circulation in Gestell. Heidegger later thinks our world as a standing reserve for use without a central will or plan from which it can be organized. Everything is available. Everything is used. But there is no longer a center and an overall plan; there is only availability and use without end and without mutuality. This will be "the essence of modern technology--‐--‐the steadily rotating recurrence of the same."6 In Hegel's view, such total leveling and exchange does not occur because the circulation of need and commodities within civil society develops its implicit limits. 4 The infinitely complex criss--‐cross movements of reciprocal production and exchange, and the equally infinite multiplicity of means therein employed, become crystallized, owing to the universality inherent in their content, and distinguished into general groups (PR 201). Civil society's circulation creates fixed posts within itself. Groups and functions come to occupy these posts whose identity cannot be easily exchanged away. Some of these derive from natural givens, others from the division of labor. The state helps to articulate civil society's movement by ratifying some of these institutions and roles and by creating others. We can see Hegel creating such fixed posts in his treatment of the agricultural class that is to embody the spirit of the nation. Despite its historical becoming, the national spirit (a people's characteristic values, styles of acting, and sense of identity) is in the experience of its citizens an immediate given. This is especially the case for those citizens living an agricultural life. The life of agricultural people, Hegel says, does "not owe much to reflection," but is centered on planning and future storage, and an immediate sense of ethical substantiality, family, and trust (PR 203). In Hegel's state, agricultural capital and property cannot be sold. It must remain in the family line; in legal terms, it is entailed and cannot be alienated. This removes agricultural capital from market exchange. Similarly, the large agricultural landowners are given political position and power by birth. These maneuvers create a fixed point that is endowed with the national values and that exists outside the circulation of commodities and power. It is tempting to view Hegel's treatment of the agricultural class as a pragmatic concession to the Prussian Junkers, but I see it as crucial to his theory. Hegel wanted farmers and landowners to embody an immediate unreflective sense of particular loyalty and values in order to anchor the community amid civil society's whirling circulation of goods and status. The state needs a basis in a feeling of particular identity that is not available for exchange. Without this states lose their particularity and blend into a world--‐wide, anonymous civil society that reduces human identity to that of mere consumers and producers. The role Hegel assigns to women provides a similar anchoring, because women are to have a special loyalty to family values and rootedness in nature. Hegel's treatment of women has been criticized for (among other things) nostalgically romanticizing an oppressive regime. But, as with the farmers, so the women fulfill a necessary function in the theory: they provide the moment of immediacy. Hegel says that the foundation of states is agriculture and marriage. "Security, consolidation, lasting satisfaction of needs, and so forth--‐--‐things which are the most obvious recommendations of marriage and agriculture--‐--‐are nothing but forms of universality, modes in which rationality, the final end and aim, asserts itself in these spheres" (PR 203z). Universality asserts itself; for Hegel these fixed posts are not artificially introduced into the circulation, for they ultimately stem from 5 Spirit's coming to itself through the dialectic of particular, universal, and individual identity. Of course, it is just these things, "security, consolidation, lasting satisfaction of needs" that thinkers from Heidegger to Baudrillard would say are impossible in that total circulation where there is no immediacy, where no form is stabilized, and where there is no ontological guarantee of self--‐return. There are still other anchors that Hegel sees developing within civil society. The division of labor becomes institutionalized into the corporations (which are more like trade associations than what we think of as single corporations).7 "One joins a corporation because of one's talent, birth, but especially one's individual particular will and desire, which receive their right, merit, dignity by this choice" (PR 206). Without being a member of a corporation a man lacks rank and dignity; without a corporation he has to try to gain recognition for himself by giving external proofs of his success in his business, and to these proofs no limit can be set. He cannot live in the manner of his class, for no class exists for him, since in civil society it is only something common to particular persons which really exists. Hence he cannot achieve for himself a way of life proper to his standing and less idiosyncratic. (PR 253z)8 The corporation provides a second family (PR 250f), yet the corporation member "belongs to a whole which is also an organ of the entire society" (PR 253). The state realizes its own form in part by taking up these groupings from civil society: "only by being authorized does an association become a corporation" (PR 253z). Hegel's corporation members are craft workers distinguished by their specialized skills. Hegel does discuss what he calls the rabble [Pöbel] which can be thought of as the beginnings of an urban proletariat. Hegel does not see these people as de--‐skilled workers, however, but rather as welfare dependents. The rabble is not a Marxist proletariat that circulates as raw labor; their problem is that they do not circulate at all, but are outside the system and fed on its surplus. "The sanctity of marriage and the dignity of corporation membership are the two fixed points around which the unorganized atoms of civil society revolve" (PR 255z). "As family was the first, so the corporation is the second ethical root of the state, one planted in civil society" (PR 255). Family and corporation provide ethical substance: ways of living and being that are not exchangeable. Social identities are to be chosen freely but do not circulate freely; the chosen becomes substantial. Yet our freedom is to be preserved. Substance becomes subject; the social can only exist through mutual recognition. It is the dependence of identity on mutual recognition (and the need for that process of recognition to possess particular content and a recognized external vehicle) that puts limits on circulation. We are not free--‐floating individuals facing some resistant structure imposed upon us. Modern society, for Hegel, involves mutual recognition of the rational structures that limit circulation in order to create social space for the exercise of civic and political freedom. 6 "[Plato] could only cope with the principle of self--‐subsistent particularity, which in his day had forced its way into Greek ethical life, by setting up in opposition to this purely substantial state" (PR 185z). In the modern state the individual is educated into free choice of those rational roles that make up the state. Legitimated differentiations within the circulation provide "the process whereby [citizens'] particularity is educated up to subjectivity" (PR 187). When this happens that freedom is achieved that is "having to do only with what it has itself produced and stamped with its seal" (PR 186z). The result is both a mutually free community and the legitimated circulation of power, status, and goods within a functionally differentiated whole. We are familiar with the problematic degree to which Hegel's state--‐--‐or any state--‐--‐can realize this goal. We should not stop at the particular institutional arrangements that Hegel is promoting, however, for these stem from a deeper claim. The circulations of civil society and state lie within a larger motion that educates and tames them. Recall that for Hegel the state originates in a violent act and expresses a national spirit, but in the end it will be judged according to the form enacted in its constitution. States begin in violence and then achieve form. It is not the form--‐giving, but the form itself that is judged. So it must be that Hegel has criteria for judging forms. But how? He cannot rely on a functional judgment because the larger functional ensemble is itself a form to be judged. Yet he needs teleology: the self--‐presence of spirit to itself. What is at stake is the being of form. A form is judged against the conditions of possibility of its being. Since being is to be thought as self--‐expression and self--‐ return, the form will be judged against its possible enactment of that self--‐return, and against its ability to express the process that gives it being. Is the form capable of holding the truth, of being a form of full community, of containing/expressing the life of Spirit? As with Plato, Hegel's criteria for judging a form come from his notion of what constitutes full being. It is on this basis that Hegel can argue that some constitutional arrangements are better than others and that there may be a final rational state. While Heidegger has nothing similar to Hegel's recommendation of particular institutions, there is a parallel in Heidegger to the deeper Hegelian claim. Heidegger talks of stepping back to the context through which forms have their being. There is, however, no legitimation discourse in Heidegger.9 In that sense, there is no foundation for judging political forms. Nor can modern exchange and circulation be confined by any social form. Yet it remains true that for Heidegger any ontic system of exchange is opened up by our relation to being and the Event, and we can ask to what extent communal arrangements help or hinder our recognition of our deep condition. 7 Our condition, however, is not to be moments in a grand Hegelian ontological circulation. Nor is the event of our relation to being infinite or self--‐returning. Paradoxically, if in the modern world the circulation of beings seems infinite and ontically unlimited that is because of our ontological finitude. On the one hand, our entrapment in the infinite technological circulation of beings is our particular finite destiny, which involves a particular temporal structure (the collapse of the dimensions of time into bland availability). On the other hand, our fascination with infinite circulation stems in part from our forgetfulness and avoidance of our deeper ontological finitude.10 The most difficult thing in Heidegger is to keep both the priority (the "nearness") of that ontological condition (so that we can find our authentic relation to being) while not letting the call or destiny in that condition become either purely formal or too particular. If the destiny becomes too formal, the finitude of temporalization is smoothed out into some kind of general ontological circulation. I think Heidegger would rightly complain that thinkers such as Baudrillard deny our finitude and undermine the relation of time and being. But if our destiny is not purely formal, it risks becoming a particular call of the sort that Heidegger thought he could discern in the 1930's. If you do not have Hegel's legitimating ontological process to provide a critical norm, can you safely talk about calls and destinies that are linked to our fundamental relation to being? In this context, we might wonder whether it is really fair to say--‐--‐as Heidegger would--‐--‐that Hegel's foundational discourse offers only the ontic relation of a big being to other beings. Going beyond particular institutional proposals, we can make these comparisons between the "deeper" events of form in Heidegger and Hegel: A) In Heidegger, the step back to the origin behind any ontic origins or definite forms takes us to the event of presencing in its finitude and withdrawal. In Hegel, the step back behind any empirical relation of entities to one another takes us to the event of the presencing of the belonging together of the logical categories. B) In Heidegger, the event of presencing founds nothing; it offers no form. Any foundational relation or structure is ontic. In Hegel, the event of presencing has its own form that is expressed in certain definite self--‐conscious structures. C) In Heidegger, the event of presencing has its own dimensions. These give us some general "formal" restrictions on what can count as an economy of presence or a world. In Hegel, the logical categories and their mutual event define the structures of what must count as a fully self--‐present world. D) In Heidegger, the step back provides no foundation and no detailed criticism, but it does question any claims to legitimacy that might be made by a particular form of society or state. In Hegel, the step back is both foundational and critical. It affirms a structure for the full being of community, justifying some institutions and criticizing others. 8 Note that this critical judgment is accomplished in Hegel by an act of letting be. We let the form develop and exhibit its limitations and its ties to its own context and to the fuller forms that ultimately make it possible. In Heidegger, too, there is a letting be. What is let be is, among other things, the process in which form has its being: but now that process is being's withdrawal in the Event. The process looks to the origin that withdraws, rather than to the goal that approaches. This can be read either as an affirmation of the finitude of the state or as the call that encourages us to obey. In the event of form, for Hegel, historical becoming and earlier versions of the state are lifted up into the current form. In the end, history is wrapped up into a total present. For Heidegger, however, the appeal to origins is an appeal to a history that cannot be capsulized, an origin of possibility that does not reduce to a present field of alternatives however rich. The distension of time opposes the dialectic of time. The past is not for Hegel an origin to be interrogated for fresh possibilities or deeper meaning. The beginning of thought and of the state do not remain rich with undeveloped possibilities. The founding act and the primitive or early forms of community make no calls to us now. History is differentiation, not retrieve. Legitimated differentiation blocks total circulation. Heidegger distrusts differentiation; he prefers interpenetration, each in every, as in his Fourfold. Hegel mistrusts that kind of undifferentiated, unsystematic interpenetration; he thinks that it is a romantic escape. Hegel wants disruption followed by a coming together in rational mediation that maintains tensions within a totality. But in Heidegger's Gestell there is no way for it all to come together in a differentiated totality. It cannot come together that way because it has no form. Gestell has movement, it opens a way beings are present, but it has no overall form or system. It is not the result or cause of differentiation. In Hegel and his successors, the immensely complex criss--‐cross movements of exchange and circulation can be contained because the system of exchange stands as a particular kind of interaction that has in--‐built movements toward universality and rationality. Ultimately this is because the current social differentiation is enclosed in a still larger movement of history and spirit. So Heidegger's Gestell poses a challenge for differentiation theories influenced by Hegel (for instance Jameson and Habermas) facing the postmodern world. For Gestell is not a totality. It is not differentiated, and it has no form. No social arrangements can be anything more than items of use within Gestell; they cannot mediate an overall movement the way political institutions do for Hegel. Yet we cannot lose sight of the fact that Heidegger does agree with Hegel that there is a deeper level of our relation to being than is obvious in the endless exchanges in modern society. 9 Those such as Baudrillard who go beyond Heidegger by rejecting his lingering metaphors of surface and depth make our ontological condition an unlimited circulation. In so doing they offer a perverse romantic organicism in which each and every thing is touching each and every thing. From a Heideggerian point of view, this is ontologically inadequate because it loses the finitude of our temporalization. From a Hegelian point of view, it is ontically misleading because it mis--‐describes the role that particularity and mediated immediacy play in individual and social life. Although everyone we have been discussing would agree that there is no simple givenness or immediacy to our social roles and arrangements, I suspect that it is the Hegelian tradition that has the resources for talking about the strange situation of particular roles and values in our world today. Even though these ways of talking about mediated immediacy are obscure and can be used clumsily, as Hegel used them in the cases of farmers and women, Hegelian thought refuses to make us totally creatures of exchange and distance and irony. Heidegger avoids this too, but he can only speak in a global way about the destiny of the times. His more extreme followers can only turn everything into an ironic play at being itself. While I have not said anything about Derrida in this essay, I think that on the issues I am discussing he belongs more in the Hegelian camp. To put the matter briefly, Derrida never claims that the unlimited economy of signs can be instituted in actual social or economic relationships, while Baudrillard seems to say that commodity society has already done so. Such volatility loses an important dimension in our situation. Many mini--‐ nations and groups within nations are currently creating themselves. In their struggles to give form to themselves they are claiming some immediate basis in national feeling or group identity. The Hegelian tradition can speak of differentiation and insertion into larger processes (which themselves do not have an immediately given form), and this allows more ways to be self--‐critical when making or meeting such claims to group identity. The rhetoric of Gestell (and its successors) provides important warnings to these groups, but does it have much to say in detail? After he abandoned his talk about the destiny of this or that people, Heidegger felt he had nothing to say about concrete situations. Baudrillardian declarations that all these new/old identities are items within the flow of simulacra put the critic in the position of the Enlightenment intellectual who understands what is really going on and can criticize the natives for clinging to outmoded superstitions about the fixity of identity. This does not encourage dialogue about the particularities of concrete situations. A properly chastened Hegel and his successors have at least the possibility of meeting today's renewed concern for identity and givenness with a combination of immediacy and mediation that may hold both criticism and complicity.11 Notes 10 1. I have argued elsewhere that this story about traditional society should be questioned and that it functions largely as a myth of origin for modern self--‐ consciousness. See Postmodern Sophistications: Philosophy, Architecture, and Tradition (Chicago: University of Chicago Press, 1990), chapter 7. 2. Martin Heidegger, An Introduction to Metaphysics, (Garden City: Doubleday, 1959), 12. References to the works of Hegel and Heidegger will be given in footnotes with the exception of references to Hegel's Philosophy of Right, which will be given in the text by paragraph number from the Knox translation (Oxford: Oxford University Press, 1967). When references are given to both German and English editions of a work, they are separated by a slash with the German edition first. Several Heideggerian terms of art are used frequently; Sein is translated as "being," Ereignis as "Event," and Gestell is left untranslated. 3. Heidegger, "The Self--‐Assertion of the German University," cited in Michael Zimmerman Heidegger's Confrontation with Modernity: Technology, Politics, Art (Bloomington: University of Indiana Press, 1990), 67. 4. "The Origin of the Work of Art," Holzwege (Frankfurt am Main: Klostermann, 1963), 50 / Poetry, Language, Thought (New York: Harper and Row, 1971), 61. 5. "Overcoming Metaphysics," Vorträge und Aufsätze I (Pfullingen: Neske, 1967), 83ff / The End of Philosophy (New York: Harper and Row, 1973), 103ff XXVI. 6. Was Heisst Denken? (Tübingen: Niemeyer, 1971), 47 / What is Called Thinking? (New York: Harper and Row, 1968), 109. 7. The state allows free choice of occupation, but the corporation arrangement militates against easy change. The model resembles the common European pattern of choosing or being measured for a status early in your life. 8. The corporation also educates members to a universal point of view. Hegel says: "the corporation [is the place] in which the particular citizen . . . emerges from his single private interest, and has a conscious activity for a comparatively universal end, just as in his legal and professional duties he has his social morality" (Encyclopedia (1830 edition) 534). 9. See Reiner Schürmann, Heidegger on Being and Acting: From Principles to Anarchy (Bloomington: University of Indiana Press, 1987), 89--‐91. 10. "Once, however, in the beginning of Western thinking, the essence of language flashed in the light of being--‐--‐once, when Heraclitus thought the logos as his guiding word, so as to think in this word the being of beings. But the lightning abruptly vanished. No one held onto its streak of light and the nearness of what it illuminated. We see this lightning only when we station ourselves in the storm of being. Yet everything today betrays the fact that we bestir ourselves only to drive storms away. We organize all available means for cloud seeding and storm dispersal in order to have calm in the face of the storm. But this calm is no 11 tranquility. It is only anesthesia; more precisely, the narcotization of anxiety in the face of thinking." ("Logos," Vorträge und Aufsätze III [Pfullingen: Neske, 1967], 25 / Early Greek Thinking [New York: Harper and Row, 1975], 78). Is total circulation another way to calm the storm? The call of the origin must not be reduced to something ontic related to something else ontic in an economy of presence. 11. "Without totality our politics become emaciated, our politics become dispersed, our politics become nothing but existential rebellion. Some heuristic (rather than ontological) notion of totality is in fact necessary if we are to talk about mediations, interrelations, interdependencies, about totalizing forces in the world" (Cornell West, "Interview with Cornell West," in Universal Abandon: The Politics of Postmodernism [Minneapolis: University of Minnesota, 1988], 270).

ÉTUDE CRITIQUE L. Brisson, Platon : Parménide, Traduction, introduction et notes, Paris, GF-Flammarion, Troisième édition revue et mise à jour, 2011. Rares sont les traducteurs que l'on traduit à leur tour. Pour euxmêmes je veux dire, plutôt que pour pallier le fait qu'on ne connaît pas la langue de l'original comme dans le fameux brocard foscolien du temps où l'ignorance du Grec ancien était encore une marque d'infamie. Luc Brisson fait partie de cette élite : quatre ans après être sortie sur le marché francophone, sa traduction du Parménide de Platon était présentée au public italien dans la version d'Amelia Riccardo1. On ne peut que saluer l'initiative des éditeurs péninsulaires : s'il y a une traduction du dialogue platonicien qui méritait d'être reprise c'est bien celle de Luc Brisson, que nous présentons ici dans sa troisième édition. Comme dans les tirages antérieurs (1994, 1998), la traduction est précédée d'une introduction de soixante pages environ (p. 9-73), d'un plan du dialogue (p. 75-76) et de trois remarques préliminaires concernant respectivement les treize lieux où Luc Brisson ne suit pas l'édition de Claudio Moreschini (p. 81), la nature et la finalité de sa traduction ainsi que de l'apparat des notes (p. 82). Celles-ci, au nombre de cinq cents onze, suivent la traduction (p. 255-282), qui est également accompagnée de trois annexes consacrées, dans l'ordre, aux interprétations anciennes (p. 285-291) et d'obédience « analytique » du dialogue (p. 293-306), de même qu'à la reconstruction proposée par Gregory Vlastos du célèbre argument que les spécialistes appellent sans doute un peu vite le « troisième homme de Platon » (p. 307-308). ÉTUDE CRITIQUE 185 1 Amelia Riccardo, Platone. Parmenide. Traduzione, introduzione e note di Luc Brisson, Napoli, Loffredo, 1998. REVUE DE PHILOSOPHIE ANCIENNE, XXX (2), 2012 Deux cartes topographiques (p. 77-78), un tableau généalogique des ascendants maternels de Platon (p. 79), une bibliographie raisonnée (p. 309-322), mise à jour pour l'occasion (p. 323-336), une chronologie (p. 337-340) et deux index – l'un thématique (p. 341-344), l'autre nominal (p. 345-347) – viennent compléter ce bel ensemble qui fait désormais un peu moins de 350 pages. Luc Brisson écarte d'emblée deux familles de lectures traditionnelles, dont l'emphase théologico-ontologisante, dans un cas, logicoformalisante, dans l'autre, lui paraissent tout aussi éloignées de l'attitude « plus neutre et foncièrement historique » (p. 10) qu'il revendique pour lui mais aussi pour l'auteur du dialogue. Luc Brisson ne le dit nulle part, mais s'il y a une interprétation dont il prend l'exact contre-pied c'est plutôt celle d'Alfred Edward Taylor2, pour qui le personnage éponyme serait le défenseur d'un monisme axé sur l'intelligible. Ses attaques ne viseraient point les Formes que l'on saisit par la pensée ; elles seraient plutôt dirigées contre l'univers sensible dont l'illusion a encore assez d'emprise sur le jeune Socrate pour qu'il hésite à lui dénier toute réalité (ce qui, bien entendu, permettrait de sortir par le haut des apories de la participation). Tout à l'opposé, Luc Brisson considère que Parménide (aussi bien celui de l'histoire ou plutôt de la préhistoire de la métaphysique que celui auquel le génie de Platon a prêté la parole dans le dialogue qui porte son nom) est le partisan d'un monisme certes rigoureux, mais dont l'unité n'est pas celle d'une réalité intelligible, la Forme, mais celle de l'univers sensible tout entier, c'està-dire du tout qui se trouve englober l'ensemble des choses sensibles, qui ne font qu'un mais que l'homme perçoit comme étant plusieurs. L'effet le plus spectaculaire de cette identification de l'Un parménidéen avec le monde tout court est que l'objet sur lequel porte l'exercice dialectique qui occupe toute la deuxième partie du dialogue ne serait pas celui que les interprètes croient le plus souvent. Contrairement à ce que Parménide semble affirmer en 137b3 (« περὶ τοῦ ἑνὸς αὐτοῦ ὑποθέμενος », que Luc Brisson traduit « en faisant porter <mon hypoLeone GAZZIERO186 2 Cf. Alfred Edward Taylor, The Parmenides of Plato, Oxford, Clarendon Press, 1934, p. 13. thèse> sur l'un lui-même »), cet objet n'est pas τὸ ἓν αὐτό, mais précisément τὸ πᾶν, le tout pris dans sa dimension cosmologique. Les conséquences qu'il s'agit de dégager à partir de l'alternative « s'il est un ou bien s'il n'est pas un » (137b3-4) ne concerneraient donc pas l'un lui-même, mais cela dont on prédique le fait d'être un ou pas un, à savoir l'univers sensible pris dans sa totalité. On a beaucoup discuté des mérites et des inconvénients de ce changement radical de paradigme herméneutique. Denis O'Brien lui a même découvert un ancêtre illustre en la personne de William Wardlaw Waddell3, chez qui on peut effectivement lire dans une note ad 137b3-4 : « we should probably be nearer the truth if we understood εἴτε ἓν ἐστιν <τὸ πᾶν> εἴτε μὴ ἕν, as in 128b, which would modify the argument a good deal »4. Luc Brisson lui-même est d'ailleurs revenu sur sa découverte à plus d'une reprise5. Par souci de brièveté et de pertinence, évoquons en quelques détails la considération d'ordre méthodologique qui est au coeur d'un dispositif qui, non sans raison, a « emballé » plus d'un lecteur6. Accordons à Luc Brisson les principaux points qu'il a entrepris de montrer et qui ont été débattus à un moment ou à un autre. Convenons avec lui qu'en dépit du fait qu'il est tout sauf naturel de lire et comprendre la thèse de Parménide comme il le fait, celle-ci se construit en 137b2-4 sur un sujet sous-entendu, à savoir τὸ πᾶν que l'on a rencontré bien avant, en 128b1, et qui brillera par son absence tout le long de ÉTUDE CRITIQUE 187 3 Denis O'Brien, « "L'hypothèse" de Parménide (Platon, Parménide, 137a7 137b4) », Revue des Études Grecques, 120, 2007, p. 418. 4 William Wardlaw Waddell, The Parmenides of Plato after the Paging of the Clarke Manuscript, Glasgow, James Maclehose and Sons, 1894, p. 110. 5 Luc Brisson, « "Is the World One ?" A New Interpretation of Plato's Parmenides », Oxford Studies in Ancient Philosophy, 22, p. 2002, p. 1-20 ; « "S'il (= le monde) est un". La seconde partie du Parménide de Platon considérée du point de vue de Parménide et de Zénon », dans M. Barbanti et F. Romano (éd.), Il Parmenide di Platone e la sua tradizione, Catania, CUECM, 2002, p. 41-57. 6 Cf. Yvon Lafrance, « Compte rendu de Parménide. Platon. Traduction inédite, introduction et notes par Luc Brisson. Collection "GF-Texte intégral", Paris, Flammarion, 1994, 333 p. », Dialogue, 35, 1996, p. 393. la deuxième partie, qui lui serait pourtant tout entière consacrée7. Admettons aussi que l'expression intervient avec la valeur que Luc Brisson lui assigne, malgré le fait qu'elle ne figure pas comme telle dans les fragments du poème de Parménide et qu'il n'est pas évident non plus que chez Platon lui-même, qui s'en sert aussi ailleurs pour résumer le point de vue de Parménide (et de Mélissos), elle inclue les réalités sensibles voire coïncide avec elles plutôt qu'elle ne les exclue en les reléguant dans le non-être de ce qui apparaît et disparaît sans le moindre frémissement de ce tout qui est sans commencement et sans fin, immobile, continu, inengendré et impérissable8. Concédons surtout que le registre matérialiste et l'approche spatio-temporelle de l'exercice dialectique de la deuxième partie du dialogue, qui a agacé Luc Brisson pendant une décennie d'étude et de réflexion9, soit une Leone GAZZIERO188 7 Le point a été soulevé avec beaucoup de force par Denis O'Brien, « Le Parménide historique et le Parménide de Platon », dans A. Havlicek et F. Karfik (éd.), Plato's Parmenides, Prague, Oikoumene, 2005, p. 234-256. On remarquera seulement que la construction que Luc Brisson impose au texte est singulièrement absconse : si vraiment il fallait interpréter le « περὶ τοῦ ἑνὸς αὐτοῦ ὑποθέμενος, εἴτε ἓν ἐστιν εἴτε μὴ ἕν, τί χρὴ συμβαίνειν; » (137b34) comme si on lisait « εἴτε ἓν <τὸ πᾶν> ἐστιν εἴτε μὴ ἕν » à la place de « εἴτε ἓν ἐστιν εἴτε μὴ ἓν », comment explique-t-on que l'énoncé en question fait pendant à « περὶ τοῦ ἑνὸς αὐτοῦ ὑποθέμενος » ? Qu'est-ce qui a bien pu empêcher Platon d'utiliser en lieu et place de cette dernière clause un « περὶ τοῦ παντὸς <αὐτοῦ ὑποθέμενος> » que l'on rencontre ailleurs sous sa plume (cf. e.g. Timaeus, 48e2 et passim) ? Au reste, est-il plausible que l'objet d'une hypothèse, en l'occurrence l'un lui-même, soit prédicat plutôt que sujet (ou sujet et prédicat à la fois) des questions que l'on pose à son égard ? 8 Cf. Francesco Fronterotta, « Fra Parmenide e Platone. Una nuova edizione francese del Parmenide », Giornale critico della filosofia italiana, 74, 1995, p. 382-390. 9 Luc Brisson, « Prefazione », dans Amelia Riccardo, op. cit., p. 12 : « Ma dopo circa dieci anni di studio e di riflessione sul Parmenide, mi infastidiva una questione irrisolta : come è possibile, se la seconda parte del dialogo è dedicata all'Uno inteso come forma per eccellenza, che Platone affermi che "tutto ciò che è si trova da qualche parte" (Parm., 151a3-4 ; cfr. 145e1) e che "deve essere e divenire nel tempo" (Parm., 151e7 152a2) ? E ancora perché bonne raison de douter que son objet soit une réalité intelligible plutôt que sensible, alors même qu'un registre et une approche en tout point similaires ne suscitent pas la moindre perplexité concernant l'objet du réquisitoire contre les Formes de la première partie du dialogue10. Il n'en reste pas moins que la lecture de Luc Brisson repose sur un transfert qui, sans être aberrant comme tel11, demande à tout le moins à être fondé en fait sinon en droit plutôt qu'à intervenir comme s'il allait de soi. « J'ai adopté – nous dit Luc Brisson d'entrée de jeu – une attitude plus neutre et foncièrement historique qui cherche, à travers le témoignage de Platon, à comprendre quelle fut la démarche philosophique de Parménide et de Zénon et quelle stratégie adopta Platon ÉTUDE CRITIQUE 189 chiedersi se l'Uno ha dei limiti o no, un principio, una metà e una fine, se la sua figura è retilinea o circolare e così via ? ». 10 N'est-il pas, à sa façon, tout aussi déconcertant de lire, dans des raisonnements qui portent de toute évidence sur les Formes, que participer c'est participer ou bien à la Forme tout entière ou bien à l'une de ses parties (131a4-5) ? N'est-il pas tout aussi vexant d'admettre qu'il est possible de regarder avec son âme de la même façon la Forme et les particuliers (132a5-6) ou encore de poser comme s'il allait sans dire que si les particuliers ressemblent aux Formes, les Formes aussi ressemblent aux particuliers (132d5-7) ? etc. Pas plus que la distinction entre sensibles et intelligibles – invoquée par le jeune Socrate en 130a1-2 – n'empêche que les intelligibles soient traités comme des sensibles, le conseil de laisser de côté les sensibles pour s'intéresser aux intelligibles – conseil que Parménide lui-même prodigue en 135e1-3 – n'empêchera pas que l'Un soit traité à son tour comme une réalité corporelle. La leçon qui se dégage des deux parties du dialogue est d'ailleurs la même : aussi longtemps que, tout en les séparant, l'on regarde la Forme et les sensibles, l'un et le multiple de la même manière, une série d'apories et de contradictions insurmontables s'ensuivront. 11 Luc Brisson a sans doute raison d'écrire que « la métaphysique aussi a une histoire » (p. 73). Même si la question demeure entière de savoir si c'est l'affaire du métaphysicien d'écrire cette histoire, qui plus est, au moment de la faire, rien n'exclut a priori qu'un penseur du calibre de Platon ait pu faire d'une pierre deux coups et qu'il ait écrit non une, mais deux pages d'histoire de la métaphysique en même temps (l'une comme métaphysicien, l'autre comme historien de la métaphysique). pour se réapproprier les résultats auxquels étaient parvenus ses prédécesseurs » (p. 10). L'ambition d'arracher l'interprétation du dialogue à l'emprise ancestrale de lectures qui ne se soucient pas assez de le replacer dans son contexte d'origine est une chose, une très bonne chose au demeurant. C'est tout autre chose que de procéder comme si Platon partageait ce même souci et qu'il faisait oeuvre d'historien à son tour. Il est à peu près certain que l'intérêt du dialogue et tout d'abord sa signification ne tiennent point au fait qu'il serait un document dont la vocation était de soustraire à l'oubli un entretien philosophique mémorable entre deux figures illustres du passé. Et pour cause : la discussion entre Parménide et Socrate, très probablement, n'a jamais eu lieu. Si tant est qu'il a adhéré à une version quelconque de la doctrine des Formes qu'on lit dans les dialogues de Platon, Socrate le fera bien des années après, comme il le suggère lui-même dans l'esquisse de bilan intellectuel qu'il brosse dans un autre dialogue de Platon, le Phédon. Eût-il pu pressentir l'existence d'entités aussi admirables que les Idées de Platon, Parménide aurait difficilement pu en discuter dans le vocabulaire technique que lui prête la première partie du dialogue, tout comme il aurait difficilement pu utiliser ce même vocabulaire au cours de l'exercice qu'il est supposé avoir mené dans sa deuxième partie. Cela ne veut pas dire que le témoignage de Platon soit biaisé ou particulièrement sujet à caution. Cela veut dire plutôt que Platon n'est tout simplement pas en train de nous offrir un témoignage, même au sens très large où un Hérodote et un Thucydide – auxquels Luc Brisson se plaît à le rapprocher12 – peuvent être considérés comme des sources relativement fiables pour remonter aux événements et aux discours qu'ils se trouvent relayer. Bien entendu, le problème n'est pas tant qu'une fois que l'on admet une dose de fiction, fût-elle homéopathique (mais en l'occurrence elle est massive), dans le récit platonicien, on est forcé de reconnaître qu'il poursuivait un autre but que celui de restituer la pensée de Parménide dans son contexte historique. Le problème n'est pas tant non plus Leone GAZZIERO190 12 Luc Brisson, « "Is the World One ?" A New Interpretation of Plato's Parmenides », art. cit., p. 20. qu'une « stratégie d'appropriation », comme l'appelle Luc Brisson, peut être perçue comme un facteur qui risque de compromettre la valeur de témoignage du document qui s'organise stratégiquement en fonction de cette appropriation. Le problème est plutôt que l'alter-ego platonicien du Parménide historique parle d'au moins deux voix, qui plus est très différentes. S'il faut le croire sur parole, comme Brisson nous encourage en gros à le faire, auquel convient-il alors de se fier ? Est-ce au Parménide pré-métaphysique qui reprend tacitement la thèse par laquelle Socrate a résumé son point de vue au début du dialogue pour la proposer en tant qu'objet de l'exercice qui va longuement l'occuper par la suite ? Ou bien est-ce au Parménide métaphysicien qui encourage Socrate à ne pas s'égarer dans les sensibles et à s'entraîner à la dialectique en se concentrant sur les Formes intelligibles13 ? La question est incontournable, mais Luc Brisson ne la pose nulle part. Elle mérite néanmoins d'entrer en ligne de compte : on pourra toujours s'interroger, après coup, au sujet de l'éventuelle convergence entre le point de vue matérialiste sur les Formes dont Platon montre qu'il finit dans une impasse et les apories du monisme parménidéen de l'Un-Univers. Faire de ce dernier problème, redoutable pour un historien, une sorte de donnée première du puzzle et le ÉTUDE CRITIQUE 191 13 Il est assez curieux de constater que deux au moins parmi les lecteurs les mieux disposés envers la perspective inaugurée par Luc Brisson – à savoir : Yvon Lafrance, art. cit., p. 394 et Sébastien Charles, « Du Parménide à Parménide », Les études philosophiques, 59, 2001, p. 543 – attribuent à Socrate plutôt qu'à Parménide la consigne qui est censée commander le τρόπος τῆς γυμνασίας qui occupera toute la deuxième partie du dialogue : « οὗτος, εἶπεν, ὅνπερ ἤκουσσας Ζήνωνος. πλὴν τοῦτό γέ [135e] σου καὶ πρὸς τοῦτον ἠγάσθην εἰπόντος, ὅτι οὐκ εἴας ἐν τοῖς ὁρωμένοις οὐδὲ περὶ ταῦτα τὴν πλάνην ἐπισκοπεῖν, ἀλλὰ περὶ ἐκεῖνα ἃ μάλιστά τις ἂν λόγῳ λάβοι καὶ εἴδη ἂν ἡγήσαιτο εἶναι [En faisant précisément ce que tu as entendu Zénon faire. Sous la réserve toutefois de ce que tu lui as dit et qui m'a ravi, à savoir qu'il faut ne laisser l'enquête s'égarer ni dans les choses visibles ni même dans ce qui les concerne, mais l'appliquer aux choses qui sont par excellence objets de la raison et dont on pourrait estimer que ce sont des Formes] » (135d7 135e3). point de départ de sa solution est, en revanche, un pari risqué. C'est peut-être d'ailleurs le plus grand mérite de l'interprétation avancée par Luc Brisson. Mieux que d'autres il nous a donné toute la mesure des enjeux d'un tel problème, dont on s'aperçoit mieux – après lui et grâce à son travail – qu'ils sont tout à fait considérables et n'intéressent pas moins les lecteurs et les spécialistes de Platon que ceux du Parménide historique. Traduire une oeuvre est un exercice éminemment ponctuel et ne se laisse apprécier qu'à même le texte. C'est pourquoi, au lieu de me contenter de quelques généralités, toutes flatteuses qu'elles seraient par ailleurs14, j'évoquerai plutôt deux choix de traduction parmi beaucoup d'autres, par lesquels cette traduction tantôt se transcende ou se surpasse elle-même tantôt s'avère moins heureuse qu'à son ordinaire. Je tirerai ces exemples de la section la plus discutée du dialogue, la seule à laquelle Luc Brisson consacre une annexe ad hoc, la dernière (p. 307-308), fort mal inspirée d'ailleurs, puisque des innombrables lectures que ce passage a suscitées, Luc Brisson a cru utile de monter en épingle celle de Gregory Vlastos qui non seulement cumule les méprises15, mais qui est aussi et surtout à l'origine d'une vaste métalittérature dont la caractéristique principale est que les épigones de Gregory Leone GAZZIERO192 14 Luc Brisson est partout à la hauteur de son ambition habituelle, qui est de fournir « une traduction claire, précise et simple » (p. 82). Cf. Luc Brisson, Platon : Lettres, Paris, Flammarion, 1987, p. 64 ; Platon : Phèdre, Paris, Flammarion, 1989, p. 71-72 ; Platon : Timée, Paris, Flammarion, 1992, p. 78. 15 Dans les nombreuses pièces que Gregory Vlastos a versées à ce dossier (à commencer par la plus tristement célèbre : « The Third Man Argument in the Parmenides », The Philosophical Review, 63, 1954, p. 319-349), il a insisté sans cesse sur la tournure déductive de l'argument, alors qu'elle est récursive ou itérative ; pour apprivoiser le texte et le forcer à épouser cette structure logique, il a mis en branle une vaste traque aux prémisses tacites ou dissimulées ; il n'a tenu aucun compte de la dimension psychologique de l'argument, pourtant évidente dans ces quelques lignes où la deuxième personne du singulier est présente partout ; il a même amputé sa traduction du « ὡσαύτως » en 132a5, en dépit du fait qu'il constitue la cheville ouvrière de tout le raisonnement. Vlastos semblent avoir mis leur point d'honneur à corriger ses erreurs par d'autres erreurs16. Platonis Parmenides, Claudio Moreschini (éd.), Roma, Edizioni dell'Ateneo, 1966, 132a1 132b2 : οἶμαί σε ἐκ τοῦ τοιοῦδε ἓν ἕκαστον εἶδος οἴεσθαι εἶναι* ὅταν [2] πόλλ' ἄττα μεγάλα δόξῃ σοι εἶναι, μία τις ἴσως δοκεῖ ἰδέα ἡ αὐτὴ [3] εἶναι ἐπὶ πάντα ἰδόντι, ὅθεν ἓν τὸ μέγα ἡγῇ εἶναι. [4] ἀληθῆ λέγεις, φάναι. [5] τί δ' αὐτὸ τὸ μέγα καὶ τἄλλα τὰ μεγάλα ἐὰν ὡσαύτως τῇ ψυχῇ ἐπὶ [6] πάντα ἴδῃς, οὐχὶ ἓν τι αὖ μέγα φανεῖται, ᾧ ταῦτα πάντα ἀνάγκη [7] μεγάλα φαίνεσθαι; [8] ἔοικεν. [9] ἄλλο ἄρα εἶδος μεγέθους ἀναφανήσεται, παρ' αὐτό τε τὸ μέγεθος [10] γεγονὸς καὶ τὰ μετέχοντα αὐτοῦ* καὶ έπὶ τούτοις αὖ πᾶσιν [132b1] ἕτερον, ᾧ ταῦτα πάντα μεγάλα ἔσται* καὶ οὐκέτι δὴ ἓν ἕκαστόν [2] σοι τῶν εἰδῶν ἔσται, ἀλλὰ ἄπειρα τὸ πλῆθος. <Parménide :> Voici, j'imagine, à partir de quelle considération tu en viens à poser que chaque Forme est une. Chaque fois que plusieurs choses te paraissent être grandes, c'est, je suppose, une seule Forme, qui t'apparaît être la même, lorsque tu les embrasses toutes du regard ; voilà pourquoi tu estimes que le Grand est unique. <Socrate :> Tu dis vrai, répondit-il. <Parménide :> Eh bien, le Grand en soi et ces autres choses que sont les choses grandes, suppose que, de la même façon, avec les yeux de l'âme, tu les embrasses toutes du regard. N'est-ce pas que de nouveau apparaîtra quelque chose d'unique qui est grand, et en vertu de quoi ces mêmes choses dans leur ensemble apparaîtront nécessairement grandes ? <Socrate :> Il semble bien. <Parménide :> ÉTUDE CRITIQUE 193 16 Tel est notamment le cas du seul exemple que mentionne Luc Brisson, mais il n'est ni le seul ni le plus extravagant. De fait Mario Mignucci, « Plato's "Third Man" Arguments in the Parmenides », Archiv für Geschichte der Philosophie, 72, 1990, p. 143-181 ne remet point en discussion la structure pseudo-syllogistique de l'argument héritée de Gregory Vlastos ni l'opportunité de boucher des trous laissés béants par Platon à grand renfort d'axiomes subsidiaires ; il fait si peu de cas du rôle de l'âme dans ce texte que le « τῇ ψυχῇ » en 132a5 disparaît de sa traduction ; si le « ὡσαύτως » en 132a5 ne connaît pas le même sort, il ne joue pas pour autant le moindre rôle dans sa reconstruction de l'argument. C'est donc une autre Forme de Grandeur qui va faire son apparition, s'étendant sur la Grandeur en soi et sur les choses qui participent de cette Forme, ce qui revient à dire que, en plus de la Grandeur en soi et des choses qui en participent, il y aura encore une Forme, différente, en vertu de laquelle la Grandeur en soi et les choses qui en participent seront grandes. Par suite, chacune de tes Formes ne sera désormais plus une, mais elle se multipliera sans limite. 132a2 (ἰδέα). La première occurrence du mot ἰδέα dans le Parménide a été diversement comprise. La plupart des interprètes considère que l'expression renvoie ici à quelque chose comme une propriété commune ou une forme « en nous », intercalée entre les particuliers et la Formes à proprement parler, dont il serait question déjà en 130b3-4 où Platon distingue entre une Ressemblance elle-même (αὐτὴ ὁμοιότης) et une ressemblance que nous possédons (ἡ ὁμοιότης ἡμεῖς ἔχομεν). De son côté, Luc Brisson ne s'est pas encombré d'une telle nature médiane et il a, au contraire, pris ἰδέα en variation synonymique avec εἶδος qui le précède et le suit à la fois dans le texte. Sinon dans l'absolu, du moins du point de vue de la traduction, cette solution est la meilleure que l'on puisse envisager. Elle présente, en effet, beaucoup d'avantages. Notamment, elle tranche dans le bon sens le problème de savoir si ἰδέα et εἶδος ont la même signification ; elle permet également d'éviter les difficultés liées au fait d'admettre une ontologie stratifiée (Formes, propriétés, particuliers) qui s'accorde mal avec la nature des objections soulevées dans la première partie du dialogue, qui lui laissent très peu d'espace (à vrai dire, aucun). Si toute traduction est solidaire d'une interprétation, on pourrait difficilement demander mieux en termes d'élégance de l'une et de sobriété de l'autre. 132a10 (γεγονός, καὶ ἐπί). Si le premier exemple constitue plutôt la règle, évoquons à présent l'exception, ne serait-ce que dans l'espoir de briser la malédiction de Vlastos qui a frappé à point nommé. En 132a10, Luc Brisson traduit le participe parfait γεγονός par « s'étendre » : « C'est donc une autre Forme de Grandeur qui va faire son apparition, s'étendant (γεγονός) sur la Grandeur en soi et sur les choses qui participent de cette Forme » (132a9-10). Il avait pourtant employé le même verbe quelques lignes plus haut, en 131b7, pour rendre – à meilleur Leone GAZZIERO194 escient – un autre participe, καταπετάσας : « ἡδέως γε, φάναι, ὦ Σώκρατες, ἓν ταὐτὸν ἅμα πολλαχοῦ ποιεῖς, οἷον εἰ ἱστίῳ καταπετάσας πολλοὺς ἀνθρώπους φαίης ἓν ἐπὶ πολλοῖς εἶναι ὅλον [jolie façon, Socrate, reprit-il, de faire que la Forme se retrouve une et identique en même temps en plusieurs endroits. C'est comme si tu étendais ( καταπετάσας) un voile sur plusieurs êtres humains et que tu disais : "le voile reste en sa totalité, lorsqu'il est étendu sur plusieurs choses"] » (131b6-8). Tout comme il faut se garder de tenir rigueur à un grand compositeur d'avoir mis trop de notes dans sa partition, il vaut mieux éviter de reprocher à un virtuose de la traduction d'avoir employé trop peu de mots. Cela dit, le fait d'affaiblir l'aspect émergent de la nouvelle Forme – péché tout à fait véniel si tant est qu'il en est un – se double dans la traduction de Luc Brisson d'une initiative plus brutale et, surtout, plus discutable : « ce qui revient à dire que (καί), en plus de la Grandeur en soi et des choses qui en participent, il y aura encore une Forme, différente, en vertu de laquelle la Grandeur en soi et les choses qui en participent seront grandes ». Luc Brisson donne au deuxième καί de la ligne 132a10 (καὶ ἐπὶ τούτοις αὖ πάσιν κτλ.) une valeur épexégétique (« ce qui revient à dire que ») qu'il n'a pas. Il ne s'agit pas de ressasser que la nouvelle Forme s'étend au-dessus des particuliers et de l'ancienne Forme. Si tel était le cas, c'est toute la structure itérative de l'argument qui serait mise à mal. Or, si chacune des étapes du raisonnement reprend la précédente, elle se projette aussi dans la suivante. Les lignes 132a9 132b1 ne font pas exception. De fait, l'« ἄλλο ἄρα εἶδος μεγέθους ἀναφανήσεται, παρ' αὐτό τε τὸ μέγεθος [10] γεγονὸς καὶ τὰ μετέχοντα αὐτοῦ », qu'on lit en 132a9-10, fait pendant au « τί δ' αὐτὸ τὸ μέγα καὶ τἆλλα τὰ μεγάλα », qu'on lit en 132a5. Aussi bien l'un que l'autre capitalisent les acquis de ce qui précède et permettent à l'argument de rebondir. Dans un cas (132a9-10), il s'agit de récapituler l'apparition d'une nouvelle Forme de Grandeur aux yeux de l'âme qui regarde de la même façon la Forme elle-même et les particuliers qui en participent (132a6-7) ; dans l'autre cas (132a5), il s'agissait de faire de même pour la Forme qui s'était révélée à l'âme au niveau de l'inspection des sensibles (132a1-3). Or, traduire, comme le fait Luc Brisson, « καὶ ἐπὶ τούτοις αὖ πᾶσιν [132b1] ἕτερον, ᾧ ταῦτα πάντα ÉTUDE CRITIQUE 195 μεγάλα ἔσται » par « ce qui revient à dire que, en plus de la Grandeur en soi et des choses qui en participent, il y aura encore une Forme, différente, en vertu de laquelle la Grandeur en soi et les choses qui en participent seront grandes » équivaut à entraver la remontée à l'infini au moment même où celle-ci prend son essor. À la place, il vaut mieux traduire « ainsi qu'une autre <Forme>, à nouveau, à part de toutes celles-là, par laquelle toutes ces choses seront grandes ». Il importe, en effet, de basculer vers la position d'une autre Forme ἐπὶ τούτοις πᾶσιν, comme l'association du καί et de l'αὖ le suggère à elle toute seule très fortement et comme la symétrie de ce même αὖ avec celui qui intervient en 132a6 le requiert aussi, dans la mesure où – dans les deux cas – il scande l'apparition récursive d'une nouvelle unité de la multiplicité des choses grandes auxquelles on intègre d'abord une première Idée, puis une deuxième et désormais une troisième. Concrètement, alors que Luc Brisson traduit comme si la Forme en 132b1 (le ἕτερον εἶδος) était toujours la même, la deuxième, celle qui a fait son apparition en 132a6-7 (οὐχὶ ἓν τι αὖ μέγα φανεῖται) et qui est rappelée également en 132a910 (ἄλλο ἄρα εἶδος μεγέθους), il est préférable de traduire comme s'il s'agissait d'une nouvelle Forme, la troisième, qui vient s'ajouter à la deuxième et poursuit ainsi la remontée à l'infini entamée par celle-ci. Puisque l'ouvrage est promis à d'autres révisions, signalons qu'un petit nombre de coquilles déparent encore l'ensemble. Au moins une s'est ajoutée aux anciennes17. Une autre est doublement relapse, puisqu'elle a été corrigée entre-temps ailleurs18. Elles sont cependant tout à fait anodines19, parfois même drôles20 ou, du moins, amusantes à Leone GAZZIERO196 17 La « mise en jour » qu'on lit dans la page de garde. 18 p. 28, note 43, ligne 4 : « en argument dans l'un et l'autre sens » au lieu d'« en argumentant dans l'un et l'autre sens » (erreur rectifiée par l'auteur lui-même dans « Une nouvelle interprétation du Parménide de Platon », dans Pierre-Marie Morel (éd.), Aristote et la notion de nature, Bordeaux, Presses universitaire de Bordeaux, 1997, p. 78, note 17). 19 p. 110, ligne 6 : « au vue de » au lieu de « au vu de » ; p. 111, ligne 7 : « qualifiées » au lieu de « qualifiés ». 20 p. 313, ligne 33 : « Mauricio » au lieu de « Maurizio » ; p. 314, ligne 18 : « Scuola Normale di Piza » au lieu de « Scuola Normale di Pisa ». expliquer21. Le supplément bibliographique omet de mentionner parmi les traductions sorties entre 1994 et 2010 celle de Mary Louise Gill et Paul Ryan, Plato : Parmenides, Indianapolis, Hackett, 1996 que Luc Brisson avait pourtant utilisée pour son « "Is the World One ?" A New Interpretation of Plato's Parmenides » (cf. note 5). Puisque le « troisième homme de Platon » a beaucoup retenu l'attention de Luc Brisson et de son lecteur, la section de la bibliographie qui lui est consacrée aurait dû comporter – à tout le moins – une entrée de plus, qui – elle – mérite d'être considérée comme la « critique la plus complète de la position de Gregory Vlastos », à savoir Henry Teloh et David James Louzecky, « Plato's Third Man Argument », Phronesis, 17, 1972, p. 80-94. Une introduction inspirée, une traduction belle et fidèle, des appendices savants, une bibliographie exhaustive. Le tome 688 de la collection GF Flammarion a sans doute encore de beaux jours devant lui. Leone GAZZIERO Département de Philosophie de l'Université de Genève Laboratoire d'études sur les monothéismes (EPHE, Paris-Sorbonne) [email protected] ÉTUDE CRITIQUE 197 21 p. 309, ligne 18 : dans l'édition de John Burnet le Parménide se lit dans le tome II, non pas dans le tome III comme le signale Luc Brisson. Le dialogue figure en revanche dans le tome III de l'édition genevoise d'Henri Estienne. Ceci explique peut-être cela.

Relations in Anatomical Ontologies∗ Fabian Neuhaus Barry Smith 1 Introduction It is now increasingly accepted that many existing biological and medical ontologies can be improved by adopting tools and methods that bring a greater degree of logical and ontological rigor. In this chapter we will focus on the merits of a logically sound approach to ontologies from a methodological point of view. As we shall see, one crucial feature of a logically sound approach is that we have clear and functional definitions of the relational expressions such as 'is a' and 'part of '. While this chapter is mainly concerned with the general issues of methodology, the chapter of this book on 'Spatial Representation and Reasoning' [1], will apply the methodology to the specific case of spatial relations. Although both chapters are self-contained, we recommend that they be seen as forming a unity. 2 The semantic content of type terms The reason why logical rigor is crucial for the development and use of biomedical ontologies becomes clear if we consider their purpose and mode of operation. The term 'ontology' is used very ambiguously, but in the life sciences 'ontology' means roughly: 'controlled vocabulary in computer interpretable form' and in this chapter we will restrict ourselves to this reading of the term. For a more detailed account of what ontologies are, see [3]. Since ontologies must be not just computer readable but also computer interpretable, an ontology is more than a list of terms stored in a computer parsable format; it comprises also the semantic content associated with these terms – or at least it is supposed to do so. Before we take a closer look at how this works, we need to make some terminological distinctions. First, we shall use the term 'type' in what follows to refer to those entities in reality which terms in ontologies designate. Second, ∗This is a preprint of the chapter 'Modelling Principles and Methodologies – Relations in Anatomical Ontologies' in Albert Burger, Duncan Davidson and Richard Baldock (Editors): Anatomy Ontologies for Bioinformatics: Principles and Practice (to appear in 2007). This work was funded by the National Institutes of Health through the NIH Roadmap for Medical Research, Grant 1 U 54 HG004028. Information on the National Centers for Biomedical Computing can be found at: http://nihroadmap.nih.gov/bioinformatics 1 it is important that we distinguish between terms (like 'female pelvis') and 'part of ' on the one hand, and the semantic content of such terms on the other. In biomedical ontologies there are two kinds of terms: those denoting types (to be more specific, biomedical types) and those denoting relations. Thus in an anatomy ontology the term 'female pelvis' denotes the type Female Pelvis and the term 'part of ' denotes the parthood relation. (Throughout this chapter, we use italics and initial capitals when using type terms to denote types, and quotation marks when we need to talk about the terms themselves. Further we will assume that all terms denote types of the human anatomy, if not explicitly stated otherwise.) Terms are linguistic entities that are created by humans; they satisfy linguistic conventions created by humans; and they can be used to create sentences that express statements about the world. It is terms that are the bearers of semantical content, which means they have denotations – which for present purposes are types and relations. Note that the types and relations that we are talking about are not the familiar entities that we know from set theory. In [10] we propose a distinction between types (for example the type Ear, which is what particular ears share in common) and the sets which are the extensions of such types (the collection of all particular ears). This distinction should be borne in mind to avoid certain sorts of confusion. The types of the biomedical domain (also sometimes called 'universals' or 'kinds') are the patterns in reality that scientists study and describe in their theories. Sets and types behave similarly in one important respect: just as sets have members, so types have instances. However, there are important differences between the two. First: for any arbitrary chosen group of individuals there is a corresponding set, but there need not be a corresponding type. For example, there is the set whose members are exactly: Barbara Bush, Bill Clinton's left foot, and a given red blood cell of my dog; but there is no corresponding type in reality of which exactly these entities are instances. Types are contrasted with such arbitrary collections by the fact that they can serve as objects of scientific investigation and play a role in scientific generalizations (some of which are then captured in ontologies). A second important distinction turns on the fact that the membership relation is timeless, whereas the instantiation of types is timedependent. For example, an animal that instantiates the type Adult Frog now used to instantiate the type Tadpole at some time in the past. Hence types like Adult Frog and Tadpole gain and lose instances over time, where sets cannot gain or lose members. (If you 'add' a member to a set, then the result will be a different set.) Individuals, similarly, can gain and lose parts. However individuals, like this blood cell or that heart are distinguished from the types Blood Cell and Heart by the following criterion: At any time of its existence an individual necessarily occupies a unique spatial location; a type, in contrast, can be (through its instances) fully present at multiple locations. For example you are necessarily present at exactly one location, whereas the type Humanity is currently located at about six billion different locations. With this background it is easy to formulate the problem that ontologies address. As mentioned above, an ontology is not just a list of syntactical strings 2 like 'pelvis', 'urinary bladder', 'body' (its type terms); its goal is to comprehend also the semantical content of these terms – that is, the types which they denote – in a machine readable form. This is more problematic than one might think. If humans do not understand the meaning of a term, we can use dictionaries. For example, assume that you don't speak German and you are wondering what the term 'Handwurzelknochen' means. If you look it up you will come to the following conclusion: 1 The German term 'Handwurzelknochen' denotes the type Carpal Bone. Note that in (1) the German expression is in quotation marks, whereas the corresponding English term on the right hand site is not. This is because in (1) we are using the English term in order to explain the semantic content of the corresponding German term. This strategy works very well – at least for those who understand the expression 'carpal bone'. But imagine that a young child were to ask you what 'Handwurzelknochen' means. Because the child does not know what a carpal bone is, (1) would not be very helpful. In the best case the child would memorize (1) and would afterwards be able to say 'carpal bone' whenever somebody asks what 'Handwurzelknochen' means. But obviously she would not know the semantic content of either term. For this reason (1) would not be an appropriate answer in the given case. It would be better to explain the term to the child for example with the help of pictures in an anatomy textbook. But while we can explain the semantical content of terms to people with the help of examples, paraphrases, pictures, and translations into other natural languages, these strategies won't work for computers. A computer has no better understanding of the term 'carpal bone' than of 'Handwurzelknochen', so statement (1) will provide the computer with no assistance at all in grasping the semantical content of the latter. Of course we could create a digital dictionary which links 'Handwurzelknochen' to the term 'carpal bone'. Such a dictionary might be useful, because it allows human users to find the appropriate translation; but in this case the computer would be in a similar situation as the child who knows that the two terms have the same meaning, but does not understand either of them. And while we can use pictures to educate the child, this strategy, too, will not work for computers. For them we need an alternative approach – ontologies. In a first, rough formulation, the main idea of ontologies is the following. The semantical content of expressions that denote types is captured in ontologies through the assertion of relations between the types. This can be easily seen if we conceive an ontology as a graph, whose nodes are labeled with type terms like 'myelin' or 'lipoprotein' and whose edges are labeled with relation terms like 'is a', 'part of ' and 'located in'. A graph with many labels is a syntactic entity, but it is important to notice that in an ontology each edge of the graph is equivalent to a statement about the corresponding entities in the biomedical domain. For example, if a node labeled 'myelin' and another node labeled 'lipoprotein' are connected by an edge labeled 'is a', then the ontology expresses the statement 'Myelin is a Lipoprotein'. Hence the ontology contains claims about the relations that hold between the types that are the denotations of the 3 type terms of the ontology – and it is here that the ontology gains semantic traction. It is important to notice that in a well-constructed ontology the type terms will be connected by many links to other terms, thus creating a semantical network each link of which represents a statement about some ontological relation between the types in reality represented by its nodes. The idea is that the semantic content of a term is not determined by any one specific link, but rathe by its connection with many other terms that charges it with semantic content. This holistic approach to semantics is not new; it is a central feature of de Saussure's structuralism [6] or of Trier's word field theory [12]. Both de Saussure and Trier identified the meaning of a term with its position in a semantic network of terms. Note however that the holistic thesis in this radical form, although accepted by many contemporary computer science ontologists, is not plausible. For consider the following statements: 2 Shkart is a Trkarp. 3 Brajhn is a Trkarp. 4 Trkarp part of Xriprg. If the semantical content of a type term in an ontology were completely determined by its relation to other terms, we would have a good understanding what 'trkarp' means by looking at (2-4). However, no semantical content is fixed by (2-4), and even adding further similar statements would not bring about a change in this respect. This does not mean that additional statements would not make a difference. On the contrary: each of them puts additional restrictions on the use of the corresponding terms, and thus helps to narrow down their semantic content. For example, (2-4) allow for the possibility that 'trkarp' denotes Pelvis, 'shkart' denotes Female Pelvis, 'brajhn' denotes Male Pelvis, and 'xriprg' denotes Body. However, if we add the additional statement (5), then this possibility is eliminated. 5 Shkart is a Brajhn. Imagine we were to add hundreds of additional statements to our list by using the terms 'shkart', 'trkarp', 'brajhn', 'xriprg' together with a few dozen other similar fantasy type terms. Each connection in the resultant semantic network would restrict the possible interpretations of 'trkarp' and the other terms and thus provide us with extra semantical content. However, even with hundreds of additional statements it would still not be possible to determine which type is denoted by 'trkarp', for there will still be many possible interpretations left. The meaning of the terms will thus not be completely determined, and so the holistic thesis in its radical version is false. It is important to realize the falsity of radical holism, because this has an important consequence: since an ontology can't completely fix the semantics of a type term, the semantic content of a term is in this sense not an all-or nothing matter but a matter of degree: the more a term is connected to other terms, 4 the more its semantical content is determined. A sparse ontology that consists of only loosely connected terms provides these terms with very little in the way of semantic content. In particular, type terms in an ontology that are not distinguished by their connections within the network of terms are semantically indistinguishable with respect to that ontology. For example, assume that we have an ontology that consists of (2-4) and (6). 6 Shkart and Brajhn are disjoint. Statement (6) guarantees that 'shkart' and 'brajhn' do not denote the same type. However, even with (6) the terms 'shkart' and 'brajhn' would still not be distinguishable: all we know about them is that they denote subtypes of Trkarp and that their denotations are disjoint. This limits the value of the given ontology for applications. For example, assume that two scientists use the ontology to annotate their data and that one of them believes that 'shkart' denotes Male Pelvis while the other believes that it denotes Female Pelvis. Since the ontology does not contain any information about the difference between male and female pelvises, an automatic reasoner would never be able to detect that the scientists are using the term 'shkart' in a crucially different way. This example shows why sparse ontologies are inferior to rich ontologies; the latter convey a greater amount of semantic content. 3 The semantic content of relation terms Let us recap the results so far. An ontology is more than a list of type terms, it is designed to encapsulate also the 'meaning' of these terms in a computer parsable form. Biomedical ontologies consist of statements that involve type terms and a relation term; since ontologies are often visualized as graphs, it is helpful to think of their type terms as labels attached to nodes and of the relation terms as labels attached to edges. Since there is no way to tell a computer directly which type is determined by a given type term, ontologies seek to do this indirectly. The basic idea is that the semantic content of a type term is captured by its position in the network of type terms of which it is a constituent. Since each statement expresses a relation between the denotations of its type terms, each statement limits the possible interpretations of its type terms. For example, (7) expresses the thesis that the denotations of 'trkarp' and 'xriprg' are related by the parthood relation, thus limiting the possible interpretations of 'trkarp' and 'xriprg'. 7 Trkarp part of Xriprg. The approach will not be sufficient to single out some specific type as denotation of each given type term, but if the term is connected to a multitude of other terms, then the possibilities will be correspondingly restricted. Note that, according to this approach, the semantical content is determined by restricting the possible interpretations of the type terms via the relations between their respective denotations – in our example the parthood relation between 'trkarp' 5 and 'xriprg'. However, while humans know that the relation term 'part of ' is supposed to express the parthood relation, computers do not. For a computer 'part of ' is just a string like any other, and for the computer (7) is itself a string which is not intrinsically different from a string such as (8): 8 Trkarp cxzc Xriprg. How, then, do we bridge the gap between relation terms and the relations themselves? When we describe the links between the types we use relation terms like 'is a' and 'part of '; but how do we capture the denotations of the latter in a machine interpretable way? This question is important for two reasons. First, it is relations which form the principal vehicle for interoperability of ontologies. Thus if the same relations can be used in all members of a given set of ontologies, then to this degree these ontologies form an interoperable family – an idea which forms one central pillar of the OBO Foundry initiative (http://www.obofoundry.org). The use of common relations when creating a system of ontologies is equivalent to the use of a common gauge when creating an international railway system. Second, the relation terms used in ontologies typically denote rather abstract relations. If we use expressions like 'part of ', 'located in', and 'develops from' as unanalyzed primitives, these expressions are semantically underspecified. As shown in [2] and [11], the result is that they are used in an ambiguous way. For example, in the FMA we find: 9 Female Pelvis part of Body. 10 Urinary Bladder part of Female Pelvis. 11 Urinary Bladder part of Body. Statement (9) is used to assert that every female pelvis is part of a human body, but it does not imply that every body has a female pelvis as part. In contrast, (10) is used to assert that every female pelvis has a urinary bladder as a part, but not that every urinary bladder is part of a female pelvis. The parthood relation between the types denoted in (11) is the strongest of the three: Every urinary bladder is a part of a body and every body has a urinary bladder as part. Another example is the use of 'contains' in GALEN, where we find: 12 Pelvic Cavity contains Ovarian Artery. 13 Male Pelvic Cavity contains Urinary Bladder. 14 Tooth Socket contains Tooth. Here the different statements express very different states of affairs, because the relation term 'contains' is used ambiguously. For every ovarian artery there is a pelvic cavity such that the pelvic cavity contains the ovarian artery. However, not every pelvic cavity contains an ovarian artery. This is expressed by 6 (12). In contrast (13) states that every male pelvic cavity contains a urinary bladder, but it does not say that every urinary bladder is contained in a male pelvic cavity. In (12) and (13) 'contains' denotes distinct relations holding, respectively, between a type of immaterial entity (a cavity) and types of material objects (arteries, urinary bladders). In both cases the material objects are completely located in the cavities. In contrast, 'contains' in (14) relates two types of material objects (tooth sockets and teeth). Further the teeth are only partially contained in the tooth sockets. Hence 'contains' in (14) expresses a relation that is very different from the relations expressed by the same term in (12) and (13). The fact that 'part of ' and 'contains' in statements (9-14) are used ambiguously would be less problematic if the statements were to appear in a text that is intended to be read by humans with some knowledge of anatomy. A human can use background knowledge to disambiguate the statements in appropriate ways. However, a computer is not able to handle ambiguity in the way a human can, so that it is crucial for an ontology that relation terms are used in a clear-cut way; otherwise automatic reasoning is bound to lead to false conclusions (see [2] for examples). In addition, since the relations are used in ontologies to determine the semantical content of the terms in the ontology, a lack of clarity with respect to the relations will contaminate the whole ontology. For this reason, too, therefore it is essential for the use of an ontology that the semantics of the relation terms be made explicit in a non-ambiguous way. The first step in solving this problem is to distinguish between relations that hold between types and those that hold between the instances of those types. Ontologies are about types: Statement (9) asserts that a parthood relation holds between the type Female Pelvis and the type Body, statement (12) that a containment relation holds between the type Pelvic Cavity and the type Ovarian Artery, etc. Since the type terms in an ontology denote types and the relation terms like 'part of ', 'is a', 'develops from' denote relations between types, instances might seem to be not important for an ontologist. However, an anatomist is not able to study the types directly. We have epistemic access to types only via their instances. Hence the only way to evaluate a statement concerning types – for example 'Pelvic Cavity contains Ovarian Artery ' – is to look at instances of Ovarian Artery and their locations; there is no way to look at the type Ovarian Artery directly. Similarly, the only way to evaluate a statement like 'Appendix part of Body ' is to look at instances of the type Appendix and to check whether they are part of some instances of the type Body. Note that the parthood relation between the instances differs from the various parthood relations on the type-level that we have considered above. One major difference is that the parthood relation between anatomical structures (i.e. between the different types of anatomical structures) holds in a time-dependent way. For example, it might be the case that Bill's appendix is part of his body at 6 am, but that it is not part of his body at 8pm on the same day. The relation expressed in 'Appendix part of Body ' is however a timeless relation between types. Arguably, another difference is that the fact that Bill's appendix is part 7 of his body at a given time entails that the location of his appendix and the location of his body overlap at this time (where it is not clear what it would mean for the type Appendix to have a location that overlaps with the location of the type Body). To capture the differences we will henceforth distinguish relations between types, for which we use italic font, from relations of other kinds, picked out by using bold. We begin with studying the part of at t relation between instances (where t stands in for times). Only by studying this and similar relations on the instance level can we gain insight into the parthood relations on the type-level ([2], [8]). Although the latter are at first importance for ontologies, they are actually secondary to the former in an epistemic sense. We can use the tight connections between part of at a given time and the parthood relations on the type-level to disambiguate the use of 'part of ' in the problematic cases mentioned above. In cases (9-11) the relation term 'part of ' can be read as denoting three different relations; hence we have to distinguish at least three different parthood relations that hold between types. In the following we will use the term 'part of ' only to denote one of these relations; for the others ones we will use the terms 'is part ' and 'integral part of '. Let C and C1 be types of anatomical entities, let x, y, z be anatomical entities (instances), and t a time. Further, let 'Cyt' be the abbreviation for 'y is an instance of C at time t' and, 'C1zt' the abbreviation for 'z is an instance of C1 at t'. We can now define:1 d 1 C part of C1 =def for all y, t, if Cyt then there is some z such that C1zt and y part of z at t. d 2 C is part C1 =def for all z, t, if C1zt then there is some y such that Cyt and y part of z at t. d 3 C integral part of C1 =def C part of C1 and C is part C1. These definitions provide an example of how we can define relations between types in terms of the relations between the corresponding instances. One major advantage of these definitions is that they provide us with a better understanding of the type-level statements that form an ontology. With the help of the definitions (d 1 d 3) it is easy to see that (15) is true, but (16) false, in virtue of the fact that there are (human) bodies that have no female pelvis (because they have a male pelvis). 15 Female Pelvis part of Body. 16 Female Pelvis is part Body. 1The terms 'part of ' and 'integral part of ' are defined as in [8], is part is the inverse of has part as defined in [8], which means that C is part C1 is logically equivalent to C1 has part C. The relations part of, is part, and integral part of are equivalent to P1, P2, and P12 as defined in [2] and in [1] in this book. 8 In addition, the definitions (d 1 d 3) allow us to check the logical properties of the type level relations and their logical connection. Without the definitions it might not be obvious whether 'A part of B' implies 'B is part A' and vice versa, or in other words whether part of is the inverse of is part. With the help of the (d 1 d 3) it is easy to see that this is not the case. Let us consider another example. Does (15) entail (17)? 17 Body is part Female Pelvis. According to the definition (d 2) the statement (17) means: For any instance of Female Pelvis at any time, there is some instance of Body such that that instance of Body is part of that instance of Female Pelvis at that time. Since human bodies are never parts of pelvises, this is obviously false – hence we have shown that part of is not the inverse of is part ([2], [8]). Let's consider two other examples. Since men have urinary bladders, some urinary bladders are not part of a female pelvis. Hence (18) is false. In contrast, (19) is true, because female pelvises have urinary bladders as parts: 18 Urinary Bladder part of Female Pelvis. 19 Urinary Bladder is part Female Pelvis. The definitions (d 1 d 3) provide us with a clear understanding of the relations which allows us to use the corresponding assertions to draw logical inferences. To give a very primitive example, from (d 1 d 3) it follows immediately that (20) entails (21) and (22). Such logical connections facilitate automatic reasoning (see [1] in this book). 20 Urinary Bladder integral part of Body. 21 Urinary Bladder part of Body. 22 Urinary Bladder is part Body. In the beginning of this section we addressed two problems: (a) Humans use relation terms like 'part of ' ambiguously, which undermines the quality of ontologies and leads automatic reasoners astray. And (b) the relations are used in ontologies to determine the semantic content of the type terms, hence we need to capture the denotation of relation terms like 'part of ' in a machine interpretable form. These problems were addressed by defining relations on the type-level (in our example integral part of, part of, and is part) with the help of a relation between individuals (part of at). The definitions (d 1 d 3) allow us to resolve the ambiguities in existing uses of the term 'part of ' and it is easy to translate the definitions above into a formal language, hence the approach that was embraced in the last section was a step in the right direction. However, it did not solve the problems completely. The definitions (d 1 d 3) involve the parthood relation between individuals. Hence the denotation of the terms 'part of ', 'is part ', and 'integral part of ' depends on the denotation 9 of the term 'part of at' that we used in these definitions. Thus in order to get a clear understanding of these terms we need to determine the denotation of 'part of at'. This can be done via an axiomatization of this relation, i.e. by providing a set of axioms which amount to a so-called 'contextual definition' of 'part of at'. A contextual definition is not really a definition in the strict sense, but the axioms serve to capture our intuitions about the logical properties of the relation that is axiomatized and thus they restrict the possible interpretations of the term 'part of at'. It would have been possible to axiomatize the various parthood relations on the type-level directly instead of defining them with the help of the parthood relation on the instance-level. There are however two reasons why it is better not to do this, but to use the part of at relation as we have done. One reason is that we could use the single parthood relation part of at on the level of individuals to define the three relations on the type-level. Thus we needed only one primitive notion instead of three. Further, since we have access to types only via their instances, our intuitions about the logical properties of the relattions on the instance-level are much more developed. Moreover, much of our digital data about anatomical and other entities in the biomedical domain comes in the form of the instance data contained, for example, in clinical records. Actually, since 'part' is not a technical term but an expression we use in daily life (we talk about engine parts, or the parts of former Yugoslavia, or about cellulose as part of wood) one might suspect that we have very strong intuitions about the parthood relation and thus that it would be easy to develop a theory of wholes and their parts. Indeed it is true that people have strong opinions on mereological questions; unfortunately the intuitions governing our daily talk about wholes and their parts are quite heterogeneous (if not plainly inconsistent). For this reason mereology is a controversial field in philosophy. Hence it is important to give an explicit account of part of at, otherwise typelevel terms like 'integral part of ', 'part of ', and 'is part ' will themselves be used ambiguously. This is not the place to present a full axiomatization (see [7]), but some examples of axioms that many people would embrace are: Ax. 1 At any time t, every x part of x at t. Ax. 2 For any x, y, t: if x is part of y at t and y part of x at t, then x = y. Ax. 3 For any x, y, z, t: if x is part of y at t and y is part of z at t, then x is part of z at t. These axioms express time-relativized versions of the reflexivity, antisymmetry, and transitivity of parthood, respectively. The approach that we have considered in this section allows us to restrict the semantical content of relation terms. This is achieved in two steps. We define the type-level relation with the help of a relation between individuals and we then give an axiomatization of the latter relation. This approach has been presented by means of by appealing to just a few examples and is still rather sketchy. In [1] in this book it will be covered systematically and in greater depth. 10 4 Canonicity So far we did not discuss one important objection to the above approach.2 Let's assume that we encounter statement (23) in a textbook on human anatomy. 23 Appendix is part Body. Statement (23) is true: the human body has an appendix. However, according to the definition (d 2) statement (23) is equivalent to (24): 24 For every x and time t, if x is a Body at t, then there is an instance of Appendix y at t such that y is part of x at t. Statement (24) is plainly false: there are plenty of people who live happily without an appendix. Each of them provides a counterexample to the claim in (24) that every body has an appendix. Since (23) is true, but (24) is false, the statements (23) and (24) cannot be equivalent. Does that mean that our analysis of statements like (23) is incorrect? Is definition (d 2) inappropriate? In order to understand the root of the problem we need to distinguish between canonical anatomy and instantiated anatomy ([5], [9]). Instantiated anatomy concerns the anatomical entities represented for example in data about actual cases generated in clinical practice. Canonical anatomy is the result of generalizations deduced from qualitative observations that are implicitly sanctioned by their accepted usage by anatomists. While instantiated anatomy and canonical anatomy are both founded in empirical observations, only instantiated anatomy contains empirical statements about human bodies and their anatomical parts. In contrast, the relation between canonical anatomy and human bodies is in some respects similar to the relation between a technical drawing and the artifacts that are built with the help of the drawing. As anybody who has assembled a piece of Swedish furniture knows, many existing artifacts do not exactly match their technical drawings. That does not make the technical drawing 'false'; a technical drawing is not an empirical description of the composition of the existing artifacts; rather it tells us how the artifacts should be composed. Analogously, a canonical anatomy gives an account of the 'prototypical' composition of the male or female human body. For example, (23) does not assert that all human bodies have an appendix, but rather that a human body is supposed to have an appendix. Thus (23) cannot be refuted by the fact that some people lack an appendix. This example shows that a canonical anatomy consists of statements that describe how the anatomical entities of a given organism are supposed to be composed (for example in light of the structure of the underlying genes); and it is this that distinguishes a canonical anatomy from an instantiated anatomy. The distinction between instantiated and canonical anatomy is important since it allows us to analyze the source of the mismatch between (23) and (24). Statement (24) would be an appropriate analysis of (23) if (23) would be an 2We thank Cristian Cocos, Alan Rector, and Cornelius Rosse for their critical remarks and suggestions. 11 assertion about instantiated anatomy – and in this case (23) would be false, since it is an empirical fact that not all human bodies have an appendix. However, we have assumed above that (23) is a statement within a textbook on canonical anatomy. One way to make the force of statements of this kind explicit is to use an adverb as in (25): 25 Canonically, Appendix is part Body. Syntactically, the expression 'canonically' in (25) works like 'necessarily', 'possibly', 'it is permissible that' and other expressions that are – from a logical perspective – logical operators. However, while the semantics of the latter is well understood, the semantics of 'canonically' is not. Thus in this form (25) is a logical black box and for this reason useless for logical reasoning. This is why we will present a logical analysis of statements like (25) in the remainder of this section. In [8] we have (implicitly) embraced the assumption that in the context of canonical anatomy the domain of discourse is restricted to canonical entities. In this case (25) would have the same meaning as (23), except for an implicit understanding that we consider only canonical entities – which can be made explicit by restricting the range of the variables in (24). Hence – according to this approach – (25) is equivalent to (26): 26 For every x and time t, if x is a Body at t, then there is an instance of Appendix y at t such that y is part of x at t; where the variables x and y range exclusively over canonical entities. The term 'canonical entity' can be defined as follows: d 4 An anatomical entity x is canonical with respect to a given anatomy A if and only if x is structured in the way it is supposed to be structured according to anatomy A. Since a human body without an appendix is not canonical, it follows that such bodies fall outside the domain of quantification, and thus the problematic cases are excluded. Unfortunately, this way of understanding 'canonically' leads to new difficulties. For example, (27) would be equivalent to (28). 27 Canonically, Appendix part of Body. 28 For every x and time t, if x is an instance of Appendix at t, then there is an instance of Body y at t such that x part of y at t, where the variables x and y range exclusively over canonical entities. Statement (28) expresses that every canonical appendix is part of a canonical body – which is obviously wrong: there are people who have a perfectly normal appendix, but are lacking teeth. Therefore the idea of restricting the domain of quantification to canonical entities does not work; we need to find an alternative way to analyze (25). 12 In order to come up with the needed analysis, we have to remember that canonical anatomy gives an account of how a male or female human body is supposed to be composed. Thus a statement that is part of a canonical anatomy expresses a requirement that a human body has to meet in order to conform to the given canonical anatomy. We can express this in the following way: Let 'IAxt' be the abbreviation for 'x is a human body that is in conformity with anatomy A at t'. (As mentioned above we assume that we deal with human anatomy; otherwise one has to modify the definition of 'IAxt' in the obvious way.) d 5 Canonically, C is part C1 =def for all x, t, necessarily, if IAxt, then for all z, if C1zt there is some y such that Cyt and y part of z at t; where y and z are anatomical entities that are part of x at t. Definition (d 5) can be paraphrased as follows: if a statement of the form 'C is part C1' is part of a canonical human anatomy, then the following holds for any human body x at any given time: necessarily, if x is in conformity with the given anatomy (at this time), then, for any anatomical part of x that is an instance of C1 (at this time), there is an anatomical part of x that is an instance of C (at this time) and the instance of C is part of the instance of C1 (at this time). Let's consider an example. Definition (d 5) entails that (29) is equivalent to (30): 29 Canonically, Carpal Bone is part Hand. 30 Necessarily, if x is a human body that is in conformity with A at time t, then for all y, if y is an instance of Hand at t, there is (at least) one entity z that is an instance of Carpal Bone at t and is part of y at t; where y and z are anatomical entities that are part of x at t. Analogously, we can define part of for canonical anatomies: d 6 Canonically, C part of C1 =def for all x, t, necessarily, if IAxt, then for all y, t, if Cyt, there is some z such that C1zt and y part of z at t; where y and z are anatomical entities that are part of x at t. Definition (d 6) expresses the following: if a statement of the form 'C part of C1' is part of a canonical human anatomy, then the following holds for any human body x at any given time: necessarily; if x is in conformity with A (at this time), then, for any anatomical part of x that is an instance of C (at this time), there is an anatomical part of x that is an instance of C1 (at this time) and the instance of C is part of the instance of C1 (at this time). The definitions (d 6) and (d 5) are closely linked to (d 1) and (d 2), respectively: the parts of the definitions that are emphasized by using small caps are the right hand sides of the definitions (d 1) and (d 2). We chose this way of presenting the definitions because it shows that the original account of the last section is preserved, it is just that it is now embedded in a context that does justice to the fact that statements like (23) are part of a canonical anatomy. 13 Let's consider another relation, where time plays a more important role than in the examples above. The human body is supposed have deciduous teeth and the human body is supposed to have androgenic hair – but obviously not at the same time. We can express this fact with the help of a relation excludes in (31), where excludes is defined in definition (d 7). 31 Canonically, Deciduous Tooth excludes Androgenic Hair. d 7 Canonically, C excludes C1 =def for all x, t, necessarily, if IAxt, then there are no y, zt, such that Cyt and C1zt; where y and z are anatomical entities that are part of x at t. The relation excludes serves here as a simple example that illustrates how time can play an important role for the definitions of type-level relations; this holds in particular for relations that concern the development of anatomical entities. Since we have focused on parthood relations in this section so far, let's consider an example that involves the contains relation between types, e.g. (32). Further, let's assume that we have an account of the corresponding contains relation on the instance-level (see [1]). We can now define contains with the help of contains as in definition (d 8). 32 Canonically, Male Pelvic Cavity contains Urinary Bladder. d 8 Canonically, C contains C1 =def for all x, t, necessarily, if IAxt, then for all y, if Cyt then there is some z such that C1zt and y contains z at t; where y and z are anatomical entities that are part of x at t. Hence (32) is equivalent to (33), which is itself a complicated way of expressing (34): 33 Necessarily, if IAxt, then for all y, if y is an instance of Pelvic Cavity at t then there is some z such that z is an instance of Urinary Bladder at t and y contains z at t; where y and z are anatomical entities that are part of x at t. 34 Necessarily, if x is a human body that is in conformity with A at t, and x has a pelvic cavity, then there is a urinary bladder that is contained in the pelvic cavity. We will now generalize our approach and define 'canonically'. In this section we have analyzed statements of the form (35), where rel stands in for 'is part ', 'part of ', 'excludes', and 'contains'. 35 Canonically, C rel C1. As we have mentioned above, the definitions (d 5) and (d 6) (where rel is is part and part of, respectively) are closely linked to the definitions (d 1) and (d 2), which define statements of the form 'C is part C1' and 'C part of C1'. Analogously, the definition (d 8) is closely linked to (d 9). (Again, the relevant parts of the definition (d 8) are in small caps.) 14 d 9 C contains C1 =def if Cyt, then there is some z such that C1zt and y contains z at t. It seems that for any definition that defines statements of the form (35), there is a corresponding definition of the statements that does not begin with 'canonically'. We will use this connection in order to define 'canonically': d 10 Let rel be any binary type-level relationship, and C, C1 any types, and assume we have a definition of the following form: C rel C1 =def φ(y, z) where φ(y, z) represents a formula that involves only relationships between individuals and that ensures that y and z are anatomical entities that are instances of C and C1, respectively. In this case we can define: Canonically, C rel C1 =def for all x, t, necessarily, if IAxt, then (φ(y, z) and y part of x at t and z part of x at t) Definition schema (d 10) provides us with a systematic link between the relations within a canonical anatomy and the use of the corresponding relations within an instantiated anatomy.The definition schema (d 10) works not only for relations such as those that we have considered in this section; it can be applied to many type-level relations and in particular to the type-level spatial relations that will be considered in [1]. 3 5 Conclusions One purpose of an ontology is to encapsulate the meanings of its terms in a computer parsable form. We analyzed how anatomical anatomies fulfill this purpose. An anatomical ontology consists of statements composed of two kind of terms denoting types and relations, respectively. Typically such statements involve two type terms, so that they are of the form 'A rel B'. We showed that there is no way to tell a computer directly, for any given type term, which type is denoted by that term. Thus ontologies must find ways to convey such information indirectly: broadly, it is the totality of the relations between the denotations of the type terms that determines the semantical content of the type terms taken individually. This works as follows. Each statement 3Note that the connection between the relations is not always as straight forward as in the examples considered above. For example, definition (d 11) does not capture the semantic content of 'excludes'. d 11 C excludes C1 =def there are no y, z, t, such that Cyt and C1zt. A more appropriate definition of 'excludes' is: d 12 C excludes C1 =def there are no u, y, z, t, such that y part of u at t, z part of u at t, Cyt, and C1zt. 15 'A rel B' asserts that the denotation of 'A' and the denotation of 'B' are linked by the relation rel. Thus any interpretation of 'A' and 'B' according to which their denotations do not meet this requirement is ruled out. The possible interpretations of the terms 'A' and 'B' are in this sense limited by the statement 'A rel B'. While this approach is not sufficient to single out any specific type as denotation of a given term, if the term is connected to a multitude of other terms, then the possibilities will be correspondingly restricted. Fortunately, in the domain of anatomy we are already in possession of high-quality representations of such multiple relations. Since the semantical content of type terms is determined by the relations that are expressed by 'is a', 'part of ', 'contains' and other relation terms, it is crucial to make explicit which relations these terms denote. This analysis is important not only because of our aim to capture the semantical content of the terms of an ontology in a machine-readable form, but also because people tend to use relation terms ambiguously, in a way which reduces the quality of ontologies. We showed that many relations between types can be defined with the help of relations that hold between instances of these types, and an approach based on this recognition has the advantage that we typically have a better understanding of the relations between instances than of the relations between the corresponding types. Further, the approach has the virtue of economy, since it is often possible to define different relations on the type-level with the help of one relation on the instance-level. On the given approach the meaning of a statement 'A rel B' in an ontology is an empirical assertion about the instances of types A and B. Thus 'Embryo develops from Zygote' is true if and only if: for any instance of Embryo x there is an instance of Zygote y such that x developed from y. Here developed from is an instance level relation that holds between individuals. 'Embryo develops from Zygote' is thus an empirical assertion that can be falsified (by discovering that at least one embryo did not develop from a zygote). In the case of canonical anatomical ontologies such as the FMA, in contrast, the situation is more complicated, since canonical anatomical ontologies do not consist of empirical assertions in this sense, but rather of statements that express how the corresponding entities are supposed to relate to each other (in virtue of the workings of the underlying structural genes). For this reason we analyzed statements of the form 'canonically, A rel B' in such a way as to show how the semantic content of such statements is systematically linked to statements without the prefix 'canonically '. Very roughly, a statement like 'canonically, A rel B' expresses that, necessarily, any human body x that is in conformity with the given anatomy meets the requirement 'A rel B', where 'A rel B' can spelled out as in the non-canonical case and the domain of discourse is restricted to the anatomical entities that are part of x. The fundamental picture then remains the same: the semantic content of the type terms is provided by the network of relations between them. A profound understanding of these relations is thus a prerequisite for a non-ambiguous use of type terms of the sort which can support automatic reasoning. The next chapter will present a deeper and more systematic analysis of those specific sorts of spatial relations that are relevant 16 for anatomical ontologies. References [1] T. Bittner, M. Donnelly, L. Goldberg and F. Neuhaus. Modelling Principles and Methodologies – Spatial Representation and Reasoning Albert Burger, Duncan Davidson and Richard Baldock (eds.): Anatomy Ontologies for Bioinformatics: Principles and Practice (in print). [2] M. Donnelly, T. Bittner, and C. Rosse. A formal theory for spatial representation and reasoning in bio-medical ontologies. Artificial Intelligence in Medicine, 36(1):1–27, 2006. [3] N Guarino. Formal Ontology and Information Systems. In N Guarino (ed.), Proceedings of the 1st International Conference on Formal Ontologies in Information Systems, FOIS'98, IOS Press, Trento, 3-15, 1998. [4] I. Johansson. On the Transitivity of the Parthood Relations. In H. Hochberg and K. Mulligan (eds.), Relations and Predicates, 161-181, 2004. [5] C Rosse, JL Mejino , BR Modayur , R Jakobovits, KP Hinshaw, JF Brinkley. Motivation and organizational principles for anatomical knowledge representation: the digital anatomist symbolic knowledge base. Journal of the American Medical Informatics Association, 5(1):17-40, 1998. [6] F de Saussure. Grundfragen der allgemeinen Sprachwissensschaft. Walter De Gruyter, Berlin/Leipzig, 1967. [7] P. Simons. Parts, A Study in Ontology. Clarendon Press, Oxford, 1987. [8] B. Smith, W. Ceusters, B. Klagges, J. Köhler, A. Kumar, J. Lomax, C. Mungall, F. Neuhaus, A. Rector, and C. Rosse. Relations in biomedical ontologies. Gnome Biology, 6(5):46, 2005. [9] B Smith, A Kumar, W Ceusters, C Rosse. On carcinomas and other pathological entities. Comparative and Functional Genomics, vol 6, 7-8, 379-387, 2005. [10] B Smith, W Kusnierczyk, D Schober, W Ceusters. Towards a Reference Terminology for Ontology Research and Development in the Biomedical Domain. Proceedings of KR-MED 2006, in press. [11] B. Smith and C. Rosse. The Role of Foundational Relations in the Alignment of Biomedical Ontologies. In M. Fieschi, et al. (eds.), Medinfo 2004, IOS Press, Amsterdam, 444-448, 2004. [12] J Trier. Aufsätze und Vorträge zur Wortfeldtheorie. Berlin: DeGruyter, 1973.

comment Three things digital ethics can learn from medical ethics Ethical codes, ethics committees, and respect for autonomy have been key to the development of medical ethics - elements that digital ethics would be advised to emulate. Carissa Véliz The past decade has been rife with data misuse, hacks, and corporate wrongdoing. As people have become more aware of the ways in which tech companies abuse their power, a 'techlash' has ensued, combined with calls for more ethics1,2. But efforts to respond to this demand have been fraught with hollow promises, oversights, and mistakes3 that have attracted further criticism against both tech companies and the professed limits of ethics. It is perhaps no surprise that the first experiments in digital ethics have misfired; the discipline is relatively new, and most of the endeavours behind digital ethics have been made by computer engineers, lawyers, journalists, or businesspeople with little or no background in ethics4. Given this context, and in order to suggest a way forward for digital ethics, it is helpful to look to another field within practical ethics that has a longer history - medical ethics. Medical and digital ethics Ethical concerns have never been foreign to medicine. It is, after all, a field that deals directly with matters of life and death. Hippocrates, often considered the father of medicine in Western culture, urged physicians to do no harm. The discipline of medical ethics, however, did not fully develop until the 1970s. For most of history, physicians were left on their own to decide what it means to do no harm, without any training or institutional support to help them avoid mistakes. Two factors contributed to the development of medical ethics. First, medical scandals highlighted the need to establish ethical standards and regulation. In 1972, for example, The New York Times revealed that subjects with syphilis had gone untreated for four decades as part of the Tuskegee Syphilis Experiment, despite treatment being available, and without the knowledge or consent of subjects5. Second, the development of technology confronted doctors with new ethical challenges they did not know how to resolve. The mechanical ventilator, for instance, forced a rethink of the concept of death and the development of the ethics of organ transplantation: doctors were now faced with warm, heart-beating bodies whose brains were no longer working but whose organs could be procured for transplantation6. In short, there were practical demands that needed to be met, and it was clear that dealing with ethical dilemmas should not be solely the job of healthcare professionals, whose expertise is in keeping people healthy, as opposed to resolving ethical conundrums. The analogy between digital ethics and medical ethics is quite close. Much like Hippocrates, Google, a company that can be considered to be one of the founders of the digital age, famously introduced the motto 'Don't be evil' as its fundamental code of conduct. Given that almost no one thinks of themselves as evil, or even capable of being evil, the dictum is hardly helpful in aiding engineers, programmers, and data scientists to identify and resolve ethical problems. Like medicine in the 1970s, digital technology companies have been the protagonists of serious controversies in the past couple of years. As more people are becoming aware that they are being affected by unethical digital practices, the need for ethical standards is becoming more apparent. Likewise, with the development of technology related to the collection, analysis, and use of personal data, as well as the design of new apps, platforms, and tools such as autonomous cars, we are confronted with new ethical dilemmas that engineers, programmers, and data analysts are not especially suited or trained to resolve. The analogy between medical and digital ethics is not perfect, however. The digital context is much more political than the medical one, as well as more dominated by private forces, and it will have to develop its own ethical practices. Despite the differences, there are three elements that have been vital for the success of medical ethics, which digital ethics would be advised to emulate: the development of ethical codes, the use of ethics committees, and respect for personal autonomy. Ethical codes Ethical codes are necessary to establish benchmarks for good practices. The digital world is in urgent need of codes analogous to the Nuremberg Code, the Declaration of Geneva, the Belmont Report, and the Declaration of Helsinki, which have shaped medical and research policies around the world despite them not being legally binding. Some might think that laws such as the European General Data Protection Regulation (GDPR) should be enough. But the GDPR only addresses issues related to personal data. It says nothing, for example, about how to programme ethical decisions in autonomous cars, or about ethical dilemmas of future technologies. Furthermore, laws are narrow in scope, as they should be; they establish minimal requirements of behaviour for social institutions to function well. Ethics goes NaturE ElEctroNics | www.nature.com/natureelectronics comment beyond that - it identifies moral issues, reflects on the kind of society we want to live in based on ideas of what a good life looks like, and makes recommendations accordingly. Laws allow us to have orderly interactions with one another within a framework of basic fairness. Ethics allows us to strive towards ways of life that will be most conducive to our own and others' wellbeing. Some companies such as Google have issued principles for their future work in artificial intelligence (AI)7. While efforts from companies to think through ethical issues and make public commitments are desirable, and a step in the right direction, businesses are driven by private interests that can get in the way of impartiality. Companies can be too vague in formulating their principles, they can change their code of conduct as they see fit, sometimes surreptitiously, and the principles proposed may not be the result of an appropriate process of consultation and agreement between relevant parties. For a digital ethics code to carry enough moral weight to have an industry-wide impact, the principles proposed must be the product of deliberation of a legitimate, independent, and neutral body that is inclusive and diverse, and above all, has the public interest as its principal concern. In 2016, the European Data Protection Supervisor announced an Ethics Advisory Group that was to "consider the wider ethical implication of how personal data is conceived and used" in an effort to lead the conversation on ethics in the digital age. While it was hoped that the result would be guidelines or recommendations for better digital practices8, the report ended up being a series of reflections about "socio-cultural shifts" that were only vaguely related to ethics and avoided attempting to "define the rights and wrongs of navigating the digital ecosystem"9. From the point of view of the success path of medical ethics, the report was a missed opportunity. A more successful attempt has been the recent European Commission's Ethics Guidelines for Trustworthy Artificial Intelligence10. The guidelines, however, have been criticized for being "lukewarm, short-sighted and deliberately vague", mostly due to the lack of professional ethicists on the high-level expert group on AI, and an overabundance of industry representatives who managed to water down what were supposed to be "red lines"11. For international ethical codes to have moral authority, they have to be shaped by actors who can represent the public good - not by the private interests of industry. A common objection against ethics is that it only amounts to self-regulation12 and that it is nothing beyond the expression of good wishes; in short, that ethics does not have teeth. But ethical codes can and do have teeth, even when they are not legally binding. If a code achieves sufficient legitimacy and recognition, it can be expected to be respected by all professionals. Penalties for the breach of the ethics code can include suspension and possible expulsion from the profession. If a doctor has a sexual relationship with a patient, for example, they can lose their licence. The analogy between medical and digital ethics points towards a need to professionalize tech jobs to hold them to high standards13. If doctors, lawyers, and architects require a licence to work, there is no reason not to have equally high standards for computer scientists and engineers - particularly given how much influence they have over our lives through their structuring of our digital world. The Association for Computing Machinery's Code of Ethics and Professional Conduct is a good start, but more needs to be done for a code to be adopted worldwide and across industry in a way that compels computing professionals and institutions to adhere to it14. Ethics committees Before the 1970s, there were virtually no ethics committees in hospitals15. Today, it is hard to find a hospital without one. Similarly, every technology company should have an ethics committee. Ethics committees have at least three roles to play. The first is education16. It is the responsibility of ethics committees to keep up with the latest relevant information that will allow them both to make good ethical decisions, and to educate the staff around them in best practices. Ethics committees also engage in community outreach, both to achieve familiarity with the community's concerns, and to share with the community the essentials of ethical practices. The second role of ethics committees is policy formation and review16. Ethics committees apply international guidelines to design ethical internal policies that are specific to the institution they work at. The third role of ethics committees is to provide ethical consultation16. Any stakeholder, from programmers to executives, as well as users and clients, should be able to turn to such a committee if they have ethical concerns. Ethics committees consider moral problems on a case-by-case basis. To resolve issues, they take into consideration international guidelines, their own policies, previous experiences, and the best interests of all stakeholders (particularly those of the most vulnerable), among other elements. Ethics committees are there to make sure people's rights are respected (the right to autonomy, among others), to help resolve value conflicts, and to assess whether a particular technology is worth being developed and deployed. No tech project should go out into the world without having been assessed by a team of qualified people with a view to bettering society and avoiding negative consequences. Ethics committees should also follow-up on projects after they have been launched to make sure there are no moral problems that had not been foreseen. Ethics committees should reflect diversity and inclusion. As a minimum, they should be composed of someone who can understand technology at a deep level and explain it to others (a programmer, engineer, or computer scientist), a practical ethicist, a lawyer, an expert on risk assessment (possibly a statistician), and at least one lay member of the public (possibly from an interested non-governmental organization). Ethical dilemmas around digital technology have a tendency to have more social and political implications than medical dilemmas, which is why it might also be important to include sociologists and political scientists. Ethics committees can help ethics have teeth by, on the one hand, having the power to block grossly unethical projects from going forward (similar to ethics committees at universities), and on the other hand, by putting more responsibility on the shoulders of those who decide to act against the recommendation of a committee in less clear-cut cases, in consultations, or in follow-up assessments of a project. For example, a doctor can act against the recommendation of her hospital's ethics committee, but if things go wrong, that doctor is likely to face more consequences (possibly even legal consequences) than if she had acted in accordance with the ethical recommendation. Ethics committees need to be relatively independent of the company they work for. Their jobs must be guaranteed, regardless of their ethical views or recommendations. In addition to having local ethics committees, it would also be desirable to have higherlevel ethics committees that can be publicly funded and can oversee lower-level committees, as well as take charge of the most difficult cases. respect for autonomy For digital projects to be ethical, they must respect the autonomy of individuals. Autonomy is the capacity to act in accordance with reason in a way that NaturE ElEctroNics | www.nature.com/natureelectronics comment responds to one's own motives17. Being autonomous means being able to choose our values for ourselves and live accordingly. One of the most important ethical changes in the history of medicine is the transition from paternalism to respect for people's autonomy. The first edition of the Code of Medical Ethics of the American Medical Association, adopted in 1847, stated18: "The obedience of a patient to the prescriptions of his physician should be prompt and implicit. He should never permit his own crude opinions as to their fitness, to influence his attention to them." In contrast, the latest edition states19 that patients have a right "to make decisions about the care the physician recommends and to have those decisions respected. A patient who has decision-making capacity may accept or refuse any recommended medical intervention." Patients should have the right to refuse treatment, among other reasons, because medical decisions are not only scientific or technical, but also value-laden. A patient who prefers to forego a painful treatment in order to enjoy a shorter, but more pleasant life, is not irrational or medically mistaken. In identical clinical scenarios, two patients may rationally and reasonably choose different treatments because their values are different. Similarly, technological decisions are not only about facts (for example, about what is more efficient), but also about the kind of life we want and the kind of society we strive to build. The beginning of the digital age has been plagued by impositions, with technology companies often including a disclaimer in their terms and conditions that "they can unilaterally change their terms of service agreement without any notice of changes to the users"20. Changes towards more respect for autonomy, however, can already be seen. With the implementation of the GDPR in Europe, for instance, tech companies are being urged to accept that people may prefer services that are less efficient or possess less functionality if that means they get to keep their privacy. One of the ways in which technology has failed to respect autonomy is through the use of persuasive technologies. Digital technologies that are designed to chronically distract us not only jeopardize our attention, but also our will, both individually and collectively21. Technologies that constantly hijack our attention threaten the resources we need to exercise our autonomy. If one were to ask people about their goals in life, most people would likely mention things such as "spending more time with family" - not many people would suggest "spending more time on Facebook". Yet most people do not accomplish their goals - we get distracted21. Collectively, we might want to have a more just and equal society, but here too, it is unclear whether technology companies are doing much to help us achieve those aims. Technology companies should be on our side, helping us attain our goals as individuals and societies - not theirs. outlook Ethics can play an important role in developing and implementing technology in a way that better contributes to peoples' rights and wellbeing. That ethics is important, however, does not deny the equally important task of legislation. To complement ethics, one proposal that appears promising is that of implementing fiduciary duties for people and institutions handling personal data22. Just like doctors owe their loyalty to their patients, tech companies should owe their loyalty to their users. Our data should never be used against us, and a person's welfare should take precedence over economic interests. Legislation, however, will always be limited. Ethics needs to step up to fill in the blanks, take care of the unexpected, and help digital tech think through the possible consequences of innovations. For these tasks, the digital world would do well to look to the successful path of medical ethics, in which the development of ethical codes, the implementation of ethics committees, and showing respect for people's autonomy played a vital role. Unethical practices breed distrust, resentment, and unnecessary conflict. Digital tools are already powerful enough that their misuse can cause great harm, and their hegemony is only going to grow in the following decades. Ordinary citizens are fast becoming the slaves of digital technologies, rather than their masters. Better digital ethics is urgently needed to reverse this trend. ❐ Carissa Véliz Uehiro Centre for Practical Ethics, Wellcome Centre for Ethics and Humanities, Faculty of Philosophy, Christ Church, University of Oxford, Oxford, UK. e-mail: [email protected] Published: xx xx xxxx https://doi.org/10.1038/s41928-019-0294-2 References 1. Botsman, R. Dawn of the techlash. The Guardian (11 February 2018); https://go.nature.com/2YroGnu 2. Smith, E. The techlash against Amazon, Facebook and Google- and what they can do. The Economist (20 January 2018); https:// go.nature.com/2K41ZAA 3. Statt, N. Google dissolves AI ethics board just one week after forming it. The Verge (4 April 2019); https://go.nature. com/2Zg727k 4. Mahieu, R., van Eck, N. J., van Putten, D. & van den Hoven, J. Ethics Inf. Technol. 20, 175–187 (2018). 5. Heller, J. Syphilis victims in U.S. study went untreated for 40 years. The New York Times (26 July 1972); https://go.nature. com/2JVJcbE 6. Harvard Ad Hoc Committee JAMA 205, 337–340 (1968). 7. Pichai, S. AI at Google: our principles. Google (7 June 2018); https://go.nature.com/2LJvzhY 8. Powles, J. & Véliz, C. How Europe is fighting to change tech companies' 'wrecking ball' ethics. The Guardian (30 January 2016); https://go.nature.com/2Omhh9b 9. Ethics Advisory Group Towards a Digital Ethics (European Data Protection Supervisor, 2018); https://go.nature.com/2K4B9Iz 10. Ethics Guidelines for Trustworthy AI (European Commission, 8 April 2019); https://go.nature.com/2K6HWBl 11. Metzinger, T. Ethics washing made in Europe. Der Tagesspiegel (8 April 2019); https://go.nature.com/2Y9pcuQ 12. Whittaker, M. et al. AI Now Report 2018 (AI Now Institute, 2018); https://ainowinstitute.org/AI_Now_2018_Report.pdf 13. Schneier, B. Click Here to Kill Everybody: Security and Survival in a Hyper-Connected World (W. W. Norton & Company, 2018). 14. ACM Code of Ethics and Professional Conduct (ACM, 2018); https://www.acm.org/code-of-ethics 15. Aulisio, M. P. AMA J. Ethics 18, 546–553 (2016). 16. Aulisio, M. P. & Arnold, R. M. Chest 134, 417–424 (2008). 17. Christman, J. Autonomy in moral and political philosophy. In The Stanford Encyclopedia of Philosophy Spring 2018 edn (ed. Zalta, E. N.) (Stanford Univ., 2018); https://go.nature.com/2Yt0WmP 18. Code of Ethics of the American Medical Association (T. K. and P. G. Collins, 1848); https://go.nature.com/2Y8vBX4 19. Patient Rights: Code of Medical Ethics Opinion 1.1.3 (American Medical Association, 2016); https://go.nature.com/2K4TYvb 20. Koepke, L. "We can change these terms at anytime": the detritus of terms of service agreements. Medium (18 January 2015); https://go.nature.com/2OiJzBi 21. Williams, J. Stand Out of Our Light: Freedom and Resistance in the Attention Economy (Cambridge Univ. Press, 2018). 22. Balkin, J. M. UC Davis Law Rev. 49, 1183–1234 (2016). Acknowledgements The work of the author has been supported by a Wellcome Trust grant (203132/Z/16/Z). NaturE ElEctroNics | www.nature.com/natureelectronics

Seeing Qualitons as Qualia A Dialogue with Wittgenstein on Private Experience, Sense Data and the Ontology of Mind Hilan Bensusan / Eros Carvalho, Brasilia, Brazil [email protected] In section 304 of the Investigations (1973), Wittgenstein responds to his interlocutor who asks him "But you will surely admit that there is a difference between painbehaviour accompanied by pain and pain-behaviour without any pain? (...) And yet you again and again reach the conclusion that the sensation itself is a nothing". The charge is that our basic qualitative sensory states such as the ones present when we have pain are irrelevant and can be dismissed – it is as if they were not present. One could read Wittgenstein's interlocutor to be pressing him to endorse the denial of qualia. "Qualia" is a frequently used expression to refer to basic qualitative sensory states. Wittgenstein, however, responds that it is as if they were not present, but still there is something present. He writes: "Not at all. It is not a something, but it is not a nothing either! The conclusion was only that a nothing would serve just as well as a something about which nothing can be said." Wittgenstein then refuses the charge of denying qualia. He is rather hinting at a way to understand qualia. Or so we argue in this work. We believe Wittgenstein considers qualia to be something like tropes. Tropes are abstract particulars1. The friend of tropes shares the nominalist dislike of universals (and, typically, of properties). There could be oneplace or many-places tropes; the former are sometimes called qualitons and the latter relatons. Instead of properties, trope theory takes every predication to involve particularity; 'x is a book' doesn't predicate the same property as 'y is a book' – our predicate 'book' does no more than point at some relevant similarity between x and y; it names no property. The green of a leaf of grass is not the same as the green of another leaf – only they can be relevantly similar, similar enough to be under the same predication. We hold that qualia can be seen as qualitons, and not as (universal) properties of a mental state (such as pain, for example). Further, we are convinced that Wittgenstein hints in this direction. Hacker's comments (1993) on section 304 of the Investigations suggest that what Wittgenstein "is doing is rejecting the grammar of name and object". Having a pain is not like to having a penny. So, pain is not concrete. Also, it does not make sense to say that we have now the same pain we have had yesterday. So, pain is not a universal. These together suggest that pain is an abstract particular. At the same time, a sensation is not a nothing, as it's not the absence of anything. It is nothing only in the sense that it cannot be used unaided in predications. We will try to make this clearer and elaborate further on the view by considering bits of his Lecture notes on "Private experience" and "Sense data" (1968). Wittgenstein writes: "What if someone asked: "How do I know that what I call seeing red isn't an entirely different experience every time?"" (p. 279). The friends of tropes would then reply: Why do we need them to be the same? Isn't enough that we use the same word for what is red? Red doesn't need to be the name of a sensation. And Wittgenstein: "We say here that a name is given to a particular impression. And this is strange and puzzling. For it seems as though the impression were too ethereal to be named (Marrying a woman's wealth)." (p. 275). It sort of escapes us until we grab it with a concept. It is as if there is nothing to be known until we find the resources to express it. In order for me to know I have a pain I need to speak English, otherwise I don't know what to do with the particular sensation I have. When I learn a language, I learn to use my qualiton as a qualitative indication of me having a pain or me sensing something red. Such a qualitative indication works only when I am familiar with the rules that govern the use of 'red' or 'pain' in English. "What could be meant by: truthfully calling a color impression 'red'? Does the word fit one impression better than another?" (p. 295). Further, "[i]f I say 'I see red' without reason, how can I distinguish between saying it with truth and saying it as a lie?" (p. 294). The expression of an impression is only true or false with respect to rules for concepts, with respect to usage in a public language. If there is no independent stance of judgment, my expression that I see red can always be a lie. It's me alone with my qualiton. A trope, seen as a quale, is private. There is nothing in a trope that makes it fit a word better than any other. Tropes have no name – they are particulars, they are re-identified only when they are clustered together by our sensory vocabulary. Sensations are red or green, qualia (as qualitons) could be anything provided that we acquire the relevant concepts and, with them, the relevant patterns of similarity. "But we are under the impression that we can point to the pain, as it were unseen by the other person, and name it." (p. 276) We can point at the pain, but not at the qualiton. When we point at the pain, we are pointing and naming a state that is identified through different indicators. The trope is the subjective and qualitative indicator. The same trope could indicate something very different. Qualia are enabling conditions for concept acquisition; a given qualiton is neither sufficient nor necessary for any concept to be acquired. What we mean with our words for sensations is not something of our own. Only the qualiton is private, but we don't talk about it. "The difficulty is that we feel we have said something about the nature of pain when we say that one person can't have another person's pain." (p. 277) We assume every pain is associated with a sensation – with a quale. Without qualia, we would hesitate to call it a pain. Yet, we can be easily fooled by pain-behaviour. As Seeing Qualitons as Qualia A Dialogue with Wittgenstein on Private Experience, Sense Data and the Ontology of Mind / Hilan Bensusan / Eros Carvalho 30 you wrote in the Investigations (257), we could not learn what is meant by pain by attending solely to our qualia. "Now whom shall we call blind? What is our criterion for blindness? A certain kind of behaviour. And if the person behaves in that particular way, we not only call him blind but teach him to call himself blind. And in this sense his behavior also determines the meaning of blindness for him. But now you will say: "surely blindness isn't a behavior; it's clear that a man can behave like a blind man and not be blind. Therefore 'blindness' means something different; his behavior only helps him to understand what we mean by 'blindness.' The outward circumstances are what both he and we know. Whenever he behaves in a certain way, we say that he sees nothing; but he notices that a certain private experience of his coincides with all these cases and so concludes that we mean this experience of his by saying that he sees nothing." (p. 285) Yes, a blind person is detected through behavior. The qualitons of a blind person could not be suitably exploited to provide the behavior we would identify as seeing. Of course there is a physiological counterpart to blindness, but the physiological tests are designed to make sense of our common sense idea of what is blindness.2 We assume also that there is a quale associated to blindness even if we cannot access it. We are constantly under the impression that we are naming brute sensations and not a complex of sensations and behaviors when we use expressions like blindness (or color-blindness, or red-blindness). "As it were: There is something further about it, only you can't say it; you can only make the general statement. It is this idea which plays hell with us." (p. 276) Indeed, we feel compelled to say that there is a quale (a qualiton) corresponding to each occurrence of, say, pain. We talk of pain in general, but there is a particular indicator of pain in each case – and we learn to see them as relevantly similar. We exploit the qualitons available to us when we are learning our sensory vocabulary. I wonder "[h]ow can we point to the color and not to the shape? Or to the feeling of toothache and not to the tooth, etc?" (p. 276) That reminds me of a case Noodhof (1998) considers. A glass is shattered as a result of a soprano singing a note. It seems tempting to say that it is the pitch and not the meaning of the singing that caused the shattering. Similarly, my feeling of pain (and not the activation of my Cfibers) is what makes me scream. It is in virtue of the pain that I scream. Gozzano (2008), for one, holds that tropes ought to be simples, that is, they must be maximally determinate. If this is so, there should be a pitch-trope and a meaning-trope. Similarly, when considering a causal process that could lead to the acquisition of two different concepts (say, tooth and toothache), one should posit two and not just one quale. The claim is that if a trope is a simple, it cannot carry two causal powers. Robb (1997), on the other hand, holds that causal powers are connected to particulars – it is a particular trope that causes the (particular) shattering of the glass. While it could be that it is better to describe the trope as having a high pitch – each trope has several similarity relations with other tropes – it is the trope qua trope that causes the shattering. Similarly, it is the trope qua trope that is part of the causal story we want to tell about learning the concept of toothache. Of course, in this case we have troubles individuating qualitons. However, the relevant causal powers are to be found not only in qualia but also on the language learning context around the process. If this is so, a single trope can be part of causal processes of concept acquisition for several different concepts. Qualitons can even have relations of similarity among them independently of their role in our vocabulary learning, but these relations play no role in our capacity to identify sensations. Learning a language involves learning a way to exploit our qualia. We propose to see qualia as abstract particulars, to be exploited in our process of language acquisition. In that process, we cluster qualia together when we learn similarity relations. Our view is therefore one where similarity relations are crucial for the acquisition of concepts. One could, however, fear that judgments of similarity cannot get off the ground if all they have to start out with are mere abstract particulars. Suppose one is learning a sensory concept like 'red' or 'bitter' and has to acquire the relevant similarity relations among her qualia. If one has the quale Q and is taught that it resembles quale R, but not quale P, how could one compare those qualia without having them somehow present in the mind? In other words, how can my past qualia be retrieved when I need them in order to learn similarity relations if they are not (from the beginning) available in a conceptual format? The question resembles the one Wittgenstein poses at section 342 at the Investigations (1973): how can a deaf-mute person recall thoughts she had before she was introduced to any language, written or otherwise? This is a troublesome area, but we believe we can sketch a way out. Consider one's attention to quale Q. We assume attention is somehow different from predication – I can attend to Q without making a predication of the sort 'Q is φ' (note that the abstract particular is the subject of the possible predication). If I can attend to Q, then we can have it present to the mind, at least sometimes, together with P or R. Notice that this procedure of attention can be thoroughly private and subjective – as it can differ completely from one to another person. Still, we believe this privacy is both enough to make sure that qualia as qualitons are useful for concept acquisition and does not violate the kernel of the assault on the given. All that is required is that we can manage to attend to two qualitons at the same time so that we can start to grasp the notions of similarity and relevant difference. This should be eventually enough to get the process off the ground – a process of gradual refinement of concepts so that what is roughly red eventually gets refined into different shades of red. We would like to finish in a less heroic tone. We assume that we can attend to more than one qualiton at a time (and register similarities in a rudimentary way) independently of our introduction to a public language. We take this to be a plausible assumption. If we are not entitled to make this assumption, maybe the account of qualia as abstract particulars loses some of its attraction. In any case, there is an interesting lesson to be learned: any talk of qualia that ascribes to them an explanatory role comes together with some commitments concerning privacy. Endnotes 1 Williams (1953) and Campbell (1990) are seminal articles on tropes.
2 Cf. Sacks (1996) on cases of blindsight. Seeing Qualitons as Qualia A Dialogue with Wittgenstein on Private Experience, Sense Data and the Ontology of Mind / Hilan Bensusan / Eros Carvalho 31 Literature Campbell, K. (1990). Abstract Particulars. Oxford: Basil Blackwell. Gozzano, S. (2008). "Tropes' simplicity and mental causation", in: Gozano & Orilia (2008), 133-154. Gozzano, S. & Orilia, F., eds. (2008). Tropes, Universals and the Philosophy of Mind, Frankfurt: Ontos-Verlag. Hacker, P. M. S. (1993). Wittgenstein: Meaning and Mind, Volume 3 of an Analytical Commentary on the Philosophical Investigations, Part II: Exegesis 243-42. Wiley Blackwell. Nordhoof, P. (1998). "Do tropes resolve the problem of mental causation?", The Philosophical Quarterly, 48, 221-226. Robb, J. (1997). "The problems of mental causation", Philosophical Quarterly, 47, 178-194. Sacks, O. (1996). An anthropologist on Mars: Seven paradoxical tales. New York: Vintage Books. Williams, D. C., 1953. "The Elements of Being," Review of Metaphysics 7: 3-18, 171-192; cited from Principles of Empirical Realism, Springfield: Charles C Thomas 1966: 74-109. Wittgenstein, L. (1968). "Notes for Lectures on 'Private Experience' and 'Sense Data'", The Philosophical Review, 77 (3), 275-320. Wittgenstein, L. (1973). Philosophical Investigations, Oxford, Blackwell.

Truthfulness and Gricean Cooperation* Andreas Stokke Forthcoming in Grazer Philosophische Studien Abstract This paper examines the Gricean view that quality maxims take priority over other conversational maxims. It is shown that Gricean conversational implicatures are routinely inferred from u erances that are recognized to be untruthful. It is argued that this observation falsifies Grice's original claim that hearers assume that speakers are obeying other maxims only if the speaker is assumed to be obeying quality maxims, and furthermore the related claim that hearers assume that speakers are being cooperative only to the extent that they assume they are being truthful. 1 Introduction It is a platitude that conversation involves cooperation. Slightly more specifically, philosophers and linguists typically take for granted that the ways we communicate with each other in conversation rely on speakers and hearers cooperating on efficiently ge ing information across, or something to the same effect. By far, the most influential account of communicative cooperation is also the one that can be said to be responsible for establishing the paradigm of understanding conversations as cooperative enterprises in the first place, namely the account given by Paul Grice (1989). Grice set out an understanding of conversation as guided by his Cooperative Principle and conversational maxims, the la er being principles the observation of *I am grateful to Ma Benton, Chris Gauker, Torfinn Huvenes, Eliot Michaelson, Anders Schoubye, and Jonas Åkerman for valuable comments and suggestions. 1 which ensure, or at least tend to ensure, cooperation in conversation. Grice also held that cooperation in conversation is closely tied to truthfulness. In particular, Grice thought that, among the maxims of conversation, certain maxims of quality – admonishing speakers to be truthful – have a special status. Briefly put, Grice claimed that speakers are assumed to be observing other maxims only if they are assumed to be obeying quality maxims. I want to argue here that this view of cooperation does not square with the facts about conversational practices. In particular, I want to target the claim that, when hearers make the kind of inferences from what speakers say that Grice was interested in, they assume that speakers are being cooperative only to the extent that they assume they are being truthful. My main motivation for rejecting this idea is the observation that inferences that have the marks of inferences of Gricean conversational implicatures are often made by hearers, even when they recognize that the speaker is violating quality maxims, and as such is being untruthful. Here is an example where this happens: Louise knows that Thelma has been drinking, but Thelma doesn't realize that Louise knows this. (1) Louise. Are you OK to drive? Thelma. I haven't been drinking. As I will argue, even though Louise knows that Thelma is being untruthful, she nevertheless infers that Thelma meant to convey that she is OK to drive. My focus is chiefly on the kind of inferences hearers draw based on observation of what speakers say, and less on what speakers intend, or can reasonably intend, to be conveyed by their u erances. It has been pointed out – e.g., by Jennifer Saul (2002) – that these twomay come apart. A speaker might want to convey a particular implicature, even if the hearer fails to infer it, and a hearer might infer unintended implicatures from what a speaker says. I confine myself to cases in which what the speaker intends to convey coincides with what the hearer infers from what she says. My argument will be that hearers often make inferences intended by speakers, based on assumptions of cooperation along Gricean lines, but which do not involve assumptions of truthfulness. Section 2 reviews Grice's view concerning the priority of quality maxims and the kind of example I take to be evidence against it. In section 3 I consider a number of ways the Gricean might respond to this evidence. I argue that there are inferences hearers make that are best explained as proceeding via the assumption that the speaker is cooperating, but which do not involve assumptions about 2 the speaker's truthfulness. Section 4 argues that the relevant inferences have the features of inferences of Gricean conversational implicatures and briefly discusses some consequences for how to understand the kind of cooperation involved in conversation. 2 Grice on the Priority of Quality 2.1 The Gricean Category of Quality Grice's Cooperative Principle was stated as follows (Grice, 1989, 26): Cooperative Principle Make your contribution such as is required, at the stage at which it occurs, by the accepted purpose or direction of the talk exchange in which you are engaged. In turn, the maxims were divided into four categories, Quantity, Quality, Relation, and Manner. Among these the category of Quality included a supermaxim and two specific maxims (Grice, 1989, 27): Supermaxim of Quality: Try to make your contribution one that is true. First Maxim of Quality: Do not say what you believe to be false. Second Maxim of Quality: Do not say that for which you lack adequate evidence. Here we will only be concerned with the Supermaxim of Quality and the First Maxim of Quality. Grice held that the First Maxim of Quality and the Supermaxim of Quality enjoy a special status in relation to the other maxims and the Cooperative Principle. In "Logic andConversation," immediately after presenting themaxims, Grice comments, [I]t might be felt that the importance of at least the first maxim of Quality is such that it should not be included in a scheme of the kind I am constructing; other maxims come into operation only on the assumption that this maxim of Quality is satisfied. While this may be correct, so far as the generation of implicatures is concerned it seems to play a role not totally different from the 3 other maxims, and it will be convenient, for the present at least, to treat it as a member of the list of maxims. (Grice, 1989, 27) Amajor task for this paper will be to examine how to understand the central claim encapsulated in this passage, that is, the suggestion that "other maxims come into operation only on the assumption that [the First Maxim of Quality] is satisfied." But it appears at first sight that Grice thinks that the kind of communication that is to be seen as guided by the maxims, in some sense, relies on the assumption that the speaker is not saying something she believes to be false. This idea is reinforced in the "Retrospective Epilogue," where Grice wrote, The maxim of Quality, enjoining the provision of contributions which are genuine rather than spurious (truthful rather than mendacious), does not seem to be just one among a number of recipes for producing contributions; it seems rather to spell out the difference between something's being and (strictly speaking) failing to be, any kind of contribution at all. (Grice, 1989, 371) By "The maxim of Quality," in this passage, Grice presumably has in mind the Supermaxim of Quality. This is suggested by his characterizaton of it as pertaining to "contributions," rather than to what is said more specifically. Similarly, the Cooperative Principle is explicitly a principle about contributions. For Grice, contributions comprised both things that are said, or asserted, and things that are conversationally implied. One way of understanding this passage from the "Retrospective Epilogue," therefore, is to see it as suggesting that unless one is aiming to make a truthful contribution, one cannot be seen to be cooperating in the sense intended by the Cooperative Principle. I will return to this idea concerning cooperation later (see section 3.2). First, I want to look at the former passage, concerning the relation between the First Maxim of Quality and the other maxims. 2.2 Violating and Flouting Quality Maxims A first stab at how to understand Grice's suggestion in the passage from "Logic and Conversation" quoted above might be as follows (where H is the hearer and S is the speaker): (I) H assumes thatS is trying to satisfy othermaxims in producing an u erance u only ifH assumes that S satisfies the First Maxim of Quality in producing u. 4 This is one way of reading Grice's suggestion that "other maxims come into operation only on the assumption that [the First Maxim of Quality] is satisfied." (1989, 27) The immediate problemwith (I) is that it is arguably in conflictwith theGricean understanding of a range of cases in which the First Maxim of Quality is exploited in order to generate conversational implicatures. These include tropes like irony, metaphor,meiosis, andhyperbole (seeGrice, 1989, 34–35). TakeGrice'swell-known example of the first of these: X, with whom A has been on close terms until now, has betrayed a secret of A's to a business rival. A and his audience both know this. A says X is a fine friend. (Grice, 1989, 34) In this case the hearer sees that the speaker is not satisfying the First Maxim of Quality. So according to (I), she should not assume that the speaker is trying to satisfy any other maxims in making the u erance. But if so, on the Gricean picture, the hearer should not infer that the speaker wants to convey something different from what she said.1 In particular, the Gricean treatment of these cases assumes that the hearer takes the speaker as trying to satisfy the Supermaxim of Quality. WhenA says that X is a fine friend, the hearer should think that the speakerwants to satisfy the Supermaxim of Quality, and in this way infer that shewants to implicate that A is not a fine friend. The obvious response to this is that, according to Grice's conception of irony, metaphor, meiosis, and hyperbole, these are all cases in which the First Maxim of Quality is flouted. For Grice, cases that give rise to inferences of implicatures are cases in which the speaker violates maxims in a way that is intended to be noticed by the speaker and thereby to trigger the kind of reasoning that is expected to terminate in the inference of an implicature. More generally, since irony and the other tropes involve flagrantly saying something one believes to be false, this is presumably not the kind of untruthfulness that a Gricean would take to be a hindrance to cooperative communication. And in particular, if the Gricean is right that hearers' inferences of implicatures turn on assumptions of truthfulness, it is reasonable to think that such assumptions are 1Grice held that, in cases like irony, the speaker has not said, but "has made as if to say," (Grice, 1989, 34) the literal content of her u erance. The reason for this is that Grice thought of what is said as akin to how some think of assertion. In this paper I use the notion of saying in a loose sense, since nothing will hang on this. So, for instance, I allow myself to assume that ironic speakers say the literal contents of their u erance, even if they do not assert it. For discussion of this, see, e.g., Neale (1992), Stokke (2013). 5 not defeated by a speaker's use of tropes such as irony, metaphor, meiosis, and hyperbole. So, since in these kinds of cases the violation of the First Maxim of Quality is done in such a way as to make the hearer notice the violation, the claim that the First Maxim of Quality is a prerequisite for the operation of the other maxims is be er understood as not pertaining to cases in which it is flouted for the purpose of generating implicatures. Let us stipulate that by disregarding a maxim, we mean violating it but not by flouting it. To disregard a maxim is to disobey it, but not in the way that is intended to call a ention to itself, which Grice identified as exploiting maxims in order to generate implicatures. Given this, one proposal for understanding the Gricean claim concerning the priority of quality maxims is as in (II).2 (II) H assumes thatS is trying to satisfy othermaxims in producing an u erance u only ifH assumes that S is not disregarding the First Maxim of Quality in producing u. According to (II), assuming that someone is trying to satisfy maxims requires assuming that they are speaking truthfully, while allowing that someone may blatantly and openly speak untruthfully in order to generate implicatures by exploiting quality maxims. 2.3 Implicatures Inferred from Detected Quality Violations A situation in which a hearer can see that a speaker is flouting a maxim is not the only kind of situation in which a hearer can see that a speaker is violating a maxim. The hearer might be able to see that the speaker is violating a maxim but is not doing so in the kind of blatant way that is intended to trigger inferences of implicatures. If a hearer can see that a speaker is violating a maxim and that the speaker thinks that the hearer will not notice, the speaker is not flouting themaxim. Consequently, (II) implies that if the hearer can see that the speaker thinks that her violation of the First Maxim of Quality will go unnoticed, the hearer will not try to reconcile that she said what she did with an assumption that she is observing maxims other than the First Maxim of Quality. However, at least at first sight, this does not square with the facts about how u erances are interpreted in this kind of situation. Even if the hearer can see that 2See Benton (in press) for a suggestion of this kind. 6 the speaker is being covertly untruthful, she may still take the speaker as having implicated something other thanwhat she said. Consider our example from above: Louise knows that Thelma has been drinking, but Thelma doesn't realize that Louise knows this. (1) Louise. Are you OK to drive? Thelma. I haven't been drinking. In this case Louise will take Thelma as having implicated that she is OK to drive. Yet Louise knows that Thelma is violating the First Maxim of Quality, even though she is not doing so by flouting themaxim.3 So, since the implicature is still inferred, this is evidence against (II). In support of this conclusion, we can note that the Gricean will agree with this description of the analogous, truthful cases, as in (2). Thelma hasn't been drinking. (2) Louise. Are you OK to drive? Thelma. I haven't been drinking. The Gricean will explain (2) as a case of conversational implicature, more precisely a case of implicature inferred via the assumption that Thelma is trying to satisfy the Maxim of Relation (Grice, 1989, 27). Maxim of Relation Be relevant. Note that, in both (1) and (2), the inference the hearer makes is also intended by the speaker. In the terminology of Saul (2002), both are case in which an "u ererimplicature" coincides with an "audience-implicature." By the same token, even if writers like Stephen Neale (1992) are right that being intended by the speaker is a necessary condition on conversational implicature, both cases qualify. (I return to this in section 4.1). 3If one prefers, one may reconstruct the case so as to specify that Louise does not know prior to Thelma's u erance that she has been drinking and hence already knows the answer to the question she is asking. For example, one can think of cases in which Louise recognizes from Thelma's utterance itself that she has been drinking, e.g., she is slurring her words, or the like. Thanks to Eliot Michaelson for this suggestion. 7 Given that the Gricean will count (2) as a case of conversational implicature, to vindicate the claim in (II), the Gricean needs to motivate that there is a relevant difference between (1) and (2). In particular, the Gricean needs to argue that, despite the apparent similarities mentioned above, (1) should not be classified as a case of conversational implicature, or more generally as the kind of inference that Grice was interested in accounting for. In the next section, I will argue that there is no convincing way of making this kind of argument. 3 Inference, Cooperation, and Truthfulness 3.1 Repairing Maxim Violations One potential way of arguing for a difference between (1) and (2) starts from the observation that the hallmark of a Gricean inference is that it aims at establishing that maxims that are violated at the level of what is said are satisfied at the level of what is meant or implied. As we might say, when maxims are violated at the level of what is said, a Gricean inference should repair such violations at the level of what is implied. In (1) Louise recognizes that Thelma is violating the First Maxim of Quality at the level of what is said. Hence, it might be objected that Louise's inference should aim at repairing this violation at the level of what is implied. In particular, Louise's inference should establish that Thelma is satisfying the Supermaxim of Quality.4 However, in (1), Louise's inference does not establish that Thelma is satisfying the Supermaxim of Quality, since she can see that what is implied is something Thelma believes to be false (that she is OK to drive). So the violation of the First Maxim of Quality is not repaired. Hence, so the objection goes, the inference involved in (1) is not of the kind the Gricean wants to associate with conversational implicatures. In reply to this objection we should note that, while inferring implicatures is indeed a process that centrally aims at repairingmaxim violations, this kind of repair strategy is directed at floutings of maxims. An implicature is inferred, according to the Gricean scheme, when the hearer can see that the speaker is blatantly and 4The quality maxims have the feature that while the First Maxim of Quality and the Second Maxim of Quality pertain to what is said, the Supermaxim of Quality is explicitly a principle concerning contributions. Since Grice took contributions to include both what is said and what is implied, the Supermaxim of Quality plays the role of the maxim that is seen to be satisfied at the level ofwhat is implied by implicatures such as those involved in irony, metaphor, etc. For othermaxims, e.g., the Maxim of Relation, the repair will target the maxim itself. 8 noticeably failing to comply with one or more maxims, and hence that she is doing so precisely with the intention of triggering the inference of the implicature. Yet, in (1), the First Maxim of Quality is not flouted but is violated in a covert way, albeit the deceit is unsuccessful. In the terminology introduced above, Thelma's u erance in (1) disregards the First Maxim of Quality. So, since the First Maxim of Quality is not flouted, in this case, why should the hearer try to repair the violation at the level of what is implied, that is, why should she assume that the speaker is trying to satisfy the Supermaxim of Quality? Moreover, Thelma does flout a maxim at the level of what is said, namely Relation, and this violation is repaired by the inference of the implicature that Thelma is OK to drive. Inferring the implicature that Thelma is OK to drive is a way of squaring what she said, i.e., that she has not been drinking, with the presumption that she is trying to obey Relation. In this case, the measure of what is relevant is naturally taken to be given by what constitutes an answer to the question that was explicitly asked, i.e., "Are you OK to drive?" Since there are only two answers to this question – yes or no – inferring the implicature is a way of establishing that while what she said is not strictly speaking relevant, Relation is nevertheless satisfied at the level of what is implied. 3.2 Obeying Maxims and Being Cooperative A more promising response to our claim that the inference involved in (1) is relevantly parallel to the inference involved in (2) is to object that the former inference cannot be seen as proceeding via an appeal to the Cooperative Principle. Grice took inferences of implicatures to centrally rely on the assumption that the speaker is cooperating. For example, towards the end of "Logic and Conversation," Grice says, [T]o calculate a conversational implicature is to calculate what has to be supposed in order to preserve the supposition that the Cooperative Principle is being observed [...]. (Grice, 1989, 39–40) Similarly, in providing his "general pa ern for the working out of a conversational implicature," Grice explicitly includes an appeal to the Cooperative Principle.5 The general pa ern is described as follows: 5Gricemakes essentially the same remark in "U erer'sMeaning and Intentions" (see Grice, 1989, 86). 9 He has said that p; there is no reason to suppose that he is not observing the maxims, or at least the Cooperative Principle; he could not be doing this unless he thought that q; he knows (and knows that I know that he knows) that I can see that the supposition that he thinks that q is required; he has done nothing to stop me thinking that q; he intends me to think, or is at least willing to allow me to think, that q; and so he has implicated that q. (Grice, 1989, 31) The same choice of words is used immediately above, when Grice states that someone implicates something only if "he is to be presumed to be observing the conversational maxims, or at least the Cooperative Principle [...]." (Grice, 1989, 30) So perhaps the suggestion thatGricean inferences involve assumptions of truthfulness should be understood as not pertaining to reasoning about whether the speaker is obeying maxims, rather than to reasoning about whether the speaker is being cooperative at all. We might try to restate the Gricean claim as in (III). (III) H assumes that S is trying to satisfy the Cooperative Principle, in producing an u erance u only ifH assumes that S is not disregarding the First Maxim of Quality in producing u. If the Gricean claim about truthfulness and cooperation is to be understood in this way, one might suggest trying to distinguish between (1) and (2) by agreeing that both are inferred via an assumption that the speaker is obeying Relation, while insisting that only in the la er case does the hearer also assume that the speaker is being cooperative. In support of this line of argument, onemight point to the fact that, even though Grice thought of themaxims as norms that implement cooperation in conversation, he is careful to qualify his statement of this idea. Grice says that the maxims are principles "the following of which will, in general, yield results in accordance with the Cooperative Principle." (1989, 26, emphasis added) Perhaps we should think of (1) as a case where following the maxims does not yield compliance with the Cooperative Principle. There are two main problems with this strategy for differentiating between (1) and (2). First, if it is agreed that the inference in (1) proceeds via the assumption that the speaker is trying to satisfy Relation, the Gricean has arguably given up the original view concerning the centrality of quality maxims. Conceding that the inference in (1) proceeds by assuming the speaker is obeying Relation is in conflict with Grice's suggestion that "other maxims come into operation only on the assumption that [the First Maxim of Quality] is satisfied." (Grice, 1989, 27) 10 Second, it is hard to see how to understand the suggestion that, in (1), Louise assumes that Thelma is trying to satisfy Relation but does not assume that she is trying to comply with the Cooperative Principle. This would amount to the claim that while Louise assumes that Thelma is trying to be relevant, she is not assuming that Thelma is trying to make her contribution such as is required, at the stage at which it occurs, by the accepted purpose or direction of the talk exchange in which she is engaged. It is at best unclear how to make sense of this idea. Hence, despite the fact that, in (1), Thelma is being untruthful, it is not easy to see this as a case inwhich compliancewith Relation does not yield compliancewith the Cooperative Principle. Instead, it might be proposed that the reason Thelma cannot be seen as being cooperative is that, since she is being untruthful, she is not speaking in a way that counts as what Grice thinks of as making a contribution at all. We have already seen evidence for this way of thinking in Grice's statement, from the "Retrospective Epilogue," that the Supermaxim of Quality marks the difference between something being a (genuine) contribution or not (see Grice, 1989, 371). However, it is question-begging to appeal to this idea at this point in the dialectics. We have pointed to evidence against the claim that the relevant inferences are made only under the assumption that the speaker is being truthful. The potential response under consideration is that, while Louise assumes that Thelma is obeying Relation, she does not assume that Thelma is obeying the Cooperative Principle, and hence the inference is sufficiently different from those involved in truthful cases. But to motivate this proposal by the claim that making a contribution at all requires being truthful – and hence that assuming that someone is making a contribution requires assuming they are being truthful – is to beg the question. It is no good to simply insist that making cooperative contributions requires being truthful. As a final option at this point, one might suggest distinguishing between different senses of cooperation. One might think that Grice's original notion of cooperation should be understood as applying to discourse that is ultimately aimed at exchanging truths.6 On the other hand, since inferences that appear to have all the trademarks of Gricean inferences are routinely drawn in spite of known untruthfulness, this might suggest looking for a way of understanding cooperativeness such that being cooperative, in the relevant sense, does not require obeying quality maxims. A view of this kind is suggested by Richmond Thomason (1990). Agreeing 6See Grice (1989, 371) for some support for this interpretation. 11 that "implicatures are possible in situations that can only be described as hostile or uncooperative" (1990, 355), Thomason suggests that we should react by understanding the kind of cooperation involved in conversations as "a shared sense of where the conversation has been and of where it is headed: the common plan of the conversation." (1990, 356) One can see the kind of cooperation that, according to the Gricean program, is necessary for generating implicatures – and for communicatingmore generally – as cooperation toward the goal of sharing truths. If so, then one cannot regard someonewho violates qualitymaxims as being cooperative. Yet one can also understand cooperation in a conversation as cooperation toward efficient information-sharing, broadly understood.7 Given this kind of distinction, contributions might be seen as "genuine," in the Gricean sense, only if the comply with the truth-oriented sense of cooperation. Hence, Thelma's u erance will be seen as not genuine, in this sense. But on the other hand, the implicature that is inferred from it might be explained in terms of cooperation in the broader, Thomasonian sense. This line of thought, however, is not a threat to the argument I am pursuing. It still amounts to accepting that implicatures may be inferred by way of a presumption that the speaker is being cooperative (in one sense), even when the hearer can see that the speaker is not being truthful in the sense of obeying qualitymaxims. To be sure, the Gricean might reserve a narrow sense of cooperation and declare that untruthful contributions are not cooperative, in this sense, and hence should be seen as "spurious." But this does not explain away implicatures that are inferred from untruthful contributions. (I return later, in 4.1, to the issue of implicaturegeneration without presumptions of cooperation.) The Gricean is in need of a convincing way of motivating that, while the inference in (2) involves an assumption of cooperation, the inference in (1) does not. However, as I argue below, there are reasons for thinking that differentiating between the cases in this way will be arbitrary. 3.3 Situated Inference How might one argue that Louise's inference in (1) does not proceed via the assumption that Thelma wants to cooperate? A promising suggestion is that, rather than seeing the inference in (1) as a Gricean inference, based on assumptions concerning the speaker's intentions, or state of mind more generally, the inference can 7See also Lepore and Stone (2015, ch. 14) for recent, relevant discussion. 12 be explained simply as a logical or probabilistic inference relying on background assumptions. The claim would be that, in (1), Louise infers that Thelma is OK to drive – or perhaps that Thelma is probably OK to drive – solely on the basis of the information provided bywhat is said, i.e., that she has not been drinking, and background assumptions such as that Thelma knows how to drive, the road conditions are not unusual, etc. To be sure, in (1), Louise does not come to believe that Thelma is OK to drive – nor that she is probablyOK to drive – on the basis of her inference. But granting this arguably does not rule out seeing Louise as inferring from what Thelma said that she is probably OK to drive. We might think of the case along the following lines. Louise notes that Thelma said that she has not been drinking. She has available the information that if Thelma has not been drinking, she is probably OK to drive. So Louise notes that what Thelma said implies that she is probably OK to drive. Indeed, it seems right to say that what Louise takes from Thelma's u erance in (1), among other things, is the observation that what she said implies that she is probably OK to drive. Hence, as long as this is all wemean by the claim that Louise infers that Thelma is probably OK to drive from what Thelma said, we are not precluded from seeing Louise as making this kind of inference.8 The main problem for the Gricean is that, if she endorses this approach to (1), it is hard to see how to motivate not applying it to cases like (2), as well. If one thinks that, in (1), Louise can infer that Thelma is probably OK to drive directly from what she says, and background assumptions, without recourse to an assumption concerning Thelma's cooperativeness, what is the argument for not accounting for (2) in the sameway? As before, there seems to be noway of differentiating between the cases without begging the question. In other words, this route appears to end up endorsing the view that, at least for cases the Gricean wants to explain as implicatures generated by Relation, most of these can be explained as ordinary inferences of the kind described above. Chris Gauker (2001) defends a position of this kind. According to Gauker, "the concept of conversational implicature is not a useful concept in the theory of language." (Gauker, 2001, 170) Instead, Gauker argues that cases the Gricean explains in terms of implicature are be er explained as cases of what he calls situated inference. That is, the inferences that the Gricean thinks are inferences of implicatures are, for 8There is another sense of "A infers q from p," i.e., the sense in which this involves deducing q from p by consciously going through a process of valid reasoning. It is less plausible to think of Louise inferring that Thelma is probably OK to drive from what she says in this sense. But even if this is the preferred way of thinking of her reasoning, this clearly does not commit one to the claim that she comes to believe that Thelma is OK to drive. 13 Gauker, simply inferences that are made on the basis of what is said, awareness of relevant contextual facts, and background assumptions. Consider, for example, how Gauker explains Grice's (1989, 32) familiar case of the motorist whose car is out of gas. The dialogue is as in (3). (3) A: I am out of gasoline. B: There is a gas station around the corner. Gauker writes, To explain the success of communication in this case, we have to explain how A is able to conclude fromwhat B explicitly says that he can get gas at the gas station around the corner. One explanation for A's drawing this conclusion might be Grice's own explanation of this, namely, that A recognizes that this is what B must be supposing if B is conforming to the Cooperative Principle. An alternative explanation is that A reasons from the truth of what B says and the character of the external situation to the conclusion that if there is a gas station around the corner then probably it is open and has gas to sell. Thoughts about what the speaker must have been thinking need not play any role whatsoever, and the speaker need not have intended the hearer to have any such thoughts. In my view, this is in fact the be er explanation. (Gauker, 2001, 174) Applied to the case of (1), the claim would be that Louise infers that Thelma is probably OK to drive on the basis of her recognition that Thelma said that she has not been drinkingwithout proceeding byway of an assumption that Thelmawants to cooperate.9 And moreover, the same explanation would be given for run of the mill cases like (2). It is not being denied that Thelma says what she does precisely with the intention that Louise should make the inference she does make. A view like Gauker's merely claims that Louise does not need the assumption that this is what Thelma intends as a premise for her inference. As Gauker says about the gas station example: B might indeed intend that A will conclude that the gas station is open and has gas to sell (or maybe not), and in order for A to reach this conclusion A 9We assume that Gauker's idea of reasoning "from the truth of" what is said can be construed along the lines suggested earlier, i.e., so as not necessarily to involve coming to believe the conclusion. In particular, we assume that an explanation alongGauker's lines can be given for cases where the hearer knows that what the speaker says is false. We can think of the hearer, in such cases, as reasoning from the supposition that what the speaker said is true. 14 will have to pay a ention to various features of the situation beyond what B explicitly said; but there is no special reason for A to pay a ention to what B might have had in mind in speaking, and B need not intend A to do so. (Gauker, 2001, 175) I take it that the kind of inference Gauker describes might occur in many cases that, according to the Gricean, involve implicatures. In particular, we should grant that, in (1) and (2), Louisemight infer fromThelma's u erance that she is probablyOK to drive without proceeding by way of a premise concerning Thelma's state of mind. Yet, even if Gauker is right about the inferences targeted by his discussion, it can be argued that there are inferences, which hearers routinely make, and which do rely on assumptions of cooperation. In both (1) and (2), Louise does not only infer that Thelma is probably OK to drive, and nor is that all that Thelma intended her to infer. In both cases Louise also infers that Thelmawanted to convey that she is OK to drive. The inference she makes is not simply one about the facts concerning whether Thelma is (probably) OK to drive. Louise also makes an inference about Thelma's goals in making her u erance. Similarly, in the motorist example of (3), that he can get gas at the gas station around the corner is not all that A infers from B's u erance. A will also infer that B wanted to convey that he can get gas at the gas station around the corner. Generalizing, in many cases, the hearer not only makes an inference fromwhat is said, but also infers something about what the speaker wanted to convey by sayingwhat she did. Andmoreover, while Gaukermight be right about the former type of inference, it is hard to see how to account for the la er kind of inference without seeing it as proceeding by way of assumptions about the speaker's efforts to cooperate. In both (1) and (2), the most natural way of explaining how Louise infers from Thelma's u erance that Thelmawants to convey that she is OK to drive is arguably to see it as an inference that proceeds by way of the assumption that Thelma wants to cooperate. That is, the assumption that Thelma wants to provide an answer to the question, "Are you OK to drive?" And similarly, to explain howA infers, in (3), that B wants to convey that A can get gas at the gas station around the corner, one arguably needs to see the inference as proceeding via the assumption that B wants to cooperate. That is, roughly, that B wants to make a contribution that is helpful in the situation where A has just informed B of being out of gas. Against Gauker's wholesale rejection of the category of implicatures, then, the Gricean can claim that at least some inferences based on observation of what is said that hearers routinely make are best explained as proceeding by way of assump15 tions concerning the hearer's a empt to cooperate. However, of course, this hardly vindicates the further Gricean claim that cooperation involves truthfulness. In (1) Louise's inference that Thelmawants to convey that she is OK to drive ismade despite Louise's recognition that Thelma is violating both the First Maxim of Quality and the Supermaxim of Quality. So if we are right that this inference is explained in terms of an assumption of cooperation, we still have reason to conclude that assuming that one's interlocutor is cooperating does not require assuming that she is being truthful. 3.4 Bald-Faced Implicature At this point the Gricean might want to shift gears and focus more on what hearers must be assuming about what speakers want them to assume, rather than focusing more directly on what hearers assume about the extent to which speakers in fact comply with norms of truthfulness. Perhaps the Gricean should grant that, in (1), Louise does not assume that Thelma is being truthful, but point out that Louise nevertheless knows that Thelma intended her to believe that she is being truthful. In other words, it might be observed that, even if the inference in (1) does not proceed via the assumption that Thelma is in fact being truthful, it does proceed via an assumption that Thelma intended Louise to assume that she is being truthful. Along these lines, the Gricean might want to recast her view as (IV). (IV) H assumes thatS is trying to satisfy othermaxims in producing an u erance u only ifH assumes thatS intended thatH assume thatS is not disregarding the First Maxim of Quality in producing u. Arguably, if something like (IV) is right, this would be a way of vindicating the original idea that a speaker's assumption that a hearer is obeying maxims relies on her assuming that the speaker is being truthful, at least in the sense that it relies on an assumption that the speaker wanted to be perceived as being truthful. (IV) is satisfied in normal cases of implicature, in which the hearer has reason to believe (or has no reason not to believe) that the speaker is saying something she believes to be true in order to implicate something she also believes to be true, as in (2) and (3). Moreover, (IV) still allows the Gricean to pursue the response to the position championed by Gauker we described above. That is, it might be granted that inferring that Thelma is probably OK to drive itself does not rely on assumptions about Thelma's intentions, but still, inferring that Thelma wanted to convey that she is probably OK to drive does rely on such assumptions. Indeed, it is reasonable to think that it does. 16 The problem with this is that we can imagine cases in which not even (IV) is satisfied. In (1) Thelma is trying to be covertly untruthful, but is found out by Louise. But there are cases in which the speaker is not even trying to be covertly untruthful, but in which the relevant kind of inference is still made by the hearer. These are cases in which an implicature is derived from so-called bald-faced lies, that is, open or undisguised lies.10 As we might say, these are cases of bald-faced implicature. Suppose, for example, that it is common knowledge between Thelma and Louise that Thelma has been drinking. That is, Thelma knows that Louise knows that Thelma knows that Louise knows, etc., that Thelma has been drinking. If our dialogue takes place in this se ing, Louise will still infer that Thelma wanted to convey that she is probably OK to drive. Of course, as before, she will not believe that Thelma is probably OK to drive. But she will still infer that Thelma wanted to convey that she is, and indeed, Louise will recognize (correctly) that Thelma wanted her to recognize that she wanted her to do so. Again, it is hard to see how to explain this inference except as based on an assumption about Thelma's wants to cooperate, and in particular, that shewants to satisfy Relation. And so, (IV) still does not manage to spell out a way of upholding the Gricean idea that a hearer is assumed to be cooperative, in the sense of obeying maxims, only to the extent that she is assumed to be truthful. 4 Implicature and Rationality 4.1 Implicature without Truthfulness I have argued that at least some inferences that are drawn on the basis of assumptions of cooperation aremade independently of assumptions of truthfulness. Should we think of these cases as inferences of conversational implicatures? I think we should. As is often emphasized, Grice stresses repeatedly that an implicature is an inference that is needed for preserving the presumption that the speaker is observing the maxims, or at least the Cooperative Principle. For example, in the passage we quoted earlier, Grice says, 10For discussion, see Carson (2006), Sorensen (2007), Fallis (2009), Stokke (2013). Some writers, e.g., Lackey (2013), object that, in the relevant examples, the speaker is nevertheless trying to hide information from the hearer. Even if such critics are right, this is irrelevant to my argument, since, in the relevant cases, the hearer is still not assuming that the speaker intended the hearer to assume that the speaker is not violating the First Maxim of Quality. 17 [T]o calculate a conversational implicature is to calculate what has to be supposed in order to preserve the supposition that the Cooperative Principle is being observed [...]. (Grice, 1989, 39–40, emphasis added) Similarly, considerGrice's statement of three necessary conditions on implicature:11 Amanwho, by (in, when) saying (ormaking as if to say) that p has implicated that q, may be said to have conversationally implicated that q, provided that (1) he is to be presumed to be observing the conversational maxims, or at least the Cooperative Principle; (2) the supposition that he is aware that, or thinks that, q is required in order tomake his saying ormaking as if to say p (or doing so in those terms) consistentwith this presumption; and (3) the speaker thinks (and would expect the hearer to think that the speaker thinks) that it is within the competence of the hearer to work out, or grasp intuitively, that the supposition mentioned in (2) is required. (Grice, 1989, 30–31, emphasis added) Louise's inference that Thelma wants to convey that she is OK to drive, in (1), fits this pa ern. As suggested earlier (see section 3.1), in (1), the assumption that Thelmawants to cooperate amounts to the assumption that she wants to convey an answer to the question, "Are you OK to drive?" Since there are only two answers to this polar question, assuming that Thelma wants to convey an answer means assuming that she wants to convey either a yes or a no. If Louise were to infer that Thelma wanted to convey that she is not OK to drive, this would make a mystery of why Thelma said that she has not been drinking. So in order to reconcile that Thelma said what she did with the assumption that she wants to convey an answer to the question, "Are you OK to drive?" Louise must infer that Thelma wanted to convey that she is OK to drive. So Louise's inference that Thelma wanted to convey that she is OK to drive satisfies Grice's characterizations of conversational implicatures. Moreover, as suggested earlier (see section 2.3), that Thelma wanted to convey that she is OK to drive is something Thelma intended Louise to infer from her u erance. As noted earlier, Saul (2002) distinguishes between u erer-implicatures and audience-implicatures, the former being, roughly, implicatures the speaker intended the audience to infer, and the la er being, roughly, implicatures the audience in fact infers from what the speaker says. The kind of examples I have appealed to – as represented by (1) – are cases where these coincide. Hence, even if one is sympathetic to this kind of distinction, one should agree that cases like (1) are cases of implicature in both senses. 11See also (Grice, 1989, 370). 18 By contrast, Stephen Neale (1992) suggests that something is an implicature only if it is intended, and he argues that this condition is grounded in Grice's view of implicatures as an aspect of speaker meaning: A necessary condition on conversational implicatures [...] is that they are intended. This follows [...] at least from the fact that (a) what U implicates is part of what U means, and (b) what U means is determined by U 's communicative intentions. (Neale, 1992, 528) To repeat, in (1), Thelma intends that Louise infer that she is OK to drive. And furthermore, the la er content is something that Thelma meant, in the Gricean sense. For Grice, to say that a speaker S meant that p, roughly, is to say that S intended that the hearerH come to believe that p partly as a result of recognizing this intention.12 In (1) Thelma wants Louise to believe both that she has not been drinking and that she is OK to drive. Both these things are part of what Thelma means, on this occasion. Finally, some writers defend the view that, contrary to Grice's own account, implicatures do not rely on cooperation. For example, Wayne Davis (1998) argues that "Conversational implicatures may exist when there is no presumption on anyone's part that the speaker is observing the Cooperative Principle." (1998, 115) Part of Davis's case for this conclusion is based on cases similar to our case of (1). Here is one of his examples (Davis, 1998, 116): (4) Karen: Were you with Jennifer last night? George: I was out drinking with the guys. Davis claims that, "George implicated that he was not with Jennifer last night. He may have implicated this even if Karen knows he is lying, having seen George and Jennifer together." (Davis, 1998, 116) For Davis, a speaker's conversationally implicating something is chiefly a matter of the speaker having certain intentions.13 He takes cases like (4) to demonstrate that speakers may have such intentions even when they are not presumed to be cooperating. Davis writes, What S implicates cannot be due even in part towhat others presume or know about S. To implicate something is to mean or imply it in a certain way. And asGrice [...] correctly observed, tomean or imply something is to have certain intentions. But S's intentions do not depend on what anyone else presumes. (Davis, 1998, 122) 12See Neale (1992, 515). 13See Saul (2002, 240-241) for criticism of this view. 19 Accordingly, Davis thinks that, in (4), George may have intended to convey that he was not with Jennifer last night, even if Karen does not take him to be cooperating. This I take to be hard to deny. But, by contrast, what I have been arguing for is a point concerning the way hearers infer, or work out, implicatures. I take the idea that such inferences rely on assumptions of cooperation, and perhaps on assumptions of truthfulness, to be a more interesting thesis than the one Davis is targeting. It is worth stressing, however, that my arguments do not conflict with Davis's case for the conclusion that speakers may intend implicatures even when they are not presumed to be cooperating. I conclude that the case of (1) has the characteristics of a case of conversational implicature, independently of how one thinks of the issues concerning the difference betweenu erer-implicature and audience-implicature, or the issue ofwhether being intended by the speaker is a necessary condition on implicature. In these respects, (1) is parallel to (2). Hence, we should conclude that conversational implicature – and more broadly, the kind of communication that Grice identified as relying on cooperation – does not rely on a presumption of truthfulness.14 4.2 Untruthfulness and the Rational Basis of Cooperation According towhat I have been arguing, the conclusion that Thelmawants to convey that she is OK to drive is arrived at via the assumption that Thelmawants to satisfy theMaxim of Relation, or at least the Cooperative Principle. We have seen that this assumption may be in place even when the hearer knows that the speaker is not observing quality maxims, i.e., even when the hearer knows the speaker is being untruthful. However, it might be asked, why should one assume that one's interlocutor is being cooperative when one can see that they are being untruthful? Familiarly, Grice was adamant that the practice of following the Cooperative Principle and the maxims has a rational basis. As he says, I would like to be able to think of the standard type of conversational practice not merely as something that all or most do in fact follow but as something 14This conclusion is compatiblewith other views according towhich truthfulness is a prerequisite for communicative practices. For example, Lewis (1969), (1975) held that a language L is used by a population P if and only if there is a convention of truthfulness and trust in L among the members of P . The fact that Gricean inferences may be drawn in the absence of assumptions of truthfulness is compatible with the view that conventions of truthfulness and trust are necessary for our practice of using a particular language to communicate with each other. 20 that it is reasonable for us to follow, that we should not abandon. (Grice, 1989, 29) Given this, one way of suggesting an explanation of the basis for Louise's assumption that Thelma wants to cooperate in (1) is to say that she does so simply because she is following a practice that is already established, and which has a rational underpinning. But it might be felt that it would be preferable if it could be established that, even in the case of (1), the assumption that Thelma wants to cooperate can be seen as rational, and not just because it constitutes following a practice that is otherwise rational. Put differently, can it be rational to assume that someone one knows is being untruthful is observing Gricean maxims, or at least the Cooperative Principle? A full response to this question is far beyond the scope of this paper. But I want to end by suggesting at least some reason for thinking that it can be given an affirmative answer. It is not unreasonable to think that we need to be able to work out what people want to convey to us, even when we know they are being untruthful. Doing so is helpful, for example, for evaluating the overall trustworthiness of the speaker, which in turn is useful for evaluating what they might tell us in the future, or indeed evaluating what they have told us in the past. More generally, discovering what someone will a empt to make us believe – even when we already know the truth of the ma er – may be useful for learning about their character, or for learning about their intentions in the more immediate situation. If this is right, it at least goes some way toward explaining why it is reasonable to take speakers as cooperating, in the sense of obeying maxims and trying to make pertinent contributions to conversations, even if one knows they are being untruthful. References Benton, M. (in press). Gricean quality. Forthcoming in Noûs. Carson, T. (2006). The definition of lying. Noûs, 40, 284–306. Davis, W. (1998). Implicature: intention, convention, and principle in the failure of Gricean theory. Cambridge and New York: Cambridge University Press. Fallis, D. (2009). What is lying? Journal of Philosophy, 106, 29–56. 21 Gauker, C. (2001). Situated inference versus conversational implicatures. Noûs, 35(2), 163-.189. Grice, H. (1989). Studies in the way of words. Cambridge, MA: Harvard University Press. Lackey, J. (2013). Lies and deception: An unhappy divorce. Analysis, 73(2), 236– 248. Lepore, E., & Stone, M. (2015). Imagination and convention: Distinguishing grammar and inference in language. Oxford and New York: Oxford University Press. Lewis, D. (1969). Convention: A philosophical study. Cambridge, MA: Harvard University Press. Lewis, D. (1975). Languages and language. In Philosophical papers (Vol. I, p. 163188). Oxford and New York: Oxford University Press, 1983. Neale, S. (1992). Paul Grice and the philosophy of language. Linguistics and Philosophy, 15, 509–559. Saul, J. (2002). Speaker meaning, what is said, and what is implied. Noûs, 36(2), 228–248. Sorensen, R. (2007). Bald-faced lies! Lying without the intent to deceive. Pacific Philosophical Quarterly, 88, 251–264. Stokke, A. (2013). Lying and asserting. Journal of Philosophy, CX(1), 33–60. Thomason, R. (1990). Accommodation, meaning, and implicature: Interdisciplinary foundations for pragmatics. In P. Cohen, J. Morgan, & M. Pollack (Eds.), Intentions in communication (pp. 325–363). Cambridge, Mass. and London: MIT Press.

Separability, Locality, and Higher Dimensions in Quantum Mechanics [A shortened version of this paper will appear in Current Controversies in Philosophy of Science, S. Dasgupta and B. Weslake, eds. Routledge.] Alyssa Ney [email protected] November 30, 2016 Abstract: This paper describes the case that can be made for a high-dimensional ontology in quantum mechanics based on the virtues of avoiding both nonseparability and nonlocality. 1. Introduction In his paper, "On the Einstein Podolsky Rosen paradox," John Bell derived a result according to which a theory capturing the statistical predictions of quantum mechanics cannot be one that avoids situations in which the result of one measurement correlates with the result of another space-like separated from it such that no prior determination could suffice to explain the correlation. He used this result to argue that: In a theory in which parameters are added to quantum mechanics to determine the results of individual measurements, without changing the statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote. Moreover, the signal involved must propagate instantaneously, so that such a theory could not be Lorentz invariant. (1964/1987, p. 20) As we know, it has since been observed that settings of a measuring device in one location may exhibit an instantaneous, thus superluminal dependence on outcomes in distant locations (Aspect et. al. 1981), thus confirming Bell's predictions. And so such nonlocal dependence seems to be a feature of our world, not merely for a quantum theory with "added parameters." Both quantum theories and our world thus seem to exhibit the kind of nonlocality Bell argued for. But is this a necessary consequence of his proof and the experimental tests that we should include in our best metaphysical interpretation of quantum systems? 2 Although some may be happy or at least resigned to accept nonlocality as a consequence of quantum theories, others seek ways to avoid the implication that nonlocality is a fundamental feature of our world. One strategy is to reject a key assumption on which Bell's derivation of his theorem is thought to rely: the separability of quantum systems. The goal of this paper is to discuss what is perhaps a more promising alternative. This is to avoid both nonlocality and nonseparability by adopting a higher-dimensional interpretation of quantum systems. This higher-dimensional interpretational framework is now commonly referred to in the literature as wave function realism (Albert 2013). It provides an interesting way to achieve a kind of local and separable metaphysics, however, as we will see, not all considerations in favor of locality and separability may apply to generate support for this interpretation. 2. Entanglement, Nonseparability, and Nonlocality Bohm's illustration of the kind of case with which Einstein, Podolsky, and Rosen (and thus Bell) were concerned considers an extremely simple entangled state (Bohm 1951). Suppose that we have a molecule containing two atoms in a state in which the total spin is zero and that the spin of each atom is ħ/2. Roughly speaking, this means that the spin of each particle points in a direction exactly opposite to that of the other, insofar as the spin may be said to have any definite direction at all. Now suppose that the molecule is disintegrated by some process that does not change the total angular momentum. The two atoms will begin to separate and will soon cease to interact appreciably... When the atoms separated, each atom would continue to have every component of its spin angular momentum opposite to that of the other. The two spin-angular-momentum vectors would 3 therefore be correlated... Suppose now that one measures the spin angular momentum of any one of the particles, say No. 1. Because of the existence of correlations, one can immediately conclude that the angular-momentum vector of the other particle (No. 2) is equal and opposite to that of No. 1. (1951, p. 614) In this scenario, our atoms are in an entangled state, the singlet state, in which two particles are entangled with respect to their spin along some particular axis. Particles in such a state may be represented by the following wave function: ψ'()*+,- = / 0 x − up 5 x − down : − / 0 x−down 5 x − up : The Born rule, the rule of quantum mechanics that allows us to infer probabilities for measurement results from such representations, will then tell us that were we to measure the spin states of these atoms, we would have a 50% chance of finding the first x-spin-up and the second x-spin down, and a 50% chance of finding the first x-spin-down and the second x-spin up. To say that the wave function of these atoms describes an entangled state is simply to say that they are in a state where due to some previous process or interaction, the expectation values of measurement results with respect to a particular variable are modally correlated. We are able to correctly describe a system as in an entangled state without yet getting into the metaphysics of the situation, in particular before asking whether the atoms are in a state that is either (a) nonseparable or (b) nonlocal, in the senses to be described.1 Let's disentangle these notions now. A. Nonseparability 1 But don't we need to know at least that e1 and e2 are numerically distinct, that they are two things, to know ysinglet is an entangled state? No, different variables of a single entity can be entangled as well (cf. Quian and Eberly 2013). 4 Separability is a feature of physical systems in which the systems' constituents individually occupy distinct regions of space-time. A system located at a space-time region R is separable when it contains subsystems located at nonoverlapping proper subregions of R and all states of the system at space-time region R are wholly determined or grounded by the states of those subsystems. A state of such a system is a separable state when it is wholly determined by states of these subsystems. Similarly, Howard writes: [Separability] is a fundamental ontological principle governing the individuation of physical systems and their associated states, a principle implicit in many classical physical theories. It asserts that the contents of any two regions of space-time separated by a nonvanishing spatiotemporal interval constitute separable physical systems, in the sense that (1) each possesses its own, distinct physical state, and (2) the joint state of the two systems is wholly determined by these separate states. (1989, pp. 225-226) The key difference here is Howard's addition of the constraint that the relevant subsystems are those separated by some "nonvanishing spatiotemporal interval." I don't think this is required to discuss separability or its failure. We may illustrate this notion using nonscientific examples. For example, that a pair of tennis balls is orange and yellow is a separable state of the pair, for that they together have these colors is determined by the colors of the individual balls, one being orange and one being yellow. On the other hand, a couple's being married is a nonseparable feature of a pair of individuals, technically, since it is not determined by the states of the individuals taken separately. A couple's being married is not a particularly interesting nonseparable feature since although it is true that it is not determined by features of the individuals taken separately, it is determined by the state of the individuals plus the features of some other things in the couple's environment. The kind of 5 nonseparability suggested by quantum entanglement is rather more interesting because it is often thought not to be reducible to individual states of subsystems combined with states of the system's environment, or anything else. Systems in quantum mechanically entangled states, like Bohm's pair of atoms, are often thought to exhibit fundamental nonseparability. The singlet state is thought to be an example of a nonseparable state because the atoms' being in such a spin state is not determined by any individual facts about them, including facts about their individual x-spins. For in such a state, it appears there is no definite fact about the atoms' individual x-spins. For each one, we know that if we conduct an x-spin measurement, there is a 50% chance of finding an x-up result and a 50% chance of finding an x-down result. There is nothing more definite we may say about their individual x-spins. And yet it is also a fact about the pair that if their x-spin state is to be measured, it is absolutely certain that they will be found to have opposite spins. This joint fact about the system as a whole is not determined by any fact about the atoms' individual x-spins. Schrödinger emphasized this seeming consequence of quantum mechanics explicitly in 1935: Maximal knowledge of a total system does not necessarily include total knowledge of all of its parts, not even when these are fully separated from each other and at the moment are not influencing each other at all. (1935, p. 160) This consequence is significant enough for Schrödinger that he places the entire sentence in italics. Systems can be and often are in entangled states of many variables, such as spin, position, momentum, and energy. For this reason, nonseparability appears to be a pervasive feature of quantum mechanics. Below I will consider ontological interpretations of quantum systems that reveal this apparent nonseparability to be a consequence of a more fundamental metaphysics that 6 is separable in higher dimensions. It is worth mentioning beforehand however that there is a way of ensuring separability, if one wants to adopt the Bohmian approach to quantum theories. Bohmian mechanics is an alternative approach to quantum mechanics. It contains a dual ontology of (a) particles that always possess determinate individual values of position and momentum and (b) a wave function, which is interpreted in various ways, sometimes as a physical wave in three-space, a so-called guiding wave that pushes the particles around (Bohm 1952), other times as something with more of a nomological status, determining like a law how the particles will behave over time (Goldstein and Zanghì 2013). In Bohmian mechanics, one may argue that there are no facts about joint states of the atoms that fail to be determined by the states of the individual atoms. There is a fact about something else, the wave function, that is not determined by the states of the atoms (taken individually or together). Bohmian mechanics interprets entanglement as a feature of states of the wave function. However, one could say the matter ontology of Bohmian mechanics is perfectly separable. Perhaps because Bohmian mechanics has this feature, it is also manifestly nonlocal. B. Nonlocality The issue of locality/nonlocality has frequently been conflated with that of separability/nonseparability in the scientific and philosophical literature. This is not really so surprising given the multiplicity of meanings our language assigns to 'local.' In one such usage, we may think of nonlocality as a matter of a system's features not being determined by features 7 of its "local" (i.e. spatially more localized) parts. However, we already have a name for that feature, 'nonseparability.'2 Unlike separability which concerns (noncausal) metaphysical determination or grounding of the features of systems, in the sense to be discussed here, locality is a causal notion. Lange has defined locality in the following way: Spatiotemporal locality: For any event E, any finite temporal interval τ > 0, and any finite distance δ > 0, there is a complete set of causes of E such that for each event C in this set, there is a location at which it occurs that is separated by a distance no greater than δ from a location at which E occurs, and there is a moment at which C occurs at the former location that is separated by an interval no greater than τ from a moment at which E occurs at the latter location. (2002, p. 15) This account captures the idea that a system is nonlocal if it manifests direct instantaneous action at a spatiotemporal distance. Thus nonlocal systems exhibit superluminal influence in violation of the special theory of relativity. As is standard in the literature then, we can understand locality as a principle that there is no superluminal influence, including instantaneous action across spatial distances. However, it is also worth noting as it will be important below that Lange's more precise and general definition does not assume what spatial background the distances δ appear in. We may ask about the preservation of locality in any spatial framework, even those less familiar than the three-dimensional spatial framework of the manifest image. Just as entangled systems are often thought to instantiate fundamental nonseparability, so they are thought to manifest fundamental nonlocality. This was famously argued to be a 2 Howard (1985) is especially direct that we should avoid this conflation of concepts: "Most importantly, it should be understood that the separability of two systems is not the same thing as the absence of an interaction between them, nor is the presence of an interaction the mark of their non-separability" (p. 173). 8 consequence of quantum entanglement in the Einstein, Podolsky, and Rosen (EPR) paper of 1935, assuming quantum mechanics is able to give a complete description of reality. To make this more vivid, let's again consider the EPRB setup and imagine the atoms' spins are measured by sending them through a Stern-Gerlach apparatus, a simple device involving a pair of magnets that deflects particles in one spatial direction or another based on their spin. Figure 1 Let's imagine atomA is sent through its apparatus slightly before atomB is sent through its. The measurement on atomA will then instantaneously affect whether atomB is deflected up or down by the magnets. And this is so no matter how far apart the two Stern-Gerlach apparatuses are. Nonlocality is widely believed to be a genuine feature of quantum systems. After the EPR paper, as we noted, Bell argued that nonlocality cannot be avoided by granting the 9 ontological incompleteness of quantum mechanics and postulating additional variables, so even Bohmian mechanics is a nonlocal theory. Subsequently, nonlocality appears to have been demonstrated in many experimental settings. The experiments by Alain Aspect in the 1980s are perhaps the most famous (Aspect et. al. 1981). However, some believe it is possible to avoid nonlocality by appealing to the nonseparability of quantum systems or by providing an even more revisionary metaphysical interpretation. 3. Avoiding Nonlocality In a later section we will explore what exactly is supposed to be so problematic about nonlocality. Presently, we will address some ways of avoiding at least fundamental nonlocality in quantum mechanics. The first involves an appeal to the failure of separability and has been defended by Howard and Teller. The second allows for a recovery of separability while simultaneously ensuring locality by way of a move to a higher-dimensional metaphysics. A. Fundamental Nonseparability Howard argues that we can ensure a local metaphysics for quantum mechanics by rejecting the separability of quantum states. He writes: The separability principle operates on a more basic level as, in effect, a principle of individuation for physical systems, a principle whereby we determine whether in a given situation we have only one system or two. If two systems are not separable, then there can be no interaction between them, because they are not really two systems at all. (1985, p. 173) And, in light of the experimental confirmation of Bell's results: 10 We must give up either separability or locality... But if these are our only alternatives, then most of us would likely prefer the former alternative, on the grounds that special relativistic locality constraints are too much a part of our physics to be sacrificed to the cause of saving separability, all the more so because we have ready at hand a highly successful non-separable quantum mechanics... In fact, I believe that Einstein himself would have followed us in this choice had he been forced to choose between these two alternatives. (1985, p. 197) Howard's view is that when one accepts nonseparability, one thereby rejects the numerical distinctness of the individuals in entangled states. In Bohm's set-up, the proposal is we reject the belief that the atoms are distinct entities and instead view them as one. Then we can explain the observed correlations without requiring nonlocality. There is no instantaneous superluminal influence of atomA on atomB because there is no distinction between atomA and atomB. One may ask why we are entitled to conclude just by the assumption that atomA and atomB are numerically identical that there cannot be causal influence between distant wings of the experiment. Surely a state of one object can cause another state of that self-same object. My being thirsty can cause my getting up to get a glass of water. What is more plausible, however, is that causal relations are irreflexive, that one state of an individual cannot cause that very state of the self-same individual. My being thirsty at a certain time t cannot cause my being thirsty at that same time t.3 And so what Howard seems to be assuming is not or not merely that the atoms are numerically identical, but that the relevant states are identical (atomA's being found x-spin up and atomB x-spin down). 3 That causal relations are irreflexive is a standard assumption in the causation and causal modeling literature, e.g. Pearl (2000). 11 Strictly speaking, this rejection of numerical distinctness of states goes beyond what nonseparability requires. We can see this by considering an alternative nonseparable approach advocated by Teller. Teller also argues that nonseparability can help us avoid nonlocality, elaborating a view he calls 'relational holism.' Teller defines this as the view that "collections of objects have physical relations which do not supervene on the non-relational physical properties of the parts" (1986, p. 73). Given skepticism in contemporary metaphysics that supervenience is adequate to representing metaphysical positions like this (e.g. Kim 1984, Fine 2001), it is probably better to construe relational holism as the view that collections of objects have fundamental physical relations which are not grounded in (i.e. metaphysically determined by) the non-relational physical properties of the parts.4 Teller argues that cases of quantum entanglement like the EPRB setup we have been considering provide genuine cases in which collections of objects have what he calls "inherent relations," relations whose instantiation are not determined by intrinsic features of the relata. They are thus genuine examples of nonseparable states. It may not be immediately clear how Teller's view avoids non-locality. According to relational holism, Bohm's atoms are numerically distinct, as are the states into which they enter on the two wings of the experiment. Thus it seems measurement on one atom does instantaneously affect the other (and the relevant expectation values) some distance away. In a later paper (1989), Teller argues how it is supposed to be that relational holism avoids nonlocality. Teller grants that there will be stable correlations between the states of the two atoms in Bohm's set-up, but argues that because of the inherent relations linking them, there is no reason to infer from these correlations to a causal mechanism linking the two wings. Because of the entanglement, the correlation may be brute: 4 Below, I will use the language of 'determination' to be neutral between Teller's supervenience formulation of relational holism and a more contemporary grounding formulation. 12 To say that causal locality has been violated most plausibly should be taken to mean that there are nonrelational properties of space-time points which are related in some other way – by action (lawlike dependencies) at a distance or through superluminal causal chains. On the other hand, when we are concerned with nonsupervening [or ungrounded] relations, this circle of ideas has no grip. There is no question of superluminal or distant action between nonrelational, definite values. (1989, p. 215) and later: The correlation – as an objective property of the pair of objects taken together – is simply a fact about the pair. This fact will arise from and give rise to other facts. But it need not itself be decomposable in terms of or supervenient upon some more basic, nonrelational facts. There need be no mechanism into which the correlation can be analysed. (1989, p. 222) Thus, Teller argues relational holism allows the Bell correlations to be brute, not requiring further explanation as the consequence of a causal relation between the distant events. It is worth noting that today, a more common nonseparable interpretation of quantum mechanics postulating irreducible relations is not framed in terms of Teller's relational holism, but rather ontic structural realism.5 Some versions are very much like Teller's in terms of positing objects bearing primitive relations not explainable in terms of intrinsic features of the relata (Esfeld). Others differ in that they aim to eliminate objects altogether (Ladyman and Ross 2007, French 2014). I will not discuss such approaches here however since advocates of such views do not typically use structural realism in order to avoid nonlocality. 5 Thanks to Michael Esfeld. 13 Returning to Teller, it is not clear to me why he thinks that the pull to explain correlations is removed once we allow there are relations that are not determined by intrinsic features of their relata. The idea seems to be that once one gives up the assumption that all relational features are determined by intrinsic features of their relata (that is, in other words, once one adopts relational holism), one will thereby give up the general assumption that correlations must have explanations. Indeed he states that the adoption of relational holism frees us generally from all common cause reasoning. But even if relational holism makes it reasonable to allow some brute relations, it is not clear why it should remove the general presumption that correlations not be brute. To do so would seem to throw the baby out with the bathwater, giving up one of the most basic assumptions of scientific reasoning. Thus, at least for now, I would argue that prima facie, Howard's nonseparable metaphysics provides a more successful metaphysical motivation for the avoidance of nonlocality. Though, as I have noted, it is involves more than a mere rejection of separability. B. Wave Function Realism A metaphysics for quantum mechanics that was considered and rejected early on by Schrödinger but more recently advocated by David Albert (1996, 2013, 2015) has the virtue of avoiding both nonseparability and nonlocality at least at the fundamental level and perhaps simpliciter. Like Howard's interpretation, this involves the view that what appear to be distinct particles are instead manifestations of one fundamental entity. This for Albert is a single field, which he labels, following the name for the mathematical object used to represent it, the quantum wave function. It is a field in the sense that it is an object whose nature is specified by an assignment of numbers (complex values of amplitude and phase) to each point in the space it inhabits. The 14 view is thus called 'wave function realism.' The key innovation of wave function realism is to allow that this field is not spread out in the three-dimensional space of our ordinary experience, but instead is spread out in the space in which wave functions are typically represented, a higherdimensional state space. So unlike in standard versions of Bohmian mechanics, where we recognize both ordinary three-dimensional matter and a wave function, in this picture, there is only what inhabits the high-dimensional space of the wavefunction.6 The matter is itself constituted by the wave function. The nature of the space the wave function inhabits is based on configuration space representations in classical mechanics. In classical mechanics, we use a configuration space of 3N dimensions to represent the possible three-dimensional locations of a system of N particles. Since classical particles always have definite locations, the locations of an entire system of N particles can be represented by a single particle at one point in configuration space. Since quantum mechanics allows individual particles to have locations that are indefinite, quantum systems will generally be represented as fields smeared out over this 3N-dimensional space. For example, in Bohm's set-up, at the start of the measurement process, the atoms will have indeterminate locations, i.e. it is indefinite whether each atom is deflected up or down by the magnetic field in its respective Stern-Gerlach apparatus. According to wave function realism, the field (the wave function) will possess nonzero amplitude at points in configuration space corresponding to each of these possibilities. For the nonrelativistic case, the ontology of quantum mechanics is a wave function spread over a high-dimensional space with the structure of a classical configuration space evolving according to the Schrödinger dynamics, supplemented 6 The qualification "standard versions of Bohmian mechanics" is needed because Albert (1996) has also proposed a wave function realist interpretation of Bohmian mechanics, where both the matter and the wave function live in the high-dimensional space required to capture the allowable states of the wave function. 15 perhaps with a collapse dynamics, depending on one's favored approach to the measurement problem.7 It should be noted it would be incorrect strictly speaking to call the space the wave function inhabits a configuration space since in this picture it is the wave function that is fundamental, not particle configurations. However, we can capture the features of the wave function and its space in a "top-down" manner by considering what sort of fundamental metaphysics would be capable of recovering the nonfundamental appearances of a system of multiple particles in three-dimensional space, and this is how the configuration space representation is useful. To visualize the wave function realist's proposal, we may consider first an image of how things would appear in a three-dimensional representation, when the locations of the atoms are indeterminate after they pass through their respective Stern-Gerlach devices but before wave function collapse (should there be collapse). The position state of the atoms is represented by the circles.8 Figure 2 7 The total space of the wave function must actually have a more complex structure. One reason is we must include additional dimensions as well corresponding to the degrees of freedom for the spin states of the particles. 8 Note I am setting to one side Bohmian mechanics, which provides an alternative strategy for ensuring separability. 16 Note that in this three-dimensional image, we do not immediately see the correlation between the first atom's being deflected upward (or downward) and the second atom's being deflected downward (or upward). We only see that each individual atom has its state spread over the two possible deflection locations. This contrasts with the image of what is found according to the higher-dimensional interpretation. Here each point in the higher-dimensional space corresponds to a total three-dimensional state of the entire system (states which include the positions of the atoms and the whole of the apparatus). The wave function in the case of the EPRB setup before measurement will be spread over multiple points. In particular, there are clusters of high amplitude at two regions in the higher-dimensional space, one corresponding to situations in which atomA is deflected upwards and atomB is deflected downwards, and the other corresponding to situations which atomA is deflected downwards and atomB upwards. Facts 17 about the entanglement of the system are thus captured directly in the 3N-dimensional interpretation.9 Figure 3 The resulting wave function metaphysics is completely separable. It is separable because all states of the wave function, including the entangled states we have been considering, are completely determined by localized assignments of amplitude and phase to each point in the space of the wave function. This is not so if we want to get facts about entanglement into the three-dimensional interpretation. This requires adopting some form of nonseparability, either by 9 For much more detail on the nature of the wave function according to the wave function realist, see Ney (2013). 18 denying the distinctness of atoms A and B, or as in Teller's picture, adding fundamental facts about the correlations between the atoms.10 Howard cites a late discussion of the EPR thought experiment in which Einstein appears to argue that if one wishes to avoid the incompleteness of quantum mechanics, one's only options are to embrace nonseparability or nonlocality (Einstein 1949, quoted in Howard 1985). The way Howard sees it, one can deny that objects at spatial distances have separate existences, or one must accept superluminal influence. Of course, he chooses to reject separability. What we are seeing here is that by shifting our conception of what the fundamental space is, one can avoid having to make this choice. Wave function realism provides a metaphysical interpretation of quantum mechanics that is compatible with EPR, Bell's theorem, and the laboratory experiments that followed while being both separable and local. The state of the total field is determined by its state at each point and there is no action at a distance. To help visualize this consequence, consider the following diagram which depicts the evolution of the quantum state from a situation in which the location of the atoms (in threespace) are indefinite to one in which atomA is measured. For concreteness, we assume a spontaneous collapse dynamics and the three-dimensional interpretation of quantum systems as grounded in a mass-density field as articulated by Ghirardi and Bassi (2003). Figure 4 10 Barry Loewer (1996) also argues that a move to a higher-dimensional ontology is needed to preserve separability for quantum systems. 19 What happens is that the measurement process (and resulting collapse) on the left-hand-side immediately causes a change in the state of the system on the right-hand-side.11 By contrast, consider an image of the measurement process in the higher-dimensional space of wave function realism. Again, we consider the evolution of the wave function according to spontaneous collapse dynamics. Figure 5 11 In the Ghirardi and Bassi framework, collapses are spontaneous, they are not caused by measurements. In situations describable as measurements, in which we are dealing with the interactive entanglement of a very high number of fundamental particles, the probabilities of such a spontaneous collapse event becomes extremely high. 20 In this case, we again see a change starting from a field that is initially spread out to one that is subsequently less spread out. However what has happened in this case is not that something on the left has caused a change in the part of the field on the right, has caused it to become more localized. Recall that each point in the wave function's space corresponds to a total configuration of a three-dimensional system. So if you want to ask where in this picture is the measurement that took place on the left wing of the experiment, the answer is: it is located all over the configuration space. So it is the state of the wave function at each point at the first time that then causes the state of the wave function to be what it is at each point at the second time. What we 21 have is simply a smooth process in which the whole wave becomes more bunched up over time in one region of its space.12 The evasion of nonlocality is maintained even more clearly on a dynamics for the wave function that does not involve collapse. For example, on an Everettian picture, the evolution of the wave function over time could instead be pictured as: Figure 6 12 Some might balk at the use of causal language here because they believe that causation has to (by definition) always involve the action of small/localized things on other small/localized things (see e.g. Field 2003), and what is happening here instead is the global state of a wave function at all points at a time determining the global state of the wave function at a later time. If so, fine, whether this is described as global causation or only determination, there is still no action at a spatial distance. 22 Again, the global state of the wave function at one time influences the global state of the wave function at all later times. What we have is simply a wave that becomes a bit more spread out over time. Again, there is no action at a spatial distance in the fundamental metaphysics. Now there is a question about whether there is nonseparability or nonlocality simpliciter on this picture, even if fundamentally everything is separable and local. First one must ask, can the wave function realist accept in addition to what is described by Figure 5, the situation depicted in Figure 4 as some kind of derivative reality. This depends on whether it is correct to say that the wave function is capable of grounding the derivative existence of three-dimensional objects. This is a contentious issue we will not get into here.13 However, if wave function realism is compatible with a derivative (but real) three-dimensional world, then there will be nonseparability in that derivative three-dimensional space, since there will be states of systems that are not determined by states of what is happening at the subregions occupied by those systems' constituents.14 (These states will only be determined by states of the wave function.) What about derivative nonlocality? This is a question about what explains the correlations between the spatially distant measurement events. On this view, it is not an interaction between the two wings of the experiment that explains the correlations, but instead the dynamical evolution of the wave function. So although there are certainly the observed correlations in threedimensional space, there fails to be nonlocal influence. One might think there is no obvious reason why one would care to avoid such derivative nonlocality. After all, what appears in the three-dimensional metaphysics as nonlocality is only a nonfundamental manifestation of a more fundamental local process. We will return to this issue in the penultimate section. But I take it to 13 But see Albert (2013, 2015) and Ney (in progress). 14 Note again, I am putting Bohmian mechanics to one side here. 23 be a virtue of the wave function realist interpretation that it may explain the Bell correlations without having to violate relativity by postulating nonlocal, superluminal influence. 4. Objection: Nonseparability Does not Ensure Locality I now want to consider arguments that the nonseparable metaphysics of Howard is not itself sufficient to avoid the nonlocality that appears to arise in the case of EPRB setups. If it is not, then wave function realism would be an even more attractive option for those hoping to avoid nonlocality. In his paper "Nonseparability does not relieve the problem of Bell's Theorem," Joe Henson argues that Howard is mistaken that one can avoid nonlocality by rejecting separability. Howard had argued that it is generally assumed that the systems on the two wings of an EPRB setup are separable, numerically distinct systems. In a formal reconstruction of Bell's argument, one that I will not reproduce here, Henson argues that Bell's argument requires only a weaker principle, that the systems on either wing of the experiment may be localized at distinct spatial regions, a principle he calls Localized Events: All events can be associated to regions of spacetime in a consistent manner. (2013, p. 1012) Since the argument doesn't require the stronger separability principle, one cannot avoid the conclusion (nonlocality) by rejecting it. As Henson puts it: If one wants to rely on one's favourite derivation of Bell's theorem for the purposes of this discussion, one needs to show that the assumptions one makes are equivalent to, or weaker than, what is used (explicitly or implicitly) in the standard versions. After all, if I added the assumption that I live in London to a derivation of Bell's theorem, that would 24 not make it reasonable for a group of angry realists to drive me out of town in the hope of saving locality. (2013, p. 1009) But does Howard's argument rest on this mistake? As we have seen, Howard's main point is that the metaphysical conclusion we should draw from experimental confirmations of the quantum predictions is that the systems A and B are not numerically distinct. "If two systems are not separable, then there can be no interaction between them, because they are not really two systems at all." And so although standard derivations of Bell's theorem may not assume a separability assumption explicitly, in describing the measurement results as distinct events that may be localized at distinct locations, we are thus assuming the negation of what Howard wants us to consider. So Howard could plausibly reject even Henson's weaker principle of Localized Events. But it seems to me that separability is a red herring here. By denying the distinctness of atomA and atomB and indeed the corresponding measurement events, Howard is denying that there even are such two subsystems localized at distinct regions. Rather, there is just one system that isn't (at least straightforwardly) localized to one or the other region. So there is no issue of whether the facts about A and B determine the facts of the system contained at the union of the space-time regions at which they are located. But whether what is at issue is nonseparability or not, there is a question about whether Howard's strategy is successful. Henson argues that his claim of numerical identity is unwarranted. He says: The suggestion ... is that one could say to the worried experimenter "don't worry, when you saw the flashing light, that actually corresponded to an event in the whole experimental region, not an event in your lab. So you see, it's all local... Actually your reaction to the flashing light didn't happen in your head either, but in the larger 25 region."... It is in no way more unreasonable to apply this kind of reasoning to a hypothetical case of superluminal signaling than it is to apply it to outcomes in the EPRB experiment. (2013, p. 1020) Henson's objection is essentially that Howard's rejection of numerical distinctness is ad hoc. I interpret him in this last sentence as pointing out that we could make a similar move in any hypothetical case of superluminal signaling and thus avoid nonlocality. But, he says, "if we rely on this, we may as well have avoided analysis of Bell's theorem by rejecting all locality assumptions except no-signaling in the first place" (2013, p. 1021). I do not agree that the move Howard makes would be "no more unreasonable" to apply in any case of superluminal signaling. And this is because the denial of numerical distinctness in situations of quantum entanglement like the EPRB setup is motivated independently of the desire to avoid nonlocality (or superluminal signaling). It is motivated also by the fact that in a threedimensional metaphysics, it is simply not possible to give a complete account of what measurement results we should expect for atomA without considering the entangled system of which atomA appears to be only a part (and similarly for atomB). The atoms thus do not appear to have distinct realities.15 This supports Howard's interpretative strategy. But ultimately I am sympathetic to Henson's skepticism. Although Howard may avoid what is strictly speaking nonlocal interaction between distinct objects or events, I don't think his strategy to avoid nonlocality is as satisfactory as the wave function realist's. For recall that according to the wave function realist, the entire three-dimensional framework is derivative. Influence only appears to be transmitted instantaneously across a spatial distance, but the source of these correlations is really a local influence in a higher-dimensional space. On the other hand, 15 Arguments to this effect, not in any way relying on a desire to evade superluminal influence are presented in other work. See, for example, Ney (2013, in progress). 26 Howard wants us to view reality as three-dimensional. We may say that what appear to be two distinct measurement events are not fundamentally distinct, but Howard will not deny that the single object and event is indeed spread out somehow in the three-dimensional space. And so although we might not have be a causal interaction between two things, but only a state? or a process? involving only one, what we are committed to is still something that will involve a mysterious coordination across two distant parts of space at a time. 5. Why Prefer a Metaphysics for Quantum Physics that is Separable and Local? It is possible to paint the demand for both separability and locality as the results of an unreasonable demand to make our interpretations of physical theories conform to our intuitions. For it can seem to us only natural that the properties of a whole all be traceable to, determined by properties of its parts, and also that actions do not have immediate effects across spatial distances. But, as Ladyman and Ross (2007) have rightly argued, there is no reason to believe that we would have been hardwired as a result of evolution to be good at reasoning about topics of fundamental physics or metaphysics. Our question here is what a quantum world would be like, and there is no good reason to think our intuitions are good guides to the nature of a world like this. Some would press back on this last point. For example, Valia Allori (2013) defends a view she finds in Einstein, that "the whole of science is nothing more than a refinement of our everyday thinking." In her view, the best physical theorizing departs as minimally as possible from the manifest image of ordinary experience, and only where it has to. However, first, it is not clear what the argument for this claim is. If theory and experiment allow for a radical departure from the manifest image that yet possesses many other theoretical virtues including fertility for 27 the development of further physics, then why would it not be worthwhile to explore what this unfamiliar metaphysics looks like? Second, because the interpretation we have seen that recovers both separability and locality also rejects as fundamental a three-dimensional spatial background, replacing it with an unfamiliar, high-dimensional background, it is not really so plausible to argue that this separable, local metaphysics is closer to the manifest image than one that would jettison one or both of separability and locality, but retain the low-dimensional spatial background of our experience.16 One might argue that it is not merely our intuitions and background assumptions that support separability and locality, but inductive reasoning. One could thus appeal to our past observations in day-to-day life. However, this sort of strategy seems hopeless, since what we are evaluating here are the conclusions we should draw from more refined experiments that do suggest the world is nonlocal or at least nonseparable. A more promising type of argument considers the sort of interpretational assumptions that will allow us to formulate inductively successful empirical theories. Howard considers several passages from Einstein that make this kind of case for separable and local theories. On separability, Einstein proposes: [I]t appears to be essential for this arrangement of the things introduced in physics that, at a specific time, these things claim an existence independent of one another, insofar as these things 'lie in different parts of space'. Without such an assumption of the mutually independent existence (the 'being-thus') of spatially distant things, an assumption which originates in everyday thought, physical thought in the sense familiar to us would not be 16 To be clear, Allori herself doesn't argue for wave function realism. She is just advocating for the methodological principle stated above. 28 possible. Nor does one see how physical laws could be formulated and tested without such a clean separation. (quoted in Howard 1985, pp. 187-188) Howard claims he doesn't know what to make of this passage; how it is to be supported. But he speculates that what makes separability useful in the construction of successful physical theories is that it gives us a sufficient condition for the individuation of physical systems: spatial separation (Howard 1985, p. 192). Without this, it is difficult to imagine what could provide an objective basis for individuating objects.17 But this doesn't seem correct. First, separability does not seem sufficient to allow for the individuation of physical systems by spatial separation. Even if we grant that all states of systems are metaphysically determined by the states of subsystems located at subregions of the region the system occupies, these subsystems (and the systems they constituent) may nonetheless fail to be clearly individuated because they possess gappy or otherwise deviant spatial trajectories. But anyway there is a more natural way to interpret Einstein's concern. As we've seen, there are at least two ways to develop a nonseparable metaphysics for quantum mechanics. The weaker version is Teller's which would have us say that entangled systems may involve distinct entities, but that these entities instantiate inherent, i.e. irreducible relations. What this entails is that if we want to have a complete description of one of these objects that will allow us to know how it will behave over time in its environment, one must bring in facts about the other entity with which it is entangled. The facts about it and how it will behave in its environment necessarily bring in facts about the other object, which may be a significant distance away. In the absence of full knowledge of entanglement relations, this makes 17 Since Howard himself rejects separability, he cannot use spatial separation then as a principle for individuating physical systems. He proposes (1985, p. 198) we instead use facts about the nonexistence of quantum correlations to individuate physical systems. 29 it challenging to predict what something in a given spatially localized set of circumstances is going to do, how it will behave. In the stronger version, Howard's, we don't know how the object will behave, for we don't even have the full object in front of us. What can be said for locality? There does not seem to be anything conceptually incoherent in the idea of a nonlocal metaphysics. However, here again we may bring in considerations about prediction and control. Indeed this is how Einstein appears to defend locality: For the relative independence of spatially distant things (A and B), this idea is characteristic: an external influence on A has no immediate effect on B; this is known as the 'principle of local action', which is applied consistently only in field theory. The complete suspension of this basic principle would make impossible the idea of the existence of (quasi-)closed systems and, thereby, the establishment of empirically testable laws in the sense familiar to us. (quoted in Howard 1985, p. 188) Although clearly distinct from the first concern about developing successful theories without a separable metaphysics, the motivation for a local metaphysics is similar. If what is nearby and observable may be affected by objects that are spatially distant, then without full knowledge of the occupants of the total space-time manifold, how are we to make predictions about how the objects we observe will behave? Locality is required to allow us to formulate testable empirical theories. As an additional empirical point, nonlocality implies a violation of special relativity. So to support locality, we may appeal to all of the considerations supporting special relativity. Although the wave function realist interpretation supports nonlocal correlations in the derivative three-dimensional space of objects, it does not support nonlocality, since these correlations are 30 not explained in terms of superluminal influence. This by itself does not mean it is compatible with special relativity, which ultimately depends on whether the theory wave function realism is an interpretation of is Lorentz covariant. Versions of quantum theory with collapse of the wave function will still include frame-dependent facts about probabilities that don't crop up for, e.g., Everettian quantum mechanics. This is an issue that runs orthogonal to the metaphysical question of wave function realism. Nonetheless, the issue of frame-dependent facts about causal influence is a metaphysical issue, and it is a virtue of wave function realism that it can avoid such facts. In addition to the empirical considerations that speak in favor of interpretations of quantum mechanics that are separable and local, there are also pure metaphysical considerations. Returning to separability, the nonseparable metaphysics we have seen seem both of them in a respect to be conceptually unstable. According to Teller's relational holism, the fundamental facts about any object A in an entangled state cannot be grasped without considering what is a distinct object B. But this suggests that these objects are not actually distinct, since one cannot really understand one apart from the other. It thus raises the question of why one should think there really are two things, not one. Thus, I would argue that relational holism isn't just problematic for the construction of empirically testable physical theories, the position threatens to collapse into some more traditional form of holism, like Howard's. Now Teller rejects a form of holism according to which the entangled objects are numerically identical, because he finds such views obscure: Holism has always seemed incoherent, for it seems to say that two distinct things can somehow be entangled or intermeshed so that they are not two distinct things after all. Yet apparent unintelligibility does not prevent holism from recurring, not only in the 31 work of philosophers of East and West, but also in what quantum mechanics seems to many of us to be saying about the world. (1986, p. 73) Relational holism is the solution that allows us an intelligible form of holism. But why is it unintelligible to say that what appear to be two things are really one? One problem may be that we know there are two of something. The data we have for the Bell correlations are numerically distinct. We observe this state here and at one time, and we observe a distinct state there at some other time.18 So how can we consistently maintain that what we know to be two states are in fact one? One answer would be to interpret holism as a fundamentality thesis: a thesis about what fundamentally exists, not what exists simpliciter. Jonathan Schaffer in his work has helpfully articulated a distinction between existence monism and priority monism (Schaffer 2010). In the former, the claim is that there is only one entity full stop. Priority monism by contrast asserts that fundamentally there is only one entity. The multiple entities (particles, states, etc.) we encounter in our experience are nonfundamental or derivative entities that are grounded in what is fundamentally just one thing. In recent work with Jenann Ismael (Ismael and Schaffer forthcoming), they argue for this priority monism as a reasonable interpretation of quantum mechanics. Howard himself clearly has in mind a more traditional kind of holism (the form that says there is just one thing simpliciter).19 However, I want to suggest that either form of monism is unstable for it is obscure on these proposals how objects (in the case of priority monism, how the fundamental objects) get assigned locations. We have seen Howard must reject Henson's Principle of Localized Events. Neither atom gets assigned to one location or the other, but rather 18 This is similar to a principle Henson takes to be essential to the derivation of Bell's theorem. He call it the Operational Consistency of Localization (2013, p. 1016). 19 Schaffer distinguishes this, what he calls 'existence monism,' from priority monism. 32 to the total region. But then it appears that there is nothing assigned to either subregion. And this would seem to indicate the region is empty. Yet it is not. I would suggest that a view that embraces holism while capturing a coherent account of locations would only be the wave function realist framework. We say there is one thing, but it is located in the higher-dimensional space of the wave function. Now unfortunately while the wave function realist proposal does provide a coherent, conceptually stable metaphysics, it cannot reap the consequences of all of the good arguments for separability and locality considered above. Since our observations represent objects in the low dimensional space, to have a successful physical theory, we will want to be able to make reliable predictions about what will happen when we observe or manipulate objects localized to threeor four-dimensional regions. But to the extent that wave function realism allows for the existence of a three-dimensional metaphysics, it will not be separable, and although it will strictly speaking be local, it will possess nonlocal correlations. The full separability and locality reside in the higher-dimensional picture which is unfortunately not the picture we use when we do experiments and manipulate objects. The case for a separable and local metaphysics for quantum mechanics then comes from more broadly philosophical considerations, special relativity, perhaps brute intuition, and additionally considerations of what provides a more coherent and stable picture. 6. Conclusion In this paper, we have seen how the higher-dimensional, wave function realist interpretation of quantum theories provides a metaphysics that is fundamentally both separable and local. If one favors separability and locality for the reasons described, then one will thereby have reason to 33 prefer wave function realism and its attendant higher dimensions over rival interpretations of quantum theories such as standard Bohmian mechanics (which may be separable but is certainly not local) or the various holist approaches (which may be local – though I am skeptical – but are certainly not separable). It is worth emphasizing however that although the fundamental metaphysics offered by the wave function realist is both separable and local, the more pragmatic, inductive arguments favoring separability and locality are unable to provide support for this position. This is because it is only the fundamental metaphysics on this picture that is separable and local. 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Holism, Separability, and the Metaphysical Implications of the Bell Experiments. Philosophical Consequences of Quantum Theory: Reflections on Bell's Theorem. J.T. Cushing and E. McMullin, eds. Notre Dame: University of Notre Dame Press, 224-253. Ismael, Jenann and Jonathan Schaffer. forthcoming. Quantum Holism: Nonseparability as Common Ground. Synthese. Kim, Jaegwon. 1984. Concepts of Supervenience. Philosophy and Phenomenological Research. 45: 153-176. 35 Ladyman, James and Don Ross. 2007. Every Thing Must Go. Oxford: Oxford University Press. Lange, Marc. 2002. An Introduction to the Philosophy of Physics. Oxford: Blackwell. Loewer, Barry. 1996. Humean Supervenience. Philosophical Topics. 24: 101-27. Ney, Alyssa. 2013. Introduction. The Wave Function: Essays in the Metaphysics of Quantum Mechanics. Oxford: Oxford University Press. Ney, Alyssa. In progress. Finding the World in the Wave Function: Some Strategies for Solving the Macro-object Problem. Synthese. Pearl, Judea. 2000. Causality. Cambridge: Cambridge University Press. Quian, Xiao-Feng and J.H. Eberly. 2013. Entanglement is Sometimes Enough. URL=<http://arxiv.org/pdf/1307.3772.pdf> Schaffer, Jonathan. 2010. Monism: The Priority of the Whole. Philosophical Review. 119(1): 3176. Schrödinger, Erwin. 1935. The Present State of Quantum Mechanics. Quantum Theory and Measurement. J. Wheeler and W. Zurek, eds. Princeton: Princeton University Press. Teller, Paul. 1986. Relational Holism and Quantum Mechanics. British Journal for the Philosophy of Science. 37(1): 71-81. Teller, Paul. 1989. Relativity, Relational Holism, and the Bell Inequalities. Philosophical Consequences of Quantum Theory: Reflections on Bell's Theorem. J.T. Cushing and E. McMullin, eds. Notre Dame: University of Notre Dame Press, 208-223.

Philosophy Pathways – Issue 230 – 28th February 2019 https://philosophypathways.com/newsletter/ 1 TORTURE WITH CONSENT by Terence Rajivan Edward Abstract. There are attempts to define torture which say that a person is only being tortured if the pain inflicted upon them is pain that they have not consented to. I recommend that we define torture without this condition. In an article on torture for the Stanford Encyclopedia of Philosophy, Seumas Miller considers three different definitions of torture (2017: §1). He challenges the first two definitions but not the third. All of these definitions include the condition that a person is only being tortured if the pain inflicted upon them is pain that they have not consented to. If a person consents, then it is not torture. This non-consent condition is never questioned. But the condition is very much open to doubt. One of the texts which Miller refers to itself disputes the condition (Twining 1978: 158-159), but in ways that Miller suggests he would disregard. I present an example below that is different to the examples offered there, but is related to a theory about how legal punishments can be justified (Nino 1983: 298). Let us imagine that someone migrates to a country and is asked to consent to certain rules being implemented while they are there. One of the rules they consent to is that if they are in possession of information which is needed to protect the security of the country and are asked by the government to communicate this information, they will either voluntarily communicate it or else they will be subject to extremely painful procedures which aim to force communication. Let us further imagine that at some point in time, this person is asked to give such information. They have the information, but they refuse to give it. They are tortured in order to extract the information. But did they not consent to this torture, given what they agreed to when entering the country? Philosophy Pathways – Issue 230 – 28th February 2019 https://philosophypathways.com/newsletter/ 2 It may be proposed that they consented earlier, but they do not consent at the time when they are tortured. But can one withdraw one's earlier consent in this way? Even if we grant that the earlier consent can later be withdrawn, what about the person who says, "I know what I agreed to, and I accept that I will be tortured now"? When we consider this person, it does not seem that torture must be without consent. They have not withdrawn their earlier consent. It makes sense to say that the person was still tortured, tortured with their consent. Furthermore, we do not want to discount this case when discussing the morality of torture or whether torture should ever be legal. (For example, if we think that torture should not be used to obtain information, because it is not reliable for this purpose, the conclusion applies to this case as well.) Therefore it is mistaken to include a non-consent condition in the definition of torture. I do not think anyone will disagree with this argument, but even if someone does, the non-consent condition is simply too questionable to form part of a definition of torture in legal and moral contexts. The United Nations definition does not include this condition, fortunately. Objections have been made to that definition (Davis 2005: 163). Even if these objections are correct, unless the dubious nature of the nonconsent condition is exposed, there is a danger that it will end up as part of the replacement. I shall end this paper with a note. Saying that a person has consented does not mean that it is morally acceptable to torture them. I think you can consent to others doing things to you which are not morally acceptable things to do, even after consent. Philosophy Pathways – Issue 230 – 28th February 2019 https://philosophypathways.com/newsletter/ 3 References Davis, M. 2005. The Moral Justifiability of Torture and other Cruel, Inhuman, or Degrading Treatment. International Journal of Applied Philosophy 19: 161-178. Miller, S. 2017. Torture. In E.N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Summer 2017 edition). Accessed on 9th February 2018 from: https://plato.stanford.edu/archives/sum2017/entries/torture/ Nino, C.S. 1983. A Consensual Theory of Punishment. Philosophy & Public Affairs 12: 289-306. Twining, W. and Paskins, B. 1978. Torture and Philosophy. Proceedings of the Aristotelian Society, Supplementary Volume 52: 143-194. Terence Rajivan Edward University of Manchester © Terence Rajivan Edward 2019 Email: [email protected]

sehepunkte 19 (2019), Nr. 4 Marion Lauschke / Franz Engel / Johanna Schiffler (Hgg.): Ikonische Formprozesse Sowohl die ikonologische als auch die formanalytische Methode, wie sie heute sehr prominent von Horst Bredekamp und Gottfried Boehm vertreten werden, waren zuletzt heftiger Kritik ausgesetzt, je eine animistische Theorie zu vertreten, die von der "Macht der Bilder" oder dem "Leben der Bilder" ausgehe. [1] Gerade diese Kritik, so zeigt sich nun, führt nach annähernd 100 Jahren Grabenkampf zwischen der auf historischen Voraussetzungen aufbauenden und der ahistorisch argumentierenden Methode zu einer Annäherung. Sie wurde durch eine von Bredekamp ausgerichtete Tagung 2015 in Berlin initiiert, deren Ergebnisse nun schriftlich vorliegen. Nach deren Lektüre ist es bemerkenswert, dass mit den 10 Beiträgen einerseits sehr wohl an der Eigenständigkeit der Bilder und damit deren Wirkungspotential sui generis festgehalten wird, zugleich jedoch andererseits auf deren Abhängigkeit von der Rezeptionsleistung der Betrachterin verwiesen wird. Der Schlüssel dazu wird in dem emotional geprägten Auslegungsprozess durch die Betrachterin gesehen, der sich weniger an den Bildmotiven entzündet als an der formalen Bildstruktur. Im Titel des von Marion Lauschke, Johanna Schiffler und Franz Engel herausgegebenen Bands Ikonische Formprozesse. Zur Philosophie des Unbestimmten in Bildern spiegelt sich dieses Doppelte wider. Über den Einbezug der Wahrnehmung als grundlegend für die Auslegung von Bildern schliessen die Beiträgerinnen entsprechend an die seit Längerem interdisziplinär ausgerichtete, lebhafte Diskussion vor allem innerhalb der Philosophie zum Enaktivismus und Embodiment bzw. zu Verkörperungstheorien an. Vor diesem Hintergrund stellt sich die Frage, gehen die vorgelegten Forschungen damit tatsächlich neue Wege, oder deuten sie am Ende die virulente lebendige Erfahrungsqualität mit Bildern erneut wie in der Tradition der formalen Ästhetik "nur" als eine rein ästhetische, die in einer Erkenntnis mündet; oder vermögen sie darin auch handlungsrelevante und insofern semiotische Aspekte zu erkennen? Wobei gerade der Anspruch, die lebendige Erfahrung in Zusammenhang mit Geschichte und Kultur zu bringen, wie er in der Einleitung formuliert wird (VII-XVII, hier X-XI), diese neue vielversprechende Perspektive eröffnet. Offensichtlich wird diese Neuausrichtung in dem Beitrag des Psychologen und Kognitionswissenschaftlers Wolfgang Prinz (101-121). Ihm zufolge lassen sich Bilder als "Handlungen und Intentionen fremder Autoren" verstehen, deren latentes ästhetisches (technisch, künstlerisch, gesellschaftlich, historisch), politisches und mediales (als Werkzeuge oder gar Waffen) Potential in Akten der Rezeption aktiviert bzw. aktualisiert wird (113-121, hier 116). Für Bredekamp selbst (123-141) äussert sich diese Handlungsrelevanz, wie er am Beispiel spätmittelalterlicher Skulpturen deutlich macht, in der Reaktion aller Beteiligten auf die Bilder. Ihre Prägnanz und Überfülle vermitteln, dass "sie nicht allein von dieser Welt" sind (131) und entsprechend am Vorabend der Reformation durch Hus Anlass für den Bildersturm wurden. Gottfried Boehm (63-84) sieht dagegen in Bildern keine Handlungsrelevanz, sondern betont den fühlbaren Erkenntniswert von Bildern. Grundlage dafür sei, dass alles was in Bildern erscheint, im Licht stehe und insofern Voraussetzung für Differenzund Kontrastphänomene sei. Durch diesen "Zustrom von Energien" seien Bilder "von Grund auf emotional gesättigt", wie er an einem Beispiel von Rubens deutlich macht (81). [2] Diesem Ansatz folgt Kerstin Thomas (85-100) im Anschluss an Meyer Schapiro nach, indem sie davon ausgeht, dass das "künstlerische Ausdrucksgeschehen" den Betrachterinnen Emotionen erfahrbar macht (86). In dieser Tradition steht auch Claudia Blümle (143-161), die an John Dewey anschliesst, dessen Ansatz auf der Vorstellung eines universellen Rhythmus der Natur gründe, von der sich der Mensch entfremdet habe. Die rhythmisch erlebbaren Ordnungen der Kunst in der ästhetischen Erfahrung der Form ermöglichen eine Wiedervereinigung mit der Natur. Auf diese Weise werde, wie sie mit Henri Maldiney betont "im rhythmischen Werden der Formen eine signifikante Gegenwart" sichtbar und für uns erfahrbar (153-161, hier 154). Denken, so Eva Schürmann (29-43), hängt von Wahrnehmen ab und letzteres kann nicht objektiv sein, da es körperlich gebunden und insofern von der zeitlichen und räumlichen Potentialität, wie sich die Dinge uns zeigen, abhänge. Wobei Originalveröffentlichung in: Sehepunkte : Rezensionsjournal für die Geschichtswissenschaften 19 (2019), Nr. 4 [15.04.2019] URL: http://www.sehepunkte.de/2019/04/32565.html [zuletzt besucht 08.05.2019] Schürmann in Anlehnung an Maurice Merleau-Ponty und am Beispiel von Kentridge und Turrell damit nicht ausdrücklich auf ein mögliches Handlungspotential verweist, wie sie es an anderer Stelle bereits einführte. [3] Eine eindeutige Anbindung von Bildern an Kultur und Geschichte auch im Sinne einer Handlungsrelevanz eröffnet dagegen Marion Lauschke (45-62, hier 57-62). Grundlage dafür sei die Lenkungskraft der ikonischen Formprozesse, wie sie im Anschluss an Dewey und am Beispiel von Klee zeigt (Stichwort Affordanzen / Enaktivismus, hier 53-60). [4] Grundüberlegungen, wie sie den Enaktivismus ausmachen, bestimmen auch den Ansatz von Henri Bergson, wie ihn Viola Nordsieck vorstellt (163-184). Demnach lassen sich die Ordnungen, wie sich Welt gibt, als Rhythmen (Prozesse) verstehen. Für unser Handeln ist das zentral, indem wir dem Rhythmus auch in den Künsten (166-169, 179-180) nicht nur nachspüren, sondern beginnen ihn "zu übernehmen, zu beherrschen, zu variieren" (179-181, hier 180). Ähnlich argumentierte bereits Alfred N. Whitehead, dem zufolge, wie es Oswald Schwemmer (185-195) herausstellt, Wirklichkeit nur als Reaktion (Handeln) auf Geschehendes zu verstehen sei, die erfasst werde und bereits auf elementarster Ebene stattfinde. Grundlage dafür sei die "Struktur des Erlebens" (188-195). Rückblickend auf die Beiträge zeigt sich, dass die Beurteilung ikonischer Formprozesse, die in enger Anbindung an körperlich erlebte Wahrnehmungsprozesse gesehen werden, nicht einhellig erfolgt. Und bemerkenswerter Weise spiegelt sich diese Ambivalenz bereits in dem an den Anfang gesetzten Impulsaufsatz des engen Wegbegleiters von Bredekamp John M. Krois wider. Denn in seinem 2005 auf Englisch verfassten Aufsatz bzw. Teilstück eines Buchprojekts mit dem Arbeitstitel "Philosophy and Iconology" (1-27) stellt er das Erleben / Fühlen zunächst in einen engen Zusammenhang mit Handlungen. Gerade in der Begegnung mit Bildern gewinne dieser Zusammenhang an Bedeutung, wie er im Anschluss an Whitehead betont, da sie verführen können ("elicit(s) dedication", 17). Wobei er weiterführend diese Verbindung jedoch nicht schlüssig bei dem für seine Forschungen wichtigen Philosophen Ernst Cassirer deutlich machen kann, der vermittelt über Erwin Panofsky mit der Etablierung der ikonologischen Methode für die Kunstgeschichte so wichtig ist. Das ist umso bedauerlicher, da Cassirer mit der "perception of expression" ("Ausdrucks-Wahrnehmung") vergleichbar mit Whitehead, wie ich mit meinen eigenen Forschungen aufzeige, indirekt ebenfalls handlungsrelevante Aspekte aufzeigt. D.h. statt die Ausdrucks-Wahrnehmung mit Cassirer als "Ursprungswahrnehmungsform" einzuführen, die vor jeder mythischen, ästhetischen oder logischen Auffassung erfolgt, bringt Krois hier die "perception of expression" mit der "symbolischen Prägnanz" bzw. konkret mit ästhetischen Urteilen in Verbindung, gemäss denen eine Linie weniger als lebendig bewegender Impuls, sondern als "schön" beurteilt wird (1-15, hier 4-8). [5] Doch ein solches Urteil hängt grundlegend vom jeweiligen Zeitgeist ab und damit von historisch-kulturell normierten Vorstellungen. Sie hat insofern keine Handlungsrelevanz. [6] Die Bindung der Erfahrung mit Bildern an die Form, wie es wegweisend von der formalen Ästhetik nahe gelegt wurde und nun von der für Geschichte und Gegenwart so wichtigen ikonologischen Forschung ausdrücklich aufgegriffen wird, eröffnet trotz erkennbarer Ambivalenzen in den Beiträgen neue vielversprechende Perspektiven für das Verständnis von Bildern in unserer Lebenswelt. Anmerkungen: [1] Vgl. hierzu Lambert Wiesing 2013, in: Sehen lassen. Die Praxis des Zeigens, Frankfurt a. M. 2013, 78-105, hier 88. [2] Vgl. ergänzend zur urspr. Konzeption der Theorie des "unbestimmten Grundes" Gottfried Boehm: Augenmass. Zur Genese der ikonischen Evidenz, in: Movens Bild. Zwischen Evidenz und Affekte, hg. v. Gottfried Boehm et al., München 2008, 15-38, hier 36. Gemäss dieser Theorie ist alles, was sich uns zeigt, als "Akte der Orientierung" auf eine Welt zu verstehen, in der wir uns schon immer bewegen ("primordiale" Welthabe) (21). [3] Vgl. hierzu Schürmanns Aussagen zu Bildern, die das Potential haben, mit den von Einzelnen und Interessengemeinschaften verfolgten Zielen zu brechen bzw. sie umzustürzen, in: dies.: Erscheinung als Ereignis. Zeittheoretische Überlegungen zur Fotografie, in: Erscheinung und Ereignis. Zur Zeitlichkeit des Bildes, hg. v. Emmanuel Alloa, München 2013, 17. [4] Affordanzen als Handlungsanweisungen zu untersuchen, steht vor allem in den archäologischen Bildwissenschaften seit Längerem im Fokus. Vgl. hierzu zuletzt die Tagung v. Johanna Fabricius und Elisabeth Günther im Rahmen des Excellenz-Clusters Topoi an der FU Berlin v. 2.-4. November 2018 (https://www.topoi.org/event/46051/). [5] Vgl. hierzu Martina Sauer: Ästhetik und Pragmatismus. Zur funktionalen Relevanz einer nicht-diskursiven Formauffassung bei Cassirer, Langer und Krois, in: IMAGE. Zeitschrift für interdisziplinäre Bildwissenschaft 20 (2014), http://www.gib.uni-tuebingen.de/own/journal/upload/16e8879566e23e0d4ca3b02dce1003c2.pdf, vgl. ferner zur wissenschaftshistorischen Aufarbeitung: http://archiv.ub.uniheidelberg.de/artdok/6246/1/Sauer_Aesthetik_versus_Kunstgeschichte_2018.pdf. [6] Vgl. zu dieser Verbindung des Urteils "schön" mit sozio-evolutionär geprägten Prozessen, die grundsätzlich nichtfunktional sind und damit nicht zu einem Handeln veranlassen, Sabine A. Döring: Ästhetischer Wert und emotionale Erfahrung, in: Kunst und Philosophie, Ästhetischer Wert und Design, hgg. v. Julian Nida-Rümelin und Jakob Steinbrenner, Ostfildern 2010, 53-73, insb. 60-73. Rezension über: Marion Lauschke / Franz Engel / Johanna Schiffler (Hgg.): Ikonische Formprozesse. Zur Philosophie des Unbestimmten in Bildern (= Image Word Action / Bild Wort Aktion / Imago Sermo Actio; 3), Berlin: de Gruyter 2018, XVII + 198 S., 75 Abb., ISBN 978-3-11-053103-9, EUR 49,95 Rezension von: Martina Sauer Bühl Empfohlene Zitierweise: Martina Sauer: Rezension von: Marion Lauschke / Franz Engel / Johanna Schiffler (Hgg.): Ikonische Formprozesse. Zur Philosophie des Unbestimmten in Bildern, Berlin: de Gruyter 2018, in: sehepunkte 19 (2019), Nr. 4 [15.04.2019], URL: http://www.sehepunkte.de/2019/04/32565.html Bitte geben Sie beim Zitieren dieser Rezension die exakte URL und das Datum Ihres letzten Besuchs dieser Online-Adresse an.

This is the accepted manuscript of an article published in Australasian Philosophical Review, Volume 1, Issue 4, 2017. Please use the published version here for citing this article. Is Future-oriented Mental Time Travel Inextricably Linked to the Self? Elena Popa Abstract Ganeri's discussion of mental time travel and the self focuses on remembering the past, but has less to say with respect to the status of future-oriented mental time travel. This paper aims to disambiguate the relation between prospection and the self from the framework of Ganeri's interpretation of three Buddhist views by Buddhaghosa, Vasubandhu, and Dignaga. Is the scope of Ganeri's discussion confined to the past, or is there a stronger assumption that future thought always entails self-representation? I argue that if mental time travel towards the past and towards the future are continuous, both past and future thought should be possible independently of selfrepresentation. An assumption of discontinuity however would enable the employment of the self as one of the defining differences between remembering the past and imagining the future. The two options can be further contrasted on the basis of distinct ways of constructing past/future scenarios (field vs. observer perspective), modes of experiencing time (known vs. lived), and the origin of mental time travel (episodic vs. semantic memory). I further assess the compatibility of future-oriented thought with the three Buddhist views on the basis of these coordinates. 1. Introduction Does mental time travel require a sense of self? Or, to use Tulving's (1985) metaphor, can there be mental time travel without a traveller? In 'Mental Time Travel and Attention', Ganeri addresses these questions from the perspective of three Buddhist views on memory and mental time travel. Owing to its negation of the self-implication condition, and its independence from representationalism, Ganeri favours Buddhaghosa's view. While Ganeri's arguments hold if mental time travel is thought of predominantly on the model of remembering the past, one may wonder where the future comes in within this picture. The range of the self-implication condition can be explored through a set of distinctions involving future thought: the debate between continuism and discontinuism, field and observer perspective, known time and lived time, and semantic versus episodic memory accounts of mental time travel. This paper aims to disambiguate the status of future-oriented time travel in relation to the self from the offset of Ganeri's investigation. The 1 motivation for this inquiry lies in the role of prospection in current psychological research on memory and mental time travel. Remembering the past as the paradigmatic case of mental time travel transpires from Ganeri's definition of mental time travel, alongside his reconstructions of two of the Buddhist views (Buddhaghosa's and Vasubandhu's). From the onset, describing Tulving's challenge, Ganeri (forthcoming) defines mental time travel from the perspective of the past: 'mental time travel is the reliving or re-experiencing of an experience one has had before, a relocating of oneself in subjective time to the past (or, equally, to the future)' (p. 3). Likewise, the reconstruction of Buddhaghosa's view focuses on the past, with no mention whether the future would by default require the self, or whether this perspective may be extended to the future: 'Buddhaghosa's contribution to this discussion is to argue that consciousness of one's past can be grounded in a type of autonoesis that does not require self-representation' (p. 16, my emphasis). Moreover, in Ganeri's description of Vasubandhu's approach, future-oriented thought only appears in conjunction with selfrepresentation: 'for Vasubandhu, however, both autobiographical episodic memory and futureoriented thought are forms of delusion...' (p. 24). While not overtly stating it, Ganeri appears to either confine his talk about time without self to the past, or, on a stronger claim, to assume that the self-implication condition always holds for future-oriented mental time travel. In what follows I show that both interpretations above – extending Ganeri's argument for the past to the future, or arguing that unlike the past, future thought requires a sense of self can find support in present psychology. 2. The importance of mental time travel towards the future Regarding the role of memory and mental time travel, one hypothesis from psychology focuses on the pursuit of future goals, rather than on remembering past events. For instance, Suddendorf and Corballis claim that 'the crux of mental time travel lies in its role in enhancing biological fitness in the future, so that mental time travel into the past is subsidiary to our ability to imagine future scenarios' (2007: 302). Klein brings forth a proposal along the same lines: 'it is possible that memory enabled humans, over the course of evolutionary history, to be aware of the future before we were able to consciously experience the past' (2013: 64). It should be noted that these claims are made from a naturalistic perspective, where the preservation of the self holds a central role. My focus, however, rests on a number of philosophical issues related to the function of future-oriented mental time travel, as well as its relation to remembering the past. One issue is the debate over continuity versus discontinuity between future and past-oriented mental time travel. Perrin and Michaelian (2017) describe discontinuism and continuism as follows: 2 'for the latter, there is a difference in kind between what we do when we remember the past and what we do when we imagine the future; for the former, there is only a difference of degree' (p. 229). For the purposes here, adherence to one of these views may bring about different consequences with respect to the relation between mental time travel towards the past and the future. Continuism would require an explanation of imagining the future consistent with the main characteristics of remembering the past. By contrast, discontinuism would leave open the possibility of completely distinct processes involved in thinking about the past and the future, and the two can, at least in principle, be investigated separately.1 The consequences of these two stances, along with the connection between semantic memory and imagining the future will be brought together with self-implication in the next section. Another issue, analyzed by Klein and Steindam (2016), is the relation between mental time travel and the subjective experience of temporarily. The authors distinguish between 'lived time' and 'known time'. As the authors put it, 'in the former case, subjective temporality is directly given as part of one's occurrent mental state, whereas in the latter, subjective temporality is the product of inferential or interpretive acts' (p. 142). For my purposes here, the key difference is that unlike 'known time', 'lived time' requires a sense of self. In continuation of Tulving's (1985) work, the authors extend 'known time' to noetic consciousness and future scenarios on the basis of semantic memory: 'by allowing that noetic consciousness can promote a form of temporal subjectivity based on conceptual analysis (i.e., "known time"), the construct can be modified to accommodate the type of subjective temporality associated with semantic-based FMTT [future mental time travel]' (p. 143). The future scenarios based on semantic memory go against the view that future-oriented mental time travel originates exclusively in episodic memory, and is illustrated by Klein by reference to patients with memory impairments who can imagine a public future, but lack the sense of ownership of the scenario (see D.B.'s case, in Klein 2013). A final distinction is between observer perspective and field perspective (Nigro and Nisser 1983). The field perspective, despite its association with the first person perspective, appears to correspond largely to memories where one remembers a view, but not one having experienced the view: '... the scene appears from one's own position; one seems to have roughly the field of view that was available in the original situation and one does not "see oneself" (Nigro and Nisser 1983: 467-468). This is contrasted with memories where one sees one's self, from a third person view, having the same experience. Research by McDermott et al. (2016) shows that these two perspectives also apply to anticipating the future. The work by Nigro and Nisser, McDermott et al., 1 On the classification by Perrin and Michaelian, extreme discontinuism holds that the two can be investigated separately. 3 and D'Argembeau and Van der Linden converges on the prevalence of the observer perspective for events going further back or forward in time. This distinction is important for the paper here for two reasons. Firstly, the field perspective may be compatible with scenarios akin to the one by Buddhagosa's, where a memory is reenacted as if one were reexperiencing it without seeing one's self as part of the memory. With the addition of insights by McDermott et al., this may provide a lead to extending Ganeri's take on Buddhagosa's view to the future. Namely, if remembering a past experience is a kind of simulation (Ganeri 2017: 414), the same paradigm could apply to anticipating an experience constituted of familiar states. Secondly, if this distinction applies to both future and past, it adds to the evidence for continuism: people can experience both the future and the past from both perspectives. 3. Future-oriented mental time travel and the self The distinctions above help disambiguate the relation between future-oriented mental time travel and the self-implication condition. Notably, the psychological evidence favouring continuity would imply that the possibility of remembering one's past without remembering one's experiencing it may equally apply to imagining the future. The distinctions between lived time and known time, as well as the observer and the field perspective would flesh out the continuist picture. By contrast, discontinuity would enable a separate treatment of remembering the past, imagining the future, and their connection to the self-implication condition. If there is continuity between past and future-oriented mental time travel, and it is possible to remember the past without a sense of ownership of the experience, then the same should hold for anticipating the future. The question is how to describe a future scenario where there is no ownership from the subject's part. There are two answers, based on the distinctions by Nigro and Nisser on the one hand, and Klein and Steindam, on the other hand. Firstly, the field perspective can be mapped onto future scenarios without including one's sense of self. Thus, it would be possible, for instance, for one to anticipate seeing a sunset from the field perspective, without conceptualizing one's self as watching the sunset, on the basis of previous acquaintance with the surroundings, having watched sunsets from different angles etc. As sketched above, connecting the field perspective to the future may provide the starting point for an extension of Ganeri's considerations on Buddhaghosa and recollecting the past. Ganeri's (2017) approach to mental time travel towards the past as simulation may as well work for future scenarios involving previously experienced components (as opposed to, say, a scenario based on recollection). Secondly, future scenarios independent from the sense of self can fall under 'known time'. As shown by Klein (2013), imagining future scenarios on the basis of semantic memory is possible under impairments of 4 episodic memory. One example is the patient D.B., who, despite remembering past events, had trouble claiming ownership of his memories. Nevertheless, he could anticipate a public future. The possibility of constructing both past and future scenarios on the basis of semantic memory is consistent with continuism. The difference between this way of imagining the future and the field perspective is that 'known time' is inferential. That is, one can reconstruct the past or project the future by putting together information which does not necessarily involve one's self. This picture is more sophisticated than the reconstruction of Buddhaghosa's view, as it requires representation, and may work as an interpretation of reflexivity under Dignaga's view. Finally, it should be noted that continuism is not completely incompatible with the self-implication condition. If the self is defined such that it accommodates differences in degree, but not in kind, between remembering the past and anticipating the future, then the self-implication condition need not stand or fall along continuity between past and future-directed mental time travel. If mental time travel towards the future and towards the past are discontinuous, then the selfimplication requirement may contribute to the difference in kind between the two. Under the assumption of discontinuity, it could be the sense of self as such, or a set of capacities that may apply to the future but not to the past constituting the self, that distinguish past from future-oriented mental time travel. In relation to the 'lived time'-'known time' distinction, a version of discontinuity reliant on self-implication would deny the possibility of future scenarios based solely on known time, and thus, even when they originate in semantic memory, the self may still be present. This appears to hold in Klein's (2013) interpretation of the situation of another patient, R.B., who can imagine a personal future on the basis of semantic memory. The self-implication condition for future-oriented mental time travel under the discontinuity assumption can be further supported by the naturalistic perspective sketched above. Namely, if the purpose of memory and mental time travel is the preservation of one's self, the sense of self may not be essential for remembering the past, but it is always present upon imagining the future. This can be strengthened by the observer perspective holding for distant future, where scenarios are more often accompanied by a vision of one's self in third person perspective. While this appears to be inferential, as in the case of known time, the representation of the time traveller is necessary. As with continuism above, discontinuism does not necessarily rule out the possibility of future-oriented mental time travel without the self. Nevertheless, since under discontinuism remembering the past and imagining the future may be underwritten by different capacities, an account of future-oriented mental time travel independent from the self-implication condition would need to employ structures different from those that hold for past-oriented time travel. 5 4. Conclusions Simply put, if Ganeri's argument mainly applies to past-oriented mental time travel, the current investigation could widen its scope, to include the future. However, if Ganeri assumes futureoriented mental time travel to entail the self, then this stance can draw either from a commitment to a version of discontinuism focusing on the self, or from a definition of the self consistent with the evidence for continuity (i.e., as a matter of degree) applying to the future, but not to the past. If future and past-oriented mental time travel are continuous, the self-implication requirement can hold for the future and not the past only if the concept of self relies on differences in degree between the two. If the past and future modes are discontinuous, the self, as a requirement for imagining the future, may be one of the features distinguishing them. If the discontinuity amounts to different capacities, then future scenarios independent from a sense of self are possible, but they should be accounted for independently of what holds for remembering the past. Another consequence is that admitting of future scenarios involving 'known time' only would enable futureoriented mental time travel without the self. Thus, tying future scenarios to the sense of self, would also imply that projecting, simulating, and imagining involve 'lived time'. This appears to be in line with Klein's and Steindam's interpretation of Tulving's original considerations. Finally, the field and observer perspectives can support both interpretations – the field perspective for future scenarios may be an instance of imagining the future without imagining one's self experiencing it, while the observer perspective for distant future events can be interpreted through the naturalistic claim that mental time travel would ultimately serve purposes linked to the self. Regarding the three Buddhist approaches, Ganeri's interpretation of Buddhaghosa's view applies to past-oriented mental time travel, and could be extended to the future under a view coalescing the field perspective and future thought. Vasubandhu's considerations on the mind would gain more support from a view explaining the connection between the future and the self (through 'lived time', or the naturalistic focus on the future). Dignaga's concept of reflexivity could support personal and impersonal interpretations in accordance with the stance of whether past or future representations are necessarily tied to self-representation. In the case of continuity, known time and semantic memory may provide impersonal future scenarios that nevertheless involve representations. In the case of discontinuity, even with the semantic memory in place, and under impairments of episodic memory there may still be a sense of self (as Klein interprets R.B.'s case). 6 References D'Argembeau, A., Van der Linden, M., 2006. Individual differences in the phenomenology of mental time travel: The effect of vivid visual imagery and emotion regulation strategies. Consciousness and cognition, 15(2). 342-350. Ganeri, J., forthcoming. Mental Time Travel and Attention, Australian Philosophical Review 1/2: Ganeri, J., 2017. Classical Indian Philosophy. in S. Bernecker & K. Michaelian, The Routledge Handbook of Philosophy of Memory, 408-415. Klein, S.B., Steindam, C.H.L.O.E., 2016. The role of subjective temporality in future-oriented mental time travel. Seeing the Future: Theoretical Perspectives on Future-Oriented Mental Time Travel. 135-153. Klein S.B., Cosmides L., Tooby J., Chance S., 2002. Decisions and the evolution of memory: multiple systems, multiple functions. Psychol Rev, 109. 306–329. Klein, S.B., 2013. The Complex Act of Projecting Oneself into the Future. WIREs Cognitive Sciences 4. 63-79. McDermott, K.B., Wooldridge, C.L., Rice, H.J., Berg, J.J. and Szpunar, K.K., 2016. Visual perspective in remembering and episodic future thought. The Quarterly Journal of Experimental Psychology, 69(2). 243-253. Nigro, G., Neisser, U., 1983. Point of view in personal memories. Cognitive Psychology, 15. 467– 482. Perrin, D., Michealian, K., 2017. Memory as Mental Time Travel. in S. Bernecker & K. Michaelian, The Routledge Handbook of Philosophy of Memory. 228-241. Suddendorf T., Corballis M.C., 2007. The evolution of foresight: what is mental time travel, and is it unique to humans?. Behavioral and Brain Science, 30. 299–313. Tulving, E., 1985. Memory and Consciousness, Canadian Psychology/Psychologie Canadienne, 26/1. 1–12.

on July 30, 2018http://rstb.royalsocietypublishing.org/Downloaded from rstb.royalsocietypublishing.orgOpinion piece Cite this article: Ward EJ. 2018 Downgraded phenomenology: how conscious overflow lost its richness. Phil. Trans. R. Soc. B 373: 20170355. http://dx.doi.org/10.1098/rstb.2017.0355 Accepted: 21 May 2018 One contribution of 17 to a theme issue 'Perceptual consciousness and cognitive access'. Subject Areas: cognition Keywords: inattentional blindness, change blindness, visual awareness, phenomenology Author for correspondence: Emily J. Ward e-mail: [email protected]& 2018 The Author(s) Published by the Royal Society. All rights reserved.Downgraded phenomenology: how conscious overflow lost its richness Emily J. Ward Department of Psychology, University of Wisconsin-Madison, Madison, WI 53706, USA EJW, 0000-0002-2789-2753 Our in-the-moment experience of the world can feel vivid and rich, even when we cannot describe our experience due to limitations of attention, memory or other cognitive processes. But the nature of visual awareness is quite sparse, as suggested by the phenomena of failures of awareness, such as change blindness and inattentional blindness. I will argue that once failures of memory or failures of comparison are ruled out as explanations for these phenomena, they present strong evidence against rich awareness. To accommodate and explain these massive failures of awareness, any theory of phenomenal consciousness must downgrade phenomenology to a degree where it is functionless or, ironically, does not reflect what we experience. This article is part of the theme issue 'Perceptual consciousness and cognitive access'.1. Introduction We seem to experience a rich visual world. As we go through our day, we encounter all types of colours, objects and events. This sense of experiencing things right in front of our eyes has inspired-and continues to inspire-many aspects of perception research. Vision scientists have a special affinity for phenomenologically convincing demonstrations of visual phenomena. When a new phenomenon 'works as a demo', it effectively and intuitively reveals an aspect of how the mind works [1]. This functionality of phenomenology carries a lot of weight when building mechanistic and theoretical accounts of perceptual processing. In short, vision scientists take phenomenology seriously. Even though we spend much of our life in a series of in-the-moment experiences (when we are not remembering, planning or sleeping), it is surprisingly difficult to assess the contents of what Block [2] has called phenomenal consciousness: 'what it's like to be in [a] state'. (p. 227). Do we experience a rich world that we simply cannot describe due to limitations of attention, memory or other cognitive processes, as proposed by Block [2,3] and others (e.g. [4–6])? Or is the nature of awareness quite sparse, as suggested by demonstrations of failures of awareness, such as change blindness [7,8] and inattentional blindness [9,10]? These questions are typically explored in the domain of visual awareness, but they are relevant questions for other domains of conscious experience as well. In the domain of olfaction, for example, there does not seem to be a distinction between what we experience and what we can access [11]. But in visual perception, whether awareness is rich and 'overflows' our ability to report about it or whether it is constrained by cognitive limitations has been debated at length [12–15]. I will argue that to accommodate and explain inattentional blindness and change blindness, theories of rich awareness downgrade phenomenology to a degree where it is functionless or, ironically, does not reflect what we experience. I will specifically argue that: (1) distinguishing inattentional blindness and change blindness is important because the two phenomena provide different evidence against rich awareness; visual display rich awareness RSVP iconic memory change blindness inattentional blindness aware of every item, but fail to compare aware of colour diversity but not individual items aware of current item, but it is not encoded into memory aware of every item, but fail to encode into memory post-cue brings items into awareness aware of unexpected event, but fail to report unaware of unexpected event sparse awareness aware of current item, but it is not encoded into memory Figure 1. Overview of experimental paradigms. The leftmost column represents the visual displays that are presented to participants. The middle column represents what a participant would see and respond if they had rich visual awareness of the display. The rightmost column represents what a participant would see and respond if they had sparse awareness of the display. rstb.royalsocietypublishing.org Phil.Trans.R.Soc.B 373:20170355 2 on July 30, 2018http://rstb.royalsocietypublishing.org/Downloaded from (2) failures to encode specific details into memory occur in many different paradigms, and such memory failures are a reasonable alternative explanation for inattentional blindness; (3) one can rule out memory failures by instructing people to immediately report what they see when they look at a visual display; (4) immediate report instruction in repeated inattentional blindness experiments demonstrates that inattentional blindness is a perceptual deficit; (5) to accommodate and explain repeated inattentional blindness and massive change blindness, theories of rich awareness downgrade phenomenology. 2. Distinguishing change blindness and inattentional blindness The claim that in-the-moment experiences have rich phenomenology is challenged by two types of evidence: change blindness and inattentional blindness. Although often discussed together, these two types of 'blindness' need to be distinguished. Change blindness is the failure to notice changes to a visual scene, even if those changes happen right before one's eyes (e.g. [8]; see figure 1 for an example from [16]). Change blindness has been demonstrated in dozens of different ways. Many examples of change blindness include avisual interruption: from the simplest demonstrations, in which an image will flash on and off with some detail changing between the two images (e.g. [17]), to more complex demonstrations, such as short movies that use careful camera work or editing to obscure a mid-scene change [18], or real-life demonstrations where an experimenter swaps places with another experimenter when a large object (e.g. a plywood board or door) temporarily blocks the subject's view [19]. The failure of participants to notice these changes is all the more surprising because built into many of the demonstrations (especially those with images that flash on and off ) is the task instruction to pay attention to and detect changes to the scene (e.g. [17]). Therefore, in many cases of change blindness, the inability to report the change is not limited by a general lack of attention. Do these failures to notice changes mean that awareness is not in fact as rich and detailed as people seem to experience? Not necessarily. First, many of the changes people fail to report are small and irrelevant to the meaning of the scene. Failing to notice some tree branches disappearing or the colour of a person's shirt changing generally has no consequences for further cognitive processing. Thus, missing small and irrelevant details is not convincing evidence against rich awareness. Second, a failure to report a change could be caused not by a failure to represent the scene richly enough, but a failure to compare the representations before and after the change1 [21]. Third, a failure to report a change could be caused by a failure to encode the rstb.royalsocietypublishing.org Phil.Trans.R.Soc.B 373:20170355 3 on July 30, 2018http://rstb.royalsocietypublishing.org/Downloaded from perceptual states into memory. Even if viewers could make the before/after comparison, they would not be able to report the change if the perceptual representations did not make it into a durable form of memory in the first place. For these reasons, failing to report a change in a change blindness demonstration does not necessarily indicate a lack of rich phenomenal experience. Inattentional blindness paradigms overcome some of the limitations of change blindness paradigms. Inattentional blindness occurs when people fail to notice an otherwise salient event when their attention is occupied [9,22,23] (see figure 1 for an example from [24]). One of the most famous demonstrations of sustained inattentional blindness [10], in which a man in a gorilla suit goes unnoticed by observers performing an attentionally demanding task, is one of the most widely recognized demonstrations in psychology, presumably because it violates people's intuitions of what they should be able to notice given the apparent richness of phenomenal consciousness. Compared to change blindness paradigms, the unexpected events in sustained inattentional blindness paradigms are usually very salient, such as a novel item appearing on screen that is a new colour or shape [23]. The events are readily visible when a participants' attention is directed toward these events, but-allegedly-become invisible when attention is directed toward another task, such as counting how many times a distractor shape bounced off the edge of the display. Because the unexpected events stay in view for several seconds, there is not an obvious need to compare the event to a pre-event representation (such as in the case of change blindness paradigms), so it seems unlikely that inattentional blindness would be due to a failure of comparison. However, because participants are only asked about their experience of the unexpected event after the fact, it has been a long-standing possibility that inattentional blindness could be due to a limitation of memory rather than a failure of visual awareness. Until recently, this seemed like an insurmountable problem with using inattentional blindness to challenge rich phenomenology.3. The problem of failures of memory If inattentional blindness can be explained as a failure of memory, then demonstrations of it, like demonstrations of change blindness, do not pose a challenge to the view that we have rich visual awareness. Therefore, it is important to be clear about what it means for something to be a failure of memory and about what approaches can be taken to rule out this possibility. A failure of memory in this case is not the same as forgetting where you parked your car or put your keys; it is subtler than that. Memory failures of this type are best illustrated by two well-known paradigms in cognitive science: the partial report paradigm used in studies of iconic memory [25] and the rapid serial visual presentation (RSVP) paradigm [26,27]. Partial report was originally used to demonstrate iconic memory [25]. In these studies, participants viewed grids of 9 or 12 letters that were presented for 50 ms (see figure 1 for a schematic). First, participants were instructed to report all the letters that had been presented. In this whole report condition, participants were only able to report about four letters. Next, participants were instructed to report only theletters that had been presented in one of the rows. In this partial report condition, they were able to report about 75% of the letters from a row of three or four letters. This indicated that the letters available to them from the whole grid was about nine (75%  12 letters), since they could report 75% of any row. Critically, participants maintained this level of performance even when the row was cued after the entire display had disappeared. So although participants did not know beforehand which row they would be asked to report, they nonetheless were able to report any subset of the letters that were post-cued. This result was taken to show the existence of a high-capacity, but fragile iconic memory store in which all the letters of the display are encoded, but that fades rapidly and cannot be fully accessed or reported. These findings provided strong empirical inspiration for distinguishing between phenomenal versus access consciousness. Although participants were unable to report much of what they saw when their reports were unconstrained (i.e. whole report), the data appear to show that they had a richer, more detailed representation of the display-if only briefly. But this rich, detailed representation was not encoded durably into memory. These experiments [25] also provided subjective inspiration for distinguishing between phenomenal versus access consciousness: in the original paper by Sperling [25], it is reported that participants felt that 'they have seen more than they can remember [or] report afterwards'. (p. 1). This statement is important because it initially established the phenomenology of iconic memory (one could imagine a case where the same results were obtained but where participants were not so sure about what they saw). However, there is perhaps too much emphasis on this one statement, because beyond it, Sperling [25] did not directly assess participants' phenomenology. In addition, it is not clear whether the statement reflects participants' initial impression of the display, or their impression after their substantial experience with the display (five participants took part in seven experiments spread across 12 sessions that were scheduled three times weekly). More recent research using a modified iconic memory paradigm has shown that feeling that you saw more than you can report does not guarantee that the in-the-moment phenomenology was of high fidelity: for example, people mistakenly perceive letter-like symbols as real letters when presented alongside normal letters [28]. Nonetheless, Sperling [25] demonstrates that an inability to report one's experience due to fragile memory encoding does not mean the experience was sparse. This can be demonstrated in another way by viewing an RSVP stream. In an RSVP paradigm, visual items are presented rapidly (usually approx. 100 ms) to the observer, one right after another (see figure 1 for a schematic). As a result, if you were to view an RSVP stream of letters, it is unlikely you would be able to report all the letters you saw in order, and perhaps you would not even be able to report any specific letter you saw in the stream. But, in the moment, your impression of the letters would be that they seemed clearly visible- though fleeting-and that you were unable to report the letters only because you were asked after the stream had been presented. In both the iconic memory and RSVP paradigms, participants' experience is queried after the display has disappeared and they cannot accurately report what they saw. This pattern of results is also what is obtained with the inattentional rstb.royalsociet 4 on July 30, 2018http://rstb.royalsocietypublishing.org/Downloaded from blindness paradigm: in most demonstrations of inattentional blindness, participants are asked about their experience after the unexpected event had come and gone. This leaves open the possibility that participants saw the unexpected event, but failed to encode it durably into memory, in much the same way as in iconic memory and RSVP paradigms. ypublishing.org Phil.Trans.R.Soc.B 373:201703554. Ruling out failures of memory Determining whether participants saw and forgot a visual display or failed to see the display in the first place can be difficult. In the case of iconic memory and RSVP, special report instructions are given to the participants to allow them to access (at least part of) their experience. The most straightforward way to distinguish between a failure to encode into memory and a failure of perception is to have participants immediately report what they see, when they see it. For example, if a participant in an RSVP experiment is told to 'press the space bar as soon as you see the letter M', they will press the space bar if they see the M and fail to press the space bar if they do not see the M. This immediate report instruction thus provides an accurate report about inthe-moment experience. Participants can accurately detect targets even at exceedingly brief presentations (possibly as fast as 13 ms, e.g. [29] but certainly as fast as 53 ms, e.g. [30]). The task does not rely on memory because a response is given based on what the participant does or does not perceive when the target is present. Therefore, immediate report instructions can identify failures of perception separate from failures of memory. However, using an immediate report task in an inattentional blindness experiment presents a problem: if participants are instructed to immediately report when they see something unexpected, they then have an expectation for the unexpected event! So while they can give an accurate report of what they saw or failed to see, their attention to the unexpected event will attenuate or eliminate inattentional blindness. Because of this dilemma, determining whether inattentional blindness was truly a perceptual deficit or simply just a failure to encode into memory had been thought to be unsolvable in principle, e.g. that 'there are serious problems with any experimental effort to directly ask subjects if something is consciously perceived without attention', and that this 'proves to be impossible because the demand to report on [an unexpected event] directs attention to [it]' [31, p. 73]. Although there has been scepticism of the inattentional amnesia account of inattentional blindness [32], it nonetheless remained a possibility, and thus did not constitute convincing evidence against rich visual awareness for the reasons described previously.5. Repeated inattentional blindness Using a new technique, my colleagues and I have found a way to escape this dilemma and have shown that inattentional blindness truly is a deficit of perception [24]. The usual account of inattentional blindness is that it is due to a lack of any expectations about the unexpected events. But what if instead of having no expectations, participants formed a specific expectation about what type of unexpected event was to occur? If this were the case, participants could be given the instruction to immediately report seeing anything unexpected, but if the unexpected event did not match their specific expectation, theywould still experience inattentional blindness paradigm (if it were in fact a perceptual deficit). Using a sustained inattentional blindness (e.g. [23,33]), we showed participants a display containing the letters L and T which could be either black or white and which moved randomly across the screen. Participants counted how many times the white Ls crossed the midline of the display. There were four trials of this sort, but on the fifth trial, an unexpected object-a dark red cross-slowly traversed the midline (bottom row, figure 1). Immediately after this trial, participants were asked if they noticed anything about the last trial, and then asked whether they noticed that a dark red cross had appeared on screen. We found that a substantial portion of the participants did not report seeing the unexpected event, demonstrating the basic inattentional blindness effect. Instead of ending the experiment there, we then gave participants one more instruction: to keep an eye out for anything else unexpected and to press the spacebar as soon as they see something unexpected. As described previously, this immediate report instruction would permit the participants to accurately report their in-the-moment experience without relying on their memory for the experience at all. The participants then completed several more trials: several trials in which nothing unexpected happened, but also several trials in which the same unexpected event (red cross) appeared. By repeating the unexpected event in this manner, we built up participants' expectation about what type of event could appear. On the final critical trial, a novel unexpected event (blue letter E moving in the opposite direction) appeared for half of the participants, while yet another occurrence of the same unexpected event as before (i.e. red cross) appeared for the other half of participants. We found that more participants missed the novel unexpected event compared to the repeated unexpected event. This demonstrated repeated inattentional blindness in the same participants in the same session. But critically, even when participants were willing and able to provide immediate report of the earlier unexpected events, they still missed the novel unexpected event. Their failure to give immediate report in this experiment thus indicates that they truly did not consciously perceive the event, rather than failing to encode it into memory. With these results, we concluded that inattentional blindness genuinely reflects a deficit in perception rather than memory, presenting a strong challenge to the thesis of rich visual awareness.6. Downgrading phenomenology To maintain that participants in inattentional blindness experiments have any in-the-moment experience of the unexpected event, proponents of rich awareness must concede that this representation cannot be used. For example, participants cannot use it to provide immediate report of any of the specifics of the unexpected event (such as colour or shape); cannot use it to pick out the encountered item from a lineup (indicating that the unexpected event does not even serve as a perceptual prime); and they cannot provide immediate report about anything at all, even as the experience occurs in front of their eyes for several seconds. Given these results, what then is their conscious experience of the unexpected event? Even if participants were able to indicate that 'something was different' about trials in which the rstb.royalsocietypublishing.org Phil.Trans.R.Soc.B 373:20170355 5 on July 30, 2018http://rstb.royalsocietypublishing.org/Downloaded from novel unexpected event occurred (there is no evidence that they feel this way), there is nothing functional and nothing rich about that phenomenology. To accommodate our results showing inattentional blindness is a perceptual deficit, the 'richness' of phenomenology must be severely downgraded. If phenomenal consciousness is 'what it's like to be in [a] state' [2] (p. 227), it does not seem like it is like anything to participants when they encounter an unexpected event when their attention is otherwise engaged. Another example of how phenomenology must be downgraded to accommodate new empirical evidence can be demonstrated by showing that people miss a large magnitude of details or changes to scenes. As discussed previously, there are several limitations to change blindness that make it problematic as evidence against rich awareness. However, it should not be dismissed entirely, especially in cases where the number or magnitude of changes is substantial. In another recent study, my colleagues and I combined change blindness and iconic memory to test whether individuals required conscious perception of all parts of a complex visual display in order to report a summary statistic about the display [16]. Our study was based on one by Bronfman and colleagues [4], which probed the content of in-the-moment experience by testing whether people could report the colour diversity of an array of coloured letters, even if their experience of the individual letters fades too quickly to access. Other examples of summary statistics in perception, such as statistics like size [34,35] and location [36], can be reported without any awareness of the individual elements that make up the statistic [37,38]. Likewise, we hypothesized that people would be able to report the colour diversity of the display, but their ability to do so would not require awareness of any of the individual letters' colours. Using a modified iconic memory paradigm, participants in this experiment were presented with a brief array of coloured letters (see figure 1 for a schematic). The colours could either be drawn from a narrow part of a colour wheel (low diversity: e.g. purples and pinks) or from the whole colour wheel (high diversity). Participants were cued to a specific row of letters and had to report the identity of one of the letters after the display had disappeared. Thus, participants attended to a specific row, but did not know beforehand which letter they would have to report (similar to [25]). Participants also had a secondary task of reporting the colour diversity of either the attended row or nonattended rows. Replicating Bronfman et al. [4], we found that participants could report the identity of the target letter, and could report the colour diversity of both the attended row and the nonattended rows. Are people able to report colour diversity because they are consciously encoding each of the letters' colours, even when the letters were unattended? Or are people able to do this because they perceive just the diversity summary statistic and not the individual elements? Other studies have demonstrated that participants mistakenly perceive letter-like symbols as real letters when presented alongside normal letters [28], suggesting that people are not consciously encoding each element with high detail. In our study, we theorized that if participants were aware of every element in the display, they should notice when at least one of the elements changes during the course of the experiment. To test this, we incorporated achange blindness component to the task [4]: during half of the trials, all of the letters' colours in the unattended rows were shuffled mid-trial (18 colour changes). Across two experiments, none of the 12 participants in each noticed any colour change during 192 change trials. This totalled 3456 missed changes. Had any participant noticed any one of these changes, our experiment could have been used to support rich awareness. But with these results, we concluded that people can be aware of and report ensemble properties, like colour diversity, without being aware of individual elements. To be clear, this experiment cannot rule out rich phenomenology in an absolute sense. It could be the case that participants saw all the colours in rich detail, but failed to encode the items into memory to compare them to their initial colour before the change. But the failure to report any of the 3456 changes highlights how functionless and impoverished in-the-moment phenomenology must be. In this experiment, proponents of rich awareness must concede that if people do experience the colour change, the representation of this experience is not being used in any way: the changes do not influence colour diversity or letter recall performance; participants cannot report any specific colour change; and they cannot report that any change happened at all, despite participating in the experiment for nearly an hour.7. Conclusion Overall, to accommodate and explain repeated inattentional blindness and massive change blindness, any theory of phenomenal consciousness must downgrade phenomenology to a degree where it is functionless or, ironically, does not reflect what we experience. If it is necessary to explain these failures by appealing to failures of comparison or memory, then proponents of rich awareness may find themselves in an uncomfortable position when these failures occur in real life. Based on the results discussed above, a driver may hit a child who has run into the street because the driver's attention was otherwise occupied and he failed to perceive the child. The alternative is that the driver saw the child in rich detail, but failed to compare his representation of the child to a previous version, or failed to encode the child properly into memory, and hit her with his car anyway. If this alternative is true, then phenomenology does not seem like one worth advocating for. Phenomenology should be functionally useful, even if what we are consciously aware of is sparse. Fortunately, there are new aspects of perception that we are learning more about as a consequence of this debate. In particular, we are learning that our perceptual system is capable of very sophisticated statistical perception, especially in the absence of awareness [16,39]. Statistical perception may help reconcile cognitive and physiological limitations with our subjective impression of a rich detailed world [39]. For example, we are only seeing rich detail and colour in central vision to begin with, and our perception of scenes arises through stitched-together fixations [40]. By better understanding the representations that result from statistical perception, we may better understand why we have a holistic experience of our visual environment. There is also much we do not understand about how cues affect visual awareness, rstb.royalsocietypubl 6 on July 30, 2018http://rstb.royalsocietypublishing.org/Downloaded from especially how cueing a stimulus after it has disappeared may bring it into awareness for the first time [41]. Exploring visual awareness through these avenues may help us understand why we feel like we experience a rich visual world when we in fact do not. Data accessibility. This article has no additional data. Competing interests. I declare I have no competing interests.Funding. I received no funding for this study.Endnote 1Although it is unclear why change-which is arguably the most prioritized feature of visual perception [20]-would not be included in phenomenal consciousness.ishing.orgReferencesPhil.Trans.R.Soc.B 373:201703551. Carbon C-C. 2014 Understanding human perception by human-made illusions. Front. Hum. Neurosci. 1, 8. 2. Block N. 1995 On a confusion about a function of consciousness. Brain Behav. 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Available online at http://journal.uny.ac.id/index.php/jrpm Jurnal Riset Pendidikan Matematika 4 (1), 2017, 120-127 Copyright © 2017, Jurnal Riset Pendidikan Matematika ISSN 2356-2684 (print), ISSN 2477-1503 (online) A Case Study of Misconceptions Students in the Learning of Mathematics; The Concept Limit Function in High School Widodo Winarso 1 *, T. Toheri 2 1, 2 Department of Mathematics Education Faculty of Education and Teaching IAIN Shyekh Nurjati Cirebon. Jalan Perjuangan Bypass, Sunyaragi, Kesambi, Kota Cirebon, Jawa Barat 45132, Indonesia * Corresponding Author. Email: [email protected], Telp: (0231) 481264 Received: 14 December 2016; Revised: 5 May 2017; Accepted: 8 May 2017 Abstract This study aims to find out how high the level and trends of student misconceptions experienced by high school students in Indonesia. The subject of research that is a class XI student of Natural Science (IPA) SMA Negeri 1 Anjatan with the subject matter limit function. Forms of research used in this study is a qualitative research, with a strategy that is descriptive qualitative research. The data analysis focused on the results of the students' answers on the test essay subject matter limit function with the number of students by 16 (sixteen). Data collection techniques used are shaped test methods essay, interview method to students who have misconceptions, and methods of documentation of the test answers. Examination of the validity of the data using a triangulation technique that compares the data written test results with data from interviews. The results of this study can be described as follows; (1) The level of misconceptions experienced by students belonging to the category of low, amounting to 12.18%. However, students who do not understand the concept quite high at 40.38%, and the others are students who understand the concept that is equal to 47.44%. (2) The misconception most experienced students lie in subconcepts explain the meaning of limit function at one point through the calculation of values around that point, that is equal to 20.51%. The tendency misconceptions experienced by students is located on errors and operating concepts that misconceptions students that there should be no limit in the completion of the writing of the emblem and misconceptions about the limit students to conclude if the limit value of 0 is no limit to the value of the function. Keywords: misconception, limit function, mathematics learning How to Cite: Winarso, W., & Toheri, T. (2017). A case study of misconceptions students in the learning of mathematics; The concept limit function in high school. Jurnal Riset Pendidikan Matematika, 4(1), 120-127. doi:http://dx.doi.org/10.21831/jrpm.v4i1.12060 Permalink/DOI: http://dx.doi.org/10.21831/jrpm.v4i1.12060 __________________________________________________________________________________________ INTRODUCTION One of the important objectives of mathematics in this first year of higher education students who learn mathematics to understand and analyze the mathematical concepts and problems resolutions procedures (RittleJohnson, Siegler, & Alibali, 2001). The concept is one of the basic mathematical studies of objects and matter. In mathematics, the concept is presented using the definition, understanding of the important role definition holds in mastering mathematics completely. Through understanding the students to understand the concept of the material being taught. Understanding math also primary purpose of any material submitted by the teacher to Achieve desired concept (Isrotun, 2013). But in fact, many student teachers consider an empty vessel to be filled and not be aware that students are human beings who have the potential to think and prejudice, as a result of the student mix prejudice occurred with a new concept. Students may be able to apply new concepts to problems with cognitive levels are low, but when it is a matter of understanding the cognitive levels already or even applications, and a higher level of the other, then Pre-concept would be devastating and can lead to students experiencing misconceptions (Inayah, 2003). Jurnal Riset Pendidikan Matematika, 4 (1), 2017 121 Widodo Winarso, T. Toheri Copyright © 2017, Jurnal Riset Pendidikan Matematika ISSN 2356-2684 (print), ISSN 2477-1503 (online) The misconception stems from the student (preconceptions) which is already one will be continued and sustained. The success of each level of education influenced the success of students to master the competencies in previous levels. A good understanding will serve as a good base or foundation for the next level (Earl, 2004). Some information states that the low mastery of concepts and misconceptions on students affect the low value of Minimum Criteria exhaustiveness (KKM) on concepts and fields of study. Similarly, the concept of limit function, the concept of limit function is new knowledge for students. The concept of limit function is an abstract concept and provides only a symbol lim ", so it can not be seen directly how the shape and purpose of the concept of limit function. Formal definitions limit function is taught in calculus lecture activities commonly known as the ɛ definition (read: epsilon) and δ (pronounced: delta) (Bahar et al, 2012). One of the teachers in the high schoolIndonesia stated that for students in the school, the math is still considered difficult subjects when national exams. He said this is because there are some materials that are less well understood by learners. One less mastered the subject matter is the limit function Therefore, the solution of student difficulties in understanding the concept of the limit function must be found so as not to impact on the understanding of matter further. Based on this background, researchers interested in knowing how high the level of misconceptions experienced by students and student misconceptions tendency in the material limit function. Literature Review Direct objects in mathematics are facts, skills, concepts, and rules (Jordaan, 2009). The concept a concept in mathematics is arranged in a hierarchical, structured, logical and systematic starts from the concept simple to complex concepts. Learning mathematics is like a chain of mutually sustainable and make it into a whole chain. The interrelated concepts in mathematics even simple concept has a role as a prerequisite for the concept toward understanding the most complex concept (Lestari, Triyono, & Joharman, 2012). According to MulyonoAbdurahman stated that the concept refers to a basic understanding (Jordaan, 2009). Students develop a concept when they were able to classify or categorize objects-objects or when they can associate a name with a certain group of objects, for example, between the concept of a triangle and the non-triangular. Various studies have been conducted to identify the understanding of concepts with reference to the criteria established. Abraham et. al. has developed criteria for classifying understanding of concepts such as in Table 1 (Jordaan, 2009). Misconceptions based grouping is done by Abraham et.al on one level of understanding of the concept of the show has not fulfilled all the components concept mastery (Jordaan, 2009). Therefore, the analysis from misconceptions that occur in students can be done through an analysis of the concept components that have not been mastered by students. The misconception is a concept that is incompatible with the concept of which is recognized by experts (Suprapto, 2013). If a student is experiencing an error when receiving an understanding of learning concepts first, will have an impact not only at the time that students learn the concept. But it would also result in further learning is the development of the concept. According to Soedjadi mathematical misconceptions can occur from several sources (Farida, 2016), (1) the meaning of the word, such as misconceptions about the term "high", (2) the practical aspects, such as the value then assume the same importance 2x5 and 5x2, (3) simplification, for example, understanding lineup that does not connect with the function or mapping, (4) singularity structure of mathematics, for example, there is a presumption in mathematics should be no contradiction without seeing a review of different systems, (5) images, for example by drawing the set of natural numbers as a subset of the set of integers that concluded integer more than the original number. Based on the description and understanding of the above misconceptions in mathematics is defined as the use of the mathematical concepts that are inconsistent with the scientific understanding or definition accepted by scientists. While the percentage of misconceptions level can be grouped into several categories as shown in the Table 2. Jurnal Riset Pendidikan Matematika, 4 (1), 2017 122 Widodo Winarso, T. Toheri Copyright © 2017, Jurnal Riset Pendidikan Matematika ISSN 2356-2684 (print), ISSN 2477-1503 (online) Table 1. Grouping Degrees Concept Training Criteria The Degree of Understanding Category  No answer / empty, replied "I do not know" No response Nounderstand  Repeating the statement, but the answer is not related to any questions or are unclear Do not understand  Answered with an explanation illogical Misconceptions Misconceptions  Answer showed no concept of controlled but no statement in reply which showed misconceptions Understandpartly withmisconceptions  Answer showed only partially mastered concepts without any misconceptions Understandpartially Understand  Answer demonstrated the concept understood by all true explanation Understanding the concept Table 2 Category Misconception (Suwarna, 2013) Percentage (%) Category 0 – 30 Low 31 – 60 Moderate 61 – 100 High Common mistakes done by children in doing the math, that is the lack of knowledge about the symbol, the lack of understanding of the value of the place, the use of the process wrong, miscalculations, and writing that can not be read so that the learner made a mistake because no longer able to read his own writing. The condition is caused by several factors. The factors that cause errors in the math homework covers the causes of fault location, cause this type of error factors, factors causing this type of error concept, factors causing this type of errorowned operations, factors causing this type of error principle (Lerner, 2003). Misconceptions about the subject matter in some way limit function, namely misconceptions regarding the existence of the limit function and relation to limit the left and right limit, limit function misconceptions various forms, and misconceptions about the limit theorem (Jordaan, 2009). Examples of misconceptions made by learners is when students were asked to give their perceptions of symbols lim ⟶c ( ) = 3, by giving them the question _ whether a function should be defined at that point to have a limit on the time and what is the relationship between the value of the function at that time with the concept of limit, then from some learners will respond that it should be defined function in c and definitely value function is equal to 3 _ contrary with the definition of limit (Larson & Edwards, 2013). METHODS The target in this study is a class XI student of Natural Sciences (IPA) SMA Negeri 1 Anjatan in Indonesia who have misconceptions based on the analysis of the test results on the subject of mathematics learning limit function. Selection of this class is based on several considerations. The consideration was partly because a class XI student of Natural Science (IPA) SMA Negeri 1 Anjatan in Indonesia experienced enough problems as it is in the study of less mastering concepts particularly well in mathematics (limit function). The design used in this research is descriptive qualitative research that describes an event in the present. This descriptive qualitative study aimed to describe, summarize a variety of conditions, different situations or phenomena that exist in the community that the object of research (Bowen, 2009). According to Miles and Huberman suggests that activity in the qualitative data analysis performed interactively and runs continuously until complete, so that the data is already saturated ( iles u er an alda a ). Activities in the analysis of the data, that is data reduction, data display, and conclusion drawing/verification. RESULTS AND DISCUSSION Result Test Description Based on the results of the research shows that the students who have misconceptions far less when compared with students who understand and do not understand the concept, meaning that misconceptions that occur in XIclasses in Natural Sciences(IPA) by category misconceptions (Suwarna, 2013) in Table 2 are low. As shown in Table 3. Jurnal Riset Pendidikan Matematika, 4 (1), 2017 123 Widodo Winarso, T. Toheri Copyright © 2017, Jurnal Riset Pendidikan Matematika ISSN 2356-2684 (print), ISSN 2477-1503 (online) Based on the results of data analysis, the average students who experience misconception is lower when compared with the categories of students who understand the concept and students who do not understand the concept of limit. The distribution of values from the Six sub-concepts is as follows. First, it can be seen on the sub-concept describes the function limit function at one point through the calculation of value around the point, the students who experienced misconception average of 20.51%, students who understand the concept of 56.41%, and do not understand the concept of 23.08%. Secondly, the sub-concept predicts the value of f(x) if x goes through the graph and calculation, the students experiencing misconception on average are 12.82%, the students who understand the concept of 66.67%, and the students who do not understand the concept of 20.51%. Thirdly, the sub-concept determines the limit value of algebraic functions based on the nature of the limit, the students experiencing misconception average of 12.82%, students who understand the concept of 30.77%, and students who do not understand the concept of 56.41%. Fourth, the sub-concept determines the limit value of the indefinite function based on the limit properties, the students who have a misconception on average are 10.25%, the students who understand the concept of 66.67%, and the students who do not understand the concept of 23.08 %. Fifth, the sub-concept determines the limit value of the root shape function based on the limit properties, the students experiencing misconception on average 7.69%, the students who understand the concept of 12.82%, and the students who do not understand the concept of 79.49%. While the latter, the sub-concept determines the limit value of the trigonometric function based on the limit properties, the students experiencing misconception average of 8.98%, the students who understand the concept of 51.28%, and students who do not understand the concept of 39.74 %. Subconcepts with the highest percentage misconception are subconcepts explain the meaning of limit function at one point through the calculation of value -value around that point, that is equal to 20.51%. This error occurs because the majority of students believe that if the results obtained = 0 means that the value f(x) does not exist. This is because of misconceptions students understand the concept of the presence or absence of a function when the limit load factor of zero makers. In general, misconception experienced by students the high school in Indonesia there on subconcepts explain the meaning of limit function at one point through the calculation of value around the point, namely (1) misconceptions concluded if the limit value of 0 is no limit value for the function. (2) misconception that there should be no limit in the completion of writing the symbol of matter limit. Table 3 Percentage of Students Understand Concepts, Misconceptions and Not Understand Concepts SubConcept Question Category Answers Students PK MK TPK Explaining the meaning of limit function at one point by calculating values around the point 1 51,28% 23,08% 25,64% 2 61,54% 17,95% 20,51% 56,41% 20,51% 23,08% Predicting the value of f(x) if x towards infinity through graphs and calculations. 3 66,67% 12,82% 20,51% Determining the value of the limit functional algebra based on the nature of limit 4 30,77% 12,82% 56,41% Determining the value of the limit function indeterminate forms based on nature the nature of limit 5 66,67% 10,25% 23,08% Determining the value of the limit function of the root by nature the nature of limit 6 12,82% 7,69% 79,49% Determining the value of the limit of trigonometric functions by nature the nature of limit 7 38,46% 7,69% 53,85% 8 64,10% 10,26% 25,64% 51,28% 8,98% 39,74% Total average 47,44% 12,18% 40,38% Jurnal Riset Pendidikan Matematika, 4 (1), 2017 124 Widodo Winarso, T. Toheri Copyright © 2017, Jurnal Riset Pendidikan Matematika ISSN 2356-2684 (print), ISSN 2477-1503 (online) Figure 1. Answer Subject 18 Problem No. 1 Figure 2. Answer Subject 3 Problem Number 2 Figure 3. Answer Subject 6 Problem Number 3 Figure 4. Answers Subjects 6 Problem Number 4 Figure 5. Answer Subject 1 Question Number 5 Figure 6. Answer Subject 2 Problem Number 6 Jurnal Riset Pendidikan Matematika, 4 (1), 2017 125 Widodo Winarso, T. Toheri Copyright © 2017, Jurnal Riset Pendidikan Matematika ISSN 2356-2684 (print), ISSN 2477-1503 (online) Figure 7 Answers Subjects 37 Problem No. 7 Figure 8. Answers Subjek 37 Problem No. 7 Data Analysis The following analysis is presented misconception of the study subjects with the possible causes of these misconceptions. Figure 1 shows that according to ZulfaAmrina errors made by the subjects 18 is a misconception that students' mistakes because they do not write emblem limit function " when using the procedure count limit (Abidin, 2012). Students also make mistakes in understanding the concept of the presence or absence of a limit of a function when the result is 0. Possible cause of the misconception is the lack of understanding of the definition of the concept of limit function and a lack of understanding of the value and place (Lerner, 2003). Figure 2 shows that according to ZulfaAmrina errors made by the subject 3 is operating errors that students make mistakes count at the time of filling the table, and isconceptions y writing " when using the procedure count limit (Abidin, 2012). This condition can occur due to teacher factor in teaching mathematics. Not a few teachers give less attention to the achievement of students' understanding of the mastery of the concept of limit. So students can experience misconceptions. The possible causes of misconceptions are the lack of teachers' emphasis on limit function theorems and the lack of understanding of the concept of existence of function limits and their relation to the left and right limits (Jordaan, 2009). Figure 3 shows that according to ZulfaAmrina errors made by the subject 6 is a misconception that students do not write e le li it " when using the procedure count limit, and operation errors that students make mistakes arithmetic operation division of the highest rank (Abidin, 2012). Possible causes are a weak factor misconception practical aspects that students only pay attention to the practical aspect without regard to the concept (Irawan, Riyadi, & Triyanto, 2012). Miscalculation, as well as the lack of understanding of the value and place (Lerner, 2003). Figure 4 shows that according to ZulfaAmrina errors made by the subject 6 is a misconception that students do not write e le li it " when using the procedure count limit, and operation errors that students make mistakes factoring arithmetic algebraic function in order to obtain these results (Abidin, 2012). Possible causes are a factor misconception practical aspects that students only pay attention to the practical aspect without regard to the concept (Irawan et al., 2012). Miscalculation, as well as the lack of understanding of the value and place (Lerner, 2003). Jurnal Riset Pendidikan Matematika, 4 (1), 2017 126 Widodo Winarso, T. Toheri Copyright © 2017, Jurnal Riset Pendidikan Matematika ISSN 2356-2684 (print), ISSN 2477-1503 (online) Figure 5 shows that according to ZulfaAmrina errors made by the subject 1 is a misconception that students misunderstood the concept of limit theorem function by writing " when using the procedure count limit, and miscalculation that students perform arithmetic operation error factoring algebraic function in order to obtain these results (Abidin, 2012). Possible causes of misconception are the lack of understanding about value And points, as well as calculation errors (Lerner, 2003). Figure 6 shows that according to ZulfaAmrina errors made by the subject 2 is the systematic errors that students make mistakes in taking steps to resolve problems in order to obtain such a result and the misconceptions that students misunderstood the concept of limit theorem function by writing " √ when using the procedure count limit (Abidin, 2012). The possible cause of the misconception is the use of the wrong process, as well as a lack of understanding about the value and place of (Lerner, 2003). Figure 7 shows that according to ZulfaAmrina errors made by the subjects 37 is the systematic errors that students make the mistake of changing the shape of trigonometric functions towards resolving the matter in order to obtain these results (Abidin, 2012). Possible causes misconception is the wrong use of the process (Lerner, 2003). Figure 8 shows that according to ZulfaAmrina errors made by the subject 37 is an operation error that one student in performing arithmetic operations division trigonometric functions, and misconceptions that students misunderstood the concept of limit theorem function y writing " when determining the value of the limit of a function (Abidin, 2012). Possible cause of the misconception is the lack of understanding of the value and place, as well as calculation errors (Lerner, 2003). CONCLUSION Based on the research result, it can be concluded that; (1) Based on the analysis written tests and interviews were conducted to class XI student of Natural Sciences (IPA) 5SMA Negeri 1 Anjatan in Indonesia. Level experienced misconceptions belong to the category of low, amounting to 12.18%. However, students who do not understand the concept quite high at 40.38% and the rest are students who understand the concept that is equal to 47.44%. (2) Based on the analysis written tests and interviews were conducted to 16 students of class XI Natural Sciences (IPA) 5 SMA Negeri 1 Anjatan in Indonesia who have misconceptions. The misconception of the most widely experienced by students lies in subconcepts explain the meaning of limit function at one point through the calculation of value-value around that point, that is equal to 20.51%. 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Session II: The Neural Mind CONSCIOUSNESS MODELED: REIFICATION AND PROMISING PLURALISM RASMUS GRØNFELDT WINTHER University of California (Santa Cruz) 1. INTRODUCTION A tremendous opportunity lies in store for those of us interested in consciousness. I here bracket the arguments of those who wish to minimize or reduce consciousness away, either up into socially-based discourses and institutional power (e.g., Luhmann or Foucault) or down into sheer neural patterns (e.g., Paul Churchland). Rather, consciousness will be understood as a phenomenon and process requiring study. What is paradoxical is that explorers of the territory of consciousness seem to be studying consciousness out of existence, from inside the field of «consciousness studies». How? Through their love of the phenomenon/process, they have developed powerful single models – lenses – through which to understand consciousness. But in doing so, they also seek to destroy the other equally useful lenses. Our opportunity lies in halting the vendettas and cross-speakings/cross-fire. The imploration is to stop the dichotomous thinking and pernicious reification of single models, and instead search for divisions of labor, complementarities, and legitimate redescriptions among the various extant models. In other words, what would happen if we reimagined the conceptual classifications of the various models of consciousness, classifications based on general philosophical dichotomies (e.g., representational/non-representational and individualist/non-individualist), as a variety of compatible and even complementary perspectives on the same complex phenomenon and process? What would happen if rather than dig in our heels vis-à-vis our favorite theory of consciousness, at the exclusion of all the others, we saw our perceived enemy as an actual, indeed necessary, friend-in-waiting? What would it take to see a battlefield as a collaborative opportunity, to see a promising pluralism rather than an endless state space of conflict? This paper is a brief exploration of, almost a prolegomenon to, these questions. In the next section, I present compact descriptions of three models of consciousness: computational, networked, and embodied. The last section fleshes out promising pluralism. I argue that multiple models are needed to explain, understand, and intervene in complex phenomena: 1. They divide (and complement) ontological labor in that multiple models focus on different parts and aspects of consciousness. 2. They divide (and complement) explanatory labor by asking different questions, employing distinct ontologies, and using distinct methods to understand consciousness and its parts and aspects. © PENSAMIENTO, ISSN 0031-4749 PENSAMIENTO, vol. 67 (2011), núm. 254, pp. 617-630 08_RasmusGRONFELDT.qxd:Maqueta.qxd 4/6/12 11:51 Página 617 3. They sometimes provide legitimate redescriptions of each other, which itself increases understanding. Let us now turn to our descriptive task. 2. THREE MODELS OF CONSCIOUSNESS: COMPUTATIONAL, NETWORKED, AND EMBODIED The three models to be investigated are each built on a «central metaphor» 1: computers, networks, and the body. Each of these three things, both abstract and concrete, provide a wealth of associations, assumptions, and dichotomies (both «productive» and «pernicious»), and an object/process of study for each model of consciousness. It is my joy here as a student of these models to engage in what I call an «assumption archeology» (following Michel Foucault, Les mots et les choses, Michael Friedman, 1999, and Ian Hacking, 2002; see also Winther, 2012), which is a sort of conceptual excavation of the hidden presuppositions and constitutive principles that stand behind or under or within a given model. The second and third models of consciousness are related, but they emphasize different missing aspects of computational consciousness: (potentially representational, but anti-individualistic) extended-ness and (potentially individualistic, but antirepresentational) materiality, respectively. 2.1. Computational Consciousness According to the first computational model of consciousness (CC, hereafter), reasoning, thinking, and reflection consist of three components: (1) representations that are (2) formal and (3) manipulated/transformed according to explicit rules (themselves internal to the representations). The mind as a computer (i.e., a Turing Machine, a Finite State Machine) is the dominant metaphor under CC. Let us see how by discussing each component in turn. What follows is a description of consciousness and the world according to CC. Representations (or models) of the world are both (i) abstracted from sense-data and (ii) programmed with in-built categories and rules (e.g., Scott Atran's cognitive modules of natural kinds, or Chomsky's universal grammar – each module or the grammar(s) is full of internal assumptions and biases about the world). The world according to CC, is a complex interplay between empirical regularities, including causal relations and robust processes and stable objects, and problems to be solved in the world. It includes the names of your grandmothers and memories of them, as well as concerns with where you will stay and what you will see when you travel to South Africa. Sec ond, representations are formal. That is, they are expressed in strings of symbols. Third, representations as strings of symbols (e.g., program 618 SESSION II: THE NEURAL MIND: RASMUS GRØNFELDT WINTHER PENSAMIENTO, vol. 67 (2011), núm. 254 pp. 617-630 1 See Galison (1988) who develops an intercalated brick-wall «central metaphor» with three layers, data, instruments, and theories. 08_RasmusGRONFELDT.qxd:Maqueta.qxd 4/6/12 11:51 Página 618 algorithms in C++ or sets of mathematical functions expressed in differential calculus, which could themselves be translated into algorithmic programs) are themselves subject to precise and highly-constrained transformation rules. But where are these rules represented? They are themselves programmed in the rich representations. Some symbol strings contain information about various aspects of the world. They are «world models». Such symbols have explicit meanings – no latent, tacit or hidden meanings or knowledge here. Others are the rules for manipulating the world models (i.e., the programs). Put differently, in terms of cognitive content and manipulation, there is nothing outside the formal representations. Let us turn from (brief) analysis to (brief) history. It is well known that Behaviorism was the leading school of thought in psychology in America from the 1930s to 1960s. This tradition had a strong effect in academia across the globe (although Psychoanalysis and Gestalt Psychology, among other alternatives, also survived and continued developing in Continental Europe, among other places). The internalist break with Behaviorism came in the late 1950s. Computers were the source of the new cogntivist metaphor of mind and consciousness. The development of computers, and the existence proof of programs demolished the cognitive allergies of the behaviorists. Back then, those who believed in internal cognitive dynamics were the underdog and their new views on the strong analogies between human abstraction, reasoning, and problem-solving, on the one hand, and computer program calculations, compiling, and problem-solving inspired a significant amount of new work and young minds. Three select quotes and two book titles will motivate this high-level history. First the quotes. In a short, elegant piece started in 1954, the Artificial Intelligence pioneer Marvin Minsky, wrote: «mental processes resemble more the kinds of processes found in computer programs: arbitrary symbol-associations, treelike storage schemes, conditional transfers, and the like» («Matter, Mind, and Models», 1965). In his influential book, Gödel, Escher, Bach, Hofstadter characterized consciousness thus: «Consciousness is that property of a system that arises whenever there exist symbols in the system which obey triggering patterns somewhat like the ones described in the past several sections [p. 385; the «triggering patterns» referred to are all low-level, structured neuronal activation networks]» 2. Third, a recent comment on the cognitivist revolution is instructive: «The cognitive revolution overcame concerns about inner representational states in large part by pointing to computers. Computers operate autonomously on the basis of inner representation states, without falling into an explanatory regress, so why not people?» (Clapin, 2002; introduction, p. 13). CC broke with behaviorism by acknowledging (or reifying?) the existence of formal representations which it simultaneously placed inside the mind and above materiality. SESSION II: THE NEURAL MIND: RASMUS GRØNFELDT WINTHER 619 PENSAMIENTO, vol. 67 (2011), núm. 254 pp. 617-630 2 GEB (as it is also known) inspired many, and became the Book for a number of undergraduates majoring in programs such as Symbolic Systems, started in the 1980s at Stanford University. As a student at Stanford in the 1990s, I directly witnessed how powerful CC was. 08_RasmusGRONFELDT.qxd:Maqueta.qxd 4/6/12 11:51 Página 619 Now recall two book titles. A well-known textbook in computer science, Algorithms + Data Structures = Programs (Wirth, 1975), says it all in the title. Programs (CC: representations) consist of algorithms (CC: transformation rules) and data structures (CC: abstracted empirical regularities). Second, Chomsky's famous transformational grammar rebuttal of behaviorism, Syntactic Structures (1957), tied representational internalism and formalism in the title itself. The computer scientist Knuth, so important to the analysis of algorithms and the creator of the typesetting program TeX, had this to say of Chomsky's book: «Here was a marvelous thing: a mathematical theory of language in which I could use a computer programmer's intuition!» (2003, preface) (Interestingly, Knuth read Syntactic Structures on his honeymoon.) In short, under CC, consciousness consists of representations that are programs, constituted by algorithms (rules) and data structures 3. All revolutions start with hope. The cognitivist revolution is no exception. But has CC ended in despair or hype? It still has extremely influential proponents (e.g., Jerry Fodor, Zenon Pylyshyn, and developers of the ongoing AI project, Cyc). But CC as a model of consciousness has weakened for two related reasons: internal difficulties and the increased availability of alternatives. Regarding the former, issues such as the «framing problem», which inquires into how a program can pick out only the relevant inputs and outputs from the myriad possibilities of sense and rule inputs, and behavioral and/or problem-solving outputs, or the exceedingly great difficulty programs have with modeling and effecting bodily movement, have made CC a rather unpopular candidate for explaining consciousness. In addition, alternative models, to which we shall now turn, have themselves ushered in their own revolutions. My plea, however, is for attempting to locate places where these models might mutually strengthen one another. 2.2. Networked Consciousness A classic exercise in analytical epistemology asks whether the instruments we use to gather sense-data (e.g., telescopes, microscopes, Geiger counters) could count as sensory apparatus if we hypothesized complex organic beings with analogous internally-built detecting capabilities 4. That is, if extraterrestrial beings were directly able to see pulsars pulsating in the deep recesses of outer space, or bacterial cells in pond water, would that sense data be «internal» or «external» to the beings? Some epistemologists (or undergraduates who have just been asked the question), argue that in this hypothetical thought experiment, the sense data is indeed internal to the beings, whereas for us humans it would remain external. Others argue that this thought experiment shows that there is 620 SESSION II: THE NEURAL MIND: RASMUS GRØNFELDT WINTHER PENSAMIENTO, vol. 67 (2011), núm. 254 pp. 617-630 3 A related, complementary analysis of CC is found in the Stanford Encyclopedia of Philosophy entry on «The Computational Theory of Mind» (CTM) (Horst, 2011). Most basically the article argues that CTM = Representational Theory of Mind + Computational Account of Reasoning. Again, Consciousness (Mind) is fleshed out in terms of the formal representations and their manipulation. 4 Thanks to Judith Baker (York University) for reminding me of this thought-experiment. 08_RasmusGRONFELDT.qxd:Maqueta.qxd 4/6/12 11:51 Página 620 no hard and fast line between internal and outer, and hence much possible sense data is either wholly internal or external (depending on whether you are an internalist or externalist). A third group claims that gedankenexperiments are effectively meaningless and useless as intuitions always differ, and thus the natural sensory apparatus of those who practice science (i.e., Homo sapiens) is what determines the boundary between internal and external. These are clearly nuanced epistemological and semantic disputes. Exactly analogous disputes are carried out in the arguments over consciousness. The Cartesian homunculus is a single, individualistic creature, somewhere inside each and every one of our (literal) heads. Typically, CC holds it to do reasoning and thinking by itself, without assistance from the outside world. But could parts of the external world, outside of the individual's head (e.g., Otto's notebook, to which we will turn below), somehow be part and parcel of consciousness itself? Networked Consciousness (NC) says yes, and is thus an externalist member of the second group referred to above. In contrast to the rationalistic and individualistic Cartesian and Enlightenment view, NC insists that «the homunculus» is neither alone nor single. S/he is not well-individuated. Rather, consciousness is extended. Abstraction and computation (or whatever your favorite version of thinking is, representational or anti-representational) is a distributed process. NC is not necessarily anti-representationalist, but it is anti-individualist. Let us explore two important contributions to NC, Clark and Chalmers' concept of «the extended mind» (1998) and Hutchins' (1995) notion of «distributed cognition». In their ground-breaking essay «The Extended Mind», Clark and Chalmers invited us to think of cognition and even consciousness as a distributed process. They still thought that there was something special about what happened inside the head (which we cannot fully elucidate in this article), but their main point was to extend mental and cognitive processes to outside the skull. Their central example is Otto and Inga, two figures who need to find their way to MoMA in New York City. They are both far away, in other parts of NYC. Otto suffers from Alzheimer's disease and must carry around a notebook in which he writes many pieces of information. Among these, are the directions to MoMA from pretty much any point in NYC. In contrast, Inga is perfectly healthy and relies on her «internal memory» to get to MoMA from any point in NYC. Clark and Chalmers ask whether the notebook is in any sense part of Otto's own memory, reasoning, and cognition. They write: «In both cases [Otto and Inga] the information is reliably there when needed, available to consciousness and available to guide action, in just the way that we expect a belief to be.» There is no in-principle difference between the two sources of information (Otto's notebook, Inga's head). Indeed, «there is nothing sacred about skull and skin.» They summarize their extended mind perspective thus: «... the human organism is linked with an external entity in a two-way interaction, creating a coupled system that can be seen as a cognitive system in its own right. All the components in the system play an active causal role, and they jointly govern behavior in the same sort of way that cognition usually does. If we remove the external component the system's behavioral competence SESSION II: THE NEURAL MIND: RASMUS GRØNFELDT WINTHER 621 PENSAMIENTO, vol. 67 (2011), núm. 254 pp. 617-630 08_RasmusGRONFELDT.qxd:Maqueta.qxd 4/6/12 11:51 Página 621 will drop, just as it would if we removed part of its brain. Our thesis is that this sort of coupled process counts equally well as a cognitive process, whether or not it is wholly in the head». Clark and Chalmers' paper was extremely influential and brought home the point that many aspects of consciousness and cognition should be understood as distributed and networked. Slightly earlier work on this topic had made a similar point by using a sort of philosophical anthropology approach. In Cognition in the Wild, Hutchins had explored how information flowed and decisions were made in airline cockpits. In a co-authored paper (Hollan et al., 2000), Hutchins summarized his view thus: «Whereas traditional views look for cognitive events in the manipulation of symbols inside individual actors, distributed cognition looks for a broader class of cognitive events and does not expect all such events to be encompassed by the skin or skull of an individual. For example, an examination of memory processes in an airline cockpit shows that memory involves a rich interaction between internal processes, the manipulation of objects, and the traffic in representations among the pilots. A complete theory of individual memory by itself is insufficient to understand how this memory system works. Furthermore, the physical environment of thinking provides more than simply additional memory available to the same processes that operate on internal memories. The material world also provides opportunities to reorganize the distributed cognitive system to make use of a different set of internal and external processes.» (pp. 175-176). Hutchins and collaborators emphasized the distributed networks aspects of cognition and consciousness. They also explored the different modalities and mechanisms of cognitive processes. That is, by distinguishing among, for instance, «internal processes», «manipulation of objects», and «traffic in representations», these scholars moved towards a taxonomy of different types of processes and objects involved in extended or distributed cognition. Not all abstraction and thinking is the same, nor are all parts of cognitive processes identical or even similar in type. The different roles actual computation plays, as compared to information stored in physical devices such as altimeters or pressure gauges, as compared to conversation and information exchange across the pilot, co-pilot, flight engineer and so forth needs to be analyzed. An important contribution of Hutchins' work is to start working towards a «typology of components of distributed consciousness.» It is not sufficient to say that consciousness is distributed, we must also explore the structure and function of its parts. NC takes issue with the individualism of CC. It extends consciousness to include what is outside the individual. That is, the scaffolding of consciousness is itself part of consciousness. This anti-individualist move problematizes the internal vs. external dichotomy, and perhaps even overcomes it 5. Moreover, NC organizes the different components of this extended system into a taxonomy. 622 SESSION II: THE NEURAL MIND: RASMUS GRØNFELDT WINTHER PENSAMIENTO, vol. 67 (2011), núm. 254 pp. 617-630 5 See Oyama (2000) for an analogous move to overcome the internal/external dichotomy in the nature/nurture debates so entrenched in the biological and psychological sciences. 08_RasmusGRONFELDT.qxd:Maqueta.qxd 4/6/12 11:51 Página 622 Distinct parts have specific structures and functions. The research program of the NC model of consciousness thus involves characterizing parts and roles of an extended consciousness system. In short, we must explore extended relationality in our efforts to understand consciousness. 2.3. Embodied Consciousness A different set of traditions, both philosophical and empirico-scientific (and their combination), have also resisted the CC model of consciousness. The Embodied Consciousness (EC) alternative model of consciousness is defended in a broad (and sometimes conflicting) variety of ways by investigations as different as (1) the diverse phenomenological tradition of Edmund Husserl, Martin Heidegger, Maurice Merleau-Ponty, and Alfred Schutz, (2) the more unified pragmatism of William James and John Dewey, and (3) the recent empirico-scientific studies of embodied robotics by Rodney Brooks and of embodied linguistics by Mark Johnson 6. One common thread across these traditions is a focus on the body, and on imagining thinking, cognition, and reasoning as embodied rather than abstract processes. In what follows, I will briefly motivate the general problem of CC, as seen from the EC model. I will then turn to three modalities or parts of consciousness highlighted by EC and strongly downplayed by CC: (1) (sophisticated) sense data, (2) movement, and (3) feelings. In order to itself embody these modalities/parts of consciousness, we will meet cutting-edge work in robotics, linguistics and kinesthetics 7. Philosophically, the incarnation of CC is the Cartesian homunculus. It is wellknown that Descartes posited an ego that cogitated. Indeed, the only principle of which Descartes could be certain in the bout of systematic doubt with which he starts his Meditations (1641), was that there was an I doing all the thinking, all the doubting. This thinking I consisted of res cogitans – i.e., cogitating thing/stuff/matter. Upon this I, Descartes built his system of knowledge. Some have since imagined this I as a sort of reified homunculus, an abstract representer of abstractions, deeply severed from his (almost invariably and importantly a «him») body, and processing raw sense-input with his abstract central processing unit. Almost needless to say, Kant built further on Descartes picture of an abstract, individuated, severed reasoning agent by postulating a rich, inner, structure rationality. Kant's philosophy gave us an understanding of the 'homunculus' (if we may) internal conceptual/rational categories, unity of apperception, and bounds of reason. SESSION II: THE NEURAL MIND: RASMUS GRØNFELDT WINTHER 623 PENSAMIENTO, vol. 67 (2011), núm. 254 pp. 617-630 6 Two other essential figures, with well-articulated research programs are Antonio Damasio (1994) and Maxine Sheets-Johnston (1999). The former explores a neuroscience focusing on embodiment and feeling; the latter investigates the literal role of movement and kinesthetics in self-awareness. Unfortunately, there is not sufficient space-time to investigate these important views (see also MacIver, 2009). 7 Wilson (1995) and Anderson (2003) provide ample discussion regarding the NC and EC models of consciousness. 08_RasmusGRONFELDT.qxd:Maqueta.qxd 4/6/12 11:51 Página 623 This homunculus has been resisted. One of the ways its existence was questioned, and its de-reification invited, was by emphasizing the corporeal parts and modalities of consciousness. Mind was not separate from body – indeed it was shaped by, constrained by, and guided by the whispers of physicality. In his monumental Being and Time (1927), Heidegger wrote: «The kind of dealing which is closest to us is as we have shown, not a bare perceptual cognition, but rather that kind of concern which manipulates things and puts them to use; and this has its own kind of 'knowledge'» (Heidegger, 1962, p. 95). As has so been so ably analyzed by the critic of AI (Artificial Intelligence) and CC, Hubert Dreyfus (e.g., 1992), Heidegger is an excellent point source for reflections about human being-in-the-world. Heidegger's work which downplay representation and abstract thinking, and instead focuses on our material activity, on our use of tools-for particular purposes, and on our literal bodies. Related resistance to Descartes' homunculus is found in William James' evolutionarily-aware and anti-«vicious abstractionist» (James' own term from The Meaning of Truth 1909/1911) meditations on psychology: «Mental facts cannot be properly studied apart from the physical environment of which they take cognizance. The great fault of the older rational psychology was to set up the soul as an absolute spiritual being with certain faculties of its own by which the several activities of remembering, imagining, reasoning, willing, etc. were explained, almost without reference to the peculiarities of the world with which these activities deal. But the richer insight of modern days perceives that our inner faculties are adapted in advance to the features of the world in which we dwell, adapted, I mean, so as to secure our safety and prosperity in its midst» (James, 1900, p. 13). James had published his justly famous Principles of Psychology in 1890. In emphasizing (his term) «the stream of consciousness» and the inseparability of sentiment and rationality, James lay the groundwork for an empiricallygrounded embodied model of consciousness, an alternative to the CC model of consciousness. Evolutionary theory entered in two ways. First, by emphasizing our continuity with animals, including continuity in sensorimotor apparatus and processes, and continuity in problems and issues-going-of-concern relevant to survival. Second, by highlighting the continuity, indeed interpenetration, of fact and value. The way we are, and behave in the world, are inseparably tied to the way we should be and behave. James did find ways to use evolutionary theory, and our adapted and material bodies, as a normative platform from which moral rules emanated. These rules neither diffused into unjustified relativism nor ware they Kantian-like idealistic and rational top-down moral imperatives 8. In short, both Heidegger and James are key figures from 624 SESSION II: THE NEURAL MIND: RASMUS GRØNFELDT WINTHER PENSAMIENTO, vol. 67 (2011), núm. 254 pp. 617-630 8 Useful essays on James can be found in Putnam 1997. I am fortunate to be a member of Lucas McGranahan's PhD committee (Department of Philosophy, University of California [Santa Cruz]). Lucas is comparing Nietzsche and James, with a particular eye on their respective 08_RasmusGRONFELDT.qxd:Maqueta.qxd 4/6/12 11:51 Página 624 Phenomenology and Pragmatism, respectively, that resist the Cartesian homunculus by adopting the EC model of consciousness. Scientific work which could be said to fall within embodiment theory includes the robotics of Rodney Brooks and the linguistics of Johnson 9. Each of these research programs studies related sets of modalities or parts of consciousness, not studied by CC (nor by NC, a non-individualist model). Brooks is interested in (sophisticated) sense data and movement, which are part of a strong robotenvironment coupling. Johnson is particularly concerned with human movement and feelings, as captured in, and determining of, our dominant metaphors. Let us briefly explore each. As discussed in his intellectual biography (2002), Brooks has worked on robotics for a number of years, using a paradigm utterly different than the representationalist one typical (ay, definitional) of CC. He decided to forego representations altogether, and instead build tight robot-environment information loops, in which the ant-like robot reacted immediately to feedback from its environment and adjusted its behavior accordingly. In Cambrian Intelligence, Brooks writes: «Essentially the idea is to set up appropriate, well conditioned, tight feedback loops between sensing and action, with the external world as the medium for the loop» (p. 109). In his now-classic paper «Intelligence without representation», he wrote: «We have reached an unexpected conclusion (C) and have a rather radical hypothesis (H): (C) When we examine very simple level intelligence we find that explicit representations and models of the world simply get in the way. It turns out to be better to use the world as its own model. (H) Representation is the wrong unit of abstraction in building the bulkiest parts of intelligent systems.» In other words, the world is its own best model and representation is an inappropriate abstraction unit for building an (artificially) intelligent system – «in order to understand something, I must build it», might be an engineer's motto. The «body» of the robot uses complex sensory data and highly reactive motor units to deal effectively with its environment. Tacit knowledge (Michael Polanyi's concept) itself gets entrenched and built in a bottom-up fashion from robot-environmental interaction. Brooks' robots are a beautiful exemplar, in the kuhnian sense, of the EC model. Analogously, our own bodies are highly sensitive SESSION II: THE NEURAL MIND: RASMUS GRØNFELDT WINTHER 625 PENSAMIENTO, vol. 67 (2011), núm. 254 pp. 617-630 views on evolution and ethics. His work on James has helped me see the founding figure of psychology in the USA with a fresh perspective (see McGranahan, 2011). 9 I also work with Alexis Mourenza, another PhD student at the University of California (Santa Cruz). Her detailed conceptual and scientific investigations of animal consciousness point to a whole other area of research closely tied to the EC model which I must also sidestep in this paper due to space limitations. See, for example, Bekoff, Allen, and Burhgardt (2002). 08_RasmusGRONFELDT.qxd:Maqueta.qxd 4/6/12 11:51 Página 625 reactive units to our environment. Our consciousness is grounded in and shaped by the embodied flow of information, sensations, and qualia. Mark Johnson is a linguist and philosopher perhaps best known for his coauthored book Metaphors We Live By (1980, Lakoff and Johnson) with George Lakoff. In a more recent book, The Meaning of the Body, Johnson explores EC, with an eye towards the process of aesthetic judgment and establishment of meaning in our lives. In one particularly constructive chapter, «From Embodied Meaning to Abstract Thought», Johnson fleshes out how abstract thought, reasoning, and consciousness could arise from the various parts, processes, and movements of the body. Drawing on his linguistic studies, Johnson argues that our materiality is often the source for our abstract concepts and thinkings. There is a «conceptual metaphor» (his term) structure built into the relations between our bodies and our languages. In this chapter, he writes: «Dewey's pragmatist continuity thesis claims that we must be able to move, without any ontological or epistemological rupture, from the body-based meaning of spatial and perceptual experience that is characterizable by image schemas and affect contours all the way up to abstract conceptualization and reasoning. This same notion of ontological continuity underlies most second-generation (embodied) cognitive science. The existence of abstract concepts thus poses a fundamental problem for any naturalistic view of meaning as grounded in the qualities and structures of sensorimotor experience. How can thinking about abstract, nonphysical entities possibly be grounded in the body?» (p. 176). ... «... abstract concepts are defined by conceptual metaphors that recruit the semantics and inference patterns of sensorimotor experience. ... AFFECTION IS WARMTH, IMPORTANT IS BIG, MORE IS UP/LESS IS DOWN, HAPPY IS UP/SAD IS DOWN, STATES ARE LOCATIONS, CAUSES ARE FORCES, ... TIME IS MOTION...» (pp. 178-179). Conceptual metaphorical structures take body experiences, feelings and movements as the source [e.g., warmth, spatial orientation, and (literal) forces], and abstract thoughts and reasonings are the target (e.g., affection, more or less, and causes/causation). Johnson holds that we could generalize conceptual metaphors to many other aspects of our abstract toolbox. He also appeals to other sorts of embodied reasoning processes, where the body constrains and shapes our thought. EC is a different, but related (to NC) way, of reacting to CC. Here the focus is on internal, material modes of having and experiencing and making consciousness. In particular we look at the body, and its (1) (sophisticated) sense data, (2) movement, and (3) feelings. By submerging into the body, we explore the relationship between body and environment, and see that body and consciousness are deeply enmeshed and intertwined. CC's view from above is hardly sufficient. In this second section, I have provided brief descriptions of the models of consciousness under our purview here. An assumption archeology has been carried out on each of them. Needless to say, much work remains to be done. 626 SESSION II: THE NEURAL MIND: RASMUS GRØNFELDT WINTHER PENSAMIENTO, vol. 67 (2011), núm. 254 pp. 617-630 08_RasmusGRONFELDT.qxd:Maqueta.qxd 4/6/12 11:51 Página 626 3. PROMISING PLURALISM The goal of our exercise has been to emphasize three models of consciousness, with an eye towards their respective division and complementation of labor, and their ultimate integration. Consciousness is a complex process, with at least three loci, as described here: (1) representations that can be described formally, (2) externalized, networked components, (3) embodied situated-ness. That is, consciousness has to be understood through at least a trichotomy which we must overcome: formal representation(above) vs. embodied(below/inside) vs. networked(outside). This turns out actually to be a trichotomy (formal representation/embodied/networked) correlated with various dichotomies both general (e.g., representational/non-representational and individualist/ non-individualist) and more specific (e.g., representational/embodied and individualist/networked). The standard position on dichotomies (or trichotomies or n-chotomies), is to see the opposing poles or ends as mutually exclusive and collectively exhaustive. However, we can also view these standard oppositions as an opportunity for dialectical overlap and interpenetration (see, e.g., Levins and Lewontin 1985, Winther 2008, 2011). No single model should be individually and imperialistically reified. Rather, there is strength in dialogue and numbers (see also Mitchell 2009). I conclude this prolegomenon on models of consciousness by stating three lessons. The first lesson is that each model is important and relevant. Each emphasizes distinct parts and aspects of the ontology of consciousness and of explanations pertinent to consciousness. Each asks different questions, and employs its own methodologies. CC is not sufficient. It lacks an account of feeling, sense-data, and movement – the three essential components of EC – which influence both the form and content of consciousness. Moreover, it is too internalistic and individualistic. It lacks an account of the variety of external processes and objects – elucidated by NC – which again necessarily modulate and shape consciousness. Each model of consciousness is necessary and insightful. Complex consciousness, like the proverbial elephant examined by the blind men of the Eastern fable, requires analyses from different points of view. A (static) pluralism is thus necessary to get a complete understanding of consciousness. But there are more and deeper lessons. In our attempts to avoid and to overcome reification, which is the way I prefer to describe the purpose of this paper, we (1) engage in dialogue, (2) are sensitive to data, (3) engage in selfreflection (our «assumption archeology»), and (4) attempt integration. Through these activities, we see that especially in this case of modeling consciousness, the three models are not only individually necessary, but they also shape and constrain one another. That is, they exhibit a definitional and dynamical dialectic. For instance, representations are vulnerable to emotions and sense-data, and to inputs being received from a broad variety of external places. The dichotomies are vulnerable and contingent. The second lesson is that dialectical thinking is useful here. Definitions as well as actual process-dynamic explanations of each SESSION II: THE NEURAL MIND: RASMUS GRØNFELDT WINTHER 627 PENSAMIENTO, vol. 67 (2011), núm. 254 pp. 617-630 08_RasmusGRONFELDT.qxd:Maqueta.qxd 4/6/12 11:51 Página 627 model of consciousness are sensitive to the definitions and explanations of the other two models. Not only is each model necessary, each model constrains and shapes the other two. A (dynamic) pluralism thus itself changes its own grounds of possibility. The third lesson is that our classification is itself unstable and vulnerable. It can change with the introduction of further models of consciousness. For instance, introducing Hameroff and Penrose's quantum consciousness, would add a whole new physical theory to the physical substrate of consciousness. This addition would invite us to see new triangulations, new aspects, of the models of consciousness we have thus far explored. The «consciousness wars» are better thought of as «consciousness collaborations». Like any part of science, work on the sciences of the mind and consciousness is a Ship of Theseus that we are constantly rebuilding at sea. Such reconstruction requires collaboration. ACKNOWLEDGEMENTS Thanks to Jens Degett and Cristina Heller del Riego for inviting me to a thought-provoking scientific and philosophical conference. Thanks to Quentin Cooper, Stuart Hameroff, Amparo García-Plaza, Juani Guerra, Lluis Oviedo, and Steen Rasmussen, among many others at the «Life, Evolution, and Complexity» conference (December 2010, Comillas University, Madrid), for conversation. Given the context of the enjoyable conference at which this paper was presented, I believe it instructive to self-critically state my assumptions regarding religion. I am not religious in any standard sense, and certainly do not believe in any monotheistic deity. I am an avid supporter of scientific methods, and of careful philosophical discussion of the empirical and ethical strengths and weaknesses of the products (e.g., models, theories, interventions) of such methods. I am generally agnostic about the possible (rational and justified) relationships between science and religion. Jácome Armas, Judith Baker, Mark Detweiler, Claus Emmeche, Peter Godfrey-Smith, Ian Hacking, Hervé Kieffel, Sergio Martínez, Amir Najmi, Susan Oyama, and Frederik Stjernfelt have provided a gentle forum of discussion for many of these ideas, for a number of years. BIBLIOGRAPHY ANDERSON, M. L. (2003), «Embodied Cognition: A field guide», Artificial Intelligence 149: 91-130. BEKOFF, M.; ALLEN, C., and BURGHARDT, G. M., The Cognitive Animal. Empirical and Theoretical Perspectives on Animal Cognition, MIT Press. BLACKMORE, S. (2005), Conversations on Consciousness, Oxford University Press. BROOKS, R. (1991), «Intelligence without representation», Artificial Intelligence 47:139159. Reprinted in Cambrian Intelligence: The Early History of the New AI, 1999, MIT Press. BROOKS, R. (2002), Flesh and Machines. How Robots Will Change Us, New York: Pantheon Books. CHOMSKY, N. (1957), Syntactic Structures, Berlin: Mouton de Gruyter. CLAPIN, H. (ed.) (2002), Philosophy of Mental Representation, Oxford University Press. 628 SESSION II: THE NEURAL MIND: RASMUS GRØNFELDT WINTHER PENSAMIENTO, vol. 67 (2011), núm. 254 pp. 617-630 08_RasmusGRONFELDT.qxd:Maqueta.qxd 4/6/12 11:51 Página 628 CLARK, A., and CHALMERS, D. J. (1998), «The Extended Mind», Analysis 58:7-19. DAMASIO, A. (1994), Descartes' Error. Emotion, Reason, and the Human Brain, New York: GP Putnam's Sons. DREYFUS, H. (1992), What Computers (Still) Can't Do: A Critique of Artificial Reason, MIT Press. FOUCAULT, M. (1970), The Order of Things. An Archeology of the Human Sciences, Vintage Books. FRIEDMAN, M. (1999), The Dynamics of Reason, Stanford, CA: CSLI Publications. GALISON, P. (1988), «History, Philosophy, and the Central Metaphor», Science in Context 2:197-212. HACKING, I. (2002), Historical Ontology, Cambridge, MA: Cambridge University Press. HAMEROFF, S. (2010), «The "conscious pilot" – dendritic synchrony moves through the brain to mediate consciousness», Journal of Biological Physics 36:71-93. HAMEROFF, S., and PENROSE, R. (Online), «Conscious Events as Orchestrated Space-Time Selections». Found at following URL: http://www.quantumconsciousness.org HEIDEGGER, M. (1962), Being and Time, Harper Collins. HOFSTADTER, D. (1979), Gödel, Escher, Bach: An Eternal Golden Braid, Basic Books. HOLLAN, J.; HUTCHINS, E., and KIRSH, D. (2000), «Distributed Cognition: Toward a New Foundation for Human-Computer Interaction Research», ACM Transactions on Computer-Human Interaction 7:174-196. HORST, S., «The Computational Theory of Mind», The Stanford Encyclopedia of Philosophy (Spring 2011 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/ archives/spr2011/entries/computational-mind/>. HUTCHINS, E. (1995), Cognition in the Wild, MIT Press. JAMES, W. (1900), Psychology. Briefer Course, American Science Series, New York: Henry Holt and Co. - (1909/1911), The Meaning of Truth. A Sequel to 'Pragmatism', New York: Longmans, Green, and Company. JOHNSON, M. (2007), The Meaning of the Body. Aesthetics of Human Understanding, University of Chicago Press. KNUTH, D. E. (2003), Selected Papers on Computer Languages, Stanford, CA: CSLI Lecture Notes, no. 139. LAKOFF, G., and JOHNSON M. (1980), Metaphors We Live By, University of Chicago Press. LEVINS, R., and LEWONTIN, R. (1985), The Dialectical Biologist, Harvard University Press. MACIVER, M. A. (2009), «Neuroethology: From Morphological Computation to Planning», The Cambridge Handbook of Situated Cognition, Robbins, P., and Aydede, M. (eds.), Cambridge University Press, pp. 480-504. MCGRANAHAN, L. (2011), «William James's Social Evolutionism in Focus», The Pluralist 6(3):80-92. MINSKY, M. L. (1965), «Matter, Mind and Models», Proc Intl Federation of Information Processing Congress 1:45-49. (Also easily found online: http://groups.csail.mit.edu/ medg/people/doyle/gallery/minsky/mmm.html) MITCHELL, S. (2009), Unsimple Truths. Science, Complexity and Policy, University of Chicago Press. OYAMA, S. (2000), The Ontogeny of Information. Developmental Systems and Evolution, 2nd ed., Duke University Press. PUTNAM, R. A. (ed.) (1997), The Cambridge Companion to William James, Cambridge University Press. SHEETS-JOHNSTONE, M. (1999), The Primacy of Movement, Amsterdam: John Benjamins. SESSION II: THE NEURAL MIND: RASMUS GRØNFELDT WINTHER 629 PENSAMIENTO, vol. 67 (2011), núm. 254 pp. 617-630 08_RasmusGRONFELDT.qxd:Maqueta.qxd 4/6/12 11:51 Página 629 WILSON, R. A. (1995), Cartesian Psychology and Physical Minds: Individualism and the Sciences of the Mind, Cambridge University Press. WINTHER, R. G. (2008), «Systemic Darwinism», Proceedings National Academy of Sciences, 105 (33): 11833-11838. - (2011), «Part-Whole Science», Synthese 178:397-427. - (2012), «Evo-Devo as a Trading Zone», in: A. LOVE (ed.), Conceptual Change in Biology: Scientific and Philosophical Perspectives on Evolution and Development, Springer Verlag (Boston Studies in the Philosophy of Science). WIRTH, N. (1975), Algorithms + Data Structures = Programs, Prentice-Hall. Philosophy Department RASMUS GRØNFELDT WINTHER Karolinska Institutet, Stockholm University of California (Santa Cruz) Center for Philosophy of Nature and Science Studies, University of Copenhagen Biocomplexity Center, Niels Bohr Institute, University of Copenhagen [email protected] www.rgwinther.com 630 SESSION II: THE NEURAL MIND: RASMUS GRØNFELDT WINTHER PENSAMIENTO, vol. 67 (2011), núm. 254 pp. 617-630 08_RasmusGRONFELDT.qxd:Maqueta.qxd 4/6/12 11:51 Página

Prajñā Vihāra RELIGION AND CREATIVE IMAGINATION: RELIGIOUS REPRESENTATION IN I. B. SINGER'S IN MY FATHER'S COURT AND THE SHADOW-THEATER (WAYANG) IN INDONESIA Andi Herawati and Andi Rachmawati Syarif Indiana University, USA and Muhammadiyah University Kendari, Indonesia ABSTRACT Even within religion, the creativity of imagination offers an invaluable defense against the tendencies towards dogma and absolutism. It also provides spaces living and experiencing life in diverse ways. This paper discusses the different facets of creative imagination in religious art and literature forms by comparing Isaac Bashevis Singer's In My Father's Court with Wayang shadow theater in Indonesia. I will show that they possess similar features demonstrating a reflection on religious law, creativity and everyday life. In Singer's work, the synagogue is a theater, and Singer's father functions in the same way the puppet master, or Dalang. operates in Wayang theater. This allows for the negotiations between religious law and the living community. Keywords: Isaac Bashevis Singer, Religious Creativity, Wayang, Religious Law Prajñā Vihāra Vol. 20 no. 2 July to December 2019, 32-47 © 2000 by Assumption University Press Andi Herawati and Andi Rachmawati Syarif 33 Scholars have often described religion as providing a rigorous set of beliefs, symbols and rituals which have been shaped by the historical dynamics of communities. Peter Beyer, for example, discusses religion both substantially and functionally. The substantive aspect relates to the supernatural, while the functional aspect of religion focuses on religion's social or psychological purposes and effects1. He pointed out that to "observe religion as a social phenomenon is to observe it as a communication"2 But this raises the question, if religion is a form of communication, might we ask if it is possible to see religion as a creative form of communication where its effective expression involves creative imagination.3 The distinctiveness of literature and drama stands out in contrast to the conceptual disciplines of theology and philosophy. The creative use of words, movement and sounds create a unique world of ideas, and enables us to express things that the theologian or philosopher might find difficult to express. Thus the expressive indeterminacy of the creative imagination can often offer an alternative and defense against the abstract rationalism often encountered in philosophy and theology, and against the tendencies towards dogma and absolutism. The creative imagination opens a space for feeling, emotion, and the mysteries of a particular experience of life. At the same time, literature, like philosophy, shares a determination to question received wisdom and to open up new modes of thought, perception and action. The creative and imaginative qualities of human nature allow people to approach, shape and perceive religious teachings, sources and traditions in very different ways. In addition, creativity naturally spreads the message of religion beyond the limits of merely theological matters, instead touching on many more practical, equally universal aspects of life and experience, and more practical, such as art, literature, dialogue, ethics, etc.4 One form of religious expression that is particularly popular in today's world is fiction in its many forms, whether poem, novel, film or other kinds of performance. 34 Prajñā Vihāra This transcending of limits through creative expression allow us to see themes and patterns repeated in various religious expressions which relate to very separate and unique lived experiences. This interpretation of religion through the creative imagination, allows one not only to recognize one's own life within an exotic culture, but it also allows a creative bridging between cultures. By taking a closer look at the contemporary study on Religion, where the qualitative aspects of study of humanities, is able to see very complex situations in social phenomena, this paper discusses the creative imagination represented by religious actors, players, at the same adherents or followers. By taking examples from two different forms of creative art, stories are represented as religious representation in different context of life and people religious life, with different ways of negotiating life, through individual, rational interpretation to mystical interpretation. While both representing different religious tradition within different social-cultural context, they both are similar in a way to show us how to negotiate life within religiousdogmatic prescription that very often are not compatible with people's real life. Isaac B. Singer's book, In My Father's Court, is a work of literature that explores a unique life while expressing universal religious ideas. It becomes a question of life and law. From the narrow religious perspective of law what Singer goes beyond scripture in order to accommodate the life of his people. The focus on lived experience is important because the narrow perspective of religious law is often removed from living society. In this work, Singer powerfully represents and re-creates an Eastern European Jewish culture that is today unfamiliar to most people. In the book, his father's synagogue, rather than simply being a place where people sought law and justice, came to represent a space for religious life to break through into the everyday world, creating a unique space for his people to learn, to contemplate, to negotiate life, and to overcome dramatically challenging conditions. Singer vividly draws on that world of his childhood to share the deeply humanistic lessons that he himself had begun to learn already in his own childhood. Andi Herawati and Andi Rachmawati Syarif 35 In these exotic tales which recount his rabbi father's life in a Polish Ghetto, Singer is able to lead his readers on their own spiritual journey, to discover deeper human truths that can be valued across religious and cultural divides. His themes center on seeking truth in the face of lust, greed, pride, obsession, misfortune, unreason, communicating all the surprises and challenges of the human condition. Such a creative art is not just about a particular life, or a particular story, but also about communicating universal themes creatively through these particular forms. The wayang shadow-puppet drama is a very old Javanese tradition going back to the ancient animist traditions and extending through the introduction of the Indian dramatic forms of the Ramayana and the Mahabharata and finally extending to the arrival of Islam with the Wali Songo the "Nine Sufi Saints", who in the fifteenth century, were able to spread Islamic spiritual teachings in Indonesia5, in ways in harmony with the older dramatic forms. The imaginative effectiveness of this puppet-theater (and associated local musical forms, such as the gamelan orchestra) meant that the Javanese sensed no contradiction between this new vehicle for Islamic teachings and the fact that these puppet-theater stories were often Hindu in origin, while some of their characters may have stemmed from even earlier local roots. Nor do the writers and directors of wayang performances-both old and new-find it odd that the Javanese philosophy has deep connections to Sufism, the mystical dimension and expressions of Islam. The creative characters of wayang are demonstrated, first, through its language. The wayang presents a very rich and creative use of language. Aside from the religious-ethical instructions and reflections on the roles of the leader and the people, the stories also communicate more complex themes through allusions, parables, jokes and social criticism6. Second, the plot in the shadow theater is not fixed, rather it fluctuates with the emergence of new problems and situations. The complex situations that people face in life – from searching for truth, to the challenges of faith, to the solving of ethical problems – involves doubt, conflict, paradox, 36 Prajñā Vihāra and unanswered questions. The purpose of the stories is not just to give solutions to the problems of life, but rather, to show new possibilities for contemplation and the negotiation of living situations. Human characters portrayed are always individuals and often are not even representations of Javanese identity, but are rather more universal. Third, wayang is a media of Islamic Education, and a creative attempt to communicate Islamic values.7. Some have argued that the highly stylized human forms in the Javanese wayang puppets were very effective in accommodating local culture and life with early Islamic teaching in Indonesia, especially throughout Java, Bali and Lombok. Yet this artistic also plays a role in communicating new, contemporary themes and issues arising in areas of religious ideology, religious law, social criticism, ethics and morality, politics. In this way wayang can be a diverse and highly creative contemporary religious representation mirroring the transformation of people's lives through stories and characters (lakon), like the strange and often exotic figures and stories represented in Singer's book. Although the two "religious arts" and representation may not have a close historical connection, I will show that they both involve similar creative features to bring out unsuspected dimensions of religious life, suggesting important insights for the study of religion. So we can ask, what is the role of creative imagination in religious expressions and forms? Specifically, how does Singer's creativity represent a unique expression of life, but at the same time allow us to see universal themes that link us with the unique expression of life found in Javanese wayang theater? First, it is important to explore the features of Singer's book to see what is shared with the shadow-puppet performance, by looking at the representation of religious life through stories which relate the everyday life in his childhood Polish Jewish community. Those common features in Singer's stories represent areas of life where life cannot be reduced to right or wrong, permitted or forbidden things, as was often understood in the learned religious texts. They do not involve preaching normative Andi Herawati and Andi Rachmawati Syarif 37 beliefs and practices, instead those stories evoke the sort of places where individuals must by themselves somehow negotiate life's mysterious and tricky situations. The plot in his stories does not unfold in a familiar, constant pace, but rather fluctuate through difficult, often inconclusive and unexpected problems and situations. The complex ethical situations that people face in life, from searching for the truth to struggling with the validity of faith, are portrayed in these stories as a process of seeking the truth, since his characters live through dramas of doubt, conflict, paradoxes, and questioning. These often inconclusive stories, echo the famous "teaching stories" of the Hasidic mystics. The purpose of such stories, unlike the rabbinic religious authority of Singer's father and his fellows, is not merely about providing a legally or scripturally valid "solution" to the problems of life. Instead, they point to a very different form of religiosity requiring a long inner process of contemplation and negotiation. Singer suggests, through these stories, that it is the individual's inner qualities (at once spiritual and ethical) which alone can confront evil in the world; that the human character is drawn to the purity of heart, clarity and courage. It is important to note that these rare, but essential inner qualities arise in characters who are always distinctly individual and not based upon any religious or ethnic identity. Moreover, the lessons conveyed by these stories are not just abstract ethical principles, but lively representations of the lives of people. Take for example, the story about a man asking whether it is proper to sleep with his dead wife. The reason is his modest crowded home where there is no place to sleep except with his dead wife's body. Instead of providing a religious-law solution, Singer's father helps him with money and neighbors help him with cleaning his house. In this case, he reconstitutes this person's situation by transforming the "strictly-legal solution" into a more personal-negotiated solution. Yet for his Rabbi father, it brings up doubts concerning his own beliefs. He felt it odd for a Jew to have only one bed in his house. Whether someone is a pious Jews is not necessarily determined by how many beds he or she might have. 38 Prajñā Vihāra On another day, an old woman comes to the Rabbi to get a divorce from her husband, not because she no longer wants to live with him, but because she loves him so much, she feels he is entitled to a new, younger wife who might bear him a child. She has already chosen her successor for him. The Rabbi facilitated their divorce and the man re-married this new wife. Not long after the wedding, the new wife did not give birth, the old man fell ill and passed away and the old wife passed away. The Rabbi questions himself, why did he facilitate their divorce? The best a man can do is to negotiate and respond creatively to life, even though the outcome is far from what we expect. Once again, he recognizes painfully that this is life. This story shows that the uncertainties of life are a lesson for everyone involved, for the old man and women, and for the rabbi himself. Another woman comes to him asking why two slaughtered geese "shriek" when they are being carried in her shopping-basket, and whether such possessed geese could possibly be kosher. While Singer's mother hints at the need for logic in encountering this seemingly mysterious and wondrous event. Her approach to religious faith requires explanation. That blind faith can be dangerous and needs to be deepened by doubt. Singer's father comments on his mother's way of "logic tearing down faith, mocking it, holding it up to ridicule and scorn." This suggests the unavoidable paradoxes in life and the importance of conflict in the development of faith and human decision making. In all those stories Singer provides an insight into life far beyond legalistic positions, determined solutions, and flat situations. They are always dynamic, unpredictable, while at the same astonishingly unavoidable. There is something about such literature that gives us space for an imaginative understanding of life in that way, and it seems that the wayang puppet theater does something similar: i.e., creating the inner space for transforming the ways that people encounter religion-not as something outwardly official or authoritative, but as something playful. By "playful" here I mean entering into a different realm of possibility, of different Andi Herawati and Andi Rachmawati Syarif 39 interpretations and representations. It allows us to play out situations across the usual limitations of dogma and social determination. It is a way for individuals and communities to express and engage themselves more freely and creatively. The use of language is particularly important in the shadow wayang theater, as it is supposed to be effectively understood by the audience, who are usually mostly Javanese. So the language used in the shadow wayang in the past was usually the ancient local kawi language. But even today, contemporary wayang puppeteers still use Javanese slang, characterized by a direct, critical and symbolic language that readily lends itself to the elaboration of myth, ritual, social interaction, and other aspects of daily life. While wayang theater represents the beliefs and the philosophy of the Javanese which fulfill the need for spiritual inquiry and meaning, it is also like Singer's book in that the characters and the plot are always in a dynamic situation. The richness of the characters in both Singer's book and the shadow wayang genres and plays allows the creative dalang (puppeteer) of the latter to re-create a world that simultaneously questions and brings to deeper life religion's meaning and teachings that people otherwise very often just take for granted, as something simply "given" in their society. In his role as artist and priest (like that of Singer's father role), a dalang is expected to introduce and reinforce the traditionally accepted social and philosophical concepts. Yet the creativity of the dalang allows those who are illiterate to become acquainted with these ideas. That is why no matter how critical the dalang and his performance is, ordinary people may get the lesson immediately. And that lesson itself goes far beyond social and current events, as the shared philosophical foundation of wayang is one of purification and edification. For the representation of the popular shadow-puppet Ramayana and Mahabharata-inspired figures-like Yushistira, Durna, Sengkuni, and local Javanese figures like Semar, Nala Gareng, Petruk, and Bagong-as a unit actually represent the distinctive characteristics of an ideal Muslim personality. The stories they represent, drawn from 40 Prajñā Vihāra Islamic spirituality and Javanese philosophical teachings, make these plays vehicles of social criticism, including highlighting the limitations of formal or "official" religion. The humorous or ironic dialogue between these striking characters is necessarily what we can find in the stories in Singer's book which have a similar critical stance in regard to his own father's relative "orthodoxy". Like the comic characters who live patiently in wayang and Javanese legend, the legends and myths that live in society are often latent (not fully conscious or codified) regulatory systems that continue to control the empirical behavior of the members of the community concerned. In the wayang tradition, the clowns' servants8 often appear as moralistic agents who offer useful suggestions to their kings in times of misery or pressure. For example, Semar is not merely a clown-servant (abdi) puppet, but he is also a legendary, semi-divine figure in the world of puppets, as well as a legend and political myth. Semar is known as a character who saves the source of leadership, who is charismatic and rational-yet in his physical appearance he looks humble, and outwardly he does not reflect power. Semar once advices Pandawa that men should not just think about what they will eat every day in seven days, and that as the leader, Pandawa should show humility which is represented by "luwe" (hunger) and "a simple life" to his people. Thus, in order to protect the kingdom and to have the power to resist, the leader should appoint a good advisor, deputy and or officer (patih), since there is no power of the leader (Ratu) without people, and the responsibility of the Ratu is to serve the will of people. The Javanese mystical-philosophical text, "Suluk Wujil", is an old Javanese text written by one of Javanese saints, Sunan Bonang. It tells of the spiritual journey of a clown-servant called Wujil, the disciple of Wahdat, to find his master. After several years performing ritual, Wujil complains to his master that he has not yet gained any mystical experiences. One of the main subjects of the dialogue in the play revolves around the importance of "intention" or "purified will", of true prayer and the importance of self-knowledge. There is always an explanation Andi Herawati and Andi Rachmawati Syarif 41 for something (like Singer's mother's response to the shrieking dead geese). Praying by itself will not give anything unless Wujil realizes it with self-knowledge and good intentions. Wahdat's long responses to Wujil's inquiry, covers his life story in a very philosophical and symbolic way. For instance, he says: "The turban was used to hit and beat inside the mosque after they were angry with each other, and prayed individually. That is the result of shirk (error), because they assume that each person's intelligence is the most important. As a result, the person does not understand pure will or sincere intentions". 9 He continues: "This problem is very difficult. People may not hold fast to the letters (written text), for the existence of text is contingent upon the existence of understanding (ideas, guesses). And there is no single understanding, but they are many, which leads to error, since there are many people who deify their understanding. People already feel happy merely by reciting the Qur'an, (and other religious text), yet that is only a whisper of understanding". 10 Here, Wahdat presents anti-scripturalism or anti-literalism. Blind adherence to the mere letters of scripture is a form of idolatry. Now the symbols of wayang become meaningful in terms of the complex hermeneutics of mystical Islam. And of course, this ideal can be discovered, and has been communicated, through the encounter of religious texts with the deeper philosophy of life of Javanese society. 42 Prajñā Vihāra The reflection of a message of religious/Islamic education appears in the narrative of Sengkuni and Durna:11 Sengkuni who is a royal officer of Kurawa, once worried about Kurawa's future, and he is advised by Durna that a man should not be discouraged. Instead she/he should be resigned and submissive to God, needing His encouragement and relief. Although the public knows that Durna and Sengkuni are antagonistic figures, in puppetry they basically never justify that a person is simply good or bad, but invite the puppet lovers to see a situation from another perspective. The message (either to Muslim or non-Muslim) is that as a creature, human beings have to continue to fight and not despair, because God always gives a beauty and responds according to the condition of his servant. In addition to the conversation between Durna and Sengkuni12, another example that describes wayang language as a medium of Islamic education is Wibisana's speech to the Pandavas. Here is a fragment of a Wibisana conversation. In this example, Wibisana is giving advice to the Pandavas to live a good life. "There are seven ways to live a good life: beciking kelakuan, akehinng kepinteran, pakolehing kagunan, sugih, ngudi singgih, buntasing sesurupan, and landheping panggahita", which means being good, intelligent, useful, rich, respectable, polite, and insightful-all of which are the forms of positive action taught in Islam. The language in wayang is very rich with comic expressions of facts and allusion or satire and jokes and humor. Along with religious, ethical and political content, there are also stories of love scandals, jokes and social criticism in wayang theatre. It, for example, appears in the dialogue between comic/clown puppets, Kenyot and Tonglang about their marriage.13 Tonglang told Kenyot that he has problem with his wife. He says, "When I give her more money, she serves me only rice. Less money no food at all, while she selfishly eats out alone. Had I known this hardship I would never have sought a wife. She just gives me a blink-of-the-eye orgasm at the expense of yearlong stressfulness." Kentot advises him that if he is no longer pleased with her, he could divorce her. Tongleng replies that he is afraid of her mother who may bewitch him, and that he might Andi Herawati and Andi Rachmawati Syarif 43 fear that bad things would happen to her after he divorced her. Kentot then simply responds, "well, divorce her and I will take her widower?." What role can strict legal rules play in a story such as this? Singer's awareness of how many-sided people are, and how complex the human condition is, does not undermine an understanding of what it means to be Jewish. Rather, through his stories, his readers can learn the many sides of religion and what it can mean to be Jewish. Likewise, with the shadow-puppets, the figures of the dalang (author/ performer) and the lakon (the characters played) are always dynamic, keeping with the changing times and issues in the midst of society, while also encouraging their audience to discover how truly complex life is, beyond just simply being a Muslim, and to recognize that we cannot avoid those conflicts that make up the very journey of life itself. Thus wayang isnot just a form of expressive creativity, it is also a collection of life stories. Here I can quote the dalang Ki Prabowo who says that the dalang should have immense experience in his life in order to be creative and in order to project the stories to the audiences. Because wayang stories are basically a portrait of ourselves as human beings. Thus the message of wayang is not just for Javanese. or what it means to be a Javanese or a Muslim, but it has relevance to people all over the world. The messages that the dalang through the wayang wants to share is that the basic values of truth and right living are really the same everywhere. Evil in all the world is manifested in the greedy and power-hungry, and that the evil will always be defeated by high ideals and virtue. In its development, wayang has been transformed into a strategic way to negotiate religion and social life and very often serves today as a public medium in addressing social and political issues, moral issues and likely critics of the country, society, government. It is even used to comment on and discuss such governmental programs as natural disaster mitigation. It cannot be denied that wayang shows have an identity or character in their own forms, and that they continue to be transformed as they are used to challenge new dilemmas. In response to the quest for the 44 Prajñā Vihāra present and in response to each person's own life situation, that is aspect of its active and highly effective creativity. Religion is often considered as a form of law based upon scripture and the enforcement of its rules. But both Singer's book and wayang show religious life is communal, and that it therefore requires voluntary communal consent and conscience. Thus it always involves negotiation, and since it is a living reality, it vibrates with life and it has both life's inconstancy and its persistence, like in the beth-din stories, farther than what just strict, formal rules can offer. It is in light of this realization that Singer wishes in his introduction to "In My Father's Court" that the "Beth-Din" can be a universal institution. Beyond being a place for small community of Jewish faithful, Singer intimates, it could become a place all humans can learn from: i.e., learning how God's Mercy and Judgment can be manifested in every situation, so that the Beth-din might become the place to learn about God's justice and Mercy. What wayang performance can offer back to Singer's stories is that in projecting this complex life situation, beyond being a Jew or Muslim, and beyond what the religious text says, life always requires a full, deepened understanding of God's Mercy (Loving kindness, hesed and rahmat) that most of the time precedes the law and the rule, and judgment. While both Judaism and Islam very often are seen as religions which emphasize religious law, then through these two forms of religious representation, the "law" might be understood more richly as how human beings come understand God's Mercy. Or as the Qur'an says: "God's Mercy has precedence over His wrath." This is the famous understanding of God's Mercy and law that has been long discussed, especially in Sufism. Finally, what I find most similar between these two creative approaches is that the synagogue can be like the wayang theater or stage, the shadows are like Singer's father (since it is through him that the stories happen), and the director is Singer as the (Polish Jewish) dalang. This theater is the "imagination" of the world, its past, present and future. And beyond this range of times, all the figures are like puppets before the Director, just as Islamic spirituality through the hadith tells us that Andi Herawati and Andi Rachmawati Syarif 45 the hearts of human beings are shifting between two God's fingers. The human individual, the human heart here is God's puppet, while the world is the play of God's imagination. ENDNOTES 1. Peter Beyer, Religions in global society (New York: Routledge, 2006) 2. Ibid, p.4. 3. For preliminary study on Religion and Creativity, see Kim-lien Thi Nguyen, "an Exploratory Study on the Relationship between Creativity, Religion, and Religiosity" (2012). Master's Theses published by San Jose State University. Online source: http:// scholarworks.sjsu.edu/etd_theses/4204. 4. Petra Kuppinger, (2017), "Religion, art and creativity in the global city", in Culture and Religion, published by Routledge, Vol. 18, No. 4, 343–352, p 433. https://doi.org/10.1080/14755610.2017.1402419 5. Nicholas Tarling, "Religion and Popular Beliefs". in The Cambridge History of Southeast Asia, ed. Jerome Chʻen and Nicholas Tarling, Vol. I. 1970. Cambridge: Cambridge University Press, p. 331; R.M. Ismunandar, Wayang: Asal Usul dan Jenisnya. 1994. Semarang: Dahara, p. 96. 6. Jan Mrázek (1999), "Javanese Wayang Kulit in the Times of Comedy: Clown Scenes, Innovation, and the Performance's Being in the Present World". Part One: Indonesia, No. 68, pp. 38-128. 7. Cf Ferdi Efendi, "Learning Islam from the Performance of Wayang Kulit (shadow puppets)", published by Hunafa: Jurnal Studia Islamika, Vol. 14, No.1 Juni 2017: 99-115. 8. In wayang mythology, the characters of comic clowns are not merely clowns; instead, in the macrocosmos, they are often identified with aspects of the Highest Being or Divine. 9. Suluk Wujil (101) Kepet kinepetaken ing masjis// awekasan padha pepurikan// asembahhyanng dhewek-dhewek// puniku palanipun//sirik gugon uajring tulis//tan wruhjatining niyat//palaning wong bingung//lanang wadon padha ngrarh//angulati niyat kang sejati-sejati//tan wruh ing dedalannya. 10. Suluk Wujil (102) Mapan angeling ujar puniku//nora kena ngukuhi aksara// kang aksara kadadine//dadining nyana iku//nyana nora among sawiji// nyana awarnawarna// dadine kapahung// akeh anyembah ing nyana//paksa hresthi sarira bisa angaji// ujare nyananira. 11. Arifin, "Learning Islam", p.104. 12. Ibid, p.106 46 Prajñā Vihāra 13. I Nyoman Sedara, Kawi Dalang : Creativity in Dalang Theater (2002), a Thesis published by University of Georgia, p. 105. Online sources : https://athenaeum. libs.uga.edu/handle/10724/29419; p. 105. BIBLIOGRAPHY Arifin, Ferdi (2017), "Learning Islam from the Performance of Wayang Kulit (Shadow Puppets)", Hunafa: Jurnal Studia Islamika, Vol. 14 (No. 1). pp. 99-115. Chʻen, Jerome and Nicholas Tarling, ed. 1970. The Cambridge History of Southeast Asia, Vol. 1.Cambridge: Cambridge University Press. Ismunandar, R.M. 1994. Wayang: Asal Usul dan Jenisnya. Semarang: Dahara. Peter Beyer, Peter. 2006. Religions in global society. New York: Routledge. Mrázek, Jan., (1999), "Javanese Wayang Kulit in the Times of Comedy: Clown Scenes, Innovation, and the Performance's Being in the Present World". Part One: Indonesia, No. 68, pp. 38-128. Kuppinger, Petra (2017), "Religion, art and creativity in the global city", in Culture and Religion, published by Routledge, Vol. 18, No. 4, 343–352, p 433. https://doi.org/10.1080/14755610.2017.1402419 Masroer (2015), "Spiritualitas Islam dalam Budaya Wayang Kulit Masyarakat Jawa dan Sunda, Sosiologi Agama, Vol. 9 (No. 1). pp. 38-61. Nguyen, Kim-lien Thi, an Exploratory Study on the Relationship between Creativity, Religion, and Religiosity" (2012). Master's Theses published by San Jose State University. Online source: http:// scholarworks.sjsu.edu/etd_theses/4204 Singer, Isaac Bashevis. 2001. In My Father's Court, Poland: Vintage. Andi Herawati and Andi Rachmawati Syarif 47 Sedara, I Nyoman, Kawi Dalang : Creativity in Dalang Theater (2002), a Thesis published by University of Georgia. Online sources : https://athenaeum.libs.uga.edu/handle/10724/29419 Tarling, Nicholas .1970, "Religion and Popular Belie The Cambridge History of Southeast Asia, ed. Cambridge: Cambridge University Press. Website: http://purl.galileo.usg.edu/uga_etd/sedana_i_nyoman_200205_phd http://achmad-suchaimi-sememi.blogspot.com/2017/05/naskah-sulukwujil-dan-terjemahnya.html

omics of Higher Education in the 21st Century J.-M. Kuczynski Table of Contents The Economics of Higher Education in the 21st Century: Part I Introduction to Part I Retention-rates and Graduation-rates: Introductory Remarks Do Graduationand Retention-rates Matter? Open-market Companies (OMC's): How they Differ from Universities Colleges Sell Degrees, not Instruction Enrollment-rates Matter: Retentionand Graduation-rates Don't Universities: Their Distinctive Business-model How Useful Do Students Really Want Their Educations To Be? The Paradox of Macroeconomic Efficiency When in Doubt, Drop Out: Dispelling the Myth that Only Losers Drop Out Knowledge-management: Introductory Remarks Why Knowledge-management Cannot be of Assistance to Universities Key Points Conclusion of Part 1: How to Maximize Revenue by Optimizing Education The Economics of Higher Education in the 21st Century Part II Introduction to Part II What is Knowledge-management (KM)? An Actual Company that Could Benefit from KM Who Exactly Sounds the KM-alarm? KM as Preventative Measure KM in Relation to Organizations that Sell Results KM in Relation to Organizations that Sell Services Service-organizations Depend on Poor KM KM Impossible Unless Employment and Pay are Performance-dependent No KM without Financial Transparency Expense-padding in Relation to KM A Corollary: Education Must be Digitized Whenever Possible DMO: A New and Better Kind of University All Accreditations Examination-based How DMO Turns Non-STEM Students into STEM Students Emphasis on Instruction as Opposed to Selection More Degree-levels at DMO Instructors Never to Function as Gatekeepers Payments to Go Straight from Student to Instructor The Inverted Payment Pyramid 25 Desiderata that DMO Must Satisfy Conclusion of Part 2 Conclusion of the Present Work The Economics of Higher Education in the 21st Century Part I Introduction to Part I In the first part of this two-part work, the economics of higher education are explained. It is made clear how a university's business model differs from that of a company that has to compete on the open market, and on this basis it is explained: (i) Why universities are in no way threatened by low retention rates and graduation rates; (ii) Why universities cannot significantly improve or otherwise alter the quality of their educational services without imperiling their very existences; (iii) Why universities do not have to improve the quality of their educational services; (iv) Why universities couldn't improve the quality of their services even if they wanted to; (v) Why the fact that many universities have low retentionand graduationrates does not a represent a business opportunity, or opportunity of any other kind, for anyone, whether inside or outside of academia; and (vi) Why principles of Knowledge Management (KM) that are so useful when it comes to helping businesses that compete on the open market are completely useless, and indeed of negative utility, when it comes to helping universities solve their problems. In the second part of this work, it is explained how to construct an online university that is both lucrative and provides instruction that is faster, better, cheaper, and more useful than the instruction provided by any existing (or possible) brick-andmortar university. It is also explained how the principles of KM can be used to optimize such a university, once it is up and running. Retention-rates and Graduation-rates: Introductory Remarks A university's retention rate is the percentage of students who stay after the first year. A university's graduation rate is the percentage of students who graduate within six years of enrollment. These numbers never coincide, but they track each other. Universities with high retention rates have high graduation rates, and universities with low retention rates have low graduation rates. Retention rates are necessarily at least as high as graduation rates and in practice are always higher. Also, even if a university has a graduation rate of 0%, as is the case with a number of institutions, it doesn't follow that nobody graduates from it, only that nobody does so within six years of enrollment. Do Graduationand Retention-rates Matter? From a financial perspective, a university has relatively cogent reasons to want a high retention rate: when a student doesn't enroll for a second year, that university loses business, and it doesn't when he does. A university has less strong reasons to want a high graduation rate. If a student comes back year after year and never graduates, the university makes more money than it does if he graduates. All of this said, universities with low graduation and retention rates almost always tend to stay in business. Great Basin College in Nevada has a retention rate of 4% and a graduation rate of 0%, but it is not in financial trouble.

Aesthetic Disobedience Jonathan A. Neufeld (penultimate draft of an article forthcoming in Journal of Aesthetics and Art Criticism) 1. Introduction: Why "Aesthetic Disobedience"? It this paper I explore a concept of artistic transgression that I call aesthetic disobedience. By using the term "aesthetic disobedience," I mean to draw a parallel with the political concept of civil disobedience. Acts of civil disobedience break some law in order publicly to draw attention to, and recommend the reform of, a conflict between the commitments of the legal system and some shared commitments of a community. Acts of aesthetic disobedience do the same in the artworld: they break an entrenched artworld norm in order publicly to draw attention to, and recommend the reform of, a conflict between artworld commitments and some shared commitments of a community. I argue that considering artistic transgressions under the concept of aesthetic disobedience highlights features of modern artworld practices that are often overlooked. Most significantly, it draws attention to the ways in which a wide variety of citizens of the artworld, including not just artists and performers but also members of the audience, can deliberatively participate in the transformation of the rules and boundaries of the artworld itself. It is almost axiomatic that breaking rules is an important engine of creativity and innovation in the modern artworld. To describe the beginnings of great art movements by pointing to transformative moments where the rules were fruitfully broken is a commonplace. Violations of the rules of harmony and resolution in tonal music by, for example, Wagner, Debussy, and Schoenberg; or the violations of established rules of perspective and representation by Manet, Cézanne, and Kandinsky have been at the center of how the history of European art 2 has been told since the 19th century. The use of political terms to describe transgressive elements of the artworld is also nothing new. One of the most common political terms, "revolution" and its cognates, is used to describe art movements, works, individual artists, or formal innovations. Uses of "revolution" range from the straightforwardly political (Andre Breton's and Diego Rivera's Manifesto for an Independent and Revolutionary Art, or Richard Wagner's Art and Revolution) to a mixture of political and artistic descriptions (the uses of "revolutionary" that were deliberately linked to "musical Bolshevism" that upset Arnold Schoenberg, for example) to the rather ordinary art-historical descriptions of, say, E. H. Gombrich who routinely uses "revolutionary" to describe stylistic and formal innovations across historical periods.1 In contemporary usage, the term is so common as to have taken on the air of cliché. A quote attributed to Paul Gauguin, "In art one is either a plagiarist or a revolutionary"2 (usually edited to the pithier, "Art is either plagiarism or revolution"), is a slogan printed on a T-shirt. It is worth remembering that, in political philosophy, "revolution" is reserved for movements that aim to overturn the existing legal order and replace it with an entirely new one.3 The application of the distinction between revolution and other forms of dissidence in practice proves to be tricky and, as we will see below, is often contested. Nevertheless, there is a clear difference, and it is important to maintain the distinction. If we take the standard for revolution at all seriously-that is, if we take it that revolution replaces a normative order with another one- then it is rarely the case that the aims of artists really are revolutionary. Schoenberg and Stravinsky, in distancing their music from "revolution," were at pains to argue just this (Schoenberg replacing "revolution" with the non-political and less radical "evolution," for example).4 Moreover, it seems to me that the term is insufficiently sensitive to capture the distinctive character of a number of rule-breaking artistic practices. 3 "Aesthetic disobedience" better sheds light on much of what is interesting in certain transgressive actions in artistic practices. It does this in part because it does not so easily lose touch with the political correlate, "civil disobedience," that motivates it and gives it sense. Retaining a connection to the structure of civil disobedience reveals an often overlooked characteristic of some of the most interesting acts of artistic transgression: the public and deliberative backdrop against which they occur and which they aim to shape. In Section 2, I set out the characteristics of civil disobedience. Roughly, an act of civil disobedience is a public communicative act that breaks a law in order to draw attention to and reform perceived conflicts between law and other shared normative commitments. In Section 3, I begin to illustrate the parallel characteristics of the concept of aesthetic disobedience with an example: Peter Handke's Sprechstück ("speak-in"), Publikumsbeschimpfung (translated as Offending the Audience). Again roughly, an act of aesthetic disobedience is characterized as a public communicative act that breaks an artworld norm in order to draw attention to and to reform perceived conflicts between an entrenched norm of the artworld and other, broadly speaking, aesthetic commitments.5 The important question of who can engage in acts of aesthetic disobedience is addressed in Section 4. I argue that aesthetic disobedience is not limited to artists and performers. Rather, audiences are also capable of acts aimed at reforming entrenched norms of the artworld. This is a shift from the usual uses of "revolutionary" that focus almost exclusively on transgressive acts of artists. In light of the diverse possibilities for participation suggested in Section 4, Section 5 raises the question of what should count as an artworld norm that could be the target of aesthetic disobedience. Looking to a limit case testing the boundary between aesthetic disobedience and revolution, I show that Tania Bruguera's participatory artwork Immigrant Movement International targets norms distinguishing the artworld from ordinary social and political 4 practice. This shows that what norms might count as candidates for aesthetic disobedience must ultimately remain open. 2. Civil Disobedience Civil disobedience is familiar from political and legal philosophy.6 John Rawls's influential definition states that an act of civil disobedience is a "public, nonviolent, conscientious yet political act contrary to law usually one with the aim of bringing about a change in the law or policies of the government... intended to addresses the public's sense of justice... within the limits of fidelity to law."7 Every one of these characteristics has been criticized in various ways. Rawls's conception of "public" includes notification of the authorities in advance of the disobedient action, along with acceptance of punishment for violating the law, for example. This would rule out, say, Pussy Riot's performance of a "punk prayer" in Moscow's Cathedral of Christ, or intersection blocking since these actions depend on sudden interruption and would be prevented by the authorities were advance notice given. The "non-violence" requirement is also stickier than Rawls makes it out to be. Not only is the definition of violence notoriously difficult to specify but violence might be seen as appropriate in the face of particularly abhorrent laws. Finally, it is not clear what, exactly, "within the limits of fidelity to law" means.8 While any discussion of the details of civil disobedience rightly becomes tangled in a web of complication, we can usefully give a broad-stroke sketch of its core features to establish a working definition. Acts of civil disobedience break some law in order publicly to draw attention to, and recommend the reform of, a conflict between the commitments of the legal system and some shared commitments of a community. This account calls attention to five central characteristics of acts of civil disobedience: 5 CD1. The acts violate the law CD2. Civil disobedients accept the risk of legal punishment for their actions9 CD3. The acts are performed publicly-they are communicative CD4. The acts aim to draw attention to a conflict, or a set of conflicts, between normative and legal commitments or authority CD5. They aim to promote a change within the legal system. I take these to be necessary conditions of civil disobedience that distinguish it from ordinary protest (which need not break any laws), from ordinary law-breakings (which need neither to happen in public, to draw attention to any deeper normative commitments, nor to be aimed at promoting change), and from revolutionary acts (which aim to overthrow a particular institutionalization of norms altogether rather than promoting specific changes within the law). In classic cases of civil disobedience from the civil rights movement in the United States, particular laws were broken to call attention to the way that laws violated a deeper shared commitment of right or wrong. This shared commitment could be a morality, a conception of rights, liberty, fairness, justice, equality, and so on. CD3's publicity requirement is broader and different in kind than Rawls's publicity requirement-it simply aims to situate civil disobedience within the context of political deliberation. Laws are broken in order to communicate to other citizens reasons to change the law. The communication of reasons in a public sphere is not simply a one-way affair. Civil disobedients, in presenting reasons in a public and deliberative context, open themselves to countervailing reasons offered by their fellow citizens. In short, while civil disobedience is extraordinary in that it violates laws, is contestatory, and is confrontational; a democratic ethos of deliberation and communication forms part of its foundation.10 This very general account of the connection of the publicity requirement with deliberation should be enough to motivate the 6 arguments that follow. I leave the precise structure of deliberation vague in the hope of avoiding controversies within and between various accounts of deliberative and radical democracy, and what counts as violence or the "the limits of fidelity to law." So far so good on civil disobedience, I hope. But why aesthetic disobedience? Before continuing to think abstractly about the parallel between the concepts of civil disobedience and aesthetic disobedience, I would like to set out the first of several examples that will both help motivate the need for a concept of aesthetic disobedience and put us in better position to start sketching some of its key characteristics. 3. Aesthetic Disobedience: Peter Handke's Offending the Audience In Offending the Audience (Publikumbeschimpfung),11 a work for theater by Peter Handke, the performers speak directly to the audience, about the audience, about what the performers are doing, and about theater in general. The performers claim that there will be no play, that the audience is the subject of the work, that the audience members are the objects of attention. Handke calls the work a Sprechstück, which has been translated into English as "speak-in", echoing "sit-in", to capture their quality of performative protest. The speak-in culminates with a series of critical assessments of the audience juxtaposing cliché evaluations with blunt insults. "You were the right ones. You were breathtaking. You did not disappoint our wildest hopes. You were born actors. Play-acting was in your blood, you butchers, you buggers, you bullshitters, you bullies, you rabbits, you fuck-offs, you farts."12 Handke's description of his aim resonates with the characteristics of civil disobedience where, rather than laws, the target of disobedience are artistic and aesthetic norms of the theater. 7 The idea was to have the spectators in the orchestra thrown back upon themselves. What mattered to me was making them feel like going to the theatre more, making them see all plays more consciously and with a different consciousness. My theatrical plan is to have the audience always look upon my play as a means of testing other plays. I first intended to write an essay, a pamphlet, against the theatre, but then I realized that a paperback isn't an effective way to publish an anti-theatre statement. And so the outcome was, paradoxically, doing something onstage against the stage, using the theatre to protest against the theatre of the moment-I don't mean theatre as such, the Absolute, I mean theatre as a historical phenomenon, as it is to this day.13 Note that, while Publikumsbeschimpfung is a work of "anti-theatre," it is not a revolutionary work that either breaks all norms of the theater or recommends throwing out all norms of the theater. Handke's stage directions insist, "The usual theatre atmosphere should prevail... The concept of what is sartorially inappropriate should be strictly applied."14 The norms of decorum, uniformity, silence, passivity and (in Handke's eyes) apathy were to be made vivid before being challenged. The work aims to bring the audience to think and act differently about the theater, to get the audience to think and act critically about their role in the theater. A transformation of theater practice demanded an extraordinary, practical, and theatrical intervention. Only public performative engagement with, and violation of, the norms of theater would sufficiently illustrate the problem and deliberatively engage the theater-going public. At the 1966 German premiere in Frankfurt, the audience took the call to act critically in the theater quite seriously. They clapped, talked back to the performers, heckled, laughed and booed. During the performance on the second night, several audience members responded directly to the dialogue that was accusing them of being apathetic. While calling out and arguing with the actors, the (scripted) dialogue continued, "Standing, you would be more effective hecklers."15 The hecklers not only stood, but eventually walked onto the stage to join the performers and disrupted the performance. When suggestions from the actors and from Claus Peymann, the director, that they leave, were ignored, Peymann actually pushed them off the stage. The 8 exchange draws attention to the norms of the theater that the performers of Publikumbeschimpfung still clearly took to be in force and actually helped physically to enforce. In the first place, the actors stuck very closely to the written text, responding to hecklers with lines from the script in such a way that made it seem spontaneous. In the second place, the director enforced the rule that the participation of audience members is not to include their physically interfering with the performers on stage. So, Publikumsbeschimpfung publicly breaks the norms of the theater while drawing attention to the conflict between those entrenched, institutionalized norms and broader shared commitments of the participants of theatrical practice in order to promote a change in practice. We are now in a position to sketch an account of the characteristics of aesthetic disobedience that runs parallel to the sketch of civil disobedience from above: AD1. Acts of aesthetic disobedience violate a deeply entrenched artworld norm, or a set of norms. AD2. Aesthetic disobedients accept the risk of sanction for their actions. AD3. Acts of aesthetic disobedience are performed publicly-they are communicative AD4. Acts of aesthetic disobedience aim to draw attention to a conflict between normative commitments and entrenched norms of the artworld. AD5. Acts of aesthetic disobedience aim to promote a change within the entrenched norms artworld. Note first that while nothing in these necessary conditions for aesthetic disobedience entails a particular conception of art, they have a deep, and I think salutary, effect on historical and institutional theories by adding a deliberative dimension to the norms structuring artworlds. In particular, aesthetic disobedience calls attention to the ways in which moments of institutional transformation, along with the conferral of institutional authority can be, and often are reflective and deliberative. Where the institutional theory of art often assumes the existence and enforcement of a particular set of norms, and assumes that structures of authority are in place to 9 sanction innovation, taking the possibility of aesthetic disobedience seriously reveals the possibility of a more active and contestatory role open to citizens of the artworld.16 These characteristics also distinguish acts of aesthetic disobedience from ordinary artistic innovation. For example, while Béla Bartók called for a number of innovative sounds to be drawn from stringed instruments (say, the "Bartók pizzicato" where the plucked string is snapped back onto the fingerboard so the tone is accompanied by a cracking sound), he did not violate the norms of string playing-making innovative sounds and timbres is part of the stock and trade of composers. The reader can easily multiply examples in all of the arts. At the other end of a spectrum of rule breaking, these characteristics also distinguish acts of aesthetic disobedience from revolutionary acts that do not promote change within entrenched artworld norms but, rather, aim to overthrow the artworld and replace it with another. As I mentioned above, distinguishing cases of revolution from cases of aesthetic disobedience is tricky, just as it is in the political realm.17 Cage's 4'33" can be fruitfully understood as a revolutionary work. Whether it itself counts as music, it aims to completely restructure the way we understand, experience, perform, and compose music. The fact that it is a revolutionary violation of norms might play some part in an explanation of why there is a debate over its status as music. One question that immediately arises concerns the nature of AD1's "deeply entrenched artworld norm" that is being broken in cases of aesthetic disobedience. In the case of civil disobedience, breaking a law is a fairly straightforward matter. Laws, especially in modern bureaucratic states, are relatively clearly (though not perfectly clearly) codified and institutionalized in a way that artworld norms are not. This is not a fatal worry, however. It is not part of the concept of law that laws be perfectly clearly codified or written down-unwritten elements of the common law are no less law for not being formally codified, for example. So 10 acts of civil disobedience do not depend on the ease of the identification of law. Still, one might rightly note that, whatever difficulty attends the identification, the norms of the artworld are not identified in the same way that laws are. The laws of the state are identified with reference to the authoritative sources of those laws.18 The sources of artworld norms are more varied and their authority is less formal than the sources of law and the sanctions for violating artworld norms are not nearly as weighty as the sanctions possible for violating the law. Nevertheless, there are clear cases of norms that have the entrenched status of law, and whose violation provoke relatively clear and significant reactions from various authorities in the artworld (critics, art institutions, academies, and other artists, for example). Examples of violations of formal artistic norms might include Schoenberg's and Stravinsky's breaking of the norms of tonality or Duchamp's or Warhol's breaking the norms of the kind of object that can count as a work of art. A widespread network of practices and institutions contribute to the entrenchment of the norms and their sanctions, which are risked by artists and meted out by critics, gallerists, museum directors, granting agencies, and a variety of educational institutions. This leads us a question that might arise with regard to AD2: what sanctions do aesthetic disobedients risk facing when they violate an entrenched norm? The sanctions for violating formal norms range from widespread negative critical reviews, denial of reviews altogether, or the loss or denial of exhibition or performance opportunities. In some cases, sanctions for violations of artworld norms can be every bit as weighty as the violation of laws. One need only think of the treatment of avant-garde art in totalitarian regimes in the middle of the 20th century to have a number of particularly vivid examples. At this point, one might worry that any violation of a norm counts as aesthetic disobedience. For example, one might wonder whether I am just pointing to what Kendall 11 Walton would call contra-standard properties that "have a tendency to disqualify a work from a category in which we nevertheless perceive it."19 Works exhibiting contra-standard properties are the bread and butter of artistic innovation and capture a far broader spectrum of artistic innovation than aesthetic disobedience does. What is the difference? It may well be that all artistic acts of aesthetic disobedience produce works with contra-standard properties. But, as we will see shortly, not all acts of aesthetic disobedience are artistic acts. But even among artistic acts, there is no requirement that works exhibiting contra-standard properties either aim to draw attention to a conflict between normative commitments and entrenched norms of the artworld (AD4) or that they promote a change within artworld norms (AD5). Works exhibiting contrastandard properties might simply break the rules without any aim to reform or call attention to the structure of artworld norms themselves. Even if they do aim to reform or call attention to norms (as Walton says they often do), they may do so either in an aesthetically disobedient or in a revolutionary manner. Even if one were to grant everything I have argued so far and admit that such deeply entrenched formal norms parallel to laws are possible, one might still think that they are a thing of the past in the contemporary anything-goes artworld. Aesthetic disobedience might do nothing more than open an explanatory space for philosophers, critics and historians as they talk about certain artistic acts and works that occurred before the end of art, to borrow a phrase from Danto.20 I would respond that the concept of aesthetic disobedience is of more than mere historical interest. The reason for this, while first appearing to be quite simple, has far-reaching ramifications: not every norm relevant to the creative movement of the artworld is a formal norm. Street art can serve as a clear example of what I am after here. The transgression that one might think central to street art is not its violation of formal norms of visual art, but the violation 12 of the norms of where, how, by whom, and for whom art is displayed. Nick Riggle argues that something is street art if and only if "its material use of the street is internal to its meaning."21 As a result, "Street art is deeply antithetical to the artworld. That is, for each part of the artworld, street art resists to some appreciable extent playing a role in it,"22 insofar as its material use of the street prevents attempts to bring it into galleries and museums as well as attempts to make it marketable and sellable. The norms broken here are not formal norms of visual artworks but a variety of institutional norms of the artworld governing the dissemination, display, and even ownership of artworks. When institutional norms are included among the possible targets of aesthetically disobedient acts, the realm of disobedient action is opened to a much wider constituency than we had initially been considering. Typically, when we speak of revolutionaries in art we speak of artists and performers. Broadening our focus, as I think we should, beyond breaking the formal norms of an art form opens the possibility of disobedience to a wider field of artworld participants. The Handke stage-stormers mentioned above show us one important possibility: acts of aesthetic disobedience committed by the audience. 4. Aesthetically Disobedient Audiences There are a number of well-known examples of audience disruption and protest in musical performance-the audience's reaction to the 1861 Parisian premiere of Wagner's Tannhäuser caused it to be pulled after only three performances and the raucous premiere of Stravinsky's Rite of Spring is notorious. I would like to focus on a more recent example. Since 1938, as a response to the atrocities committed by the Nazis, there had been an unofficial ban on 13 the live performance of Wagner in Israel. In 1984, Zubin Mehta attempted to perform the prelude from Tristan und Isolde as an encore at an Israel Philharmonic Orchestra concert. Before performing the encore, Mehta turned to the audience and suggested that those for whom the music was disturbing could leave. There were boos and several people did leave. The ones that remained continued to boo, but ultimately fell silent as the music rose in volume. The second night, however, there was a more concerted protest. The boos were more unrelenting and Mehta was forced to stop the performance. Several versions of the story have a survivor climbing on stage and touching Mehta's arm to stop him.23 The audience members violated two powerful norms that govern the space of classical music concerts. The first is the norm of silence in the concert hall during the performance. By vocally expressing their displeasure about what was being performed, the audience communicated their condemnation of Mehta's choice of encore. Moreover, there was no mistaking what they were objecting to-it was the choice to perform a work by Wagner in Israel. During the first performance, the disobedients submitted to the authority of Mehta's continued performance of the work, aided by the sheer power of the sounding work itself. In the second performance, this authority was defied and the performance was stopped. What was at stake were deep competing commitments within music. On the one hand, Mehta, along with Daniel Barenboim years later,24 clearly believed that Wagner's music itself did not embody the values that the Nazi's used it to support. If he thought it did, he would doubtless refuse to perform Wagner. This commitment to a kind of formal purity or autonomy of music still runs deep in contemporary "classical" musical practice-so deep that Barenboim later referred to it explicitly in his arguments in support of performing Wagner.25 To interrupt a performance for moral and political reasons is to make a statement about the relationship between those commitments and 14 the commitment to music's purity. It either calls this purity into question altogether, or it allows it to have some pro-tanto value that is defeasible by countervailing moral and political values. One need not focus exclusively on political examples, however. Even mundane acts of booing, noisemaking, tomato or turnip throwing, when sufficiently disruptive and aimed at a sufficiently entrenched norm, could count as aesthetic disobedience. For example, in the 2000/2001 season at La Scala, Salvatore Licitra played Manrico in Il Travatore. In two places in the well-known 3rd act cabaletta, "Di quella pira," tenors traditionally interpolate high C's for the written G's below high C in a bravado show of virtuosity. In deference to the score, and against the operatic performance tradition, Ricardo Muti instructed Licitra to follow the score.26 The decision was vigorously catcalled and booed when Licitra sang the less impressive, but actually specified G's. Here again the authority of the conductor and performer is challenged, along with the very strong commitment to obeying the score. The audience was committed to the countervailing commitment to virtuosic display in La Scala. These instances of audience-led aesthetic disobedience publicly and communicatively broke prominent norms of the European musical artworld. They shed critical light on a conflict between normative commitments of the musical public, and it called for a change within the institutionalization those commitments. An objection might be raised here that an audience's disobedience in these cases does not amount to aesthetic disobedience since, unlike the acts of artists, the audience's acts are not themselves aesthetic, or do not produce relevantly aesthetic results. I do not think this is the case, as long as we have a sufficiently nuanced characterization of the act and of performances. The stage-stormers' intervention during the performance of Publikumsbeschimpfung is, I think, a clear case that supports my view. The audience members claimed the mantle of performer, taking the argument of the script very seriously. They performed their understanding of the conclusion 15 of the speak-in: they, too, could be authoritative speakers in the space of the theater. In standing and arguing, and then climbing onto stage, they restructured the space of aesthetic appreciation while drawing attention to and criticizing aspects of that very space.27 In a similar way the noisy and vocal intervention during the performance of instrumental music dramatically reshapes the structure of the performance event. Even if the music continues during the booing (as it did in the case of Mehta's first performance of the Prelude of Tristan, or in the three 1861 Paris performances of Tannhäuser before its cancellation), it is continuing in the face of or in spite of the boos. The boos come to mark the performance and, depending on the effectiveness of the protest, they can come to mark the work and to shape its future performances.28 More important, though, acts of aesthetic disobedience led by audiences, as we see in the case of attempts to perform Wagner in Israel or an even more recent case of the cancellation of Burkhard Kosminski's Nazi-themed Tannhäuser in Düsseldorf, can have a deep impact on artistic practice.29 The aesthetic nature of audience-led disobedience is more vivid in cases in the visual arts where the marks left on the work by protest are often literal. In 2001, Jake and Dinos Chapman bought a rare complete set of prints from Goya's famous and influential Disasters of War series. The artists "rectified" the prints by painting puppies and clown faces over the faces of the victims of war and titled their series 2003 Insult to Injury.30 The defacing of a revered work was condemned by a number critics as nothing but artistically shallow, merely shocking violation, desecration, and vandalism. Upon learning about what the Chapmans had done, but before seeing the works, the art critic Jonathan Jones thought the project was "nasty, insane, deviant."31 The artists claimed that the paintings were not vandalism, and were not simply aimed to shock. Rather they meant to "kick the underbelly" of what they took to be Goya's portrayal of 16 Enlightenment struggle with irrationality. "Because he has a predilection for violence under the aegis of a moral framework. There's so much pleasure in his work."32 In the context of the onset of the Iraq war, the critique of the violence of moralizing has a broader political significance. More important for my purposes here, though, the Chapmans aimed to highlight and lampoon what they take to be the easy, unreflective, and uncritical humanistic moralizing of contemporary museum patrons, audiences, and critics. After Jonathan Jones actually saw the rectified prints, he "fell into [the artists'] trap" and deemed them "nasty, psychotic and value free; not so much a travesty of Goya as an extension of his despair."33 In working both with and against Goya, the Chapmans are at the same time aesthetically disobedient audience and artist. Distinguishing between acts of aesthetic disobedience and acts of mere vandalism is not always easy in practice, as the dispute between critics of Insult to Injury shows. On the one hand, there are clear cases of mere vandalism. For example, the security guard who drew a heart and wrote "Reggie + Crystal, I Love you Tushee Love Buns" on Roy Lichtenstein's Curtains merely vandalized the painting-neither AD4 nor AD5 is satisfied. On the other hand, there are cases that are unproblematically identified as acts of aesthetic disobedience but that are perhaps unjustified. If we take an act to be unjustified, it might tempt us to label an act of aesthetic disobedience as mere vandalism. But this would be a mistake. For example, in February, 2014 at an exhibition of Ai Weiwei's work at the Pérez Art Museum in Miami, Florida. The exhibition included several 2000 year old Han Dynasty vases that Weiwei had dipped in paint. The vases were accompanied by the famous photo triptych of Weiwei dropping and smashing a similar vase. Florida artist Maximo Caminero, to protest what he saw as the museum's failure to support local artists while spending enormous sums on international exhibitions, performed a vase smashing of his own using Weiwei's work. He picked up and dropped one of the painted vases to 17 mimic Weiwei's actions depicted in the triptych.34 Weiwei, the museum, and much of the artworld have condemned the act as mere vandalism, though I take it to be a clear, if perhaps incompetent and unjustified, act of aesthetic disobedience. The question of whether or not an act is justified is distinct from its categorization as aesthetically disobedient. Since it is part of the nature of aesthetically disobedient acts that they violate a deeply held norm and this violation evokes shock and condemnation, it should come as no surprise that the acts are characterized by their opponents as merely transgressive, merely disruptive, merely shocking, or mere vandalism. Nevertheless, even though categorizing individual cases is bound in practice to be contested and to be tangled with the distinct questions of an act's justification, the public commitment to reforming artistic practice-AD3-AD5 taken together-should be sufficient to distinguish vandalism from aesthetic disobedience. In this section we have seen that, while artists often provide us with the clearest cases of aesthetic disobedience when they break deeply held formal norms with the aim of reforming artworld practices, audiences can also engage in acts of aesthetic disobedience. This broadly deliberative and participatory core of aesthetic disobedience draws our attention to the ways that audiences shape our aesthetic and artistic practices in much the same manner as artists do. Whereas the agents of revolution are generally taken to be artists and perhaps artworks, aesthetic disobedience countenances, and even foregrounds, the participation of a broader cross-section of the citizens of the artworld. What distinguishes these audience-led acts of aesthetic disobedience from mere disruption or mere vandalism, which they can closely resemble, is the public commitment to reforming artworld practice. When participation in aesthetic disobedience is opened to such a wide variety of actors the question arises, what counts as an artworld norm that an aesthetic disobedient might target? 18 So far, we have seen that formal norms of artworks and genres might be targeted, along with any number of norms of presentation and reception within a particular artistic practice. Rather than attempting to canvass or categorize all types of artworld norms that might be contested, I will consider what I take to be a limit case of a work that calls the very boundaries of the artworld itself into question. A case of aesthetic disobedience on the border with revolution, the example will ultimately show that the question of which norms might be appropriate targets for aesthetic disobedience must not only remain open, but inevitably remains a matter of contestation. 5. The Limits of Aesthetic Disobedience: Tania Bruguera's Immigrant Movement International Tania Bruguera has worked with Creative Time and the Queens Museum of Art in New York on a long-term work of public art entitled Immigrant Movement International (IMI). IMI's website describes the work as follows: "...[IMI] took on the form of a community center, paying respect to the tradition and victories of U.S. civic movements. [IMI] is an art project implementing the concept of Useful Art, in which artists actively implement the merger of art into society's urgent social, political, and scientific issues."35 The artwork "took the form of a community center" that provides workshops giving legal advice, cooking and urban gardening advice, health classes from a variety of cultures, English through Art History workshops, music, art and dance classes for both children and adults. IMI produces occasional "performances" that mostly advocate "progressive immigration reform," and look very much like, in fact they are often indiscernible from, political demonstrations. Finally, the members of IMI have recently drafted a "Migrant Manifesto" that has been published online.36 19 What artworld norms are challenged here? The answer lies in the most frequently raised question about IMI. Tom Finkelpearl, the director of the Queens Museum, has said that the predictable question, "Why does IMI qualify as art?" is frequently raised to him by donors, board members, critics, newspaper reporters, and even by participants in the work itself.37 What really distinguishes IMI from a community center? One might reach for an institutional answer: it is a work because Bruguera, Creative Time, and the Queens Museum say it is. As much as Bruguera is committed to moving within and using the institutional authority of the artworld to achieve her own ends, she does not simply assume that this authority is what constitutes her art as art. Instead, she advocates for a view that art and artist's obligations, as artists, reach beyond the boundaries of the artworld as they are institutionalized by museums, galleries, and funding agencies. In her "Introduction to Useful Art," Bruguera makes the following suggestive claim: All art is useful, yes, but the usefulness we are talking about is the immersion of art directly into society with all our resources. It has been too long since we have made the gesture of the French Revolution the epitome of the democratization of art... We need to focus on the quality of the exchange between art and its audience.38 Bruguera's goal, then, is to highlight the sources of the norms of the artworld, drawing them closer to the audience in a gesture toward democratization of artistic practices. The "quality of exchange" to be focused on draws on the activity and creativity of the audience, bringing them into the creative act of art-making. IMI is an exemplary work of Useful Art, and shows why Useful Art is often included under the broader category of Participatory Art.39 Being directly immersed in society means that the aims of participatory works depend on what the participants do as co-creators of the works. These aims often call into question the artworld authority that allowed them to be pursued in the first place. In Bruguera's IMI, the particular structure of the institutional authority of the artworld is made to serve works of Useful Art, which themselves 20 contest the particular institutionalization of the authority of the artworld. It is no wonder, then, that the central question concerning IMI is whether it counts as art. At its very heart, the work challenges what it is to be a work of art along with the obligations and responsibilities that accompany that designation.40 Is IMI, therefore, simply a revolutionary gesture that aims to abolish the boundary between art and non-art? I think not. Bruguera's work achieves many of its ends because it claims for itself the mantle of art and through the very questioning of why it counts as art. The institutionally maintained boundaries of the artworld are retained, taken advantage of, and valued, even as the particular shape of the institutionalization is criticized. Had Bruguera advocated or performed an erasure of the boundaries between artworld and the social and political worlds, hers would be an act of aesthetic or artistic revolution. Aesthetic disobedience, by contrast, pairs the defense of broad artworld commitments with criticism of their specific institutionalization. So while it does not seem to me that Bruguera's work amounts to a revolutionary gesture, others might disagree. That there might be disagreement about radical cases, especially at or near the moment of disobedience, should not be surprising. This is more than a mere fog-of-war epistemic worry, though. It points to what I take to be interesting about aesthetic disobedience. Namely, it shows that aesthetic disobedience, as a deliberative gesture, involves moving citizens of the artworld to see that they share certain normative commitments that are being violated. An aesthetically disobedient act draws attention to a conflict in normative commitments that the citizens of the artworld may not have noticed, and about which they may need convincing. It is perfectly natural, then, for an aesthetically disobedient act to look like many things to different citizens of the artworld especially in the midst of deliberation over what is, in its essence, a 21 contested topic. Where structuring norms of an artworld are questioned, as in IMI, it might be a matter of reasonable dispute whether it is a case of revolution or aesthetic disobedience. We are now in a position to answer more fully the question asked at the end of the previous section, "What qualifies as an artworld norm that an act of aesthetic disobedience might target?" On the one hand, works of aesthetic disobedience like Bruguera's draw on and performatively affirm at least some of the established authority of artworld practices. On the other hand, part of the point of aesthetic disobedience, and Bruguera's work is an example of this, can be to call into question the shape of the boundaries of the artworld. This can range from straightforward cases of formal innovation and violation in which artists like Stravinsky and Schoenberg trafficked; it can include a variety of disruptive audience interventions; or, in extreme cases like Bruguera's, the targeted norm might involve the distinction between artworld and the contemporary social-political world. In practice, this makes it difficult to distinguish sharply between acts of aesthetic disobedience and aesthetic revolution. Since some of the very boundaries of an artworld might be called into question, we need to leave open a characterization of the norms that might be targeted by acts of aesthetic disobedience.41 1 Breton, Andre, "Manifesto for an Independent and Revolutionary Art," in What is Surrealism: Selected Writings, ed. Rosemont, Franklin (New York: Pathfinder Press, 1979); Wagner, Richard, "Art and Revolution," Richard Wagner's Prose Works Vol. 1: The Art-Work of the Future, etc., ed. Ashton Ellis, William (London: Kegan Paul, Trench, Trübner 1892); selections indicating Schoenberg's reluctance to identify his music with "revolution" can be found in Auner, John Henry, A Schoenberg Reader (New Haven: Yale University Press 2003), Ch. 4; Gombrich, E.H. The Story of Art, 16th Edition (New York: Phaedon Press 1995). For a brief history of political language in art criticism, see Francis Haskell's "Art and the Language of Politics," Journal of European Studies, 4.3 September 1, 1974, pp. 215-232 (many thanks to Ivan Gaskell for pointing me to this article). 2 Attributed to Gauguin by James Huneker in The Pathos of Distance: A Book of a Thousand and One Moments, (New York: Charles Scribner's Sone 1913), p 128. 3 See Hannah Arendt's classic work On Revolution, pp. 34-35, and Chapters by Bedau, Rawls, Morreal, and Smart in Civil Disobedience in Focus, Hugo A. Bedau (ed.), (London: Routledge, 1991). 4 Stravinsky, The Poetics of Music in the Form of Six Lessons (Cambridge, MA: Harvard University Press 1947), p. 9; Schoenberg op cit. p. 165. 5 While I don't carefully distinguish between aesthetic and artistic commitments, for the purposes of this paper I take the former as a broader term that includes the latter as a subset. Nothing in this paper turns on this. 6 See: Rawls, Theory of Justice (Cambridge: Harvard University Press, 1971) and Political Liberalism, Second Edition, (New York: Columbia University Press, 1996); Raz, The Authority of Law (Oxford: Clarendon Press 1979) and Ethics in the Public Domain (Oxford: OUP 1994); Greenawalt, Conflicts in Law and Morality (Oxford: 22 Clarendon Press 1987); Brownlee, Conscience and Conviction: The Case for Civil Disobedience (Oxford: OUP 2012); Dworkin, Taking Rights Seriously (Cambridge: Harvard University Press 1977) and Law's Empire (Cambridge: Harvard 1986); Markovits, "Democratic Disobedience," Yale Law Journal, 114: 1897-1952; 7 Rawls, Theory of Justice, p. 364-67. 8 See, Smart, Brian, 'Defining Civil Disobedience,' and Morreall, John, 'The Justifiability of Violent Civil Disobedience,' in Bedau op cit. For a succinct critical account of Rawls, see Robin Celikates, "Civil Disobedience and the Practice of Civil Freedom," in David Owen (ed.), Global Citizenship: James Tully in Dialogue (London: Bloomsbury Press 2014), Ch. 6. 9 I borrow the language of "risk" from Brownlee's "non-evasion condition" in Brownlee, 2012, p. 37ff. 10 See Brownlee, "The Communicative Aspects of Civil Disobedience and Lawful Punishment," Criminal Law and Philosophy, 1 (2), 2007: 179-192 and Conscience and Conviction, pp. 29-47; and Smith Civil Disobedience and Deliberative Democracy, (New York: Routledge 2013). 11 Handke, Peter. Kaspar and Other Plays. Translated by Michael Roloff. New York: Farrar, Straus and Giroux, 1969. Thanks to Morgan Koerner for suggesting this example. 12 Kaspar and other Plays., p. 31 13 Joseph, Artur and Handke, Peter. "Nauseated by Language: From an Interview with Peter Handke." The Drama Review 15, no. 1 (Autumn, 1970): p. 58. 14 Kaspar and Other Plays, p. 5. 15 Kaspar and Other Plays, p. 19 16 Thanks to an Anonymous Referee 2 for encouraging me to show more clearly the connection of aesthetic disobedience to institutional theories. Graham McFee's use of T. J. Diffey's term "The Republic of Art" in his institutional account comes closest to the sort of reflective deliberation about artworld norms I am pointing to here. McFee, Artistic Judgement: A Framework for Philosophical Aesthetics (London: Springer 2011), Chapter 6; Diffey "The Republic of Art," British Journal of Aesthetics, Vol. 9, No 2. (1969) pp. 145-156. 17 See Celikates, "Civil Disobedience and the Practice of Civil Freedom," op cit. 18 I borrow the legal positivist terms of Joseph Raz here, but there is no need to commit to any particular view of the nature of those sources, their independence from or entanglement with moral norms and so on. See The Authority of Law, (Oxford: Clarendon Press 1979), pp. 48ff. 19 Kendall Walton, "Categories of Art," The Philosophical Review, Vol. 79, No. 3 (July 1970), p. 352. Again, I thank Anonymous Referee 2 for pushing me on this point. 20 Arthur Danto, "The End of Art," The Philosophical Disenfranchisement of Art, (New York: Columbia University Press 1986, pp. 81-116; and After the End of Art, (Princeton: Princeton University Press 1997). Danto, of course, borrows the phrase from Hegel. 21 Riggle, Nicholas Alden, "Street Art," Journal of Aesthetics and Art Criticism, Vol. 68, No. 3. (Summer, 2010), p. 246. 22 Ibid, p. 248. 23 Mehta's account does not include this. See his interview with Dorit Gabai / Ma'ariv 29th July 2005, http://www.zubinmehta.net/44.0.html accessed January 11, 2013. 24 See my, "Musical Formalism and Political Performance," Contemporary Aesthetics, Volume 7 (2009). 25 Barenboim, David, and Edward W. Said. Parallels and Paradoxes. New York: Pantheon Books, 2002. 26 See Diana Burgwyn's, "Why opera audiences boo," in Broad Street Review 12/24/2006, http://www.broadstreetreview.com/index.php/main/article/Why_opera_audiences_boo accessed January 14, 2013. 27 This argument partially resonates with Jacques Rancière's account of politics and aesthetics. But while Rancière seems to argue that any aesthetic gesture sensuously restructures what we take for granted such that all aesthetic acts are acts of "dissensus," my claim is much narrower. Maintaining a distinction not only between aesthetics and politics, as Rancière does, but also between disobedient and ordinary aesthetic gestures in art is explanatorily helpful. See Rancière's Dissensus: On Politics and Aesthetics, (New York: Continuum Press 2010), and The Politics of Aesthetics, (London: Bloomsbury Academic 2004). 28 See my "Critical Performances," teorema, Volume XXXI/3, (Autumn, 2012), pp. 89-104. 29 "Wagner Controversy: Opera Cancels Holocaust Staging of Tannhäuser," Spiegel Online International, May 9, 2013, http://www.spiegel.de/international/zeitgeist/wagner-opera-cancelled-due-to-holocaust-staging-oftannhaeuser-a-898937.html 30 Jonathan Jones, "Look What We Did," The Guardian March 30, 2003. 31 Ibid. 23 32 Ibid. 33 Ibid. 34 Caminero had no idea what the vase really was. "Local Artist Apologizes for Smashing Priceless Vase at Ai Weiwei Exhibit at Pérez Art Museum Miami," Miami Herald, Feb. 19, 2014, http://www.miamiherald.com/2014/02/18/3943647/local-artist-apologizes-for-smashing.html. 35 http://immigrant-movement.us/wordpress/about/ accessed on January 14, 2013 36 Ibid. 37 Relayed by Tom Finkelpearl in discussion at the American Society for Aesthetics annual meeting in 2012. 38 Bruguera, "Introduction to Useful Art," http://www.taniabruguera.com/cms/528-0Introduction+on+Useful+Art.htm, accessed January 14. 2013 39 For recent examples of participatory art see Nato Thompson's Living as Form (Cambridge: MIT, 2012); Claire Bishop's Artificial Hells: Participatory Art and the Politics of Spectatorship (NY: Verso, 2012); Pablo Helguera's Education for Socially Engaged Art (NY, Jorge Pinto Books 2011); and Grant Kester's The One and the Many: Contemporary Collaborative Art in a Global Context (Durham: Duke University Press, 2011). 40 This is compatible with non-institutional conceptions of art. A claim within the artworld that something is a work of art, even though it hasn't been acknowledged as such by artworld institutions, is a claim that artworld institutions have misapplied the concept of art (whatever that is). Any shift in our institutionally sanctioned conception of art might then be understood in terms of discovery (Bruguera helped us to discover that our conception of art was overly constrained because the concept of art in fact includes works like IMI). This might involve a more radical and interesting shift for institutional or historical theories where I take it the very concept of art is challenged when the particular institutionalization is challenged. 41 I would like to thank the members of the Aesthetics Work Group at the College of Charleston, the participants of the Ästhetischer Ungehorsam conference at the Internationales Forschungszentrum Kulturwissenschaften in Vienna, Christine Abbt, Hanne Appelqvist, Jennifer Bestman, Kimberly Brownlee, John Brunero, Kathleen Eamon, Tom Finkelpearl, Lydia Goehr, Gregg Horowitz, Michael Kelly, Morgan Koerner, Bernadette Meyler and the participants of the Law and Humanities Colloquium at Cornell Law School, Brian Soucek, Robert Talisse and Tyler Zimmer. Finally, the anonymous referees (particularly Anonymous Referee 2) for this Journal provided very generous and helpful comments.

From eye to machine: Shifting authority in color measurement Sean F. Johnston Introduction Given a subject so imbued with contention and conflicting theoretical stances, it is remarkable that automated instruments ever came to replace the human eye as sensitive arbiters of color specification. Yet, dramatic shifts in assumptions and practice did occur in the first half of the twentieth century. How and why was confidence transferred from careful observers to mechanized devices when the property being measured – color – had become so closely identified with human physiology and psychology? A fertile perspective on the problem is via the history of science and technology, paying particular attention to social groups and disciplinary identity to determine how those factors affected their communities' cognitive territory. There were both common and discordant threads motivating the various technical groups that took on the problems of measuring light and color from the late nineteenth century onwards, and leading them towards the development of appropriate instruments for themselves.1 The transition from visual to photoelectric methods could be portrayed as a natural evolution, replacing the eye by an alternative providing 290 Theories, Technologies, Instrumentalities of Colour more sensitivity and convenience – indeed, this is the conventional positivist view propounded by technical histories.2 However, as other case studies have demonstrated, the adoption of new measurement technologies seldom is simple, and frequently has a significant cultural component.3 Beneath this slide towards automation lay a raft of implicit assumptions about objectivity, the nature of the observer, the role of instruments, and the trade-offs between standardization and descriptive power. While espousing rational arguments for a physical detector of color, its proponents weighted their views with tacit considerations. The reassignment of trust from the eye to automated instruments was influenced as much by the historical context as by intellectual factors. I will argue that several distinct aspects were involved, which include the reductive view of color provided by the trichromatic theory; the impetus provided by its association with photometry; the expanding mood for a quantitative and objective approach to scientific observation; and, the pressures for commercial standardization. As suggested by these factors, there was another shift of authority at play: from one technical specialism to another. The regularization of color involved appropriation of the subject by a particular set of social interests: communities of physicists and engineers espousing a 'physicalist' interpretation, rather than psychologists and physiologists for whom color was conceived as a more complex phenomenon. Moreover, the sources for automated color measurement, and instrumentation for measuring color, were primarily from the industrial sphere rather than from academic science. To understand these shifts, then, it is necessary to explore differing views of the importance of observers, machines and automation. The nineteenth-century context: the questionable centrality of the observer The judgement and description of color was based traditionally on visual observation. A dyed fabric, painted wall, glazed ceramic or brewed potion would be compared by its producer to memory or to an available example of the desired color. For such applications, color was seen as unproblematic; the observer merely confirmed what evidently was there. This understanding of color as being a property of objects, and external to the human senses, was promoted by another implicit assumpFrom eye to machine: Shifting authority in colour measurment 291 tion: that the only 'proper' light was daylight. The candle, gas and kerosene lighting of the mid nineteenth century were commonly considered to be imperfect substitutes for the rich, balanced tones of sunlight.4 Hence, variations in perceived color were attributed to improper viewing conditions rather than to complexities of the visual process. Color, and its potential variability, were externalized. Stabilize the viewing conditions, it was argued, and one rendered color judgement reliable. Such assumptions supported straightforward color descriptions and routine evaluation. In short, standardization avoided problems. Scientific investigations supported this utilitarian view of color by promoting a 'physicalist' interpretation of color perception, linking the perceived color to the wavelengths of the light source and to the spectral characteristics of the illuminated object. According to the trichromatic theory elaborated successively by Thomas Young, Hermann von Helmholtz and James Clerk Maxwell, the eye itself could be understood as a three-component sensor responding to red, green and blue components of light.5 There was, however, discordance in this straightforward acceptance of visual observations. The very notion of measuring color attracted criticism, focused initially on criticisms of photometry, and centered on undesirable human factors in the evaluation of brightness. Color description, in the physicalist interpretation, was a simple generalization of the unambiguous technique of determining brightness. The measurement of any color could be reduced to three photometric measurements: a measurement of intensity through a red, a green and a blue filter. Yet some practitioners of the photometric art questioned the reliability of their measurements. Photometry itself appeared intrinsically to be an imprecize demi-science, owing to the vagaries of the human eye. They concluded that they could be misled by inadvertent prejudice, and that the matching of two lights by eye was prone to psychological bias. One of the first to voice this concern was Benjamin Thompson, who in 1794 had employed a double-blind method to avoid the problem of being 'led into temptation'.6 Helmholtz later wrote of visual measurement that the whole region is closely entangled with physiological problems of the utmost difficulty, and moreover the investigators who can make advances are necessarily limited, because they must have long practice in the observation of subjective phenomena before they are qualified to do more than see what others have seen before them.7 292 Theories, Technologies, Instrumentalities of Colour Even careful attention to technique by meticulous observers resulted in measurements that were of doubtful accuracy. Measurements were affected by several subtle considerations that could be easily missed by a novice investigator. 'Bare directions will not suffice', wrote the author of another guide, but the practitioner must bring to the task a judgement trained for instrumental manipulation and an appreciation for the many modifying influences that the measurements which he obtains may possess in value.8 When differently colored lights were to be compared, even this care was not enough. Because of the differences in the color responses of different observers, no amount of repetition or control of viewing conditions could remove the inherent personal bias. The industrialization of color Despite these unsettled and unsettling foundations, practitioners of colorimetry continued to rely implicitly on visual photometry. Most practitioners by the late nineteenth century saw themselves as engineers rather than as scientists. Gas inspectors, in fact, a common feature of towns in the second half of the century, became the principal users and developers of photometry during that period. Indeed, most research on the subject became associated with the lighting industry: the rise of electric lighting from the 1880s led to immediate competition with gas illumination. The measurement of color was inextricably part of a growing system of standardization and testing. Gas and electric lighting were of distinctly different colors (as were different gas mantle and electric filament technologies). Inter-comparison therefore required the resolution of what was termed the 'heterochromatic photometry' problem: how to determine a quantity called 'brightness' for such different light sources, when their colors complicated matters? Like the measurement of illumination, interest in the measurement of color had strong utilitarian motivations. Dye production had expanded dramatically after the development of synthetic dyes in the second half of the nineteenth century, and by the turn of the twentieth century dye chemistry was a major industry, accompanied by the growth of research laboratories.9 In the printing industry, colorprinting processes had been much developed and were commonplace by the 1890s. Both of these applications demanded high-quality matching of colors and routine, rapid measurements. The demands from industry From eye to machine: Shifting authority in colour measurment 293 for color standards for dyes and inks required research into the perception of color, the effects of lighting, lamp characteristics and surface finish. These applications also promoted a simple reductionist description of color: colors were to be evaluated in isolation, or by comparison only with a reference standard; they were interpreted as static properties that did not change with time; and, they were seen as intrinsic characteristics of the products being manufactured. Proponents of gas and electric lighting both appropriated photometry as a tool to support their claims about the stability and cost efficiency of their products. This, in turn, drove further refinement of color measurement to make it better able to detect subtle differences between light sources. From a handful of consulting engineers, such research moved to industrial laboratories from the turn of the century. Important research on color was undertaken, for example, at the United Gas Improvement Company in London; the National Electric Light Association (NELA) Research Laboratory in Cleveland, Ohio; at the Westinghouse laboratory in Pittsburgh, Pennsylvania; the Eastman Kodak lab in Rochester, New York; and the British General Electric Company in London. Governments, too, were developing an interest in more precise measurement of light and color. Photometric and colorimetric standards became a responsibility of the new national labs at the same time: the Physicalisch Technische Reichsanstalt (PTR) in Germany, the National Physical Laboratory (NPL) in England and the National Bureau of Standards (NBS) in America. Such institutions sought to refine measurement techniques based on human observers. By the First World War, it was not unusual to repeat visual photometric observations several hundred times to obtain adequate precision.10 But even careful attention to time-consuming technique by meticulous observers resulted in measurements that were often of doubtful accuracy because of the differences in the color responses of different observers' eyes. This proved to be a serious problem in evaluating standard lamps, which varied in yellowness of tint. The comparison of the pentane standard – the late-Victorian national intensity standard adopted in Britain – with a carbon filament electric lamp, necessitated the drafting of all available technical staff at the National Physical Laboratory as observers to obtain an unbiased mean.11 These energetic and costly programmes to normalize the observer were only possible in large institutions. For industrial applications, a biased visual judgement appeared unavoidable. The reputed imprecision 294 Theories, Technologies, Instrumentalities of Colour of photometry and colorimetry restricted the usages to which they were applied; in turn, the undemanding usages placed little pressure on practitioners to improve their technique. This circle of low expectations – imprecise results – poor reputation – low expectations thus relegated the measurement of light and color to the depths of the scientific toolbox. By the opening decade of the new century, then, the measurement of color was commonly seen as fundamentally limited, owing to the treacherous human eye. What could machines do? Assumptions – often implicit, and not necessarily shared by all practitioners – that color was a property external to human perception, that color judgement demanded a standardization of observing conditions, and that the eye itself was problematic – paved the way for the acceptance of automated methods. Several communities were concerned with the 'control' of color at the turn of the twentieth century: engineers and industrialists tasked with judging the color of products, physicists responsible for national standards, and astronomers characterising stars. Each drew its expertise from the physical sciences; each implicitly accepted the physicalist view of color. These groups sporadically considered the replacement of the human observer by a more reliable alternative. Three attributes, only weakly coupled, were behind this: the desires for (i) quantification, (ii) objectivity and (iii) automation. Quantification The rise of a quantitative perspective in science, peaking in the late nineteenth century, has been well documented.12 Lord Kelvin's view, that only 'when you can measure what you are speaking about, and express it in numbers' can you 'know something about it' and 'advance to the stage of science', soon attained the status of unquestioned truth. The dictum also related quantification to occupational status. Scientific and engineering professions were emerging with increasing frequency at the turn of the twentieth century, in parallel with a rise of technical employment in industry. Measurement served both a disciplinary and social function, providing a cachet of scientific respectability and progress for subjects that did not yet have a disciplinary focus. From eye to machine: Shifting authority in colour measurment 295 Objectivity From the turn of the century, there was an increasingly pervasive mood in the physics community for 'objective' measurements (as opposed to the 'subjectivity' of the human eye as an observational tool). Visual observations, it was argued, even when stabilized by elaborate experimental protocols, were too reliant on indefinable factors – psychological bias, personal variation and fatigue – to allow precision adequate for commercial purposes. 'Observer-independent' methods were claimed by many engineers and physicists to be objective because they would be free from the distortions and complications of human vision, influences that were suspected even if not entirely elucidated. By removing the difficult-to-control human contribution, the quantification would be rendered simpler and intrinsically more trustworthy.13 And having recently tamed inanimate standards such as resistance, technologists were confident that the measurement of brightness and color could be controlled equally satisfactorily by concerted effort.14 But scientific fashion played an important role in promoting this view. Indeed, photodetectors based on physical effects were imbued with very different characteristics (if mostly unconfirmed at this time). Yet, physical detectors had no shortage of 'distortions' and 'complications' of their own. Indeed, the very definition of a 'distortion' hides an underlying definition of normality. The changing fashion was aided by the appropriation of the subject by influential technical communities. There was a two-fold claim to objectivity: first, that color itself is objective, a property of objects rather than a perception constructed by the human visual system; and second, that measurement, too, should be objective, by using physical apparatus. The eye, it was argued, was objective only in principle.15 And the properties of instruments could themselves be measured and regularized in a way that human observations never could be. By the First World War, for example, American investigators claimed to have developed a physical alternative to the eye. Consisting of the combination of a thermopile and a filter to screen out invisible radiation, they touted it as an 'artificial eye'.16 The central problem was to transform the spectral response of the radiometer (which responded almost equally to wavelengths over a very broad range) into a close approximation of the very uneven color response of the human eye. Practical problems, however, centered on the feeble response of such a system to visible light. 'The degree of sensibility required is very high', wrote the inventor, suggesting that the refinement of thermopile design and 296 Theories, Technologies, Instrumentalities of Colour galvanometer sensitivity was severely limited.17 He was to write sixteen years later that 'the possibility of using some form of radiometer as a substitute for the eye has been a long-standing dream', but evidently one not yet realized satisfactorily.18 Automation As with quantification and objective measurement, the argument for automating color measurements was part of a general trend in engineering and industry.19 This was supported by economic factors: the deskilling of measurement, for example, enabled mass production of standardized products, and automated measurement promised greater speed and lower labour cost. For practitioners trained in the physical sciences, then, machine measurement of color promised distinct advantages such as better precision, objectivity or speed than the eye could provide. Along with these practical promises, however, physical methods required a shift of epistemology. The physical scientists who took it up saw colorimetry no longer as a common-sense procedure intimately tied to human vision, but as a branch of energy measurement closely linked with spectrophotometry. By re-interpreting it in this way, they reclassified the eye, making it merely one of the more unreliable detectors of radiant energy, rather than as the central element in a perception-oriented technique. Thus, instruments had the capacity to do things human observers could not. They could regularize the measurement of color, and regulate it both numerically and legally. They could, in fact, validate human observations, serving as a standard that normalized visual experience. In so doing, instruments de-privileged visual observation, reclassifying it as individualistic and second-best. This tailoring of colorimetry to the conceptions of physical scientists proved irresistible in the commercial world. The trajectory of automation While the intellectual environment was favourable for the advance of automation at the turn of the century, there were deep practical roots for this technological inclination. From eye to machine: Shifting authority in colour measurment 297 Early measurement technologies The human observer of color had long been assisted by various aids, intended to enhance discrimination or to bolster memory. The earliest of these could be termed 'paper-based' technologies. In the first half of the nineteenth century, several systems had been proposed for describing or mapping colors. These were usually based on color charts that divided color into a few distinct dimensions, constructions pursued earlier by Newton and Goethe. But these systems were devized for, and of interest to, distinctly separate groups: artists, bird fanciers, flower enthusiasts and industrialists – all having distinct ideas of color measurement.20 Color-measuring instruments appearing through the late nineteenth century, such as those devized by Hermann von Helmholtz, James Clerk Maxwell, William de Wiveleslie Abney, could be described as adjustable or interactive color charts, because they permitted mixing two or three colors to create the perception of another.21 These devices had two effects. First, they promoted the trichromatic theory, demonstrating that the colors perceived by humans could be synthesized from three primaries. This had a stronger intellectual basis than many of the ad hoc divisions of color space earlier in the century, and attracted physical scientists particularly. Second, these instruments argued persuasively that color could be usefully expressed by the measurement of a few numbers.22 Such devices gave credence to colorimetry as a quantitative study. This was a limited and highly reductive sort of analysis, to be sure, but still one that allowed practical applications and great scope for research.23 Attractive alternatives New varieties of so-called 'photocells' and 'photoelectric tubes' proliferated between the 1870s and 1920s and were proposed periodically as solutions for routine photometry and colorimetry.24 The photosensitivity of selenium had been discovered in 1872 and was repeatedly proposed as a close electrical analogue of the eye, notably by the industrialist Werner Siemens.25 A turn-of-the-century practitioner was optimistic but not entirely accurate, reporting that 'light of all refrangibilities from red to violet is effective' and that 'a mere pin point of sensitive surface is as effective as a square centimètre'.26 Samuel Langley invented the bolometer in 1880, a detector consisting of a thin metal strip that changed resistance with temperature. This 298 Theories, Technologies, Instrumentalities of Colour joined the thermocouple and thermopile as a sensitive detector of heat, and light radiation. The quantitative use of such electrical devices was made more practicable by the development in 1882 of the D'Arsonval galvanometer. The selenium cell was joined, in the second decade of the twentieth century, by the phototube. This thermionic valve having a photosensitive cathode was developed into a variety of sizes, materials and constructions. Physicists, in particular, were strongly drawn to phototubes for the same reasons that they rejected the human eye: such tubes could be understood. While contemporary theory was inadequate to explain the behavior of selenium, phototubes were based on the photoelectric effect, a phenomenon amenable to concerted research. Phototubes were part of the new physics, elevating photoelectric devices from mere components for inventors to the subjects of research in their own right. These new devices were both a fascinating technical challenge and means to advancement for physicists in industry. Making machines work The shift towards objective measurements was consolidated by technological change between the world wars. The inter-war period was a turning point, characterized by active development of non-visual, quantitative and automated instruments for color measurement. Engineering practice, centring on visual methods, had remained little changed from the 1870s until the 1920s for the vast majority of colorimetric work. By the Great War, however, there was an independent trend by astronomers towards physical methods of stellar measurement that were based principally on photography. Laboratory spectroscopists also took up these photographic methods after the war.27 Physicists, on the other hand, increasingly investigated photoelectric measurement techniques. As they gradually resolved many of the technical limitations of these detectors, other scientists began to adopt them for light and color measurement by the late 1920s. This merging of method saw the newly categorized 'subjectivity' of visual photometry decisively rejected for the 'objectivity' of physical techniques. This gradual process, repeated in each technical community, involved the recasting of colorimetry into a less problematic form. In the process, the human component of the measurement chain was minimized, and the observer was made ever more remote. Nevertheless, the limited successes of the first decade of photoelectric instrumentation highlighted an earlier concern: how reliable were the measurements, and From eye to machine: Shifting authority in colour measurment 299 how did they relate to human perception? The new technologies proved, in their own ways, to be as troublesome as visual methods had been. The superiority of physical detectors over the eye was a matter of scientific faith rather than reality; photodetectors of the 1920s were fickle and highly fallible. The mapping of what were, at the time, largely illusory properties onto these physical detectors was begun by the astronomers, for whom visual observation of star magnitudes and color were particularly difficult. Most astronomers designed their own. In England, A. F. and F. A. Lindemann published the first account of the details of photoelectric apparatus and methods for astronomical photometry in 1919.28 The potassium phototube responded most strongly to blue-violet light, while the response of the caesium type peaked in the yellow portion of the spectrum. That the photocells responded differently to light than did the eye did not deter them; indeed, the Lindemanns marshalled it as a demonstration of the success for the new technology. They described the fabrication of photocells having potassium and caesium sensitive surfaces, noting that the two types could be used to measure a 'color index' for stars. Thus the astronomers recast the stumbling block that had dissuaded lighting engineers into a pedestal to extend their own observational grasp. Physical detectors had other disadvantages besides responding to colors differently than did the eye. They tended to produce erratic or drifting signals because of temperature and chemical change, ageing and instability, and nonlinearity of the early electronic circuitry. By the mid 1920s systems of compensating for the (very different) intrinsic defects of such devices had been devized, making them roughly the equal of the human eye for some photometric applications.29 Enthusiastic proselytising by early proponents was also important, as illustrated by the American physicist Arthur Hardy. Hardy had begun to study problems in the field of color printing when he joined MIT in the early 1920s. Realizing that 'a great mass of spectrophotometric data would be required', he sought an alternative to visual color analysers, which typically were used to make measurements at thirty discrete wavelengths in the visible spectrum. The available 'Thalofide' cells, a compound of thallium sulphide that changed resistance when illuminated, gave erratic results. Like the Lindemanns before him, Hardy did not judge this extreme variability to be a disadvantage. He noted that 'this erratic behavior was not altogether unexpected. Neither was it a 300 Theories, Technologies, Instrumentalities of Colour great disappointment because of the almost certain necessity of employing vacuum tube amplifiers, which at that time were almost as erratic.' Hardy's first automatic instrument could yield good visible-range spectra in as little as 30 seconds. The prototype was soon being used to record as many as 3000 spectra in a single month.30 Hardy's enthusiasm was contagious. His recording photoelectric 'color analyser' was widely publicized. The instrument was adapted and commercialized by General Electric in 1935 as the first automated recording spectrophotometer.31 His later production of the Handbook of Colorimetry argued for the superiority of automated devices over the eye by sheer quantity of data. Through such convincing demonstrations, colorimetry became closely allied with, and directed by, the disciplinary and occupational rise of spectrophotometry and physical photometry, themselves the construction of physicists, chemists and astronomers. So automation symbolically removed the problematic observer from the measurement, making this an attractive and highly visible benefit of physical methods. By relegating the operator to interpreting graphs or numerical lists – an activity seemingly free of physiological and psychological factors – automated instruments appeared to redraw the boundaries to position colorimetry firmly within the realms of physical science. That such a demarcation entailed the adoption of new light detectors having their own complexities, and requiring a definition of how the visual sensation related to their replacements, was not at first an issue. The growing acceptance of the photoelectric detection of light and color were promoted on several fronts. In Britain, for example, members of the NPL photometry department, gradually convinced of the practical superiority of such detectors to the eye, cautiously endorsed the use of physical photometers in 1930; their collaborators at the GEC Research Laboratory were demonstrating prototypes of commercial instruments; and small firms were introducing photoelectric colorimeters. In 1933, the Science Museum recognized this commercial wave by mounting a three-month exhibition of photoelectric equipment.32 Such public demonstrations rode on a wave of technological enthusiasm for quantification and objective measurement, driven by commercial forces for high-throughput measurement of color. By the late 1930s, the adjectives 'photoelectric' and 'automatic' had become a short-hand for 'modernity'. From eye to machine: Shifting authority in colour measurment 301 Opposing the machine As discussed above, the movement towards automated instruments suggests a gradual and largely unopposed transition. However, once detectors other than the eye were proposed seriously, opposition began to be voiced in several quarters. Critics included a handful of physicists and a larger, and growing, community of physiologists and psychologists. Disciplinary perspectives As we have seen, the communities developing instrumental practices in color measurement were firmly aligned with physical science. During and after the First World War, the new links between colorimetry, national laboratories and industry were consolidated by the formation of optical societies. The Optical Society of America, for example, was founded in 1916 principally by a group at Eastman Kodak, and brought together researchers and engineers concerned with all aspects of optics, including photometry and colorimetry. Its Journal of the Optical Society of America and Review of Scientific Instruments became the principal English-language organ for scientific optics in the 1920s. Unlike continental optical journals, JOSA dealt with subjects such as color measurement and the physical principles of light detectors. In Britain, the Journal of Scientific Instruments (founded in 1923) covered similar subjects, notably electrical measuring devices. In both countries, societies of 'illuminating engineering', comprizing mainly engineers and scientists, provided another important outlet for research papers.33 The new technology was being embraced through a broadening and redefinition of optics through such publication channels. A few physicists argued that the eye was an essential component in any measurement purporting to quantify visual attributes. The inventors of the most popular visual photometer, Otto Lummer and Eugen Brodhun of the PTR, noted at the turn of the century: The purpose of practical photometry is to compare the total intensities of light sources as they are perceived by our eyes. In such a measurement of the purely physiological effect of flames only the eye can therefore be used; all other measuring instruments, such as the radiometer, selenium cell, bolometer and many more of the kind, are to be discarded in so far as these indicate physical effects of light sources.34 302 Theories, Technologies, Instrumentalities of Colour If the one-dimensional measurement of brightness could not be entrusted to physical detectors, so the argument went, how could the more subtle three-dimensional measurement of trichromatic color be done by anything but the human eye? Physiologists, too – particularly at the National Electric Lamp Association Laboratory in Ohio, which boasted of its research on 'the physics of illumination and its physiological and psychological effects on the human organism' – argued for the indispensability of the human eye in any color-measuring instrument.35 But as I have argued elsewhere, the measurement of color provoked strongest criticism in post WWI committees tasked with standardization.36 Psychologists contended that color is a subjective sensation difficult to quantify and accord between different observers, let alone 'physical' instruments. In 1931, the Commission Internationale de L'Éclairage (CIE), the only international forum for light and color, defined a specification of the 'standard observer' – an 'average' human color response based on fewer than two dozen British males – along with standardized color filters and light sources. The specification was engineered by John Guild and Irwin Priest, physical scientists at the national laboratories in Britain and America, respectively. The instruments embodied the theoretical perspective of a particular intellectual group.37 The accepted artificiality of this averaged, mathematized human response made the acceptance of a non-human observer that much easier, and promoted the use of physical colorimeters. While the CIE standard triggered some complaints, these centered on the issue of domination of the research program by certain countries, rather than on the cognitive aspects of color.38 The CIE membership was top-heavy with physicists and engineers, the Commission itself having developed from a pre-war international photometric commission. For many of the CIE members, color was reduced to the problem of heterochromatic photometry.39 A better opportunity for debate was the Committee on Colorimetry formed by the Optical Society of America, and operating during 19191922 and 1932-1953. In these committees, psychologists gained first a foothold and then equality of representation. There, they argued that the trichromatic definition was founded on a dearth of experience and had a paucity of descriptive capacity. They emphasized that perceived colors were a combination of physiological mechanisms and psychological constructs, often bearing no simple relationship to the wavelengths of light involved. Nevertheless, by the time the differing theoretical stances From eye to machine: Shifting authority in colour measurment 303 of the two academic communities were confronted, commercial colorimetry was well advanced. The combination of standards and new instrumentation promoted the view of color as essentially a physical phenomenon through the early 1930s. The standard made possible the numerical expression of some color attributes, but did not make color matching any easier. Complications in practice The adoption of physical instruments eventually could assure more repeatable measurements, but at the expense of generality: the machines did not always do an adequate job of mimicking the human eye. To cope with the more awkward visual characteristics such as surface gloss and the angular dependence of color, firms developed specialized photoelectric instruments. These proliferated in variety and number through the 1930s. But separating the subjective and physical characteristics of color remained a problem faced daily and directly on the factory floor. Writing of his mixed experiences with colorimetric instruments, a representative of the Printing and Allied Trades Research Association (London) observed: Unfortunately, the spectrophotometer is a costly instrument and requires skilled operation: as a result, many so-called reflectometers, whitenessand brightness-meters have made their appearance...It is not generally realized, however, that papers are not necessarily a good match even when the 'red', 'green' and 'blue' readings are the same; conversely, papers may be a good visual match and yet give different readings. . . it is not commonly appreciated in the trade that color is 'three-dimensional', and that consequently no single instrument reading can define a color.40 Two options were available: either to use human observers and visual colorimeters – i.e. to revert to conventional but tedious color matching – or to employ physical colorimeters. The demand for rapid and reliable testing of products during the 1930s argued for physical methods, just as the testing of incandescent electric lamps had done in the national laboratories a decade earlier. Again, practitioners made the shift from physiological to physical methods. Their pragmatic solution was to continue with the development of specialized instruments to measure more of the awkward visual characteristics, while sharpening the specification of standardization of color comparison. 304 Theories, Technologies, Instrumentalities of Colour Conclusion The seemingly inexorable advance of automated color instruments was, in fact, contentious and fragile. Development was pressed by the three rising fashions of quantification, objectivity and automation by physicists and engineers. Only when machines were presented as a credible alternative was the human eye vaunted as indispensable; the rise of machines came in parallel with a rise of the psychological understanding of color. As the non-physicalist perspective became more vocal, the very serious limitations of physical detectors were tamed quietly by narrowing the scope for their use and claims for their utility. One color text of 1952 spoke of the 'simplification' and 'subduing' of color to the requirements of measurement.41 Yoked to its intended applications, color measurement has, today, become a technological workhorse. And while the physicalist theory has lost its luster, the machines that embody it have more authority today than any pair of eyes. Notes 1. Johnston (1996a; 1996b; 2000). 2. See, for example, the implicit technological determinism and positivism in successive editions of Walsh (1926; 1953; 1958; 1965). 3. The case of instruments for the detection of ionising radiation has been discussed in Hughes (1993); for radio astronomy, see Agar (1994). 4. See Schivelbusch (1986). 5. Helmholtz (1924); Maxwell (1857). 6. Thompson (1794, 362). 7. Helmholtz (1924, viii). 8. P. Stiles, Photometrical Measurements, quoted in Walsh (1926). 9. Homburg (1992). 10. Walsh (1926, 175-80). 11. NPL (1911, 39). 12. On the changing social value of quantification, see Kruger et al (1987) and Kuhn (1962). 13. The importance of 'observation without an observing subject' as a precondition for non-subjective reasoning is discussed in Swijtink (1987). See also Porter (1996). 14. E.g. Hunt (1995). This opinion pervaded the early national laboratories and was actively pursued at the PTR, where an 'absolute' standard of brightness, the so-called Violle standard, was under development at the turn of the century. From eye to machine: Shifting authority in colour measurment 305 15. Via trichromatic perception. 16. E.g. Coblentz (1915) and Ives (1915). The thermopile, a high-sensitivity variant of the thermocouple, had been in use since the middle of the previous century to detect heat. 17. Ives (1915). 18. Ives and Kingsbury (1931). 19. On the attractions of automation, see Bennett (1991). For technical histories, see Bennett (1979; 1993). 20. E.g. Hay (1846); Chevreul (1858); Ridgway (1886); Chrysanthémists, Société des (1905); Munsell (1907). 21. See, for example, Maxwell (1857); Helmholtz (1924); Abney (1913). 22. E.g. Abney (1891) and Lovibond (1897). 23. E.g. Luckiesh (1915). 24. Langley (1881; 14); Hempstead (1977). For the background context, see also Johnston (2001). 25. Siemens (1875a; 1875b). 26. Minchin (1892). 27. E.g. Dobson et al (1926); Harrison (1934). 28. Lindemann (1919). 29. Selenium cells, by contrast, produced inadequate voltage to deflect even a sensitive electrometer when illuminated with violet light. This made them unsuitable for colorimetric measurement, because researchers had established the importance of these extreme wavelengths on color perception. Unable to respond to a color to which the eye responded, selenium failed as a viable replacement for colorimetric applications, but found a place in photometry: such cells were at the center of commercial developments from the 1930s, when firms such as Weston marketed selenium-based instruments as light meters. 30. Hardy (1929; 1935; 1938). 31. Michaelson (1938). 32. This included displays of the major types of photocell and their principles, and industrial examples such as package counters, burglar alarms, street lamp switching and daylight brightness meters. 33. The original membership of the Illuminating Engineering Society of London included only 4% medical doctors; its New York counterpart listed none. 34. Lummer and Brodhun (1899), quoted in Kangro (1976). 35. Fleming and Pearce (1922); Hyde (1909). The NELA lab was established in 1908, and became a wholly-owned part of General Electric in 1911. 36. See Johnston (1996a.) for the confrontation of these views. 37. The instrument as 'reified theory' was first described in Bachelard (1933), and has subsequently been taken up by many commentators. 38. Britain and America had dominated the post-war CIE, when Germany and its former allies were excluded from international scientific conferences. 306 Theories, Technologies, Instrumentalities of Colour Germany, through the research lines pioneered by Helmholtz and Ewald Hering, had hitherto dominated color research. 39. The President of the heterochromatic photometry committee, Charles Fabry, admitted himself 'a little frightened at the size and difficulty of colorimetric questions', and argued that the Commission should concern itself solely with the physical side of color, ignoring its psychological aspects. See Fabry (1924). 40. Harrison (1941). 41. Murray (1952).

Author Query Form Iconic semiosis and representational efficiency in the London Underground Diagram Article: COGSEM-2014-0012 Query No Page No Query Q1 1 Please provide e-mail address (if required) for "Pedro Atã and Breno Bitarello." Q2 1 Please check and confirm whether the added details (author names and affiliations) are correct. Q3 3 Please check whether the inserted short title is OK as typeset. Q4 5 Please confirm whether permission has been obtained to publish this image. If so, please provide details of the permission information, to be included in the figure caption. Q5 6 Please confirm whether permission has been obtained to publish this image. If so, please provide details of the permission information, to be included in the figure caption. Q6 13 The reference "Peirce, 1931–1935/1958" is listed in the references list but is not cited in the text. Please either cite the reference or remove it from the references list. Q7 13 The reference "Peirce, 1931–1935/1958" is listed in the references list but is not cited in the text. Please either cite the reference or remove it from the references list. Pedro Atã, Breno Bitarello and João Queiroz* Iconic semiosis and representational efficiency in the London Underground Diagram Q1 Q2 Abstract: The icon is the type of sign connected to efficient representational features, and its manipulation reveals more information about its object. The London Underground Diagram (LUD) is an iconic artifact and a well-known example of representational efficiency, having been copied by urban transportation systems worldwide. This paper investigates the efficiency of the LUD in the light of different conceptions of iconicity. We stress that a specialized representation is an icon of the formal structure of the problem for which it has been specialized. By embedding such rules of action and behavior, the icon acts as a semiotic artifact distributing cognitive effort and participating in niche construction. Keywords: iconicity, operational iconicity, optimal iconicity, representational efficiency, diagrammatic reasoning, London Underground Diagram, Peirce DOI 10.1515/cogsem-2014-0012 1 Introduction The design of the London Underground Diagram (LUD) is a well-known example of representational efficiency, facilitating urban transportation for thousands of everyday users, copied by urban transportation systems worldwide. Present in virtually every major city in the world, it has established an international paradigm on how to perform simple decision-making tasks regarding networks of stations and lines. Its origins date back to 1933, when the engineer draughtsman Henry C. (Harry) Beck proposed several innovative features to the old Underground Map, sacrificing geographic accuracy in favor of specialization in particular tasks (see Walker 1979). *Corresponding author: João Queiroz, Institute of Arts and Design, Federal University of Juiz de Fora, Juiz de Fora, Brazil, E-mail: [email protected] Pedro Atã, Institute of Arts and Design, Federal University of Juiz de Fora, Juiz de Fora, Brazil Breno Bitarello, Education, Arts and History of Culture Program, Mackenzie University, São Paulo – SP, Brazil Cognitive Semiotics 2014; aop 1 5 10 15 20 25 30 35 40 This paper explores the design of the London Underground Diagram identifying the semiotic basis of its representational efficiency. Efficiency in a representation is a matter of iconic semiosis.1 Several conceptions of iconicity have been acknowledged: the icon is operationally defined as a sign whose manipulation reveals, by direct observation of its intrinsic property, some information on its object (operational iconicity) (CP 2.2792; Stjernfelt 2011: 397); but it has also been connected to representational features involved in the specialization of signs for certain purposes (optimal iconicity) (Stjernfelt 2011: 415). It is the type of sign whose signification is S-dependent (that means, dependent on the sign itself) and, more traditionally, it has been defined as similarity between the sign and its object. These different conceptions of iconicity sometimes appear to generate contradictory claims regarding representational efficiency. To solve such contradictions, we stress that a specialized representation is an icon of the formal structure of the problem for which it has been specialized. Icons are cognitive artifacts, material tools that embed cognition and shape our minds. The London Underground Map is a remarkable example of a cognitive artifact, providing a niche3 built for extraction and manipulation of relations, capable of generating overall changes in the behavior of the users and influencing in the understanding of the city itself. In the following sections, we (i) introduce Peirce's concept of iconic sign, (ii) describe the London Underground Diagram and its representational features, (iii) investigate the LUD's efficiency by examining its relevant innovations in the light of different conceptions of iconicity, (iv) describe its role in cognitive niche construction. Our conclusions relate cognitive distribution and niche construction with representational efficiency as a matter of iconicity. 1 We employ the term "representational efficiency" in the sense used by Zhang (1997), meaning the easiness of use of representations in problem-solving tasks, which can be empirically measured through the comparison of cognitive performances on isomorphic representations (see Zhang and Norman 1994; Zhang 1997; Chuah et al. 2000). In this sense, representational efficiency is an influence that is directed from the material features of the representation to the cognitive performance. This process is identified as iconic: signification is determined by the sign materiality (criterion of relative dependence of the sign process) and problem-solving involves the discovery of information about an object (operational definition of the sign) (see Atã and Queiroz 2014). This is not to say that indexical and symbolic signification is absent, but rather that the decisive element for efficiency is iconicity. 2 Following a scholarship tradition, Peirce's work will be referred to as CP (followed by volume and paragraph number for quotes from The Collected Papers of Charles S. Peirce). 3 In Ecology, the concept of niche means the environmental conditions required for a certain species to live. Cognitive niche construction is related to the transformation of problem spaces in order to aid thinking (see Clark 2006a). 2 Pedro Atã et al. 1 5 10 15 20 25 30 35 40 2 Peirce's iconic semiosis Peirce defined semiosis (sign-mediated processes) as an irreducible triadic relation between a sign (S), its object (O) and its interpretant (I). We will hereafter refer to this triad as S-O-I. That is, according to Peirce, any description of semiosis involves a relation constituted by three irreducibly connected terms (CP 2.242), S-O-I. As it is well known, sign-mediated processes show a notable variety. There are three fundamental kinds of signs underlying meaning processes – icons, indexes, and symbols. Respectively, a sign may be analogous to its object, spatio-temporally connected to it, or might represent it by means of a law, rule, or norm. These classes correspond to relations of similarity, contiguity, and law between sign and object (see Table 1). Icons are signs that stand for their objects through similarity or resemblance, irrespective of any spatio-temporal physical correlation that sign S may have with an existent O. If a determinative relation of the S by the O is a relation of analogy, that is, if S is a sign of O in virtue of a certain quality that S and O share, then S is an icon of O. S and O are related due to the identity of some aspect they share. Icons are very dependent on the material, form, and structure of which they are made – "An Icon is a sign which refers to the Object that it denotes merely by virtue of characters of its own, and which it possesses, just the same, whether any such Object actually exists or not" (CP 2.247). In contrast, if S is a sign of O by reason of "a direct physical connection" (CP 1.372) between them, S is said to be an index of O. In that case, S is really determined by O, in such a way that both must exist as events – "An Index is a sign which refers to the Object that it denotes by virtue of being really affected by that Object" (CP 2.248). The notion of spatio-temporal co-variation is the most characteristic property of indexical processes. The examples range from a pronoun demonstrative or relative, which "forces the attention to the particular object intended without describing it" (CP 1.369), to physical symptoms of diseases, photographs, weathercocks, thermometers. Finally, in a symbol, the relation between S and O is logically dependent on the third term, I. In a symbolic relation, the interpretant stands for "the object through the sign" by a determinative relation of law, rule or convention (CP 2.276). Table 1: The fundamental types of signs underlying meaning processes – icons, indexes, and symbols. They are characterized in terms of relative dependence of sign-object-interpretant (S-O-I) components in triadic relation. Sign S-O relation S-O-I dependence Icon Similarity Monadic (S) Dependent of intrinsic properties of S Index Contiguity Dyadic (S-O) Dependent of S-O spatio-temporal correlation Symbol Law Triadic (S-O-I) S-O dependent of I mediation Iconic semiosis and representational efficiencyQ3 3 1 5 10 15 20 25 30 35 40 The icon is the only type of sign that involves a direct presentation of qualities that pertain to its object. Analogies depend on icons. When manipulated, the icon "reveals" aspects or qualities of its object. The key of iconicity is not perceived resemblance between the sign and what it signifies but rather the possibility of making new discoveries about the object of a sign through observing features of the sign itself. Thus a mathematical model of a physical system is an iconic representation because its use provides new information about the physical system. This is the distinctive feature and value of iconic representation: a sign resembles its object if, and only if, study of the sign can yield new information about the object (Hookway 2002: 102). The icon is not just the only type of sign involving a direct presentation of qualities that pertain to its object; it is also the only sign through which, by its direct observation, it is possible to discover something about its object. Maps, graphs and diagrams are special types of icons. As soon as an icon can be considered as consisting of interrelated parts, and since these relations are subject to experimental manipulation governed by laws, we are working with diagrams (see Stjernfelt 2007: 92). Diagrams are the principal way of acquiring new knowledge about relations. They represent, through the relations between its parts, the relations that constitute the related parts of the object it represents. The object of the diagram is always a relationship, and the related parts of the diagram represent the relationships that constitute the object represented. The prototypical diagram is described as the manipulation of a geometric figure for the observation of a theorem. But the idea is quite general. An example taken from algebra is enlightening: "In fact, every algebraic equation is an icon, since that shows, through their algebraic signs (which are not themselves icons) relations of the quantities involved" (CP 2.282, emphasis added). Indeed, if a sign is observed as a whole consisting of interrelated parts, and these related parts are subject to experimental modification governed by rules, we are operating with a diagram. The London Underground Diagram is an example of a diagrammatic cognitive artifact, providing a niche built for extraction of relational properties. 3 London Underground Diagram (LUD): A cognitive tool for its users The London Underground Diagram (LUD) is a hallmark of information design that influenced many other public transportation diagrams, a "form of representation judged to be so effective that it is now employed by virtually every transportation authority in the world" (Spence 2007: 77). 4 Pedro Atã et al. 1 5 10 15 20 25 30 35 40 The original version of the LUD was created by the Henry C. (Harry) Beck in 1933. Previously to the LUD, maps of the London Underground System adhered to geographically more accurate representations of the lines and station locations (see Figure 1). Beck produced his first sketch for the London UndergroundMap in 1931. The design was based upon and adapted from an electrical circuit diagram (with which Beck was familiar as he was an engineer draughtsman). Such diagrams omit or falsify the relative physical position of wires in order to convey the information about connectivity. Beck saw a similarity with the underground railway network in that it was possible to ignore the geographical information altogether and remove some of the sources of confusion in the previous, more literal maps (Whitby 1996: 70). Figure 1:Q4 A route guide of the Underground System made by F.H. Stingmore, published circa 1932 (this is an overall equivalent version of Stingmore's 1919 guide shown in Garland [1994], the only difference being the addition of a few stations and lines). The background is blank and the different lines are color-coded. Although the concern for geographic accuracy diminished in comparison with the previous maps, it is a central component of the design. © TfL from the London Transport Museum collection. Iconic semiosis and representational efficiency 5 1 5 10 15 20 25 30 35 40 Beck's initial sketch was transformed into a properly labeled and colorcoded diagram (Figure 2) where he compressed the outlying portions of lines. The central area of the network appears to be viewed through a convex lens so as to enlarge its scale, and route lines are simplified in verticals, horizontals and diagonals (45°) (Garland 1994: 16). In later versions of the London Underground Diagram based on the last of Beck's diagrams (published in 1959), his successors retained the essential structure from the original: octagonal grid and colored lines meeting at angles of 90° or 45°; stations arranged to show the position of each one to the next instead of the real geographic distance between them; the presence of the simplified River Thames along the bottom of the diagram helping the notion of position and scale; noninterchange stations represented by ticks and interchange stations represented sometimes by rings sometimes by diamonds (Garland 1994). Graphical changes such as changing the color of the lines and the fonts used in the names of the stations in order to improve the grasping of information by the users and reduce Figure 2:Q5 Beck's original Underground Diagram, from 1933. © TfL from the London Transport Museum collection. 6 Pedro Atã et al. 1 5 10 15 20 25 30 35 40 their possibility of confusion were made, also to accommodate the expansion of the transport system. As a result of the adaptations and modifications made by Beck and his successors, we have the diagram as we know it today. 4 Representational efficiency and iconic semiosis in the London Underground Diagram The LUD (Figure 2) has been recognized as more efficient than a geographically more accuratemap (such as Figure 1). We assume that the type of semiosis involved in the signification of the efficient properties of a representation is the iconic semiosis (see Atã and Queiroz 2014; Zhang and Norman 1994; Zhang 1997). Efficiency corresponds to advantage in the material manipulation of the sign for a certain goal. Iconicity is involved whenever signification is dependent on the materiality and structure of the sign. However, to say that difference in efficiency is due to iconicity is not enough to clarify what happened in the transition from the oldmap to the LUD that has shaped the cognitive niche of the users. In the following paragraphs, we further analyze the notion of iconicity and the representational differences between the two representations of London Underground System. The notion of iconicity can be understood in different ways. Traditionally, it has been defined as "similarity" between sign and object. It has also been defined as relative dependence on S in the S-O relation (see Queiroz 2012). Stjernfelt (2011) identifies two different contrasting conceptions of icon and iconicity in Peirce's work: first, the icon can be operationally defined as any sign whose manipulation is able to reveal more information about its object. This operational definition of the icon focuses solely on the capability of a sign to enclose information about its object. Following the author, we use the term "operational iconicity" to refer to the conception of iconicity arising solely from this operational definition. Operational iconicity contrasts with a stricter notion that considers factors such as immediacy of the information presented and economy of elements. We refer to the conception arising from these stricter criteria as "optimal iconicity" (Stjernfelt 2011: 400). Stjernfelt (2011: 414) exemplifies the distinction between operational and optimal iconicity through the example of a digital picture. A picture can be digitally represented as pixels on a screen or as a linear sequence of digital information. If we only take into account the operational definition of the icon, the two representations are equally iconic: they are informationally equivalent (i.e. enclose the same amount of information), and one can be algorithmically transformed into the other. However, this operational definition alone ignores some representational features that are decisive for the S-O relationship in each Iconic semiosis and representational efficiency 7 1 5 10 15 20 25 30 35 40 sign: in the pictorial image, for example, object contours are represented as continuous lines while in the linear digital representation this information is scattered throughout the code. A single object contour is materially closer to a single continuous line than several scattered pieces of information, regardless of the interpreter (see Stjernfelt 2011: 414). Therefore, it is more iconic. Put in another words, a one-to-one correspondence holds some kind of logical and phenomenological intrinsic iconic value that is shattered by a one-to-several correspondence.4 This is an example of the optimal notion of iconicity. In the LUD, the operational iconicity criterion is able to unambiguously identify the diagram as an icon. It must be iconic semiosis, since a user manipulating the LUD is able to discover implicit information about the Underground System, e.g. on which line to embark to get to a specific station. It does not differentiate, however, between the LUD and older maps. On the other hand, the optimal iconicity criterion is able to stress the LUD's specialization as a problem-solving tool, thus differentiating it from other representations equally capable of revealing information about lines and stations. The LUD has proved to be more efficient for navigation in the Underground System than a geographically more accurate map (such as Figure 1), even though the latter contains more information about the Underground System than the former (see Table 2). 4 Stjernfelt (2011) has related the development from a more operational to an optimal conception of iconicity to the transition to a more realist stance in Peirce's philosophy. Table 2: A comparison between information of O (the Underground System) contained in S (maps and diagrams) for the LUD and a geographically accurate map of the Underground System. The LUD contains less information about the Underground System than the map. Therefore, it is less iconic for operational iconicity, suggesting it to be less similar to the Object. However, it is more efficient, therefore more iconic for optimal iconicity, suggesting it to be more similar to the Object. Information of O accurately contained in S Geographically accurate Map London Underground Diagram Stations Yes Yes Connections between stations (tube lines) Yes Yes Connections between lines (interchange stations) Yes Yes Distance between stations Yes No Geographic location of stations Yes No Length of lines Yes No Specific directions and changes of directions of lines Yes No 8 Pedro Atã et al. 1 5 10 15 20 25 30 35 40 There is more information to be discovered about the Underground System in a geographically accurate map than in the LUD. In this sense, we should conclude that the map is more iconic than the LUD with regard to operational iconicity. Since operational iconicity is a detrivialization of the psychological notion of similarity (see Stjernfelt 2011: 397), we can also conclude that a geographically accurate map is more similar to the Underground System than the LUD. The same conclusion might be reached intuitively: an observer, looking at the map which shows the real trajectories of the lines through the city might say that "it looks more like" the real Underground System than a simplified diagram. The above conclusion appears to inflict a contradiction between similarity and representational efficiency. A geographically accurate map is more iconic (operational iconicity) and "looks more like" the Underground System itself, and yet it is less efficient for navigation in the same Underground System than a simplified diagram. The contradiction can also be understood in terms of opposing operational and optimal iconicity. Compared to a geographically accurate map, the LUD is simultaneously less iconic for operational iconicity, thus, less similar, and more iconic for optimal iconicity, thus, more similar. This, we argue, is a false contradiction, that points to what is relevant in the transition from the old maps to the LUD: while the geographically accurate map might actually be more similar to the London Underground System understood as a whole, the LUD, with the rules of manipulation and behavior it entails, is more similar to the particular experience of the Underground users and the most relevant variables involved in the choices they need to make. This experience of orientation and navigation in the Underground System can be modeled as a game (see Walker 1979) with a formal structure that comprises an initial state (the user's current station), a final state (destination), intermediate states and a set of rules (see Table 3). The LUD is a more efficient representation because it embeds this formal structure more directly than a geographically accurate map.5 It is easier to locate the user current location (initial state) and destination (final state). It is also easier to grasp the overall structure of possible lines and connections among which to choose (intermediate states), with no superfluous information such as changes of directions or specific distances between stations. There are others notable factors why the LUD, with regard to its rules of action and behavior, can be seem as more similar to the experience of a user in 5 A similar argument is presented by Zhang and Norman (1994): in one of their experiments, the authors argue that the more efficient isomorph of the Tower of Hanoi puzzle game is the one that externalizes most rules of the game, so that the performance of the players is efficiently constrained. This process of externalization of constraints has been characterized as iconic (Atã and Queiroz 2014). Iconic semiosis and representational efficiency 9 1 5 10 15 20 25 30 35 40 the Underground System than a geographically accurate map. The concrete experience a user has on an Underground trip is one of no visible landscape or landmarks with which to mark and be conscious of the specific changes of direction of the lines or the specific distances traveled. Since there is also no traffic and the trains move in high speed, the differences in distance can be less significant for the amount of time a train will spend to get to the destination than the number of stops it will need to make. The experience the user has is, arguably, of a continuous homogeneous movement interrupted only by the stops in the stations, just like a straight line undisturbed by topographic issues and interrupted only by the chain of blobs or ticks that represent the stations. In this sense, a hypothetical user that is completely unaware of the geography of the city of London above the ground and is familiar only with the experience of the Underground might agree that, even intuitively, the LUD looks more like the Underground System than a geographically accurate map. In comparison to its predecessor, Beck's diagram has diminished the amount of implicit reachable information in the map, reducing the number of possible operations to be performed (to know about real distances, for example). Beck has added features that do not increase the amount of information, but rather decrease the difficulty of the search for the proper information, which influences in the whole process of problem-solving. That means to say that the behavior of the user as well as the task itself are constrained and, to a certain extent, defined by the material iconic features of the representation. A problem solver behaves Table 3: The formal structure of the game-like experience a user has when trying to solve problems related to navigation in the Underground System. The Underground User Game: Formal Structure Initial State the user's current station Final state the user's goal station Intermediate states every the stations the user is going to access in order to go from the initial to the final state. Rules for moving between states In order to move the user embarks on a train, following its path on the line until the station (final or intermediate) she wants to disembark. The train will follow its path on one particular orientation until the end of the line. It will not change its trajectory, orientation or line while traveling. There are two types of stations: normal stations only allow for embarking or disembarking on one line. Interchange stations allow for changing lines. 10 Pedro Atã et al. 1 5 10 15 20 25 30 35 40 according to a problem space that corresponds to a formal structure of states and rules; this problem space is made available through iconic features of the representations involved in the cognitive process of solving the problem, so that this material representational features shape the behavior of the solvers. Change in efficiency in the transition from the geographically more accurate maps to the LUD corresponds to iconicity in the LUD putting the users in direct touch with rules that are really part of the experience of using the Underground System. 5 The London Underground Diagram as a cognitive artifact Peirce can be considered an important precursor of the situated mind and distributed cognition thesis (Atã and Queiroz 2014). Recently, the distributed cognition and extended mind approach have questioned the legitimacy of skin and skull to serve as criteria for the demarcation of the boundaries between mind and the outside world (see Clark and Chalmers 1998; Clark 1998, Clark 2006b). For Peirce, mind is semiosis (i.e., sign action) in a materially embedded form and cognition is the development of available semiotic artifacts, in which is embodied a power to produce interpretants (see Skagestad 2004). From this perspective, the fundamental unit of cognitive interest is reconceived and replaced by an environmentally embedded space of semiotic skills and artifacts. As we adapt the environment to facilitate our purposes, deploying our mind in external representations, we participate in the construction of cognitive (or semiotic) niches, which fundamentally alter our cognitive capabilities (see Clark 2006a). Cognitive niche construction transforms the environment in which cognition takes place, through the selection of environmental features capable of mediating and controlling behavior (see Magnani 2009; Clark 2008: 61–63). Beck's design has reduced the similarity of the LUD to the geographical identity of the Underground System and instead increased its similarity to a specific structure of rules and goals that characterizes a particular experience of urban transportation and urban space. It has selected a habit – a set of relations and rules of action – and materialized it through iconicity so that it manifests itself again as iconic semiosis in the behavior of the users. This formal structure thus becomes a coupled part of the mind of Londoners, now hybrid beings embedded with a particular set of rules of action. For them, the LUD stands as a common familiar model, a specialized environment built for extraction and manipulation of relations. In this sense, the impact of the efficiency of the Iconic semiosis and representational efficiency 11 1 5 10 15 20 25 30 35 40 LUD goes beyond the scope of discrete particular problem-solving tasks. It becomes part of the semiotic niche of urban dwellers, making them more suited to the urban environment and influencing in their overall behavior and perception towards the city. It is "more than a simplification of Underground railway routes [...] it is an essential simplification of the city itself" (Garland 1994: 5). 6 Some conclusions In our approach, while it may be of little relevance whether cognition is happening inside or outside the head, it is decisive that it must happen in representations: writing tools, modeling artifacts, notational systems, languages, and so forth. This conception neither restricts representations to symbolic semiosis as would orthodox representationalism nor rejects representations as would anti-representationalism. The study of distributed cognition benefits from the system proposed by Peirce in the sense that it offers a model of how and by virtue of what the mind semiotically unfolds itself. As the study of the representations and its functioning becomes a necessary part of the study of cognition, Peirce's conception of icon arises as an important tool for the investigation of thought processes. Iconicity is a central idea that connects cognitive distribution, niche construction and representational efficiency. An efficient representation is an icon of a structure of habits (rules of action) that foster certain kinds of cognitive behavior that are appropriate for an objective (here conceived as a game-like activity with an initial state and a goal state). Iconicity helps to clarify how it is possible for a habit to be embedded on a representation and be forced upon the user. Representational features act themselves as rules of action because of the interrelatedness of its parts being analogous to certain effects of the environment that allow: (i) the embedding of extractable information in the sign about the object (related to operational iconicity) and (ii) the direct manipulation of this information (related to optimal iconicity). Through iconicity, cognition is distributed. As representations mold cognitive behavior, they become part of an ongoing process of niche construction, where the cognitive potentialities of groups of individuals are expanded or directed towards certain purposes. In our example, a particular experience of urban transportation, partly determined by the technology itself of Underground transportation, materializes itself on a sign that causes urban dwellers to adapt to it, thus participating in niche construction. The most decisive step of the process happens through iconic semiosis. The reduction of the amount of information in a representation by 12 Pedro Atã et al. 1 5 10 15 20 25 30 35 40 virtue of its specialization for specific tasks does not oppose different conceptions about iconicity, but rather redefines the object of the sign, clarifying its role as the materialization of a problem space optimized to function as an environment where cognition develops through manipulation of diagrams. References Atã, P., & J. Queiroz. 2014. Icon and abduction: Situatedness in Peircean cognitive semiotics. In L. Magnani (ed.), Studies in applied philosophy, epistemology and rational ethics 8, model-based reasoning in science and technology, 301–313. Berlin & Heidelberg: Springer. Chuah, J., J. Zhang & T. R. Johnson. 2000. The representational effect in complex systems: A distributed representation approach. Proceedings of the 22nd Annual Conference of the Cognitive Science Society, 633–638. Clark, A. 1998. Being there: Putting brain, body, and world together again. Cambridge: MIT Press. Clark, A. 2006a. Language, embodiment, and the cognitive niche. Trends in Cognitive Sciences 10(8). 370–374. Clark, A. 2006b. Memento's revenge: The extended mind, extended. In R. Menary (ed.), Objections and replies to the extended mind, 1–43. Oxford: Ashgate. Clark, A. 2008. Supersizing the mind. Oxford: Oxford University Press. Clark, A. & D. Chalmers. 1998. The extended mind. Analysis 58. 7–19. Garland, K. 1994. Mr. Beck's underground map. London: Capital Transport Publishing. Hookway, C. 2002. Truth, rationality, and pragmatism – Themes from Peirce. Oxford: Oxford University Press. Magnani, L. 2009. Abductive cognition: The epistemological and eco-cognitive dimensions of hypothetical reasoning. Berlin & Heidelberg: Springer. Peirce, C. S. 1931–1935/1958Q6 . The collected papers of Charles Sanders Peirce, volumes I–VI. C. Hartshorne and P. Weiss, eds. Cambridge: Harvard University Press. Peirce, C. S. 1931–1935/1958Q7 . The collected papers of Charles Sanders Peirce, volumes VII–VIII. A. W. Burks, ed. Cambridge & Charlottesville, VA: Harvard University Press & Intelex Corporation. Queiroz, J. 2012. Complexification. In D. Favareau, P. Cobley & K. Kull (eds.), A more developed sign – Interpreting the work of Jesper Hoffmeyer, 67–70. Tartu: Tartu University Press. Skagestad, P. 2004. Peirce's semeiotic model of the mind. In C. Misak (ed.), The Cambridge companion to Peirce, 241–246. Cambridge: Cambridge University Press. Spence, R. 2007. Information visualization: Design for interaction. Upper Saddle River, NJ: Prentice Hall. Stjernfelt, F. 2007. Diagrammatology – An investigation on the borderlines of phenomenology, ontology, and semiotics. Heidelberg: Springer. Stjernfelt, F. 2011. On operational and optimal iconicity in Peirce's diagrammatology. Semiotica 186. 395–419. Walker, J. A. 1979. The London Underground Diagram: A semiotic analysis. Icographic 14–15. 2–4. Iconic semiosis and representational efficiency 13 1 5 10 15 20 25 30 35 40 Whitby, B. 1996. Multiple knowledge representations: Maps and aeronautical navigation. In D. Peterson (ed.), Forms of representation, 67–79. Wiltshire: Intellect Books. Zhang, J. 1997. Distributed representation as a principle for the analysis of cockpit information displays. The International Journal of Aviation 7. 105–121. Zhang, J. & D. A. Norman. 1994. Representations in distributed cognitive tasks. Cognitive Science 18. 87–122. Bionotes Pedro Atã is a student at the Post-Graduate program in Languages, Culture, and Arts at the Federal University of Juiz de Fora, Brazil. He is a researcher of the Iconicity Research Group and assistant editor of the Commens Digital Companion to Charles S. Peirce. His interests include distributed cognition, semiotic theory of mind, creativity, and abduction. Breno Bitarello is a tattoo artist and a Ph.D student in the Education, Arts, and History of Culture program at Mackenzie University in Brazil. His research interests are tattoo and tattooing, interactive arts, and process art. João Queiroz is a professor at the Institute of Arts and Design, at the Federal University of Juiz de Fora, Brazil. He co-edited two special issues of the journal Semiotica: "Diagrammatical reasoning and Peircean logic representations" (2011) and "Abductive inference" (2005). He is co-editor of the Commens Digital Companion to Charles S. Peirce together with Mats Bergman and Sami Paavola. His academic interests include Peirce's philosophy and semiotics, situated and embodied cognition, and abductive inference. 14 Pedro Atã et al. 1 5 10 15 20 25 30 35

La Iglesia doliente Un largo invierno en Cracovia Miriam Dolly Arancibia Libro publicado en Ediciones Plaza, San Juan, 2013 ISBN: 978987-1899-89-0 (197 páginas) Extractos del libro original ¡Libertad – una conquista continua, simplemente no puede ser poseída! Se trata de un regalo, pero mantenerla es una lucha. Regalo y lucha inscriben sus páginas, ocultas todavía abiertas. Por la libertad pagas con todo tu ser, por lo tanto llama a tu libertad que te permite pagar el precio, de poseerte a ti mismo nuevamente cada vez. A ese precio entramos en la historia y toca sus épocas. ¿Dónde está la línea divisoria entre aquellas generaciones que pagaban muy poco y aquellas que pagan demasiado? ¿En qué lado de esa línea estamos? (Thinking my country, Karol Wojtyła, Poemas, 1998, 212) 2 Tabla de Contenidos 1. Antecedentes históricos de la ciudad de Cracovia La Universidad La edad de oro La Academia de Bellas Artes El siglo XX El período entreguerras (19181939) Cracovia bajo el dominio nazi (1939-1945) 2. Edith Stein 3. La posguerra, inicio del dominio soviético 4. La Iglesia Católica en Polonia durante el régimen soviético 5. Jerzy Popiełuszko Su filosofía de vida 6. Conclusión Referencias bibliográficas Perfil biográfico e intelectual de la autora Prólogo Crecí en la Fe católica gracias a los cuidados de mis padres. De sus labios, de sus plegarias y de sus gestos aprendí desde muy niña a confiar en Jesús. La formación religiosa fue un elemento muy importante en nuestra familia. La práctica del culto, las escuelas religiosas, moldearon una parte importante de mi niñez. A esto se sumaba la existencia de una pequeña biblioteca que fue creciendo a la par de nuestras vidas colegiales. Mi madre conservaba el afán de sabiduría de sus años de formación docente, se mantenía actualizada en muchos aspectos, la compra de libros formaban parte del presupuesto familiar. Desde muy niña aprendí a valorar las horas de lectura, me sumergía en mundos asombrosos, aprendía a soñar y volar con mi imaginación a países muy lejanos, poseedores de otras culturas y de diversas lenguas. De todos aquellos libros, que permanecen en un rincón de mis recuerdos, hubo uno cuyo título quedó grabado en mi memoria: "He visto el rostro doliente de la Iglesia" de Vicente D'Agostino. Con toda la inmadurez propia de una niña que apenas comenzaba los años escolares, no alcanzaba a dibujar en mi mente el mapa de aquel mundo desconocido. Recuerdo que escuchaba conversaciones sobre los sufrimientos de nuestra Iglesia en países 3 dominados por el comunismo. Corría el año 1968, yo contaba con apenas 8 años, las convulsiones sociales del mundo me eran ajenas dada mi corta edad. Pero quedaban grabadas en mí aquellas impresiones transmitidas en charlas de mayores. Transcurrieron los años y ese libro permanecía en nuestra biblioteca o en la mesa de noche de mi madre. Con el tiempo pude leerlo, aunque muchas de aquellas cosas eras inexplicables para mí. En una ocasión recibimos la visita de una familia emigrada de Hungría, escapaban de un sistema que los aterrorizaba y lo hicieron corriendo el riesgo de perder su vida. Conservo nítidos recuerdos de aquellas conversaciones de adultos, relatos sobre sufrimientos padecidos, sobre la falta de libertad, sobre las persecuciones y finalmente sobre la huida de la patria. Más de cuarenta años después, tuve la ocasión de vivir durante seis meses en la ciudad de Cracovia, Polonia, desde el otoño de 2012 hasta la llegada de la primavera del 2013. La imaginación alimentada por aquellas lecturas infantiles finalmente alcanzaba el nivel de la realidad. Así, recorriendo las calles de Cracovia, los campos de concentración de Oświęcim (Auschwitz I y Auschwitz II, Birkenau), y especialmente cuando me detuve a orillas del Danubio desde el Puente de las Cadenas en la ciudad de Budapest, Hungría, tornaron a mi mente con diáfana claridad los recuerdos de infancia: aquel libro que por alguna extraña razón me impactara tan profundamente, las comidas, los juegos y conversaciones con aquella familia húngara. Mi estancia en Cracovia fue posible por una beca de estudios postdoctorales, ello me permitió conocer la historia de aquellos pueblos no sólo a través de sus libros sino también conviviendo con su gente y aprendiendo su lengua. Fui descubriendo la cruda realidad de aquellas palabras: "He visto el rostro doliente de la Iglesia", sí, efectivamente, las historias allí ocurridas por causa del totalitarismo nazi y comunista fueron el rostro doliente de una Iglesia perseguida. Desde hace veinte años llegó la democracia a aquellos lares, muchas cosas han cambiado, tuvieron grandes victorias como la colosal figura de Karol Wojtyła. La Iglesia católica en Polonia enfrenta ahora nuevos problemas, nuevos desafíos, el período de transición postcomunista los impulsa a replantearse muchos aspectos de su identidad. Sin embargo, las heridas profundas 4 causadas durante los años del terror no cicatrizaron del todo todavía. Es imposible pasar por Cracovia y luego pretender seguir como si nada. Su historia deja huellas penetrantes. Las víctimas fueron innumerables, murieron miles y miles de personas, de pobladores civiles, religiosos, judíos, y por cuanto motivo encontraron los represores de turno. De todas aquellas víctimas, me centraré en las figuras de Edith Stein y de Jerzy Popiełuszko pues creo que sus vidas heroicas no pueden quedar en el olvido, merecen que sus nombres sean recordados por muchas generaciones de diversas partes del mundo. Con ellos y a través de ellos, en este libro busco rendir homenaje, con profunda admiración, a quienes vivieron la Fe hasta el límite de morir por ella. Muchos de los distinguidos y eminentes polacos quedarán ausentes tanto en referencias como en citas bibliográficas, espero que el lector sepa comprender, pues el impulso que me mueve a escribir no obedece a afanes de rigurosidad dictados por la academia sino, sencilla y principalmente, a la necesidad íntima de narrar un relato subjetivo a partir de una experiencia vital de aprendizaje, acaecida durante un largo invierno en la bellísima ciudad de Cracovia. Introducción Edith Stein y Jerzy Popiełuszko, dos vidas, dos mártires, dos víctimas del terror totalitario, uno en tiempos del nazismo, el otro bajo el dominio comunista. Ambos germinaron, resplandecieron y murieron en tierra polaca, si bien Edith Stein nació en Breslau, una ciudad de la región de Baja Silesia que en ese entonces pertenecía a Alemania. Luego de la Segunda Guerra Mundial, dicha ciudad pasó a formar parte de Polonia, actualmente se la conoce con el nombre de Wrocław. Geográficamente, Polonia está ubicada en Europa Central, no en Europa del Este como generalmente se la identifica debido a su período histórico bajo el régimen soviético. Su posición geográfica le ha valido quedar a merced de los intereses de potencias siempre acechantes. Fue sucesivamente dividida, en el siglo XVIII, por el Tratado de 1772 que repartía la República de las Dos Naciones entre Rusia, Austria y Prusia. 5 La segunda partición tuvo lugar en 1793 por obra de Rusia y Prusia, entre los cuales se distribuyeron grandes extensiones del territorio polaco. En 1795 Rusia, Prusia y Austria se repartieron nuevamente Polonia luego de luchas, acuerdos y tratados. De este modo desapareció Polonia como estado independiente hasta 1807, fecha en la que se erigió el Gran Ducado de Varsovia aunque el Estado polaco independiente se lograría recién en 1918, luego de la Primera Guerra Mundial. La ciudad de Cracovia, o Kraków, es una de las más antiguas de Polonia, actualmente es la segunda luego de su capital, Varsovia. Situada a orillas del río Vístula en la Baja Región de Polonia, la ciudad data del siglo VII d.C. Tradicionalmente fue uno de los principales centros de la vida académica, cultural y artística, así como uno de los centros económicos más importantes. Fue capital de Polonia desde 1038 a 1569; capital de la Comunidad lituano-polaca desde 1569 a 1596; constituyó el Gran Ducado de Cracovia desde 1846 a 1918; y Región de Cracovia desde el siglo XIV hasta 1999. Actualmente es la capital de la Región de Baja Polonia. Durante muchas centurias Cracovia fue conocida como "la Roma polaca" debido a sus numerosas iglesias y sus sitios sagrados. Probablemente alrededor del año 1044 el rey Kazimierz llevó los benedictinos a Cracovia. En el año 1222 llegaron los dominicos, y poco después los jesuitas, quienes se establecieron en la iglesia de San Pedro y San Pablo, la cual con el tiempo fue transformada en parroquia trasladándose los jesuitas a nuevos terrenos situados en la calle Copérnico. Cracovia es tierra de santos polacos y beatos de la Iglesia católica 1 . Desde tiempos remotos acuden a ella peregrinos para venerar pinturas milagrosas ubicadas en diversos sitios de la ciudad. Una de ellas es la imagen de Nuestra Señora de las Arenas, coronada ceremonialmente en 1883; o también la de Nuestra Señora de los Dolores en la Iglesia de los franciscanos. Actualmente existe una cierta tolerancia religiosa que permite la convivencia de católicos, judíos y ortodoxos ucranianos entre otras religiones, sin embargo, la fuerte presencia de la Iglesia católica 1 MALECKI, Jan M, A history of Kraków for everyone, WYDAWNICTWO LITERACKIE, Kraków, 2008p.202 6 continúa siendo un hecho significativo en la vida cotidiana de Cracovia. En comparación con otras ciudades de Europa, las iglesias no son meros museos con tesoros artísticos sino que reciben una permanente concurrencia de personas practicantes de los ritos y del culto católico. Así por ejemplo, son llamativas las oraciones y visitas diarias a la imagen de Nuestra Señora de Jasna Góra, réplica de la que se encuentra en el Monasterio de Czestohowa, situada en la entrada de la Basílica de Santa María, frente a la plaza principal de Cracovia. También recibe el nombre de reina y madre de Polonia. Las celebraciones litúrgicas se realizan en diferentes idiomas tales como inglés, francés, italiano y español aunque la lengua polaca es todavía predominante. Otros aspectos de la religión católica presentes en Cracovia son la participación de las familias, la cantidad de jóvenes en las celebraciones, el número de candidatos al sacerdocio y de mujeres consagradas a la vida religiosa. En este contexto, no es extraño el recuerdo permanente de Karol Wojtyła quien vivió allí muchos años, como sacerdote, profesor, obispo, arzobispo y cardenal antes de que se convirtiera en el Papa Juan Pablo II. A primera vista, Cracovia es una ciudad conservadora de sus tradiciones, de sus cuentos y leyendas, de sus actividades comerciales, de su música, de su religión católica, como también de la cultura judía y de los recuerdos de las guerras mundiales, especialmente los de la segunda guerra mundial. En esta ciudad, que fuera el foco cultural de Europa central, acaecieron algunos de los grandes genocidios de la historia. Un cuarto de su población fue exterminada por los nazis, y muchos miles continuaron desapareciendo bajo el régimen soviético. Por esta ciudad pasaron los trenes que transportaban condenados a los campos de la muerte. Las historias de lucha, muerte y supervivencia tejidas en sus entrañas, conforman todavía hoy el halo de misterio en torno a la pregunta por la naturaleza humana y el acecho constante del mal. Frente a ello, Edith Stein y Jerzy Popieluszko levantaron la cruz como símbolo de libertad, de vida, de eternidad. 7 1. Antecedentes históricos de la ciudad de Cracovia Llegué a Cracovia hacia fines de otoño, dispuesta a absorber todos los conocimientos que el contacto con otra cultura me proporcionaría. No alcanzaba a imaginar el mundo nuevo que se abría a mis pies ofreciéndome un sinnúmero de enriquecedoras experiencias. Puesto que en mi ciudad no existen comunidades polacas, partí a Polonia desconociendo su lengua, su cultura y una gran parte de su historia. Esta carencia me preparaba positivamente, me entusiasmaba la posibilidad de conocer lo que hasta ese momento ignoraba. No me atemorizaba el famoso invierno polaco pues sería también una experiencia nueva para mí, habituada a 45° de calor durante las largas jornadas veraniegas de la región cuyana. Había obtenido una beca Erasmus de investigación postdoctoral en Filosofía en la Universidad Jagiellonian, por lo cual me encontraba exenta de la asistencia a clases. Ello me permitía organizar mi propio proyecto de investigación, con mayor flexibilidad y libertad que los jóvenes llegados desde diversas partes del mundo para iniciar sus estudios de maestría. No encontré otros profesores con becas similares a la mía, comprendí entonces que mi investigación sería un camino muy solitario pero al mismo tiempo me encontraba gozando de independencia, libertad y una biblioteca poseedora de incalculables tesoros. Curiosamente, esa soledad que me obligó a valerme por mí misma en muchas ocasiones me ayudó a imaginar con mayor cercanía los trágicos momentos que marcaron la historia polaca. Muy pronto me enfrenté con la barrera del idioma ya que el inglés requerido para la beca no suplía el idioma polaco, por el contrario, necesitaba aprenderlo con apremio a fin de resolver las urgencias de la vida cotidiana. Perdurarán en mi memoria las callecitas que atravesaba desde mi departamento hasta un instituto de idioma ubicado frente al castillo Wawel donde tomaba clases de idioma polaco tres veces por semana. Había alcanzado las condiciones ideales para un largo y fructífero período de estudio, estaba decidida a no desperdiciar la oportunidad y con todo el entusiasmo me lancé a conocer la ciudad, su gente, sus costumbres, su historia. Los días todavía eran cálidos y soleados; las tardes eran animadas por turistas que recorrían las atracciones organizadas para el 8 tiempo otoñal; los estudiantes comenzaban a llegar y a instalarse en las numerosas residencias estudiantiles. La plaza de Cracovia sin lugar a dudas es hasta hoy el corazón de la ciudad. Todos convergen a ella. Siempre hay motivo para festejar y celebrar: el final del otoño, el de las vacaciones, el tiempo de Navidad, de Pascuas. Todo es ocasión para instalar el mercado, un escenario, o para comer platos típicos o presenciar espectáculos de danzas tradicionales. Los carruajes tirados por caballos, rememorando un lejanísimo tiempo de nobles y reyes, atraen a turistas al mismo tiempo que engalanan la ciudad. Los acordes de los nocturnos de Chopin, que resonaban frecuentemente en calles, teatros, radios, me acompañaban como telón de fondo otorgando un aire romántico y melancólico a mis primeras impresiones. A cada paso descubría un pedacito de historia celosamente guardado, un trozo de leyenda, todo ello me llevaba gradualmente por un recorrido en el tiempo como en un espiral que llegaba hasta los orígenes. Los primeros rastros de asentamientos descubiertos por los arqueólogos en la región de Cracovia datan de tiempos muy lejanos, de muchos miles de años atrás 2 . Los primeros en asentarse fueron cazadores (de osos y mamuts), seguidos por los productores y agricultores primitivos. Más recientemente, en la escala de los últimos 2000 años, los pueblos de origen celta se establecieron en el siglo VI d.C. 3 Con el tiempo, el medio ambiente natural favorable alentó a los asentamientos humanos a crecer y multiplicarse en el área metropolitana de la actual Cracovia. Las tierras alrededor del Vístula (Wisła) eran fértiles, regadas por los afluentes más pequeños: el Rudawa, el Prądnik y el Wilga. Había colinas que preveía un eficaz potencial de defensa; la más importante entre ellas era Wawel, cuyo nombre ha evolucionado de la palabra polaca wąwel, que significa un lugar seco y elevado en medio de pantanos. Todos estos factores permitieron el desarrollo y crecimiento de viviendas constituyéndose con el tiempo en centro de la vida religiosa y económica 4 . La tradición popular remite los orígenes de la ciudad a la leyenda en torno a su fundador el rey Krak, su hija la princesa Wanda y al 2 Idem, p.7 3 Ver: HALECKI, O., A History of Poland, London and Henley: Routledge and Kegan Paul, 1978 4 Idem, p. 8 9 dragón de Wawel. Esta historia, era especialmente adorada en la época romántica, ampliamente explotada por artistas de la época, y confirmada gracias a algunas reliquias conservadas. El antiguo montículo en la orilla derecha del Vístula llegó a ser conocido como el montículo de Krak (o Krakus) y otro en las cercanías de la aldea de Mogila, estuvo ligado a Wanda. Se creía en la existencia de un dragón en la cueva rocosa en la colina de Wawel que fue derrotado y sus enormes huesos fueron colgados fuera de la entrada a la Catedral. La leyenda de Krak forma parte de los cuentos legendarios de Polonia. Las leyendas de otras regiones polacas incluían la historia de Popiel y Piast. Esas historias son muy antiguas, fueron elaboradas y publicadas por Wincenty un maestro llamado Kadłebek (o Vincent Kadłubek de Cracovia, alrededor del 11501223) en sus crónicas de los reyes y príncipes de Polonia. En su narración, Grakch o Krak, llegó a las tierras del Vístula bañado por la gloria de muchas batallas victoriosas, una vez proclamado rey por los polacos se dedicó a organizar su estado. Sin embargo, sus súbditos, se encontraban aterrorizados por un monstruo horrible, al cual mantenían constantemente saciado con ganado para impedir el secuestro de seres humanos. Dos hijos de Grakch idearon una treta: atrajeron a la bestia con un cadáver relleno de azufre, lo devoró y el azufre le provocó la muerte. El menor de los dos, en un intento de reunir todos los elogios para sí mismo, asesinó a su hermano y se convirtió en sucesor al trono de su padre. Fue él quien fundó la ciudad en la colina sobre la guarida del dragón. Cuando el crimen de Krak el joven fue expuesto, fue expulsado del país y le sucedió en el trono su hermosa hermana Wanda. Sin embargo, puesto que ella había jurado permanecer virgen, su muerte dejó al país sin un heredero. Este cuento fue reelaborado por sucesivas crónicas. En ellas surgen nuevas interpretaciones de la leyenda: del dragón se afirma que para saciar la sed terrible causada por el azufre bebió tanta agua del Vístula hasta que el animal reventó. Se habla de Skuba, el zapatero astuto que ideó como aprisionar el extremo del dragón; o sobre el príncipe alemán Rytygier quien se enamoró perdidamente de Wanda y quería casarse con ella, pero se frustró porque ella misma se ahogó en el Vístula en lugar de ceder la corona a un alemán. 10 Las investigaciones científicas contemporáneas explican estas historias. Es difícil confirmar algo de verdad en la historia de Wanda ya que los primeros eslavos no permitían que sus mujeres ocuparan puestos de autoridad. El nombre "Cracovia" se pensaba que era de origen posesivo, es decir, derivado del nombre del propietario de la ciudad o poblado, por ello se afirmaba la existencia de un Krak (o Krok) conectado con los inicios de la ciudad, de quien no se sabe nada más con certeza. Según Małecki, todas estas leyendas sobre el origen de Cracovia, albergan al menos algo de verdad, indican que ya en el período tribal la ciudad debe haber sido un importante centro de poder y probablemente también de comercio y culto religioso 5 . El siglo IX fue un período de integración en las tierras de los eslavos. La importancia de Cracovia como centro de poder está confirmada por descubrimientos arqueológicos. Debajo de la superficie del patio en el castillo Wawel las bases han sido desenterradas de un edificio rectangular que data de alrededor del mismo tiempo como más antiguo dejando al descubierto edificios sacros. Se piensa que pertenecen a lo que fuera el paladio, el asiento del príncipe o su gobernador y probablemente data del siglo X. La historia de Cracovia en ese siglo continúa en gran medida envuelta en el misterio. Sin embargo, es probable que estuviera todavía bajo dominio Checo, algo que indirectamente se sugiere por una fuente árabe. Esto sugeriría que en el momento en que se estaba creando el estado polaco, Cracovia no era parte de él y que fue posiblemente adherida recién hacia el final del reinado de Mieszko, en 990, o unos años antes, cuando su hijo Bolesław el valiente, se estableció en Cracovia y comenzó a gobernar allí independientemente 6 . En el famoso Congreso de Gniezno en el año 1000, cuando Bolesław recibió al emperador romano Otto III en la tumba de San Adalberto, se estableció la jerarquía de la iglesia en Polonia: además del Obispado de Poznan, un Arzobispado, o metrópolis, fue instaurado en Gniezno y obispados subordinados a él en Kołobrzeg, Wrocław y Cracovia. No cabe duda de que el cristianismo llegó a Cracovia en el siglo X, en algún momento 5 Idem, p.12 6 Idem, p.16 11 entre 930 y 965, y que su llegada debe vincularse con el período de gobierno checo, por lo tanto antes del bautismo de Mieszko I. El 25 de Abril del año 1333 tuvo lugar la segunda coronación real en la Catedral de Wawel. En esa ocasión, accedió al trono el hijo de Łokietek, Kazimierz, quien luego fue llamado "el Grande". A lo largo de su reinado (13331370), no sólo sostuvo el pequeño estado que había heredado, sino que también aseguró las fronteras y extendió su territorio. Sin embargo, su mérito principal fue el de llevar estabilidad interna al país y le permitió la prosperidad económica y cultural 7 . El prestigio económico de Cracovia fue favorecido por la proximidad de las minas de sal en Wieliczka y en Bochnia, a las que Kazimierz el Grande les prestó particular atención. También extendió su apoyo a la misma ciudad de Cracovia con nuevos privilegios comerciales. Con ese fin, obligaban a pasar por la ciudad a los comerciantes extranjeros procedentes del sur, protegiéndola de este modo, de la competencia con Torun. La capital del reino no sólo prosperaba económicamente, rápidamente se convirtió en centro de la política de alto nivel realizada por Kazimierz, quien en ocasiones actuó como mediador en conflictos entre otros monarcas. Durante el reinado de Kazimierz Magno dos ciudades nuevas fueron creadas en las inmediaciones de Cracovia, una de ellas recibió el mismo nombre del rey: Kazimierz, la otra, Kleparz, fue ubicada en el lado opuesto de la ciudad. La Universidad Cracovia debe una de sus mejores posesiones a Kazimierz el Grande: su Universidad. Convertirse en una ciudad universitaria tenía inmensa importancia, no sólo en términos de cultura y de ciencia 8 . En Europa occidental había varias universidades, pero en Europa Central (incluyendo las tierras alemanas) había sólo una, en Praga, fundada en 1348 por Carlos IV. Kazimierz trabajó por esta meta por más de una década. Uno de los pasos necesarios era el consentimiento del Papa. A principios de 1364 el Papa Urbano V emitió la bula aprobando los planes del 7 Idem, p. 49 8 Idem, p. 58 12 rey. El 12 de mayo de 1364 el rey hizo el documento oficial de fundación. La Universidad de Cracovia o la Academia, contemplaba un studium generale, una escuela general modelada en los institutos italianos de aprendizaje. Debía ser organizada como una asociación de estudiantes quienes elegirían a un rector, al mismo tiempo, era también una institución del estado pues el rey financiaba su mantenimiento. Los fondos provenían de los ingresos de las minas de sal. La universidad disfrutó de una cierta autonomía dentro de la ciudad, los miembros de la asociación de estudiantes estaban sujetos a la jurisdicción del rector. Fueron creadas tres facultades: la de artes liberales, la de medicina y la de derecho. Probablemente se ubicó en el Castillo real en el monte Wawel 9 . La muerte prematura del rey Kazimierz, en 1370, sumada a la total falta de interés por parte de su sucesor, el rey Luis de Anjou (rey de Polonia y Hungría), la llevó gradualmente a su colapso 10 . La universidad fue restablecida en 1400 gracias a la intercesión de la reina Jadwiga, quien llevó el caso ante el Papa en Avignon. La estructura correspondió a la de las universidades medievales, basadas en un sistema jerárquico, de allí que la facultad de teología era considerada la más importante y que ofrecía mejor remuneración. La edad de oro En la segunda mitad del siglo XV florecieron las escuelas de Matemáticas y Astrología. Sus representantes más eminentes fueron: Marcin Krol de Zurawica (1422 1460); Marcin Bylica de Olkusz (1433 1493), quien más tarde se convirtió en el jefe astrólogo del rey Matthias Corvinus en Buda; Marcin Biem (ca. 1470 1540), quien ideó la reforma del calendario Juliano; Jan of Glogow (1445 1507), el autor de numerosos tratados matemáticos y astronómicos conocidos en toda Europa. Entre los años 1491-1495, Mikolaj Kopernik (Nicolaus Copernicus) estudió artes liberales en Cracovia. El Alma Mater de Cracovia fue también un destacado centro para el estudio de la Geografía. Su más prominente geógrafo fue Maciej Miechowita, también médico e historiador. Fue el autor de la 9 WALTOS, Stanisław, http://www. Jagiellonianhomepage 10 Idem 13 notable y ampliamente traducida Tractatus de duabus Sarmatiis (1517), que proporcionó la primera descripción sistemática de las tierras entre el Vístula, el Don y el mar Caspio. Desde la segunda mitad del siglo XV Cracovia comenzó a atraer a empresarios extranjeros exitosos e innovadores. Esta nueva tendencia llegó en un momento de reactivación económica significativa en Europa occidental, cuando se separaban nuevas formas de producción y comercio, se experimentaba una verdadera revolución gracias a importantes descubrimientos geográficos 11 . Sus principales líneas de negocio fueron el comercio a larga distancia, las operaciones de banca, industria y minería. En el primer cuarto del siglo XVI la Academia de Cracovia estaba todavía en su apogeo, atrayendo estudiantes de Polonia y del exterior. El impacto del renacimiento se reflejó en la Universidad por el aumento de interés en la cultura de los antiguos, con énfasis en el uso del latín clásico y la introducción de griego e incluso hebreo 12 . El advenimiento de las ideas humanistas en Cracovia fue acompañado por la difusión del nuevo estilo del renacimiento italiano. Cambió también el aspecto edilicio de la ciudad, su estilo gótico: casas burguesas, torres defensivas, altísimas torres de iglesia y muros fortificados como los del castillo de Wawel, fue reemplazado dando lugar a una ciudad más abierta, llena de luz. El blanco de su piedra arenisca mezclado armoniosamente con la policromía, reflejó el culto humanista de la belleza y la alegría de vivir 13 . Sin embargo, desde mediados del siglo, la atracción por estudiar en Cracovia comenzó a decaer debido a la moda entre jóvenes polacos de estudiar en universidades italianas. En la primera mitad del siglo XVI, la Academia de Cracovia rechazó las ideas de la Reforma 14 . La censura por parte del obispo y del rector eliminó todos los impresos considerados heréticos. En consecuencia, se cerraron las residencias para estudiantes alemanes y húngaros, sólo los estudiantes polacos y lituanos continuaron estudiando en Cracovia. Además, el número de jóvenes nobles en la Universidad declinó constantemente, dado que la nobleza polaca había adquirido los derechos para importantes cargos independientemente de los requisitos académicos. Algunos jóvenes 11 MALECKI, J, Op. cit. p.92 12 Idem, p.97 13 Ibidem 14 WALTOS, Stanisław, http://www. Jagiellonianhomepage 14 nobles se interesaban por estudiar pero lo hacían fuera de Polonia, principalmente en Boloña y en Padua. La Academia de Cracovia aún contaba con un número de eminentes eruditos polacos y extranjeros, los cuales daban conferencias sobre el trabajo fundamental de Copérnico De Revolutionibus, pero la edad de oro se acercaba a su fin. En el siglo XVIII la Universidad continuó declinando. Se introdujo la enseñanza sistemática del alemán y del francés, así como cursos en Derecho polaco, Geografía e Ingeniería militar. Se designaron profesores entre los eruditos educados en universidades extranjeras bajo el espíritu de la ilustración, fueron ellos quienes las transmitieron entre los estudiantes. Por otra parte, la comunidad universitaria participó activamente en la insurrección de Kosciuszko de 1794, las autoridades universitarias donaron prácticamente todos sus objetos de valor a la causa nacional 15 . La tercera y última partición de Polonia planteó una grave amenaza para la existencia misma de la Universidad. Fue sometida borrando su carácter polaco y reduciéndola gradualmente al rango de escuela secundaria. Esta amenaza desapareció después de la derrota de Austria en la guerra con Francia en 1809, cuando Cracovia fue incorporada al ducado de Varsovia. Al proceso de liberalización política dentro de Austria le siguió la concesión de autonomía a Galicia, la parte de Polonia que se encontraba bajo dominio austríaco. Fue el comienzo de otra época dorada para la Universidad, rebautizada como Universidad Jagiellonian en 1817. Una vez más, la Universidad se convirtió en un importante centro académico. Los logros científicos de la época incluyeron el trabajo de los siguientes profesores: el químico Karol Olszewski (18461915), el físico Zygmunt Wroblewski (1845-1888), quienes fueron los primeros en licuar oxígeno y nitrógeno del aire en 1883 y más adelante también otros gases; el fisiólogo Napoleón Cybulski (1854-1919), quien explicó el funcionamiento de la adrenalina; el patólogo anatómico Tadeusz Browicz (1847-1928), quien identificó el microbio de la fiebre tifoidea; el físico Marian Smoluchowski (1872-1917), autor del mayor trabajo sobre la teoría cinética de la materia; el químico León Marchlewski (1869-1946), quien realizó investigaciones sobre la clorofila; Paulin Kazimierz 15 Idem 15 Zurawski (1866-1953) y Stanislaw Zaremba (1863-1942), cuya investigación excepcional dio origen a una nueva escuela de matemáticas; su trabajo fue desarrollado por sus discípulos eminentes. La conciencia de los polacos de su propia historia fue formada en gran parte por las obras de los historiadores ilustres de Cracovia, particularmente por Michal Bobrzynski (1849-1935) y Jozef Szujski (1835-1883). Otros famosos eruditos fueron Kazimierz Morawski (1852-1925), que se especializó en estudios clásicos y Leon Sternbach (1864-1940), especialista en estudios bizantinos. La mayoría de los estudiantes eran hombres, pero en 1897 fueron admitidas las primeras mujeres para estudiar Farmacia. Poco a poco fueron aceptadas por otras facultades; la última en hacerlo fue la facultad de Derecho en 1918. La Academia de Bellas Artes La segunda entre las escuela más antiguas de educación superior en Cracovia, después de la universidad, fue la Academia de Bellas Artes 16 . Comenzó con un grupo de sillas de pintura y dibujo en la universidad en 1818 y en 1873 finalmente logró su autonomía como la Escuela de Bellas Artes. Unos años más tarde contaba con su propio edificio en lo que es ahora la Plaza Matejki y donde continúa hasta hoy. El primer director de la escuela independiente de Bellas Artes fue el mejor artista nacional de Polonia, Jan Matejko (1838-1893), quien fue responsable de su desarrollo. Matejko fue el creador de numerosos retratos y pinturas sobre temas históricos, incluyendo enormes lienzos que representaban acontecimientos y figuras de la historia de su nación, lo cual tuvo un gran impacto en la conciencia histórica de los polacos. Aunque muy criticado como artista, fue famoso y respetado por su contribución al fortalecimiento de la conciencia nacional y por generar un vínculo con la tradición histórica. El siglo XX ............................................................................................. 16 Małecki, J, Op. cit. p. 197 16 2.Edith Stein Edith Stein fue parte de ese triste período nazi en la historia de Polonia. De orígenes hebreos, ella misma nos relata su vida en tiempos tan oscuros. De ese modo, nos ha legado un gran tesoro, el ejemplo de la riqueza espiritual y la entereza frente a la adversidad que sólo las almas nobles pueden alcanzar. Fue precisamente en la primavera del año 1933 cuando escribió el libro Historia de una familia hebrea 17 , por expreso pedido de un sacerdote, a fin de reflexionar sobre la condición hebrea, dado que se introducía en Alemania una inexplicable lucha contra el pueblo judío. Con dicha obra no se proponía llevar a cabo una apología del hebraísmo ni una exposición de la religión hebraica, de un modo más sencillo pero profundo, buscaba relatar en qué consiste la humanidad hebrea desde su propia experiencia de vida. Es muy profundo el término que utiliza, "humanidad", ya que se encontraba inmersa en una época en la que ésta era negada a los hebreos. Se sentía también impactada por la educación racista de la época y quizás buscaba rendir un tributo a su madre, quien no le perdonaría fácilmente el hecho de haberse convertido al cristianismo. El árbol genealógico de Edith nos permite trazar líneas imaginarias por tierras polacas. No debe olvidarse que Polonia había sufrido tres particiones, por ese motivo, algunas ciudades formaron parte, ora de Alemania ora de Polonia, según los avatares políticos. Puesto que se proponía narrar al mismo tiempo la biografía de su madre, Edith nos remite principalmente a la línea de parentesco materna. Sus bisabuelos eran oriundos de Poznan, sus abuelos de Lublin. Por su relato sabemos que sus ascendientes se dedicaban al comercio y que las plegarias eran parte importante en la vida familiar. Lublin era el lugar de descanso, donde Edith crecía rodeada de lazos familiares muy estrechos. En sus líneas se trasluce el clima de intercambio cultural en el que fue creciendo. Los lugares por los que transcurría su infancia eran aquellos que cambiaban su signo de pertenencia, según los cambios políticos durante las particiones de Polonia. Fueron esas experiencias las que le permitieron entrar 17 STEIN, E., Storia di una familia ebrea, Lineamenti autobiografici: l'infanzia e gli anni giovanili, Roma: Cittá Nuova, 1992 17 en contacto con la lengua y cultura germana pero también con la polaca. "Cómo nos sentíamos orgullosos cuando reteníamos alguna palabra del dialecto polaco para hacernos comprender por los campesinos o cuando además eran confiados al servicio de la casa" 18 . Sin embargo, cuando Lublin fue recuperada por Polonia, los familiares de Edith lo sintieron como la pérdida de la patria, en ellos indudablemente era más fuerte el sentido de pertenencia a Alemania, al punto de emigrar para nunca regresar. Más tarde sus padres tomarían una decisión similar y es así que la familia se trasladó a Breslau (actual Wrocław). Fue allí donde nació Edith un 12 de Octubre de 1891. En Julio de 1893 murió su padre, "mi madre me tenía en brazos cuando él nos saludó antes de emprender el viaje del cual no regresaría, yo lo había llamado todavía una vez más cuando se volvió y partió" 19 . En la actual ciudad de Wrocław se encuentra todavía la casa donde Edith vivió con su madre y sus hermanos, desde 1910 hasta 1933, ubicada en la calle Nowoiejska, número 38, (antes llamada Michaelisstrasse). En 1939 las autoridades nazis expropiaron la casa a la familia Stein. Hoy alberga la sede de la Sociedad Edith Stein. Desde muy niña Edith se sentía inclinada por la docencia. Educada en un hogar donde las lecturas eran importantes, disfrutó de sus años escolares, "casi me parece que en la escuela me sentía más cómoda que en mi propia casa" 20 . En cambio, sentía poca inclinación por las tareas domésticas. Edith se describía a sí misma como una muchacha que iba pasando por transformaciones incomprensibles y discontinuas. Se definía con un carácter de mucha vivacidad durante sus primeros años de vida, estaba siempre en movimiento, llena de ideas extrañas, impertinente, invenciblemente obstinada y llena de ira cuando algo contrariaba su voluntad 21 . Sin embargo, poseía una sensibilidad tan profunda que los pesares de las demás personas la afectaban en alto grado y sufría en silencio por ellos. 18 Idem, p. 31 19 Idem, p. 67 20 Idem, p. 60 21 Idem, p. 67 18 A la edad de 7 años experimentó un gran cambio, Edith afirmaba que a esa edad predominó en ella la razón tornándose más dócil, aplacada y cortés. A los 14 años finalizó el noveno grado de la escuela superior, era la Pascua de 1906, no habiendo encontrado motivación suficiente para continuar, dejó la escuela y se fue hacia Hamburgo. Edith confiesa que no había establecido lazos de afecto con ningún profesor en particular ni con amigas, de modo que no le resultó difícil despegarse de la escuela. Su adolescencia no fue fácil y esa falta de lazos lo demostraba. El círculo de personas en el cual Edith se desenvolvía era muy estrecho, lo cual empeoraba por el modo en que se cerraba en sí misma, en su mundo interior. Por tales razones no le resultó difícil alejarse de su hogar. Diez meses permaneció en Hamburgo, retornó a Breslau pero su crisis adolescente le impedía continuar los estudios en el Liceo. Hasta los 16 años permaneció ausente de la vida escolar, una vez recobrada la necesaria motivación para retomar sus estudios, debió prepararse tomando clases particulares. Superó ampliamente el examen de admisión, comenzaba así una nueva etapa en su vida. El 3 de Marzo de 1911 Edith pasó su examen de madurez finalizando el Liceo. Continuó estudiando en la Universidad de Breslau. Allí asistió a los cursos de Introducción a la Psicología con William Stern y el seminario de Filosofía de la naturaleza con Richard Hönigswald 22 . Del seminario con Stern sacó mucho provecho, de allí se formó un grupo pedagógico al cual asistía Edith sosteniendo largas tertulias sobre problemas referidos a la enseñanza y a la práctica escolar. Permaneció en la Universidad de Breslau cuatro semestres, si bien parecía muy apegada a la institución, una vez finalizados, le embargaba la impresión de que no recibiría nada más allí. Sentía la necesidad de nuevos estímulos y casi repentinamente decidió continuar sus estudios en otra universidad lejos de Breslau. Sin dudas influyó en su decisión la lectura de las Investigaciones lógicas de Edmund Husserl sugeridas por Moskiewicz 23 . Partió entonces a Gottinga que por ese entonces era el centro por excelencia de los estudios fenomenológicos. Llevaba consigo un 22 Idem, p.178 23 Idem, p. 198 19 posible tema de tesis sugerido por Stern, pero pronto lo abandonaría convencida de que ése no era su camino. A medida que yo avanzaba en la lectura de Edith Stein era inevitable que no me sintiera cada vez más atraída por los avatares de su vida. Como mujer apasionada por la Filosofía y habiendo dedicado mi vida a mi familia y a la carrera académica universitaria, por momentos me sentía muy identificada. La vitalidad de Edith y su deseo por aprender durante los años de vida universitaria reflejan el típico perfil de una alumna entusiasta. De modo admirable, pues no la detenían los obstáculos, desde los más simples hasta los tristes períodos de la Primera Guerra Mundial. Su entusiasmo no disminuía a pesar de la incomprensión de algunas compañeras, las duras respuestas de algunos maestros, especialmente de Husserl. Nada la desanimaba, ella continuaba adelante pese a todos. La relación con Husserl fue de hecho muy fluctuante, la lectura de la gran obra Investigaciones Lógicas había conquistado el espíritu inquisidor de Edith. Fue, de hecho, el elemento motivador que la animó a abandonar las comodidades de la cercanía familiar para aventurarse en los pormenores de la vida universitaria en Gottinga. Allí disfrutó plenamente de todo lo que un auténtico ambiente académico podía ofrecer, no sólo asistía a clases sino que además se nutría de ricas conversaciones en grupos y sociedades filosóficas. No perdía el tiempo, a medida que cumplía con las obligaciones académicas elaboraba en su mente su futura propuesta de tesis. En Husserl encontró un maestro, la relación de afecto fue creciendo muy paulatinamente, alimentada por intervención de la esposa de Husserl, Malwine. Husserl era muy exigente y Edith era mujer, en un mundo universitario ofrecido casi exclusivamente a los varones, ella debió lidiar contra los prejuicios demostrando constantemente su capacidad. Pese a su permanente disposición y comprobada capacidad intelectual, así como a su afán por lograr aportes creativos a la Filosofía, su tesis se vio demorada ante la falta de lectura de su maestro. Husserl no le dedicaba la suficiente atención y el tiempo transcurría inexorablemente. ............................................................................ 20 5.Jerzy Popiełuszko Una de las impresiones más fuertes que recibí al comienzo de mi estadía en Cracovia fue la historia acerca de Jerzy Popiełuszko. Nada conocía sobre él, ni sobre su vida ni sobre su heroica muerte. Todavía recuerdo vívidamente el impacto que causó en mí cuando por vez primera supe de su martirio. Transcurrían los primeros días de mi estancia en aquella ciudad. Todo era motivo de un continuo descubrimiento. Recién había comenzado mi estudio del idioma polaco, por lo que apenas podía balbucear el alfabeto. Entre las múltiples dificultades prácticas que ello significaba, se encontraba la imposibilidad de comprender lo que se transmitía por la radio o la televisión. Sin embargo, y con el afán de aprenderlo todo absorbiendo cada rasgo de aquella cultura, el televisor se encontraba encendido en aquella tarde del 19 de Octubre de 2013. Intentaba con mucho esfuerzo incorporar algunos mínimos vocablos de un idioma tan extraño. De repente comenzó la proyección de un documental. No podía comprender lo que allí se relataba pero advertía, por la fuerza de las imágenes, que se trataba de algo muy importante. Los escasos subtítulos me ayudaban a reafirmar la idea de que era un acontecimiento de especial significado en la historia de Polonia. Rápidamente tomé un lápiz y escribí sobre un papel aquel nombre todavía impronunciable para mí: Jerzy Popiełuszko. Sabía que se trataba de algo grande. Seguí con vivo interés aquellas imágenes en blanco y negro extraídas de documentales de años pasados. Intenté captar los detalles. Apenas finalizó la proyección, busqué rápidamente información en internet en los idiomas que me resultaban más familiares y comprensibles. Fue así que conocí la historia de Jerzy Popiełuszko. Me sentí fuertemente conmovida. A partir de aquella noche, no dejé de buscarlo, deseaba conocer más sobre su vida, su padecimiento y su martirio. Me encontraba en el lugar ideal para realizar mi búsqueda y yo sabría sacar provecho de aquella circunstancia. Comencé en la biblioteca, en cuanto documento escrito podía encontrar, terminé explorando los caminos recorridos por el padre Jerzy el día de su martirio. Visité entonces la Iglesia de San Stanislaw Kostka en Varsovia, la Iglesia en donde celebró su última misa en la ciudad de Bydgoszcz y finalmente Torun. 21 Fui tras sus huellas, intentado reconstruir mentalmente aquellos momentos de angustia y de sufrimiento extremo. Sentía que su testimonio debía ser conocido, su vida, su sacrificio, sus luchas en nombre de la cruz merecían ser honrados con nuestro recuerdo. Sus palabras debían perdurar más allá de las homilías pronunciadas en los duros tiempos del régimen soviético. Su memoria debía trascender las fronteras del tiempo y de los límites geográficos. Jerzy Popiełuszko nació el 14 de septiembre de 1947 en un pueblo llamado Okopy cerca de Suchowola en Podlasie. Sus padres, Marianna y Władysław, se dedicaban al trabajo agrícola 24 . Desde 1961 estudió en la escuela secundaria en Suchowola. Sus maestros lo describían como un alumno hábil y ambicioso. Desde los primeros años sirvió como monaguillo. En el baile de graduación expresó su vocación para el sacerdocio de modo que, tras los exámenes finales en la secundaria en 1965, ingresó en el seminario en Varsovia. Al principio del segundo año de estudios se unió a las fuerzas armadas, debido al tratado de 1950 por el que el estado reclutaba clérigos como un modo de desafiar a la Iglesia. Los funcionarios estatales esperaban que bajo un sistema cuidadosamente planificado de adoctrinamiento, los jóvenes aspirantes abandonarían sus estudios en los seminarios. Desde 1966 a 1968 cumplió el servicio militar en una unidad especial para los estudiantes del seminario en Bartoszyce. Como clérigo y soldado, Jerzy Popiełuszko manifestó gran valor en la defensa de sus puntos de vista, lo cual sin embargo lo dejaba a merced de diversas formas de discriminación. Una vez finalizado el servicio militar, contrajo una grave enfermedad que acentuaría los rasgos frágiles de su aspecto físico. El 28 de mayo de 1972 fue ordenado sacerdote por el cardenal Stefan Wyszyński. En la parte trasera del cuadro conmemorando su primera misa escribió palabras muy significativas: "Dios me envía a predicar el santo Evangelio y tratar las heridas del corazón miserable". Desarrolló su sacerdocio en las siguientes parroquias: la iglesia de la Santísima Trinidad en Ząbki, la iglesia de la Santísima Virgen Reina de Polonia en Anin y la iglesia del Niño Jesús en Żoliborz. 24 http://www.popieluszko.net.pl/english/index_en.php, Parafia Św. Stanisława Kostki w Warszawie 22 En su sacerdocio se sintió especialmente aficionado por el trabajo con niños y adolescentes. Hacia fines de 1978 fue nombrado líder de la comunidad del personal médico. A partir de esa fecha celebró mensualmente la misa en la capilla de Sacra Miser. Gracias a su compromiso en estas reuniones de oración nació la idea de unificar a muchas personas en círculos médicos, especialmente a las enfermeras. Uno de los frutos de este intenso trabajo pastoral fue la creación de un servicio médico voluntario durante la primera peregrinación del Papa a Polonia en 1979. El 20 de mayo de 1980 comenzó su ministerio en la iglesia parroquial de San Stanislaw Kostka 25 , el que iba a ser el último lugar de su trabajo. El verano de 1980 fue un período de avance en su vida. El domingo 31 de agosto una delegación de trabajadores siderúrgicos preguntó por el cardenal Stefan Wyszyński para pedirle la asistencia de un sacerdote en la planta de acería. El capellán del cardenal Wyszynski se dirigió a la iglesia de San Stanislaw Kostka. En la sacristía preguntó al padre Jerzy Czarnota si podría ir a la acería, pero éste había contraído un compromiso anterior. En ese momento el padre Jerzy Popiełuszko se ofreció a ir. El mismo Jerzy recordó los acontecimientos de ese día: "Nunca olvidaré ese día y esa misa. Cuando recorría mi camino a la acería, sentía muchos temores. Era una experiencia totalmente nueva. Seguía haciéndome muchas preguntas. ¿Qué encontraré? ¿Cómo me van a recibir? ¿Habrá suficiente espacio para la misa? ¿Quién hará la lectura de las escrituras y cantará? Estas pueden parecer preguntas muy ingenuas hoy, pero en ese momento me acompañaban y me punzaban todo el camino. Cuando me acerqué a la puerta de las acerías tuve la primera sorpresa. Vi una gruesa fila de personas que estaban sonriendo y llorando al mismo tiempo. Cuando aplaudieron, pensé que alguien importante me estaba siguiendo. Pronto descubrí que el aplauso era para mí, el primer sacerdote que atravesaba la puerta de la acería. En ese momento me di cuenta de que esta alegre bienvenida en realidad estaba dirigida a la iglesia, que había ido tocando pacientemente a las puertas de las fábricas polacas durante treinta largos años. Mis dudas eran innecesarias, todo había quedado listo 25 Cfr. HELD, J., Dictionary of East European history since 1945, Westport, Connecticut: Greenwood press, 1994, p. 357-359 23 para la misa. En medio de la plaza de la fábrica había un altar con una cruz pero un confesionario improvisado. Dicha Cruz fue clavada más tarde cerca de la puerta de la acería. Ha sobrevivido a tiempos difíciles y se ha mantenido firme hasta ahora, siempre adornada con flores frescas. Algunos hombres se ofrecieron como lectores. Deberías oír las voces de aquellos hombres que a menudo habían utilizado un lenguaje vulgar y ahora estaban leyendo las sagradas escrituras con piedad. Luego exclamaron miles de personas. ¡Loado sea Dios!" 26 . La amistad espiritual entre el padre Jerzy y los acereros surgió de esa reunión alrededor del altar al aire libre. Después de ese evento celebró misa para ellos cada domingo a las 10 en la iglesia de San Stanislaw Kostka. Sostuvo reuniones mensuales regulares. Creó una especie de escuela para obreros donde él predicó el Evangelio, pero también organizó una serie de cursos para ayudarles a obtener conocimientos básicos. Especialistas en distintas materias enseñaron a obreros los conceptos elementales de literatura polaca, historia, ciencias sociales de la iglesia, derecho, economía y hasta técnicas de negocios. La mayoría de los obreros de las plantas más grandes en Varsovia eran los participantes de estos inusuales cursos. Tenían libros de estudiantes especiales e incluso pasaron los exámenes. El padre Jerzy compartió penas y alegrías con los trabajadores siderúrgicos. El 25 de abril de 1981 la bandera del movimiento de Solidaridad de las acerías de Varsovia fue consagrada en una ceremonia festiva. El obispo Zbigniew Kraszewski, celebró la misa y visitó a los trabajadores en su planta una semana más tarde. Después se declaró la ley marcial, el padre Jerzy realizó varias actividades de caridad. Proporcionó ayuda a los perseguidos y a aquellos que sufrían la injusticia 27 . A partir del 28 de febrero de 1982 celebró la misa por la patria pronunciando sermones religiosos y patrióticos (26 en total). En su predicación explicó aspectos morales de la realidad dolorosa a la luz del santo Evangelio y las enseñanzas de la iglesia. En septiembre de 1983 organizó una peregrinación de obreros de las acerías de Varsovia a Jasna Góra. Un año más tarde obreros de diferentes partes de Polonia se unieron a la peregrinación. Así la 26 Idem 27 http://www.popieluszko.net.pl/english/index_en.php, Parafia Św. Stanisława Kostki w Warszawie 24 idea del padre Jerzy tomó la forma de peregrinación anual de los obreros polacos a Jasna Góra cada tercer domingo de septiembre. Debido a sus actividades, el padre Jerzy se convirtió en blanco de duros ataques por parte de las autoridades estatales. Hubo un número creciente de incidentes a fin de intimidarlo. Su casa fue robada dos veces, su auto fue destruido y él mismo fue supervisado permanentemente. Incluso alguien arrojó un explosivo en su apartamento. En dos ocasiones el padre Popiełuszko fue víctima de un accidente de coche que parecía haber sido planeado. Comenzaron a llegar cartas oficiales a las autoridades de la iglesia diciendo que los sermones en la iglesia de San Stanislaw Kostka amenazaban a los intereses de la República popular polaca. En septiembre de 1983, el fiscal de vice provincia Anna Jackowska inició una investigación sobre presunto abuso de la libertad de conciencia y fe por presuntas amenazas a los intereses de la República popular polaca, y el 12 de diciembre el padre Jerzy debió enfrentar las acusaciones. Este evento marcó el inicio de un período extremadamente difícil en su vida. Desde enero hasta julio de 1984 fue interrogado 13 veces. También fue arrestado pero liberado por intervención de las autoridades de la iglesia. Fue acusado y liberado de castigo sobre la base de la amnistía de 1984. Obispos, amigos y feligreses lo acompañaban todos los días con la oración. Una campaña calumniosa dirigida por el portavoz del gobierno Jerzy Urban (conocido por el apodo de Jan Rem) dio lugar a ataques especialmente brutales. En el otoño de 1984 la situación del padre Jerzy se tornaba cada vez más difícil. Aunque él creía profundamente en el sentido de su servicio, estaba cansado de los continuos ataques y tenía el presentimiento de que podía morir. Problemas relacionados con su salud y la persistente tensión psicológica le hicieron considerar algún tipo de descanso, o la idea de estudiar en Roma. Sin embargo, decidió quedarse en Varsovia. El 13 de octubre en el camino de Gdansk a Varsovia ocurrió el primer atentado contra la vida del padre Jerzy. El segundo fue planeado para el 19 de octubre. El padre Jerzy regresaba a Varsovia luego de sus tareas pastorales en Bydgoszcz cuando su auto fue detenido por tres agentes de la seguridad del estado. Le 25 golpearon hasta la muerte y arrojaron su cuerpo maltrecho y atado en el río Vístula cerca de Włocławek 28 . El 20 de octubre, la radio estatal anunció que Popiełuszko había desaparecido y que se presumía que había sido secuestrado por desconocidos 29 . Al día siguiente, miles de personas de todo el país comenzaron a reunirse en la iglesia en Żoliborg, y se celebraron misas a cada hora. 28 WEIGEL, G., Op. cit. p. 149 29http://elpais.com/diario/1984/10/27/ultima/467679604_850215.html DIARIO EL PAIS, ARCHIVO SÁBADO, 27 de octubre de 1984 "Jerzy Popieluszko, sacerdote polaco, militante de la oposición, que ha sido secuestrado y del que se teme que haya podido ser asesinado" JOSÉ COMAS 27 OCT

John Bell on Subject and Object Hans Halvorson July 3, 2020 It's quite amazing that in the span of four short pages, John Bell can make the pioneers of quantum mechanics seem collectively like just so many addle-brains. I'm speaking here of Bell's article "Subject and object" (1987). I cannot deny the rhetorical effectiveness of this article. In fact, I consider it a model for how one can - with the effective application of insinuation and rhetorical question - render a view seemingly unworthy of serious consideration. Nonetheless, I cannot hold Bell's paper up as a paradigm of philosophical inquiry, because he gives so little effort to understanding what others were saying. We can do better, and we must do better, if we're ever going to make progress with the foundations of quantum physics. Bell begins his article by claiming that: 1. Quantum mechanics is fundamentally about the results of 'measurements'. 2. The subject-object distinction is needed for quantum mechanics, but 3. "Exactly where or when to make it [i.e. the subject-object distinction] is not prescribed." (p 40) Bell then says that (3) is a serious defect that makes quantum mechanics "vague" and "intrinsically ambiguous" and "only approximately selfconsistent." Let me begin by saying that I simply deny (1), i.e. that quantum mechanics is fundamentally about the results of measurements. I'm afraid that Bell has himself made a logical leap from "the quantum mechanical formalism needs a user" to "quantum mechanics is fundamentally about the results of measurements." There is a wide range of possibilities between these two extremes - e.g. that the quantum-mechanical formalism provides a means for 1 translating facts about subatomic reality into a language that human beings can understand. I will grant that Bell is correct about (2), that the subject-object distinction is needed for quantum mechanics, but unfortunately, Bell has misunderstood the sense in which it is needed. He seems to think that quantum mechanics must describe the world as bifurcated into two parts - subject and object. If that were correct, then I would completely understand Bell's unease with the distinction. If the theory describes a world with two parts, then the theory should offer some guidance about what belongs to each part. But if you think about the meaning the word "subject", it quickly becomes obvious that it's not supposed to play the role of a predicate in the theory (unlike, say, "electron"). Rather, the idea is that a subject uses the theory to describe objects - and in the case at hand, these objects fall under the laws of quantum mechanics. The theory sees no subjects, it sees only objects, and so it has no need for specifying where and when the subject-object split occurs. Such a split is a necessary prerequisite to physical theorizing, when a subject decides to use a theory to try to say something true about the world. Now what about the complaint that quantum mechanics does not specify who the subject is, or when and where and how she decides to use the theory? But wait a minute. Is there any theory that does that? What an amazing theory it would be! Indeed, such a theory would fulfill Hegel's aspiration of finally unifying the subject and object. In other words, such a theory would "theorize itself." Is Bell suggesting that quantum mechanics is defective because it doesn't yet achieve the Hegelian Aufhebung of the subject-object distinction? So, in short, Bell is correct that quantum mechanics, as it stands, needs a subject. But that is true of every theory that has ever appeared in physics - i.e. these theories need subjects to decide when and where and how to describe things. Bell's subsequent rhetoric in the article is effective only against the backdrop of his false assumption that the subject must appear in the quantummechanical description. For example, Bell raises a question for which quantum mechanics doesn't appear to have an answer. "Now must this subject include a person? Or was there already some such subject-object distinction before the appearance of life in the universe?" (p 40) 2 But quantum mechanics is simply not interested in the question of what counts as a subject. If you ask me what counts as a subject, then my answer is that anyone who can use a theory to describe things is a subject - no other qualifications are necessary! If your dog can theorize, then he is a subject, and if an artificial intelligence could theorize, then it would also be a subject. And to Bell's second question, I suspect that before the appearance of "life" in the universe, there were no things that could describe other things, and hence no subjects. But that doesn't mean that we subjects, living today, cannot describe the universe as it was before the existence of any subjects. In fact, the entire point of the subject-object distinction is that when a subject S is treating some X as an object, then it is indifferent to S whether X is also a subject - because as far as S is concerned, X is merely an object. If you now ask me, but is X really a subject or an object? Here I say that the question is misguided. Those two categories are not mutually exclusive. Without a doubt, each subject in our world can be an object of some subject's description. So perhaps what you want is a more comprehensive theory that answers the question of who or what can be a subject. But then who would be the subject who uses that theory, and must she wait for the theory to tell her that she is a subject before she can make use of it? I feel that we have now swum into deep metaphysical waters. For the business of physics, is it not enough that the subjects know who they are? Due to misunderstanding the role of the subject in quantum mechanics, Bell also falsely accuses quantum mechanics of being "intrinsically ambiguous and approximate" (p 41, emphasis in original). If quantum mechanics does not describe a world split into subject and object, then where is the ambiguity supposed to appear? If Bell says that the ambiguity arises in what quantum mechanics is intended to describe - i.e. what counts as the object - then I would ask how that is different from any other physical theory. Take one of Bell's favorite theories: Bohmian mechanics. What is Bohmian mechanics supposed to describe? You might say: it describes particles following deterministic trajectories. But then I would ask: which particles, and which trajectories? You see, even in Bohmian mechanics, it's left to the discrimination of the theoretical physicist to decide how many particles, which Hamiltonian, when the interaction turns on and off, etc.1 So, if stan1Consider, for example, the Bohmian description of a momentum measurement: According to Norsen, "one could 'turn off' the potential energy V (x) which confines the electron to the vicinity of the origin . . . " (Norsen, 2017, p 196). To echo Bell's question, exactly where and when is the potential energy turned off? 3 dard quantum mechanics is "intrinsically ambiguous and approximate" how is that not also the case for Bohmian mechanics? In "Subject and object", Bell slices and dices his opponent - a straw person of Bell's own making. The real problem, I think, is that Bell wants a theory that has no need for a subject. References Bell, J. S. (1987). Subject and object. In Speakable and unspeakable in quantum mechanics, pp. 40–44. Cambridge University Press. Norsen, T. (2017). Foundations of quantum mechanics. Springer.

Table of Contents Prematter Introduction Using the Geometry Applet About the text Euclid A quick trip through the Elements References to Euclid's Elements on the Web Subject index Book I. The fundamentals of geometry: theories of triangles, parallels, and area. Definitions (23) Postulates (5) Common Notions (5) Propositions (48) Book II. Geometric algebra. Definitions (2) Propositions (13) Book III. Theory of circles. Book VII. Fundamentals of number theory. Definitions (22) Propositions (39) Book VIII. Continued proportions in number theory. Propositions (27) Book IX. Number theory. Propositions (36) Book X. Classification of incommensurables. Definitions (11) Propositions (37) Book IV. Constructions for inscribed and circumscribed figures. Definitions (7) Propositions (16) Book V. Theory of abstract proportions. Definitions (18) Propositions (25) Book VI. Similar figures and proportions in geometry. Definitions (11) Propositions (37) Definitions I (4) Propositions 1-47 Definitions II (6) Propositions 48-84 Definitions III (6) Propositions 85-115 Book XI. Solid geometry. Definitions (28) Propositions (39) Book XII. Measurement of figures. Propositions (18) Book XIII. Regular solids. Propositions (18) Table of contents ● Propositions (18) Propositions Proposition 1. If a straight line is cut in extreme and mean ratio, then the square on the greater segment added to the half of the whole is five times the square on the half. Proposition 2. If the square on a straight line is five times the square on a segment on it, then, when the double of the said segment is cut in extreme and mean ratio, the greater segment is the remaining part of the original straight line. Lemma for XIII.2. Proposition 3. If a straight line is cut in extreme and mean ratio, then the square on the sum of the lesser segment and the half of the greater segment is five times the square on the half of the greater segment. Proposition 4. If a straight line is cut in extreme and mean ratio, then the sum of the squares on the whole and on the lesser segment is triple the square on the greater segment. Proposition 5. If a straight line is cut in extreme and mean ratio, and a straight line equal to the greater segment is added to it, then the whole straight line has been cut in extreme and mean ratio, and the original straight line is the greater segment. Proposition 6. If a rational straight line is cut in extreme and mean ratio, then each of the segments is the irrational straight line called apotome. Proposition 7. If three angles of an equilateral pentagon, taken either in order or not in order, are equal, then the pentagon is equiangular. Proposition 8. If in an equilateral and equiangular pentagon straight lines subtend two angles are taken in order, then they cut one another in extreme and mean ratio, and their greater segments equal the side of the pentagon. Proposition 9. If the side of the hexagon and that of the decagon inscribed in the same circle are added together, then the whole straight line has been cut in extreme and mean ratio, and its greater segment is the side of the hexagon. Proposition 10. If an equilateral pentagon is inscribed ina circle, then the square on the side of the pentagon equals the sum of the squares on the sides of the hexagon and the decagon inscribed in the same circle. Proposition 11. If an equilateral pentagon is inscribed in a circle which has its diameter rational, then the side of the pentagon is the irrational straight line called minor. Proposition 12. If an equilateral triangle is inscribed in a circle, then the square on the side of the triangle is triple the square on the radius of the circle. Proposition 13. To construct a pyramid, to comprehend it in a given sphere; and to prove that the square on the diameter of the sphere is one and a half times the square on the side of the pyramid. Lemma for XIII.13. Proposition 14. To construct an octahedron and comprehend it in a sphere, as in the preceding case; and to prove that the square on the diameter of the sphere is double the square on the side of the octahedron. Proposition 15. To construct a cube and comprehend it in a sphere, like the pyramid; and to prove that the square on the diameter of the sphere is triple the square on the side of the cube. Proposition 16. To construct an icosahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the square on the side of the icosahedron is the irrational straight line called minor. Corollary. The square on the diameter of the sphere is five times the square on the radius of the circle from which the icosahedron has been described, and the diameter of the sphere is composed of the side of the hexagon and two of the sides of the decagon inscribed in the same circle. Proposition 17. To construct a dodecahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the square on the side of the dodecahedron is the irrational straight line called apotome. Corollary. When the side of the cube is cut in extreme and mean ratio, the greater segment is the side of the dodecahedron. Proposition 18. To set out the sides of the five figures and compare them with one another. Remark. No other figure, besides the said five figures, can be constructed by equilateral and equiangular figures equal to one another. Lemma. The angle of the equilateral and equiangular pentagon is a right angle and a fifth. Elements Introduction Book XII. Table of contents ● Propositions (18) Propositions Proposition 1. Similar polygons inscribed in circles are to one another as the squares on their diameters. Proposition 2. Circles are to one another as the squares on their diameters. Lemma for XII.2. Proposition 3. Any pyramid with a triangular base is divided into two pyramids equal and similar to one another, similar to the whole, and having triangular bases, and into two equal prisms, and the two prisms are greater than half of the whole pyramid. Proposition 4. If there are two pyramids of the same height with triangular bases, and each of them is divided into two pyramids equal and similar to one another and similar to the whole, and into two equal prisms, then the base of the one pyramid is to the base of the other pyramid as all the prisms in the one pyramid are to all the prisms, being equal in multitude, in the other pyramid. Lemma for XII.4. Proposition 5. Pyramids of the same height with triangular bases are to one another as their bases. Proposition 6. Pyramids of the same height with polygonal bases are to one another as their bases. Proposition 7. Any prism with a triangular base is divided into three pyramids equal to one another with triangular bases. Corollary. Any pyramid is a third part of the prism with the same base and equal height. Proposition 8. Similar pyramids with triangular bases are in triplicate ratio of their corresponding sides. Corollary. Similar pyramids with polygonal bases are also to one another in triplicate ratio of their corresponding sides. Proposition 9. In equal pyramids with triangular bases the bases are reciprocally proportional to the heights; and those pyramids are equal in which the bases are reciprocally proportional to the heights. Proposition 10. Any cone is a third part of the cylinder with the same base and equal height. Proposition 11. Cones and cylinders of the same height are to one another as their bases. Proposition 12. Similar cones and cylinders are to one another in triplicate ratio of the diameters of their bases. Proposition 13. If a cylinder is cut by a plane parallel to its opposite planes, then the cylinder is to the cylinder as the axis is to the axis. Proposition 14. Cones and cylinders on equal bases are to one another as their heights. Proposition 15. In equal cones and cylinders the bases are reciprocally proportional to the heights; and those cones and cylinders in which the bases are reciprocally proportional to the heights are equal. Proposition 16. Given two circles about the same center, to inscribe in the greater circle an equilateral polygon with an even number of sides which does not touch the lesser circle. Proposition 17. Given two spheres about the same center, to inscribe in the greater sphere a polyhedral solid which does not touch the lesser sphere at its surface. Corollary to XII.17. Proposition 18. Spheres are to one another in triplicate ratio of their respective diameters. Next book: Book XIII Previous: Book XI Elements Introduction Table of contents ● Definitions (28) ● Propositions (39) Definitions Definition 1. A solid is that which has length, breadth, and depth. Definition 2. A face of a solid is a surface. Definition 3. A straight line is at right angles to a plane when it makes right angles with all the straight lines which meet it and are in the plane. Definition 4. A plane is at right angles to a plane when the straight lines drawn in one of the planes at right angles to the intersection of the planes are at right angles to the remaining plane. Definition 5. The inclination of a straight line to a plane is, assuming a perpendicular drawn from the end of the straight line which is elevated above the plane to the plane, and a straight line joined from the point thus arising to the end of the straight line which is in the plane, the angle contained by the straight line so drawn and the straight line standing up. Definition 6. The inclination of a plane to a plane is the acute angle contained by the straight lines drawn at right angles to the intersection at the same point, one in each of the planes. Definition 7. A plane is said to be similarly inclined to a plane as another is to another when the said angles of the inclinations equal one another. Definition 8. Parallel planes are those which do not meet. Definition 9. Similar solid figures are those contained by similar planes equal in multitude. Definition 10. Equal and similar solid figures are those contained by similar planes equal in multitude and magnitude. Definition 11. A solid angle is the inclination constituted by more than two lines which meet one another and are not in the same surface, towards all the lines, that is, a solid angle is that which is contained by more than two plane angles which are not in the same plane and are constructed to one point. Definition 12. A pyramid is a solid figure contained by planes which is constructed from one plane to one point. Definition 13. A prism is a solid figure contained by planes two of which, namely those which are opposite, are equal, similar, and parallel, while the rest are parallelograms. Definition 14. When a semicircle with fixed diameter is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a sphere. Definition 15. The axis of the sphere is the straight line which remains fixed and about which the semicircle is turned. Definition 16. The center of the sphere is the same as that of the semicircle. Definition 17. A diameter of the sphere is any straight line drawn through the center and terminated in both directions by the surface of the sphere. Definition 18. When a right triangle with one side of those about the right angle remains fixed is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a cone. And, if the straight line which remains fixed equals the remaining side about the right angle which is carried round, the cone will be right-angled; if less, obtuseangled; and if greater, acute-angled. Definition 19. The axis of the cone is the straight line which remains fixed and about which the triangle is turned. Definition 20. And the base is the circle described by the straight in which is carried round. Definition 21. When a rectangular parallelogram with one side of those about the right angle remains fixed is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a cylinder. Definition 22. The axis of the cylinder is the straight line which remains fixed and about which the parallelogram is turned. Definition 23. And the bases are the circles described by the two sides opposite to one another which are carried round. Definition 24. Similar cones and cylinders are those in which the axes and the diameters of the bases are proportional. Definition 25. A cube is a solid figure contained by six equal squares. Definition 26. An octahedron is a solid figure contained by eight equal and equilateral triangles. Definition 27. An icosahedron is a solid figure contained by twenty equal and equilateral triangles. Definition 28. A dodecahedron is a solid figure contained by twelve equal, equilateral and equiangular pentagons. Propositions Proposition 1. A part of a straight line cannot be in the plane of reference and a part in plane more elevated. Proposition 2. If two straight lines cut one another, then they lie in one plane; and every triangle lies in one plane. Proposition 3. If two planes cut one another, then their intersection is a straight line. Proposition 4. If a straight line is set up at right angles to two straight lines which cut one another at their common point of section, then it is also at right angles to the plane passing through them. Proposition 5. If a straight line is set up at right angles to three straight lines which meet one another at their common point of section, then the three straight lines lie in one plane. Proposition 6. If two straight lines are at right angles to the same plane, then the straight lines are parallel. Proposition 7. If two straight lines are parallel and points are taken at random on each of them, then the straight line joining the points is in the same plane with the parallel straight lines. Proposition 8. If two straight lines are parallel, and one of them is at right angles to any plane, then the remaining one is also at right angles to the same plane. Proposition 9 Straight lines which are parallel to the same straight line but do not lie in the same plane with it are also parallel to each other. Proposition 10. If two straight lines meeting one another are parallel to two straight lines meeting one another not in the same plane, then they contain equal angles. Proposition 11. To draw a straight line perpendicular to a given plane from a given elevated point. Proposition 12. To set up a straight line at right angles to a give plane from a given point in it. Proposition 13. From the same point two straight lines cannot be set up at right angles to the same plane on the same side. Proposition 14. Planes to which the same straight line is at right angles are parallel. Proposition 15. If two straight lines meeting one another are parallel to two straight lines meeting one another not in the same plane, then the planes through them are parallel. Proposition 16. If two parallel planes are cut by any plane, then their intersections are parallel. Proposition 17. If two straight lines are cut by parallel planes, then they are cut in the same ratios. Proposition 18. If a straight line is at right angles to any plane, then all the planes through it are also at right angles to the same plane. Proposition 19. If two planes which cut one another are at right angles to any plane, then their intersection is also at right angles to the same plane. Proposition 20. If a solid angle is contained by three plane angles, then the sum of any two is greater than the remaining one. Proposition 21. Any solid angle is contained by plane angles whose sum is less than four right angles. Proposition 22 If there are three plane angles such that the sum of any two is greater than the remaining one, and they are contained by equal straight lines, then it is possible to construct a triangle out of the straight lines joining the ends of the equal straight lines. Proposition 23. To construct a solid angles out of three plane angles such that the sum of any two is greater than the remaining one: thus the sum of the three angles must be less than four right angles. Lemma for XI.23. Proposition 24. If a solid is contained by parallel planes, then the opposite planes in it are equal and parallelogrammic. Proposition 25. If a parallelepipedal solid is cut by a plane parallel to the opposite planes, then the base is to the base as the solid is to the solid. Proposition 26. To construct a solid angle equal to a given solid angle on a given straight line at a given point on it. Proposition 27. To describe a parallelepipedal solid similar and similarly situated to a given parallelepipedal solid on a given straight line. Proposition 28. If a parallelepipedal solid is cut by a plane through the diagonals of the opposite planes, then the solid is bisected by the plane. Proposition 29. Parallelepipedal solids which are on the same base and of the same height, and in which the ends of their edges which stand up are on the same straight lines, equal one another. Proposition 30. Parallelepipedal solids which are on the same base and of the same height, and in which the ends of their edges which stand up are not on the same straight lines, equal one another. Proposition 31. Parallelepipedal solids which are on equal bases and of the same height equal one another. Proposition 32. Parallelepipedal solids which are of the same height are to one another as their bases. Proposition 33. Similar parallelepipedal solids are to one another in the triplicate ratio of their corresponding sides. Corollary. If four straight lines are continuously proportional, then the first is to the fourth as a parallelepipedal solid on the first is to the similar and similarly situated parallelepipedal solid on the second, in as much as the first has to the fourth the ratio triplicate of that which it has to the second. Proposition 34. In equal parallelepipedal solids the bases are reciprocally proportional to the heights; and those parallelepipedal solids in which the bases are reciprocally proportional to the heights are equal. Proposition 35. If there are two equal plane angles, and on their vertices there are set up elevated straight lines containing equal angles with the original straight lines respectively, if on the elevated straight lines points are taken at random and perpendiculars are drawn from them to the planes in which the original angles are, and if from the points so arising in the planes straight lines are joined to the vertices of the original angles, then they contain with the elevated straight lines equal angles. Proposition 36. If three straight lines are proportional, then the parallelepipedal solid formed out of the three equals the parallelepipedal solid on the mean which is equilateral, but equiangular with the aforesaid solid. Proposition 37. If four straight lines are proportional, then parallelepipedal solids on them which are similar and similarly described are also proportional; and, if the parallelepipedal solids on them which are similar and similarly described are proportional, then the straight lines themselves are also proportional. Proposition 38. If the sides of the opposite planes of a cube are bisected, and the planes are carried through the points of section, then the intersection of the planes and the diameter of the cube bisect one another. Proposition 39. If there are two prisms of equal height, and one has a parallelogram as base and the other a triangle, and if the parallelogram is double the triangle, then the prisms are equal. Elements Introduction Book X Book XII. Table of contents ● Definitions I (4) ● Propositions 1-47 ● Definitions II (6) ● Propositions 48-84 ● Definitions III (6) ● Propositions 85-115 Definitions I Definition 1. Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure. Definition 2. Straight lines are commensurable in square when the squares on them are measured by the same area, and incommensurable in square when the squares on them cannot possibly have any area as a common measure. Definition 3. With these hypotheses, it is proved that there exist straight lines infinite in multitude which are commensurable and incommensurable respectively, some in length only, and others in square also, with an assigned straight line. Let then the assigned straight line be called rational, and those straight lines which are commensurable with it, whether in length and in square, or in square only, rational, but those that are incommensurable with it irrational. Definition 4. And the let the square on the assigned straight line be called rational, and those areas which are commensurable with it rational, but those which are incommensurable with it irrational, and the straight lines which produce them irrational, that is, in case the areas are squares, the sides themselves, but in case they are any other rectilineal figures, the straight lines on which are described squares equal to them. Propositions 1-47 Proposition 1. Two unequal magnitudes being set out, if from the greater there is subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process is repeated continually, then there will be left some magnitude less than the lesser magnitude set out. And the theorem can similarly be proven even if the parts subtracted are halves. Proposition 2. If, when the less of two unequal magnitudes is continually subtracted in turn from the greater that which is left never measures the one before it, then the two magnitudes are incommensurable. Proposition 3. To find the greatest common measure of two given commensurable magnitudes. Corollary. If a magnitude measures two magnitudes, then it also measures their greatest common measure. Proposition 4. To find the greatest common measure of three given commensurable magnitudes. Corollary. If a magnitude measures three magnitudes, then it also measures their greatest common measure. The greatest common measure can be found similarly for more magnitudes, and the corollary extended. Proposition 5. Commensurable magnitudes have to one another the ratio which a number has to a number. Proposition 6. If two magnitudes have to one another the ratio which a number has to a number, then the magnitudes are commensurable. Corollary. Proposition 7. Incommensurable magnitudes do not have to one another the ratio which a number has to a number. Proposition 8. If two magnitudes do not have to one another the ratio which a number has to a number, then the magnitudes are incommensurable. Proposition 9. The squares on straight lines commensurable in length have to one another the ratio which a square number has to a square number; and squares which have to one another the ratio which a square number has to a square number also have their sides commensurable in length. But the squares on straight lines incommensurable in length do not have to one another the ratio which a square number has to a square number; and squares which do not have to one another the ratio which a square number has to a square number also do not have their sides commensurable in length either. Corollary. Straight lines commensurable in length are always commensurable in square also, but those commensurable in square are not always also commensurable in length. Lemma. Similar plane numbers have to one another the ratio which a square number has to a square number, and if two numbers have to one another the ratio which a square number has to a square number, then they are similar plane numbers. Corollary 2. Numbers which are not similar plane numbers, that is, those which do not have their sides proportional, do not have to one another the ratio which a square number has to a square number Proposition 10. To find two straight lines incommensurable, the one in length only, and the other in square also, with an assigned straight line. Proposition 11. If four magnitudes are proportional, and the first is commensurable with the second, then the third also is commensurable with the fourth; but, if the first is incommensurable with the second, then the third also is incommensurable with the fourth. Proposition 12. Magnitudes commensurable with the same magnitude are also commensurable with one another. Proposition 13. If two magnitudes are commensurable, and one of them is incommensurable with any magnitude, then the remaining one is also incommensurable with the same. Proposition 14. Lemma. Given two unequal straight lines, to find by what square the square on the greater is greater than the square on the less. And, given two straight lines, to find the straight line the square on which equals the sum of the squares on them. Proposition 14. If four straight lines are proportional, and the square on the first is greater than the square on the second by the square on a straight line commensurable with the first, then the square on the third is also greater than the square on the fourth by the square on a third line commensurable with the third. And, if the square on the first is greater than the square on the second by the square on a straight line incommensurable with the first, then the square on the third is also greater than the square on the fourth by the square on a third line incommensurable with the third. Proposition 15. If two commensurable magnitudes are added together, then the whole is also commensurable with each of them; and, if the whole is commensurable with one of them, then the original magnitudes are also commensurable. Proposition 16. If two incommensurable magnitudes are added together, the sum is also incommensurable with each of them; but, if the sum is incommensurable with one of them, then the original magnitudes are also incommensurable. Proposition 17. Lemma. If to any straight line there is applied a parallelogram but falling short by a square, then the applied parallelogram equals the rectangle contained by the segments of the straight line resulting from the application. Proposition 17. If there are two unequal straight lines, and to the greater there is applied a parallelogram equal to the fourth part of the square on the less but falling short by a square, and if it divides it into parts commensurable in length, then the square on the greater is greater than the square on the less by the square on a straight line commensurable with the greater. And if the square on the greater is greater than the square on the less by the square on a straight line commensurable with the greater, and if there is applied to the greater a parallelogram equal to the fourth part of the square on the less falling short by a square, then it divides it into parts commensurable in length. Proposition 18. If there are two unequal straight lines, and to the greater there is applied a parallelogram equal to the fourth part of the square on the less but falling short by a square, and if it divides it into incommensurable parts, then the square on the greater is greater than the square on the less by the square on a straight line incommensurable with the greater. And if the square on the greater is greater than the square on the less by the square on a straight line incommensurable with the greater, and if there is applied to the greater a parallelogram equal to the fourth part of the square on the less but falling short by a square, then it divides it into incommensurable parts. Proposition 19. Lemma. Proposition 19. The rectangle contained by rational straight lines commensurable in length is rational. Proposition 20. If a rational area is applied to a rational straight line, then it produces as breadth a straight line rational and commensurable in length with the straight line to which it is applied. Proposition 21. The rectangle contained by rational straight lines commensurable in square only is irrational, and the side of the square equal to it is irrational. Let the latter be called medial. Proposition 22. Lemma. If there are two straight lines, then the first is to the second as the square on the first is to the rectangle contained by the two straight lines. Proposition 22. The square on a medial straight line, if applied to a rational straight line, produces as breadth a straight line rational and incommensurable in length with that to which it is applied. Proposition 23. A straight line commensurable with a medial straight line is medial. Corollary. An area commensurable with a medial area is medial. Proposition 24. The rectangle contained by medial straight lines commensurable in length is medial. Proposition 25. The rectangle contained by medial straight lines commensurable in square only is either rational or medial. Proposition 26. A medial area does not exceed a medial area by a rational area. Proposition 27. To find medial straight lines commensurable in square only which contain a rational rectangle. Proposition 28. To find medial straight lines commensurable in square only which contain a medial rectangle. Proposition 29. Lemma 1. To find two square numbers such that their sum is also square. Lemma 2. To find two square numbers such that their sum is not square. Proposition 29. To find two rational straight lines commensurable in square only such that the square on the greater is greater than the square on the less by the square on a straight line commensurable in length with the greater. Proposition 30. To find two rational straight lines commensurable in square only such that the square on the greater is greater than the square on the less by the square on a straight line incommensurable in length with the greater. Proposition 31. To find two medial straight lines commensurable in square only, containing a rational rectangle, such that the square on the greater is greater than the square on the less by the square on a straight line commensurable in length with the greater. Proposition 32. To find two medial straight lines commensurable in square only, containing a medial rectangle, such that the square on the greater is greater than the square on the less by the square on a straight line commensurable with the greater. Proposition 33. Lemma. Proposition 33. To find two straight lines incommensurable in square which make the sum of the squares on them rational but the rectangle contained by them medial. Proposition 34. To find two straight lines incommensurable in square which make the sum of the squares on them medial but the rectangle contained by them rational. Proposition 35. To find two straight lines incommensurable in square which make the sum of the squares on them medial and the rectangle contained by them medial and moreover incommensurable with the sum of the squares on them. Proposition 36. If two rational straight lines commensurable in square only are added together, then the whole is irrational; let it be called binomial. Proposition 37. If two medial straight lines commensurable in square only and containing a rational rectangle are added together, the whole is irrational; let it be called the first bimedial straight line. Proposition 38. If two medial straight lines commensurable in square only and containing a medial rectangle are added together, then the whole is irrational; let it be called the second bimedial straight line. Proposition 39. If two straight lines incommensurable in square which make the sum of the squares on them rational but the rectangle contained by them medial are added together, then the whole straight line is irrational; let it be called major. Proposition 40. If two straight lines incommensurable in square which make the sum of the squares on them medial but the rectangle contained by them rational are added together, then the whole straight line is irrational; let it be called the side of a rational plus a medial area. Proposition 41. If two straight lines incommensurable in square which make the sum of the squares on them medial and the rectangle contained by them medial and also incommensurable with the sum of the squares on them are added together, then the whole straight line is irrational; let it be called the side of the sum of two medial areas. Lemma. Proposition 42. A binomial straight line is divided into its terms at one point only. Proposition 43. A first bimedial straight line is divided at one and the same point only. Proposition 44. A second bimedial straight line is divided at one point only. Proposition 45. A major straight line is divided at one point only. Proposition 46. The side of a rational plus a medial area is divided at one point only. Proposition 47. The side of the sum of two medial areas is divided at one point only. Definitions II Definition 1. Given a rational straight line and a binomial, divided into its terms, such that the square on the greater term is greater than the square on the lesser by the square on a straight line commensurable in length with the greater, then, if the greater term is commensurable in length with the rational straight line set out, let the whole be called a first binomial straight line; Definition 2. But if the lesser term is commensurable in length with the rational straight line set out, let the whole be called a second binomial; Definition 3. And if neither of the terms is commensurable in length with the rational straight line set out, let the whole be called a third binomial. Definition 4. Again, if the square on the greater term is greater than the square on the lesser by the square on a straight line incommensurable in length with the greater, then, if the greater term is commensurable in length with the rational straight line set out, let the whole be called a fourth binomial; Definition 5. If the lesser, a fifth binomial; Definition 6. And, if neither, a sixth binomial. Propositions 48-84 Proposition 48. To find the first binomial line. Proposition 49. To find the second binomial line. Proposition 50. To find the third binomial line. Proposition 51. To find the fourth binomial line. Proposition 52. To find the fifth binomial line. Proposition 53. To find the sixth binomial line. Proposition 54. Lemma. Proposition 54. If an area is contained by a rational straight line and the first binomial, then the side of the area is the irrational straight line which is called binomial. Proposition 55. If an area is contained by a rational straight line and the second binomial, then the side of the area is the irrational straight line which is called a first bimedial. Proposition 56. If an area is contained by a rational straight line and the third binomial, then the side of the area is the irrational straight line called a second bimedial. Proposition 57. If an area is contained by a rational straight line and the fourth binomial, then the side of the area is the irrational straight line called major. Proposition 58. If an area is contained by a rational straight line and the fifth binomial, then the side of the area is the irrational straight line called the side of a rational plus a medial area. Proposition 59. If an area is contained by a rational straight line and the sixth binomial, then the side of the area is the irrational straight line called the side of the sum of two medial areas. Proposition 60. Lemma. If a straight line is cut into unequal parts, then the sum of the squares on the unequal parts is greater than twice the rectangle contained by the unequal parts. Proposition 60. The square on the binomial straight line applied to a rational straight line produces as breadth the first binomial. Proposition 61. The square on the first bimedial straight line applied to a rational straight line produces as breadth the second binomial. Proposition 62. The square on the second bimedial straight line applied to a rational straight line produces as breadth the third binomial. Proposition 63. The square on the major straight line applied to a rational straight line produces as breadth the fourth binomial. Proposition 64. The square on the side of a rational plus a medial area applied to a rational straight line produces as breadth the fifth binomial. Proposition 65. The square on the side of the sum of two medial areas applied to a rational straight line produces as breadth the sixth binomial. Proposition 66. A straight line commensurable with a binomial straight line is itself also binomial and the same in order. Proposition 67. A straight line commensurable with a bimedial straight line is itself also bimedial and the same in order. Proposition 68. A straight line commensurable with a major straight line is itself also major. Proposition 69. A straight line commensurable with the side of a rational plus a medial area is itself also the side of a rational plus a medial area. Proposition 70. A straight line commensurable with the side of the sum of two medial areas is the side of the sum of two medial areas. Proposition 71. If a rational and a medial are added together, then four irrational straight lines arise, namely a binomial or a first bimedial or a major or a side of a rational plus a medial area. Proposition 72. If two medial areas incommensurable with one another are added together, then the remaining two irrational straight lines arise, namely either a second bimedial or a side of the sum of two medial areas. Proposition. The binomial straight line and the irrational straight lines after it are neither the same with the medial nor with one another. Proposition 73. If from a rational straight line there is subtracted a rational straight line commensurable with the whole in square only, then the remainder is irrational; let it be called an apotome. Proposition 74. If from a medial straight line there is subtracted a medial straight line which is commensurable with the whole in square only, and which contains with the whole a rational rectangle, then the remainder is irrational; let it be called first apotome of a medial straight line. Proposition 75. If from a medial straight line there is subtracted a medial straight line which is commensurable with the whole in square only, and which contains with the whole a medial rectangle, then the remainder is irrational; let it be called second apotome of a medial straight line. Proposition 76. If from a straight line there is subtracted a straight line which is incommensurable in square with the whole and which with the whole makes the sum of the squares on them added together rational, but the rectangle contained by them medial, then the remainder is irrational; let it be called minor. Proposition 77. If from a straight line there is subtracted a straight line which is incommensurable in square with the whole, and which with the whole makes the sum of the squares on them medial but twice the rectangle contained by them rational, then the remainder is irrational; let it be called that which produces with a rational area a medial whole. Proposition 78. If from a straight line there is subtracted a straight line which is incommensurable in square with the whole and which with the whole makes the sum of the squares on them medial, twice the rectangle contained by them medial, and further the squares on them incommensurable with twice the rectangle contained by them, then the remainder is irrational; let it be called that which produces with a medial area a medial whole. Proposition 79. To an apotome only one rational straight line can be annexed which is commensurable with the whole in square only. Proposition 80. To a first apotome of a medial straight line only one medial straight line can be annexed which is commensurable with the whole in square only and which contains with the whole a rational rectangle. Proposition 81. To a second apotome of a medial straight line only one medial straight line can be annexed which is commensurable with the whole in square only and which contains with the whole a medial rectangle. Proposition 82. To a minor straight line only one straight line can be annexed which is incommensurable in square with the whole and which makes, with the whole, the sum of squares on them rational but twice the rectangle contained by them medial. Proposition 83. To a straight line which produces with a rational area a medial whole only one straight line can be annexed which is incommensurable in square with the whole straight line and which with the whole straight line makes the sum of squares on them medial but twice the rectangle contained by them rational. Proposition 84. To a straight line which produces with a medial area a medial whole only one straight line can be annexed which is incommensurable in square with the whole straight line and which with the whole straight line makes the sum of squares on them medial and twice the rectangle contained by them both medial and also incommensurable with the sum of the squares on them. Definitions III Definition 1. Given a rational straight line and an apotome, if the square on the whole is greater than the square on the annex by the square on a straight line commensurable in length with the whole, and the whole is commensurable in length with the rational line set out, let the apotome be called a first apotome. Definition 2. But if the annex is commensurable with the rational straight line set out, and the square on the whole is greater than that on the annex by the square on a straight line commensurable with the whole, let the apotome be called a second apotome. Definition 3. But if neither is commensurable in length with the rational straight line set out, and the square on the whole is greater than the square on the annex by the square on a straight line commensurable with the whole, let the apotome be called a third apotome. Definition 4. Again, if the square on the whole is greater than the square on the annex by the square on a straight line incommensurable with the whole, then, if the whole is commensurable in length with the rational straight line set out, let the apotome be called a fourth apotome; Definition 5. If the annex be so commensurable, a fifth; Definition 6. And, if neither, a sixth. Propositions 85-115 Proposition 85. To find the first apotome. Proposition 86. To find the second apotome. Proposition 87. To find the third apotome. Proposition 88. To find the fourth apotome. Proposition 89. To find the fifth apotome. Proposition 90. To find the sixth apotome. Proposition 91. If an area is contained by a rational straight line and a first apotome, then the side of the area is an apotome. Proposition 92. If an area is contained by a rational straight line and a second apotome, then the side of the area is a first apotome of a medial straight line. Proposition 93. If an area is contained by a rational straight line and a third apotome, then the side of the area is a second apotome of a medial straight line. Proposition 94. If an area is contained by a rational straight line and a fourth apotome, then the side of the area is minor. Proposition 95. If an area is contained by a rational straight line and a fifth apotome, then the side of the area is a straight line which produces with a rational area a medial whole. Proposition 96. If an area is contained by a rational straight line and a sixth apotome, then the side of the area is a straight line which produces with a medial area a medial whole. Proposition 97. The square on an apotome of a medial straight line applied to a rational straight line produces as breadth a first apotome. Proposition 98. The square on a first apotome of a medial straight line applied to a rational straight line produces as breadth a second apotome. Proposition 99. The square on a second apotome of a medial straight line applied to a rational straight line produces as breadth a third apotome. Proposition 100. The square on a minor straight line applied to a rational straight line produces as breadth a fourth apotome. Proposition 101. The square on the straight line which produces with a rational area a medial whole, if applied to a rational straight line, produces as breadth a fifth apotome. Proposition 102. The square on the straight line which produces with a medial area a medial whole, if applied to a rational straight line, produces as breadth a sixth apotome. Proposition 103. A straight line commensurable in length with an apotome is an apotome and the same in order. Proposition 104. A straight line commensurable with an apotome of a medial straight line is an apotome of a medial straight line and the same in order. Proposition 105. A straight line commensurable with a minor straight line is minor. Proposition 106. A straight line commensurable with that which produces with a rational area a medial whole is a straight line which produces with a rational area a medial whole. Proposition 107. A straight line commensurable with that which produces a medial area and a medial whole is itself also a straight line which produces with a medial area a medial whole. Proposition 108. If from a rational area a medial area is subtracted, the side of the remaining area becomes one of two irrational straight lines, either an apotome or a minor straight line. Proposition 109. If from a medial area a rational area is subtracted, then there arise two other irrational straight lines, either a first apotome of a medial straight line or a straight line which produces with a rational area a medial whole. Proposition 110. If from a medial area there is subtracted a medial area incommensurable with the whole, then the two remaining irrational straight lines arise, either a second apotome of a medial straight line or a straight line which produce with a medial area a medial whole. Proposition 111. The apotome is not the same with the binomial straight line. Proposition. The apotome and the irrational straight lines following it are neither the same with the medial straight line nor with one another. There are, in order, thirteen irrational straight lines in all: Medial Binomial First bimedial Second bimedial Major Side of a rational plus a medial area Side of the sum of two medial areas Apotome First apotome of a medial straight line Second apotome of a medial straight line Minor Producing with a rational area a medial whole Producing with a medial area a medial whole Proposition 112. The square on a rational straight line applied to the binomial straight line produces as breadth an apotome the terms of which are commensurable with the terms of the binomial straight line and moreover in the same ratio; and further the apotome so arising has the same order as the binomial straight line. Proposition 113. The square on a rational straight line, if applied to an apotome, produces as breadth the binomial straight line the terms of which are commensurable with the terms of the apotome and in the same ratio; and further the binomial so arising has the same order as the apotome. Proposition 114. If an area is contained by an apotome and the binomial straight line the terms of which are commensurable with the terms of the apotome and in the same ratio, then the side of the area is rational. Corollary. It is possible for a rational area to be contained by irrational straight lines. Proposition 115. From a medial straight line there arise irrational straight lines infinite in number, and none of them is the same as any preceding. Elements Introduction Book IX Book XI. Table of contents ● Propositions (36) Propositions Proposition 1. If two similar plane numbers multiplied by one another make some number, then the product is square. Proposition 2. If two numbers multiplied by one another make a square number, then they are similar plane numbers. Proposition 3. If a cubic number multiplied by itself makes some number, then the product is a cube. Proposition 4. If a cubic number multiplied by a cubic number makes some number, then the product is a cube. Proposition 5. If a cubic number multiplied by any number makes a cubic number, then the multiplied number is also cubic. Proposition 6. If a number multiplied by itself makes a cubic number, then it itself is also cubic. Proposition 7. If a composite number multiplied by any number makes some number, then the product is solid. Proposition 8. If as many numbers as we please beginning from a unit are in continued proportion, then the third from the unit is square as are also those which successively leave out one, the fourth is cubic as are also all those which leave out two, and the seventh is at once cubic and square are also those which leave out five. Proposition 9. If as many numbers as we please beginning from a unit are in continued proportion, and the number after the unit is square, then all the rest are also square; and if the number after the unit is cubic, then all the rest are also cubic. Proposition 10. If as many numbers as we please beginning from a unit are in continued proportion, and the number after the unit is not square, then neither is any other square except the third from the unit and all those which leave out one; and, if the number after the unit is not cubic, then neither is any other cubic except the fourth from the unit and all those which leave out two. Proposition 11. If as many numbers as we please beginning from a unit are in continued proportion, then the less measures the greater according to some one of the numbers which appear among the proportional numbers. Corollary. Whatever place the measuring number has, reckoned from the unit, the same place also has the number according to which it measures, reckoned from the number measured, in the direction of the number before it. Proposition 12. If as many numbers as we please beginning from a unit are in continued proportion, then by whatever prime numbers the last is measured, the next to the unit is also measured by the same. Proposition 13. If as many numbers as we please beginning from a unit are in continued proportion, and the number after the unit is prime, then the greatest is not measured by any except those which have a place among the proportional numbers. Proposition 14. If a number is the least that is measured by prime numbers, then it is not measured by any other prime number except those originally measuring it. Proposition 15. If three numbers in continued proportion are the least of those which have the same ratio with them, then the sum of any two is relatively prime to the remaining number. Proposition 16. If two numbers are relatively prime, then the second is not to any other number as the first is to the second. Proposition 17. If there are as many numbers as we please in continued proportion, and the extremes of them are relatively prime, then the last is not to any other number as the first is to the second. Proposition 18. Given two numbers, to investigate whether it is possible to find a third proportional to them. Proposition 19. Given three numbers, to investigate when it is possible to find a fourth proportional to them. Proposition 20. Prime numbers are more than any assigned multitude of prime numbers. Proposition 21. If as many even numbers as we please are added together, then the sum is even. Proposition 22. If as many odd numbers as we please are added together, and their multitude is even, then the sum is even. Proposition 23. If as many odd numbers as we please are added together, and their multitude is odd, then the sum is also odd. Proposition 24. If an even number is subtracted from an even number, then the remainder is even. Proposition 25. If an odd number is subtracted from an even number, then the remainder is odd. Proposition 26. If an odd number is subtracted from an odd number, then the remainder is even. Proposition 27. If an even number is subtracted from an odd number, then the remainder is odd. Proposition 28. If an odd number is multiplied by an even number, then the product is even. Proposition 29. If an odd number is multiplied by an odd number, then the product is odd. Proposition 30. If an odd number measures an even number, then it also measures half of it. Proposition 31. If an odd number is relatively prime to any number, then it is also relatively prime to double it. Proposition 32. Each of the numbers which are continually doubled beginning from a dyad is even-times even only. Proposition 33. If a number has its half odd, then it is even-times odd only. Proposition 34. If an [even] number neither is one of those which is continually doubled from a dyad, nor has its half odd, then it is both even-times even and even-times odd. Proposition 35. If as many numbers as we please are in continued proportion, and there is subtracted from the second and the last numbers equal to the first, then the excess of the second is to the first as the excess of the last is to the sum of all those before it. Proposition 36. If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if the sum multiplied into the last makes some number, then the product is perfect. Next book: Book X Previous: Book VIII Elements Introduction – Table of contents ● Propositions (27) Propositions Proposition 1. If there are as many numbers as we please in continued proportion, and the extremes of them are relatively prime, then the numbers are the least of those which have the same ratio with them. Proposition 2. To find as many numbers as are prescribed in continued proportion, and the least that are in a given ratio. Corollary. If three numbers in continued proportion are the least of those which have the same ratio with them, then the extremes are squares, and, if four numbers, cubes. Proposition 3. If as many numbers as we please in continued proportion are the least of those which have the same ratio with them, then the extremes of them are relatively prime. Proposition 4. Given as many ratios as we please in least numbers, to find numbers in continued proportion which are the least in the given ratios. Proposition 5. Plane numbers have to one another the ratio compounded of the ratios of their sides. Proposition 6. If there are as many numbers as we please in continued proportion, and the first does not measure the second, then neither does any other measure any other. Proposition 7. If there are as many numbers as we please in continued proportion, and the first measures the last, then it also measures the second. Proposition 8. If between two numbers there fall numbers in continued proportion with them, then, however many numbers fall between them in continued proportion, so many also fall in continued proportion between the numbers which have the same ratios with the original numbers. Proposition 9. If two numbers are relatively prime, and numbers fall between them in continued proportion, then, however many numbers fall between them in continued proportion, so many also fall between each of them and a unit in continued proportion. Proposition 10. If numbers fall between two numbers and a unit in continued proportion, then however many numbers fall between each of them and a unit in continued proportion, so many also fall between the numbers themselves in continued proportion. Proposition 11. Between two square numbers there is one mean proportional number, and the square has to the square the duplicate ratio of that which the side has to the side. Proposition 12. Between two cubic numbers there are two mean proportional numbers, and the cube has to the cube the triplicate ratio of that which the side has to the side. Proposition 13. If there are as many numbers as we please in continued proportion, and each multiplied by itself makes some number, then the products are proportional; and, if the original numbers multiplied by the products make certain numbers, then the latter are also proportional. Proposition 14. If a square measures a square, then the side also measures the side; and, if the side measures the side, then the square also measures the square. Proposition 15. If a cubic number measures a cubic number, then the side also measures the side; and, if the side measures the side, then the cube also measures the cube. Proposition 16. If a square does not measure a square, then neither does the side measure the side; and, if the side does not measure the side, then neither does the square measure the square. Proposition 17. If a cubic number does not measure a cubic number, then neither does the side measure the side; and, if the side does not measure the side, then neither does the cube measure the cube. Proposition 18. Between two similar plane numbers there is one mean proportional number, and the plane number has to the plane number the ratio duplicate of that which the corresponding side has to the corresponding side. Proposition 19. Between two similar solid numbers there fall two mean proportional numbers, and the solid number has to the solid number the ratio triplicate of that which the corresponding side has to the corresponding side. Proposition 20. If one mean proportional number falls between two numbers, then the numbers are similar plane numbers. Proposition 21. If two mean proportional numbers fall between two numbers, then the numbers are similar solid numbers. Proposition 22. If three numbers are in continued proportion, and the first is square, then the third is also square. Proposition 23. If four numbers are in continued proportion, and the first is a cube, then the fourth is also a cube. Proposition 24. If two numbers have to one another the ratio which a square number has to a square number, and the first is square, then the second is also a square. Proposition 25. If two numbers have to one another the ratio which a cubic number has to a cubic number, and the first is a cube, then the second is also a cube. Proposition 26. Similar plane numbers have to one another the ratio which a square number has to a square number. Proposition 27. Similar solid numbers have to one another the ratio which a cubic number has to a cubic number. Next book: Book IX Previous: Book VII Book VIII introduction Table of contents ● Definitions (22) ● Propositions (39) Guide Definitions Definition 1 A unit is that by virtue of which each of the things that exist is called one. Definition 2 A number is a multitude composed of units. Definition 3 A number is a part of a number, the less of the greater, when it measures the greater; Definition 4 But parts when it does not measure it. Definition 5 The greater number is a multiple of the less when it is measured by the less. Definition 6 An even number is that which is divisible into two equal parts. Definition 7 An odd number is that which is not divisible into two equal parts, or that which differs by a unit from an even number. Definition 8 An even-times even number is that which is measured by an even number according to an even number. Definition 9 An even-times odd number is that which is measured by an even number according to an odd number. Definition 10 An odd-times odd number is that which is measured by an odd number according to an odd number. Definition 11 A prime number is that which is measured by a unit alone. Definition 12 Numbers relatively prime are those which are measured by a unit alone as a common measure. Definition 13 A composite number is that which is measured by some number. Definition 14 Numbers relatively composite are those which are measured by some number as a common measure. Definition 15 A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other. Definition 16 And, when two numbers having multiplied one another make some number, the number so produced be called plane, and its sides are the numbers which have multiplied one another. Definition 17 And, when three numbers having multiplied one another make some number, the number so produced be called solid, and its sides are the numbers which have multiplied one another. Definition 18 A square number is equal multiplied by equal, or a number which is contained by two equal numbers. Definition 19 And a cube is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers. Definition 20 Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth. Definition 21 Similar plane and solid numbers are those which have their sides proportional. Definition 22 A perfect number is that which is equal to the sum its own parts. Propositions Proposition 1 When two unequal numbers are set out, and the less is continually subtracted in turn from the greater, if the number which is left never measures the one before it until a unit is left, then the original numbers are relatively prime. Proposition 2 To find the greatest common measure of two given numbers not relatively prime. Corollary. If a number measures two numbers, then it also measures their greatest common measure. Proposition 3 To find the greatest common measure of three given numbers not relatively prime. Proposition 4 Any number is either a part or parts of any number, the less of the greater. Proposition 5 If a number is part of a number, and another is the same part of another, then the sum is also the same part of the sum that the one is of the one. Proposition 6 If a number is parts of a number, and another is the same parts of another, then the sum is also the same parts of the sum that the one is of the one. Proposition 7 If a number is that part of a number which a subtracted number is of a subtracted number, then the remainder is also the same part of the remainder that the whole is of the whole. Proposition 8 If a number is the same parts of a number that a subtracted number is of a subtracted number, then the remainder is also the same parts of the remainder that the whole is of the whole. Proposition 9 If a number is a part of a number, and another is the same part of another, then alternately, whatever part of parts the first is of the third, the same part, or the same parts, the second is of the fourth. Proposition 10 If a number is a parts of a number, and another is the same parts of another, then alternately, whatever part of parts the first is of the third, the same part, or the same parts, the second is of the fourth. Proposition 11 If a whole is to a whole as a subtracted number is to a subtracted number, then the remainder is to the remainder as the whole is to the whole. Proposition 12 If any number of numbers are proportional, then one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents. Proposition 13 If four numbers are proportional, then they are also proportional alternately. Proposition 14 If there are any number of numbers, and others equal to them in multitude, which taken two and two together are in the same ratio, then they are also in the same ratio ex aequali. Proposition 15 If a unit number measures any number, and another number measures any other number the same number of times, then alternately, the unit measures the third number the same number of times that the second measures the fourth. Proposition 16 If two numbers multiplied by one another make certain numbers, then the numbers so produced equal one another. Proposition 17 If a number multiplied by two numbers makes certain numbers, then the numbers so produced have the same ratio as the numbers multiplied. Proposition 18 If two number multiplied by any number make certain numbers, then the numbers so produced have the same ratio as the multipliers. Proposition 19 If four numbers are proportional, then the number produced from the first and fourth equals the number produced from the second and third; and, if the number produced from the first and fourth equals that produced from the second and third, then the four numbers are proportional. Proposition 20 The least numbers of those which have the same ratio with them measure those which have the same ratio with them the same number of times; the greater the greater; and the less the less. Proposition 21 Numbers relatively prime are the least of those which have the same ratio with them. Proposition 22 The least numbers of those which have the same ratio with them are relatively prime. Proposition 23 If two numbers are relatively prime, then any number which measures one of them is relatively prime to the remaining number. Proposition 24 If two numbers are relatively prime to any number, then their product is also relatively prime to the same. Proposition 25 If two numbers are relatively prime, then the product of one of them with itself is relatively prime to the remaining one. Proposition 26 If two numbers are relatively prime to two numbers, both to each, then their products are also relatively prime. Proposition 27 If two numbers are relatively prime, and each multiplied by itself makes a certain number, then the products are relatively prime; and, if the original numbers multiplied by the products make certain numbers, then the latter are also relatively prime. Proposition 28 If two numbers are relatively prime, then their sum is also prime to each of them; and, if the sum of two numbers is relatively prime to either of them, then the original numbers are also relatively prime. Proposition 29 Any prime number is relatively prime to any number which it does not measure. Proposition 30 If two numbers, multiplied by one another make some number, and any prime number measures the product, then it also measures one of the original numbers. Proposition 31 Any composite number is measured by some prime number. Proposition 32 Any number is either prime or is measured by some prime number. Proposition 33 Given as many numbers as we please, to find the least of those which have the same ratio with them. Proposition 34 To find the least number which two given numbers measure. Proposition 35 If two numbers measure any number, then the least number measured by them also measures the same. Proposition 36 To find the least number which three given numbers measure. Proposition 37 If a number is measured by any number, then the number which is measured has a part called by the same name as the measuring number. Proposition 38 If a number has any part whatever, then it is measured by a number called by the same name as the part. Proposition 39 To find the number which is the least that has given parts. Book VII is the first of the three books on number theory. It begins with the 22 definitions used in these books. The important definitions being those for unit and number, part and multiple, even and odd, prime and relatively prime, proportion, and perfect number. The topics in Book VII are antenaresis and the greatest common divisor, proportions of numbers, relatively prime numbers and prime numbers, and the least common multiple. The basic construction for Book VII is antenaresis, also called the Euclidean algorithm, a kind of reciprocal subtraction. Beginning with two numbers, the smaller, whichever it is, is repeatedly subtracted from the larger until a single number is left. This algorithm, studied in propositions VII.1 througth VII.3, results in the greatest common divisor of two or more numbers. Propositions V.5 through V.10 develop properties of fractions, that is, they study how many parts one number is of another in preparation for ratios and proportions. The next group of propositions VII.11 through VII.19 develop the theory of proportions for numbers. Propositions VII.20 through VII.29 discuss representing ratios in lowest terms as relatively prime numbers and properties of relatively prime numbers. Properties of prime numbers are presented in propositions VII.30 through VII.32. Book VII finishes with least common multiples in propositions VII.33 through VII.39. Postulates for numbers Postulates are as necessary for numbers as they are for geometry. Missing postulates occurs as early as proposition VII.2. In its proof, Euclid constructs a decreasing sequence of whole positive numbers, and, apparently, uses a principle that conclude that the sequence must stop, that is, there cannot be an infinite decreasing sequence of numbers. If that is the principle he uses, then it ought to be stated as a postulate for numbers. Numbers are so familiar that it hardly occurs to us that the theory of numbers needs axioms, too. In fact, that field was one of the last to receive a careful scrutiny, and axioms for numbers weren't developed until the late 19th century. By that time foundations for the rest of mathematics were laid upon either geometry or number theory or both, and only geometry had axioms. About the same time that foundations for number theory were developed, a new subject, set theory, was created by Cantor, and mathematics was refounded in terms of set theory. The foundations of number theory will be discussed in the Guides to the various definitions and propositions. Next book: Book VIII Previous: Book VI Book VII introduction Table of contents ● definitions (4) ● propositions (33) Definitions Definition 1. Similar rectilinear figures are such as have their angles severally equal and the sides about the equal angles proportional. Definition 2. Two figures are reciprocally related when the sides about corresponding angles are reciprocally proportional. Definition 3. A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less. Definition 4. The height of any figure is the perpendicular drawn from the vertex to the base. Propositions Proposition 1. Triangles and parallelograms which are under the same height are to one another as their bases. Proposition 2. If a straight line is drawn parallel to one of the sides of a triangle, then it cuts the sides of the triangle proportionally; and, if the sides of the triangle are cut proportionally, then the line joining the points of section is parallel to the remaining side of the triangle. Proposition 3. If an angle of a triangle is bisected by a straight line cutting the base, then the segments of the base have the same ratio as the remaining sides of the triangle; and, if segments of the base have the same ratio as the remaining sides of the triangle, then the straight line joining the vertex to the point of section bisects the angle of the triangle. Proposition 4. In equiangular triangles the sides about the equal angles are proportional where the corresponding sides are opposite the equal angles. Proposition 5. If two triangles have their sides proportional, then the triangles are equiangular with the equal angles opposite the corresponding sides. Proposition 6. If two triangles have one angle equal to one angle and the sides about the equal angles proportional, then the triangles are equiangular and have those angles equal opposite the corresponding sides. Proposition 7. If two triangles have one angle equal to one angle, the sides about other angles proportional, and the remaining angles either both less or both not less than a right angle, then the triangles are equiangular and have those angles equal the sides about which are proportional. Proposition 8. If in a right-angled triangle a perpendicular is drawn from the right angle to the base, then the triangles adjoining the perpendicular are similar both to the whole and to one another. Corollary. If in a right-angled triangle a perpendicular is drawn from the right angle to the base, then the straight line so drawn is a mean proportional between the segments of the base. Proposition 9. To cut off a prescribed part from a given straight line. Proposition 10. To cut a given uncut straight line similarly to a given cut straight line. Proposition 11. To find a third proportional to two given straight lines. Proposition 12. To find a fourth proportional to three given straight lines. Proposition 13. To find a mean proportional to two given straight lines. Proposition 14. In equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional; and equiangular parallelograms in which the sides about the equal angles are reciprocally proportional are equal. Proposition 15. In equal triangles which have one angle equal to one angle the sides about the equal angles are reciprocally proportional; and those triangles which have one angle equal to one angle, and in which the sides about the equal angles are reciprocally proportional, are equal. Proposition 16. If four straight lines are proportional, then the rectangle contained by the extremes equals the rectangle contained by the means; and, if the rectangle contained by the extremes equals the rectangle contained by the means, then the four straight lines are proportional. Proposition 17. If three straight lines are proportional, then the rectangle contained by the extremes equals the square on the mean; and, if the rectangle contained by the extremes equals the square on the mean, then the three straight lines are proportional. Proposition 18. To describe a rectilinear figure similar and similarly situated to a given rectilinear figure on a given straight line. Proposition 19. Similar triangles are to one another in the duplicate ratio of the corresponding sides. Corollary. If three straight lines are proportional, then the first is to the third as the figure described on the first is to that which is similar and similarly described on the second. Proposition 20. Similar polygons are divided into similar triangles, and into triangles equal in multitude and in the same ratio as the wholes, and the polygon has to the polygon a ratio duplicate of that which the corresponding side has to the corresponding side. Corollary. Similar rectilinear figures are to one another in the duplicate ratio of the corresponding sides. Proposition 21. Figures which are similar to the same rectilinear figure are also similar to one another. Proposition 22. If four straight lines are proportional, then the rectilinear figures similar and similarly described upon them are also proportional; and, if the rectilinear figures similar and similarly described upon them are proportional, then the straight lines are themselves also proportional. Proposition 23. Equiangular parallelograms have to one another the ratio compounded of the ratios of their sides. Proposition 24. In any parallelogram the parallelograms about the diameter are similar both to the whole and to one another. Proposition 25. To construct a figure similar to one given rectilinear figure and equal to another. Proposition 26. If from a parallelogram there is taken away a parallelogram similar and similarly situated to the whole and having a common angle with it, then it is about the same diameter with the whole. Proposition 27. Of all the parallelograms applied to the same straight line falling short by parallelogrammic figures similar and similarly situated to that described on the half of the straight line, that parallelogram is greatest which is applied to the half of the straight line and is similar to the difference. Proposition 28. To apply a parallelogram equal to a given rectilinear figure to a given straight line but falling short by a parallelogram similar to a given one; thus the given rectilinear figure must not be greater than the parallelogram described on the half of the straight line and similar to the given parallelogram. Proposition 29. To apply a parallelogram equal to a given rectilinear figure to a given straight line but exceeding it by a parallelogram similar to a given one. Proposition 30. To cut a given finite straight line in extreme and mean ratio. Proposition 31. In right-angled triangles the figure on the side opposite the right angle equals the sum of the similar and similarly described figures on the sides containing the right angle. Proposition 32. If two triangles having two sides proportional to two sides are placed together at one angle so that their corresponding sides are also parallel, then the remaining sides of the triangles are in a straight line. Proposition 33. Angles in equal circles have the same ratio as the circumferences on which they stand whether they stand at the centers or at the circumferences. Logical structure of Book VI Proposition VI.1 is the basis for the entire of Book VI except the last proposition VI.33. Only these two propositions directly use the definition of proportion in Book V. Proposition VI.1 constructs a proportion between lines and figures while VI.33 constructs a proportion between angles and circumferences. The intervening propositions use other properties of proportions developed in Book V, but they do not construct new proportions using the definition of proportion. Next book: Book VII Previous: Book V Book VI introduction Table of contents ● Definitions (18) ● Propositions (25) ● Guide to Book V ● Logical structure of Book V Definitions Definition 1 A magnitude is a part of a magnitude, the less of the greater, when it measures the greater. Definition 2 The greater is a multiple of the less when it is measured by the less. Definition 3 A ratio is a sort of relation in respect of size between two magnitudes of the same kind. Definition 4 Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another. Definition 5 Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever are taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order. Definition 6 Let magnitudes which have the same ratio be called proportional. Definition 7 When, of the equimultiples, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then the first is said to have a greater ratio to the second than the third has to the fourth. Definition 8 A proportion in three terms is the least possible. Definition 9 When three magnitudes are proportional, the first is said to have to the third the duplicate ratio of that which it has to the second. Definition 10 When four magnitudes are continuously proportional, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on continually, whatever be the proportion. Definition 11 Antecedents are said to correspond to antecedents, and consequents to consequents. Definition 12 Alternate ratio means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent. Definition 13 Inverse ratio means taking the consequent as antecedent in relation to the antecedent as consequent. Definition 14 A ratio taken jointly means taking the antecedent together with the consequent as one in relation to the consequent by itself. Definition 15 A ratio taken separately means taking the excess by which the antecedent exceeds the consequent in relation to the consequent by itself. Definition 16 Conversion of a ratio means taking the antecedent in relation to the excess by which the antecedent exceeds the consequent. Definition 17 A ratio ex aequali arises when, there being several magnitudes and another set equal to them in multitude which taken two and two are in the same proportion, the first is to the last among the first magnitudes as the first is to the last among the second magnitudes. Or, in other words, it means taking the extreme terms by virtue of the removal of the intermediate terms. Definition 18 A perturbed proportion arises when, there being three magnitudes and another set equal to them in multitude, antecedent is to consequent among the first magnitudes as antecedent is to consequent among the second magnitudes, while, the consequent is to a third among the first magnitudes as a third is to the antecedent among the second magnitudes. Propositions Proposition 1 If any number of magnitudes are each the same multiple of the same number of other magnitudes, then the sum is that multiple of the sum. Proposition 2 If a first magnitude is the same multiple of a second that a third is of a fourth, and a fifth also is the same multiple of the second that a sixth is of the fourth, then the sum of the first and fifth also is the same multiple of the second that the sum of the third and sixth is of the fourth. Proposition 3 If a first magnitude is the same multiple of a second that a third is of a fourth, and if equimultiples are taken of the first and third, then the magnitudes taken also are equimultiples respectively, the one of the second and the other of the fourth. Proposition 4 If a first magnitude has to a second the same ratio as a third to a fourth, then any equimultiples whatever of the first and third also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order. Proposition 5 If a magnitude is the same multiple of a magnitude that a subtracted part is of a subtracted part, then the remainder also is the same multiple of the remainder that the whole is of the whole. Proposition 6 If two magnitudes are equimultiples of two magnitudes, and any magnitudes subtracted from them are equimultiples of the same, then the remainders either equal the same or are equimultiples of them. Proposition 7 Equal magnitudes have to the same the same ratio; and the same has to equal magnitudes the same ratio. Corollary If any magnitudes are proportional, then they are also proportional inversely. Proposition 8 Of unequal magnitudes, the greater has to the same a greater ratio than the less has; and the same has to the less a greater ratio than it has to the greater. Proposition 9 Magnitudes which have the same ratio to the same equal one another; and magnitudes to which the same has the same ratio are equal. Proposition 10 Of magnitudes which have a ratio to the same, that which has a greater ratio is greater; and that to which the same has a greater ratio is less. Proposition 11 Ratios which are the same with the same ratio are also the same with one another. Proposition 12 If any number of magnitudes are proportional, then one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents. Proposition 13 If a first magnitude has to a second the same ratio as a third to a fourth, and the third has to the fourth a greater ratio than a fifth has to a sixth, then the first also has to the second a greater ratio than the fifth to the sixth. Proposition 14 If a first magnitude has to a second the same ratio as a third has to a fourth, and the first is greater than the third, then the second is also greater than the fourth; if equal, equal; and if less, less. Proposition 15 Parts have the same ratio as their equimultiples. Proposition 16 If four magnitudes are proportional, then they are also proportional alternately. Proposition 17 If magnitudes are proportional taken jointly, then they are also proportional taken separately. Proposition 18 If magnitudes are proportional taken separately, then they are also proportional taken jointly. Proposition 19 If a whole is to a whole as a part subtracted is to a part subtracted, then the remainder is also to the remainder as the whole is to the whole. Corollary. If magnitudes are proportional taken jointly, then they are also proportional in conversion. Proposition 20 If there are three magnitudes, and others equal to them in multitude, which taken two and two are in the same ratio, and if ex aequali the first is greater than the third, then the fourth is also greater than the sixth; if equal, equal, and; if less, less. Proposition 21 If there are three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them is perturbed, then, if ex aequali the first magnitude is greater than the third, then the fourth is also greater than the sixth; if equal, equal; and if less, less. Proposition 22 If there are any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, then they are also in the same ratio ex aequali. Proposition 23 If there are three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, then they are also in the same ratio ex aequali. Proposition 24 If a first magnitude has to a second the same ratio as a third has to a fourth, and also a fifth has to the second the same ratio as a sixth to the fourth, then the sum of the first and fifth has to the second the same ratio as the sum of the third and sixth has to the fourth. Proposition 25 If four magnitudes are proportional, then the sum of the greatest and the least is greater than the sum of the remaining two. for Book V Background on ratio and proportion Book V covers the abstract theory of ratio and proportion. A ratio is an indication of the relative size of two magnitudes. The propositions in the following book, Book VI, are all geometric and depend on ratios, so the theory of ratios needs to be developed first. To get a better understanding of what ratios are in geometry, consider the first proposition VI.1. It states that triangles of the same height are proportional to their bases, that is to say, one triangle is to another as one base is to the other. (A proportion is simply an equality of two ratios.) A simple example is when one base is twice the other, therefore the triangle on that base is also twice the triangle on the other base. This ratio of 2:1 is fairly easy to comprehend. Indeed, any ratio equal to a ratio of two numbers is easy to comprehend. Given a proportion that says a ratio of lines equals a ratio of numbers, for instance, A:B = 8:5, we have two interpretations. One is that there is a shorter line CA = 8C while B = 5C. This interpretation is the definition of proportion that appears in Book VII. A second interpretation is that 5 A = 8 B. Either interpretation will do if one of the ratios is a ratio of numbers, and if A:B equals a ratio of numbers that A and B are commensurable, that is, both are measured by a common measure. Many straight lines, however, are not commensurable. If A is the side of a square and B its diagonal, then A and B are not commensurable; the ratio A:B is not the ratio of numbers. This fact seems to have been discovered by the Pythagoreans, perhaps Hippasus of Metapontum, some time before 400 B.C.E., a hundred years before Euclid's Elements. The difficulty is one of foundations: what is an adequate definition of proportion that includes the incommensurable case? The solution is that in V.Def.5. That definition, and the whole theory of ratio and proportion in Book V, are attributed to Eudoxus of Cnidus (died. ca. 355 B.C.E.) Summary of the propositions The first group of propositions, 1, 2, 3, 5, and 6 only mention multitudes of magnitudes, not ratios. They each either state, or depend strongly on, a distributivity or an associativity. In the following identities, m and n refer to numbers (that is, multitudes) while letters near the end of the alphabet refer to magnitudes. V.1. Multiplication by numbers distributes over addition of magnitudes. m(x1 + x2 + ... + xn) = m x1 + m x2 + ... + m xn. V.2. Multiplication by magnitudes distributes over addition of numbers. (m + n)x = mx + nx. V.3. An associativity of multiplication. m(nx) = (mn)x. V.5. Multiplication by numbers distributes over subtraction of magnitudes. m(x y) = mx my. V.6. Uses multiplication by magnitudes distributes over subtraction of numbers. (m n)x = mx nx. The rest of the propositions develop the theory of ratios and proportions starting with basic properties and progressively becoming more advanced. V.4. If w:x = y:z, then for any numbers m and n, mw:mx = ny:nz. V.7. Substitution of equals in ratios. If x = y, then x:z = y:z and z:x = z:y. V.7.Cor. Inverse proportions. If w:x = y:z, then x:w = z:y. V.8. If x < y, then x:z < y:z but z:x > z:y. V.9. (A converse to V.7.) If x:z = y:z, then x = y. Also, if z:x = z:y, then x = y. V.10. (A converse to V.8.) If x:z < y:z, then x < y. But if z:x < z:y, then x > y V.11. Transitivity of equal ratios. If u:v = w:x and w:x = y:z, then u:v = y:z. V.12. If x1:y1 = x2:y2 = ... = xn:yn, then each of these ratios also equals the ratio (x1 + x2 + ... + xn) : (y1 + y2 + ... + yn). V.13. Substitution of equal ratios in inequalities of ratios. If u:v = w:x and w:x > y:z, then u:v > y:z. V.14. If w:x = y:z and w > y, then x > z. V.15. x:y = nx:ny. V.16. Alternate proportions. If w:x = y:z, then w:y = x:z. V.17. Proportional taken jointly implies proportional taken separately. If (w + x):x = (y + z):z, then w:x = y:z. V.18. Proportional taken separately implies proportional taken jointly. (A converse to V.17.) If w:x = y:z, then (w + x):x = (y + z):z. V.19. If (w + x):(y + z) = w:y, then (w + x):(y + z) = x:z, too. V.19.Cor. Proportions in conversion. If (u + v):(x + y) = v:y, then (u + v):(x + y) = u:x. V.20 is just a preliminary proposition to V.22, and V.21 is just a preliminary proposition to V.23. V.22. Ratios ex aequali. If x1:x2 = y1:y2, x2:x3 = y2:y3, ... , and xn-1:xn = yn-1:yn, then x1:xn = y1:yn. V.23. Perturbed ratios ex aequali. If u:v = y:z and v:w = x:y, then u:w = x:z. V.24. If u:v = w:x and y:v = z:x, then (u + y):v = (w + z):x. V.25. If w:x = y:z and w is the greatest of the four magnitudes while z is the least, then w + z > x + y. Logical structure of Book V Book V is on the foundations of ratios and proportions and in no way depends on any of the previous Books. Book VI contains the propositions on plane geometry that depend on ratios, and the proofs there frequently depend on the results in Book V. Also Book X on irrational lines and the books on solid geometry, XI through XIII, discuss ratios and depend on Book V. The books on number theory, VII through IX, do not directly depend on Book V since there is a different definition for ratios of numbers. Although Euclid is fairly careful to prove the results on ratios that he uses later, there are some that he didn't notice he used, for instance, the law of trichotomy for ratios. These are described in the Guides to definitions V.Def.4 through V.Def.7. * Some of the propositions in Book V require treating definition V.Def.4 as an axiom of comparison. One side of the law of trichotomy for ratios depends on it as well as propositions 8, 9, 14, 16, 21, 23, and 25. Some of Euclid's proofs of the remaining propositions rely on these propositions, but alternate proofs that don't depend on an axiom of comparison can be given for them. Propositions 1, 2, 7, 11, and 13 are proved without invoking other propositions. There are moderately long chains of deductions, but not so long as those in Book I. The first six propositions excepting 4 have to do with arithmetic of magnitudes and build on the Common Notions. The next group of propositions, 4 and 7 through 15, use the earlier propositions and defintions 4 through 7 to develop the more basic properties of ratios. And the last 10 propositions depend on most of the preceeding ones to Dependencies within Book V 2 3, 6 3 4 1 5, 8*, 12 8* 9* 7, 8* 10 8*, 10, 13 14* 7, 12 15 11, 14*, 15 16* 1, 2 17 11, 14*, 17 18 11, 16*, 17 19 7.Cor, 8, 10, 13 20, 21* 4, 20 22 11, 15, 16*, 21* 23* 7.Cor, 18, 22 24 7, 11, (14), 19 25* develop advanced properties. Next book: Book VI Previous: Book IV Table of contents ● Definitions (7) ● Propositions (16) ● Guide to Book IV ● Logical structure of Book IV Definitions Definition 1. A rectilinear figure is said to be inscribed in a rectilinear figure when the respective angles of the inscribed figure lie on the respective sides of that in which it is inscribed. Definition 2. Similarly a figure is said to be circumscribed about a figure when the respective sides of the circumscribed figure pass through the respective angles of that about which it is circumscribed. Definition 3. A rectilinear figure is said to be inscribed in a circle when each angle of the inscribed figure lies on the circumference of the circle. Definition 4. A rectilinear figure is said to be circumscribed about a circle when each side of the circumscribed figure touches the circumference of the circle. Definition 5. Similarly a circle is said to be inscribed in a figure when the circumference of the circle touches each side of the figure in which it is inscribed. Definition 6. A circle is said to be circumscribed about a figure when the circumference of the circle passes through each angle of the figure about which it is circumscribed. Definition 7. A straight line is said to be fitted into a circle when its ends are on the circumference of the circle. Propositions Proposition 1. To fit into a given circle a straight line equal to a given straight line which is not greater than the diameter of the circle. Proposition 2. To inscribe in a given circle a triangle equiangular with a given triangle. Proposition 3. To circumscribe about a given circle a triangle equiangular with a given triangle. Proposition 4. To inscribe a circle in a given triangle. Proposition 5. To circumscribe a circle about a given triangle. Corollary. When the center of the circle falls within the triangle, the triangle is acute-angled; when the center falls on a side, the triangle is right-angled; and when the center of the circle falls outside the triangle, the triangle is obtuse-angled. Proposition 6. To inscribe a square in a given circle. Proposition 7. To circumscribe a square about a given circle. Proposition 8. To inscribe a circle in a given square. Proposition 9. To circumscribe a circle about a given square. Proposition 10. To construct an isosceles triangle having each of the angles at the base double the remaining one. Proposition 11. To inscribe an equilateral and equiangular pentagon in a given circle. Proposition 12. To circumscribe an equilateral and equiangular pentagon about a given circle. Proposition 13. To inscribe a circle in a given equilateral and equiangular pentagon. Proposition 14. To circumscribe a circle about a given equilateral and equiangular pentagon. Proposition 15. To inscribe an equilateral and equiangular hexagon in a given circle. Corollary. The side of the hexagon equals the radius of the circle. And, in like manner as in the case of the pentagon, if through the points of division on the circle we draw tangents to the circle, there will be circumscribed about the circle an equilateral and equiangular hexagon in conformity with what was explained in the case of the pentagon. And further by means similar to those explained in the case of the pentagon we can both inscribe a circle in a given hexagon and circumscribe one about it. Proposition 16. To inscribe an equilateral and equiangular fifteen-angled figure in a given circle. Corollary. And, in like manner as in the case of the pentagon, if through the points of division on the circle we draw tangents to the circle, there will be circumscribed about the circle a fifteen-angled figure which is equilateral and equiangular. And further, by proofs similar to those in the case of the pentagon, we can both inscribe a circle in the given fifteen-angled figure and circumscribe one about it. Guide to Book IV All but two of the propositions in this book are constructions to inscribe or circumscribe figures. Figure Inscribe figure in circle Circumscribe figure about circle Inscribe circle in figure Circumscribe circle about figure Triangle IV.2 IV.3 IV.4 IV.5 Square IV.6 IV.7 IV.8 IV.9 Regular pentagon IV.11 IV.12 IV.13 IV.14 Regular hexagon IV.15 IV.15,Cor IV.15,Cor IV.15,Cor Regular 15gon IV.16 IV.16,Cor IV.16,Cor IV.16,Cor There are only two other propositions. Proposition IV.1 is a basic construction to fit a line in a circle, and proposition IV.10 constructs a particular triangle needed in the construction of a regular pentagon. Logical structure of Book IV The proofs of the propositions in Book IV rely heavily on the propositions in Books I and III. Only one proposition from Book II is used and that is the construction in II.11 used in proposition IV.10 to construct a particular triangle needed in the construction of a regular pentagon. Most of the propositions of Book IV are logically independent of each other. There is a short chain of deductions, however, involving the construction of regular pentagons. Dependencies within Book IV 1, 5 10 2, 10 11 11 12 1, 2, 11 16 Next book: Book V Previous: Book III Table of contents ● Definitions (11) ● Propositions (37) Definitions Definition 1. Equal circles are those whose diameters are equal, or whose radii are equal. Definition 2. A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle. Definition 3. Circles are said to touch one another which meet one another but do not cut one another. Definition 4. Straight lines in a circle are said to be equally distant from the center when the perpendiculars drawn to them from the center are equal. Definition 5. And that straight line is said to be at a greater distance on which the greater perpendicular falls. Definition 6. A segment of a circle is the figure contained by a straight line and a circumference of a circle. Definition 7. An angle of a segment is that contained by a straight line and a circumference of a circle. Definition 8. An angle in a segment is the angle which, when a point is taken on the circumference of the segment and straight lines are joined from it to the ends of the straight line which is the base of the segment, is contained by the straight lines so joined. Definition 9. And, when the straight lines containing the angle cut off a circumference, the angle is said to stand upon that circumference. Definition 10. A sector of a circle is the figure which, when an angle is constructed at the center of the circle, is contained by the straight lines containing the angle and the circumference cut off by them. Definition 11. Similar segments of circles are those which admit equal angles, or in which the angles equal one another. Propositions Proposition 1. To find the center of a given circle. Corollary. If in a circle a straight line cuts a straight line into two equal parts and at right angles, then the center of the circle lies on the cutting straight line. Proposition 2. If two points are taken at random on the circumference of a circle, then the straight line joining the points falls within the circle. Proposition 3. If a straight line passing through the center of a circle bisects a straight line not passing through the center, then it also cuts it at right angles; and if it cuts it at right angles, then it also bisects it. Proposition 4. If in a circle two straight lines which do not pass through the center cut one another, then they do not bisect one another. Proposition 5. If two circles cut one another, then they do not have the same center. Proposition 6. If two circles touch one another, then they do not have the same center. Proposition 7. If on the diameter of a circle a point is taken which is not the center of the circle, and from the point straight lines fall upon the circle, then that is greatest on which passes through the center, the remainder of the same diameter is least, and of the rest the nearer to the straight line through the center is always greater than the more remote; and only two equal straight lines fall from the point on the circle, one on each side of the least straight line. Proposition 8. If a point is taken outside a circle and from the point straight lines are drawn through to the circle, one of which is through the center and the others are drawn at random, then, of the straight lines which fall on the concave circumference, that through the center is greatest, while of the rest the nearer to that through the center is always greater than the more remote, but, of the straight lines falling on the convex circumference, that between the point and the diameter is least, while of the rest the nearer to the least is always less than the more remote; and only two equal straight lines fall on the circle from the point, one on each side of the least. Proposition 9. If a point is taken within a circle, and more than two equal straight lines fall from the point on the circle, then the point taken is the center of the circle. Proposition 10. A circle does not cut a circle at more than two points. Proposition 11. If two circles touch one another internally, and their centers are taken, then the straight line joining their centers, being produced, falls on the point of contact of the circles. Proposition 12. If two circles touch one another externally, then the straight line joining their centers passes through the point of contact. Proposition 13. A circle does not touch another circle at more than one point whether it touches it internally or externally.. Proposition 14. Equal straight lines in a circle are equally distant from the center, and those which are equally distant from the center equal one another. Proposition 15. Of straight lines in a circle the diameter is greatest, and of the rest the nearer to the center is always greater than the more remote. Proposition 16. The straight line drawn at right angles to the diameter of a circle from its end will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed, further the angle of the semicircle is greater, and the remaining angle less, than any acute rectilinear angle. Corollary. From this it is manifest that the straight line drawn at right angles to the diameter of a circle from its end touches the circle. Proposition 17. From a given point to draw a straight line touching a given circle. Proposition 18. If a straight line touches a circle, and a straight line is joined from the center to the point of contact, the straight line so joined will be perpendicular to the tangent. Proposition 19. If a straight line touches a circle, and from the point of contact a straight line is drawn at right angles to the tangent, the center of the circle will be on the straight line so drawn. Proposition 20. In a circle the angle at the center is double the angle at the circumference when the angles have the same circumference as base. Proposition 21. In a circle the angles in the same segment equal one another. Proposition 22. The sum of the opposite angles of quadrilaterals in circles equals two right angles. Proposition 23. On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side. Proposition 24. Similar segments of circles on equal straight lines equal one another. Proposition 25. Given a segment of a circle, to describe the complete circle of which it is a segment. Proposition 26. In equal circles equal angles stand on equal circumferences whether they stand at the centers or at the circumferences. Proposition 27. In equal circles angles standing on equal circumferences equal one another whether they stand at the centers or at the circumferences. Proposition 28. In equal circles equal straight lines cut off equal circumferences, the greater circumference equals the greater and the less equals the less. Proposition 29. In equal circles straight lines that cut off equal circumferences are equal. Proposition 30. To bisect a given circumference. Proposition 31. In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle; further the angle of the greater segment is greater than a right angle, and the angle of the less segment is less than a right angle. Proposition 32. If a straight line touches a circle, and from the point of contact there is drawn across, in the circle, a straight line cutting the circle, then the angles which it makes with the tangent equal the angles in the alternate segments of the circle. Proposition 33. On a given straight line to describe a segment of a circle admitting an angle equal to a given rectilinear angle. Proposition 34. From a given circle to cut off a segment admitting an angle equal to a given rectilinear angle. Proposition 35. If in a circle two straight lines cut one another, then the rectangle contained by the segments of the one equals the rectangle contained by the segments of the other. Proposition 36. If a point is taken outside a circle and two straight lines fall from it on the circle, and if one of them cuts the circle and the other touches it, then the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference equals the square on the tangent. Proposition 37. If a point is taken outside a circle and from the point there fall on the circle two straight lines, if one of them cuts the circle, and the other falls on it, and if further the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference equals the square on the straight line which falls on the circle, then the straight line which falls on it touches the circle. Next book: Book IV Previous: Book II Table of contents ● Definitions (2) ● Propositions (14) ● Guide to Book II ● Logical structure of Book II Definitions Definition 1. Any rectangular parallelogram is said to be contained by the two straight lines containing the right angle. Definition 2 And in any parallelogrammic area let any one whatever of the parallelograms about its diameter with the two complements be called a gnomon. Propositions Proposition 1. If there are two straight lines, and one of them is cut into any number of segments whatever, then the rectangle contained by the two straight lines equals the sum of the rectangles contained by the uncut straight line and each of the segments. Proposition 2. If a straight line is cut at random, then the sum of the rectangles contained by the whole and each of the segments equals the square on the whole. Proposition 3. If a straight line is cut at random, then the rectangle contained by the whole and one of the segments equals the sum of the rectangle contained by the segments and the square on the aforesaid segment. Proposition 4. If a straight line is cut at random, the square on the whole equals the squares on the segments plus twice the rectangle contained by the segments. Proposition 5. If a straight line is cut into equal and unequal segments, then the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section equals the square on the half. Proposition 6. If a straight line is bisected and a straight line is added to it in a straight line, then the rectangle contained by the whole with the added straight line and the added straight line together with the square on the half equals the square on the straight line made up of the half and the added straight line. Proposition 7. If a straight line is cut at random, then the sum of the square on the whole and that on one of the segments equals twice the rectangle contained by the whole and the said segment plus the square on the remaining segment. Proposition 8. If a straight line is cut at random, then four times the rectangle contained by the whole and one of the segments plus the square on the remaining segment equals the square described on the whole and the aforesaid segment as on one straight line. Proposition 9. If a straight line is cut into equal and unequal segments, then the sum of the squares on the unequal segments of the whole is double the sum of the square on the half and the square on the straight line between the points of section. Proposition 10. If a straight line is bisected, and a straight line is added to it in a straight line, then the square on the whole with the added straight line and the square on the added straight line both together are double the sum of the square on the half and the square described on the straight line made up of the half and the added straight line as on one straight line. Proposition 11. To cut a given straight line so that the rectangle contained by the whole and one of the segments equals the square on the remaining segment. Proposition 12. In obtuse-angled triangles the square on the side opposite the obtuse angle is greater than the sum of the squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides about the obtuse angle, namely that on which the perpendicular falls, and the straight line cut off outside by the perpendicular towards the obtuse angle. Proposition 13. In acute-angled triangles the square on the side opposite the acute angle is less than the sum of the squares on the sides containing the acute angle by twice the rectangle contained by one of the sides about the acute angle, namely that on which the perpendicular falls, and the straight line cut off within by the perpendicular towards the acutc angle. Proposition 14. To construct a square equal to a given rectilinear figure. Guide to Book II The subject matter of Book II is usually called "geometric algebra." The first ten propositions of Book II can be easily interpreted in modern algebraic notation. Of course, in doing so the geometric flavor of the propositions is lost. Nonetheless, restating them algebraically can aid in understanding them. The equations are all quadratic equations since the geometry is plane geometry. II.1. If y = y1 + y2 + ... + yn, then xy = x y1 + x y2 + ... + x yn. This can be stated in a single identity as x (y1 + y2 + ... + yn) = x y1 + x y2 + ... + x yn. II.2. If x = y + z, then x2 = xy + xz. This can be stated in various ways in an identity of two variables. For instance, (y + z)2 = (y + z) y + (y + z) z, or x2 = xy + x (x – y). II.3. If x = y + z, then xy = yz + y2. Equivalent identities are (y + z)y = yz + y2, and xy = y(x – y) + y2. II.4. If x = y + z, then x2 = y2 + z2 + 2yz. As an identity, (y + z)2 = y2 + z2 + 2yz. II.5 and II.6. (y + z) (y – z) + z2 = y2. II.7. if x = y + z, then x2 + z2 = 2xz + y2. As an identity, x2 + z2 = 2xz + (x – z)2. II.8. If x = y + z, then 4xy + z2 = (x + y)2. As an identity, 4xy + (x – y)2 = (x + y)2. II.9 and II.10. (y + z)2 + (y – z)2 = 2 (y2 + z2). The remaining four propositions are of a slightly different nature. Proposition II.11 cuts a line into two parts which solves the equation a (a – x) = x2 geometrically. Propositions II.12 and II.13 are recognizable as geometric forms of the law of cosines which is a generalization of I.47. The last propostion II.14 constructs a square equal to a given rectilinear figure thereby completeing the theory of areas begun in Book I. Logical structure of Book II The proofs of the propositions in Book II heavily rely on the propositions in Book I involving right angles and parallel lines, but few others. For instance, the important congruence theorems for triangles, namely I.4, I.8, and I.26, are not invoked even once. This is understandable considering Book II is mostly algebra interpreted in the theory of geometry. The first ten propositions in Book II were written to be logically independent, but they could have easily been written in logical chains which, perhaps, would have shortened the exposition a little. The remaining four propositions each depend on one of the first ten. Dependencies within Book II 6 11 4 12 7 13 5 14 Next book: Book III Previous book: Book I Elements Introduction Table of contents ● Definitions (23) ● Postulates (5) ● Common Notions (5) ● Propositions (48) ● Guide to Book I Definitions Definition 1. A point is that which has no part. Definition 2. A line is breadthless length. Definition 3. The ends of a line are points. Definition 4. A straight line is a line which lies evenly with the points on itself. Definition 5. A surface is that which has length and breadth only. Definition 6. The edges of a surface are lines. Definition 7. A plane surface is a surface which lies evenly with the straight lines on itself. Definition 8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line. Definition 9. And when the lines containing the angle are straight, the angle is called rectilinear. Definition 10. When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands. Definition 11. An obtuse angle is an angle greater than a right angle. Definition 12. An acute angle is an angle less than a right angle. Definition 13. A boundary is that which is an extremity of anything. Definition 14. A figure is that which is contained by any boundary or boundaries. Definition 15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another. Definition 16. And the point is called the center of the circle. Definition 17. A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle. Definition 18. A semicircle is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle. Definition 19. Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines. Definition 20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal. Definition 21. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuseangled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute. Definition 22. Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not rightangled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia. Definition 23 Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction. Postulates Let the following be postulated: Postulate 1. To draw a straight line from any point to any point. Postulate 2. To produce a finite straight line continuously in a straight line. Postulate 3. To describe a circle with any center and radius. Postulate 4. That all right angles equal one another. Postulate 5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. Common Notions Common notion 1. Things which equal the same thing also equal one another. Common notion 2. If equals are added to equals, then the wholes are equal. Common notion 3. If equals are subtracted from equals, then the remainders are equal. Common notion 4. Things which coincide with one another equal one another. Common notion 5. The whole is greater than the part. Propositions Proposition 1. To construct an equilateral triangle on a given finite straight line. Proposition 2. To place a straight line equal to a given straight line with one end at a given point. Proposition 3. To cut off from the greater of two given unequal straight lines a straight line equal to the less. Proposition 4. If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides. Proposition 5. In isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another. Proposition 6. If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another. Proposition 7. Given two straight lines constructed from the ends of a straight line and meeting in a point, there cannot be constructed from the ends of the same straight line, and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each equal to that from the same end. Proposition 8. If two triangles have the two sides equal to two sides respectively, and also have the base equal to the base, then they also have the angles equal which are contained by the equal straight lines. Proposition 9. To bisect a given rectilinear angle. Proposition 10. To bisect a given finite straight line. Proposition 11. To draw a straight line at right angles to a given straight line from a given point on it. Proposition 12. To draw a straight line perpendicular to a given infinite straight line from a given point not on it. Proposition 13. If a straight line stands on a straight line, then it makes either two right angles or angles whose sum equals two right angles. Proposition 14. If with any straight line, and at a point on it, two straight lines not lying on the same side make the sum of the adjacent angles equal to two right angles, then the two straight lines are in a straight line with one another. Proposition 15. If two straight lines cut one another, then they make the vertical angles equal to one another. Corollary. If two straight lines cut one another, then they will make the angles at the point of section equal to four right angles. Proposition 16. In any triangle, if one of the sides is produced, then the exterior angle is greater than either of the interior and opposite angles. Proposition 17. In any triangle the sum of any two angles is less than two right angles. Proposition 18. In any triangle the angle opposite the greater side is greater. Proposition 19. In any triangle the side opposite the greater angle is greater. Proposition 20. In any triangle the sum of any two sides is greater than the remaining one. Proposition 21. If from the ends of one of the sides of a triangle two straight lines are constructed meeting within the triangle, then the sum of the straight lines so constructed is less than the sum of the remaining two sides of the triangle, but the constructed straight lines contain a greater angle than the angle contained by the remaining two sides. Proposition 22. To construct a triangle out of three straight lines which equal three given straight lines: thus it is necessary that the sum of any two of the straight lines should be greater than the remaining one. Proposition 23. To construct a rectilinear angle equal to a given rectilinear angle on a given straight line and at a point on it. Proposition 24. If two triangles have two sides equal to two sides respectively, but have one of the angles contained by the equal straight lines greater than the other, then they also have the base greater than the base. Proposition 25. If two triangles have two sides equal to two sides respectively, but have the base greater than the base, then they also have the one of the angles contained by the equal straight lines greater than the other. Proposition 26. If two triangles have two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that opposite one of the equal angles, then the remaining sides equal the remaining sides and the remaining angle equals the remaining angle. Proposition 27. If a straight line falling on two straight lines makes the alternate angles equal to one another, then the straight lines are parallel to one another. Proposition 28. If a straight line falling on two straight lines makes the exterior angle equal to the interior and opposite angle on the same side, or the sum of the interior angles on the same side equal to two right angles, then the straight lines are parallel to one another. Proposition 29. A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the sum of the interior angles on the same side equal to two right angles. Proposition 30. Straight lines parallel to the same straight line are also parallel to one another. Proposition 31. To draw a straight line through a given point parallel to a given straight line. Proposition 32. In any triangle, if one of the sides is produced, then the exterior angle equals the sum of the two interior and opposite angles, and the sum of the three interior angles of the triangle equals two right angles. Proposition 33. Straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel. Proposition 34. In parallelogrammic areas the opposite sides and angles equal one another, and the diameter bisects the areas. Proposition 35. Parallelograms which are on the same base and in the same parallels equal one another. Proposition 36. Parallelograms which are on equal bases and in the same parallels equal one another. Proposition 37. Triangles which are on the same base and in the same parallels equal one another. Proposition 38. Triangles which are on equal bases and in the same parallels equal one another. Proposition 39. Equal triangles which are on the same base and on the same side are also in the same parallels. Proposition 40. Equal triangles which are on equal bases and on the same side are also in the same parallels. Proposition 41. If a parallelogram has the same base with a triangle and is in the same parallels, then the parallelogram is double the triangle. Proposition 42. To construct a parallelogram equal to a given triangle in a given rectilinear angle. Proposition 43. In any parallelogram the complements of the parallelograms about the diameter equal one another. Proposition 44. To a given straight line in a given rectilinear angle, to apply a parallelogram equal to a given triangle. Proposition 45. To construct a parallelogram equal to a given rectilinear figure in a given rectilinear angle. Proposition 46. To describe a square on a given straight line. Proposition 47. In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle. Proposition 48. If in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right. About the Definitions The Elements begins with a list of definitions. Some of these indicate little more than certain concepts will be discussed, such as Def.I.1, Def.I.2, and Def.I.5, which introduce the terms point, line, and surface. (Note that for Euclid, the concept of line includes curved lines.) Others are substantial definitions which actually describe new concepts in terms of old ones. For example, Def.I.10 defines a right angle as one of two equal adjacent angles made when one straight line meets another. Other definitions look like they're substantial, but actually are not. For instance, Def.I.4 says a straight line "is a line which lies evenly with the points on itself." No where in the Elements is the defining phrase "which lies evenly with the points on itself" applicable. Thus, this definition indicates, at most, that some lines under discussion will be straight lines. It has been suggested that the definitions were added to the Elements sometime after Euclid wrote them. Another possibility is that they are actually from a different work, perhaps older. In Def.I.22 special kinds of quadrilaterals are defined including square, oblong (a rectangle that are not squares), rhombus (equilateral but not a square), and rhomboid (parallelogram but not a rhombus). Except for squares, these other shapes are not mentioned in the Elements. Euclid does use parallelograms, but they're not defined in this definition. Also, the exclusive nature of some of these terms-the part that indicates not a square-is contrary to Euclid's practice of accepting squares and rectangles as kinds of parallelograms. About the Postulates Following the list of definitions is a list of postulates. Each postulate is an axiom-which means a statement which is accepted without proof- specific to the subject matter, in this case, plane geometry. Most of them are constructions. For instance, Post.I.1 says a straight line can be drawn between two points, and Post.I.3 says a circle can be drawn given a specified point to be the center and another point to be on the circumference. The fourth postulate, Post.I.4, is not a constuction, but says that all right angles are equal. About magnitudes and the Common Notions The Common Notions are also axioms, but they refer to magnitudes of various kinds. The kind of magnitude that appears most frequently is that of straight line. Other important kinds are rectilinear angles and areas (plane figures). Later books include other kinds. In proposition III.16 (but nowhere else) angles with curved sides are compared with rectilinear angles which shows that rectilinear angles are to be considered as a special kind of plane angle. That agrees with Euclid's definition of them in I.Def.9 and I.Def.8. Also in Book III, parts of circumferences of circles, that is, arcs, appear as magnitudes. Only arcs of equal circles can be compared or added, so arcs of equal circles comprise a kind of magnitude, while arcs of unequal circles are magnitudes of different kinds. These kinds are all different from straight lines. Whereas areas of figures are comparable, different kinds of curves are not. Book V includes the general theory of ratios. No particular kind of magnitude is specified in that book. It may come as a surprise that ratios do not themselves form a kind of magnitude since they can be compared, but they cannot be added. See the guide on Book V for more information. Number theory is treated in Books VII through IX. It could be considered that numbers form a kind of magnitude as pointed out by Aristotle. Beginning in Book XI, solids are considered, and they form the last kind of magnitude discussed in the Elements. The propositions Following the definitions, postulates, and common notions, there are 48 propositions. Each of these propositions includes a statement followed by a proof of the statement. Each statement of the proof is logically justified by a definition, postulate, common notion, or an earlier proposition that has already been proven. There are gaps in the logic of some of the proofs, and these are mentioned in the commenaries after the propositions. Also included in the proof is a diagram illustrating the proof. Some of the propositions are constructions. A construction depends, ultimately, on the constructive postulates about drawing lines and circles. The first part of a proof for a constuctive proposition is how to perform the construction. The rest of the proof (usually the longer part), shows that the proposed construction actually satisfies the goal of the proposition. In the list of propositions in each book, the constructions are displayed in red. Most of the propositions, however, are not constructions. Their statements say that under certain conditions, certain other conditions logically follow. For example, Prop.I.5 says that if a triangle has the property that two of its sides are equal, then it follows that the angles opposite these sides (called the "base angles") are also equal. Even the propositions that are not constructions may have constructions included in their proofs since auxillary lines or circles may be needed in the explanation. But the bulk of the proof is, as for the constructive propositions, a sequence of statements that are logically justified and which culminates in the statement of the proposition. Logical structure of Book I The various postulates and common notions are frequently used in Book I. Only two of the propositions rely solely on the postulates and axioms, namely, I.1 and I.4. The logical chains of propositions in Book I are longer than in the other books; there are long sequences of propositions each relying on the previous. Dependencies within Book I 1 2 3 3, 4 5, 6 5 7 8 1, 3, 8 9, 11 1, 4, 9 10 8, 10 12 11 13 14, 15 3, 4, 10, 15 16 27 13, 16 17 3, 5, 16 18 5, 18 19 3, 5, 19 20 16, 20 21 3, 20 22 8, 22 23 3, 4, 5, 19, 23 24 4, 24 25 3, 4, 16 26 13, 15, 27 28, 29 29 30 23, 27 31 13, 29, 31 32 4, 27, 29 33 4, 26, 29 34 43 4, 29, 34 35 33, 34, 35 36 31, 34, 35 37 31, 34, 36 38 31, 37 39 31, 38 40 34, 37 41 10, 23, 31, 38, 41 42 15, 29, 31, 42, 43 44 14, 29, 30, 33, 34, 42, 44 45 3, 11, 29, 31, 34 46 4, 14, 31, 41, 46 47 3, 8, 11, 47 48 Next book: Book II Proposition 1 If a straight line is cut in extreme and mean ratio, then the square on the greater segment added to the half of the whole is five times the square on the half. Let the straight line AB be cut in extreme and mean ratio at the point C, and let AC be the greater segment. Produce the straight line AD in a straight line with CA, and make AD half of AB. I say that the square on CD is five times the square on AD. Describe the squares AE and DF on AB and DC, draw the figure in DF, and carry FC through to G. I.46 Now, since AB is cut in extreme and mean ratio at C, therefore the rectangle AB by BC equals the square on AC. And CE is the rectangle AB by BC, and FH is the square on AC, therefore CE equals FH. VI.Def.3 VI.17 And, since BA is double AD, while BA equals KA, and AD equals AH, therefore KA is also double AH. But KA is to AH as CK is to CH, therefore CK is double CH. But the sum of LH and HC is also double CH. Therefore KC equals the sum of LH and HC. VI.1 But CE was also proved equal to HF, therefore the whole square AE equals the gnomon MNO. And, since BA is double AD, therefore the square on BA is quadruple the square on AD, that is, AE is quadruple DH. But AE equals the gnomon MNO, therefore the gnomon MNO is also quadruple AP. Therefore the whole DF is five times AP. And DF is the square on DC, and AP the square on DP, therefore the square on CD is five times the square on DA. Therefore, if a straight line is cut in extreme and mean ratio, then the square on the greater segment added to the half of the whole is five times the square on the half. Q.E.D. Use of this theorem This proposition is used in the proofs of XIII.6 and XIII.11. Those propositions are in turn used to make conclusions about the sides of the icosahedron and dodecahedron constructed in propositions XIII.16 and XIII.17. Next proposition: XIII.2 Book XIII introduction Proposition 2 If the square on a straight line is five times the square on a segment on it, then, when the double of the said segment is cut in extreme and mean ratio, the greater segment is the remaining part of the original straight line. Let the square on the straight line AB be five times the square on the segment AC of it, and let CD be double AC. I say that, when CD is cut in extreme and mean ratio, then the greater segment is CB. Describe the squares AF and CG on AB and CD respectively, draw the figure in AF, and draw BE through. I.46 Now, since the square on BA is five times the square on AC, therefore AF is five times AH. Therefore the gnomon MNO is quadruple AH. And, since DC is double CA, therefore the square on DC is quadruple the square on CA, that is, CG is quadruple AH. But the gnomon MNO is also quadruple AH, therefore the gnomon MNO equals CG. And, since DC is double CA, while DC equals CK, and AC equals CH, therefore KB is also double BH. VI.1 But the sum of LH and HB is also double HB, therefore KB equals the sum of LH and HB. But the whole gnomon MNO was also proved equal to the whole CG, therefore the remainder HF equals BG. And BG is the rectangle CD by DB, for CD equals DG, and HF is the square on CB, therefore the rectangle CD by DB equals the square on CB. Therefore DC is to CB as CB is to BD. But DC is greater than CB, therefore CB is also greater than BD. Therefore, when the straight line CD is cut in extreme and mean ratio, CB is the greater segment. Q.E.D. Lemma That the double AC is greater than BC is to be proved thus. If not, let BC be, if possible, double CA. Therefore the square on BC is quadruple the square on CA. Therefore the sum of the squares on BC and CA is five times the square on CA. But, by hypothesis, the square on BA is also five times the square on CA Therefore the square on BA equals the sum of the squares on BC and CA, which is impossible. II.4 Therefore CB is not double AC. Similarly we can prove that neither is a straight line less than CB double CA, for the absurdity is much greater. Therefore the double AC is greater than CB. Therefore, if the square on a straight line is five times the square on a segment on it, then, when the double of the said segment is cut in extreme and mean ratio, the greater segment is the remaining part of the original straight line. Q.E.D. This proposition is not used in the rest of the Elements. Apparently, it is only included because it is the converse of the previous proposition XIII.1. Next proposition: XIII.3 Previous: XIII.1 Book XIII introduction Proposition 3 If a straight line is cut in extreme and mean ratio, then the square on the sum of the lesser segment and the half of the greater segment is five times the square on the half of the greater segment. Cut any straight line AB in extreme and mean ratio at the point C, and let AC be the greater segment. Bisect AC at D. I say that the square on BD is five times the square on DC. Describe the square AE on AB, and draw the figure. I.46 Since AC is double DC, therefore the square on AC is quadruple the square on DC, that is, RS is quadruple FG. And, since the rectangle AB by BC equals the square on AC, and CE is the rectangle AB by BC, therefore CE equals RS. But RS is quadruple FG, therefore CE is also quadruple FG. Again, since AD equals DC, therefore HK also equals KF. Hence the square GF equals the square HL. Therefore GK equals KL, that is MN equals NE, hence MF equals FE. But MF equals CG, therefore CG equals FE. Add CN to each, therefore the gnomon OPQ equals CE. But CE was proved quadruple GF, therefore the gnomon OPQ is also quadruple the square FG. Therefore the sum of the gnomon OPQ and the square FG is five times FG. But the sum of the gnomon OPQ and the square FG is the square DN. And DN is the square on DB, and GF is the square on DC. Therefore the square on DB is five times the square on DC. Therefore, if a straight line is cut in extreme and mean ratio, then the square on the sum of the lesser segment and the half of the greater segment is five times the square on the half of the greater segment. Q.E.D. Use of this proposition This result is needed in proposition XIII.16 to show the icosahedron is inscribed in the given sphere. Next proposition: XIII.4 Previous: XIII.2 Book XIII introduction Proposition 4 If a straight line is cut in extreme and mean ratio, then the sum of the squares on the whole and on the lesser segment is triple the square on the greater segment. Let AB be a straight line cut in extreme and mean ratio at C, and let AC be the greater segment. I say that the sum of the squares on AB and BC is triple the square on CA. Describe the square ADEB on AB, and draw the figure. I.46 Since, then, AB is cut in extreme and mean ratio at C, and AC is the greater segment, therefore the rectangle AB by BC equals the square on AC. VI.Def.3 VI.17 And AK is the rectangle AB by BC, and HG is the square on AC, therefore AK equals HG. And, since AF equals FE, add CK to each, therefore the whole AK equals the whole CE. Therefore the sum of AK and CE is double AK. But the sum of AK and CE is the sum of the gnomon LMN and the square CK, therefore the sum of the gnomon LMN and the square CK is double AK. But, further, AK was also proved equal to HG, therefore the sum of the gnomon LMN and the squares CK and HG is triple the square HG. And the sum of the gnomon LMN and the squares CK and HG is the sum of the whole square AE and CK, which are the squares on AB and BC, while HG is the square on AC. Therefore the sum of the squares on AB and BC is triple the square on AC. Therefore, if a straight line is cut in extreme and mean ratio, then the sum of the squares on the whole and on the lesser segment is triple the square on the greater segment. Q.E.D. Use of this proposition This and the next three propositions are all preparatory to the construction of a dodecahedron in proposition XIII.17. Next proposition: XIII.5 Previous: XIII.3 Book XIII introduction © 1997 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 5 If a straight line is cut in extreme and mean ratio, and a straight line equal to the greater segment is added to it, then the whole straight line has been cut in extreme and mean ratio, and the original straight line is the greater segment. Let the straight line AB be cut in extreme and mean ratio at the point C, let AC be the greater segment, and let AD be equal to AC. I say that the straight line DB is cut in extreme and mean ratio at A, and the original straight line AB is the greater segment. Describe the square AE on AB, and draw the figure. I.46 Since AB is cut in extreme and mean ratio at C, therefore the rectangle AB by BC equals the square on AC. VI.Def.3 VI.17 And CE is the rectangle AB by BC, and CH is the square on AC, therefore CE equals HC. But HE equals CE, and DH equals HC, therefore DH also equals HE. Therefore the whole DK is equal to the whole AE. And DK is the rectangle BD by DA, for AD equals DL, and AE is the square on AB, therefore the rectangle BD by DA equals the square on AB. Therefore DB is to BA as BA is to AD. And DB is greater than BA, therefore BA is also greater than AD. VI.17 V.14 Therefore DB has been cut in extreme and mean ratio at A, and AB is the greater segment. Therefore, if a straight line is cut in extreme and mean ratio, and a straight line equal to the greater segment is added to it, then the whole straight line has been cut in extreme and mean ratio, and the original straight line is the greater segment. Q.E.D. Use of this proposition This proposition is used XIII.17 where a dodecahedron is constructed. Next proposition: XIII.6 Previous: XIII.4 Book XIII introduction © 1997 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 6 If a rational straight line is cut in extreme and mean ratio, then each of the segments is the irrational straight line called apotome. Let AB be a rational straight line cut in extreme and mean ratio at C, and let AC be the greater segment. I say that each of the straight lines AC and CB is the irrational straight line called apotome. Produce BA, and make AD half of BA. Since, then, the straight line AB is cut in extreme and mean ratio, and to the greater segment AC is added AD which is half of AB, therefore the square on CD is five times the square on DA. XIII.1 Therefore the square on CD has to the square on DA the ratio which a number has to a number, therefore the square on CD is commensurable with the square on DA X.6 But the square on DA is rational, for DA is rational being half of AB which is rational, therefore the square on CD is also rational. Therefore CD is also rational. X.Def.4 And, since the square on CD has not to the square on DA the ratio which a square number has to a square number, therefore CD is incommensurable in length with DA. Therefore CD and DA are rational straight lines commensurable in square only. Therefore AC is an apotome. X.9 X.73 Again, since AB is cut in extreme and mean ratio, and AC is the greater segment, therefore the rectangle AB by BC equals the square on AC. VI.Def.3 VI.17 Therefore the square on the apotome AC, if applied to the rational straight line AB, produces BC as breadth. But the square on an apotome, if applied to a rational straight line, produces as breadth a first apotome, therefore CB is a first apotome. And CA was also proved to be an apotome. X.97 Therefore, if a rational straight line is cut in extreme and mean ratio, then each of the segments is the irrational straight line called apotome. Q.E.D. Heath argues that this proposition was interpolated. Use of this proposition This proposition is used after the construction of a dodecahedron in XIII.17 to show that the side of a pentagonal face is the irrational straight line called apotome. Next proposition: XIII.7 Previous: XIII.5 Book XIII introduction © 1997 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 7 If three angles of an equilateral pentagon, taken either in order or not in order, are equal, then the pentagon is equiangular. First, let three angles A, B, and C taken in order in the equilateral pentagon ABCDE be equal to one another. I say that the pentagon ABCDE is equiangular. Join AC, BE, and FD. Now, since the two sides CB and BA equal the two sides BA and AE respectively, and the angle CBA equals the angle BAE, therefore the base AC equals the base BE, the triangle ABC equals the triangle ABE,and the remaining angles equal the remaining angles, namely those opposite the equal sides, that is, the angle BCA equals the angle BEA, and the angle ABE equals the angle CAB. I.4 Hence the side AF also equals the side BF. I.6 But the whole AC equals the whole BE, therefore the remainder FC equals the remainder FE. But CD also equals DE. Therefore the two sides FC and CD equal the two sides FE and ED, and the base FD is common to them, therefore the angle FCD equals the angle FED. I.8 But the angle BCA was also proved equal to the angle AEB, therefore the whole angle BCD equals the whole angle AED. And, by hypothesis, the angle BCD equals the angles at A and B, therefore the angle AED also equals the angles at A and B. Similarly we can prove that the angle CDE also equals the angles at A, B, and C. Therefore the pentagon ABCDE is equiangular. Next, let the given equal angles not be angles taken in order, but let the angles at the points A, C, and D be equal. I say that in this case too the pentagon ABCDE is equiangular. Join BD. Then, since the two sides BA and AE equal the two sides BC and CD, and they contain equal angles, therefore the base BE equals the base BD, the triangle ABE equals the triangle BCD, and the remaining angles equal the remaining angles, namely those opposite the equal sides. Therefore the angle AEB equals the angle CDB. I.4 But the angle BED also equals the angle BDE, since the side BE equals the side BD. I.5 Therefore the whole angle AED equals the whole angle CDE. But the angle CDE is, by hypothesis, equal to the angles at A and C, therefore the angle AED also equals the angles at A and C. For the same reason the angle ABC also equals the angles at A, C, and D. Therefore the pentagon ABCDE is equiangular. Therefore, if three angles of an equilateral pentagon, taken either in order or not in order, are equal, then the pentagon is equiangular. Q.E.D. Use of this proposition This proposition is needed in XIII.17 to show that the dodecahedron constructed there has equiangular pentagons as faces. Next proposition: XIII.8 Previous: XIII.6 Book XIII introduction © 1997 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 8 If in an equilateral and equiangular pentagon straight lines subtend two angles are taken in order, then they cut one another in extreme and mean ratio, and their greater segments equal the side of the pentagon. In the equilateral and equiangular pentagon ABCDE let the straight lines AC and BE, cutting one another at the point H, subtend two angles taken in order, the angles at A and B. I say that each of them has been cut in extreme and mean ratio at the point H, and their greater segments equal the side of the pentagon. Circumscribe the circle ABCDE about the pentagon ABCDE. IV.14 Then, since the two straight lines EA and AB equal the two lines AB and BC, and they contain equal angles, therefore the base BE equals the base AC, the triangle ABE equals the triangle ABC, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides. I.4 Therefore the angle BAC equals the angle ABE. Therefore the angle AHE is double the angle BAH. I.32 But the angle EAC is also double the angle BAC, for the circumference EDC is also double the circumference CB. III.28 VI.33 Therefore the angle HAE equals the angle AHE. Hence the straight line HE also equals EA, that is, AB. I.6 And, since the straight line BA equals AE, therefore the angle ABE also equals the angle AEB. I.5 But the angle ABE was proved equal to the angle BAH, therefore the angle BEA also equals the angle BAH. And the angle ABE is common to the two triangles ABE and ABH, therefore the remaining angle BAE equals the remaining angle AHB. Therefore the triangle ABE is equiangular with the triangle ABH. I.32 Therefore, proportionally EB is to BA as AB is to BH. VI.4 But BA equals EH, therefore BE is to EH as EH is to HB. And BE is greater than EH, therefore EH is also greater than HB. VI.14 Therefore BE has been cut in extreme and mean ratio at H, and the greater segment HE equals the side of the pentagon. Similarly we can prove that AC has also been cut in extreme and mean ratio at H, and its greater segment CH equals the side of the pentagon. Therefore, if in an equilateral and equiangular pentagon straight lines subtend two angles are taken in order, then they cut one another in extreme and mean ratio, and their greater segments equal the side of the pentagon. Q.E.D. Use of this proposition This proposition is used in the proof of XIII.11 to establish that the side of a regular pentagon inscribed in a circle with rational diameter is the irrational straight line called minor. Next proposition: XIII.9 Previous: XIII.7 Book XIII introduction © 1997 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 9 If the side of the hexagon and that of the decagon inscribed in the same circle are added together, then the whole straight line has been cut in extreme and mean ratio, and its greater segment is the side of the hexagon. Let ABC be a circle, and of the figures inscribed in the circle ABC let BC be the side of a decagon, and CD that of a hexagon, and let them be in a straight line. I say that the whole straight line BD is cut in extreme and mean ratio, and CD is its greater segment. Take the center E of the circle, join EB, EC, and ED, and carry BE through to A. III.1 Since BC is the side of an equilateral decagon, therefore the circumference ACB is five times the circumference BC. Therefore the circumference AC is quadruple CB. But the circumference AC is to CB as the angle AEC is to the angle CEB. Therefore the angle AEC is quadruple the angle CEB. VI.33 And, since the angle EBC equals the angle ECB, therefore the angle AEC is double the angle ECB. I.5 I.32 And, since the straight line EC equals CD, for each of them equals the side of the hexagon inscribed in the circle ABC. Therefore the angle CED also equals the angle CDE. Therefore the angle ECB is double the angle EDC. IV.15,Cor. I.5 I.32 But the angle AEC was proved double the angle ECB, therefore the angle AEC is quadruple the angle EDC. And the angle AEC was also proved quadruple the angle BEC, therefore the angle EDC equals the angle BEC. But the angle EBD is common to the two triangles BEC and BED, therefore the remaining angle BED equals the remaining angle ECB. Therefore the triangle EBD is equiangular with the triangle EBC. I.32 Therefore, proportionally DB is to BE as EB is to BC. VI.4 But EB equals CD. Therefore BD is to DC as DC is to CB. And BD is greater than DC, therefore DC is also greater than CB. Therefore the straight line BD is cut in extreme and mean ratio, and DC is its greater segment. Therefore, if the side of the hexagon and that of the decagon inscribed in the same circle are added together, then the whole straight line has been cut in extreme and mean ratio, and its greater segment is the side of the hexagon. Q.E.D. Use of this proposition This result is used in the construction of an icosahedron in propositions XIII.16 and XIII.18. Next proposition: XIII.10 Previous: XIII.8 Book XIII introduction © 1997 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 10 If an equilateral pentagon is inscribed in a circle, then the square on the side of the pentagon equals the sum of the squares on the sides of the hexagon and the decagon inscribed in the same circle. Let ABCDE be a circle, and let the equilateral pentagon ABCDE be inscribed in the circle ABCDE. I say that the square on the side of the pentagon ABCDE equals the sum of the squares on the side of the hexagon and on that of the decagon inscribed in the circle ABCDE. Take the center F of the circle, join AF and carry it through to the point G, and join FB. Draw FH from F perpendicular to AB and carry it through to K, join AK and KB, draw FL from F perpendicular to AK, carry it through to M, and join KN. III.1 I.12 Since the circumference ABCG equals the circumference AEDG, and in them ABC equals AED, therefore the remainder, the circumference CG, equals the remainder GD. But CD belongs to a pentagon, therefore CG belongs to a decagon. And, since FA equals FB, and FH is perpendicular, therefore the angle AFK equals the angle KFB. I.5 I.26 Hence the circumference AK equals KB. Therefore the circumference AB is double the circumference BK. Therefore the straight line AK is a side of a decagon. For the same reason AK is double KM. III.26 Now, since the circumference AB is double the circumference BK, while the circumference CD equals the circumference AB, therefore the circumference CD is also double the circumference BK. But the circumference CD is also double CG, therefore the circumference CG equals the circumference BK. But BK is double KM, since KA is so also, therefore CG is also double KM. But, further, the circumference CB is also double the circumference BK, for the circumference CB equals BA. Therefore the whole circumference GB is also double BM. Hence the angle GFB is double the angle BFM. VI.33 But the angle GFB is double the angle FAB, for the angle FAB equals the angle ABF. Therefore the angle BFN equals the angle FAB. But the angle ABF is common to the two triangles ABF and BFN, therefore the remaining angle AFB equals the remaining angle BNF. Therefore the triangle ABF is equiangular with the triangle BFN. I.32 Therefore, proportionally the straight line AB is to BF as FB is to BN. Therefore the rectangle AB by BN equals the square on BF. VI.4 VI.17 Again, since AL equals LK, while LN is common and at right angles, therefore the base KN equals the base AN. Therefore the angle LKN also equals the angle LAN. I.4 But the angle LAN equals the angle KBN, therefore the angle LKN also equals the angle KBN. And the angle at A is common to the two triangles AKB and AKN. Therefore the remaining angle AKB equals the remaining angle KNA. I.32 Therefore the triangle KBA is equiangular with the triangle KNA. Therefore, proportionally the straight line BA is to AK as KA is to AN. VI.1 Therefore the rectangle BA by AN equals the square on AK. VI.17 But the rectangle AB by BN was also proved equal to the square on BF, therefore the sum of the rectangle AB by BN and the rectangle BA by AN, that is, the square on BA, equals the sum of the squares on BF and AK. II.2 And BA is a side of the pentagon, BF of the hexagon, and AK of the decagon. IV.15,Cor. Therefore, if an equilateral pentagon is inscribed in a circle, then the square on the side of the pentagon equals the sum of the squares on the sides of the hexagon and the decagon inscribed in the same circle. Q.E.D. Use of this proposition This result is used in XIII.16 for the construction of an icosahedron and later in XIII.18 when an icosahedron is compared to the other four regular polyhedra. Next proposition: XIII.11 Previous: XIII.9 Book XIII introduction © 1997 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 11 If an equilateral pentagon is inscribed in a circle which has its diameter rational, then the side of the pentagon is the irrational straight line called minor. In the circle ABCDE which has its diameter rational let the equilateral pentagon ABCDE be inscribed. I say that the side of the pentagon is the irrational straight line called minor. Take the center F of the circle, join AF and FB and carry them through to the points G and H, join AC, and make FK a fourth part of AF. III.1 VI.9 Now AF is rational, therefore FK is also rational. But BF is also rational, therefore the whole BK is rational. And, since the circumference ACG equals the circumference ADG, and in them ABC equals AED, therefore the remainder CG equals the remainder GD. And, if we join AD, then we conclude that the angles at L are right, and CD is double CL. For the same reason the angles at M are also right, and AC is double CM. Since then the angle ALC equals the angle AMF, and the angle LAC is common to the two triangles ACL and AMF, therefore the remaining angle ACL equals the remaining angle MFA. I.32 Therefore the triangle ACL is equiangular with the triangle AMF. Therefore, proportionally LC is to CA as MF is to FA. Taking the doubles of the antecedents, therefore double LC is to CA as double MF to FA. But double MF is to FA as MF is to the half of FA, therefore also double LC is to CA as MF is to the half of FA. Taking the halves of the consequents, therefore double LC is to the half of CA as MF to the fourth of FA. And DC is double LC, CM is half of CA, and FK is a fourth part of FA, therefore DC is to CM as MF to FK. Taken together, the sum of DC and CM is to CM as MK to KF. Therefore the square on the sum of DC and CM is to the square on CM as the square on MK is to the square on KF. V.18 And since, when the straight line opposite two sides of the pentagon AC is cut in extreme and mean ratio, the greater segment equals the side of the pentagon, that is, DC, while the square on the greater segment added to the half of the whole is five times the square on the half of the whole, and CM is half of the whole AC, therefore the square on DC and CM taken as one straight line is five times the square on CM. XIII.8 XIII.1 But it was proved that the square on DC and CM taken as one straight line is to the square on CM as the square on MK to the square on KF, therefore the square on MK is five times the square on KF. But the square on KF is rational, for the diameter is rational, therefore the square on MK is also rational. Therefore MK is rational. And, since BF is quadruple FK, therefore BK is five times KF. Therefore the square on BK is twenty-five times the square on KF. But the square on MK is five times the square on KF, therefore the square on BK is five times the square on KM. Therefore the square on BK has not to the square on KM the ratio which a square number has to a square number. Therefore BK is incommensurable in length with KM. X.9 And each of them is rational. Therefore BK and KM are rational straight lines commensurable in square only. But, if from a rational straight line there is subtracted a rational straight line which is commensurable with the whole in square only, then the remainder is irrational, namely an apotome, therefore MB is an apotome and MK the annex to it. X.73 I say next that MB is also a fourth apotome. Let the square on N be equal to that by which the square on BK is greater than the square on KM. Therefore the square on BK is greater than the square on KM by the square on N. And, since KF is commensurable with FB, taken together, KB is commensurable with FB. But BF is commensurable with BH, therefore BK is also commensurable with BH. X.15 X.12 And, since the square on BK is five times the square on KM, therefore the square on BK has to the square on KM the ratio which 5 has to 1. Therefore, in conversion, the square on BK has to the square on N the ratio which 5 has to 4, and this is not the ratio which a square number has to a square number. Therefore BK is incommensurable with N. Therefore the square on BK is greater than the square on KM by the square on a straight line incommensurable with BK. V.19,Cor. X.9 Since then the square on the whole BK is greater than the square on the annex KM by the square on a straight line incommensurable with BK, and the whole BK is commensurable with the rational straight line, BH, set out, therefore MB is a fourth apotome. X.Def.III.4 But the rectangle contained by a rational straight line and a fourth apotome is irrational, and its square root is irrational, and is called minor. X.94 But the square on AB equals the rectangle HB by BM, because, when AH is joined, the triangle ABH is equiangular with the triangle ABM, and HB is to BA as AB is to BM. Therefore the side AB of the pentagon is the irrational straight line called minor. Therefore, if an equilateral pentagon is inscribed in a circle which has its diameter rational, then the side of the pentagon is the irrational straight line called minor. Q.E.D. Use of this proposition This proposition is needed in XIII.16 after the construction of a dodecahedron to show the side of a pentagonal face is the irrational straight line called minor. Next proposition: XIII.12 Previous: XIII.10 Book XIII introduction © 1997 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 12 If an equilateral triangle is inscribed in a circle, then the square on the side of the triangle is triple the square on the radius of the circle. Let ABC be a circle, and let the equilateral triangle ABC be inscribed in it. I say that the square on one side of the triangle ABC is triple the square on the radius of the circle. Take the center D of the circle ABC, join AD and carry it through to E, and join BE. III.1 Then, since the triangle ABC is equilateral, therefore the circumference BEC is a third part of the circumference of the circle ABC. Therefore the circumference BE is a sixth part of the circumference of the circle. Therefore the straight line BE belongs to a hexagon. Therefore it equals the radius DE. IV.15,Cor. And, since AE is double DE, therefore the square on AE is quadruple the square on ED, that is, of the square on BE. But the square on AE equals the sum of the squares on AB and BE. Therefore the sum of the squares on AB and BE is quadruple the square on BE. III.31 I.47 Therefore, taken separately, the square on AB is triple the square on BE. But BE equals DE, therefore the square on AB is triple the square on DE. Therefore the square on the side of the triangle is triple the square on the radius. Therefore, if an equilateral triangle is inscribed in a circle, then the square on the side of the triangle is triple the square on the radius of the circle. Q.E.D. Use of this proposition This result is needed in proposition XIII.13 coming next to show that the construction given there produces a regular tetrahedron. Next proposition: XIII.13 Previous: XIII.11 Book XIII introduction © 1997 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 13 To construct a pyramid, to comprehend it in a given sphere; and to prove that the square on the diameter of the sphere is one and a half times the square on the side of the pyramid. Set out the diameter AB of the given sphere, cut it at the point C so that AC is double CB, describe the semicircle ADB on AB, draw CD from the point C at right angles to AB, and join DA. VI.9 I.11 Set out the circle EFG with radius equal to DC, inscribe the equilateral triangle EFG in the circle EFG, take the center H of the circle, and join EH, HF, and HG. I.1 IV.2 Set HK up from the point H at right angles to the plane of the circle EFG, cut off HK equal to the straight line AC from HK, and join KE, KF, and KG. XI.12 I.3 Now, since KH is at right angles to the plane of the circle EFG, therefore it makes right angles with all the straight lines which meet it and are in the plane of the circle EFG. But each of the straight lines HE, HF, and HG meets it, therefore HK is at right angles to each of the straight lines HE, HF, and HG. XI.Def.3 And, since AC equals HK, and CD equals HE, and they contain right angles, therefore the base DA equals the base KE. For the same reason each of the straight lines KF and KG also equals DA. Therefore the three straight lines KE, KF, and KG equal one another. I.4 And, since AC is double CB, therefore AB is triple BC. But that AB is to BC as the square on AD is to the square on DC will be proved afterwards. Therefore the square on AD is triple the square on DC. But the square on FE is also triple the square on EH, and DC equals EH, therefore DA also equals EF. XIII.12 But DA was proved equal to each of the straight lines KE, KF, and KG, therefore each of the straight lines EF, FG, and GE also equals each of the straight lines KE, KF, and KG. Therefore the four triangles EFG, KEF, KFG, and KEG are equilateral. Therefore a pyramid has been constructed out of four equilateral triangles, the triangle EFG being its base and the point K its vertex. It is next required to comprehend it in the given sphere and to prove that the square on the diameter of the sphere is one and a half times the square on the side of the pyramid. Produce the straight line HL in a straight line with KH, and make HL equal to CB. I.3 Now, since AC is to CD as CD is to CB, while AC equals KH, CD equals HE, and CB equals HL, therefore KH is to HE as EH is to HL. Therefore the rectangle KH by HL equals the square on EH. VI.8,Cor. VI.17 And each of the angles KHE, EHL is right, therefore the semicircle described on KL passes through E also. cf. VI.8 III.31 If then, KL remaining fixed, the semicircle is carried round and restored to the same position from which it began to be moved, then it also passes through the points F and G, since, if FL and LG are joined, then the angles at F and G similarly become right angles, and the pyramid is comprehended in the given sphere. For KL, the diameter of the sphere, equals the diameter AB of the given sphere, since KH was made equal to AC, and HL to CB. I say next that the square on the diameter of the sphere is one and a half times the square on the side of the pyramid. Since AC is double CB, therefore AB is triple BC, and, in conversion, BA is one and a half times AC. But BA is to AC as the square on BA is to the square on AD. Therefore the square on BA is also one and a half times the square on AD. And BA is the diameter of the given sphere, and AD equals the side of the pyramid. Therefore the square on the diameter of the sphere is one and a half times the square on the side of the pyramid. Q.E.F. Lemma It is to be proved that AB is to BC as the square on AD is to the square on DC. Set out the figure of the semicircle, join DB, describe the square EC on AC, and complete the parallelogram FB. I.46 Since the triangle DAB is equiangular with the triangle DAC, therefore BA is to AD as DA is to AC. Therefore the rectangle BA by AC equals the square on AD. VI.8 VI.4 VI.17 And since AB is to BC as EB is to BF, and EB is the rectangle BA by AC, for EA equals AC, and BF is the rectangle AC by CB, therefore AB is to BC as the rectangle BA by AC is to the rectangle AC by CB. VI.1 And the rectangle BA by AC equals the square on AD, and the rectangle AC by CB equals the square on DC, for the perpendicular DC is a mean proportional between the segments AC and CB of the base, because the angle ADB is right. Therefore AB is to BC as the square on AD is to the square on DC. VI.8,Cor. Q.E.D. This figure is usually called a regular tetrahedron, that is, a solid figure contained by four equal and equilateral triangles. Euclid simply calls it "a pyramid" with the understanding that by that he means not just any pyramid, but a regular tetrahedron. A similar ambiguity occured in ancient Greek when the word "tetragon" was used. It meant either any four-angled figure or specifically a square, depending on the context. Summary of the construction Standardize the radius of the sphere at 1 unit, so that AB = 2. Then cut AB at C so that AC = 4/3 and BC = 2/3. Let DC be their mean proportional (2/3) 2. Then AD = (2/3) 6. This line AD will end up being the length of the side of the tetrahedron. Note that it has the correct value so that "the square on the diameter of the sphere is one and a half times the square on the side of the pyramid." Set out the circle EFG of radius EH = (2/3) 2, and inscribe in that circle an equilateral triangle. Then each side of the triangle will be (2/3) 6, the same as AD (XIII.12). Make HK of length 4/3 and perpendicular to the plane of the triangle, and connect KE, KF, and KG. Then K lies on the surface of the sphere. And since the triangle HKE is a right triangle, therefore its hypotenuse KE = (2/3) 6, the same as AD. Likewise KF and KG have the same length. That constructs the tetrahedron in the sphere. Coordinates for the vertices of the tetrahedron A cube can be easily constructed from a tetrahedron since the four vertices of a tetrahedron are four of the eight vertices of a cube. See proposition XIII.15. That being the case, an obvious coordinate system will make eight vertices of a cube have the coordinates (1,1,1) (1,1,–1) (1,–1,1) (1,–1,–1) (–1,1,1) (–1,1,–1) (–1,–1,1) (–1,–1,–1) that is, all eight combinations of –1 and 1 in all three coordinates. The radius for such a cube is 3, so if a unit sphere is desired, then all the coordinates would have to be divided by 3. The tetrahedron has only half of these eight vertices, and they can be chosen to be (1,1,1) (1,–1,–1) (–1,1,–1) (–1,–1,1) that is, the points which have an odd number of positive coordinates. There is another tetrahedron which has as its vertices the remaining four points which have an even number of positive coordinates. Use of this construction Constructing this regular tetrahedron is an end in itself. In the last proposition of the Elements XIII.18, the five regular polyhedra are compared, and this construction is needed there as well as constructions of the other four regular polyhedra. Next proposition: XIII.14 Previous: XIII.12 Book XIII introduction © 1997, 2002 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 14 To construct an octahedron and comprehend it in a sphere, as in the preceding case; and to prove that the square on the diameter of the sphere is double the square on the side of the octahedron. Set out the diameter AB of the given sphere, bisect it at C, describe the semicircle ADB on AB, draw CD from C at right angles to AB, and join DB. I.11 Set out the square EFGH, having each of its sides equal to DB, join HF and EG, set up the straight line KL from the point K at right angles to the plane of the square EFGH, and carry it through to the other side of the plane KM. I.46 XI.12 Cut off KL and KM from the straight lines KL and KM respectively equal to one of the straight lines EK, FK, GK, or HK, and join LE, LF, LG, LH, ME, MF, MG, and MH. I.3 Then, since KE equals KH, and the angle EKH is right, therefore the square on HE is double the square on EK. Again, since LK equals KE, and the angle LKE is right, therefore the square on EL is double the square on EK. I.47 But the square on HE was also proved double the square on EK, therefore the square on LE equals the square on EH. Therefore LE equals EH. For the same reason LH also equals HE. Therefore the triangle LEH is equilateral. Similarly we can prove that each of the remaining triangles of which the sides of the square EFGH are the bases and the points L and M are the vertices, is equilateral, therefore an octahedron has been constructed which is contained by eight equilateral triangles. XI.Def.26 It is next required to comprehend it in the given sphere, and to prove that the square on the diameter of the sphere is double the square on the side of the octahedron. Since the three straight lines LK, KM, and KE equal one another, therefore the semicircle described on LM passes through E. And for the same reason, if, LM remaining fixed, the semicircle be carried round and restored to the same position from which it began to be moved, then it also passes through the points F, G, and H, and the octahedron will be comprehended in a sphere. I say next that it is also comprehended in the given sphere. For, since LK equals KM, while KE is common, and they contain right angles, therefore the base LE equals the base EM. I.4 And, since the angle LEM is right, for it is in a semicircle, therefore the square on LM is double the square on LE. III.31 I.47 Again, since AC equals CB, therefore AB is double BC. But AB is to BC as the square on AB is to the square on BD, therefore the square on AB is double the square on BD. But the square on LM was also proved double the square on LE. And the square on DB equals the square on LE, for EH was made equal to DB. Therefore the square on AB equals the square on LM. Therefore AB equals LM. And AB is the diameter of the given sphere, therefore LM equals the diameter of the given sphere. Therefore the octahedron has been comprehended in the given sphere, and it has been demonstrated at the same time that the square on the diameter of the sphere is double the square on the side of the octahedron. Q.E.F. Of the five regular polyhedra to be constructed in a sphere, the octahedron has the easiest construction. Relative to the center of the sphere K, the lines to the six verices KE, KF, KG, KH, KL, and KM form three mutually perpendicular diameters. Also, the 12 sides group into three groups of four lines, each group forming the vertices of a square-EFGH, EMGL, and FMHL. Since the center of each square is the center of the sphere, therefore two sides, EF and FG, along with the one diameter EG of the octahedron form a 45°-45°-90° triangle. Thus, the square on the diameter of the sphere is twice the square on the side of the octahedron. Coordinates for the vertices of the octahedron If the sphere circumscribing the octahedron is the unit sphere, then a natural coordinate system to impose would have the three coordinate axes be the three perpendicular diameters. Then the points a unit distance from the origin are the six vertices of the octahedron, namely, (1,0,0) (–1,0,0) (0,1,0) (0,–1,0) (0,0,1) (0,0,–1) Duals of the regular polyhedra As will be shown in proposition XIII.18, there are exactly five regular polyhedra. The accompanying table lists these five polyhedra along with the numbers of the their faces, edges, and vertices. Their names are taken from the number of their faces, except, of course, the cube, which otherwise would be called a hexahedron. Polyhedron Faces Edges Vertices tetrahedron 4 6 4 octahedron 8 12 6 cube 6 12 8 icosahedron 20 30 12 dodecahedron 12 30 20 Note that there are two pairs of polyhedra in this table where the numbers are related. One pair is the octahedron and cube, the other is the icosahedron and dodecahedron. For these pairs the number of faces of one of the pair equals the number of vertices of the other, and both of the pair have the same number of edges. These are the pairs of "duals." The numbers for the tetrahedron indicate that it dual to itself. We can see the correspondence between the parts of one of these polyhedra and the parts of its dual. Consider the octahedron. Place a point in the circumcenter of each of the eight faces. Connect two of these points if the faces that contain them share an edge. For each of the six vertices of the octahedron, connect the four circumcenters of the adjacent faces to make a square. What results is a cube with six vertices, 12 edges, and eight faces. java applet or image An analogous construction for the cube yields an octahedron. Likewise the constructions for the icosahedron and dodecahedron yield each other, and the construction for a tetrahedron yields another tetrahedron. Use of this construction Constructing this octahedron is an end in itself. The construction is also used in proposition XIII.18 where the five regular polyhedra are compared. Next proposition: XIII.15 Previous: XIII.13 Book XIII introduction © 1997, 2002 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 15 To construct a cube and comprehend it in a sphere, like the pyramid; and to prove that the square on the diameter of the sphere is triple the square on the side of the cube. Set out the diameter AB of the given sphere, and cut it at C so that AC is double CB. Describe the semicircle ADB on AB, draw CD from C at right angles to AB, and join DB. Set out the square EFGH having its side equal to DB, draw EK, FL, GM, and HN from E, F, G, and H at right angles to the plane of the square EFGH, and cut EK, FL, GM, and HN off from EK, FL, GM, and HN respectively equal to one of the straight lines EF, FG, GH, or HE. Join KL, LM, MN, and NK. VI.9 I.11 I.46 XI.12 I.3 Therefore the cube FN has been constructed which is contained by six equal squares. XI.Def.25 It is then required to comprehend it in the given sphere, and to prove that the square on the diameter of the sphere is triple the square on the side of the cube. Join KG and EG. Then, since the angle KEG is right, for KE is also at right angles to the plane EG and of course to the straight line EG also, therefore the semicircle described on KG passes through the point E. XI.Def.3 Again, since GF is at right angles to each of the straight lines FL and FE, therefore GF is also at right angles to the plane FK. Hence also, if we join FK, then GF will be at right angles to FK. For this reason the semicircle described on GK also passes through F. Similarly it also passes through the remaining angular points of the cube. If then, KG remaining fixed, the semicircle is carried round and restored to the same position from which it began to be moved, then the cube is comprehended in a sphere. I say next that it is also comprehended in the given sphere. For, since GF equals FE, and the angle at F is right, therefore the square on EG is double the square on EF. But EF equals EK, therefore the square on EG is double the square on EK. Hence the sum of the squares on GE and EK, that is the square on GK, is triple the square on EK. I.47 And, since AB is triple BC, while AB is to BC as the square on AB is to the square on BD, therefore the square on AB is triple the square on BD. But the square on GK was also proved triple the square on KE. And KE was made equal to DB, therefore KG also equals AB. And AB is the diameter of the given sphere, therefore KG also equals the diameter of the given sphere. Therefore the cube has been comprehended in the given sphere, and it has been demonstrated at the same time that the square on the diameter of the sphere is triple the square on the side of the cube. Q.E.F. Special relationships between regular tetrahedra and cubes Note that the beginning of this construction of a cube is the same as that for the tetrahedron in proposition XIII.13, namely, the points C and D are the same. The difference is that the line AD is the edge of a tetrahedron while the line BD is the edge of a cube. Following through the construction, you will see that four of the eight vertices of the cube are the four vertices of the tetrahedron. Using the labelling of this proposition, they may be taken as E, G, L, and N. Alternatively, the other four vertices of the cube, F, H, K, and M, form the vertices of a regular tetrahedron. See the Guide to XIII.13 for more on this connection which involves placing coordinates on the vertices of the cube and tetrahedron. The volumes of the tetrahedron and cube are easily compared. When the tetrahedron is removed from the cube, there are four remaining pyramids, EGHN is one of them. By proposition XII.9 the volume of each pyramid is one-third of the volume of a prism, for instance, EGHN is one-third of the triangular prism EFKHGN, which in turn is half of the cube. Therefore each pyramid is one-sixth of the cube. Since the four pyramids together make four-sixths of the cube, that leaves one-third of the cube for the regular tetrahedron EGLN. Use of this construction Constructing a cube is an end in itself, but Euclid also starts with a cube to construct a dodecahedron in proposition XIII.17. Finally, this construction is used in XIII.18 where the five regular polyhedra are compared. Next proposition: XIII.16 Previous: XIII.14 Book XIII introduction © 1997, 1998 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 16 To construct an icosahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the square on the side of the icosahedron is the irrational straight line called minor. Set out the diameter AB of the given sphere, and cut it at C so that AC is quadruple CB, describe the semicircle ADB on AB, draw the straight line CD from C at right angles to AB, and join DB. VI.9 I.11 Set out the circle EFGHK, and let its radius be equal to DB. Inscribe the equilateral and equiangular pentagon EFGHK in the circle EFGHK, bisect the circumferences EF, FG, GH, HK, and KE at the points L, M, N, O, and P, and join LM, MN, NO, OP, PL, and EP. IV.11 I.9 Therefore the pentagon LMNOP is also equilateral, and the straight line EP belongs to a decagon. Now from the points E, F, G, H, and K set up the straight lines EQ, FR, GS, HT, and KU at right angles to the plane of the circle, and make them equal to the radius of the circle EFGHK. Join QR, RS, ST, TU, UQ, QL, LR, RM, MS, SN, NT, TO, OU, UP, and PQ. XI.12 I.3 Now, since each of the straight lines EQ and KU is at right angles to the same plane, therefore EQ is parallel to KU. But it is also equal to it, and the straight lines joining those ends of equal and parallel straight lines which are in the same direction are equal and parallel. Therefore QU is equal and parallel to EK. I.33 But EK belongs to an equilateral pentagon, therefore QU also belongs to the equilateral pentagon inscribed in the circle EFGHK. For the same reason each of the straight lines QR, RS, ST, and TU also belongs to the equilateral pentagon inscribed in the circle EFGHK. Therefore the pentagon QRSTU is equilateral. And, since QE belongs to a hexagon, and EP to a decagon, and the angle QEP is right, therefore QP belongs to a pentagon, for the square on the side of the pentagon equals the sum of the square on the side of the hexagon and the square on the side of the decagon inscribed in the same circle. XIII.10 For the same reason PU is also a side of a pentagon. But QU also belongs to a pentagon, therefore the triangle QPU is equilateral. For the same reason each of the triangles QLR, RMS, SNT, and TOU is also equilateral. And, since each of the straight lines QL and QP was proved to belong to a pentagon, and LP also belongs to a pentagon, therefore the triangle QLP is equilateral. For the same reason each of the triangles LRM, MSN, NTO, and OUP is also equilateral. Take the center V of the circle EFGHK, set VZ up from V at right angles to the plane of the circle, and produce it in the other direction VX. Cut off VW, the side of a hexagon, and each of the straight lines VX and WZ, sides of a decagon. Join QZ, QW, UZ, EV, LV, LX, and XM. III.1 XI.12 Now, since each of the straight lines VW and QE is at right angles to the plane of the circle, therefore VW is parallel to QE. But they are also equal, therefore EV and QW are also equal and parallel. XI.6 I.33 But EV belongs to a hexagon, therefore QW also belongs to a hexagon. And, since QW belongs to a hexagon, and WZ to a decagon, and the angle QWZ is right, therefore QZ belongs to a pentagon. XIII.10 For the same reason UZ also belongs to a pentagon, for if we join VK and WU, then they will be equal and opposite, and VK, being a radius, belongs to a hexagon, therefore WU also belongs to a hexagon. But WZ belongs to a decagon, and the angle UWZ is right, therefore UZ belongs to a pentagon. IV.15,Cor. XIII.10 But QU also belongs to a pentagon, therefore the triangle QUZ is equilateral. For the same reason each of the remaining triangles of which the straight lines QR, RS, ST, and TU are the bases, and the point Z the vertex, is also equilateral. Again, since VL belongs to a hexagon, and VX to a decagon, and the angle LVX is right, therefore LX belongs to a pentagon. XIII.10 For the same reason, if we join MV, which belongs to a hexagon, MX is also inferred to belong to a pentagon. But LM also belongs to a pentagon, therefore the triangle LMX is equilateral. Similarly it can be proved that each of the remaining triangles of which MN, NO, OP, and PL are the bases and the point X the vertex, is also equilateral. Therefore an icosahedron has been constructed which is contained by twenty equilateral triangles. XI.Def.27 It is next required to comprehend it in the given sphere, and to prove that the side of the icosahedron is the irrational straight line called minor. Since VW belongs to a hexagon, and WZ to a decagon, therefore VZ is cut in extreme and mean ratio at W, and VW is its greater segment. Therefore as ZV is to VW as VW is to WZ. XIII.9 But VW equals VE, and WZ equals VX, therefore ZV is to VE as EV is to VX. And the angles ZVE and EVX are right, therefore, if we join the straight line EZ, then the angle XEZ will be right since the triangles XEZ and VEZ are similar. For the same reason, since ZV is to VW as VW is to WZ, and ZV equals XW, and VW equals WQ, therefore XW is to WQ as QW is to WZ. And for this reason again, if we join QX, then the angle at Q will be right, therefore the semicircle described on XZ will also pass through Q. VI.8 III.31 And if, XZ remaining fixed, the semicircle is carried round and restored to the same position from which it began to be moved, then it will pass through Q and the remaining angular points of the icosahedron, and the icosahedron will have been comprehended in a sphere. I say next that it is also comprehended in the given sphere. Bisect VW at A'. I.9 Then, since the straight line VZ is cut in extreme and mean ratio at W, and ZW is its lesser segment, therefore the square on ZW added to the half of the greater segment, that is WA', is five times the square on the half of the greater segment. Therefore the square on ZA' is five times the square on A'W. XIII.3 And ZX is double ZA', and VW is double A'W, therefore the square on ZX is five times the square on WV. And, since AC is quadruple CB, therefore AB is five times BC. But AB is to BC as the square on AB is to the square on BD, therefore the square on AB is five times the square on BD. VI.8 V.Def.9 But the square on ZX was also proved to be five times the square on VW. And DB equals VW, for each of them equals the radius of the circle EFGHK, therefore AB also equals XZ. And AB is the diameter of the given sphere, therefore XZ also equals the diameter of the given sphere. Therefore the icosahedron has been comprehended in the given sphere. I say next that the side of the icosahedron is the irrational straight line called minor. Since the diameter of the sphere is rational, and the square on it is five times the square on the radius of the circle EFGHK, therefore the radius of the circle EFGHK is also rational, hence its diameter is also rational. But, if an equilateral pentagon is inscribed in a circle which has its diameter rational, then the side of the pentagon is the irrational straight line called minor. XIII.11 And the side of the pentagon EFGHK is the side of the icosahedron. Therefore the side of the icosahedron is the irrational straight line called minor. Corollary. From this it is clear that the square on the diameter of the sphere is five times the square on the radius of the circle from which the icosahedron has been described, and that the diameter of the sphere is composed of the side of the hexagon and two of the sides of the decagon inscribed in the same circle. Q.E.F. The icosahedron The regular icosahedron is composed of 20 faces, each face an equilateral triangle, with five triangles meeting at each vertex. There are 12 vertices, and there are 30 edges. Unlike most of the Euclid's illustrations, the diagram he used for this proposition is highly schematic; it is not intended to be an accurate projection of the icosahedron. Of course, it could be that the diagram changed over the centuries of copying, but his diagram has the advantage of spreading out the vertices to be readable. The figure shown in the proof above is a standard orthogonal projection of the icosahedron. Directly below the same icosahedron is shown without all the auxiliary lines. Use of this construction Constructing a icosahedron is an end in itself. This construction and the corollary are also used in XIII.18 where the five regular polyhedra are compared. Next proposition: XIII.17 Previous: XIII.15 Book XIII introduction © 1997 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 17 To construct a dodecahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the square on the side of the dodecahedron is the irrational straight line called apotome. Let ABCD and CBEF, two planes of the aforesaid cube at right angles to one another, be set out. Bisect the sides AB, BC, CD, DA, EF, EB, and FC at G, H, K, L, M, N, and O respectively, and join GK, HL, MH, and NO. Cut the straight lines NP, PO, and HQ in extreme and mean ratio at the points R, S, and T respectively, and let RP, PS, and TQ be their greater segments. Set up RU, SV, and TW from the points R, S, and T at right angles to the planes of the cube towards the outside of the cube, and make them equal to RP, PS, and TQ. Join UB, BW, WC, CV, and VU. XIII.15 I.10 II.11/VI.30 XI.11 I.3 I say that the pentagon UBWCV is equilateral, in one plane, and equiangular. Join RB, SB, and VB. Then, since the straight line NP is cut in extreme and mean ratio at R, and RP is the greater segment, therefore the sum of the squares on PN and NR is triple the square on RP. XIII.4 But PN equals NB, and PR equals RU, therefore the sum of the squares on BN and NR is triple the square on RU. But the square on BR equals the sum of the squares on BN and NR, therefore the square on BR is triple the square on RU. Hence the sum of the squares on BR and RU is quadruple the square on RU. I.47 But the square on BU equals the sum of the squares on BR and RU, therefore the square on BU is quadruple the square on RU. Therefore BU is double RU. But VU is also double UR, for SR is also double PR, that is, of RU, therefore BU equals UV. Similarly it can be proved that each of the straight lines BW, WC, and CV also equals each of the straight lines BU and UV. Therefore the pentagon BUVCW is equilateral. I say next that it is also in one plane. Draw PX from P parallel to each of the straight lines RU and SV and toward the outside of the cube, and join XH and HW. I.31 I say that XHW is a straight line. Since HQ is cut in extreme and mean ratio at T, and QT is its greater segment, therefore HQ is to QT as QT is to TH. But HQ equals HP, and QT equals each of the straight lines TW and PX, therefore HP is to PX as WT is to TH. And HP is parallel to TW, for each of them is at right angles to the plane BD, and TH is parallel to PX, for each of them is at right angles to the plane BF. XI.6 But if two triangles XPH and HTW, which have two sides proportional to two sides are placed together at one angle so that their corresponding sides are also parallel, then the remaining straight lines are in a straight line, therefore XH is in a straight line with HW. VI.32 But every straight line is in one plane, therefore the pentagon UBWCV is in one plane. XI.1 I say next that it is also equiangular. Since the straight line NP is cut in extreme and mean ratio at R, and PR is the greater segment, while PR equals PS, therefore NS is also cut in extreme and mean ratio at P, and NP is the greater segment. Therefore the sum of the squares on NS and SP is triple the square on NP. XIII.5 XIII.4 But NP equals NB, and PS equals SV, therefore the squares on NS and SV is triple the square on NB. Hence the sum of the squares on VS, SN, and NB is quadruple the square on NB. But the square on SB equals the sum of the squares on SN and NB, therefore the sum of the squares on BS and SV, that is, the square on BV, for the angle VSB is right, is quadruple the square on NB. Therefore VB is double BN. But BC is also double BN, therefore BV equals BC. And, since the two sides BU and UV equal the two sides BW and WC, and the base BV equals the base BC, therefore the angle BUV equals the angle BWC. I.8 Similarly we can prove that the angle UVC also equals the angle BWC. Therefore the three angles BWC, BUV, and UVC equal one another. But if in an equilateral pentagon three angles equal one another, then the pentagon is equiangular, therefore the pentagon BUVCW is equiangular. XIII.7 And it was also proved equilateral, therefore the pentagon BUVCW is equilateral and equiangular, and it is on one side BC of the cube. Therefore, if we make the same construction in the case of each of the twelve sides of the cube, a solid figure will be constructed which is contained by twelve equilateral and equiangular pentagons, and which is called a dodecahedron. XI.Def.28 It is now required to comprehend it in the given sphere, and to prove that the side of the dodecahedron is the irrational straight line called apotome. Produce XP, and let the produced straight line be XZ. Therefore PZ meets the diameter of the cube, and they bisect one another, for this has been proved in the last theorem but one of the eleventh book. XI.38 Let them cut at Z. Therefore Z is the center of the sphere which comprehends the cube, and ZP is half of the side of the cube. Join UZ. Now, since the straight line NS is cut in extreme and mean ratio at P, and NP is its greater segment, therefore the sum of the squares on NS and SP is triple the square on NP. XIII.4 But NS equals XZ, for NP also equals PZ, and XP equals PS. But PS also equals XU, since it also equals RP. Therefore the sum of the squares on ZX and XU is triple the square on NP. But the square on UZ equals the sum of the squares on ZX and XU, therefore the square on UZ is triple the square on NP. But the square on the radius of the sphere which comprehends the cube is also triple the square on the half of the side of the cube, for it has previously been shown how to construct a cube and comprehend it in a sphere, and to prove that the square on the diameter of the sphere is triple the square on the side of the cube. XIII.15 But, if the whole is so related to the whole as the half to the half also, and NP is half of the side of the cube, therefore UZ equals the radius of the sphere which comprehends the cube. And Z is the center of the sphere which comprehends the cube, therefore the point U is on the surface of the sphere. Similarly we can prove that each of the remaining angles of the dodecahedron is also on the surface of the sphere, therefore the dodecahedron has been comprehended in the given sphere. I say next that the side of the dodecahedron is the irrational straight line called apotome. Since, when NP is cut in extreme and mean ratio, RP is the greater segment, and, when PO is cut in extreme and mean ratio, PS is the greater segment, therefore, when the whole NO is cut in extreme and mean ratio, RS is the great er segment. Thus, since NP is to PR as PR is to RN, the same is true of the doubles also, for parts have the same ratio as their equimultiples, therefore NO is to RS as RS is to the sum of NR and SO. But NO is greater than RS, therefore RS is also greater than the sum of NR and SO, therefore NO is cut in extreme and mean ratio, and RS is its greater segment. V.15 But RS equals UV, therefore, when NO is cut in extreme and mean ratio, UV is the greater segment. And, since the diameter of the sphere is rational, and the square on it is triple the square on the side of the cube, therefore NO, being a side of the cube, is rational. But if a rational line is cut in extreme and mean ratio, each of the segments is an irrational apotome. Therefore UV, being a side of the dodecahedron, is an irrational apotome. XIII.6 Q.E.F. Corollary. From this it is clear that when the side of the cube is cut in extreme and mean ratio, the greater segment is the side of the dodecahedron. Q.E.D. Cubes and regular dodecahedra Euclid's construction of a dodecahedron is particularly easy because he circumscribed his dodecahedron about a cube. Just as a regular tetrahedron can be circumscribed by a cube, a cube can be circumscribed by a regular dodecahedron, indeed, two regular dodecahedra. Also, each cube circumscribes two regular tetrahedra, and a regular dodecahedron circumscribes five cubes, and also ten tetrahedra. Coordinates for the vertices of the dodecahedron We can specify a coordinate system so that the center of the sphere is located at the origin and the eight vertices of the cube are located at (1,1,1) (1,1,–1) (1,–1,1) (1,–1,–1) (–1,1,1) (–1,1,–1) (–1,–1,1) (–1,–1,–1) The points A a through F in Euclid's construction may be assigned six of these coordinates. A = (–1,–1,–1), B = (–1,1,–1), C = (1,1,–1), D = (1,–1,–1), E = (–1,1,1), F = (1,1,1). After bisecting the sides, the points G through Q receive the following coordinates. G = (–1,0,–1), H = (0,1,–1), K = (1,0,–1), L = (0,–1,–1), M = (0,1,1), N = (–1,1,0), O = (1,1,0), P = (0,1,0), Q = (0,0,–1). The points R, S, and T cut the lines they're on into extreme and mean ratios, so they have these coordinates: R = (–x,1,0), S = (x,1,0), T = (0,x,–1), where x equals ( 5 – 1)/2. Finally, points U, V, and W are outside the original cube with the coordinates U = (–x,1+x,0), V = (x,1+x,0), W = (0,x,–1–x), Use of this construction Constructing a dodecahedron is an end in itself. This construction and the corollary are also used in XIII.18 where the five regular polyhedra are compared. Next proposition: XIII.18 Previous: XIII.16 Book XIII introduction © 1997 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 18 To set out the sides of the five figures and compare them with one another. Set out AB the diameter of the given sphere, and cut it at C so that AC equals CB, and at D so that AD is double DB. Describe the semicircle AEB on AB, draw CE and DF from C and D at right angles to AB, and join AF, FB, and EB. I.11 Then, since AD is double DB, therefore AB is triple BD. In conversion, therefore, BA is one and a half times AD. But BA is to AD as the square on BA is to the square on AF, for the triangle AFB is equiangular with the triangle AFD. Therefore the square on BA is one and a half times the square on AF. V.Def.9 VI.8 But the square on the diameter of the sphere is also one and a half times the square on the side of the pyramid. And AB is the diameter of the sphere, therefore AF equals the side of the pyramid. XIII.13 Again, since AD is double DB, therefore AB is triple BD. But AB is to BD as the square on AB to the square on BF, therefore the square on AB is triple the square on BF. V.Def.9 VI.8 But the square on the diameter of the sphere is also triple the square on the side of the cube. And AB is the diameter of the sphere, therefore BF is the side of the cube. XIII.15 And, since AC equals CB, therefore AB is double BC. But AB is to BC as the square on AB to the square on BE, therefore the square on AB is double the square on BE. But the square on the diameter of the sphere is also double the square on the side of the octahedron. And AB is the diameter of the given sphere, therefore BE is the side of the octahedron. XIII.14 Next, draw AG from the point A at right angles to the straight line AB, make AG equal to AB, join GC, and draw HK from H perpendicular to AB. I.11 I.3 I.12 Then, since GA is double AC, for GA equals AB and GA is to AC as HK is to KC, therefore HK is also double KC. Therefore the square on HK is quadruple the square on KC, therefore the sum of the squares on HK and KC, that is, the square on HC, is five times the square on KC. But HC equals CB, therefore the square on BC is five times the square on CK. And, since AB is double CB, and, in them, AD is double DB, therefore the remainder BD is double the remainder DC. Therefore BC is triple CD, therefore the square on BC is nine times the square on CD. But the square on BC is five times the square on CK, therefore the square on CK is greater than the square on CD. Therefore CK is greater than CD. Make CL equal to CK, draw LM from L at right angles to AB, and join MB. I.3 I.11 Now, since the square on BC is five times the square on CK, and AB is double BC, and KL is double CK, therefore the square on AB is five times the square on KL. But the square on the diameter of the sphere is also five times the square on the radius of the circle from which the icosahedron has been described. And AB is the diameter of the sphere, therefore KL is the radius of the circle from which the icosahedron has been described. Therefore KL is a side of the hexagon in the said circle. XIII.16,Cor. IV.15,Cor. And, since the diameter of the sphere is made up of the side of the hexagon and two of the sides of the decagon inscribed in the same circle, and AB is the diameter of the sphere, while KL is a side of the hexagon, and AK equals LB, therefore each of the straight lines AK and LB is a side of the decagon inscribed in the circle from which the icosahedron has been described. XIII.16,Cor. And, since LB belongs to a decagon, and ML to a hexagon, for ML equals KL, since it also equals HK being the same distance from the center and each of the straight lines HK and KL is double KC, therefore MB belongs to a pentagon. XIII.10 But the side of the pentagon is the side of the icosahedron, therefore MB belongs to the icosahedron. XIII.16 Now, since FB is a side of the cube, cut it in extreme and mean ratio at N, and let NB be the greater segment. Therefore NB is a side of the dodecahedron. XIII.17,Cor. And, since the square on the diameter of the sphere was proved to be one and a half times the square on the side AF of the pyramid, double the square on the side BE of the octahedron and triple the side FB of the cube, therefore, of parts of which the square on the diameter of the sphere contains six, the square on the side of the pyramid contains four, the square on the side of the octahedron three, and the square on the side of the cube two. Therefore the square on the side of the pyramid is four-thirds of the square on the side of the octahedron, and double the square on the side of the cube, and the square on the side of the octahedron is one and a half times the square on the side of the cube. The said sides, therefore, of the three figures, I mean the pyramid, the octahedron and the cube, are to one another in rational ratios. But the remaining two, I mean the side of the icosahedron and the side of the dodecahedron, are not in rational ratios either to one another or to the aforesaid sides, for they are irrational, the one being minor and the other an apotome. XIII.16 XIII.17 That the side MB of the icosahedron is greater than the side NB of the dodecahedron we can prove thus. Since the triangle FDB is equiangular with the triangle FAB, proportionally DB is to BF as BF is to BA. VI.8 VI.4 And, since the three straight lines are proportional, the first is to the third as the square on the first is to the square on the second, therefore DB is to BA as the square on DB is to the square on BF. Therefore, inversely AB is to BD as the square on FB is to the square on BD. V.Def.9 VI.20,Cor. But AB is triple BD, therefore the square on FB is triple the square on BD. But the square on AD is also quadruple the square on DB, for AD is double DB, therefore the square on AD is greater than the square on FB. Therefore AD is greater than FB. Therefore AL is by far greater than FB. And, when AL is cut in extreme and mean ratio, KL is the greater segment, for LK belongs to a hexagon, and KA to a decagon, and, when FB is cut in extreme and mean ratio, NB is the greater segment, therefore KL is greater than NB. XIII.9 But KL equals LM, therefore LM is greater than NB. Therefore MB, which is a side of the icosahedron, is by far greater than NB which is a side of the dodecahedron. Q.E.F. Remark I say next that no other figure, besides the said five figures, can be constructed which is contained by equilateral and equiangular figures equal to one another. For a solid angle cannot be constructed with two triangles, or indeed planes. With three triangles the angle of the pyramid is constructed, with four the angle of the octahedron, and with five the angle of the icosahedron, but a solid angle cannot be formed by six equilateral and equiangular triangles placed together at one point, for, the angle of the equilateral triangle being two-thirds of a right angle, the six would be equal to four right angles, which is impossible, for any solid angle is contained by angles less than four right angles. XI.21 For the same reason, neither can a solid angle be constructed by more than six plane angles. By three squares the angle of the cube is contained, but by four it is impossible for a solid angle to be contained, for they would again be four right angles. By three equilateral and equiangular pentagons the angle of the dodecahedron is contained, but by four such it is impossible for any solid angle to be contained, for, the angle of the equilateral pentagon being a right angle and a fifth, the four angles would be greater than four right angles, which is impossible. Neither again will a solid angle be contained by other polygonal figures by reason of the same absurdity. Q.E.D. Lemma But that the angle of the equilateral and equiangular pentagon is a right angle and a fifth we must prove thus. Let ABCDE be an equilateral and equiangular pentagon. Circumscribe the circle ABCDE about it, take its center F, and join FA, FB, FC, FD, and FE. IV.14 Therefore they bisect the angles of the pentagon at A, B, C, D, and E. And, since the angles at F equal four right angles and are equal, therefore one of them, as the angle AFB, is one right angle less a fifth. Therefore the remaining angles FAB and ABF consist of one right angle and a fifth. But the angle FAB equals the angle FBC, therefore the whole angle ABC of the pentagon consists of one right angle and a fifth. Q.E.D. Summary of the regular polyhedra In the following table, d is the diameter of the sphere in which each regular polygon is inscribed while s is the side of the polygon. Then d2/s2 is the ratio of the square of the diameter to the square of the side. Polyhedron construction d2/s2 tetrahedron XIII.13 3/2 octahedron XIII.14 2 cube XIII.15 3 icosahedron XIII.16 (2 5)/( 5–1) dodecahedron XIII.17 (3– 5)/6 Previous: XIII.17 Book XIII introduction © 1997, 2002 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Introduction New: Jaume Domenech Larraz has translated the Elements into Catalan at http://www.euclides.org/. Euclid's Elements form one of the most beautiful and influential works of science in the history of humankind. Its beauty lies in its logical development of geometry and other branches of mathematics. It has influenced all branches of science but none so much as mathematics and the exact sciences. The Elements have been studied 24 centuries in many languages starting, of course, in the original Greek, then in Arabic, Latin, and many modern languages. I'm creating this version of Euclid's Elements for a couple of reasons. The main one is to rekindle an interest in the Elements, and the web is a great way to do that. Another reason is to show how Java applets can be used to illustrate geometry. That also helps to bring the Elements alive. The text of all 13 Books is complete, and all of the figures are illustrated using the Geometry Applet, even those in the last three books on solid geometry that are three-dimensional. I still have a lot to write in the guide sections and that will keep me busy for quite a while. This edition of Euclid's Elements uses a Java applet called the Geometry Applet to illustrate the diagrams. If you enable Java on your browser, then you'll be able to dynamically change the diagrams. In order to see how, please read Using the Geometry Applet before moving on to the Table of Contents. Copyright © 1996, 1997, 1998. http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University These pages are located at http://aleph0.clarku.edu/~djoyce/java/elements/elements.html. Copyright information 1996, 1997, 2002 Documents and files covered This copyright notice covers all documents and files served by the web server aleph0.clarku.edu within the folder ~djoyce/java/elements or its subfolders. Basic Permissions Currently, all rights are reserved. Web links, however, may be freely made to Euclid's Elements with a reference to the introduction at http://aleph0.clarku.edu/~djoyce/ja va/elements/elements.html. Mirror sites I am formulating a policy on mirroring Euclid's Elements. Euclid's Elements Introduction D.E.Joyce Dept. Math. & Comp. Sci. Clark University These pages are located at http://aleph0.clarku.edu/~djoyce/java/elements/elements.html. Proposition 1 Similar polygons inscribed in circles are to one another as the squares on their diameters. Let ABC and FGH be circles, let ABCDE and FGHKL be similar polygons inscribed in them, and let BM and GN be diameters of the circles. I say that the square on BM is to the square on GN as the polygon ABCDE is to the polygon FGHKL. Join BE, AM, GL, and FN. Now, since the polygon ABCDE is similar to the polygon FGHKL, therefore the angle BAE equals the angle GFL, and BA is to AE as GF is to FL. VI.Def.1 Thus BAE and GFL are two triangles which have one angle equal to one angle, namely the angle BAE equal to the angle GFL, and the sides about the equal angles proportional, therefore the triangle ABE is equiangular with the triangle FGL. Therefore the angle AEB equals the angle FLG. VI.6 But the angle AEB equals the angle AMB, for they stand on the same circumference, and the angle FLG equals the angle FNG, therefore the angle AMB also equals the angle FNG. III.27 But the right angle BAM also equals the right angle GFN, therefore the remaining angle equals the remaining angle. Therefore the triangle ABM is equiangular with the triangle FGN. III.31 I.32 Therefore, proportionally BM is to GN as BA is to GF. VI.4 But the ratio of the square on BM to the square on GN is duplicate of the ratio of BM to GN, and the ratio of the polygon ABCDE to the polygon FGHKL is duplicate of the ratio of BA to GF. VI.20 Therefore the square on BM is to the square on GN as the polygon ABCDE is to the polygon FGHKL. Therefore, similar polygons inscribed in circles are to one another as the squares on their diameters. Q.E.D. Proposition VI.20 states that the ratio of similar polygons is duplicate the ratio of their corresponding sides, so all that is needed is that the corresponding sides are proportional to the diameters of the circumscribed circles, a result that constitutes the bulk of the straightforward proof. This proposition is in preparation for the next in which it is shown that circles are proportional to the squares on their diameters. The connection is that the circles can be arbitrarily closely approximated by polygons, so that if the polygons are proportional to the squares, then so will the circles be proportional to the squares. The difficulty in that proof coming up is to make that argument rigorous. Next proposition: XII.2 Book XII introduction © 1997 D.E.Joyce Clark University Proposition 2 Circles are to one another as the squares on their diameters. Let ABCD and EFGH be circles, and let BD and FH be their diameters. I say that the circle ABCD is to the circle EFGH as the square on BD is to the square on FH. For, if the square on BD is not to the square on FH as the circle ABCD is to the circle EFGH, then as the square on BD is to the square on FH, the circle ABCD is either to some less area than the circle EFGH, or to a greater area. First, let it be in that ratio to a less area S. Inscribe the square EFGH in the circle EFGH. Then the inscribed square is greater than the half of the circle EFGH, for if through the points E, F, G, and H we draw tangents to the circle, then the square EFGH is half the square circumscribed about the circle, and the circle is less than the circumscribed square, hence the inscribed square EFGH is greater than the half of the circle EFGH. IV.6 III.17 Bisect the circumferences EF, FG, GH, and HE at the points K, L, M, and N. Join EK, KF, FL, LG, GM, MH, HN, and NE. Therefore each of the triangles EKF, FLG, GMH, and HNE is also greater than the half of the segment of the circle about it, for if through the points K, L, M, and N we draw tangents to the circle and complete the parallelograms on the straight lines EF, FG, GH, and HE, then each of the triangles EKF, FLG, GMH, and HNE is half of the parallelogram about it, while the segment about it is less than the parallelogram, hence each of the triangles EKF, FLG, GMH, and HNE is greater than the half of the segment of the circle about it. III.17 Thus, by bisecting the remaining circumferences and joining straight lines, and by doing this repeatedly, we shall leave some segments of the circle which will be less than the excess by which the circle EFGH exceeds the area S. For it was proved in the first theorem of the tenth book that if two unequal magnitudes are set out, and if from the greater there is subtracted a magnitude greater than the half, and from that which is left a greater than the half, and if this is done repeatedly, then there will be left some magnitude which is less than the lesser magnitude set out. X.1 Let segments be left such as described, and let the segments of the circle EFGH on EK, KF, FL, LG, GM, MH, HN, and NE be less than the excess by which the circle EFGH exceeds the area S. Therefore the remainder, the polygon EKFLGMHN, is greater than the area S. Now inscribe in the circle ABCD the polygon AOBPCQDR similar to the polygon EKFLGMHN. Therefore the square on BD is to the square on FH as the polygon AOBPCQDR is to the polygon EKFLGMHN. XII.1 But the square on BD is to the square on FH as the circle ABCD to the area S, therefore the circle ABCD is to the area S as the polygon AOBPCQDR is to the polygon EKFLGMHN. Therefore, alternately the circle ABCD is to the polygon inscribed in it as the area S is to the polygon EKFLGMHN. V.11 V.16 But the circle ABCD is greater than the polygon inscribed in it, therefore the area S is also greater than the polygon EKFLGMHN. But it is also less, which is impossible. Therefore the square on BD is to the square on FH not as the circle ABCD is to any area less than the circle EFGH. Similarly we can prove that the circle EFGH is to any area less than the circle ABCD not as the square on FH is to the square on BD. I say next that neither is the circle ABCD to any area greater than the circle EFGH as the square on BD is to the square on FH. For, if possible, let it be in that ratio to a greater area S. Therefore, inversely the square on FH is to the square on DB as the area S is to the circle ABCD. But the area S is to the circle ABCD as the circle EFGH is to some area less than the circle ABCD, therefore the square on FH is to the square on BD as the circle EFGH is to some area less than the circle ABCD, which was proved impossible. Therefore the square on BD is to the square on FH not as the circle ABCD to any area greater than the circle EFGH. Lemma V.11 And it was proved that neither is it in that ratio to any area less than the circle EFGH, therefore the square on BD is to the square on FH as the circle ABCD is to the circle EFGH. Q.E.D. Lemma I say that, the area S being greater than the circle EFGH the area S is to the circle ABCD as the circle EFGH is to some area less than the circle ABCD. For let it be contrived that the area S is to the circle ABCD as the circle EFGH to the area T. I say that the area T is less than the circle ABCD. Since the area S is to the circle ABCD as the circle EFGH is to the area T, therefore, alternately the area S is to the circle EFGH as the circle ABCD is to the area T. V.16 But the area S is greater than the circle EFGH, therefore the circle ABCD is also greater than the area T. Hence the area S is to the circle ABCD as the circle EFGH is to some area less than the circle ABCD. Therefore, circles are to one another as the squares on their diameters. Q.E.D. In the last proposition it was shown that similar polygons inscribed in circles are proportional to the squares on the diameters of the circles. By approximating circles closely by similar polygons, the proportion is carried over to the circles. The form of the proof is a double proof by contradiction. There are three cases when comparing the ratio of the squares BD:FH to the ratio of the circles ABCD:EFGH. One case is that the ratio of the squares BD:FH equals ABCD:S where S is some area less than circle EFGH. Most of the proof is spent refuting this case. The second case is that the ratio of the squares BD:FH equals ABCD:S where this time S is some area greater than circle EFGH. This is inverted to a statement that the ratio of the squares FH:BD equals EFGH to some area less than circle ABCD, which is the first case already already shown not to occur. That leaves only the third case that the ratio of the squares BD:FH equals the ratio of the circles ABCD:EFGH. (Actually, there is a gap in the proof at this last step; it was never shown that these are the only three cases. It may be true that there are three cases, that the ratio of the squares is greater, less, or equal to the ratio of the circles, but the three cases of in the proof are one step removed from these three cases.) Approximation by polygons The first case is disposed of by approximating the circles by very close polygons. To begin with a square EFGH is inscribed in the circle EFGH, and it is shown that the remainder is less than half the circle. Next the circumferences are bisected to construct an octagon EKFLGMHN, and the remainder of the circle is shown to be less than half the old remainder. Continuing, polygons of 16, 32, 64, etc., sides are constructed and each one leaves a remainder less than half the previous remainder. Now the circle EFGH exceeds the area S by some finite amount, and by the principle of proposition X.1, at some stage mentioned above, the remainder will be less than the excess of circle EFGH over S. For the rest of the proof, that stage is taken as that of the polygon EKFLGMHN, that is, circle EFGH – polygon EKFLGMHN < circle EFGH – area S, and so area S < polygon EKFLGMHN. The similar polygon AOBPCQDR is inscribed in the other circle ABCD. Then circle ABCD : area S = BC2 : FH2 = polygon AOBPCQDR : polygon EKFLGMHN. Alternately, circle ABCD : polygon AOBPCQDR = area S : polygon EKFLGMHN. But circle ABCD > polygon AOBPCQDR, so area S > polygon EKFLGMHN, contradicting the statement above that the area S is less than the polygon. Principle of exhaustion The approximation of a figure by a sequence of figures inside it is sometimes called the "principle of exhaustion." The important point of this principle is that the sequence of approximations can be made so that the difference between the original figure and the inscribed figure decreases by at least half at each step of the sequence. This principle is used in several later propositions in this book. Proposition XII.5 uses it to show that pyramids of the same height with triangular bases are proportional to their bases. The pyramids are approximated by a union of similar triangular prisms. Proposition XII.10 uses the principle of exhaustion to show that a cone inscribed in a cylinder is one-third of the cylinder. The cone is approximated by inscribed pyramids while the cylinder is approximated by inscribed prisms. Pyramids inscribed in cones are similarly used in XII.11 and XII.12. Finally, the principle of exhaustion is used in proposition XII.18 to show spheres are to one another in triplicate ratio of their diameters. There the spheres are exhausted by inscribed polyhedra. Next proposition: XII.3 Previous: XII.1 Book XII introduction © 1997 D.E.Joyce Clark University Proposition 3 Any pyramid with a triangular base is divided into two pyramids equal and similar to one another, similar to the whole, and having triangular bases, and into two equal prisms, and the two prisms are greater than half of the whole pyramid. Let there be a pyramid of with the triangular base ABC and vertex D. I say that the pyramid ABCD is divided into two pyramids equal to one another, having triangular bases and similar to the whole pyramid, and into two equal prisms, and the two prisms are greater than the half of the whole pyramid. Bisect AB, BC, CA, AD, DB, and DC at the points E, F, G, H, K, and L. Join HE, EG, GH, HK, KL, LH, KF, and FG be joined. Since AE equals EB, and AH equals DH, therefore EH is parallel to DB. For the same reason HK is also parallel to AB. Therefore HEBK is a parallelogram. Therefore HK equals EB. VI.2 I.34 But EB equals EA, therefore AE also equals HK. But AH also equals HD, therefore the two sides EA and AH equal the two sides KH, HD respectively, and the angle EAH equals the angle KHD, therefore the base EH equals the base KD. I.4 Therefore the triangle AEH equals and is similar to the triangle HKD. For the same reason the triangle AHG also equals and is similar to the triangle HLD. Now, since two straight lines EH and HG meeting one another are parallel to two straight lines KD and DL meeting one another and are not in the same plane, therefore they contain equal angles. Therefore the angle EHG equals the angle KDL. XI.10 And, since the two straight lines EH and HG equal the two KD and DL respectively, and the angle EHG equals the angle KDL, therefore the base EG equals the base KL. Therefore the triangle EHG equals and is similar to the triangle KDL. For the same reason the triangle AEG also equals and is similar to the triangle HKL. I.4 Therefore the pyramid with triangular base AEG and vertex H equals and is similar to the pyramid with triangular base HKL and the vertex D. XI.Def.10 And, since HK is parallel to AB, one of the sides of the triangle ADB, the triangle ADB is equiangular to the triangle DHK, and they have their sides proportional, therefore the triangle ADB is similar to the triangle DHK. For the same reason the triangle DBC is also similar to the triangle DKL, and the triangle ADC is similar to the triangle DLH. I.29 VI.Def.1 Now, since the two straight lines BA and AC meeting one another are parallel to the two straight lines KH and HL meeting one another not in the same plane, therefore they contain equal angles. Therefore the angle BAC equals the angle KHL. XI.10 And BA is to AC as KH is to HL, therefore the triangle ABC is similar to the triangle HKL. Therefore the pyramid with the triangular base ABC and vertex D is similar to the pyramid with the triangular base HKL and vertex D. But the pyramid with the triangular base HKL and vertex D was proved similar to the pyramid with the triangular base AEG and the vertex H. Therefore each of the pyramids AEGH and HKLD is similar to the whole pyramid ABCD. Next, since BF equals FC, therefore the parallelogram EBFG is double the triangle GFC. And since, if there are two prisms of equal height, and one has a parallelogram as base and the other a triangle, and if the parallelogram is double the triangle, then the prisms are equal. Therefore the prism contained by the two triangles BKF and EHG, and the three parallelograms EBFG, EBKH, and HKFG equals the prism contained by the two triangles GFC and HKL and the three parallelograms KFCL, LCGH, and HKFG. XI.39 And it is clear that each of the prisms, namely that with the parallelogram EBFG the base and the straight line HK its opposite, and that with the triangle GFC the base and the triangle HKL its opposite, is greater than each of the pyramids with the triangular bases AEG and HKL and vertices H and D, for, if we join the straight lines EF and EK, the prism with the parallelogram EBFG the base and the straight line HK opposite is greater than the pyramid with the triangular base EBF and vertex K. But the pyramid with the triangular base EBF and vertex A equals the pyramid with the triangular base AE and the vertex H, for they are contained by equal and similar planes. Hence the prism with the parallelogram EBF the base and the straight line HK opposite is greater than the pyramid with the triangular base AE and vertex H. But the prism with the parallelogram EBF the base and the straight line HK opposite equals the prism with the triangle GFC the base and the triangle HKL opposite, and the pyramid with the triangular base AEG and vertex H equals the pyramid with the triangular base HKL and vertex D. Therefore the said two prisms are greater than the said two pyramids with the triangular bases AEG and HKL and vertices H and D. Therefore the whole pyramid with the triangular base ABC and vertex D has been divided into two pyramids equal to one another and into two equal prisms, and the two prisms are greater than the half of the whole pyramid. Therefore, any pyramid with a triangular base is divided into two pyramids equal and similar to one another, similar to the whole, and having triangular bases, and into two equal prisms, and the two prisms are greater than half of the whole pyramid. Q.E.D. This and the next six propositions deal with volumes of pyramids. The first two of these lay the foundations for XII.5 (pyramids are proportional to their bases). In the last book it was shown in XI.32 that parallelepipeds of the same height are proportional to their bases, and XI.28 (a triangular prism is half a parallelepiped) implies that this proportionality can be carried over to prisms with triangular bases. It is not so easy to carry the proportionality over to pyramids with triangular bases. But that is what is done in XII.3 through XII.5. The basic observation is in this proposition: most of a triangular based pyramid can be filled up by two congruent prisms leaving less than half to two smaller similar pyramids. Next, if each of these two smaller pyramids are filled up by two smaller prisms leaving two even smaller pyramids in each, then the four even smaller pyramids that remain are less then 1/4 of the original pyramid. Partitioning those four again yields eight with a total volume less then 1/8 of the original pyramid. And so on. Since the desired proportionality holds for prisms, and pyramids can be partitioned nearly all into prisms, therefore the desired proportionality will hold for pyramids. This process is used and clarified in XII.5. The intermediate proposition XII.4 supplies a important technical result needed in XII.5. Next proposition: XII.4 Previous: XII.2 Book XII introduction © 1997 D.E.Joyce Clark University Proposition 4 If there are two pyramids of the same height with triangular bases, and each of them is divided into two pyramids equal and similar to one another and similar to the whole, and into two equal prisms, then the base of the one pyramid is to the base of the other pyramid as all the prisms in the one pyramid are to all the prisms, being equal in multitude, in the other pyramid. Let there be two pyramids of the same height with triangular bases ABC and DEF the points G and H the vertices, and let each of them be divided into two pyramids equal to one another and similar to the whole and into two equal prisms. XII.3 I say that the base ABC is to the base DEF as all the prisms in the pyramid ABCG to all the prisms, being equal in multitude, in the pyramid DEFH. Since BO equals OC, and AL equals LC, therefore LO is parallel to AB, and the triangle ABC is similar to the triangle LOC. For the same reason the triangle DEF is also similar to the triangle RVF. And, since BC is double CO, and EF double FV, therefore BC is to CO as EF is to FV. And on BC and CO are described the similar and similarly situated rectilinear figures ABC and LOC, and on EF and FV the similar and similarly situated figures DEF and RVF, therefore the triangle ABC is to the triangle LOC as the triangle DEF is to the triangle RVF. VI.22 Therefore, alternately the triangle ABC is to the triangle DEF as the triangle LOC is to the triangle RVF. But the triangle LOC is to the triangle RVF as the prism with the triangle LOC the base and PMN opposite is to the prism with the triangle RVF the base and STU opposite. V.16 Lemma below Therefore the triangle ABC is to the triangle DEF as the prism with the triangle LOC the base and PMN opposite is to the prism with the triangle RVF the base and STU opposite. But the said prisms are to one another as the prism with the parallelogram KBOL the base and the straight line PM opposite is to the prism with the parallelogram QEVR the base and the straight line ST opposite. XI.39 Therefore the two prisms, that with the parallelogram KBOL the base and PM opposite, and that with the triangle LOC the base and PMN opposite, are to the prisms with QEVR the base and the straight line ST opposite and with the triangle RVF the base and STU opposite in the same ratio. V.12 Therefore the base ABC is to the base DEF as the said two prisms are to the said two prisms. And similarly, if the pyramids PMNG and STUH are divided into two prisms and two pyramids, then the base PMN is to the base STU as the two prisms in the pyramid PMNG are to the two prisms in the pyramid STUH. But the base PMN is to the base STU as the base ABC is to the base DEF, for the triangles PMN and STU equal the triangles LOC and RVF respectively. Therefore the base ABC is to the base DEF as the four prisms are to the four prisms. And similarly, if we divide the remaining pyramids into two pyramids and into two prisms, then the base ABC is to base the DEF as all the prisms in the pyramid ABCG are to all the prisms, being equal in multitude, in the pyramid DEFH. Lemma But that the triangle LOC is to the triangle RVF as the prism with the triangle LOC the base and PMN opposite is to the prism with the triangle RVF the base and STU opposite, we must prove as follows. In the same figure draw perpendiculars from G and H to the planes ABC and DEF. These are, of course, equal since the pyramids are of equal height by hypothesis. XI.11 Now, since the two straight lines GC and the perpendicular from G are cut by the parallel planes ABC and PMN, therefore they are cut in the same ratios. XI.17 And GC is bisected by the plane PMN at N, therefore the perpendicular from G to the plane ABC is also bisected by the plane PMN. For the same reason the perpendicular from H to the plane DEF is also bisected by the plane STU. And the perpendiculars from G and H to the planes ABC and DEF are equal, therefore the perpendiculars from the triangles PMN and STU to the planes ABC and DEF are also equal. Therefore the prisms with the triangles LOC and RVF the bases, and PMN and STU opposite, are of equal height. Hence also the parallelepipedal solids described from the said prisms are of equal height and are to one another as their bases. Therefore their halves, namely the said prisms, are to one another as the base LOC is to the base RVF. XI.32 XI.28 Therefore, if there are two pyramids of the same height with triangular bases, and each of them is divided into two pyramids equal and similar to one another and similar to the whole, and into two equal prisms, then the base of the one pyramid is to the base of the other pyramid as all the prisms in the one pyramid are to all the prisms, being equal in multitude, in the other pyramid. Q.E.D. This proposition is subordinate to the next, XII.5, in which two pyramids with triangular bases and the same height are shown to be proportional to their bases. Its proof proceeds by partitioning each of the two original pyramids into the twopyramid-two-prism division of the previous proposition, then doing the same partition to the two smaller pyramids, then to the four even smaller pyramids, until a sufficiently small part of each original pyramid remains in whatever tiny pyramids there are while a sufficiently large part of each is composed of various sized prisms. This proposition, at least in the last paragraph, considers that situation and concludes that the base of the first pyramid is to the second as the union of the various sized prisms in the first pyramid is to the union of the various sized prisms in the second pyramid. This is the crucial step in the proof of XII.5. Next proposition: XII.5 Previous: XII.3 Book XII introduction © 1997 D.E.Joyce Clark University Proposition 5 Pyramids of the same height with triangular bases are to one another as their bases. Let there be pyramids of the same height with triangular bases ABC and DEF and vertices G and H. I say that the base ABC is to the base DEF as the pyramid ABCG is to the pyramid DEFH. For, if the pyramid ABCG is not to the pyramid DEFH as the base ABC is to the base DEF, then the base ABC is to the base DEF as the pyramid ABCG is either to some solid less than the pyramid DEFH or to a greater solid. Let it, first, be in that ratio to a less solid W. Divide the pyramid DEFH into two pyramids equal to one another and similar to the whole and into two equal prisms. Then the two prisms are greater than the half of the whole pyramid. XII.3 Again, divide the pyramids arising from the division similarly, and let this be done repeatedly until there are left over from the pyramid DEFH some pyramids which are less than the excess by which the pyramid DEFH exceeds the solid W. X.1 Let such be left, and let them be, for the sake of argument, DQRS and STUH. Therefore the remainders, the prisms in the pyramid DEFH, are greater than the solid W. Divide the pyramid ABCG similarly, and a same number of times, with the pyramid DEFH. Therefore the base ABC is to the base DEF as the prisms in the pyramid ABCG are to the prisms in the pyramid DEFH. XII.4 But the base ABC is to the base DEF as the pyramid ABCG is to the solid W, therefore the pyramid ABCG is to the solid W as the prisms in the pyramid ABCG are to the prisms in the pyramid DEFH. Therefore, alternately the pyramid ABCG is to the prisms in it as the solid W is to the prisms in the pyramid DEFH. V.11 V.16 But the pyramid ABCG is greater than the prisms in it, therefore the solid W is also greater than the prisms in the pyramid DEFH. But it is also less, which is impossible. Therefore the prism ABCG is not to any solid less than the pyramid DEFH as the base ABC is to the base DEF. Similarly it can be proved that neither is the pyramid DEFH to any solid less than the pyramid ABCG as the base DEF is to the base ABC. I say next that neither is the pyramid ABCG to any solid greater than the pyramid DEFH as the base ABC is to the base DEF. For, if possible, let it be in that ratio to a greater solid W. Therefore, inversely the base DEF is to the base ABC as the solid W is to the pyramid ABCG. But it was proved before that the solid W is to the solid ABCG as the pyramid DEFH is to some solid less than the pyramid ABCG. Therefore the base DEF is to the base ABC as the pyramid DEFH is to some solid less than the pyramid ABCG, which was proved absurd. XII.2,Lemma V.11 Therefore the pyramid ABCG is not to any solid greater than the pyramid DEFH as the base ABC is to the base DEF. But it was proved that neither is it in that ratio to a less solid. Therefore the base ABC is to the base DEF as the pyramid ABCG is to the pyramid DEFH. Therefore, pyramids of the same height with triangular bases are to one another as their bases. Q.E.D. Use of this theorem The next proposition generalizes this one so that the bases of the pyramids may be any polygons, not just triangles. In the following proposition, this proposition is used to show that a prism can be dissected into three equal (but not congruent) prisms. Next proposition: XII.6 Previous: XII.4 Book XII introduction © 1997 D.E.Joyce Clark University Proposition 6 Pyramids of the same height with polygonal bases are to one another as their bases. Let there be pyramids of the same height with the polygonal bases ABCDE and FGHKL and vertices M and N. I say that the base ABCDE is to the base FGHKL as the pyramid ABCDEM is to the pyramid FGHKLN. Join AC, AD, FH, and FK. Since then ABCM and ACDM are two pyramids with triangular bases and equal height, therefore they are to one another as their bases. Therefore the base ABC is to the base ACD as the pyramid ABCM is to the pyramid ACDM. And, taken together, the base ABCD is to the base ACD as the pyramid ABCDM is to the pyramid ACDM. XII.5 V.18 But the base ACD is to the base ADE as the pyramid ACDM is to the pyramid ADEM. XII.5 Therefore, ex aequali, the base ABCD is to the base ADE as the pyramid ABCDM is to the pyramid ADEM. V.22 And again, taken together, the base ABCDE is to the base ADE as the pyramid ABCDEM is to the pyramid ADEM. Similarly also it can be proved that the base FGHKL is to the base FGH as the pyramid FGHKLN is to the pyramid FGHN. V.18 And, since ADEM and FGHN are two pyramids with triangular bases and equal heights, therefore the base ADE is to the base FGH as the pyramid ADEM is to the pyramid FGHN. XII.5 But the base ADE is to the base ABCDE as the pyramid ADEM is to the pyramid ABCDEM. Therefore, ex aequali, the base ABCDE is to the base FGH as the pyramid ABCDEM is to the pyramid FGHN. V.22 But further the base FGH is to the base FGHKL as the pyramid FGHN is to the pyramid FGHKLN. Therefore also, ex aequali, the base ABCDE is to the base FGHKL as the pyramid ABCDEM is to the pyramid FGHKLN V.22 Therefore, pyramids of the same height with polygonal bases are to one another as their bases. Q.E.D. It is important to notice that the bases of the pyramids under consideration need not be similar, indeed they may have different numbers of sides. Use of this proposition This proposition will be used in XII.10 and XII.11 which concern the volumes of cones and cylinders. Next proposition: XII.7 Previous: XII.5 Book XII introduction © 1997 D.E.Joyce Clark University Proposition 7 Any prism with a triangular base is divided into three pyramids equal to one another with triangular bases. Let there be a prism with the triangular base ABC and DEF opposite. I say that the prism ABCDEF is divided into three pyramids equal to one another, which have triangular bases. Join BD, EC, and CD. Since ABED is a parallelogram, and BD is its diameter, therefore the triangle ABD equals the triangle EBD. Therefore the pyramid with triangular base ABD and vertex C equals the pyramid with triangular base DEB and vertex C. I.34 XII.5 But the pyramid with triangular base DEB and vertex C is identical with the pyramid with triangular base EBD and vertex D, for they are contained by the same planes. Therefore the pyramid with triangular base ABD vertex C is also equal to the pyramid with triangular base EBC and vertex D. Again, since FCBE is a parallelogram, and CE is its diameter, therefore the triangle CEF equals the triangle CBE. I.34 Therefore the pyramid with triangular base BCE and vertex D equals the pyramid with triangular base ECF and vertex D. XII.5 But the pyramid with triangular base BCE and vertex D was proved equal to the pyramid with triangular base ABD and vertex C, therefore the pyramid with triangular base CEF and vertex D equals the pyramid with triangular base ABD and vertex C. Therefore the prism ABCDEF is divided into three pyramids equal to one another which have triangular bases. And, since the pyramid with triangular base ABD and vertex C is identical with the pyramid with triangular base CAB and vertex D, for they are contained by the same planes, while the pyramid with triangular base ABD vertex C was proved to be a third of the prism with triangular base ABC and DEF opposite, therefore the pyramid with triangular base ABC and vertex D is a third of the prism with the same base ABC, and DEF opposite. Therefore, any prism with a triangular base is divided into three pyramids equal to one another with triangular bases. Corollary. From this it is clear that any pyramid is a third part of the prism with the same base and equal height. Q.E.D. The proof of this proposition is easier than it looks. The triangles ABD and EBD are equal. Now since the pyramids ABDC and DEBC have equal bases and the same altitude, by XII.5, they are equal pyramids. A similar argument shows pyramids BCEDand ECFD are equal. But DEBC and BCED are the same pyramid named differently. So the prism is divided into three equal pyramids. Use of this proposition This proposition is used in the next two propositions about volumes of pyramids and in XII.10 following them about volumes of cones and cylinders. Next proposition: XII.8 Previous: XII.6 Book XII introduction © 1997 D.E.Joyce Clark University Proposition 8 Similar pyramids with triangular bases are in triplicate ratio of their corresponding sides. Let there be similar and similarly situated pyramids with triangular bases AB and DEF vertices G and H. I say that the pyramid ABCG has to the pyramid DEFH the ratio triplicate of that which BC has to EF. Complete the parallelepipedal solids BGML and EHQP. Now, since the pyramid ABCG is similar to the pyramid DEFH, therefore the angle ABC equals the angle DEF, the angle GBC equals the angle HEF, the angle ABG equals the angle DEH, and AB is to DE as BC is to EF, and as BG is to EH. And since AB is to DE as BC is to EF, and the sides are proportional about equal angles, therefore the parallelogram BM is similar to the parallelogram EQ. For the same reason BN is also similar to ER, and BR similar to EO. Therefore the three parallelograms MB, BK, and BN are similar to the three EQ, EO, and ER. But the three parallelograms MB, BK, and BN are equal and similar to their three opposites, and the three EQ, EO, and ER are equal and similar to their three opposites. XI.24 Therefore the solids BGML and EHQP are contained by similar planes equal in multitude. Therefore the solid BGML is similar to the solid EHQP. But similar parallelepipedal solids are in the triplicate ratio of their corresponding sides. Therefore the solid BGML has to the solid EHQP the ratio triplicate of that which the corresponding side BC has to the corresponding side EF. XI.33 But the solid BGML is to the solid EHQP as the pyramid ABCG is to the pyramid DEFH, for the pyramid is a sixth part of the solid, because the prism which is half of the parallelepipedal solid is also triple the pyramid. Therefore the pyramid ABCG has to the pyramid DEFH the ratio triplicate of that which BC has to EF. XI.28 XII.7 Q.E.D. Corollary. From this it is clear that similar pyramids with polygonal bases are also to one another in the triplicate ratio of their corresponding sides. For, if they are divided into the pyramids contained in them which have triangular bases, by virtue of the fact that the similar polygons forming their bases are also divided into similar triangles equal in multitude and corresponding to the wholes, then the one pyramid with a triangular base in the one complete pyramid is to the one pyramid with a triangular base in the other complete pyramid as all the pyramids with triangular bases contained in the one pyramid is to all the pyramids with triangular bases contained in the other pyramid, that is, the pyramid itself with a polygonal base, to the pyramid with a polygonal base. VI.20 V.12 But the pyramid with a triangular base is to the pyramid with a triangular base in the triplicate ratio of the corresponding sides, therefore also the pyramid with a polygonal base has to the pyramid with a similar base the ratio triplicate of that which the side has to the side. Therefore, similar pyramids with triangular bases are in triplicate ratio of their corresponding sides. Heath gives a good argument that the corollary was added later. Use of this propostion This proposition is used to show similar cones are in triplicate ratio of the diameters of their bases in proposition XII.12. Also, the corollary justifies a statement in the corollary of XII.17 concerning similar solids. Next proposition: XII.9 Previous: XII.7 Book XII introduction © 1997 D.E.Joyce Clark University Proposition 9 In equal pyramids with triangular bases the bases are reciprocally proportional to the heights; and those pyramids are equal in which the bases are reciprocally proportional to the heights. Let there be equal pyramids with triangular bases ABC and DEF and vertices G and H. I say that in the pyramids ABCG and DEFH the bases are reciprocally proportional to the heights, that is the base ABC is to the base DEF as the height of the pyramid DEFH is to the height of the pyramid ABCG. Complete the parallelepipedal solids BGML and EHQP. Now, since the pyramid ABCG equals the pyramid DEFH, and the solid BGML is six times the pyramid ABCG, and the solid EHQP six times the pyramid DEFH, therefore the solid BGML equals the solid EHQP. XII.7,Cor But in equal parallelepipedal solids the bases are reciprocally proportional to the heights, therefore the base BM is to the base EQ as the height of the solid EHQP is to the height of the solid BGML. XI.34 But the base BM is to EQ as the triangle ABC is to the triangle DEF. Therefore the triangle ABC is to the triangle DEF as the height of the solid EHQP is to the height of the solid BGML. I.34 V.11 But the height of the solid EHQP is identical with the height of the pyramid DEFH, and the height of the solid BGML is identical with the height of the pyramid ABCG, therefore the base ABC is to the base DEF as the height of the pyramid DEFH is to the height of the pyramid ABCG. Therefore in the pyramids ABCG and DEFH the bases are reciprocally proportional to the heights. Next, in the pyramids ABCG and DEFH let the bases be reciprocally proportional to the heights, that is, as the base ABC is to the base DEF, so let the height of the pyramid DEFH be to the height of the pyramid ABCG. I say that the pyramid ABCG equals the pyramid DEFH. With the same construction, since the base ABC is to the base DEF as the height of the pyramid DEFH is to the height of the pyramid ABCG, while the base ABC is to the base DEF as the parallelogram BM is to the parallelogram EQ, therefore the parallelogram BM is to the parallelogram EQ as the height of the pyramid DEFH is to the height of the pyramid ABCG. V.11 But the height of the pyramid DEFH is identical with the height of the parallelepiped EHQP, and the height of the pyramid ABCG is identical with the height of the parallelepiped BGML, therefore the base BM is to the base EQ as the height of the parallelepiped EHQP is to the height of the parallelepiped BGML. But those parallelepipedal solids in which the bases are reciprocally proportional to the heights are equal, therefore the parallelepipedal solid BGML equals the parallelepipedal solid EHQP. XI.34 And the pyramid ABCG is a sixth part of BGML, and the pyramid DEFH a sixth part of the parallelepiped EHQP, therefore the pyramid ABCG equals the pyramid DEFH. Therefore, in equal pyramids with triangular bases the bases are reciprocally proportional to the heights; and those pyramids are equal in which the bases are reciprocally proportional to the heights. Q.E.D. The pyramids with triangular bases are one-third of the prisms with triangular bases, which aren't drawn, and the prisms are half of the parallelepipeds. Since the analogous proposition XI.34 holds for parallelepipeds, this proposition holds for pyramids. When a similar situation appears later where cones are one-third of cylinders, Euclid simply says the same holds for cones, too, with no details whatsoever. This proposition completes the theory of volumes for pyramids. The next few propositions treat the theory of volumes of cones and cylinders. Next proposition: XII.10 Previous: XII.8 Book XII introduction © 1997 D.E.Joyce Clark University Proposition 10 Any cone is a third part of the cylinder with the same base and equal height. Let a cone have the same base, namely the circle ABCD, with a cylinder and equal height. I say that the cone is a third part of the cylinder, that is, that the cylinder is triple the cone. For if the cylinder is not triple the cone, then the cylinder will be either greater than triple or less than triple the cone. First let it be greater than triple. Inscribe the square ABCD in the circle ABCD. Then the square ABCD is greater than half of the circle ABCD. From the square ABCD set up a prism of equal height with the cylinder. IV.6 Then the prism so set up is greater than the half of the cylinder, for if we also circumscribe a square about the circle ABCD, the square inscribed in the circle ABCD is half of that circumscribed about it, and the solids set up from them are parallelepipedal prisms of equal height, while parallelepipedal solids of the same height are to one another as their bases, therefore also the prism set up on the square ABCD is half of the prism set up from the square circumscribed about the circle ABCD, and the cylinder is less than the prism set up from the square circumscribed about the circle ABCD, therefore the prism set up from the square ABCD and of equal height with the cylinder is greater than the half of the cylinder. IV.7 XI.32 XI.28 or XII.6 and XII.7.Cor Bisect the circumferences AB, BC, CD, and DA at the points E, F, G, and H, and join AE, EB, BF, FC, CG, GD, DH, and HA. Then each of the triangles AEB, BFC, CGD, and DHA is greater than the half of that segment of the circle ABCD about it, as we proved before. XII.2 On each of the triangles AEB, BFC, CGD, and DHA set prisms up of equal height with the cylinder. Then each of the prisms so set up is greater than the half part of that segment of the cylinder about it, for if we draw through the points E, F, G, and H parallels to AB, BC, CD, and DA, complete the parallelograms on AB, BC, CD, and DA, and set up from them parallelepipedal solids of equal height with the cylinder, then the prisms on the triangles AEB, BFC, CGD, and DHA are halves of the several solids set up, and the segments of the cylinder are less than the parallelepipedal solids set up, hence also the prisms on the triangles AEB, BFC, CGD, and DHA are greater than half of the segments of the cylinder about them. I.31 Thus, bisecting the circumferences that are left, joining straight lines, setting up on each of the triangles prisms of equal height with the cylinder, and doing this repeatedly, we shall leave some segments of the cylinder which are less than the excess by which the cylinder exceeds the triple the cone. X.1 Let such segments be left, and let them be AE, EB, BF, FC, CG, GD, DH, and HA. Therefore the remainder, the prism with polygonal base AEBFCGDH and the same height as that of the cylinder, is greater than triple the cone. But the prism with polygonal base AEBFCGDH and the same height as that of the cylinder is triple the pyramid with polygonal base AEBFCGDH and the same vertex as that of the cone. Therefore the pyramid with the polygonal base AEBFCGDH and the same vertex as that of the cone is greater than the cone with circular base ABCD. XII.7,Cor But it is also less, for it is enclosed by it, which is impossible. Therefore the cylinder is not greater than triple the cone. I say next that neither is the cylinder less than triple the cone, For, if possible, let the cylinder be less than triple the cone. Therefore, inversely, the cone is greater than a third part of the cylinder. Inscribe the square ABCD in the circle ABCD. Therefore the square ABCD is greater than the half of the circle ABCD. IV.6 Now set up from the square ABCD a pyramid with the same vertex as the cone. Therefore the pyramid so set up is greater than half of the cone, for, as we proved before, if we circumscribe a square about the circle, then the square ABCD is half of the square circumscribed about the circle, and if we set up from the squares parallelepipedal solids of equal height with the cone, which are also called prisms, then the solid set up from the square ABCD is half of that set up from the square circumscribed about the circle, for they are to one another as their bases. XI.32 Hence the thirds of them are also in that ratio. Therefore the pyramid with the square base ABCD is half of the pyramid set up from the square circumscribed about the circle. And the pyramid set up from the square about the circle is greater than the cone, for it encloses it. Therefore the pyramid with the square base ABCD and the same vertex as that of the cone is greater than the half of the cone. Bisect the circumferences AB, BC, CD, and DA at the points E, F, G, and H, and join AE, EB, BF, FC, CG, GD, DH, and HA be joined. Then each of the triangles AEB, BFC, CGD, and DHA is greater than the half part of that segment of the circle ABCD about it. Now, on each of the triangles AEB, BFC, CGD, and DHA set pyramids up with the same vertex as the cone. Therefore each of the pyramids so set up is, in the same manner, greater than the half part of that segment of the cone about it. Thus, by bisecting the circumferences that are left, joining straight lines, setting up pyramids on each of the triangles with the same vertex as the cone, and doing this repeatedly, we shall leave some segments of the cone which will be less than the excess by which the cone exceeds the third part of the cylinder. X.1 Let such be left, and let them be the segments on AE, EB, BF, FC, CG, GD, DH, and HA. Therefore the remainder, the pyramid with the polygonal base AEBFCGDH and the same vertex as that of the cone, is greater than a third part of the cylinder. But the pyramid with the polygonal AEBFCGDH base and the same vertex as that of the cone is a third part of the prism with the polygonal base AEBFCGDH and the same height as that of the cylinder, therefore the prism with the polygonal base AEBFCGDH and the same height as that of the cylinder is greater than the cylinder with the circular base ABCD. But it is also less, for it is enclosed by it, which is impossible. Therefore the cylinder is not less than triple the cone. But it was proved that neither is it greater than triple. Therefore the cylinder is triple the cone, hence the cone is a third part of the cylinder. Therefore, any cone is a third part of the cylinder with the same base and equal height. Q.E.D. This and the next five propositions deal with the volumes of cones and cylinders. This proposition is fundamental in that it relates the volume of a cone to that of the circumscribed cylinder so that whatever is said about the volumes cylinder can be converted into a statement about volumes of cones and vice versa. In XII.11, the next proposition, cones of the same height are shown to be proportional to their bases, and therefore cylinders of the same height are proportional to their bases. In XII.12 similar cones are shown to be in the triplicate ratio of the diameters of their bases, therefore the analogous statement holds for cylinders. In XII.14 cylinders on equal bases are shown to be proportional to their heights, therefore the analogous statement holds for cones. And in XII.15 it is shown that equal cylinders are those whose bases are reciprocally proportional to their heights, and as Euclid says, "the same it true for the cones also." Next proposition: XII.11 Previous: XII.9 Book XII introduction © 1997 D.E.Joyce Clark University Proposition 11 Cones and cylinders of the same height are to one another as their bases. Let there be cones and cylinders of the same height, let the circles ABCD and EFGH be their bases, KL and MN their axes, and AC and EG the diameters of their bases. I say that the circle ABCD is to the circle EFGH as the cone AL is to the cone EN. For, if not, then the circle ABCD is to the circle EFGH as the cone AL is either to some solid less than the cone EN or to a greater. First, let it be in that ratio to a less solid O, and let the solid X be equal to that by which the solid O is less than the cone EN. Therefore the cone EN equals the sum of the solids O and X. Inscribe the square EFGH in the circle EFGH. Therefore the square is greater than the half of the circle. IV.6 Set up from the square EFGH a pyramid of equal height with the cone. Therefore the pyramid so set up is greater than the half of the cone, for if we circumscribe a square about the circle, and set up from it a pyramid of equal height with the cone, then the inscribed pyramid is half of the circumscribed pyramid, for they are to one another as their bases, while the cone is less than the circumscribed pyramid. XII.6 Bisect the circumferences EF, FG, GH, and HE at the points P, Q, R, and S, and join HP, PE, EQ, QF, FR, RG, GS, and SH. Therefore each of the triangles HPE, EQF, FRG, and GSH is greater than the half of that segment of the circle about it. Set up on each of the triangles HPE, EQF, FRG, and GSH a pyramid of equal height with the cone. Therefore each of the pyramids so set up is also greater than the half of that segment of the cone about it. Thus, bisecting the circumferences which are left, joining straight lines, setting up on each of the triangles pyramids of equal height with the cone, and doing this repeatedly, we shall leave some segments of the cone which are less than the solid X. X.1 Let such be left, and let them be the segments on HP, PE, EQ, QF, FR, RG, GS, and SH. Therefore the remainder, the pyramid with the polygonal base HPEQFRGS and the same height as that of the cone, is greater than the solid O. Now inscribe in the circle ABCD the polygon DTAUBVCW similar and similarly situated to the polygon HPEQFRGS, and on it set up a pyramid of equal height with the cone AL. Since then the square on AC is to the square on EG as the polygon DTAUBVCW is to the polygon HPEQFRGS, while the square on AC is to the square on EG as the circle ABCD is to the circle EFGH, therefore the circle ABCD is to the circle EFGH as the polygon DTAUBVCW is to the polygon HPEQFRGS. XII.1 XII.2 But the circle ABCD is to the circle EFGH as the cone AL is to the solid 0, and the polygon DTAUBVCW is to the polygon HPEQFRGS as the pyramid with the polygonal base DTAUBVCW and the vertex L is to the pyramid with the polygonal base HPEQFRGS and the vertex N. XII.6 Therefore the cone AL is to the solid O as the pyramid with the polygonal base DTAUBVCW and vertex L is to the pyramid with the polygonal base HPEQFRGS and vertex N. Therefore, alternately the cone AL is to the pyramid in it as the solid O is to the pyramid in the cone EN. V.11 V.16 But the cone AL is greater than the pyramid in it, therefore the solid O is also greater than the pyramid in the cone EN. But it is also less, which is absurd. Therefore the cone AL is not to any solid less than the cone EN as the circle ABCD is to the circle EFGH. Similarly we can prove that neither is the cone EN to any solid less than the cone AL as the circle EFGH is to the circle ABCD. I say next that neither is the cone AL to any solid greater than the cone EN as the circle ABCD is to the circle EFGH. For, if possible, let it be in that ratio to a greater solid O. Therefore, inversely the circle EFGH is to the circle ABCD as the solid O is to the cone AL. But the solid O is to the cone AL as the cone EN to some solid less than the cone AL, therefore the circle EFGH is to the circle ABCD as the cone EN is to some solid less than the cone AL, which was proved impossible. Therefore the cone AL is not to any solid greater than the cone EN as the circle ABCD is to the circle EFGH. But it was proved that neither is it in this ratio to a less solid, therefore the circle ABCD is to the circle EFGH as the cone AL is to the cone EN. But the cone is to the cone as the cylinder is to the cylinder, for each is triple each. Therefore the circle ABCD is to the circle EFGH as are the cylinders on them of equal height. XII.10 Therefore, cones and cylinders of the same height are to one another as their bases. Q.E.D. Use of this proposition This proposition is used in the proofs of XII.13 and XII.14 when the cylinders under question have the same height and equal bases, and in the proof of XII.15 for cylinders of different heights. Next proposition: XII.12 Previous: XII.10 Book XII introduction © 1997 D.E.Joyce Clark University Proposition 12 Similar cones and cylinders are to one another in triplicate ratio of the diameters of their bases. Let there be similar cones and cylinders, let the circles ABCD and EFGH be their bases, BD and FH the diameters of the bases, and KL and MN the axes of the cones and cylinders. I say that the cone with circular base ABCD and vertex L has to the cone with circular base EFGH and vertex N the ratio triplicate of that which BD has to FH. For, if the cone ABCDL does not have to the cone EFGHN the ratio triplicate of that which BD has to FH, then the cone ABCDL has that triplicate ratio either to some solid less than the cone EFGHN or to a greater. First, let it have that triplicate ratio to a less solid O. Inscribe the square EFGH in the circle EFGH. Therefore the square EFGH is greater than the half of the circle EFGH. IV.6 Now set up on the square EFGH a pyramid with the same vertex as the cone. Therefore the pyramid so set up is greater than the half part of the cone.Bisect the circumferences EF, FG, GH, and HE at the points P, Q, R, and S, and join EP, PF, FQ, QG, GR, RH, HS, and SE. Therefore each of the triangles EPF, FQG, GRH, and HSE is also greater than the half part of that segment of the circle EFGH about it. Now set up on each of the triangles EPF, FQG, GRH, and HSE a pyramid with the same vertex as the cone. Therefore each of the pyramids so set up is also greater than the half part of that segment of the cone about it. Thus, bisecting the circumferences so left, joining straight lines, setting up on each of the triangles pyramids with the same vertex as the cone, and doing this repeatedly, we shall leave some segments of the cone which are less than the excess by which the cone EFGHN exceeds the solid O. X.1 Let such be left, and let them be the segments on EP, PF, FQ, QG, GR, RH, HS, and SE. Therefore the remainder, the pyramid with the polygonal base EPFQGRHS and vertex N, is greater than the solid O. Now inscribe in the circle ABCD the polygon ATBUCVDW similar and similarly situated to the polygon EPFQGRHS, and set up on the polygon ATBUCVDW a pyramid with the same vertex as the cone. Let LBT be one off the triangles containing the pyramid with polygonal base ATBUCVDW and vertex L, and let NFP be one of the triangles containing the pyramid with polygonal base EPFQGRHS and vertex N. Join KT and MP. Now, since the cone ABCDL is similar to the cone EFGHN, therefore BD is to FH as the axis KL is to the axis MN. XI.Def.24 But BD is to FH as BK is to FM, therefore BK is to FM as KL to MN. And, alternately BK is to KL as FM is to MN. V.16 And the sides are proportional about equal angles, namely the angles BKL and FMN, therefore the triangle BKL is similar to the triangle FMN. VI.6 Again, since BK is to KT as FM is to MP, and they are about equal angles, namely the angles BKT and FMP, for whatever part the angle BKT is of the four right angles at the center K, it is the same part as the angle FMP of the four right angles at the center M. Then, since the sides are proportional about equal angles, therefore the triangle BKT is similar to the triangle FMP. VI.6 Again, since it was proved that BK is to KL as FM is to MN, while BK equals KT, and FM equals PM, therefore TK is to KL as PM is to MN. And the sides are proportional about equal angles, namely the angles TKL and PMN, for they are right, therefore the triangle LKT is similar to the triangle NMP. VI.6 And since the triangles LKB and NMF are similar, therefore LB is to BK as NF is to FM. And since the triangles BKT and FMP are similar, therefore KB is to BT as MF is to FP. Therefore, ex aequali, LB is to BT as NF is to FP. VI.6 Again, since the triangles LTK and NPM are similar, therefore LT is to TK as NP is to PM, and since the triangles TKB and PMF are similar, therefore KT is to TB as MP is to PF. Therefore, ex aequali, LT is to TB as NP is to PF. VI.6 But it was also proved that TB is to BL as PF is to FN. Therefore, ex aequali, TL is to LB as PN is to NF. V.22 Therefore in the triangles LTB and NPF the sides are proportional. Therefore the triangles LTB and NPF are equiangular, hence they are also similar. VI.5 VI.Def.1 Therefore the pyramid with triangular base BKT and vertex L is similar to the pyramid with triangular base FMP and vertex N, for they are contained by similar planes equal in multitude. XI.Def.9 But similar pyramids with triangular bases are to one another in the triplicate ratio of their corresponding sides. XII.8 Therefore the pyramid BKTL has to the pyramid FMPN the ratio triplicate of that which BK has to FM. Similarly, by joining straight lines from A, W, D, V, C, and U to K, and from E, S, H, R, G, and Q to M, and setting up on each of the triangles pyramids with the same vertex as the cones, we can prove that each of the similarly arranged pyramids also has to each similarly arranged pyramid the ratio triplicate of that which the corresponding side BK has to the corresponding side FM, that is, which BD has to FH. And one of the antecedents is to one of the consequents as all the antecedents are to all the consequents, therefore the pyramid BKTL is to the pyramid FMPN as the whole pyramid with polygonal base ATBUCVDW and vertex L is to the whole pyramid with polygonal base EPFQGRHS and vertex N, hence the pyramid with base ATBUCVDW and vertex L has to the pyramid with polygonal base EPFQGRHS and vertex N the ratio triplicate of that which BD has to FH. V.12 But, by hypothesis, the cone with circular base ABCD and vertex L also has to the solid O the ratio triplicate of that which BD has to FH, therefore the cone with circular base ABCD and vertex L is to the solid 0 as the pyramid with polygonal base ATBUCVDW and vertex L is to the pyramid with polygonal base EPFQGRHS and vertex N. Therefore, alternately the cone with circular base ABCD and vertex L is to the pyramid contained in it with polygonal base ATBUCVDW and vertex L as the solid O is to the pyramid with the polygonal base EPFQGRHS and vertex N. V.16 But the said cone is greater than the pyramid in it, for it encloses it. Therefore the solid O is also greater than the pyramid with polygonal base EPFQGRHS and vertex N. But it is also less, which is impossible. Therefore the cone with circular base ABCD and vertex L does not have to any solid less than the cone of with circular base EFGH and vertex N the ratio triplicate of that which BD has to FH. Similarly we can prove that neither has the cone EFGHN to any solid less than the cone ABCDL the ratio triplicate of that which FH has to BD. I say next that neither has the cone ABCDL to any solid greater than the cone EFGHN the ratio triplicate of that which BD has to FH. For, if possible, let it have that ratio to a greater solidO. Therefore, inversely, the solid O has to the cone ABCDL the ratio triplicate of that which FH has to BD. But the solid O is to the cone ABCDL as the cone EFGHN is to some solid less than the cone ABCDL. Therefore the cone EFGHN also has to some solid less than the cone ABCDL the ratio triplicate of that which FH has to BD, which was proved impossible. Therefore the cone ABCDL does not have to any solid greater than the cone EFGHN the ratio triplicate of that which BD has to FH. But it was proved that neither has it this ratio to a less solid than the cone EFGHN. Therefore the cone ABCDL has to the cone EFGHN the ratio triplicate of that which BD has to FH. But the cone is to the cone as the cylinder is to the cylinder, for the cylinder with the same base as the cone and of equal height with it is triple the cone. Therefore the cylinder also has to the cylinder the ratio triplicate of that which BD has to FH. XII.10 Therefore, similar cones and cylinders are to one another in triplicate ratio of the diameters of their bases. Q.E.D. An alternate proof would use the previous proposition (cylinders of the same height are proportional to their bases) and XII.14 (cylinders on equal bases are proportional to their heights), which doesn't depend on this one. Instead Euclid proves this proposition afresh in a manner like that of the previous proposition but necessarily more complicated. This proposition is not used in later ones. Next proposition: XII.13 Previous: XII.11 Book XII introduction © 1997 D.E.Joyce Clark University Proposition 13 If a cylinder is cut by a plane parallel to its opposite planes, then the cylinder is to the cylinder as the axis is to the axis. Let the cylinder AD be cut by the plane GH parallel to the opposite planes AB and CD. Let the plane GH meet the axis at the point K. I say that the cylinder BG is to the cylinder GD as the axis EK is to the axis KF. Produce the axis EF in both directions to the points L and M. Set out any number whatever of axes EN and NL equal to the axis EK, and any number whatever FO and OM equal to FK. Construct the cylinder PW on the axis LM with the circles PQ and VW as bases. Carry the planes through the points N and O parallel to AB and CD and to the bases of the cylinder PW, and let them produce the circles RS and TU about the centers N, O. Then, since the axes LN, NE, and EK equal one another, therefore the cylinders QR, RB, and BG are to one another as their bases. XII.11 But the bases are equal, therefore the cylinders QR, RB, and BG also equal one another. Since then the axes LN, NE, and EK equal one another, and the cylinders QR, RB, and BG also equal one another, and the multitude of the former equals the multitude of the latter, therefore, the multiple the axis KL is of the axis EK is the same multiple the cylinder QG is of the cylinder GB. For the same reason, the multiple the axis MK is of the axis KF is the same multiple the cylinder WG is of the cylinder GD. And, if the axis KL equals the axis KM, then the cylinder QG also equals the cylinder GW; if the axis is greater than the axis, then the cylinder is also greater than the cylinder; and if less, less. Thus, there being four magnitudes, the axes EK and KF and the cylinders BG and GD, there have been taken equimultiples of the axis EK and of the cylinder BG, namely the axis LK and the cylinder QG, and equimultiples of the axis KF and of the cylinder GD, namely the axis KM and the cylinder GW, and it has been proved that, if the axis KL is in excess of the axis KM, the cylinder QG is also in excess of the cylinder GW; if equal, equal; and if less, less. Therefore the axis EK is to the axis KF as the cylinder BG is to the cylinder GD. V.Def.5 Therefore, if a cylinder is cut by a plane parallel to its opposite planes, then the cylinder is to the cylinder as the axis is to the axis. Q.E.D. Use of this proposition This proposition is preliminary to the next in which it is shown that cylinders on equal bases are proportional to their heights. It is also used in the proposition following that. Next proposition: XII.14 Previous: XII.12 Book XII introduction © 1997 D.E.Joyce Clark University Proposition 14 Cones and cylinders on equal bases are to one another as their heights. Let EB and FD be cylinders on equal bases, the circles AB and CD. I say that the cylinder EB is to the cylinder FD as the axis GH is to the axis KL. Produce the axis KL to the point N, make LN equal to the axis GH, and construct the cylinder CM about LN as the axis. I.3 Then, since the cylinders EB and CM are of the same height, therefore they are to one another as their bases. XII.11 But the bases equal one another, therefore the cylinders EB and CM are also equal. And, since the cylinder FM has been cut by the plane CD parallel to its opposite planes, therefore the cylinder CM is to the cylinder FD as the axis LN is to the axis KL. XII.13 But the cylinder CM equals the cylinder EB, and the axis LN equals the axis GH, therefore the cylinder EB is to the cylinder FD as the axis GH is to the axis KL. But the cylinder EB is to the cylinder FD as the cone ABG is to the cone CDK. Therefore the axis GH is to the axis KL as the cone ABG is to the cone CDK and as the cylinder EB is to the cylinder FD. XII.10 Therefore, cones and cylinders on equal bases are to one another as their heights. Q.E.D. Back in proposition XII.11 cones and cylinders were shown to be proportional to their bases, and this proposition shows that they are proportional to their heights. The next proposition relates the volume to the base and height in a different way by fixing the volume so that the base and height are reciprocally proportional. Next proposition: XII.15 Previous: XII.13 Book XII introduction © 1997 D.E.Joyce Clark University Proposition 15 In equal cones and cylinders the bases are reciprocally proportional to the heights; and those cones and cylinders in which the bases are reciprocally proportional to the heights are equal. Let there be equal cones and cylinders with the circular bases ABCD and EFGH. Let AC and EG be the diameters of the bases, and KL and MN the axes, which are also the heights of the cones or cylinders Complete the cylinders AO and EP. I say that in the cylinders AO and EP the bases are reciprocally proportional to the heights, that is, the base ABCD is to the base EFGH as the height MN is to the height KL. For the height LK is either equal to the height MN or unequal. First, let it be equal. Now the cylinder AO also equals the cylinder EP. But cones and cylinders of the same height are to one another as their bases, therefore the base ABCD equals the base EFGH. XII.11 Hence, reciprocally, the base ABCD is to the base EFGH as the height MN is to the height KL. Next, let the height LK be unequal to MN, and let MN be greater. Cut QN off the height MN equal to KL. Through the point Q let the cylinder EP be cut by the plane TUS parallel to the planes of the circles EFGH and RP. Erect the cylinder ES from the circle EFGH as base and with height NQ. Now, since the cylinder AO equals the cylinder EP, therefore the cylinder AO is to the cylinder ES as the cylinder EP is to the cylinder ES. V.7 But the cylinder AO is to the cylinder ES as the base ABCD is to the base EFGH, for the cylinders AO and ES are of the same height. And the cylinder EP is to the cylinder ES as the height MN is to the height QN, for the cylinder EP is cut by a plane parallel to its opposite planes. Therefore the base ABCD is to the base EFGH as the height MN is to the height QN. XII.11 XII.13 V.11 But the height QN equals the height KL, therefore the base ABCD is to the base EFGH as the height MN is to the height KL. Therefore in the cylinders AO and EP the bases are reciprocally proportional to the heights. Next, in the cylinders AO and EP let the bases be reciprocally proportional to the heights, that is, as the base ABCD is to the base EFGH, so let the height MN be to the height KL. I say that the cylinder AO equals the cylinder EP. With the same construction, since the base ABCD is to the base EFGH as the height MN is to the height KL, and the height KL equals the height QN, therefore the base ABCD is to the base EFGH as the height MN is to the height QN. But the base ABCD is to the base EFGH as the cylinder AO is to the cylinder ES, for they have the same height. And the height MN is to QN as the cylinder EP is to the cylinder ES, therefore the cylinder AO is to the cylinder ES as the cylinder EP is to the cylinder ES. XII.11 XII.13 V.11 Therefore the cylinder AO equals the cylinder EP. V.9 And the same is true for the cones also. XII.10 Therefore, in equal cones and cylinders the bases are reciprocally proportional to the heights; and those cones and cylinders in which the bases are reciprocally proportional to the heights are equal. Q.E.D. This proof of this proposition applies to a more general situation than cones and cylinders. Whenever a magnitude x is proportional to two other magnitudes y and z, that is to say when y is fixed then x is proportional to z and when x is fixed then y is proportional to z, it follows that when x is fixed then y and z are reciprocally proportional. This proposition completes the theory of the volumes of cones and cylinders. The remaining three propositions in this book concern the volume of spheres. Next proposition: XII.16 Previous: XII.14 Book XII introduction © 1997 D.E.Joyce Clark University Proposition 16 Given two circles about the same center, to inscribe in the greater circle an equilateral polygon with an even number of sides which does not touch the lesser circle. Let ABCD and EFGH be the two given circles about the same center K. It is required to inscribe in the greater circle ABCD an equilateral polygon with an even number of sides which does not touch the circle EFGH. Draw the straight line BKD through the center K, and draw GA from the point G at right angles to the straight line BD, and carry it through to C. I.11 Therefore AC touches the circle EFGH. III.16,Cor Then, bisecting the circumference BAD, bisecting the half of it, and doing this repeatedly, we shall leave a circumference less than AD. X.1 Let such be left, and let it be LD. Draw LM from L perpendicular to BD, and carry it through to N. Join LD and DN. I.12 Therefore LD equals DN. III.3 I.4 Now, since LN is parallel to AC, and AC touches the circle EFGH, therefore LN does not touch the circle EFGH. Therefore LD and DN are far from touching the circle EFGH. If, then, we fit into the circle ABCD straight lines equal to the straight line LD and placed repeatedly, then there is inscribed in the circle ABCD an equilateral polygon with an even number of sides which does not touch the lesser circle EFGH. Q.E.F. The purpose of this construction is to separate the two concentric circles by a polygon so that a threedimensional construction can be made in the next proposition to separate two concentric spheres. This construction will actually generate a polygon whose number of sides is a power of 2 such as 8, 16, 32, etc. The next proposition requires a polygon where the number of sides is not just even, but a multiple of 4, which conveniently this construction generates. Furthermore, the next proposition requires not just that the polygon not touch the inner circle, but the chords joining alternate vertices also not touch the inner circle, which again this construction satisfies. Next proposition: XII.17 Previous: XII.15 Book XII introduction © 1997 D.E.Joyce Clark University Proposition 17 Given two spheres about the same center, to inscribe in the greater sphere a polyhedral solid which does not touch the lesser sphere at its surface. Let there be two spheres about the same center A. It is required to inscribe in the greater sphere a polyhedral solid which does not touch the lesser sphere at its surface. Cut the spheres by any plane through the center. Then the sections are circles, for as a sphere is produced by the diameter remaining fixed and the semicircle being carried round it, hence, in whatever position we conceive the semicircle to be, the plane carried through it produces a circle on the circumference of the sphere. XI.Def.14 And it is clear that this circle is the greatest possible, for the diameter of the sphere, which is of course the diameter both of the semicircle and of the circle, is greater than all the straight lines drawn across in the circle or the sphere. Let then BCDE be the circle in the greater sphere, and FGH the circle in the lesser sphere. Draw two diameters in them, BD and CE, at right angles to one another. I.11 Then, given the two circles BCDE and FGH about the same center, inscribe in the greater circle BCDE an equilateral polygon with an even number of sides which does not touch the lesser circle FGH. XII.16 Let BK, KL, LM, and ME be its sides in the quadrant BE. Join KA and carry it through to N. Set AO up from the point A at right angles to the plane of the circle BCDE, and let it meet the surface of the sphere atO. XI.12 Carry planes through AO and each of the straight lines BD and KN. They make the greatest circles on the surface of the sphere for the reason stated. Let them make such, and in them let BOD and KON be the semicircles on BD and KN. Now, since OA is at right angles to the plane of the circle BCDE, therefore all the planes through OA are also at right angles to the plane of the circle BCDE. Hence the semicircles BOD and KON are also at right angles to the plane of the circle BCDE. XI.18 And, since the semicircles BED, BOD, and KON are equal, for they are on equal diameters BD and KN, therefore the quadrants BE, BO, and KO equal one another. Therefore there are as many straight lines in the quadrants BO and KO equal to the straight lines BK, KL, LM, and ME as there are sides of the polygon in the quadrant BE. Inscribe them as BP, PQ, QR, and RO and as KS, ST, TU, and UO. Join SP, TQ, UR, and draw perpendiculars from P and S to the plane of the circle BCDE. IV.1 XI.11 These will fall on BD and KN, the common sections of the planes, for the planes of BOD and KON are also at right angles to the plane of the circle BCDE. cf.XI.Def.4 Let them so fall as PV and SW, and join WV. Now since, in the equal semicircles BOD and KON, equal straight lines BP and KS have been cut off, and the perpendiculars PV and SW have been drawn, therefore PV equals SW, and BV equals KW. III.27 I.26 But the whole BA also equals the whole KA, therefore the remainder VA also equals the remainder WA. Therefore BV is to VA as KW is to WA. Therefore WV is parallel to KB. VI.2 And, since each of the straight lines PV and SW is at right angles to the plane of the circle BCDE, therefore PV is parallel to SW. XI.6 But it was also proved equal to it, therefore WV and SP are equal and parallel. I.33 And, since WV is parallel to SP, and WV is parallel to KB, therefore SP is also parallel to KB. XI.9 And BP and KS join their ends, therefore the quadrilateral KBPS is in one plane, for if two straight lines are parallel, and points are taken at random on each of them, then the straight line joining the points is in the same plane with the parallels. For the same reason each of the quadrilaterals SPQT and TQRU is also in one plane. XI.7 But the triangle URO is also in one plane. If then we join straight lines from the points P, S, Q, T, R, and U to A, then there will be constructed a certain polyhedral solid figure between the circumferences BO and KO consisting of pyramids of which the quadrilaterals KBPS, SPQT, and TQRU and the triangle URO are the bases and the point A is the vertex. XI.2 And, if we make the same construction in the case of each of the sides KL, LM, and ME as in the case of BK, and further, in the case of the remaining three quadrants, then there will be constructed a certain polyhedral figure inscribed in the sphere and contained by pyramids, of which the said quadrilaterals and the triangle URO, and the others corresponding to them, are the bases and the point A is the vertex. I say that the said polyhedron does not touch the lesser sphere at the surface on which the circle FGH is. Draw AX from the point A perpendicular to the plane of the quadrilateral KBPS, and let it meet the plane at the point X. Join XB and XK. XI.11 Then, since AX is at right angles to the plane of the quadrilateral KBPS, therefore it is also at right angles to all the straight lines which meet it and are in the plane of the quadrilateral. Therefore AX is at right angles to each of the straight lines BX and XK. XI.Def.3 And, since AB equals AK, therefore the square on AB equals the square on AK. And the sum of the squares on AX and XB equals the square on AB, for the angle at X is right, and the sum of the squares on AX and XK equals the square on AK. I.47 Therefore the sum of the squares on AX and XB equals the sum of the squares on AX and XK. Subtract the square on AX from each, therefore the remainder, the square on BX, equals the remainder, the square on XK. Therefore BX equals XK. Similarly we can prove that the straight lines joined from X to P and S are equal to each of the straight lines BX and XK. Therefore the circle with center X and radius on of the straight lines XB or XK passes through P and S also, and KBPS is a quadrilateral in a circle. Now, since KB is greater than WV, and WV equals SP, therefore KB is greater than SP. But KB equals each of the straight lines KS and BP, therefore each of the straight lines KS and BP is greater than SP. And, since KBPS is a quadrilateral in a circle, and KB, BP, and KS are equal, and PS less, and BX is the radius of the circle, therefore the square on KB is greater than double the square on BX. Draw KZ from K perpendicular to BV. I.12 Then, since BD is less than double DZ, and BD is to DZ as the rectangle DB by BZ is to the rectangle DZ by ZB, therefore if a square is described on BZ and the parallelogram on ZD is completed, then the rectangle DB by BZ is also less than double the rectangle DZ by ZB. And, if KD is joined, then the rectangle DB by BZ equals the square on BK, and the rectangle DZ by ZB equals the square on KZ. Therefore the square on KB is less than double the square on KZ. I.46 III.31 VI.18,Cor But the square on KB is greater than double the square on BX, therefore the square on KZ is greater than the square on BX. And, since BA equals KA, therefore the square on BA equals the square on AK. And the sum of the squares on BX and XA equals the square on BA, and the sum of the squares on KZ and ZA equals the square on KA, therefore the sum of the squares on BX and XA equals the sum of the squares on KZ and ZA, and of these the square on KZ is greater than the square on BX, therefore the remainder, the square on ZA, is less than the square on XA. I.47 Therefore AX is greater than AZ. Therefore AX is much greater than AG. And AX is the perpendicular on one base of the polyhedron, and AG on the surface of the lesser sphere, hence the polyhedron does not touch the lesser sphere on its surface. Therefore, given two spheres about the same center, a polyhedral solid has been inscribed in the greater sphere which does not touch the lesser sphere at its surface. Corollary. But if in another sphere a polyhedral solid is inscribed similar to the solid in the sphere BCDE, then the polyhedral solid in the sphere BCDE has to the polyhedral solid in the other sphere the ratio triplicate of that which the diameter of the sphere BCDE has to the diameter of the other sphere. For, the solids being divided into their pyramids similar in multitude and arrangement, the pyramids will be similar. But similar pyramids are to one another in the triplicate ratio of their corresponding sides, therefore the pyramid with the quadrilateral base KBPS and the vertex A has to the similarly arranged pyramid in the other sphere the ratio triplicate of that which the corresponding side has to the corresponding side, that is, of that which the radius AB of the sphere about A as center has to the radius of the other sphere. XII.18,Cor. Similarly each pyramid of those in the sphere about A as center has to each similarly arranged pyramid of those in the other sphere the ratio triplicate of that which AB has to the radius of the other sphere. And one of the antecedents is to one of the consequents as all the antecedents are to all the consequents, hence the whole polyhedral solid in the sphere about A as center has to the whole polyhedral solid in the other sphere the ratio triplicate of that which AB has to the radius of the other sphere, that is, of that which the diameter BD has to the diameter of the other sphere. V.12 Q.E.F. The purpose of this proposition and its corollary is to separate concentric spheres so that it can be proved in the next proposition XII.18 that spheres are to each other in triplicate ratios of their diameters. The argument that the intersection of a sphere and a plane through its center is a circle is weak. It has not been shown that the sphere is generated by taking any of its diameters and rotating a semicircle on that diameter about the diameter. Even the very concept of rotation about an axis has not been formalized. Next proposition: XII.18 Previous: XII.16 Book XII introduction © 1997 D.E.Joyce Clark University Proposition 18 Spheres are to one another in triplicate ratio of their respective diameters. Let the ABC and DEF be spheres, and let BC and EF be their diameters. I say that the sphere ABC has to the sphere DEF the ratio triplicate of that which BC has to EF. For, if the sphere ABC has not to the sphere DEF the ratio triplicate of that which BC has to EF, then the sphere ABC has either to some less sphere than the sphere DEF, or to a greater, the ratio triplicate of that which BC has to EF. First, let it have that ratio to a less sphere GHK. Let DEF be about the same center with GHK. Inscribe in the greater sphere DEF a polyhedral solid which does not touch the lesser sphere GHK at its surface. XII.17 Also inscribe in the sphere ABC a polyhedral solid similar to the polyhedral solid in the sphere DEF. Therefore the polyhedral solid in ABC has to the polyhedral solid in DEF the ratio triplicate of that which BC has to EF. XII.17,Cor. But the sphere ABC also has to the sphere GHK the ratio triplicate of that which BC has to EF, therefore the sphere ABC is to the sphere GHK as the polyhedral solid in the sphere ABC is to the polyhedral solid in the sphere DEF, and, alternately the sphere ABC is to the polyhedron in it as the sphere GHK is to the polyhedral solid in the sphere DEF. V.16 But the sphere ABC is greater than the polyhedron in it, therefore the sphere GHK is also greater than the polyhedron in the sphere DEF. V.14 But it is also less, for it is enclosed by it. Therefore the sphere ABC has not to a less sphere than the sphere DEF the ratio triplicate of that which the diameter BC has to EF. Similarly we can prove that neither has the sphere DEF to a less sphere than the sphere ABC the ratio triplicate of that which EF has to BC. I say next that neither has the sphere ABC to any greater sphere than the sphere DEF the ratio triplicate of that which BC has to EF. For, if possible, let it have that ratio to a greater, LMN. Therefore, inversely, the sphere LMN has to the sphere ABC the ratio triplicate of that which the diameter EF has to the diameter BC. But, since LMN is greater than DEF, therefore the sphere LMN is to the sphere ABC as the sphere DEF is to some less sphere than the sphere ABC, as was before proved. XII.2,Lemma Therefore the sphere DEF also has to some less sphere than the sphere ABC the ratio triplicate of that which EF has to BC, which was proved impossible. Therefore the sphere ABC has not to any sphere greater than the sphere DEF the ratio triplicate of that which BC has to EF. But it was proved that neither has it that ratio to a less sphere. Therefore the sphere ABC has to the sphere DEF the ratio triplicate of that which BC has to EF. Therefore, spheres are to one another in triplicate ratio of their respective diameters. Q.E.D. This proposition completes Book XII. Although this is an important proposition, it is just the beginning of the study of volumes of spheres. The arguments given in this proof are fairly convincing that any two similar solids are to each other in triplicate ratio of their linear parts. One difficulty is defining just what similar solids are. The volume of a sphere Euclid proved in proposition XII.10 that the cone with the same base and height as a cylinder was one third of the cylinder, but he could not find the ratio of a sphere to the circumscribed cylinder. In the century after Euclid, Archimedes solved this problem as well as the much more difficult problem of the surface area of a sphere. He showed that the ratio of the sphere to the cylinder is 4:3. Since the volume of the cylinder is proportional to its base and its height, it follows that the volumes of spheres, cylinders, and cones can be found in terms of areas of circles. In algebraic terms, if we let pi stand for the ratio of a circle to the square on its radius, then the volume of a cylinder of radius r and height h is pi r2h; the volume of an inscribed cone is pi r2h/3; and the volume of a sphere of radius r is 4 pi r3/3. Next book: Book XIII Previous: XII.17 Book XII introduction © 1997 D.E.Joyce Clark University Definitions 1 and 2 Def. 1. A solid is that which has length, breadth, and depth. Def. 2. A face of a solid is a surface. The 28 definitions at the beginning of Book XI serve Books XII and XIII as well. The first two definitions correspond to definitions I.Def.2 and I.Def.3 for a line and its ends, and definitions I.Def.5 and I.Def.6 for a surface and its edges. Some examples of solids that appear in Books XI through XII are parallelepipedal solids (see proposition XI.24 and the following propositions), prisms (XI.Def.13 and proposition XI.39), pyramids (XI.Def.12, XII.3 and the following propositions), cones and cylinders (XI.Def.18 through XI.Def.24, XII.10 and the following propositions), spheres (XI.Def.14 through XI.Def.17, and propositions XII.17 and XII.18), octahedra (XI.Def.26, XIII.14), cubes (XI.Def.25, XIII.15), icosahedra (XI.Def.27, XIII.16), and dodecahedra (XI.Def.28, XIII.17). Next definition: XI.Def.3-5 Book XI introduction © 1998 D.E.Joyce Clark University Definitions 3 through 5 Def. 3. A straight line is at right angles to a plane when it makes right angles with all the straight lines which meet it and are in the plane. Def. 4. A plane is at right angles to a plane when the straight lines drawn in one of the planes at right angles to the intersection of the planes are at right angles to the remaining plane. Def. 5. The inclination of a straight line to a plane is, assuming a perpendicular drawn from the end of the straight line which is elevated above the plane to the plane, and a straight line joined from the point thus arising to the end of the straight line which is in the plane, the angle contained by the straight line so drawn and the straight line standing up. Although definition 3 states that a line needs to be at right angles with all of the straight lines which meet it and lie in the plane, proposition XI.4 states that it is only necessary that a straight line be at right angles to two lines in the plane in order that it be at right angles to all the rest. There is an implicit assumption in definition 3 as it speaks of a straight line making right angles with straight lines which meet it and are in the plane. The concept of two lines making a right angle assumes that the two sides of the angles lie in one plane, that is, that two intersecting lines lie in a plane, a statement that is supposedly verified in proposition XI.2. The concept of a line being perpendicular to a plane is central to solid geometry. It is developed and used in many propositions in Book XI, starting with XI.4. There is also an implicit assumption in definition 4, namely that the intersection of the two planes is a straight line, a statement that is supposedly verified in proposition XI.3. The concept of planes perpendicular to planes first appears inproposition XI.18 which states that if one straight line drawn in one of the planes is at right angles to the other plane, then the two planes are at perpendicular. Definition 5 is meant to define the inclination (angle) between a line and a plane as the angle between that line and the projection of it in the plane. This requires that there is a line at right angles to a plane from a point not on the plane which is assured by proposition XI.11. It also requires that the angle constructed in the definition is independent of the construction. Next defintion: XI.Def.6-8 Previous: XI.Def.1-2 Book XI introduction © 1998, 2002 D.E.Joyce Clark University Definitions 6 through 8 Def. 6. The inclination of a plane to a plane is the acute angle contained by the straight lines drawn at right angles to the intersection at the same point, one in each of the planes. Def. 7. A plane is said to be similarly inclined to a plane as another is to another when the said angles of the inclinations equal one another. Def. 8. Parallel planes are those which do not meet. As the previous definition requires certain assumptions, so does definition 6. It assumes that any two such acute angles are equal, something Euclid does not prove but could have in the course of Book XI. Definition 8 is analogous to definition I.23 for parallel lines in a plane. There is no proposition in Book XI which states that parallelism of planes is a transitive relation, but that is not difficult to prove given the rest of the propositions in the book. The first appearance of parallel planes is in proposition XI.14. When two planes are not parallel, then, by this definition, they intersect. Proposition XI.3 proclaims that this intersection is a straight line. Note that it is not defined when a line is parallel to a plane, but that would be when they don't meet. Next defintion: XI.Def.9-10 Previous: XI.Def.3-5 Book XI introduction © 1998, 2002 D.E.Joyce Clark University Definitions 9 and 10 Def. 9. Similar solid figures are those contained by similar planes equal in multitude. Def. 10. Equal and similar solid figures are those contained by similar planes equal in multitude and magnitude. While definition 9 defines similar solid figures, definition 10 describes what is commonly called "congruent" solid figures. Euclid uses "equal and similar" plane figures for "congruent" plane figures in these later books, but that could not be done for plane figures before Book VI where similarity of plane figures was defined. These definitions are incomplete as was VI.Def.1 for similar rectilinear figures. The notion of similarity for plane figures implicitly assumed a correspondence of consecutive vertices and sides. This notion of similar solid figures assumes a correspondence of of adjacent edges and faces. Different solid figures can sometimes be constructed with the same faces but with different adjacencies. For example, there are two distinct ways to attach two pyramids to two of the square faces of a cube. They could be attached to opposite faces of the cube or to adjacent faces of the cube. The resulting solids both have four remaining square faces and eight new triangular faces, but the positioning of the squares and triangles is different. Heron's definition of similar solid figures, "those which are contained by equal and similarly situated planes, equal in number and magnitude," is a bit more explicit than Euclid's. It is also apparent that Euclid did not consider the possibility of concave solids and the problems they cause his definition, problems that Simson noticed. Take a cube and first add a pyramid on the outside of one square face, and second subtract the same pyramid from the inside of a square face. The resulting two solids both have five square faces and four triangular faces, and the adjacencies are the same, but they are very different solids. To eliminate this problem, further conditions must be made on the definition of similar solids. For instance, conditions on dihedral angles between faces, or conditions on distances between all corresponding pairs of vertices. Next defintion: XI.Def.11 Previous: XI.Def.6-8 Book XI introduction © 1998 D.E.Joyce Clark University Definition 11 A solid angle is the inclination constituted by more than two lines which meet one another and are not in the same surface, towards all the lines, that is, a solid angle is that which is contained by more than two plane angles which are not in the same plane and are constructed to one point. A solid angle is intended to be bounded by three or more planes meeting at a point. The solid angle at A is bounded by the three planes ABC, ACD, and ADB. The figure ABCD is a triangular pyramid. Pyramids are defined in definition XI.Def.12 coming next. The two definitions given here for solid angle are not strictly equivalent. In the first the lines mentioned are not specified as being straight, and the surfaces are not specified as being planes. In the second the surfaces are specified as being planes, and since planes meet in straight lines (XI.3), the lines must be straight. The difference, however, may well be an oversight. Next definition: XI.Def.12-13 Previous: XI0.Def.9-10 Book XI introduction © 1998 D.E.Joyce Clark University Definitions12 and 13 Def. 12. A pyramid is a solid figure contained by planes which is constructed from one plane to one point. Def. 13. A prism is a solid figure contained by planes two of which, namely those which are opposite, are equal, similar, and parallel, while the rest are parallelograms. In the diagram below, ABCD is a pyramid with vertex D and triangular base ABC. Since it's a tetrahedron, it's still a triangular pyramid when any of the other three sides is considered the base. Also, EFGHKL is a prism with opposite triangular sides EFG and HKL. Definition 12 for pyramids is rather abbreviated, but the intention is clear. Note that in definition 13 the term "equal and similar" is used for congruence of plane figures. Prisms, by that name, are first discussed in proposition XI.39. Pyramids are treated in propositions XII.3 through XII.9 in Book XII. Next definition: XI.Def.14-17 Previous: XI.Def.11 Book XI introduction © 1998 D.E.Joyce Clark University Definitions 14 through 17 Def. 14. When a semicircle with fixed diameter is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a sphere. Def. 15. The axis of the sphere is the straight line which remains fixed and about which the semicircle is turned. Def. 16. The center of the sphere is the same as that of the semicircle. Def. 17. A diameter of the sphere is any straight line drawn through the center and terminated in both directions by the surface of the sphere. There are alternative definitions for a sphere, but Euclid chose this one, perhaps, to be analogous to the definitions of cone in XI.Def.18 and cylinder in XI.Def.22. These are all defined as solids of revolution, that is, solids generated by rotating a plane figure around a straight line called the axis of revolution. Another book would probably be required to develop the theory of spheres to the degree that Euclid developed the theory of circles in Book III, but that, apparently, was not his goal. The lack of propositions is so severe that it is not even shown that any two points on the surface of a sphere are equidistant from the center. (Any point on the surface of the sphere is a point on the circumference of one of the rotated semicircles, and all the points on any of these semicircles are equidistant from the center of the semicircles.) In the illustration at the right there is a semicircle ADB with center C and diameter AB in a plane. When the semicircle is revolved around AB, a sphere results. The sphere's axis is AB, and its center is C. If E is any point on the sphere and F the antipodal point, then the line EF is a diameter of the sphere. There are very few propositions about spheres in the Elements. Proposition XII.17 allows a kind of approximation of spheres by polyhedra preliminary to proposition XII.18 on the ratio of volumes of spheres. Also, regular polyhedra are inscribed in spheres in Book XIII With so few propositions there are gaps in the proofs. For instance, in XII.17 it is claimed that the the intersection of a plane and a sphere is a circle, but a justification is lacking. Next definition: XI.Def.18-20 Previous: XI.Def.12-13 Book XI introduction © 1998, 2002 D.E.Joyce Clark University Definitions 18 through 20 Def. 18. When a right triangle with one side of those about the right angle remains fixed is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a cone. And, if the straight line which remains fixed equals the remaining side about the right angle which is carried round, the cone will be right-angled; if less, obtuseangled; and if greater, acute-angled. Def. 19. The axis of the cone is the straight line which remains fixed and about which the triangle is turned. Def. 20. And the base is the circle described by the straight line which is carried round. The right triangle ABC with right angle at A is rotated about the side AC to produce a cone. The axis of the cone is AC, and its base is the circle with center at A and radius AB. The three different kinds of cone are not used by Euclid in the Elements, but they were important in the theory of conic sections until Apollonius' work Conics. In Euclid's time conic sections were taken as the intersections of a plane at right angles to an edge (straight line from the vertex) of a cone. When the cone is acute-angled, the section is an ellipse; when right-angled, a parabola; and when obtuse-angle, a hyperbola. Even the names of these three curves were given by the kind of angle, so, for instance, Euclid knew a parabola as a "section of a rightangled cone." It was Apollonius who named them ellipse, parabola, and hyperbola. Next definition: XI.Def.21-23 Previous: XI.Def.14-17 Book XI introduction © 1998 D.E.Joyce Clark University Definitions 21 through 23 Def. 21. When a rectangular parallelogram with one side of those about the right angle remains fixed is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a cylinder. Def. 22. The axis of the cylinder is the straight line which remains fixed and about which the parallelogram is turned. Def. 23. And the bases are the circles described by the two sides opposite to one another which are carried round. Rectangle ABEC is rotated around the side AC to produce a cylinder. Its axis is AC and it has two circles for bases. The concept of cylinder has been generalized since Euclid's time as have so many ancient mathematical concepts. Euclid's cylinders are right, circular cylinders since their axes are at right angles to their bases and their bases are circles. Next definition: XI.Def.24 Previous: XI.Def.18-20 Book XI introduction © 1998 D.E.Joyce Clark University Definition 24 Similar cones and cylinders are those in which the axes and the diameters of the bases are proportional. Two cones are similar if the axis of the first is to the axis of the second as the base diameter of the first is to the base diameter of the second. Likewise, two cylinders are similar if the axis of the first is to the axis of the second as the base diameter of the first is to the base diameter of the second. It can be shown that an equivalent condition is that the vertex angles of the cones are equal. Thus, all right-angled cones are similar. Next defintion: XI.Def.25-28 Previous: XI.Def.21-23 Book XI introduction © 1998 D.E.Joyce Clark University Definitions 25 through 28 Def. 25. A cube is a solid figure contained by six equal squares. Def. 26. An octahedron is a solid figure contained by eight equal and equilateral triangles. Def. 27. An icosahedron is a solid figure contained by twenty equal and equilateral triangles. Def. 28. A dodecahedron is a solid figure contained by twelve equal, equilateral and equiangular pentagons. These are four of the five regular solids. The tetrahedron is not mentioned here since it is a certain triangular pyramid. It's called simply the "pyramid" in Book XIII. The regular tetrahedron is constructed in proposition XIII.13, the cube in XIII.15, the octahedron in XIII.14, the icosahedron in XIII.16, and the dodecahedron in XIII.17. These five are shown to be the only regular solids in proposition XIII.18. Next proposition: XI.1 Previous: XI.Def.24 Book XI introduction © 1998 D.E.Joyce Clark University Proposition 1 A part of a straight line cannot be in the plane of reference and a part in plane more elevated. For, if possible, let a part AB of the straight line ABC be in the plane of reference, and a part BC be in a plane more elevated. Then there is in the plane of reference some straight line continuous with AB in a straight line. Let it be BD. Therefore AB is a common segment of the two straight lines ABC and ABD, which is impossible, since, if we describe a circle with center B and radius AB, then the diameters cut off unequal circumferences of the circle. Therefore, a part of a straight line cannot be in the plane of reference and a part in plane more elevated. Q. E. D. Not only is the proof of this proposition unclear, so is the statement of it. The meaning of the "plane of reference" and the role it is to play in solid geometry are unclear. Is the intent of the statement that if part of a line lies in a plane, then all of it does? At least that would be a meaningful statement. The proof of this proposition is unclear for more than one reason. Before a circle with center B and radius AB can be described, a plane has to be specified in which to describe the circle. In space there are infinitely many circles that have the same center B, the same radius AB, and even contain the point A. Indeed, this possibility of many circles with with same diameter was used to define a sphere in definition XI.Def.14. The last statement about unequal circumferences is incomprehensible. The problem is that there are no postulates for solid geometry. The postulates in Book I apparently refer to an ambient plane. Certainly Post.3, "to describe a circle with any center and radius," and Post.5 (which refers to interior angles when one line crosses two others) do. Without any postulates for nonplanar geometry it is impossible for solid geometry to get off the ground. Use of this proposition This proposition is used in the proof of the next one as well as others in the last three books of the Elements. Next proposition: XI.2 Previous: XI.Def.25-28 Book XI introduction © 1997, 2002 D.E.Joyce Clark University Proposition 2 If two straight lines cut one another, then they lie in one plane; and every triangle lies in one plane. For let the two straight lines AB and CD cut one another at the point E. I say that AB and CD lie in one plane, and that every triangle lies in one plane. Take the points F and G at random on EC and EB, join CB and FG, and draw FH and GK across. I say first that the triangle ECB lies in one plane. For, if part of the triangle ECB, either FHC or GBK, is in the plane of reference, and the rest in another, then a part also of one of the straight lines EC or EB is in the plane of reference, and a part in another. But, if the part FCBG of the triangle ECB is in the plane of reference, and the rest in another, then a part also of both the straight lines EC and EB is in the plane of reference and a part in another, which was proved absurd. XI.1 Therefore the triangle ECB lies in one plane. But, in whatever plane the triangle ECB lies, each of the straight lines EC and EB also lies, and in whatever plane each of the straight lines EC and EB lies, AB and CD also lie. XI.1 Therefore the straight lines AB and CD lie in one plane; and every triangle lies in one plane. Therefore, if two straight lines cut one another, then they lie in one plane; and every triangle lies in one plane. Q. E. D. The goal of the proof in this proposition is to produce a plane for the two lines AB and CD to lie in. Yet the proof fails to produce any plane at all. Near the beginning is the phrase "the plane of reference" occurs, but there is no reference as no planes have been mentioned. As the two lines AB and CD could be placed anywhere in space, any previously conceived plane would be irrelevant to them. Postulates of some sort are needed to justify the existance of planes. One could state that three noncollinear points determine a plane. Another might be that there are four noncoplanar points. Use of this proposition This proposition is used in the proofs of propositions XI.4, XI.6, and XII.17. Next proposition: XI.3 Previous: XI.1 Book XI introduction © 1997, 2002 D.E.Joyce Clark University Proposition 3 If two planes cut one another, then their intersection is a straight line. Let two planes AB and BC cut one another, and let the line DB be their intersection. I say that the line DB is a straight line. For, if not, join the straight line DEB from D to B in the plane AB, and the straight line DFB in the plane BC. Then the two straight lines DEB and DFB have the same ends and clearly enclose an area, which is absurd. Therefore DEB and DFB are not straight lines. Similarly we can prove that neither is there any other straight line joined from D to B except DB, the intersection of the planes AB and BC. Therefore, if two planes cut one another, then their intersection is a straight line. Q. E. D. The proof of this proposition has some flaws. Postulate I (from Book I) states that a straight line can be drawn from any point to any point. It seems to be interpreted as saying that for any plane from any point in that plane to any point in that plane a straight line in that plane can be drawn. Next it is stated that the lines in those two planes "clearly enclose an area, which is absurd." But the two lines do not lie in the same plane, so it is unclear that they enclose an area. Furthermore, the statement that two straight lines cannot enclose an area did not appear in the original elements, although it was later appended to Post.1. A more serious criticism of the proof is that it fails to prove the statement of the proposition. At most it shows that if two planes intersect at more than one point, then the line that joins them also lies in their intersection. But the possibility that their intersection consists of only one point is ignored. This is important as this proposition is used in XI.5 from two planes known to intersect at one point, a line of intersection is generated. The real problem is that there is no postulate limiting space to three dimensions. In four or or more dimensions two planes may intersect in only one point. Neither Euclid nor anyone else before the nineteenth century recognized the possibility of higher dimensional geometry, but the flaws in this proof are apparent nonetheless. There are alternative postulates to limit the geometry to three dimensions. For instance, one is based on the idea that a plane divides space into two sides. Use of this proposition This proposition is used frequently, first in the proof of XI.5. Next proposition: XI.4 Previous: XI.2 Book XI introduction © 1997, 1998 D.E.Joyce Clark University Proposition 4 If a straight line is set up at right angles to two straight lines which cut one another at their common point of section, then it is also at right angles to the plane passing through them. For let a straight line EF be set up at right angles to the two straight lines AB and CD at E, the point at which the lines cut one another. I say that EF is also at right angles to the plane passing through AB and CD. Cut off AE, EB, CE, and ED equal to one another. Draw any straight line GEH across through E at random. Join AD and CB, and join FA, FG, FD, FC, FH, and FB from a point F taken at random on EF. XI.2 I.3 Now, since the two straight lines AE and ED equal the two straight lines CE and EB and contain equal angles, therefore the base AD equals the base CB, and the triangle AED equals the triangle CEB, so that the angle DAE equals the angle EBC. I.15 I.4 But the angle AEG also equals the angle BEH, therefore AGE and BEH are two triangles which have two angles equal to two angles respectively, and one side equal to one side, namely that adjacent to the equal angles, that is to say, AE equals EB. Therefore they also have the remaining sides equals to the remaining sides, that is, GE equals EH, and AG equals BH. I.15 I.26 And, since AE equals EB, while FE is common and at right angles, therefore the base FA equals the base FB. I.4 For the same reason, FC equals FD. And, since AD equals CB, and FA also equals FB, the two sides FA and AD equal the two sides FB and BC respectively, and the base FD was proved equal to the the base FC, therefore the angle FAD also equals the angle FBC. I.8 And since, again, AG was proved equal to BH, and further, FA also equal to FB, the two sides FA and AG equal the two sides FB and BH, and the angle FAG was proved equal to the angle FBH, therefore the base FG equals the base FH. I.4 Again, since GE was proved equal to EH, and EF is common, the two sides GE and EF equal the two sides HE and EF, and the base FG equals the base FH, therefore the angle GEF equals the angle HEF. I.8 Therefore each of the angles GEF and HEF is right. Therefore FE is at right angles to GH drawn at random through E. Similarly we can prove that FE also makes right angles with all the straight lines which meet it and are in the plane of reference. But a straight line is at right angles to a plane when it makes right angles with all the straight lines which meet it and are in that same plane, therefore FE is at right angles to the plane of reference. XI.Def.3 But the plane of reference is the plane through the straight lines AB and CD. Therefore FE is at right angles to the plane through AB and CD. Therefore if a straight line is set up at right angles to two straight lines which cut one another at their common point of section, then it is also at right angles to the plane passing through them. Q. E. D. This proposition says that if a line passing through a point is perpendicular to two other lines passing through that point, then it is perpendicular to all the lines which pass through that point and which lie in the plane of those two other lines. After the preceding three dubious proofs, this one is a relief. It is a little long, but it is clear. Near the beginning of the proof, proposition XI.2 is needed to conclude that the two lines AB and CD determine a plane. The line GH is to be any line that passes through E and lies in that plane. Then, by XI.2 again, the lines AD and BC lie in this plane. Use of this proposition This proposition is used frequently starting with the proof of the next proposition. Next proposition: XI.5 Previous: XI.3 Book XI introduction © 1997, 2002 D.E.Joyce Clark University Proposition 5 If a straight line is set up at right angles to three straight lines which meet one another at their common point of section, then the three straight lines lie in one plane. Let a straight line AB be set up at right angles to the three straight lines BC, BD, and BE at their intersection B. I say that BC, BD, and BE lie in one plane. For suppose that they do not, but, if possible, let BD and BE lie in the plane of reference and BC in one more elevated. Produce the plane through AB and BC. XI.3 It intersects the plane of reference in a straight line. Let the intersection be BF. Therefore the three straight lines AB, BC, and BF lie in one plane, namely that drawn through AB and BC. Now, since AB is at right angles to each of the straight lines BD and BE, therefore AB is also at right angles to the plane through BD and BE. XI.4 But the plane through BD and BE is the plane of reference, therefore AB is at right angles to the plane of reference. Thus AB also makes right angles with all the straight lines which meet it and lie in the plane of reference. XI.Def.3 But BF, which is the plane of reference, meets it, therefore the angle ABF is right. And, by hypothesis, the angle ABC is also right, therefore the angle ABF equals the angle ABC, and they lie in one plane, which is impossible. Therefore the straight line BC is not in a more elevated plane. Therefore the three straight lines BC, BC, and BE lie in one plane. Therefore, if a straight line is set up at right angles to three straight lines which meet one another at their common point of section, then the three straight lines lie in one plane. Q. E. D. This proposition is used in the proof of the next. Next proposition: XI.6 Previous: XI.4 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 6 If two straight lines are at right angles to the same plane, then the straight lines are parallel. Let the two straight lines AB and CD be at right angles to the plane of reference. I say that AB is parallel to CD. Let them meet the plane of reference at the points B and D. Join the straight line BD. Draw DE in the plane of reference at right angles to BD, and make DE equal to AB. I.11 I.3 Now, since AB is at right angles to the plane of reference, it also makes right angles with all the straight lines which meet it and lie in the plane of reference. XI.Def.3 But each of the straight lines BD and BE lies in the plane of reference and meets AB, therefore each of the angles ABD and ABE is right. For the same reason each of the angles CDB and CDE is also right. And since AB equals DE, and BD is common, therefore the two sides AB and BD equal the two sides ED and DB. And they include right angles, therefore the base AD equals the base BE I.4 And, since AB equals DE while AD equals BE, the two sides AB and BE equal the two sides ED and DA, and AE is their common base, therefore the angle ABE equals the angle EDA. I.8 But the angle ABE is right, therefore the angle EDA is also right. Therefore ED is at right angles to DA. But it is also at right angles to each of the straight lines BD and DC, therefore ED is set up at right angles to the three straight lines BD, DA, and DC at their intersection. Therefore the three straight lines BD, DA, and DC lie in one plane. XI.5 But in whatever plane DB and DA lie, AB also lies, for every triangle lies in one plane. XI.2 Therefore the straight lines AB, BD, and DC lie in one plane. And each of the angles ABD and BDC is right, therefore AB is parallel to CD. I.28 Therefore, if two straight lines are at right angles to the same plane, then the straight lines are parallel. Q.E.D. Euclid does not consider the possibility that the two lines meet the plane at one point, but that possibility can easily be eliminated. Indeed, that is the statement of proposition XI.13 which, therefore, should preceed this one. A converse of this proposition is XI.8. Use of this proposition This proposition is used in the proofs of propositions XII.17, XIII.16, and XIII.17. Next proposition: XI.7 Previous: XI.5 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 7 If two straight lines are parallel and points are taken at random on each of them, then the straight line joining the points is in the same plane with the parallel straight lines. Let AB and CD be two parallel straight lines, and let points E and F be taken at random on them respectively. I say that the straight line joining the points E and F lies in the same plane with the parallel straight lines. For suppose it is not, but, if possible, let it be in a more elevated plane as EGF. Draw a plane through EGF. Its intersection with the plane of reference is a straight line. Let it be EF. XI.3 Therefore the two straight lines EGF and EF enclose an area, which is impossible. Therefore the straight line joined from E to F is not in a plane more elevated. Therefore the straight line joined from E to F lies in the plane through the parallel straight lines AB and CD. Therefore, if two straight lines are parallel and points are taken at random on each of them, then the straight line joining the points is in the same plane with the parallel straight lines. Q. E. D. The existence of this proposition is a good argument that Euclid's definition I.Def.7 of a plane (it lies evenly with the straight lines on itself) does not mean that if two points lie in a plane, then the line joining them also lies in the plane. If it did, then this proposition would be true by definition, and no proof would be required at all. Note that this proof assumes that every line lies in a plane, a conclusion that has not been justified. Use of this proposition This proposition is used in the proof of the next as well as proposition XII.17. Next proposition: XI.8 Previous: XI.6 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 8 If two straight lines are parallel, and one of them is at right angles to any plane, then the remaining one is also at right angles to the same plane. Let AB and CD be two parallel straight lines, and let one of them, AB, be at right angles to the plane of reference. I say that the remaining one, CD, is also at right angles to the same plane. Let AB and CD meet the plane of reference at the points B and D. Join BD. Then AB, CD, and BD lie in one plane. XI.7 Draw DE in the plane of reference at right angles to BD, make DE equal to AB, and join BE, AE, and AD I.11 I.3 Now, since AB is at right angles to the plane of reference, therefore AB is also at right angles to all the straight lines which meet it and lie in the plane of reference. Therefore each of the angles ABD and ABE is right. XI.Def.3 And, since the straight line BD falls on the parallels AB and CD, therefore the sum of the angles ABD and CDB equals two right angles. I.29 But the angle ABD is right, therefore the angle CDB is also right. Therefore CD is at right angles to BC. And since AB equals DE, and BD is common, the two sides AB and BD equal the two sides ED and DB, and the angle ABD equals the angle EDB, for each is right, therefore the base AD equals the base BE. I.4 And since AB equals DE, and BE equals AD, the two sides AB and BE equal the two sides ED and DA respectively, and AE is their common base, therefore the angle ABE equals the angle EDA. I.8 But the angle ABE is right, therefore the angle EDA is also right. Therefore ED is at right angles to AD. But it is also at right angles to DB. Therefore ED is also at right angles to the plane through BD and DA. XI.4 Therefore ED also makes right angles with all the straight lines which meet it and lie in the plane through BD and DA. But DC lies in the plane through BD and DA inasmuch as AB and BD lie in the plane through BD and DA, and DC also lies in the plane in which AB and BD lie. Therefore ED is at right angles to DC, so that CD is also at right angles to DE. But CD is also at right angles to BD. Therefore CD is set up at right angles to the two straight lines DE and DB so that CD is also at right angles to the plane through DE and DB. XI.4 But the plane through DE and DB is the plane of reference, therefore CD is at right angles to the plane of reference. Therefore, if two straight lines are parallel, and one of them is at right angles to any plane, then the remaining one is also at right angles to the same plane. Q.E.D. This is a converse of proposition XI.6. This proposition is used in the proof of the next one as well as several others in this book. Next proposition: XI.9 Previous: XI.7 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 9 Straight lines which are parallel to the same straight line but do not lie in the same plane with it are also parallel to each other. Let each of the straight lines AB and CD be parallel to EF, but not in the same plane with it. I say that AB is parallel to CD. Let a point G be taken at random on EF, and from it draw GH in the plane through EF and AB at right angles to EF, and GK in the plane through EF and CD again at right angles to EF. I.11 Now, since EF is at right angles to each of the straight lines GH and GK, therefore EF is also at right angles to the plane through GH and GK. XI.4 And EF is parallel to AB, therefore AB is also at right angles to the plane through HG and GK. XI.8 For the same reason CD is also at right angles to the plane through HG and GK. Therefore each of the straight lines AB and CD is at right angles to the plane through HG and GK. But if two straight lines are at right angles to the same plane, then the straight lines are parallel. Therefore AB is parallel to CD. XI.6 Therefore, straight lines which are parallel to the same straight line but do not lie in the same plane with it are also parallel to each other. Q. E. D. Note that this proposition is the three-dimensional analogue to proposition I.30. The Varignon parallelogram of space quadrilaterals Consider a quadrilateral ABCD whose four vertices may or may not lie in a plane. Let E, F, G, and H be the midpoints of the sides AB, BC, CD, and DA, respectively. Then the quadrilateral EFGH lies in a plane and is a parallelogram, called the Varignon parallelogram. Varignon (1654-1722) showed the area of a planar quadrilateral is twice the area of this parallelogram. The proof that EFGH is a parallelogram relies on this proposition XI.9 to show the sides are parallel, since it is readily shown that both EF and HG are parallel to the line AC, and both FG and EH are parallel to the line BD. As a corollary, it follows that the lines joining the midpoints of an arbitrary quadrilateral are concurrent and bisect each other, even if the four sides of the quadrilateral do not lie in a plane. (These are the lines EG and FH which are not drawn in the diagram.) Use of this proposition This proposition is used in the proof of the next proposition as well as others in this and the next book. Next proposition: XI.10 Previous: XI.8 Book XI introduction © 1997, 1999 D.E.Joyce Clark University Proposition 10 If two straight lines meeting one another are parallel to two straight lines meeting one another not in the same plane, then they contain equal angles. Let the two straight lines AB and BC meeting one another be parallel to the two straight lines DE and EF meeting one another not in the same plane. I say that the angle ABC equals the angle DEF. Cut BA, BC, ED, and EF off equal to one another, and join AD, CF, BE, AC, and DF. I.3 Now, since BA equals and is parallel to ED, therefore AD also equals and is parallel to BE. For the same reason CF also equals and is parallel to BE. I.33 Therefore each of the straight lines AD and CF equals and is parallel to BE. But straight lines which are parallel to the same straight line and are not in the same plane with it are parallel to one another, therefore AD is parallel and equal to CF. XI.9 And AC and DF join them, therefore AC also equals and is parallel to DF. I.33 Now, since the two sides AB and BC equal the two sides DE and EF, and the base AC equals the base DF, therefore the angle ABC equals the angle DEF. I.8 Therefore, if two straight lines meeting one another are parallel to two straight lines meeting one another not in the same plane, then they contain equal angles. Q. E. D. Of course it is necessary to be careful about in which directions the lines head. If one is changed to head into the opposite direction, then the angles won't be equal but supplementary instead. Use of this proposition This proposition is used in the proofs of propositions XI.24 and XII.3. Next proposition: XI.11 Previous: XI.9 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 11 To draw a straight line perpendicular to a given plane from a given elevated point. Let A be the given elevated point, and the plane of reference the given plane. It is required to draw from the point A a straight line perpendicular to the plane of reference. Draw any straight line BC at random in the plane of reference, and draw AD from the point A perpendicular to BC. I.12 Then if AD is also perpendicular to the plane of reference, then that which was proposed is done. But if not, draw DE from the point D at right angles to BC and in the plane of reference, draw AF from A perpendicular to DE, and draw GH through the point F parallel to BC. I.11 I.12 I.31 Now, since BC is at right angles to each of the straight lines DA and DE, therefore BC is also at right angles to the plane through ED and DA. XI.4 And GH is parallel to it, but if two straight lines are parallel, and one of them is at right angles to any plane, then the remaining one is also at right angles to the same plane, therefore GH is also at right angles to the plane through ED and DA. XI.8 And GH is parallel to it, but if two straight lines are parallel, and one of them is at right angles to any plane, then the remaining one is also at right angles to the same plane, therefore GH is also at right angles to the plane through ED and DA. XI.8 Therefore GH is also at right angles to all the straight lines which meet it and are in the plane through ED and DA. XI.Def.3 But AF meets it and lies in the plane through ED and DA, therefore GH is at right angles to FA, so that FA is also at right angles to GH. But AF is also at right angles to DE, therefore AF is at right angles to each of the straight lines GH and DE. But if a straight is set up at right angles to two straight lines which cut one another at their intersection point, then it also is at right angles to the plane through them. Therefore FA is at right angles to the plane through ED and GH. XI.4 But the plane through ED and GH is the plane of reference, therefore AF is at right angles to the plane of reference. Therefore from the given elevated point A the straight line AF has been drawn perpendicular to the plane of reference. Q. E. F. In the proof, before the line AD can be drawn from the point A perpendicular to the line BC, it is necesary to know that the point and line belong to the same plane. Such a plane can be specified by taking the line BC and a line from A to any point on BC since two intersecting lines determine a plane (XI.2). Use of this proposition The construction in this proposition is used frequently in the last three books of the Elements. Next proposition: XI.12 Previous: XI.10 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 12 To set up a straight line at right angles to a given plane from a given point in it. Let the plane of reference be the given plane and A the point in it. It is required to set up from the point A a straight line at right angles to the plane of reference. From an elevated point B draw BC perpendicular to the plane of reference, and draw AD parallel to to BC through the point A. XI.11 I.31 Then since AD and BC are two parallel straight lines, and one of them, BC, is at right angles to the plane of reference, therefore the remaining one, AD, is also at right angles to the plane of reference. XI.8 Therefore AD is set up at right angles to the given plane from the point A in it. Q. E. F. This proposition, like the last, is used frequently in the rest of the Elements to construct lines perpendicular to planes. Next proposition: XI.13 Previous: XI.11 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 13 From the same point two straight lines cannot be set up at right angles to the same plane on the same side. For, if possible, from the same point A let the two straight lines AB and AC be set up at right angles to the plane of reference and on the same side. Draw a plane through BA and AC. It intersects the plane of reference in a straight line through A. Let the line be DAE. XI.3 Therefore the straight lines AB, AC, and DAE lie in one plane. And, since CA is at right angles to the plane of reference, it also makes right angles with all the straight lines which meet it and lie in the plane of reference. XI.Def.3 But DAE meets it and lies in the plane of reference, therefore the angle CAE is right. For the same reason the angle BAE is also right. Therefore the angle CAE equals the angle BAE. And they lie in one plane, which is impossible. Therefore, from the same point two straight lines cannot be set up at right angles to the same plane on the same side. Q. E. D. This proposition is used in the proof of proposition XI.19. Also, the result of this proposition is implicitly used in the proof of XI.6. Next proposition: XI.14 Previous: XI.12 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 14 Planes at right angles to the same straight line are parallel. Let any straight line AB be at right angles to each of the planes CD and EF. I say that the planes are parallel. For, if not, then they meet when produced. Let them meet. Then they intersect as a straight line. Let it be GH. XI.Def.8 XI.3 Take a point K at random on GH, and join AK and BK. Now, since AB is at right angles to the plane EF, therefore AB is also at right angle to BK which is a straight line in the plane EF produced. Therefore the angle ABK is right. For the same reason the angle BAK is also right. XI.Def.3 Thus, in the triangle ABK the sum of the two angles ABK and BAK equals two right angles, which is impossible. I.17 Therefore the planes CD and EF do not meet when produced. Therefore the planes CD and EF are parallel. XI.Def.8 Therefore, planes at right angles to the same straight line are parallel. Q. E. D. Part of this proof is unnecessary. The line GH is irrelevant. If the two points meet, then they meet at some point, so the point K might just as well be taken as a point where they meet. The proof should, however, include another case, and that is where the given line meets both given planes at a point common to both planes, so that the three points A, B, and K are all the same point. Use of this proposition This proposition is used in the proof of the next one. Next proposition: XI.15 Previous: XI.13 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 15 If two straight lines meeting one another are parallel to two straight lines meeting one another not in the same plane, then the planes through them are parallel. Let the two straight lines AB and BC meeting one another be parallel to the two straight lines DE and EF meeting one another not in the same plane. I say that the plane produced through AB and BC and the plane produced through DE and EF do not meet one another. Draw BG from the point B perpendicular to the plane through DE and EF to where it meets the plane at the point G. XI.11 Draw GH through G parallel to ED, and GK parallel to EF. I.31 Now, since BG is at right angles to the plane through DE and EF, therefore it makes right angles with all the straight lines which meet it and lie in the plane through DE and EF. XI.Def.3 But each of the straight lines GH and GK meets it and lies in the plane through DE and EF, therefore each of the angles BGH and BGK is right. And, since BA is parallel to GH, therefore the sum of the angles GBA and BGH equals two right angles. XI.9 I.29 But the angle BGH is right, therefore the angle GBA is also right. Therefore GB is at right angles to BA. For the same reason GB is also at right angles to BC. Since then the straight line GB is set up at right angles to the two straight lines BA and BC which cut one another, therefore GB is also at right angles to the plane through BA and BC. XI.4 But planes to which the same straight line is at right angles are parallel, therefore the plane through AB and BC is parallel to the plane through DE and EF. XI.14 Therefore, if two straight lines meeting one another are parallel to two straight lines meeting one another not in the same plane, then the planes through them are parallel. Q. E. D. This proposition is not used in the rest of the Elements. Next proposition: XI.16 Previous: XI.14 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 16 If two parallel planes are cut by any plane, then their intersections are parallel. Let the two parallel planes AB and CD be cut by the plane EFGH, and let EF and GH be their intersections. XI.3 I say that EF is parallel to GH. If not, then EF and GH will, when produced, meet either in the direction of F and H or in the direction of E and G. First, let them meet when produced in the direction of F and H at K. Now, since EFK lies in the plane AB, therefore all the points on EFK also lie in the plane AB. But K is one of the points on the straight line EFK, therefore K lies in the plane AB. For the same reason K also lies in the plane CD. Therefore the planes AB and CD will meet when produced. XI.1 But they do not meet, because, by hypothesis, they are parallel. Therefore the straight lines EF and GH do not meet when produced in the direction of F and H. Similarly we can prove that neither do the straight lines EF and GH meet when produced in the direction of E and G. But straight lines which do not meet in either direction are parallel. Therefore EF is parallel to GH. Therefore, if two parallel planes are cut by any plane, then their intersections are parallel. Q. E. D. This proposition is used in the proof of the next proposition as well as proposition XI.24. Next proposition: XI.17 Previous: XI.15 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 17 If two straight lines are cut by parallel planes, then they are cut in the same ratios. Let the two straight lines AB and CD be cut by the parallel planes GH, KL, and MN at the points A, E, and B, and at the points C, F, and D, respectively. I say that the straight line AE is to EB as CF is to FD. Join AC, BD, and AD. Let AD meet the plane KL at the point O. Join EO and FO. Now, since the two parallel planes KL and MN are cut by the plane EBDO, therefore their intersections EO and BD are parallel. For the same reason, since the two parallel planes GH and KL are cut by the plane AOFC, their intersections AC and OF are parallel. XI.16 And, since the straight line EO is parallel to BC, one of the sides of the triangle ABD, therefore proportionally AE is to EB as AO is to OD. Again, since the straight line FO is parallel to CA, one of the sides of the triangle ADC, therefore proportionally AO is to OD as CF is to FD. VI.2 But it was prove that AO is to OD as AE is to EB, therefore AE is to EB as CF is to FD. V.11 Therefore, if two straight lines are cut by parallel planes, then they are cut in the same ratios. Q. E. D. This proposition is used in the proof of proposition XII.4. Next proposition: XI.18 Previous: XI.16 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 18 If a straight line is at right angles to any plane, then all the planes through it are also at right angles to the same plane. Let any straight line AB be at right angles to the plane of reference. I say that all the planes through AB are also at right angles to the plane of reference. Let the plane DE be drawn through AB. Let CE be the intersection of the plane DE and the plane of reference. Take a point F at random on CE, and draw FG from F at right angles to CE in the plane DE. I.11 Now, since AB is at right angles to the plane of reference, therefore AB is also at right angles to all the straight lines which meet it and lie in the plane of reference, so that it is also at right angles to CE. Therefore the angle ABF is right. XI.Def.3 But the angle GFB is also right, therefore AB is parallel to FG. I.28 But AB is at right angles to the plane of reference, therefore FG is also at right angles to the plane of reference. XI.8 Now a plane is at right angles to a plane when the straight lines drawn in one of the planes at right angles to the intersection of the planes are at right angles to the remaining plane. And FG, drawn in one of the planes DE at right angles to CE, the intersection of the planes, was proved to be at right angles to the plane of reference. Therefore the plane DE is at right angles to the plane of reference. XI.Def.4 Similarly it can also be proved that all the planes through AB are at right angles to the plane of reference. Therefore, if a straight line is at right angles to any plane, then all the planes through it are also at right angles to the same plane. Q. E. D. This proposition is used in the proof of proposition XII.17. Next proposition: XI.19 Previous: XI.17 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 19 If two planes which cut one another are at right angles to any plane, then their intersection is also at right angles to the same plane. Let the two planes AB and BC be at right angles to the plane of reference, and let BC be their intersection. I say that BD is at right angles to the plane of reference. Suppose it is not. From the point D draw DE at right angles to the straight line AD in the plane AB, and draw DF at right angles to CD in the plane BC. I.11 Now, since the plane AB is at right angles to the plane of reference, and DE is at right angles in the plane AB to AD, their intersection, therefore DE is at right angles to the plane of reference. XI.Def.4 Similarly we can prove that DF is also at right angles to the plane of reference. Therefore from the same point D two straight lines have been set up at right angles to the plane of reference on the same side, which is impossible. XI.13 Therefore no straight line except the intersection DB of the planes AB and BC can be set up from the point D at right angles to the plane of reference. Therefore, if two planes which cut one another are at right angles to any plane, then their intersection is also at right angles to the same plane. Q. E. D. This proposition is not used in the rest of the Elements. Next proposition: XI.20 Previous: XI.18 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 20 If a solid angle is contained by three plane angles, then the sum of any two is greater than the remaining one. Let the solid angle at A be contained by the three plane angles BAC, CAD, and DAB. I say that the sum of any two of the angles BAC, CAD, and DAB is greater than the remaining one. If the angles BAC, CAD, and DAB are equal to one another, then it is clear that the sum of any two is greater than the remaining one. But, if not, let BAC be greater. In the plane through BA and AC, construct the angle BAE equal to the angle DAB at the point A on the straight line AB. Make AE equal to AD, draw BEC across through the point E cutting the straight lines AB and AC at the points B and C, and join DB and DC. I.23 I.3 Now, since DA equals AE, and AB is common, therefore two sides are equal to two sides. And the angle DAB equals the angle BAE, therefore the base DB equals the base BE. I.4 And, since the sum of the two sides BD and DC is greater than BC, and of these DB was proved equal to BE, therefore the remainder DC is greater than the remainder EC. I.20 Now, since DA equals AE, and AC is common, and the base DC is greater than the base EC, therefore the angle DAC is greater than the angle EAC. I.25 But the angle BAE equals the angle DAB, therefore the sum of the angles DAB and DAC is greater than the angle BAC. Similarly we can prove that the sum of any two of the remaining angles is greater than the remaining one. Therefore, if a solid angle is contained by three plane angles, then the sum of any two is greater than the remaining one. Q. E. D. This is one of two necessary conditions for constructing a solid angle out of three plane angles. The next necessary condition is stated in the next proposition, and the two conditions together are shown to be sufficient in XI.23. About the proof The structure of the proof is not entirely clear. The goal is to show that the sum of any two of the angles is greater than the third. Notice is make that if they are all equal, then the goal is clearly satisfied. Then, under the assumption that if one is greater than a second, then the sum of the second and third is greater than the first. Then the other cases are declared to be similarly provable. Various interpretations have been made of the intent of the form of proof, but all require minor changes to clarify the stucture. About three-dimensional analogues of two-dimensional constructions Up until this proposition, each construction in Book XI takes place within a plane, although different constructions in the same proposition may occur in different planes. One of the constructions here, however, takes place in two different planes. The angle BAE is constructed in one plane to equal a given angle BAE in a different plane. The construction in I.23 to construct one angle equal to a given angle, strictly speaking, takes place in only one plane. Tracing that construction back through Book I leads through proposition I.22 to I.3. Proposition I.3 cuts one line off equal to another line. That basic construction can easily be modified so that the two lines are in different planes. Once that's done, the rest of the constructions in Book I also apply when their components lie in different planes. Still, the details should be verified before applying the results as done in the proof of this proposition. Next proposition: XI.21 Previous: XI.19 Book XI introduction © 1997, 2002 D.E.Joyce Clark University Proposition 21 Any solid angle is contained by plane angles whose sum is less than four right angles. Let the angle at A be a solid angle contained by the plane angles BAC, CAD, and DAB. I say that the sum of the angles BAC, CAD, and DAB is less than four right angles. Take points B, C, and D at random on the straight lines AB, AC, and AD respectively, and join BC, CD, and DB. Now, since the solid angle at B is contained by the three plane angles CBA, ABD, and CBD, and the sum of any two is greater than the remaining one, therefore the sum of the angles CBA and ABD is greater than the angle CBD. XI.20 For the same reason the sum of the angles BCA and ACD is greater than the angle BCD, and the sum of the angles CDA and ADB is greater than the angle CDB. Therefore the sum of the six angles CBA, ABD, BCA, ACD, CDA, and ADB is greater than the sum of the three angles CBD, BCD, and CDB. But the sum of the three angles CBD, BDC, and BCD equals two right angles, therefore the sum of the six angles CBA, ABD, BCA, ACD, CDA, and ADB is greater than two right angles. I.32 And, since the each sum of the three angles of the triangles ABC, ACD, and ADB equals two right angles, therefore the sum of the nine angles of the three triangles, the angles CBA, ACB, BAC, ACD, CDA, CAD, ADB, DBA, and BAD equals six right angles. Of them the sum of the six angles ABC, BCA, ACD, CDA, ADB, and DBA are greater than two right angles, therefore the sum of the remaining three angles BAC, CAD, and DAB containing the solid angle is less than four right angles. Therefore, any solid angle is contained by plane angles whose sum is less than four right angles. Q.E.D. In proposition XI.23 the condition stated here and the condition in XI.20 (the sum of any two plane angles is less than the third) together are shown to be sufficient to construct a solid angle. About the proof The proof only shows that the sum of the plane angles in all cases is less than four right angles when there are three plane angles, not when there are more than three. When there are four ore more plane angles, the proof is analogous, but it is necessary to invoke Proclus' first corollary to I.32, which states that "the sum of the interior angles of a convex rectilinear figure equals twice as many angles as the figure has sides, less four." Use of this proposition This proposition is used in the proof of remark after proposition XIII.18 to show that the five regular polyhedra constructed in Book XIII are the only five possible. Next proposition: XI.22 Previous: XI.20 Book XI introduction © 1997, 2002 D.E.Joyce Clark University Proposition 22 If there are three plane angles such that the sum of any two is greater than the remaining one, and they are contained by equal straight lines, then it is possible to construct a triangle out of the straight lines joining the ends of the equal straight lines. Let there be three plane angles ABC, DEF, and GHK, of which the sum of any two is greater than the remaining one, so that the sum of the angles ABC and DEF is greater than the angle GHK, the sum of the angles DEF and GHK is greater than the angle ABC, and, further, the sum of the angles GHK and ABC is greater than the angle DEF. Also let the straight lines AB, BC, DE, EF, GH, and HK be equal. Join AC, DF, and GK. I say that it is possible to construct a triangle out of straight lines equal to AC, DF, and GK, that is, that the sum of any two of the straight lines AC, DF, and GK is greater than the remaining one. Now, if the angles ABC, DEF, and GHK equal one another, then it is clear that, AC, DF, and GK also being equal, it is possible to construct a triangle out of straight lines equal to AC, DF, and GK. I.4 I.1 But, if not, let them be unequal. Construct the angle KHL equal to the angle ABC at the point H on the straight line HK. Make HL equal to any one of the straight lines AB, BC, DE, EF, GH, or HK. Join KL and GL. I.23 I.3 Now, since the two sides AB and BC equal the two sides KH and HL, and the angle at B equals the angle KHL, therefore the base AC equals the base KL. I.4 And, since the sum of the angles ABC and GHK is greater than the angle DEF, while the angle ABC equals the angle KHL, therefore the angle GHL is greater than the angle DEF. And, since the two sides GH and HL equal the two sides DE and EF, and the angle GHL is greater than the angle DEF, therefore the base GL is greater than the base DF. I.24 But the sum of GK and KL is greater than GL. Therefore the sum of GK and KL is much greater than DF. But KL equals AC, therefore the sum of AC and GK is greater than the remaining straight line DF. Similarly we can prove that the sum of AC and DF is greater than GK, and further, the sum of DF and GK is greater than AC. Therefore it is possible to construct a triangle out of straight lines equal to AC, DF, and GK. (I.22) Therefore, if there are three plane angles such that the sum of any two is greater than the remaining one, and they are contained by equal straight lines, then it is possible to construct a triangle out of the straight lines joining the ends of the equal straight lines. Q. E. D. This construction is the first stage of the construction in the next proposition to make a solid angle given three plane angles. The proof succeeds in showing that if each of the three plane angles is less than the sum of the other two, then each of the three lines AC, DF, and DK is less than the sum of the other two. The latter is a necessary condition for a triangle to be made with its three sides equal to those three lines according to I.20. But it was never shown to be sufficient to make such a triangle in I.22, and it is that sufficiency which is being invoked in this proof. Thus, there is a serious flaw in the proof. Next proposition: XI.23 Previous: XI.21 Book XI introduction © 1997, 2002 D.E.Joyce Clark University Proposition 23 To construct a solid angles out of three plane angles such that the sum of any two is greater than the remaining one: thus the sum of the three angles must be less than four right angles. XI.20 XI.21 Let the angles ABC, DEF, and GHK be the three given plane angles, and let the sum of any two of them be greater than the remaining one, and further, let the sum of all three be less than four right angles. It is required to construct a solid angle out of angles equal to the angles ABC, DEF, and GHK. Cut off AB, BC, DE, EF, GH, and HK equal to one another, and join AC, DF, and GK. I.3 It is therefore possible to construct a triangle out of straight lines equal to AC, DF, and GK. Construct LMN so that AC equals LM, DF equals MN, and GK equals NL. XI.22 Describe the circle LMN about the triangle LMN, and take its center O. Join LO, MO, and NO. IV.5 III.1 I say that AB is greater than LO. For, if not, AB either equals LO, or is less. First, let it be equal. Then, since AB equals LO, while AB equals BC, and LO equals OM, therefore the two sides AB and BC equal the two sides LO and OM respectively. And, by hypothesis, the base AC equals the base LM, therefore the angle ABC equals the angle LOM. I.8 For the same reason the angle DEF also equals the angle MON, and the angle GHK equals the angle NOL. Therefore the sum of the three angles ABC, DEF, and GHK equals the sum of the three angles LOM, MON, and NOL. But the sum of the three angles LOM, MON, and NOL equals four right angles, therefore the sum of the three angles ABC, DEF, and GHK equals four right angles. But the sum is also, by hypothesis, less than four right angles, which is absurd. Therefore AB is not equal to LO. I say next that neither is AB less than LO. For, if possible, let it be so. Make OP equal to AB, and OQ equal to BC, and join PQ. I.3 Then, since AB equals BC, therefore OP also equals OQ, so that the remainder LP equals QM. Therefore LM is parallel to PQ, and LMO is equiangular with PQO. VI.2 I.29 Therefore, OL is to LM as OP is to PQ, and alternately, LO is to OP as LM is to PQ. VI.4 V.16 But LO is greater than OP, therefore LM is greater than PQ. And LM equals AC, therefore AC is greater than PQ. Since, then, the two sides AB and BC equal the two sides PO and OQ, and the base AC is greater than the base PQ, therefore the angle ABC is greater than the angle POQ. I.25 Similarly we can prove that the angle DEF is also greater than the angle MON, and the angle GHK is greater than the angle NOL. Therefore the sum of the three angles ABC, DEF, and GHK is greater than the sum of the three angles LOM, MON, and NOL. But, by hypothesis, the sum of the angles ABC, DEF, and GHK is less than four right angles, therefore the sum of the angles LOM, MON, and NOL is much less than four right angles. But the sum also equals four right angles, which is absurd. Therefore AB is not less than LO. And it was proved that neither is it equal, therefore AB is greater than LO. Next set up OR from the point O at right angles to the plane of the circle LMN so that the square on OR equals the square on AB minus the square on LO. Join RL, RM, and RN. XI.12 Lemma below Then, since RO is at right angles to the plane of the circle LMN, therefore RO is also at right angles to each of the straight lines LO, MO, and NO. And, since LO equals OM, and OR is common and at right angles, therefore the base RL equals the base RM. XI.Def.3 I.4 For the same reason RN also equals each of the straight lines RL and RM. Therefore the three straight lines RL, RM, and RN equal one another. Next, since by hypothesis the square on OR equals equals the square on AB minus the square on LO, therefore the square on AB equals the sum of the squares on LO and OR. But the square on LR equals the sum of the squares on LO and OR, for the angle LOR is right, therefore the square on AB equals the square on RL. Therefore AB equals RL. I.47 But each of the straight lines BC, DE, EF, GH, and HK equals AB, while each of the straight lines RM and RN equals RL, therefore each of the straight lines AB, BC, DE, EF, GH, and HK equals each of the straight lines RL, RM, and RN. Since the two sides LR and RM equal the two sides AB and BC, and, by hypothesis, the base LM equals the base AC, therefore the angle LRM equals the angle ABC. For the same reason the angle MRN equals the angle DEF, and the angle LRN equals the angle GHK. I.8 Therefore, out of the three plane angles LRM, MRN, and LRN, which equal the three given angles ABC, DEF, and GHK, the solid angle at R has been constructed, which is contained by the angles LRM, MRN, and LRN. Q.E.F. Lemma But how it is possible to take the square on OR equal to the square on AB minus the square on LO we can show as follows. Set out the straight lines AB and LO, and let AB be the greater. Describe the semicircle ABC on AB. Fit AC into the semicircle ABC equal to the straight line LO, not being greater than the diameter AB. Join CB. IV.1 Since the angle ACB is an angle in the semicircle ACB, therefore the angle ACB is right. III.31 Therefore the square on AB equals the sum of the squares on AC and CB. I.47 Hence the square on AB equals the square on AC minus the square on CB. But AC equals LO. Therefore the square on AB equals the square on LO minus the square on CB. Therefore if we cut off OR equal to BC, then the square on AB will equal the square on LO minus the square on OR. Q. E. F. This proposition shows that the necessary conditions for constructing a solid angle found in XI.20 (the sum any two angles must be less than the third) and XI.21 (the sum of the three angles must be less than four right angles) are, in fact, sufficient. It is interesting to see how parts of the construction disappear as the angles B, E, and H grow so that these conditions fail. This proposition completes the introductory portion of Book XI. Most of the remainder deals with parallelepipedal solids and their properties. About the proof This is a rather long proof that has several stages. First, the base LMN for the proposed solid angle is constructed. This first stage has been set off as the provious proposition XI.24. After the circumcircle for this base is constructed, it is shown that the proposed edges for the solid angle, which are all equal, are greater than the radius of the circle. That part of the demonstration takes some time, and it is separated into two parts to show, first, that the edges can't equal the radius, and, second, that the edges can't be less than the radius. The next stage is to place the proposed vertex R for the solid angle. It is placed above the center O of the circumcircle so that OR2 is the difference of the square of the edge and the square of the radius. A separate lemma appears after the proposition to construct a line of this particular length. This lemma is the same as the lemma for proposition X.14 in Book X. The remainder of the proof is the verification that the proposed solid angle satisfies the requirements of the construction. The proof only covers the case when the circumcenter O of the triangle LMN lies within that triangle. Two other cases need to be considered as well-when O lies outside the triangle and when O lies on the boundary of the triangle. The three different cases need only be considered in the stage which shows that the proposed edges are greater than the radius of the circumcircle; the proof doesn't have to be split into three cases for the other stages of the proof. Next proposition: XI.24 Previous: XI.22 Book XI introduction © 1997, 2002 D.E.Joyce Clark University Proposition 24 If a solid is contained by parallel planes, then the opposite planes in it are equal and parallelogrammic. Let the solid CDHG be contained by the parallel planes AC, GF, AH, DF, BF, and AE. I say that the opposite planes in it are equal and parallelogrammic. Since the two parallel planes BG and CE are cut by the plane AC, therefore their common sections are parallel. Therefore AB is parallel to DC. Again, since the two parallel planes BF and AE are cut by the plane AC, therefore their intersections are parallel. Therefore BC is parallel to AD. XI.16 But AB was proved parallel to DC, therefore AC is a parallelogram. Similarly we can prove that each of the planes DF, FG, GB, BF, and AE is a parallelogram. Join AH and DF. Then, since AB is parallel to DC, and BH is parallel to CF, therefore the two straight lines AB and BH, which meet one another, are parallel to the two straight lines DC and CF, which meet one another, not in the same plane. Therefore they contain equal angles. Therefore the angle ABH equals the angle DCF. XI.10 And, since the two sides AB and BH equal the two sides DC and CF, and the angle ABH equals the angle DCF, therefore the base AH equals the base DF, and the triangle ABH equals the triangle DCF. I.34 I.4 And the parallelogram BG is double the triangle ABH, and the parallelogram CE is double the triangle DCF, therefore the parallelogram BG equals the parallelogram CE. I.34 Similarly we can prove that AC equals GF, and AE equals BF. Therefore, if a solid is contained by parallel planes, then the opposite planes in it are equal and parallelogrammic. Q. E. D. The statement of the theorem is not sufficiently detailed. All three of the octahedron, icosahedron, and dodecahedron (see XI.Def.26-28) are contained by parallel planes, but their faces are triangles or pentagons, not parallelograms. They do not have six faces, however, but eight, twenty, or twelve. The correct hypothesis for this proposition is that the solid is contained by three pairs of parallel planes. Then the intersection of each plane with the other four nonparallel planes can be shown to be sides of a parallelgram, and the parallelograms on opposite planes can be shown to be congruent, what Euclid would call similar and equal parallelograms. That the opposite parallelgrams are not just equal but also similar should stated in the conclusion of the proposition. Parallelepipeds The term "parallelepipedal solid," abbreviated as "parallelepiped," is used for the solid treated by this proposition. It can be defined as a solid bounded by three pairs of parallel faces. Then this proposition shows that a parallelepiped has the further properties that each face is a parallelogram, and opposite parallelograms have parallel and equal corresponding sides, and equal corresponding angles. Parallelepipeds are to solid geometry what parallelograms are to plane geometry. This proposition is the analogue of proposition I.34 which introduces parallelograms just as this proposition introduces parallelepipeds. It is likely that both are the product of Euclid's own research. Use of this proposition This proposition is used in the next as well as others in this book and the next. Next proposition: XI.25 Previous: XI.23 Book XI introduction © 1997, 2002 D.E.Joyce Clark University Proposition 25 If a parallelepipedal solid is cut by a plane parallel to the opposite planes, then the base is to the base as the solid is to the solid. Let the parallelepipedal solid ABCD be cut by the plane FG which is parallel to the opposite planes RA and DH. I say that the base AEFV is to the base FHCF as the solid ABFU is to the solid EGCD. Produce AH in each direction. Make any number of straight lines AK and KL equal to AE, and any number HM and MN equal to EH. Complete the parallelograms LP, KV, HW, and MS and the solids LQ, KR, DM, and MT. I.3 I.31 Then, since the straight lines LK, KA, and AE equal one another, therefore the parallelograms LP, KV, and AF equal one another, KO, KB, and AG equal one another, and further LX, KQ, and AR equal one another, for they are opposite. For the same reason the parallelograms EC, HW, and MS are equal one another, HG, HI, and IN equal one another, and further, DH, MY, NT and equal one another. XI.24 Therefore in the solids LQ, KR, and AU three planes equal three planes. But the three planes equal the three opposite, therefore the three solids LQ, KR, and AU equal one another. For the same reason the three solids ED, DM, and MT also equal one another. Therefore, the solid LU is the same multiple of the solid AU that the base LF is of the base AF. For the same reason, the solid NU is the same multiple of the solid HU that the base NF is of the base FH. XI.Def.10 And, if the base LF equals the base NF, then the solid LU also equals the solid NU; if the base LF exceeds the base NF, then the solid LU also exceeds the solid NU; and, if one falls short, then the other falls short. Therefore, there being four magnitudes, the two bases AF and FH, and the two solids AU and UH, equimultiples have been taken of the base AF and the solid AU, namely the base LF and the solid LU, and equimultiples of the base HF and the solid HU, namely the base NF and the solid NU, and it has been proved that, if the base LF exceeds the base FN, then the solid LU also exceeds the solid NU; if the bases are equal, then the solids are equal; and if the base falls short, then the solid falls short. Therefore, the base AF is to the base FH as the solid AU is to the solid UH. V.Def.5 Therefore If a parallelepipedal solid is cut by a plane parallel to the opposite planes, then the base is to the base as the solid is to the solid. Q. E. D. This is the first of the propositions on volumes of solids. Most of the rest of this book deals with volumes of parallelepipeds, and Book XII develops the theory of volumes for pyramids, prisms, cones, cylinders, and spheres. Euclid's foundations for volume are (1) his definition XI.Def.10 which says that if two solid figures have congruent faces, then the solids are equal, and (2) solids are magnitudes for which cut and paste principles hold. See the comments on XI.Def.10 for details. Outline of the proof The analogous proposition for two dimensions is proposition VI.1. In both propositions the ratio of two figures in one dimension is shown to be equal to the ratio of two figures in another dimension. In this proposition, the dimensions are 2 and 3, while in VI.1, the dimensions are 1 and 2. Eudoxus' definition of proportion, V.Def.5, allows these ratios of different kinds to be compared. The goal of this proof is to show that the ratio of the bases of the the two parallelepipeds is the same as the ratio of the two parallelepipeds themselves. The parallelpiped AU has the parallelogram AF as its base, while the parallelepiped HU has the parallelogram HF as its base. Thus, the goal is to derive the proportion AU:HU = AF:HF. By the definition of proportion, V.Def.5, that means for any number m and any number n that m AF >=< n HF when m AU >=< n HU. Note that Euclid takes both m and n to be 3 in his proof, just as he did in VI.1. Now m AU equals the parallelepiped LU, n HU equals the parallelepiped NU, m AF equals the parallelogram LF, and n HF equals the parallelogram NF. So what has to be shown is that LF >=< NF when LU >=< NU. Euclid makes no attempt to show that; he just states it as fact: And, if the base LFequals the base NF,then the solid LUalso equals the solid NU; if the base LFexceeds the base NF,then the solid LUalso exceeds the solid NU; and, if one falls short, then the other falls short. The case of equality is based directly on defintion XI.Def.10, for if the bases are equal, then all the planes bounding the solids are equal, which is the defintion of the solids being equal. The two cases when one base exceeds or falls short of the other implicitly depend on finding a part of one solid equal to the whole of the other solid, equality again using XI.Def.10, then concluding that one whole solid is greater than the other whole solid (C.N.5). Use of this proposition This proposition is used for the proofs of propositions XI.31, XI.32, and XI.34. Next proposition: XI.26 Previous: XI.24 Book XI introduction © 1997, 2002 D.E.Joyce Clark University Proposition 26 To construct a solid angle equal to a given solid angle on a given straight line at a given point on it. Let A be the given point on the given straight line AB, and let the angle at D be the given solid angle contained by the angles EDC, EDF, and FDC. It is required to construct at the point A on the straight line AB a solid angle equal to the solid angle at D. Take a point F at random on DF, draw FG from F perpendicular to the plane through ED and DC, and let it meet the plane at G. Join DG. XI.11 At the point A on the straight line AB construct the angle BAL equal to the angle EDC, and construct the angle BAK equal to the angle EDG. I.23 Make AK equal to DG. Set KH up from the point K at right angles to the plane through BA and AL. Make KH equal to GF, and join HA. XI.12 I say that the solid angle at A contained by the angles BAL, BAH, and HAL equals the solid angle at D contained by the angles EDC, EDF, and FDC. Cut AB and DE off equal to one another, and join HB, KB, FE, and GE. Then, since FG is at right angles to the plane of reference, therefore it is also at right angles with all the straight lines which meet it and are in the plane of reference. Therefore each of the angles FGD and FGE is right. For the same reason each of the angles HKA and HKB is also right. XI.Def.3 And, since the two sides KA and AB equal the two sides GD and DE respectively, and they contain equal angles, therefore the base KB equals the base GE. But KH also equals GF, and they contain right angles, therefore HB also equals FE. Again, since the two sides AK and KH equal the two sides DG and GF, and they contain right angles, therefore the base AH equals the base FD. I.4 But AB also equals DE, therefore the two sides HA and AB are equal to the two sides DF and DE. And the base HB is equal to the base FE, therefore the angle BAH equals the angle EDF. For the same reason the angle HAL also equals the angle FDC. I.8 And the angle BAL also equals the angle EDC. Therefore at the point A on the straight line AB a solid angle has been constructed equal to the given solid angle at D. Q. E. F. This construction is used in the next one to construct similar parallelepipeds. Next proposition: XI.27 Previous: XI.25 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 27 To describe a parallelepipedal solid similar and similarly situated to a given parallelepipedal solid on a given straight line. Let AB be the given straight line and CD the given parallelepipedal solid. It is required to describe on the given straight line AB a parallelepipedal solid similar and similarly situated to the given parallelepipedal solid CD. Construct the solid angle contained by the angles BAH, HAK, and KAB at the point A on the straight line AB equal to the solid angle C so that the angle BAH equals the angle ECF, the angle BAK equals the angle ECG, and the angle KAH equals the angle GCF,so that EC is to CG as BA is to AK, and GC is to CF as KA is to AH. XI.26 VI.12 Therefore, ex aequali, EC is to CF as BA is to AH. V.22 Complete the parallelogram HB and the solid AL. Now since EC is to CG as BA is to AK, and the sides about the equal angles ECG and BAK are thus proportional, therefore the parallelogram GE is similar to the parallelogram KB. For the same reason the parallelogram KH is similar to the parallelogram GF, and also FE is similar to HB. Therefore three parallelograms of the solid CD are similar to three parallelograms of the solid AL. But the former three are both equal and similar to the three opposite parallelograms, and the latter three are both equal and similar to the three opposite parallelograms, therefore the whole solid CD is similar to the whole solid AL. XI.Def.9 Therefore on the given straight line AB there has been described AL similar and similarly situated to the given parallelepipedal solid CD. Q. E. F. This proposition is analogous to proposition VI.18 which constructs a similar plane figure on a line, but it is not as general since it applies only to parallelepipeds and not all polyhedra. It is not used later in the Elements. Next proposition: XI.28 Previous: XI.26 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 28 If a parallelepipedal solid is cut by a plane through the diagonals of the opposite planes, then the solid is bisected by the plane. Let the parallelepipedal solid AB be cut by the plane CDEF through the diagonals CF and DE of opposite planes. I say that the solid AB is bisected by the plane CDEF. Since the triangle CGF equals the triangle CFB, and ADE equals DEH, while the parallelogram CA equals the parallelogram EB, for they are opposite, and GE equals CH, therefore the prism contained by the two triangles CGF and ADE and the three parallelograms GE, AC, and CE equals the prism contained by the two triangles CFB and DEH and the three parallelograms CH, BE, and CE, for they are contained by planes equal both in multitude and in magnitude. I.34 XI.Def.10 Hence the whole solid AB is bisected by the plane CDEF. Therefore, if a parallelepipedal solid is cut by a plane through the diagonals of the opposite planes, then the solid is bisected by the plane. Q. E. D. A minor point missing from the beginning of the proof of is that the two diagonals CF and DE lie in one plane, but it is easy to show that they lie in the lines CD and EF are parallel, and therefore, by XI.7, CF and DE lie in the plane spanned by CD and EF. This is the second proposition concerning volumes. (The first was XI.25.) The final conclusion of the proof here is justified by XI.Def.10: since the faces of the two prisms are congruent, therefore the prisms are equal and similar (that is, congruent). Several authors have criticized this conclusion because the two prisms are mirror images of each other and cannot be applied to each other in the sense of moving one in space to coincide with the other. From some points of view this criticism is valid. But the method of superposition is subject to even greater criticism. In modern geometry, depending on the style of geometry, superposition is either eliminated entirely or else completely formalized using the theory of group transformations. Use of this proposition Although this proposition is not used in the rest of this book, it is used for several propositions in the next book that deal with triangular prisms. Next proposition: XI.29 Previous: XI.27 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 29 Parallelepipedal solids which are on the same base and of the same height, and in which the ends of their edges which stand up are on the same straight lines, equal one another. Let CM and CN be parallelepipedal solids on the same base AB and of the same height, and let the ends of their edges which stand up, namely AG, AF, LM, LN, CD, CE, BH, and BK, be on the same straight lines FN and DK. I say that the solid CM equals the solid CN. Since each of the figures CH and CK is a parallelogram, therefore CB equals each of the straight lines DH and EK. Therefore DH also equals EK. I.34 Subtract EH from each, therefore the remainder DE equals the remainder HK. Therefore the triangle DCE also equals the triangle HBK, and the parallelogram DG equals the parallelogram HN. For the same reason the triangle AFG equals the triangle MLN. I.8 I.4 I.36 But the parallelogram CF equals the parallelogram BM, and CG equals BN, for they are opposite, therefore the prism contained by the two triangles AFG and DCE and the three parallelograms AD, DG, and CG equals the prism contained by the two triangles MLN and HBK and the three parallelograms BM, HN, and BN. XI.Def.10 Add to each the solid of which the parallelogram AB is the base and GEHM its opposite, therefore the whole parallelepipedal solid CM equals the whole parallelepipedal solid CN. Therefore, parallelepipedal solids which are on the same base and of the same height, and in which the ends of their edges which stand up are on the same straight lines, equal one another. Q. E. D. This proposition is the first step in the theory of volume for parallelepipeds. It is used in the proof of three of the next five propositions. Next proposition: XI.30 Previous: XI.28 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 30 Parallelepipedal solids which are on the same base and of the same height, and in which the ends of their edges which stand up are not on the same straight lines, equal one another. Let CM and CN be parallelepipedal solids on the same base AB and of the same height, and let the ends of their edges which stand up, namely AF, AG, LM, LN, CD, CE, BH, and BK, not be on the same straight lines. I say that the solid CM equals the solid CN. Produce NK and DH to meet one another at R, and produce FM and GE to P and Q. Join AO, LP, CQ, and BR. Then the solid CM, of which the parallelogram ACBL is the base and FDHM its opposite, equals the solid CP, of which the parallelogram ACBL is the base and OQRP its opposite, for they are on the same base ACBL and of the same height, and the ends of their edges which stand up, namely AF, AO, LM, LP, CD, CQ, BH, and BR, are on the same straight lines FP and DR. XI.29 But the solid CP, of which the parallelogram ACBL is the base and OQRP its opposite, equals the solid CN, of which the parallelogram ACBL is the base and GEKN its opposite, for they are again on the same base ACBL and of the same height, and the ends of their edges which stand up, namely AG, AO, CE, CQ, LN, LP, BK, and BR, are on the same straight lines GQ and NR. XI.29 Hence the solid CM also equals the solid CN. Therefore, parallelepipedal solids which are on the same base and of the same height, and in which the ends of their edges which stand up are not on the same straight lines, equal one another. Q. E. D. Two applications of the previous proposition allow its generalization to the present proposition. It is generalized one step further in the next proposition. Next proposition: XI.31 Previous: XI.29 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 31 Parallelepipedal solids which are on equal bases and of the same height equal one another. Let the parallelepipedal solids AE and CF of the same height be on equal bases AB and CD. I say that the solid AE equals the solid CF. First, let the sides which stand up, HK, BE, AG, LM, PQ, DF, CO, and RS, be at right angles to the bases AB and CD. Produce the straight line RT in a straight line with CR. Construct the angle TRU equal to the angle ALB at the point R on the straight line RT. Make RT equal to A, and RU equal to LB. Complete the base RW and the solid XU. I.23 I.3 I.31 Now, since the two sides TR and RU equal the two sides AL and LB, and they contain equal angles, therefore the parallelogram RW equals and is similar to the parallelogram HL. Since again AL equals RT, and LM equals RS, and they contain right angles, therefore the parallelogram RX equals and is similar to the parallelogram AM. For the same reason LE also equals and is similar to SU. Therefore three parallelograms of the solid AE equal and are similar to three parallelograms of the solid XU. But the former three equal and are similar to the three opposite, and the latter three equal and are similar the three opposite, therefore the whole parallelepipedal solid AE equals the whole parallelepipedal solid XU. XI.24 XI.Def.10 Draw DR and WU through to meet one another at Y, draw aTb through T parallel to DY, produce PD to a, and complete the solids YX and RI. I.31 Then the solid XY, of which the parallelogram RX is the base and Yc its opposite, equals the solid XU, of which the parallelogram RX is the base and UV its opposite, for they are on the same base RX and of the same height, and the ends of their edges which stand up, namely RY, RU, Tb, TW, Se, Sd, Xc, and XV, are on the same straight lines YW and eV. But the solid XU equals AE, therefore the solid XY also equals the solid AE. XI.29 And, since the parallelogram RUWT equals the parallelogram YT, for they are on the same base RT and in the same parallels RT and YW, and RUWT equals CD, since it also equals AB, therefore the parallelogram YT also equals CD. I.35 But DT is another parallelogram, therefore the base CD is to DT as YT is to DT. V.7 And, since the parallelepipedal solid CI is cut by the plane RF which is parallel to opposite planes, therefore the base CD is to the base DT as the solid CF is to the solid RI. For the same reason, since the parallelepipedal solid YI is cut by the plane RX which is parallel to opposite planes, therefore the base YT is to the base TD as the solid YX is to the solid RI. XI.25 But the base CD is to DT as YT is to DT, therefore the solid CF is to the solid RI as the solid YX is to RI. V.11 Therefore each of the solids CF and YX has to RI the same ratio. Therefore the solid CF equals the solid YX. But YX was proved equal to AE, therefore AE also equals CF. V.9 Next, let the sides standing up, AG, HK, BE, LM, CN, PQ, DF, and RS, not be at right angles to the bases AB and CD. I say again that the solid AE equals the solid CF. Draw KO, ET, GU, MV, QW, FX, NY and SI from the points K, E, G, M, Q, F, N, and S perpendicular to the plane of reference, and let them meet the plane at the points O, T, U, V, W, X, Y, and I. XI.11 Then the solid KV equals the solid QI, for they are on the equal bases KM and QS and of the same height, and their sides which stand up are at right angles to their bases. Above But the solid KV equals the solid AE, and QI equals CF, for they are on the same base and of the same height, while the ends of their edges which stand up are not on the same straight lines. XI.30 Therefore the solid AE also equals the solid CF. Therefore, parallelepipedal solids which are on equal bases and of the same height equal one another. Q.E.D. The statement that started with XI.29 has now been generalized two steps. In the next proposition the heights of the two parallelepipeds remain equal, but the bases vary. The present proposition is used not only in the proof of the next, but also in three more of the remaining propositions of Book XI. Next proposition: XI.32 Previous: XI.30 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 32 Parallelepipedal solids which are of the same height are to one another as their bases. Let AB and CD be parallelepipedal solids of the same height. I say that the parallelepipedal solids AB and CD are to one another as their bases, that is, that the solid AB to the solid CD as the base AE is to the base CF . Apply FH equal to AE to FG. Complete the parallelepipedal solid GK with the same height as that of CD on FH as base. I.45 I.31 Then the solid AB equals the solid GK for they are on equal bases AE and FH and of the same height. XI.31 And, since the parallelepipedal solid CK is cut by the plane DG which is parallel to opposite planes, therefore the solid CD is to the solid DH as the base CF is to the base FH. XI.25 But the base FH equals the base AE, and the solid GK equals the solid AB, therefore the solid AB to the solid CD as the base AE is to the base CF. Therefore, parallelepipedal solids which are of the same height are to one another as their bases. Q.E.D. This completes the sequence of generalizations of proposition XI.29. Euclid does not specifically have a corresponding proposition "parallelepipedal solids with equal bases are to one another as their heights," but in the next two propositions, which depend on this one, he investigates other aspects of volumes of parallelepipeds. Next proposition: XI.33 Previous: XI.31 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 33 Similar parallelepipedal solids are to one another in the triplicate ratio of their corresponding sides. Let AB and CD be similar parallelepipedal solids, and let AE be the side corresponding to CF. I say that the solid AB has to the solid CD the ratio triplicate of that which AE has to CF. Produce EK, EL, and EM in a straight line with AE, GE, and HE. Make EK equal to CF, EL equal to FN, and EM equal to FR. Complete the parallelogram KL and the solid KP. I.3 I.31 Now, since the two sides KE and EL equal the two sides CF and FN, while the angle KEL equals the angle CFN, for the angle AEG also equals the angle CFN because AB and CD are similar solids, therefore the parallelogram KL equals and is similar to the parallelogram CN. For the same reason the parallelogram KM equals and is similar to CR, and EP equals and is similar to DF. Therefore three parallelograms of the solid KP equal and are similar to three parallelograms of the solid CD. But the former three parallelograms equal and are similar to their opposites, and the latter three equal and are similar to their opposites, therefore the whole solid KP equals and is similar to the whole solid CD. XI.24 XI.Def.10 Complete the parallelogram GK, and complete the solids EO and LQ on the parallelograms GK and KL as bases with the same height as that of AB. I.31 Then since the solids AB and CD are similar, therefore AE is to CF as EG is to FN, and as EH is to FR. And CF equals EK, FN equals EL, and FR equals EM, therefore AE is to EK as GE is to EL, and as HE is to EM. XI.Def.9 But AE is to EK as AG is to the parallelogram GK, therefore GE is to EL as GK is to KL, and HE is to EM as QE is to KM. Therefore the parallelogram AG is to GK as GK to is KL, and as QE is to KM. VI.1 But AG is to GK as the solid AB is to the solid EO, GK is to KL as the solid OE is to the solid QL, and QE is to KM as the solid QL is to the solid KP, therefore the solid AB is to EO as EO is to QL, and as QL is to KP. XI.32 But, if four magnitudes are continuously proportional, then the first has to the fourth the ratio triplicate of that which it has to the second, therefore the solid AB has to KP the ratio triplicate of that which AB has to EO. V.Def.10 But AB is to EO as the parallelogram AG is to GK, and as the straight line AE is to EK, hence the solid AB also has to KP the ratio triplicate of that which AE has to EK. VI.1 But the solid KP equals the solid CD, and the straight line EK equals CF, therefore the solid AB has also to the solid CD the ratio triplicate of that which the corresponding side of it, AE, has to the corresponding side CF. Therefore, Similar parallelepipedal solids are to one another in the triplicate ratio of their corresponding sides. Q. E. D. Corollary. If four straight lines are continuously proportional, then the first is to the fourth as a parallelepipedal solid on the first is to the similar and similarly situated parallelepipedal solid on the second, in as much as the first has to the fourth the ratio triplicate of that which it has to the second. This proposition is used in the proof of proposition XI.37 and later in XII.8, an analogous proposition about similar pyramids. Next proposition: XI.34 Previous: XI.32 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 34 In equal parallelepipedal solids the bases are reciprocally proportional to the heights; and those parallelepipedal solids in which the bases are reciprocally proportional to the heights are equal. Let AB and CD be equal parallelepipedal solids. I say that in the parallelepipedal solids AB and CD the bases are reciprocally proportional to the heights, that is the base EH is to the base NQ as the height of the solid CD is to the height of the solid AB. First, let the sides which stand up, namely AG, EF, LB, HK, CM, NO, PD, and QR, be at right angles to their bases. I say that the base EH is to the base NQ as CM is to AG. If now the base EH equals the base NQ, and the solid AB equals the solid CD, then CM equal AG. For parallelepipedal solids of the same height are to one another as the bases, and the base EH is to NQ as CM is to AG, and it is clear that in the parallelepipedal solids AB and CD the bases are reciprocally proportional to the heights. XI.32 Next, let the base EH not be equal to the base NQ, but let EH be greater. Now the solid AB equals the solid CD, therefore CM is also greater than AG. Make CT equal to AG, complete the parallelepipedal solid VC on NQ as base with CT as height. I.3 I.31 Now, since the solid AB equals the solid CD, and CV is outside them, and equals have to the same the same ratio, therefore the solid AB is to the solid CV as the solid CD is to the solid CV. V.7 But the solid AB is to the solid CY as the base EH is to the base NQ, for the solids AB and CV are of equal height, and the solid CD is to the solid CV as the base MQ is to the base TQ and CM is to CT, therefore the base EH is to the base NQ as MC is to CT. XI.32 XI.25 VI.1 But CT equals AG, therefore the base EH is to the base NQ as MC is to AG. Therefore in the parallelepipedal solids AB and CD the bases are reciprocally proportional to the heights. Again, in the parallelepipedal solids AB and CD let the bases be reciprocally proportional to the heights, that is the base EH is to the base NQ, so let the height of the solid CD be to the height of the solid AB. I say that the solid AB equals the solid CD. Let the sides which stand up be at right angles to the bases. Now, if the base EH equals the base NQ, and the base EH is to the base NQ as the height of the solid CD is to the height of the solid AB, therefore the height of the solid CD also equals the height of the solid AB. But parallelepipedal solids on equal bases and of the same height equal one another, therefore the solid AB equals the solid CD. XI.31 Next, let the base EH not be equal to the base NQ, but let EH be greater. Therefore the height of the solid CD is also greater than the height of the solid AB, that is, CM is greater than AG. Make CT equal to AG again, and complete the solid CV. I.3 I.31 Since the base EH is to the base NQ as MC is to AG, and AG equals CT, therefore the base EH is to the base NQ as CM is to CT. But the base EH is to the base NQ as the solid AB is to the solid CV, for the solids AB and CV are of equal height, and CM is to CT as the base MQ is to the base QT and as the solid CD is to the solid CV. XI.32 VI.1 XI.25 Therefore the solid AB is to the solid CV as the solid CD to the solid CV. Therefore each of the solids AB and CD has to CV the same ratio. Therefore the solid AB equals the solid CD. V.9 Now let the sides which stand up, FE, BL, GA, HK, ON, DP, MC, and RQ, not be at right angles to their bases. Draw perpendiculars from the points F, G, B, K, O, M, D, and R to the planes through EH and NQ, and let them meet the planes at S, T, U, V, W, X, Y, and a. Complete the solids FV and Oa. X.11 I say that, in this case too, if the solids AB and CD are equal, then the bases are reciprocally proportional to the heights, that is, the base EH is to the base NQ as the height of the solid CD to the height of the solid AB. Since the solid AB equals the solid CD, and AB equals BT, for they are on the same base FK and of the same height, and the solid CD equals DX, for they are again on the same base RO and of the same height, therefore the solid BT also equals the solid DX. XI.29 XI.30 Therefore the base FK is to the base OR as the height of the solid DX is to the height of the solid BT. But the base FK equals the base EH, and the base OR equals the base NQ, therefore the base EH is to the base NQ as the height of the solid DX is to the height of the solid BT. Above But the solids DX and BT and the solids DC and BA have the same heights respectively, therefore the base EH is to the base NQ as the height of the solid DC is to the height of the solid AB. Therefore in the parallelepipedal solids AB and CD the bases are reciprocally proportional to the heights. Next, in the parallelepipedal solids AB and CD let the bases be reciprocally proportional to the heights, that is, as the base EH is to the base NQ, so let the height of the solid CD be to the height of the solid AB. I say that the solid AB equals the solid CD. With the same construction, since the base EH is to the base NQ as the height of the solid CD is to the height of the solid AB, and the base EH equals the base FK, and NQ equals OR, therefore the base FK is to the base OR as the height of the solid CD is to the height of the solid AB. But the solids AB and CD and the solids BT and DX have the same heights respectively, therefore the base FK is to the base OR as the height of the solid DX is to the height of the solid BT. Therefore in the parallelepipedal solids BT and DX the bases are reciprocally proportional to the heights. Therefore the solid BT equals the solid DX. Above But BT equals BA, for they are on the same base FK and of the same height, and the solid DX equals the solid DC. Therefore the solid AB also equals the solid CD. XI.29 XI.30 Therefore, in equal parallelepipedal solids the bases are reciprocally proportional to the heights; and those parallelepipedal solids in which the bases are reciprocally proportional to the heights are equal. Q. E. D. A proof of this proposition could be made with very little regard to geometry but almost entirely in terms of abstract proportions. The volume of a parallelepiped is proportional to its base for equal heights (XI.32), and proportional to its height for equal bases (not actually stated by Euclid), therefore the base and height are inversely proportional for equal volumes. Such a proof, although simpler, is not in Euclid's style. This proposition is used in the proof of proposition XII.9, an analogous statement about pyramids. Next proposition: XI.35 Previous: XI.33 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 35 If there are two equal plane angles, and on their vertices there are set up elevated straight lines containing equal angles with the original straight lines respectively, if on the elevated straight lines points are taken at random and perpendiculars are drawn from them to the planes in which the original angles are, and if from the points so arising in the planes straight lines are joined to the vertices of the original angles, then they contain with the elevated straight lines equal angles. Let the angles BAC and EDF be two equal rectilinear angles, and from the points A and D let the elevated straight lines AG and DM be set up containing, with the original straight lines, equal angles respectively, namely, the angle MDE equal to the angle GAB and the angle MDF equal to the angle GAC. Take the points G and M at random on AG and DM. Draw GL and MN from the points G and M perpendicular to the plane through BA and AC and the plane through ED and DF, and let them meet the planes at L and N. Join LA and ND. XI.11 I say that the angle GAL equals the angle MDN. Make AH equal to DM, and draw HK through the point H parallel to GL. I.3 I.31 Since GL is perpendicular to the plane through BA and AC, therefore HK is also perpendicular to the plane through BA and AC. XI.8 Draw KC, NF, KB and NE from the points K and N perpendicular to the straight lines AC, DF, AB, and DE. Join HC, CB, MF, and FE. I.12 Since the square on HA equals the sum of the squares on HK and KA, and the sum of the squares on KC and CA equals the square on KA, therefore the square on HA equals the sum of the squares on HK, KC, and CA. I.47 But the square on HC equals the sum of the squares on HK and KC, therefore the square on HA equals the sum of the squares on HC and CA. Therefore the angle HCA is right. For the same reason the angle DFM is also right. I.47 I.48 Therefore the angle ACH equals the angle DFM. But the angle HAC equals the angle MDF. Therefore MDF and HAC are two triangles which have two angles equal to two angles respectively, and one side equal to one side, namely, that opposite one of the equal angles, that is, HA equals MD, therefore they also have the remaining sides equal to the remaining sides respectively. Therefore AC equals DF. I.26 Similarly we can prove that AB also equals DE. Since then AC equals DF, and AB equals DE, the two sides CA and AB equal the two sides FD and DE. But the angle CAB also equals the angle FDE, therefore the base BC equals the base EF, the triangle equals the triangle, and the remaining angles to the remaining angles. Therefore the angle ACB equals the angle DPE. I.4 But the right angle ACK equals the right angle DFN, therefore the remaining angle BCK equals the remaining angle EFN. For the same reason the angle CBK also equals the angle FEN. Therefore BCK and EFN are two triangles which have two angles equal to two angles respectively, and one side equal to one side, namely, that adjacent to the equal angles, that is, BC equals EF, therefore the remaining sides equal the remaining sides. Therefore CK equals FN. I.26 But AC also equals DF, therefore the two sides AC and CK equal the two sides DF and FN, and they contain right angles. Therefore the base AK equals the base DN. I.4 And, since AH equals DM, therefore the square on AH equals the square on DM. But the sum of the squares on AK and KH equals the square on AH, for the angle AKH is right, and the sum of the squares on DN and NM equals the square on DM, for the angle DNM is right, therefore the sum of the squares on AK and KH equals the sum of the squares on DN and NM. And of these the square on AK equals the square on DN, therefore the remaining square on KH equals the square on NM. Therefore HK equals MN. I.47 And, since the two sides HA and AK equal the two sides MD and DN respectively, and the base HK equals the base MN, therefore the angle HAK equals the angle MDN. I.8 Therefore, if there are two equal plane angles, and on their vertices there are set up elevated straight lines containing equal angles with the original straight lines respectively, if on the elevated straight lines points are taken at random and perpendiculars are drawn from them to the planes in which the original angles are, and if from the points so arising in the planes straight lines are joined to the vertices of the original angles, then they contain with the elevated straight lines equal angles. Corollary. From this it is clear that, if there are two equal plane angles, and if elevated straight lines set up on them which are equal and contain equal angles with the original straight lines respectively, then the perpendiculars drawn from their ends to the planes in which are the original angles equal one another. Q. E. D. The situation described here occurs in the next proposition, and the corollary is used there to show two parallelepipeds have the same height. Euclid's proof is quite long. Various authors have substituted shorter ones. Next proposition: XI.36 Previous: XI.34 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 36 If three straight lines are proportional, then the parallelepipedal solid formed out of the three equals the parallelepipedal solid on the mean which is equilateral, but equiangular with the aforesaid solid. Let A, B, and C be three straight lines in proportion, so that A is to B as B is to C. I say that the solid formed out of A, B, and C equals the solid on B which is equilateral, but equiangular with the aforesaid solid. Set out the solid angle at E contained by the angles DEG, GEF, and FED, and make each of the straight lines DE, GE, and EF equal to B. Complete the parallelepipedal solid EK. Make LM equal to A. Construct a solid angle at the point L on the straight line LM equal to the solid angle at E, namely that contained by NLO, OLM, and MLN. Make LO equal to B, and LN equal to C. I.3 Now, since A is to B as B is to C, while A equals LM, and B equals each of the straight lines LO, ED, and C to LN, therefore LM is to EF as DE is to LN. Thus the sides about the equal angles NLM, DEF are reciprocally proportional, therefore the parallelogram MN equals the parallelogram DF. VI.4 And, since the angles DEF and NLM are two plane rectilinear angles, and on them the elevated straight lines LO and EG are set up which equal one an other and contain equal angles with the original straight lines respectively, therefore the perpendiculars drawn from the points G and O to the planes through NL and LM and through DE and EF equal one another, therefore the solids LH and EK are of the same height. XI.35,Cor But parallelepipedal solids on equal bases and of the same height equal one another, therefore the solid HL equals the solid EK. XI.31 And LH is the solid formed out of A, B, and C, and EK is the solid on B, therefore the parallelepipedal solid formed out of A, B, and C equals the solid on B which is equilateral, but equiangular with the aforesaid solid. Therefore, if three straight lines are proportional, then the parallelepipedal solid formed out of the three equals the parallelepipedal solid on the mean which is equilateral, but equiangular with the aforesaid solid. Q.E.D. This straightforward proof depends on viewing the two parallelepipeds as having the same height, which they do if the base of the first is taken to be the parallelogram MN and the base of the second the parallelogram DF. Next proposition: XI.37 Previous: XI.35 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 37 If four straight lines are proportional, then parallelepipedal solids on them which are similar and similarly described are also proportional; and, if the parallelepipedal solids on them which are similar and similarly described are proportional, then the straight lines themselves are also proportional. Let AB, CD, EF, and GH be four straight lines in proportion, so that AB is to CD as EF is to GH, and let there be described on AB, CD, EF, and GH the similar and similarly situated parallelepipedal solids KA, LC, ME, NG. I say that KA is to LC as ME is to NG. Since the parallelepipedal solid KA is similar to LC, therefore KA has to LC the ratio triplicate of that which AB has to CD. For the same reason ME has to NG the ratio triplicate of that which EF has to GH. XI.33 And AB is to CD as EF is to GH. Therefore AK is to LC as ME is to NG. Next as the solid AK is to the solid LC, so let the solid ME be to the solid NG. I say that the straight line AB is to CD as EF is to GH. Since, again, KA has to LC the ratio triplicate of that which AB has to CD, and ME also has to NG the ratio triplicate of that which EF has to GH, and KA is to LC as ME is to NG, therefore AB is to CD as EF is to GH. XI.33 Therefore, if four straight lines are proportional, then parallelepipedal solids on them which are similar and similarly described are also proportional; and, if the parallelepipedal solids on them which are similar and similarly described are proportional, then the straight lines themselves are also proportional. Q. E. D. This proposition completes the theory of volumes of parallelepipeds. In the proof of this proposition it is assumed that two ratios are equal if and only if their triplicate ratios are equal. The required proof is long and detailed, but not difficult. Next proposition: XI.38 Previous: XI.36 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 38 If the sides of the opposite planes of a cube are bisected, and the planes are carried through the points of section, then the intersection of the planes and the diameter of the cube bisect one another. Let the sides of the opposite planes CF and AH of the cube AF be bisected at the points K, L, M, N, O, Q, P, and R, and through the points of section let the planes KN and OR be carried. Let US be the common section of the planes, and DG the diameter of the cube AF. I say that UT equals TS, and DT equals TG. Join DU, UE, BS, and SG. Then, since DO is parallel to PE, therefore the alternate angles DOU and UPE equal one another. I.29 Since DO equals PE, and OU equals UP, and they contain equal angles, therefore the base DU equals the base UE, the triangle DOU equals the triangle PUE, and the remaining angles equal the remaining angles. Therefore the angle OUD equals the angle PUE. I.4 Therefore DUE is a straight line. For the same reason BSG is also a straight line, and BS equals SG. I.14 Now, since CA equals and is parallel to DB, while CA also equals and is parallel to EG, therefore DB equals and is parallel to EG. XI.9 And the straight lines DE and BG join their ends, therefore DE is parallel to BG. I.33 Therefore the angle EDT equals the angle BGT, for they are alternate, and the angle DTU equals the angle GTS. I.29 I.15 Therefore DTU and GTS are two triangles which have two angles equal to two angles and one side equal to one side, namely that opposite one of the equal angles, that is, DU equals GS, for they are the halves of DE and BG, therefore the remaining sides equal the remaining sides. Therefore DT equals TG, and UT equals TS. I.26 Therefore, if the sides of the opposite planes of a cube are bisected, and the planes are carried through the points of section, then the intersection of the planes and the diameter of the cube bisect one another. Q. E. D. This proposition takes care of a specific situation that occurs in proposition XIII.17. In XIII.17 a dodecahedron is constructed based on a cube, and the fact proven here in XI.38 is needed to show that the point T where SU intersects DG is the center of a sphere circumscribing the cube. There are a couple of details missing from this proof. For instance, it is not shown that KL and MN actually lie in one plane, and that the line SU actually intersects the line DG. Next proposition: XI.39 Previous: XI.37 Book XI introduction © 1997 D.E.Joyce Clark University Proposition 39 If there are two prisms of equal height, and one has a parallelogram as base and the other a triangle, and if the parallelogram is double the triangle, then the prisms are equal. Let ABCDEF and GHKLMN be two prisms of equal height, let one have the parallelogram AF as base, and the other the triangle GHK, and let the parallelogram AF be double the triangle GHK. I say that the prism ABCDEF equals the prism GHKLMN. Complete the solids AO and GP. Since the parallelogram AF is double the triangle GHK, and the parallelogram HK is also double the triangle GHK, therefore the parallelogram AF equals the parallelogram HK. I.34 But parallelepipedal solids on equal bases of the same height equal one another, therefore the solid AO equals the solid GP. XI.31 And the prism ABCDEF is half of the solid AO, and the prism GHKLMN is half of the solid GP, therefore the prism ABCDEF equals the prism GMKLMN. XI.28 Therefore, if there are two prisms of equal height, and one has a parallelogram as base and the other a triangle, and if the parallelogram is double the triangle, then the prisms are equal. Q. E. D. This proposition is designed specifically to take care of a situation that occurs in propositions XII.3 and XII.4 on the way to proving XII.5 concerning the volume of a pyramid. Both of the prisms in this proposition are triangular, but the base of the first is taken to be one of the parallelograms ACFE on its side while the base of the second is a triangular end GHK. To say that they have the height means the distance from the vertex B to the plane of the parallelogram ACEF is the same as the distance from the vertex M to the plane of the triangle GHK. When the solids are completed, they are doubled to create two parallelepipeds of the same height and equal bases, which therefore are equal, and so are their halves, the original prisms. Next book: Book XII introduction Previous proposition: XI.38 Book XI introduction © 1997 D.E.Joyce Clark University Definitions I Definition 1 Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure. Two magnitudes A and B of the same kind are commensurable if there is another magnitude C of the same kind such that both are multiples of C, that is, there are numbers m and n such that nC = A and mC = B. See definition V.Def.5 for the defintion of equality of ratios (also known as a proportion). If the two magnitudes are not commensurable, then they're called incommensurable. Propositions X.2 through X.8 and several later ones deal with commensurable and incommensurable magnitudes. In particular X.5 and X.6 state that two magnitudes are commensurable if and only if their ratio is the ratio of a number to a number. For example, if nC = A and mC = B, then the ratio of magnitudes A:B is the same as the ratio of numbers m:n. And conversely, if A:B = m:n, then the 1/nth part of A equals the 1/mth part of B. Ratios of numbers are known to modern mathematicians as rational numbers while other ratios are known as irrational numbers. Unfortunately, Euclid used the words "rational" and "irrational" in a different way in Definition 3, see below. Definition 2 Straight lines are commensurable in square when the squares on them are measured by the same area, and incommensurable in square when the squares on them cannot possibly have any area as a common measure. Note that this definition only applies to lines, that is, only lines are ever said to be "commensurable in square." Certainly, commensurable lines are also commensurable in square, but lines can be commensurable in square but not commensurable, in other words, "commensurable is square only." The most famous example of this phenomenon consists of the side A and the diagonal B of a square. They are commensurable in square since the square on B is twice the square on A, by I.47. But they are not commensurable lines. In modern terms we would say that the square root of 2 is not a rational number. Definition 3 With these hypotheses, it is proved that there exist straight lines infinite in multitude which are commensurable and incommensurable respectively, some in length only, and others in square also, with an assigned straight line. Let then the assigned straight line be called rational, and those straight lines which are commensurable with it, whether in length and in square, or in square only, rational, but those that are incommensurable with it irrational. The proof referred to at the beginning of this definition is that of X.10 which finds lines commensurable in square only, and lines incommensurable in square. Euclid uses the words "rational" and "irrational" differently than mathematicians both before and after him. The usual uses of these words correspond to commensurable and incommensurable, respectively. But when applied to lines Euclid makes them correspond to commensurable in square and incommensurable in square. First, one line is chosen as a standard, then another line is called rational if it is commensurable in square, and irrational if not. Thus, the diagonal on the square on the standard line is rational, even though it's incommensurable with the standard line, since it's commensurable in square with it. Definition 4 And the let the square on the assigned straight line be called rational, and those areas which are commensurable with it rational, but those which are incommensurable with it irrational, and the straight lines which produce them irrational, that is, in case the areas are squares, the sides themselves, but in case they are any other rectilineal figures, the straight lines on which are described squares equal to them. Although Euclid uses "rational" in an unusual way for lines, he uses it in the usual way for areas, so that an area is rational, according to Euclid, if it's commensurable with the standard square, and irrational otherwise. Book X Introduction Proposition X.1. © 1996 D.E.Joyce Clark University Proposition 1 Two unequal magnitudes being set out, if from the greater there is subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process is repeated continually, then there will be left some magnitude less than the lesser magnitude set out. Let AB and C be two unequal magnitudes of which AB is the greater. I say that, if from AB there is subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process is repeated continually, then there will be left some magnitude which is less than the magnitude C. Some multiple DE of C is greater than AB. cf. V.Def.4 Divide DE into the parts DF, FG, and GE equal to C. From AB subtract BH greater than its half, and from AH subtract HK greater than its half, and repeat this process continually until the divisions in AB are equal in multitude with the divisions in DE. Let, then, AK, KH, and HB be divisions equal in multitude with DF, FG, and GE. Now, since DE is greater than AB, and from DE there has been subtracted EG less than its half, and, from AB, BH greater than its half, therefore the remainder GD is greater than the remainder HA. And, since GD is greater than HA, and there has been subtracted from GD the half GF, and from HA, HK greater than its half, therefore the remainder DF is greater than the remainder AK. But DF equals C, therefore C is also greater than AK. Therefore AK is less than C. Therefore there is left of the magnitude AB the magnitude AK which is less than the lesser magnitude set out, namely C. Therefore, two unequal magnitudes being set out, if from the greater there is subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process is repeated continually, then there will be left some magnitude less than the lesser magnitude set out. Q.E.D. And the theorem can similarly be proven even if the parts subtracted are halves. The proof begins with two magnitudes C and AB and claims that some multiple of C is greater then AB. Definition V.Def.4 is not a justification for this statement. Euclid himself proved that a horn angle is less than any rectilinear angle in proposition III.16 and must have recognized that if the magnitude C is a horn angle, and the magnitude AB is a rectilinear angle, then no multiple of C is greater than AB. Nonetheless, he did not qualify this proposition to say that it only holds for certain kinds of magnitudes. Use of this proposition This proposition is the foundation of the method of exhaustion of Book XII. It is not used in the rest of Book X and would, perhaps, be better placed at the beginning of Book XII. This method is used in the propositions concerning areas of circles and volumes of solids. It is specifically used in propositions XII.2, XII.5, XII.10, XII.11, XII.12, and XII.16. Book X Introduction Definitions X.I Proposition X.2. © 1996 D.E.Joyce Clark University Proposition 2 If, when the less of two unequal magnitudes is continually subtracted in turn from the greater that which is left never measures the one before it, then the two magnitudes are incommensurable. There being two unequal magnitudes AB and CD, with AB being the less, when the less is continually subtracted in turn from the greater, let that which is left over never measure the one before it. I say that the magnitudes AB and CD are incommensurable. If they are commensurable, then some magnitude E measures them. Let AB, measuring FD, leave CF less than itself, let CF measuring BG, leave AG less than itself, and let this process be repeated continually, until there is left some magnitude which is less than E. Suppose this done, and let there be left AG less than E. Then, since E measures AB, while AB measures DF, therefore E also measures FD. But it measures the whole CD also, therefore it also measures the remainder CF. But CF measures BG, therefore E also measures BG. But it measures the whole AB also, therefore it also measures the remainder AG, the greater the less, which is impossible. Therefore no magnitude measures the magnitudes AB and CD. Therefore the magnitudes AB and CD are incommensurable. X.Def.1 Therefore, if, when the less of two unequal magnitudes is continually subtracted in turn from the greater that which is left never measures the one before it, then the two magnitudes are incommensurable. Q.E.D. Antenaresis (also called the Euclidean algorithm), first used in proposition VII.1, is again used in this proposition. Beginning with two magnitudes, the smaller, whichever it is, is repeated subtracted from the larger. Proposition VII.1 concerns relatively prime numbers. It is similar to this proposition, but its conclusion is different. Heath claims that Euclid uses X.1 to prove this proposition, in particular, to show that antenaresis eventually leaves some magnitude which is less than E. It is hard to tell what Euclid thought his justification was. Since both magnitudes are multiples of E, whatever justification Euclid intended back in proposition VII.2 works just as well here. Euclid did, however, put X.1 just before this proposition, perhaps for an intended logical connection. If so, there is a missing statement to the effect that GB is greater than half of AB, and so forth, so that X.1 might be invoked. An example of incommensurable magnitudes Consider the 36°-72°-72° triangle constructed ABC in proposition IV.10. This triangle was used in the following proposition IV.11 to construct regular pentagons. When its base BC is subtracted from a side AC then the remainder CD is the base of a similar triangle BCD. Likewise, when the base CD of this new triangle is subtracted from its side BD then the remainder DE is the base of yet another smaller similar triangle CDE. And so forth. Thus, when we begin with the two lines AB and BC and apply the algorithm of antenaresis to them, we get a series of lines which never ends AB, BC, CD, DE, EF, and so forth, and these lines form a never-ending continued proportion. AB:BC = BC:CD = CD:DE = DE:EF = ... Thus, according to this proposition, the two quantities AB and BC are incommensurable. Cutting the line AB at C to make this ratio AB:BC is called in VI.Def.3 cutting AB into extreme and mean ratio. A more modern name for this ratio is the "golden ratio." a series of triangles Use of this proposition This proposition is used in the next one. Book X Introduction Proposition X.1 Proposition X.3. © 1996 D.E.Joyce Clark University Proposition 3 To find the greatest common measure of two given commensurable magnitudes. Let the two given commensurable magnitudes be AB and CD with AB the less. It is required to find the greatest common measure of AB and CD. Now the magnitude AB either measures CD or it does not. If it measures it, and it does measures itself, then AB is a common measure of AB and CD. And it is manifest that it is also the greatest, for a greater magnitude than the magnitude AB does not measure AB. Next, let AB not measure CD. Then, if the less is continually subtracted in turn from the greater, then that which is left over will sometime measure the one before it, because AB and CD are not incommensurable. X.2 Let AB, measuring ED, leave EC less than itself, let EC, measuring FB, leave AF less than itself, and let AF measure CE. Since, then, AF measures CE, while CE measures FB, therefore AF also measures FB. But it measures itself also, therefore AF also measures the whole AB. But AB measures DE, therefore AF also measures ED. But it measures CE also, therefore it also measures the whole CD. Therefore AF is a common measure of AB and CD. I say next that it is also the greatest. If not, then there there is some magnitude G greater than AF which measures AB and CD. Since then G measures AB, while AB measures ED, therefore G also measures ED. But it measures the whole CD also, therefore G measures the remainder CE. But CE measures FB, therefore G also measures FB. But it measures the whole AB also, and it therefore measures the remainder AF, the greater the less, which is impossible. Therefore no magnitude greater than AF measures AB and CD. Therefore AF is the greatest common measure of AB and CD. Therefore the greatest common measure of the two given commensurable magnitudes AB and CD has been found. Q.E.D. Corollary. From this it is manifest that, if a magnitude measures two magnitudes, then it also measures their greatest common measure. This is the same proposition as VII.3 with the same diagram and the same corollary, only the terminology is slightly different. This proposition and its corollary are used in the next proposition. Book X Introduction Proposition X.2 Proposition X.4. © 1996 D.E.Joyce Clark University Proposition 4 To find the greatest common measure of three given commensurable magnitudes. Let A, B, and C be the three given commensurable magnitudes. It is required to find the greatest common measure of A, B, and C. Take the greatest common measure D of the two magnitudes A and B. X.3 Either D measures C, or it does not measure it. First, let it measure it. Since then D measures C, while it also measures A and B, therefore D is a common measure of A, B, and C. And it is manifest that it is also the greatest, for a greater magnitude than the magnitude D does not measure A and B. Next, let D not measure C. I say first that C and D are commensurable. Since A, B, and C are commensurable, some magnitude measures them, and this of course measures A and B also, so that it also measures the greatest common measure of A and B, namely D. X.3.Cor. But it also measures C, so that the said magnitude measures C and D, therefore C and D are commensurable. Now take their greatest common measure E. X.3 Since E measures D, while D measures A and B, therefore E also measures A and B. But it measures C also, therefore E measures A, B, and C. Therefore E is a common measure of A, B, and C. I say next that it is also the greatest. For, if possible, let there be some magnitude F greater than E, and let it measure A, B, and C. Now, since F measures A, B, and C, it also measures A and B, and therefore measures the greatest common measure of A and B. X.3,Cor. But the greatest common measure of A and B is D, therefore F measures D. But it measures C also, therefore F measures C and D. Therefore F also measures the greatest common measure of C and D. But that is E, therefore F measures E, the greater the less, which is impossible. X.3,Cor. Therefore no magnitude greater than the magnitude E measures A, B, and C. Therefore E is the greatest common measure of A, B, and C if D does not measure C, but if it measures it, then D is itself the greatest common measure. Therefore the greatest common measure of the three given commensurable magnitudes has been found. Corollary. From this it is manifest that, if a magnitude measures three magnitudes, then it also measures their greatest common measure. The greatest common measure can be found similarly for more magnitudes, and the corollary extended. Q.E.D. This is the same proposition as VII.3. This proposition and the last explain how to find the common measure of commensurable magnitudes. Although not explicitly invoked, they bear on the succeeding propositions which use common measures of commensurable magnitudes. Book X Introduction Proposition X.3 Proposition X.5. © 1996 D.E.Joyce Clark University Proposition 5 Commensurable magnitudes have to one another the ratio which a number has to a number. Let A and B be commensurable magnitudes. I say that A has to B the ratio which a number has to a number. Since A and B are commensurable, some magnitude C measures them. As many times as C measures A, let so many units be in D, and, as many times as C measures B, let so many units be in E. Since C measures A according to the units in D, while the unit also measures D according to the units in it, therefore the unit measures the number D the same number of times as the magnitude C measures A. Therefore C is to A as the unit is to D. Therefore, inversely, A is to C as D is to the unit. VII.Def.20 V.7.Cor Again, since C measures B according to the units in E, while the unit also measures E according to the units in it, therefore the unit measures E the same number of times as C measures B. Therefore C is to B as the unit is to E. But it was also proved that A is to C as D is to the unit, therefore, ex aequali, A is to B as the number D is to E. V.22 Therefore the commensurable magnitudes A and B have to one another the ratio which the number D has to the number E. Therefore, commensurable magnitudes have to one another the ratio which a number has to a number. Q.E.D. If A = mC and B = nC, then A:B = m:n. The proof here assumes that numbers are magnitudes, that is to say, the two definitions of proportion V.Def.5 and VII.Def.20 are compatible. This proposition is used in X.8, its contrapositive, and a few propositions after that. The next proposition is the converse of this one, and the two following that are its contrapositive, and the contrapositive of this one. It is not clear why these four statements are separated into four propositions, but the following four statements are bundled together into one proposition X.9 instead of being separate. Perhaps originally each group of four was bundled together, but later the first group was separated. Book X Introduction Proposition X.4 Proposition X.6. © 1996 D.E.Joyce Clark University Proposition 6 If two magnitudes have to one another the ratio which a number has to a number, then the magnitudes are commensurable. Let the two magnitudes A and B have to one another the ratio which the number D has to the number E. I say that the magnitudes A and B are commensurable. Divide A into as many equal parts as there are units in D, and let C equal one of them, and let F be made up of as many magnitudes equal to C as there are units in E. Since then there are in A as many magnitudes equal to C as there are units in D, whatever part the unit is of D, the same part is C of A also. Therefore C is to A as the unit is to D. VII.Def.20 But the unit measures the number D, therefore C also measures A. And since C is to A as the unit is to D, therefore, inversely, A is to C as the number D is to the unit. V.7,Cor. Again, since there are in F as many magnitudes equal to C as there are units in E, therefore C is to F as the unit is to E. VII.Def.20 But it was also proved that A is to C as D is to the unit, therefore, ex aequali, A is to F as D is to E. V.22 But D is to E as A is to B, therefore A is to B as it is to F also. V.11 Therefore A has the same ratio to each of the magnitudes B and F. Therefore B equals F. V.9 But C measures F, therefore it measures B also. Further it measures A also, therefore C measures A and B. Therefore A is commensurable with B. Therefore, if two magnitudes have to one another the ratio which a number has to a number, then the magnitudes are commensurable. Q.E.D. Corollary. From this it is manifest that, if there are two numbers as D and E, and a straight line as A, then it is possible to make a straight line F such that the given straight line is to it as the number D is to the number E. And if a mean proportional is also taken between A and F, as B, then A is to F as the square on A is to the square on B, that is, the first is to the third as the figure on the first is to that which is similar and similarly described on the second. V.19,Cor. But A is to F as the number D is to the number E, therefore the number D is to the number E as the figure on the straight line A is to the figure on the straight line B. If A:B = m:n, then, with C equal to A/m, it follows that A = mC and B = nC. The proof assumes that magnitudes are divisible. Not all magnitudes, however, are constructively divisible. For instance, a 60° angle cannot be trisected by a Euclidean construction. An alternate proof which does not depend on divisibility of magnitudes can be based on antenaresis. Use of this propostion The proposition is used in very frequently in Book X starting with the next proposition, its contrapositive. It is also used in proposition XIII.6. The corollary is also used frequently in Book X starting with X.10. Book X Introduction Proposition X.5 Proposition X.7. © 1996 D.E.Joyce Clark University Proposition 7 Incommensurable magnitudes do not have to one another the ratio which a number has to a number. Let A and B be incommensurable magnitudes. I say that A does not have to B the ratio which a number has to a number. If A does have to B the ratio which a number has to a number, then A is commensurable with B. X.6 But it is not, therefore A does not have to B the ratio which a number has to a number. Therefore, incommensurable magnitudes do not have to one another the ratio which a number has to a number. Q.E.D. This proposition is the contrapositive of the last one. It is used in X.11. Book X Introduction Proposition X.6 Proposition X.8. © 1996 D.E.Joyce Clark University Proposition 8 If two magnitudes do not have to one another the ratio which a number has to a number, then the magnitudes are incommensurable. Let the two magnitudes A and B not have to one another the ratio which a number has to a number. I say that the magnitudes A and B are incommensurable. For, if they are commensurable, then A has to B the ratio which a number has to a number. X.5 But it does not, therefore the magnitudes A and B are incommensurable. Therefore, if two magnitudes do not have to one another the ratio which a number has to a number, then the magnitudes are incommensurable. Q.E.D. This proposition is the contrapositive of X.5. It is used in frequently in X.11. Book X Introduction Proposition X.7 Proposition X.9. © 1996 D.E.Joyce Clark University Proposition 9 The squares on straight lines commensurable in length have to one another the ratio which a square number has to a square number; and squares which have to one another the ratio which a square number has to a square number also have their sides commensurable in length. But the squares on straight lines incommensurable in length do not have to one another the ratio which a square number has to a square number; and squares which do not have to one another the ratio which a square number has to a square number also do not have their sides commensurable in length either. Let A and B be commensurable in length. I say that the square on A has to the square on B the ratio which a square number has to a square number. Since A is commensurable in length with B, therefore A has to B the ratio which a number has to a number. Let it have to it the ratio which C has to D. X.5 Since then A is to B as C is to D, while the ratio of the square on A to the square on B is duplicate of the ratio of A to B, for similar figures are in the duplicate ratio of their corresponding sides, and the ratio of the square on C to the square on D is duplicate of the ratio of C to D, for between two square numbers there is one mean proportional number, and the square number has to the square number the ratio duplicate of that which the side has to the side, therefore the square on A is to the square on B as the square on C is to the square on D. VI.20,Cor. VIII.11 Next, as the square on A is to the square on B, so let the square on C be to the square on D. I say that A is commensurable in length with B. Since the square on A is to the square on B as the square on C is to the square on D, while the ratio of the square on A to the square on B is duplicate of the ratio of A to B, and the ratio of the square on C to the square on D is duplicate of the ratio of C to D, therefore A is to B as C is to D. Therefore A has to B the ratio which the number C has to the number D. Therefore A is commensurable in length with B. X.6 Next, let A be incommensurable in length with B. I say that the square on A does not have to the square on B the ratio which a square number has to a square number. If the square on A does have to the square on B the ratio which a square number has to a square number, then A is commensurable with B. Above But it is not, therefore the square on A does not have to the square on B the ratio which a square number has to a square number. Finally, let the square on A not have to the square on B the ratio which a square number has to a square number. I say that A is incommensurable in length with B. For, if A is commensurable with B, then the square on A has to the square on B the ratio which a square number has to a square number. Above But it does not, therefore A is not commensurable in length with B. Therefore, the squares on straight lines commensurable in length have to one another the ratio which a square number has to a square number; and squares which have to one another the ratio which a square number has to a square number also have their sides commensurable in length. But the squares on straight lines incommensurable in length do not have to one another the ratio which a square number has to a square number; and squares which do not have to one another the ratio which a square number has to a square number also do not have their sides commensurable in length either. Q.E.D. Corollary. And it is manifest from what has been proved that straight lines commensurable in length are always commensurable in square also, but those commensurable in square are not always also commensurable in length. Lemma. It has been proved in the arithmetical books that similar plane numbers have to one another the ratio which a square number has to a square number, and that, if two numbers have to one another the ratio which a square number has to a square number, then they are similar plane numbers. VIII.26 and converse Corollary 2. And it is manifest from these propositions that numbers which are not similar plane numbers, that is, those which do not have their sides proportional, do not have to one another the ratio which a square number has to a square number. For, if they have, then they are similar plane numbers, which is contrary to the hypothesis. Therefore numbers which are not similar plane numbers do not have to one another the ratio which a square number has to a square number. This proposition has a statement, its converse, and its and its converse's contrapositives. It says lines are commensurable if and only if the squares on them are in the ratio of a square number to a square number. For example, the diagonal of a square and the side of the square are not commensurable since the squares on them are in the ratio 2:1, and 2:1 is not the ratio of a square number to a square number, see the guide to proposition VIII.8. The proposition is used repeatedly in Book X starting with the next. It is also used in Book XIII in propositions XIII.6 and XIII.11. Book X Introduction Proposition X.8 Proposition X.10. © 1996, 1998 D.E.Joyce Clark University Proposition 10 To find two straight lines incommensurable, the one in length only, and the other in square also, with an assigned straight line. Let A be the assigned straight line. It is required to find two straight lines incommensurable, the one in length only, and the other in square also, with A. Set out two numbers B and C which do not have to one another the ratio which a square number has to a square number, that is, which are not similar plane numbers, and let it be contrived that B is to C as the square on A is to the square on D, for we have learned how to do this. X.6,Cor. Therefore the square on A is commensurable with the square on D. X.6 And, since B does not have to C the ratio which a square number has to a square number, therefore neither has the square on A to the square on D the ratio which a square number has to a square number, therefore A is incommensurable in length with D. X.9 Take a mean proportional E between A and D. Then A is to D as the square on A is to the square on E. V.Def.9 But A is incommensurable in length with D, therefore the square on A is also incommensurable with the square on E. Therefore A is incommensurable in square with E. X.11 Therefore two straight lines D and E have been found incommensurable, D in length only, and E in square and of course in length also, with the assigned straight line A. Q.E.D. This proposition exhibits the lines promised in X.Def.I.3. Just take a line D so that the square on A to the square on D is the ratio of two numbers which are not a square number to a square number. For instance, if A is the side of a square and D the diagonal of that square, then the square on A to the square on D is in the ratio 1:2, which is not the ratio of square number to a square number. Therefore D is commensurable in square only with A. (The ratio D:A is the square root of 2.) Next, if E is the mean proportional between A and D, then E is incommensurable in square with A. (The ratio E:A is the fourth root of 2.) It is certain that this proposition is not genuine. For one thing, its proof uses the next proposition. Also, the phrase "for we have learned how to do this" is the sort of thing a student would write. Finally, in the manuscript P (the primary one used by Peyrard and Heiberg) this proposition is not numbered and the next one is numbered 10. Although not genuine, this proposition ought to be, since it is used in proposition X.27 and others. Book X Introduction Proposition X.9 Proposition X.11. © 1996 D.E.Joyce Clark University Proposition 11 If four magnitudes are proportional, and the first is commensurable with the second, then the third also is commensurable with the fourth; but, if the first is incommensurable with the second, then the third also is incommensurable with the fourth. Let A, B, C, and D be four magnitudes in proportion, so that A is to B as C is to D, and let A be commensurable with B. I say that C is also commensurable with D. Since A is commensurable with B, therefore A has to B the ratio which a number has to a number. X.5 And A is to B as C is to D, therefore C also has to D the ratio which a number has to a number. Therefore C is commensurable with D. V.11 X.6 Next, let A be incommensurable with B. I say that C is also incommensurable with D. Since A is incommensurable with B, therefore A does not have to B the ratio which a number has to a number. X.7 And A is to B as C is to D, therefore neither has C to D the ratio which a number has to a number. Therefore C is incommensurable with D. V.11 X.8 Therefore, if four magnitudes are proportional, and the first is commensurable with the second, then the third also is commensurable with the fourth; but, if the first is incommensurable with the second, then the third also is incommensurable with the fourth. Q.E.D. The proof if very direct. If A:B = C:D, and the first ratio equals a numeric ratio, then the second equals that, too, but if the first is not a numeric ratio, then neither is the second. This proposition is used in repeatedly in Book X starting with X.14. It is also used in the previous proposition which was, no doubt, not in the original Elements. Book X Introduction Proposition X.10 Proposition X.12. © 1996 D.E.Joyce Clark University Proposition 12 Magnitudes commensurable with the same magnitude are also commensurable with one another. Let each of the magnitudes A and B be commensurable with C. I say that A is also commensurable with B. Since A is commensurable with C, therefore A has to C the ratio which a number has to a number. Let it have the ratio which D has to E. Again, since C is commensurable with B, therefore C has to B the ratio which a number has to a number. Let it have the ratio which F has to G. X.5 And, given any number of ratios we please, namely the ratio which D has to E and that which F has to G, take the numbers H, K, and L continuously in the given ratios, so that D is to E as H is to K, and F is to G as K is to L. VIII.4 Since A is to C as D is to E, while D is to E as H is to K, therefore A is to C as H is to K. Again, since C is to B as F is to G, while F is to G as K is to L, therefore C is to B as K is to L. V.11 But A is to C as H is to K, therefore, ex aequali, A is to B as H is to L. V.22 Therefore A has to B the ratio which a number has to a number. Therefore A is commensurable with B. X.6 Therefore, magnitudes commensurable with the same magnitude are also commensurable with one another. Q.E.D. The proof is primarily an application of VIII.4. This proposition is used in frequently in Book X starting with the next proposition. It is also used in XIII.11. Book X Introduction Proposition X.11 Proposition X.13. © 1996 D.E.Joyce Clark University Proposition 13 If two magnitudes are commensurable, and one of them is incommensurable with any magnitude, then the remaining one is also incommensurable with the same. Let A and B be two commensurable magnitudes, and let one of them, A, be incommensurable with some other magnitude C. I say that the remaining one, B, is also incommensurable with C. If B is commensurable with C, while A is also commensurable with B, then A is also commensurable with C. X.12 But it is also incommensurable with it, which is impossible. Therefore B is not commensurable with C. Therefore it is incommensurable with it. Therefore, if two magnitudes are commensurable, and one of them is incommensurable with any magnitude, then the remaining one is also incommensurable with the same. Q.E.D. The proposition is a logical variant of the previous. It is used in very frequently in Book X starting with X.18. Book X Introduction Proposition X.12 Proposition X.14. © 1996 D.E.Joyce Clark University Proposition 14 Lemma. Given two unequal straight lines, to find by what square the square on the greater is greater than the square on the less. Let AB and C be the given two unequal straight lines, and let AB be the greater of them. It is required to find by what square the square on AB is greater than the square on C. Describe the semicircle ADK on AB, fit AD into it equal to C, and join DB. IV.1 It is then manifest that the angle ADB is right, and that the square on AB is greater than the square on AD, that is, C, by the square on DB. III.31 I.47 Similarly also, if two straight lines are given, then the straight line the square on which equals the sum of the squares on them is found in this manner. Let AD and DB be the given two straight lines, and let it be required to find the straight line the square on which equals the sum of the squares on them. Place them so as to contain a right angle ADB, and join AB. It is again manifest that the straight line the square on which equals the sum of the squares on AD and DB is AB. I.47 Proposition 14 If four straight lines are proportional, and the square on the first is greater than the square on the second by the square on a straight line commensurable with the first, then the square on the third is also greater than the square on the fourth by the square on a third line commensurable with the third. And, if the square on the first is greater than the square on the second by the square on a straight line incommensurable with the first, then the square on the third is also greater than the square on the fourth by the square on a third line incommensurable with the third. Let A, B, C, and D be four straight lines in proportion, so that A is to B as C is to D, and let the square on A be greater than the square on B by the square on E, and let the square on C be greater than the square on D by the square on F. Lemma I say that, if A is commensurable with E, then C is also commensurable with F, and, if A is incommensurable with E, then C is also incommensurable with F. Since A is to B as C is to D, therefore the square on A is to the square on B as the square on C is to the square on D. VI.22 But the sum of the squares on E and B equals the square on A, and the sum of the squares on D and F equals the square on C. Therefore the sum of the squares on E and B is to the square on B as the sum of the squares on D and F is to the square on D. Therefore, taken separately, the square on E is to the square on B as the square on F is to the square on D. Therefore E is to B as F is to D. Therefore, inversely, B is to E as D is to F. V.17 VI.22 V.7.Cor But A is to B as C is to D, therefore, ex aequali, A is to E as C is to F. V.22 Therefore, if A is commensurable with E, then C is also commensurable with F, but if A is incommensurable with E, then C is also incommensurable with F. X.11 Therefore, if four straight lines are proportional, and the square on the first is greater than the square on the second by the square on a straight line commensurable with the first, then the square on the third is also greater than the square on the fourth by the square on a third line commensurable with the third. And, if the square on the first is greater than the square on the second by the square on a straight line incommensurable with the first, then the square on the third is also greater than the square on the fourth by the square on a third line incommensurable with the third. Q.E.D. A little modern algebra clarifies the situation. We assume A:B = C:D. Then if (A2 B2) : A is a numeric ratio, then so is (C2 D2) : C. It's simply because (A2 B2) : A = (C2 D2) : C. The lemma is the same as the lemma for proposition XI.23. The proposition is used in several propositions in Book X starting with X.31. Book X Introduction Proposition X.13 Proposition X.15. © 1996 D.E.Joyce Clark University Proposition 15 If two commensurable magnitudes are added together, then the whole is also commensurable with each of them; and, if the whole is commensurable with one of them, then the original magnitudes are also commensurable. Let the two commensurable magnitudes AB and BC be added together. I say that the whole AC is also commensurable with each of the magnitudes AB and BC. Since AB and BC are commensurable, some magnitude D measures them. Since then D measures AB and BC, therefore it also measures the whole AC. But it measures AB and BC also, therefore D measures AB, BC, and AC. Therefore AC is commensurable with each of the magnitudes AB and BC. X.Def.1 Next, let AC be commensurable with AB. I say that AB and BC are also commensurable. Since AC and AB are commensurable, some magnitude D measures them. Since then D measures CA and AB, therefore it also measures the remainder BC. But it measures AB also, therefore D measures AB and BC. Therefore AB and BC are commensurable. X.Def.1 Therefore, if two commensurable magnitudes are added together, then the whole is also commensurable with each of them; and, if the whole is commensurable with one of them, then the original magnitudes are also commensurable. Q.E.D. This fundamental proposition on commensurability of sums and differences is used in very frequently in Book X starting with X.17. It is also used in XIII.11. Book X Introduction Proposition X.14 Proposition X.16. © 1996 D.E.Joyce Clark University Proposition 16 If two incommensurable magnitudes are added together, the sum is also incommensurable with each of them; but, if the sum is incommensurable with one of them, then the original magnitudes are also incommensurable. Let the two incommensurable magnitudes AB and BC be added together. I say that the whole AC is also incommensurable with each of the magnitudes AB and BC. For, if CA and AB are not incommensurable, then some magnitude D measures them. Since then D measures CA and AB, therefore it also measures the remainder BC. But it also measures AB, therefore D measures AB and BC. Therefore AB and BC are commensurable, but they were also, by hypothesis, incommensurable, which is impossible. Therefore no magnitude measures CA and AB. Therefore CA and AB are incommensurable. X.Def.1 Similarly we can prove that AC and CB are also incommensurable. Therefore AC is incommensurable with each of the magnitudes AB and BC. Next, let AC be incommensurable with one of the magnitudes AB or BC. First, let it be incommensurable with AB. I say that AB and BC are also incommensurable. For, if they are commensurable, then some magnitude D measures them. Since, then, D measures AB and BC, therefore it also measures the whole AC. But it also measures AB, therefore D measures CA and AB. Therefore CA and AB are commensurable, but they were also, by hypothesis, incommensurable, which is impossible. Therefore no magnitude measures AB and BC. Therefore AB and BC are incommensurable. X.Def.1 Therefore, if two incommensurable magnitudes are added together, the sum is also incommensurable with each of them; but, if the sum is incommensurable with one of them, then the original magnitudes are also incommensurable. Q.E.D. This proposition is a logical variant of the previous one, but it is proved afresh. It is used in several others in Book X starting with X.18. Book X Introduction Proposition X.15 Proposition X.17. © 1996 D.E.Joyce Clark University Proposition 17 Lemma. If to any straight line there is applied a parallelogram but falling short by a square, then the applied parallelogram equals the rectangle contained by the segments of the straight line resulting from the application. Apply to the straight line AB the parallelogram AD but falling short by the square DB. I say that AD equals the rectangle AC by CB. This is indeed at once manifest, for, since DB is a square, DC equals CB, and AD is the rectangle AC by CD, that is, the rectangle AC by CB. Proposition 17 If there are two unequal straight lines, and to the greater there is applied a parallelogram equal to the fourth part of the square on the less minus a square figure, and if it divides it into parts commensurable in length, then the square on the greater is greater than the square on the less by the square on a straight line commensurable with the greater. And if the square on the greater is greater than the square on the less by the square on a straight line commensurable with the greater, and if there is applied to the greater a parallelogram equal to the fourth part of the square on the less minus a square figure, then it divides it into parts commensurable in length. Let A and BC be two unequal straight lines, of which BC is the greater, and let there be applied to BC a parallelogram equal to the fourth part of the square on the less, A, that is, equal to the square on the half of A but falling short by a square figure. Let this be the rectangle BD by DC, and let BD be commensurable in length with DC. Lemma I say that the square on BC is greater than the square on A by the square on a straight line commensurable with BC. Bisect BC at the point E, and make EF equal to DE. I.10 I.3 Therefore the remainder DC equals BF. And, since the straight line BC was cut into equal parts at E, and into unequal parts at D, therefore the rectangle BD by DC, together with the square on ED, equals the square on EC. II.5 And the same is true of their quadruples, therefore four times the rectangle BD by DC, together with four times the square on DE, equals four times the square on EC. But the square on A equals four times the rectangle BD by DC, and the square on DF equals four times the square on DE, for DF is double DE. And the square on BC equals four times the square on EC, for again BC is double CE. Therefore the sum of the squares on A and DF equals the square on BC, so that the square on BC is greater than the square on A by the square on DF. It is to be proved that BC is also commensurable with DF. Since BD is commensurable in length with DC, therefore BC is also commensurable in length with CD. X.15 But CD is commensurable in length with CD and BF, for CD equals BF. X.6 Therefore BC is also commensurable in length with BF and CD, so that BC is also commensurable in length with the remainder FD. Therefore the square on BC is greater than the square on A by the square on a straight line commensurable with BC. X.12 X.15 Next, let the square on BC be greater than the square on A by the square on a straight line commensurable with BC. Apply to BC a parallelogram equal to the fourth part of the square on A but falling short by a square figure, and let it be the rectangle BD by DC. It is to be proved that BD is commensurable in length with DC. With the same construction, we can prove similarly that the square on BC is greater than the square on A by the square on FD. But the square on BC is greater than the square on A by the square on a straight line commensurable with BC. X.15 Therefore BC is commensurable in length with FD, so that BC is also commensurable in length with the remainder, the sum of BF and DC. But the sum of BF and DC is commensurable with DC, so that BC is also commensurable in length with CD, and therefore, taken separately, BD is commensurable in length with DC. X.6 X.12 X.15 Therefore, if there are two unequal straight lines, and to the greater there is applied a parallelogram equal to the fourth part of the square on the less minus a square figure, and if it divides it into parts commensurable in length, then the square on the greater is greater than the square on the less by the square on a straight line commensurable with the greater. And if the square on the greater is greater than the square on the less by the square on a straight line commensurable with the greater, and if there is applied to the greater a parallelogram equal to the fourth part of the square on the less minus a square figure, then it divides it into parts commensurable in length. Q.E.D. Here is an algebraic description. Let b denote BC. Then DC is (b - (b2 A2))/2. Then the proposition asserts that the ratio b : (b - (b2 A2))/2 is a numeric ratio if and only if the ratio (b2 A2 ) : A is a numeric ratio. The lemma is also used in the next proposition. The proposition is used in several times in Book X starting with X.54. Book X Introduction Proposition X.16 Proposition X.18. © 1996 D.E.Joyce Clark University Proposition 18 If there are two unequal straight lines, and to the greater there is applied a parallelogram equal to the fourth part of the square on the less but falling short by a square, and if it divides it into incommensurable parts, then the square on the greater is greater than the square on the less by the square on a straight line incommensurable with the greater. And if the square on the greater is greater than the square on the less by the square on a straight line incommensurable with the greater, and if there is applied to the greater a parallelogram equal to the fourth part of the square on the less but falling short by a square, then it divides it into incommensurable parts. Let A and BC be two unequal straight lines, of which BC is the greater, and to BC let there be applied a parallelogram equal to the fourth part of the square on the less, A, but falling short by a square. Let this be the rectangle BD by DC, and let BD be incommensurable in length with DC. X.17,Lemma I say that the square on BC is greater than the square on A by the square on a straight line incommensurable with BC. With the same construction as before, we can prove similarly that the square on BC is greater than the square on A by the square on FD. It is to be proved that BC is incommensurable in length with DF. Since BD is incommensurable in length with DC, therefore BC is also incommensurable in length with CD. X.16 But DC is commensurable with the sum of BF and DC, therefore BC is incommensurable with the sum of BF and DC, so that BC is also incommensurable in length with the remainder FD. X.6 X.13 X.16 And the square on BC is greater than the square on A by the square on FD, therefore the square on BC is greater than the square on A by the square on a straight line incommensurable with BC. Next, let the square on BC be greater than the square on A by the square on a straight line incommensurable with BC. Apply to BC a parallelogram equal to the fourth part of the square on A but falling short by a square. Let this be the rectangle BD by DC. It is to be proved that BD is incommensurable in length with DC. With the same construction, we can prove similarly that the square on BC is greater than the square on A by the square on FD. But the square on BC is greater than the square on A by the square on a straight line incommensurable with BC, therefore BC is incommensurable in length with FD, so that BC is also incommensurable with the remainder, the sum of BF and DC. X.16 But the sum of BF and DC is commensurable in length with DC, therefore BC is also incommensurable in length with DC, so that, taken separately, BD is also incommensurable in length with DC. X.6 X.13 X.16 Therefore, if there are two unequal straight lines, and to the greater there is applied a parallelogram equal to the fourth part of the square on the less but falling short by a square, and if it divides it into incommensurable parts, then the square on the greater is greater than the square on the less by the square on a straight line incommensurable with the greater. And if the square on the greater is greater than the square on the less by the square on a straight line incommensurable with the greater, and if there is applied to the greater a parallelogram equal to the fourth part of the square on the less but falling short by a square, then it divides it into incommensurable parts. Q.E.D. This proposition is a logical variant of the last. It is used in frequently in Book X starting with X.33. Book X Introduction Proposition X.17 Proposition X.19. © 1996 D.E.Joyce Clark University Proposition 19 Lemma. Since it has been proved that straight lines commensurable in length are always commensurable in square also, while those commensurable in square are not always commensurable in length also, but can of course be either commensurable or incommensurable in length, it is manifest that, if any straight line is commensurable in length with a given rational straight line, it is called rational and commensurable with the other not only in length but in square also, since straight lines commensurable in length are always commensurable in square also. But, if any straight line is commensurable in square with a given rational straight line, then, if it is also commensurable in length with it, in this case it is also called rational and commensurable with it both in length and in square, but, if again any straight line, being commensurable in square with a given rational straight line, is incommensurable in length with it, in this case it is also called rational but commensurable in square only. Proposition 19 The rectangle contained by rational straight lines commensurable in length is rational. Let the rectangle AC be contained by the rational straight lines AB and BC commensurable in length. I say that AC is rational. Describe the square AD on AB. Then AD is rational. I.46 X.Def.4 And, since AB is commensurable in length with BC, while AB equals BD, therefore BD is commensurable in length with BC. And BD is to BC as DA is to AC. VI.1 Therefore DA is commensurable with AC. X.11 But DA is rational, therefore AC is also rational. X.Def.4 Therefore, the rectangle contained by rational straight lines commensurable in length is rational. Q.E.D. This is the first proposition that deals with rational lines and rational squares. As required by definitions X.Def.I.3 and X.Def.I.3, there is some assigned straight line to act as a standard to which other lines and squares are compared for rationality. That line is usually not mentioned in the propositions. In this proposition, it is assumed that both sides of the rectangle AB and BC are rational lines. That means these lines are commensurable in square to the standard line, that is, their squares are commensurable with the standard square. It is also assumed that AB and BC are commensurable with each other. Therefore the rectangle AC is commensurable with the square on AB, but that's commensurable with the standard square, so the rectangle AC is too. The proposition is used several times starting with X.25. The lemma is used in X.23. The next proposition is a converse of this one, but the language obscures that from notice. Book X Introduction Proposition X.18 Proposition X.20. © 1996 D.E.Joyce Clark University Proposition 20 If a rational area is applied to a rational straight line, then it produces as breadth a straight line rational and commensurable in length with the straight line to which it is applied. Let the rational area AC be applied to AB, a straight line once more rational in any of the aforesaid ways, producing BC as breadth. I say that BC is rational and commensurable in length with BA. Describe the square AD on AB. Then AD is rational. I.46 X.Def.4 But AC is also rational, therefore DA is commensurable with AC. And DA is to AC as DB is to BC. Therefore DB is also commensurable with BC, and DB equals BA. Therefore AB is also commensurable with BC. VI.1 X.11 But AB is rational, C therefore BC is also rational and commensurable in length with AB. Therefore, if a rational area is applied to a rational straight line, then it produces as breadth a straight line rational and commensurable in length with the straight line to which it is applied. Q.E.D. This proposition is a converse of the last, except that it's preceded by applying an area to a straight line to get the rectangle. That would be more evident if it read "If one side of a rational rectangle is rational, then the other side is rational and commensurable with the first. This proposition is used frequently in Book X starting with X.26. Book X Introduction Proposition X.19 Proposition X.21. © 1996 D.E.Joyce Clark University Proposition 21 The rectangle contained by rational straight lines commensurable in square only is irrational, and the side of the square equal to it is irrational. Let the latter be called medial. Let the rectangle AC be contained by the rational straight lines AB and BC commensurable in square only. I say that AC is irrational, and the side of the square equal to it is irrational, and let the latter be called medial. Describe the square AD on AB. Then AD is rational. X.Def.4 And, since AB is incommensurable in length with BC, for by hypothesis they are commensurable in square only, while AB equals BD, therefore DB is also incommensurable in length with BC. And DB is to BC as AD is to AC, therefore DA is incommensurable with AC. VI.1 X.11 But DA is rational, therefore AC is irrational, so that the side of the square AC is also irrational. X.Def.4 Let the latter be called medial. Q.E.D. This proposition is used frequently in Book X starting with the next propostition. Book X Introduction Proposition X.20 Proposition X.22. © 1996 D.E.Joyce Clark University Proposition 22 Lemma. If there are two straight lines, then the first is to the second as the square on the first is to the rectangle contained by the two straight lines. Let FE and EG be two straight lines. I say that FE is to EG as the square on FE is to the rectangle FE by EG. Describe the square DF on FE, and complete GD. Since then FE is to EG as FD is to DG, and FD is the square on FE, and DG the rectangle DE by EG, that is, the rectangle FE by EG, therefore FE is to EG as the square on FE is to the rectangle FE by EG. Similarly the rectangle GE by EF is to the square on EF, that is GD is to FD, as GE is to EF. VI.1 Q.E.D. Proposition 22 The square on a medial straight line, if applied to a rational straight line, produces as breadth a straight line rational and incommensurable in length with that to which it is applied. Let A be medial and CB rational, and let a rectangular area BD equal to the square on A be applied to BC, producing CD as breadth. I say that CD is rational and incommensurable in length with CB. Since A is medial, the square on it equals a rectangular area contained by rational straight lines commensurable in square only. X.21 Let the square on it equal GF. But the square on it also equals BD, therefore BD equals GF. But it is also equiangular with it, and in equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional, therefore, BC is to EG as EF is to CD. VI.14 Therefore the square on BC is to the square on EG as the square on EF is to the square on CD. VI.22 But the square on CB is commensurable with the square on EG, for each of these straight lines is rational, therefore the square on EF is also commensurable with the square on CD. X.11 But the square on EF is rational, therefore the square on CD is also rational. Therefore CD is rational. X.Def.4 And since EF is incommensurable in length with EG, for they are commensurable in square only, while EF is to EG as the square on EF is to the rectangle FE by EG, therefore the square on EF is incommensurable with the rectangle FE by EG. Lemma X.11 But the square on CD is commensurable with the square on EF, for the straight lines are rational in square, and the rectangle DC by CB is commensurable with the rectangle FE by EG, for they equal the square on A, therefore the square on CD is incommensurable with the rectangle DC by CB. X.13 But the square on CD is to the rectangle DC by CB as DC is to CB, therefore DC is incommensurable in length with CB. Lemma X.11 Therefore CD is rational and incommensurable in length with CB. Therefore, the square on a medial straight line, if applied to a rational straight line, produces as breadth a straight line rational and incommensurable in length with that to which it is applied. Q.E.D. This proposition is used frequently in Book X starting with the next propostition. Book X Introduction Proposition X.21 Proposition X.23. © 1996 D.E.Joyce Clark University Proposition 23 A straight line commensurable with a medial straight line is medial. Let A be medial, and let B be commensurable with A. I say that B is also medial. Set out a rational straight line CD. Apply the rectangular area CE to CD equal to the square on A, producing ED as breadth. Then ED is rational and incommensurable in length with CD. And apply the rectangular area CF to CD equal to the square on B, producing DF as breadth. X.22 Since A is commensurable with B, therefore the square on A is also commensurable with the square on B. But EC equals the square on A, and CF equals the square on B, therefore EC is commensurable with CF. And EC is to CF as ED is to DF, therefore ED is commensurable in length with DF. VI.1 X.11 But ED is rational and incommensurable in length with DC, therefore DF is also rational and incommensurable in length with DC. X.13 X.Def.3 Therefore CD and DF are rational and commensurable in square only. But the straight line is medial on which the square equals the rectangle contained by rational straight lines commensurable in square only, therefore the side of the square equals the rectangle CD by DF is medial. X.21 And B is the side of the square equal the rectangle CD by DF, therefore B is medial. Therefore, a straight line commensurable with a medial straight line is medial. Q.E.D. Corollary From this it is manifest that an area commensurable with a medial area is medial. Note And in the same way as was explained in the case of rationals it follows regards medials, that a straight line commensurable in length with a medial straight line is called medial and commensurable with it not only in length but in square also, since, in general, straight lines commensurable in length are always commensurable in square also. X.18,Lemma But, if any straight line is commensurable in square with a medial straight line, then if it is also commensurable in length with it, the straight lines are called, in this case too, medial and commensurable in length and in square, but, if in square only, they are called medial straight lines commensurable in square only. The proposition is used in X.67 and X.104, the corollary in X.33 and many others, and the note in X.27 and a few others. Book X Introduction Proposition X.22 Proposition X.24. © 1996 D.E.Joyce Clark University Proposition 24 The rectangle contained by medial straight lines commensurable in length is medial. Let the rectangle AC be contained by the medial straight lines AB and BC which are commensurable in length. I say that AC is medial. Describe the square AD on AB. Then AD is medial. And, since AB is commensurable in length with BC, while AB equals BD, therefore DB is commensurable in length with BC, so that DA is commensurable with AC. VI.1 X.11 But DA is medial, therefore AC is also medial. X.23,Cor. Therefore, the rectangle contained by medial straight lines commensurable in length is medial. Q.E.D. (Forthcoming) Book X Introduction Proposition X.23 Proposition X.25. © 1996 D.E.Joyce Clark University Proposition 25 The rectangle contained by medial straight lines commensurable in square only is either rational or medial. Let the rectangle AC be contained by the medial straight lines AB and BC which are commensurable in square only. I say that AC is either rational or medial. Describe the squares AD and BE on AB and BC. Then each of the squares AD and BE is medial. Set out a rational straight line FG. Apply the rectangular parallelogram GH to FG equal to AD, producing FH as breadth, apply the rectangular parallelogram MK to HM equal to AC, producing HK as breadth, and further apply similarly NL to KN equal to BE, producing KL as breadth. Then FH, HK, and KL are in a straight line. Since each of the squares AD and BE is medial, and AD equals GH while BE equals NL, therefore each of the rectangles GH and NL is also medial. And they are applied to the rational straight line FG, therefore each of the straight lines FH and KL is rational and incommensurable in length with FG. X.22 And, since AD is commensurable with BE, therefore GH is commensurable with NL. And GH is to NL as FH is to KL, therefore FH is commensurable in length with KL. VI.1 X.11 Therefore FH and KL are rational straight lines commensurable in length, therefore the rectangle FH by KL is rational. X.19 And, since DB equals BA while OB equals BC, therefore DB is to BC as AB is to BO. But DB is to BC as DA is to AC, and AB is to BO as AC is to CO, therefore DA is to AC as AC is to CO. VI.1 But AD equals GH, AC equals MK, and CO equals NL, therefore GH is to MK as MK is to NL. Therefore FH is to HK as HK is to KL. Therefore the rectangle FH by KL equals the square on HK. VI.1 V.11 VI.17 But the rectangle FH by KL is rational, therefore the square on HK is also rational. Therefore HK is rational. And, if it is commensurable in length with FG, then HN is rational, but, if it is incommensurable in length with FG, then KH and HM are rational straight lines commensurable in square only, and therefore HN is medial. X.19 X.21 Therefore HN is either rational or medial. But HN equals AC, therefore AC is either rational or medial. Therefore, the rectangle contained by medial straight lines commensurable in square only is either rational or medial. Q.E.D. (Forthcoming) Book X Introduction Proposition X.24 Proposition X.26. © 1996 D.E.Joyce Clark University Proposition 26 A medial area does not exceed a medial area by a rational area. If possible, let the medial area AB exceed the medial area AC by the rational area DB. Set out a rational straight line EF. Apply to EF the rectangular parallelogram FH equal to AB producing EH as breadth. Subtract the rectangle FG equal to AC. Then the remainder BD equals the remainder KH. But DB is rational, therefore KH is also rational. Since each of the rectangles AB and AC is medial, and AB equals FH while AC equals FG, therefore each of the rectangles FH and FG is also medial. They are applied to the rational straight line EF, therefore each of the straight lines HE and EG is rational and incommensurable in length with EF. X.22 Since DB is rational and equals KH, therefore KH is rational. And it is applied to the rational straight line EF, therefore GH is rational and commensurable in length with EF. X.20 But EG is also rational, and is incommensurable in length with EF, therefore EG is incommensurable in length with GH. X.13 And EG is to GH as the square on EG is to the rectangle EG by GH, therefore the square on EG is incommensurable with the rectangle EG by GH. X.11 But the squares on EG and GH are commensurable with the square on EG, for both are rational, and twice the rectangle EG by GH is commensurable with the rectangle EG by GH, for it is double it, therefore the sum of the squares on EG and GH is incommensurable with twice the rectangle EG by GH. X.6 X.13 Therefore the sum of the squares on EG and GH plus twice the rectangle EG by GH, that is, the square on EH is incommensurable with the sum of the squares on EG and GH. II.4 X.16 But the squares on EG and GH are rational, therefore the square on EH is irrational. X.Def.4 Therefore EH is irrational. But it is also rational, which is impossible. Therefore, a medial area does not exceed a medial area by a rational area. Q.E.D. This proposition is used in several others in Book X starting with X.42. Book X Introduction Proposition X.25 Proposition X.27. © 1996 D.E.Joyce Clark University Proposition 27 To find medial straight lines commensurable in square only which contain a rational rectangle. Set out two rational straight lines A and B commensurable in square only. Take a mean proportional C between A and B. Let it be contrived that A is to B as C is to D. X.10 VI.13 VI.12 Then, since A and B are rational and commensurable in square only, therefore the rectangle A by B, that is, the square on C, is medial. Therefore C is medial. VI.17 X.21 And since A is to B as C is to D, and A and B are commensurable in square only, therefore C and D are also commensurable in square only. X.11 And C is medial, therefore D is also medial. X.23.Note Therefore C and D are medial and commensurable in square only. I say that they also contain a rational rectangle. Since A is to B as C is to D, therefore, alternately, A is to C as B is to D. V.16 But A is to C as C is to B, therefore C is to B as B is to D. Therefore the rectangle C by D equals the square on B. But the square on B is rational, therefore the rectangle C by D is also rational. Therefore medial straight lines commensurable in square only have been found which contain a rational rectangle. Q.E.D. (Forthcoming) Book X Introduction Proposition X.26 Proposition X.28. © 1996 D.E.Joyce Clark University Proposition 28 To find medial straight lines commensurable in square only which contain a medial rectangle. Set out the rational straight lines A, B, and C commensurable in square only. Take a mean proportional D between A and B. Let it be contrived that B is to C as D is to E. X.10 VI.13 VI.12 java applet or image Since A and B are rational straight lines commensurable in square only, therefore the rectangle A by B, that is, the square on D, is medial. Therefore D is medial. VI.17 X.21 And since B and C are commensurable in square only, and B is to C as D is to E, therefore D and E are also commensurable in square only. X.11 But D is medial, therefore E is also medial. X.23,Note Therefore D and E are medial straight lines commensurable in square only. I say next that they also contain a medial rectangle. Since B is to C as D is to E, therefore, alternately, B is to D as C is to E. V.16 But B is to D as D is to A, therefore D is to A as C is to E. Therefore the rectangle A by C equals the rectangle D by E. VI.16 But the rectangle A by C is medial, therefore the rectangle D by E is also medial. X.21 Therefore medial straight lines commensurable in square only have been found which contain a medial rectangle. Q.E.D. Lemma 1 is used in X.48, and the proposition itself is used in X.75. Book X Introduction Proposition X.27 Proposition X.29. © 1996 D.E.Joyce Clark University Proposition 30 To find two rational straight lines commensurable in square only such that the square on the greater is greater than the square on the less by the square on a straight line incommensurable in length with the greater. Set out a rational straight line AB, and two square numbers CE and ED such that their sum CD is not square. Describe the semicircle AFB on AB. Let it be contrived that DC is to CE as the square on BA is to the square on AF, and join FB. X.29.Lemma 2 X.6,Cor. Then, in a similar manner to the preceding, we can prove that BA and AF are rational straight lines commensurable in square only. Since DC is to CE as the square on BA is to the square on AF, therefore, in conversion, CD is to DE as the square on AB is to the square on BF. V.19,Cor. III.31 I.47 But CD does not have to DE the ratio which a square number has to a square number, therefore neither has the square on AB to the square on BF the ratio which a square number has to a square number. Therefore AB is incommensurable in length with BF. X.9 And the square on AB is greater than the square on AF by the square on FB incommensurable with AB. Therefore AB and AF are rational straight lines commensurable in square only, and the square on AB is greater than the square on AF by the square on FB incommensurable in length with AB. Q.E.D. This proposition is used in the next three propositions. Book X Introduction Proposition X.29 Proposition X.31. © 1996 D.E.Joyce Clark University Proposition 31 To find two medial straight lines commensurable in square only, containing a rational rectangle, such that the square on the greater is greater than the square on the less by the square on a straight line commensurable in length with the greater. Set out two rational straight lines A and B commensurable in square only such that the square on A, being the greater, is greater than the square on B the less by the square on a straight line commensurable in length with A. X.29 Let the square on C equal the rectangle A by B. Now the rectangle A by B is medial, therefore the square on C is also medial. Therefore C is also medial. X.21 Let the rectangle C by D equal the square on B. Now the square on B is rational, therefore the rectangle C by D is also rational. And since A is to B as the rectangle A by B is to the square on B, while the square on C equals the rectangle A by B, and the rectangle C by D equals the square on B, therefore A is to B as the square on C is to the rectangle C by D. But the square on C is to the rectangle C by D as C is to D, therefore A is to B as C is to D. But A is commensurable with B in square only, therefore C is also commensurable with D in square only. X.11 And C is medial, therefore D is also medial. X.23,Note Since A is to B as C is to D, and the square on A is greater than the square on B by the square on a straight line commensurable with A, therefore the square on C is greater than the square on D by the square on a straight line commensurable with C. X.14 Therefore two medial straight lines C and D, commensurable in square only and containing a rational rectangle, have been found, and the square on C is greater than the square on D by the square on a straight line commensurable in length with C. Similarly also it can be proved that the square on C exceeds the square on D by the square on a straight line incommensurable with C, when the square on A is greater than the square on B by the square on a straight line incommensurable with A. X.30 Q.E.D. This proposition is used in X.34 and X.35. Book X Introduction Proposition X.30 Proposition X.32. © 1996 D.E.Joyce Clark University Proposition 32 To find two medial straight lines commensurable in square only, containing a medial rectangle, such that the square on the greater is greater than the square on the less by the square on a straight line commensurable with the greater. Set out three rational straight lines A, B, and C commensurable in square only, such that the square on A is greater than the square on C by the square on a straight line commensurable with A. Let the square on D equal the rectangle A by B. X.29 Then the square on D is medial. Therefore D is also medial. X.21 Let the rectangle D by E equal the rectangle B by C. Then since as the rectangle A by is is to the rectangle B by C as A is to C, while the square on D equals the rectangle A by B, and the rectangle D by E equals the rectangle B by C, therefore A is to C as the square on D is to the rectangle D by E. java applet or image But the square on D is to the rectangle D by E as D is to E, therefore A is to C as D is to E. But A is commensurable with C in square only, therefore D is also commensurable with E in square only. X.11 But D is medial, therefore E is also medial. X.23,Note And, since A is to C as D is to E, while the square on A is greater than the square on C by the square on a straight line commensurable with A, therefore the square on D is greater than the square on E by the square on a straight line commensurable with D. X.14 I say next that the rectangle D by E is also medial. Since the rectangle B by C equals the rectangle D by E, while the rectangle B by C is medial, therefore the rectangle D by E is also medial. X.21 Therefore two medial straight lines D and E, commensurable in square only, and containing a medial rectangle, have been found such that the square on the greater is greater than the square on the less by the square on a straight line commensurable with the greater. Similarly again it can be proved that the square on D is greater than the square on E by the square on a straight line incommensurable with D when the square on A is greater than the square on C by the square on a straight line incommensurable with A. X.30 Q.E.D. (Forthcoming) Book X Introduction Proposition X.31 Proposition X.33. © 1996 D.E.Joyce Clark University Proposition 33 Lemma. Let ABC be a right-angled triangle having the angle A right, and let the perpendicular AD be drawn. I say that the rectangle CB by BD equals the square on BA, the rectangle BC by CD equals the square on CA, the rectangle BD by DC equals the square on AD, and the rectangle BC by AD equals the rectangle BA by AC, and first that the rectangle CB by BD equals the square on BA. Since in a right-angled triangle AD has been drawn from the right angle perpendicular to the base, therefore the triangles ABD and ADC are similar both to the whole ABC and to one another. VI.8 Since the triangle ABC is similar to the triangle ABD, therefore CB is to BA as BA is to BD. Therefore the rectangle CB by BD equals the square on AB. For the same reason the rectangle BC by CD also equals the square on AC. VI.4 VI.17 Since, if in a right-angled triangle a perpendicular is drawn from the right angle to the base, then the perpendicular so drawn is a mean proportional between the segments of the base, therefore BD is to DA as AD is to DC. Therefore the rectangle BD by DC equals the square on AD. VI.8,Cor. VI.17 I say that the rectangle BC by AD also equals the rectangle BA by AC. Since we said, ABC is similar to ABD, therefore BC is to CA as BA is to AD. VI.4 Therefore the rectangle BC by AD equals the rectangle BA by AC. VI.16 Q.E.D. Proposition 33 To find two straight lines incommensurable in square which make the sum of the squares on them rational but the rectangle contained by them medial. Set out two rational straight lines AB and BC commensurable in square only such that the square on the greater AB is greater than the square on the less BC by the square on a straight line incommensurable with AB. X.30 Bisect BC at D. Apply to AB a parallelogram equal to the square on either of the straight lines BD or DC and deficient by a square figure, and let it be the rectangle AE by EB. VI.28 Describe the semicircle AFB on AB, draw EF at right angles to AB, and join AF and FB. Since AB and BC are unequal straight lines, and the square on AB is greater than the square on BC by the square on a straight line incommensurable with AB, while there was applied to AB a parallelogram equal to the fourth part of the square on BC, that is, to the square on half of it, and deficient by a square figure, making the rectangle AE by EB, therefore AE is incommensurable with EB. X.18 And AE is to EB as the rectangle BA by AE is to the rectangle AB by BE, while the rectangle BA by AE equals the square on AF, and the rectangle AB by BE is to the square on BF, therefore the square on AF is incommensurable with the square on FB. Therefore AF and FB are incommensurable in square. Since AB is rational, therefore the square on AB is also rational, so that the sum of the squares on AF and FB is also rational. I.47 Since, again, the rectangle AE by EB equals the square on EF, and, by hypothesis, the rectangle AE by EB also equals the square on BD, therefore FE equals BD. Therefore BC is double FE, so that the rectangle AB by BC is also commensurable with the rectangle ABEF. But the rectangle AB by BC is medial, therefore the rectangle AB by EF is also medial. X.21 X.23,Cor. But the rectangle AB by EF equals the rectangle AF by FB, therefore the rectangle AF by FB is also medial. Lemma But it was also proved that the sum of the squares on these straight lines is rational. Therefore two straight lines AF and FB incommensurable in square have been found which make the sum of the squares on them rational, but the rectangle contained by them medial. Q.E.D. The first part of the lemma encompasses proposition I.47, but the proof of it depends on the theory of similar triangles developed in Book VI, unlike Euclid's proof of I.47. This proposition is used in propositions X.39 and X.76. Book X Introduction Proposition X.32 Proposition X.34. © 1996 D.E.Joyce Clark University Proposition 34 To find two straight lines incommensurable in square which make the sum of the squares on them medial but the rectangle contained by them rational. Set out two medial straight lines AB and BC, commensurable in square only, such that the rectangle which they contain is rational and the square on AB is greater than the square on BC by the square on a straight line incommensurable with AB. X.31 ad fin. Describe the semicircle ADB on AB. Apply a parallelogram to AB equal to the square on BE and deficient by a square figure, namely the rectangle AF by FB. Then AF is incommensurable in length with FB. VI.28 X.18 Draw FD from F at right angles to AB, and join AD and DB. Since AF is incommensurable in length with FB, therefore the rectangle BA by AF is also incommensurable with the rectangle AB by BF. X.11 But the rectangle BA by AF equals the square on AD, and the rectangle AB by BF equals the square on DB, therefore the square on AD is also incommensurable with the square on DB. And, since the square on AB is medial, therefore the sum of the squares on AD and DB is also medial. III.31 I.47 And, since BC is double DF, therefore the rectangle AB by BC is also double the rectangle AB by FD. But the rectangle AB by BC is rational, therefore the rectangle AB by FD is also rational. X.6 But the rectangle AB by FD equals the rectangle AD by DB, so that the rectangle AD by DB is also rational. Lemma Therefore two straight lines AD and DB incommensurable in square have been found which make the sum of the squares on them medial, but the rectangle contained by them rational. Q.E.D. This proposition is used in proposition X.40. Book X Introduction Proposition X.33 Proposition X.35. © 1996 D.E.Joyce Clark University Proposition 35 To find two straight lines incommensurable in square which make the sum of the squares on them medial and the rectangle contained by them medial and moreover incommensurable with the sum of the squares on them. Set out two medial straight lines AB and BC commensurable in square only, containing a medial rectangle, such that the square on AB is greater than the square on BC by the square on a straight line incommensurable with AB. Describe the semicircle ADB on AK, and make the rest of the construction as above. X.31 ad fin. Since AF is incommensurable in length with FB, therefore AD is also incommensurable in square with DB. X.18 X.11 Since the square on AB is medial, therefore the sum of the squares on AD and DB is also medial. III.31 I.47 Since the rectangle AF by FB equals the square on each of the straight lines BE and DF, therefore BE equals DF. Therefore BC is double FD, so that the rectangle AB by BC is also double the rectangle AB by FD. But the rectangle AB by BC is medial, therefore the rectangle AB by FD is also medial. X.32,Cor. And it equals the rectangle AD by DB, therefore the rectangle AD by DB is also medial. X.33,Lemma Since AB is incommensurable in length with BC, while CB is commensurable with BE, therefore AB is also incommensurable in length with BE, so that the square on AB is also incommensurable with the rectangle AB by BE. X.13 X.11 But the sum of the squares on AD and DB equals the square on AB, and the rectangle AB by FD, that is, the rectangle AD by DB, equals the rectangle AB by BE, therefore the sum of the squares on AD and DB is incommensurable with the rectangle AD by DB. I.47 Therefore two straight lines AD and DB incommensurable in square have been found which make the sum of the squares on them medial and the rectangle contained by them medial and moreover incommensurable with the sum of the squares on them. Q.E.D. This proposition is used in propositions X.41 and X.78. Book X Introduction Proposition X.34 Proposition X.36. © 1996 D.E.Joyce Clark University Proposition 36 If two rational straight lines commensurable in square only are added together, then the whole is irrational; let it be called binomial. Let two rational straight lines AB and BC commensurable in square only be added together. I say that the whole AC is irrational. Since AB is incommensurable in length with BC, for they are commensurable in square only, and AB is to BC as the rectangle AB by BC is to the square on BC, therefore the rectangle AB by BE is incommensurable with the square on BC. X.11 But twice the rectangle AB by BC is commensurable with the rectangle AB by BC, and the sum of the squares on AB and BC is commensurable with the square on BC, for AB and BC are rational straight lines commensurable in square only, therefore twice the rectangle AB by BC is incommensurable with the sum of the squares on AB and BC. X.6 X.15 X.13 And, taken jointly, twice the rectangle AB by BC together with the squares on AB and BC, that is, the square on AC, is incommensurable with the sum of the squares on AB and BC. II.4 X.16 But the sum of the squares on AB and BC is rational, therefore the square on AC is irrational, so that AC is also irrational. Let it be called binomial. X.Def.4 Q.E.D. This proposition is used very frequently in Book X starting with the next proposition. Book X Introduction Proposition X.35 Proposition X.37. © 1996 D.E.Joyce Clark University Proposition 37 If two medial straight lines commensurable in square only and containing a rational rectangle are added together, the whole is irrational; let it be called the first bimedial straight line. Let two medial straight lines AB and BC commensurable in square only and containing a rational rectangle be added together. I say that the whole AC is irrational. Since AB is incommensurable in length with BC, therefore the sum of the squares on AB and BC is also incommensurable with twice the rectangle AB by BC, and, taken jointly, the sum of the squares on AB and BC together with twice the rectangle AB by BC, that is, the square on AC, is incommensurable with the rectangle AB by BC. X.36 II.4 X.16 But the rectangle AB by BC is rational, for, by hypothesis, AB and BC are straight lines containing a rational rectangle, therefore the square on AC is irrational. Therefore AC is irrational. And let it be called a first bimedial straight line. X.Def.4 Q.E.D. This proposition is used in X.43 and a few others in Book X. Book X Introduction Proposition X.36 Proposition X.38. © 1996 D.E.Joyce Clark University Proposition 38 If two medial straight lines commensurable in square only and containing a medial rectangle are added together, then the whole is irrational; let it be called the second bimedial straight line. Let two medial straight lines AB and BC commensurable in square only and containing a medial rectangle be added together. I say that AC is irrational. Set out a rational straight line DE, and apply parallelogram DF to DE equal to the square on AC, producing DG as breadth. I.41 Since the square on AC equals the sum of the squares on AB and BC and twice the rectangle AB by BC, apply EH, equal to the sum of the squares on AB and BC, to DE. Then the remainder HF equals twice the rectangle AB by BC. II.4 Since each of the straight lines AB and BC is medial, therefore the squares on AB and BC are also medial. But, by hypothesis, twice the rectangle AB by BC is also medial. And EH equals the sum of the squares on AB and BC, while FH equals twice the rectangle AB by BC, therefore each of the rectangles EH and HF is medial. And they are applied to the rational straight line DE, therefore each of the straight lines DH and HG is rational and incommensurable in length with DE. X.22 Since AB is incommensurable in length with BC, and AB is to BC as the square on AB is to the rectangle AB by BC, therefore the square on AB is incommensurable with the rectangle AB by BC. X.11 But the sum of the squares on AB and BC is commensurable with the square on AB, and twice the rectangle AB by BC is commensurable with the rectangle AB by BC. X.15 X.6 Therefore the sum of the squares on AB and BC is incommensurable with twice the rectangle AB by BC. X.13 But EH equals the sum of the squares on AB and BC, and HF equals twice the rectangle AB by BC. Therefore EH is incommensurable with HF, so that DH is also incommensurable in length with HG. VI.1 X.11 Therefore DH and HG are rational straight lines commensurable in square only, so that DG is irrational. X.36 But DE is rational, and the rectangle contained by an irrational and a rational straight line is irrational, therefore the area DF is irrational, and the side of the square equal to it is irrational. cf. X.20 X.Def.4 But AC is the side of the square equal to DF, therefore AC is irrational. Let it be called a second bimedial straight line. Q.E.D. This proposition is used in X.44 and a few others in Book X. Book X Introduction Proposition X.37 Proposition X.39. © 1996 D.E.Joyce Clark University Proposition 39 If two straight lines incommensurable in square which make the sum of the squares on them rational but the rectangle contained by them medial are added together, then the whole straight line is irrational; let it be called major. Let two straight lines AB and BC incommensurable in square, and fulfilling the given conditions, be added together. X.33 I say that AC is irrational. Since the rectangle AB by BC is medial, therefore twice the rectangle AB by BC is also medial. X.6 X.23,Cor. But the sum of the squares on AB and BC is rational, therefore twice the rectangle AB by BC is incommensurable with the sum of the squares on AB and BC, so that the sum of the squares on AB and BC together with twice the rectangle AB by BC, that is, the square on AC, is also incommensurable with the sum of the squares on AB and BC. Therefore the square on AC is irrational, so that AC is also irrational. Let it be called major. X.16 X.Def.4 Q.E.D. This proposition is used in X.57 and a few others in Book X. Book X Introduction Proposition X.38 Proposition X.40. © 1996 D.E.Joyce Clark University Proposition 40 If two straight lines incommensurable in square which make the sum of the squares on them medial but the rectangle contained by them rational are added together, then the whole straight line is irrational; let it be called the side of a rational plus a medial area. Let two straight lines AB and BC incommensurable in square, and fulfilling the given conditions, be added together. X.34 I say that AC is irrational. Since the sum of the squares on AB and BC is medial, while twice the rectangle AB by BC is rational, therefore the sum of the squares on AB and BC is incommensurable with twice the rectangle AB by BC, so that the square on AC is also incommensurable with twice the rectangle AB by BC. X.16 But twice the rectangle AB by BC is rational, therefore the square on AC is irrational. Therefore AC is irrational. Let it be called the side of a rational plus a medial area. X.Def.4 Q.E.D. This proposition is used in X.46 and a few others in Book X. Book X Introduction Proposition X.39 Proposition X.41. © 1996 D.E.Joyce Clark University Proposition 41 If two straight lines incommensurable in square which make the sum of the squares on them medial and the rectangle contained by them medial and also incommensurable with the sum of the squares on them are added together, then the whole straight line is irrational; let it be called the side of the sum of two medial areas. Let two straight lines AB and BC incommensurable in square and satisfying the given conditions be added together. X.35 I say that AC is irrational. Set out a rational straight line DE. Apply to DE the rectangle DF equal to the sum of the squares on AB and BC, and apply to DE the rectangle GH equal to twice the rectangle AB by BC. Then the whole DH equals the square on AC. II.4 Now, since the sum of the squares on AB and BC is medial, and equals DF, therefore DF is also medial. And it is applied to the rational straight line DE, therefore DG is rational and incommensurable in length with DE. For the same reason GK is also rational and incommensurable in length with GF, that is, DE. X.22 Since the sum of the squares on AB and BC is incommensurable with twice the rectangle AB by BC, therefore DF is incommensurable with GH, so that DG is also incommensurable with GK. VI.1 X.11 And they are rational, therefore DG and GK are rational straight lines commensurable in square only. Therefore DK is irrational and what is called binomial. X.36 But DE is rational, therefore DH is irrational, and the side of the square which equals it is irrational. X.Def.4 But AC is the side of the square equal to HD, therefore AC is irrational. Let it be called the side of the sum two medial areas. Q.E.D. Lemma. And that the aforesaid irrational straight lines are divided only in one way into the straight lines of which they are the sum and which produce the types in question we will now prove after premising the following lemma. Set out the straight line AB, cut the whole into unequal parts at each of the points C and D, and let AC be supposed greater than DB. I say that the sum of the squares on AC and CB is greater than the sum of the squares on AD and DB. Bisect AB at E. Since AC is greater than DB, subtract DC from each, therefore the remainder AD is greater than the remainder CB. But AE equals EB, therefore DE is less than EC. Therefore the points C and D are not equidistant from the point of bisection. Since the rectangle AC by CB together with the square on EC equals the square on EB, and, further, the rectangle AD by DB together with the square on DE equals the square on EB, therefore the rectangle AC by CB together with the square on EC equals the rectangle AD by DB together with the square on DE. II.5 And of these the square on DE is less than the square on EC, therefore the remainder, the rectangle AC by CB, is also less than the rectangle AD by DB so that twice the rectangle AC by CB is also less than twice the rectangle AD by DB. Therefore the remainder, the sum of the squares on AC and CB, is greater than the sum of the squares on AD and DB. Q.E.D. This proposition is used in X.65 and a couple of others in Book X. Book X Introduction Proposition X.40 Proposition X.42. © 1996 D.E.Joyce Clark University Proposition 42 A binomial straight line is divided into its terms at one point only. Let AB be a binomial straight line divided into its terms at C. Then AC and CB are rational straight lines commensurable in square only. I say that AB is not divided at another point into two rational straight lines commensurable in square only. For, if possible, let it be divided at D also, so that AD and DB are also rational straight lines commensurable in square only. It is then manifest that AC is not the same as DB. If possible, let it be so. Then AB is also the same as CB, and AC is to CB as BD is to DA. Thus AB is divided at D also in the same way as by the division at C, which is contrary to the hypothesis. Therefore AC is not the same with DB. For this reason also the points C and D are not equidistant from the point of bisection. Therefore that by which the sum of the squares on AC and CB differs from the sum of the squares on AD and DB is also that by which twice the rectangle AD by DB differs from twice the rectangle AC by CB, because both the squares on AC and CB together with twice the rectangle AC by CB, and the squares on AD and DB together with twice the rectangle AD by DB, equal the square on AB. II.4 But the sum of the squares on AC and CB differs from the sum of the squares on AD and DB by a rational area, for both are rational, therefore twice the rectangle AD by DB also differs from twice the rectangle AC by CB by a rational area, though they are medial, which is absurd, for a medial area does not exceed a medial by a rational area. X.21 X.26 Therefore a binomial straight line is not divided at different points. Therefore it is divided at one point only. Therefore, a binomial straight line is divided into its terms at one point only. Q.E.D. This proposition is used in X.47. Book X Introduction Proposition X.41 Proposition X.43. © 1996 D.E.Joyce Clark University Proposition 43 A first bimedial straight line is divided at one and the same point only. Let AB be a first bimedial straight line divided at C, so that AC and CB are medial straight lines commensurable in square only and containing a rational rectangle. X.37 I say that AB is not so divided at another point. If possible, let it also be divided at D, so that AD and DB are also medial straight lines commensurable in square only and containing a rational rectangle. Since, then, that by which twice the rectangle AD by DB differs from twice the rectangle AC by CB is that by which the sum of the squares on AC and CB differs from the sum of the squares on AD and DB, while twice the rectangle AD by DB differs from twice the rectangle AC by CB by a rational area, for both are rational, therefore the sum of the squares on AC and CB also differs from the sum of the squares on AD and DB by a rational area, though they are medial, which is absurd. X.26 Therefore a first bimedial straight line is not divided into its terms at different points. Therefore it is so divided at one point only. Therefore, a first bimedial straight line is divided at one and the same point only. Q.E.D. (Forthcoming) Book X Introduction Proposition X.42 Proposition X.44. © 1996 D.E.Joyce Clark University Proposition 44 A second bimedial straight line is divided at one point only. Let AB be a second bimedial straight line divided at C, so that AC and CB are medial straight lines commensurable in square only and containing a medial rectangle. It is then manifest that C is not at the point of bisection, because the segments are not commensurable in length. X.38 I say that AB is not so divided at another point. If possible, let it also be divided at D, so that AC is not the same with DB, but AC is supposed greater. It is then clear that the sum of the squares on AD and DB is also, as we proved above, less than the sum of the squares on AC and CB. Suppose that AD and DB are medial straight lines commensurable in square only and containing a medial rectangle. Lemma Set out a rational straight line EF, apply to EF the rectangular parallelogram EK equal to the square on AB, and subtract EG, equal to the sum of the squares on AC and CB. Then the remainder HK equals twice the rectangle AC by CB. II.4 Again, subtract EL, equal to the sum of the squares on AD and DB, which were proved less than the sum of the squares on AC and CB. Then the remainder MK also equals twice the rectangle AD by DB. Lemma Now, since the squares on AC and CB are medial, therefore EG is medial. And it is applied to the rational straight line EF, therefore EH is rational and incommensurable in length with EF. X.22 For the same reason HN is also rational and incommensurable in length with EF. And, since AC and CB are medial straight lines commensurable in square only, therefore AC is incommensurable in length with CB. But AC is to CB as the square on AC is to the rectangle AC by CB, therefore the square on AC is incommensurable with the rectangle AC by CB. X.11 But the sum of the squares on AC and CB is commensurable with the square on AC, for AC and CB are commensurable in square. X.15 And twice the rectangle AC by CB is commensurable with the rectangle AC by CB. X.6 Therefore the sum of the squares on AC and CB is also incommensurable with twice the rectangle AC by CB. X.13 But EG equals the sum of the squares on AC and CB, and HK equals twice the rectangle AC by CB, therefore EG is incommensurable with HK, so that EH is also incommensurable in length with HN. VI.1 X.11 And they are rational, therefore EH and HN are rational straight lines commensurable in square only. But, if two rational straight lines commensurable in square only are added together, then the whole is the irrational which is called binomial. X.36 Therefore EN is a binomial straight line divided at H. In the same way EM and MN is also proved to be rational straight lines commensurable in square only, and EN is a binomial straight line divided at different points, H and M. And EH is not the same with MN, for the sum of the squares on AC and CB is greater than the sum of the squares on AD and DB. But the sum of the squares on AD and DB is greater than twice the rectangle AD by DB, therefore the sum of the squares on AC and CB, that is, EG, is much greater than twice the rectangle AD by DB, that is, MK, so that EH is also greater than MN. Therefore EH is not the same with MN. Therefore, a second bimedial straight line is divided at one point only. Q.E.D. (Forthcoming) Book X Introduction Proposition X.43 Proposition X.45. © 1996 D.E.Joyce Clark University Proposition 45 A major straight line is divided at one point only. Let AB be a major straight line divided at C, so that AC and CB are incommensurable in square, and let the sum of the squares on AC and CB be rational, but the rectangle AC by CB medial. I say that AB is not so divided at another point. If possible, let it also be divided at D, so that AD and DB are incommensurable in square and the sum of the squares on AD and DB is rational, but the rectangle contained by them medial. Then, since that by which the sum of the squares on AC and CB differs from the sum of the squares on AD and DB is also that by which twice the rectangle AD by DB differs from twice the rectangle AC by CB, while the sum of the squares on AC and CB exceeds the sum of the squares on AD and DB by a rational area, for both are rational, therefore twice the rectangle AD by DB also exceeds twice the rectangle AC by CB by a rational area, though they are medial, which is impossible. X.26 Therefore a major straight line is not divided at different points. Therefore it is only divided at one and the same point. Therefore, a major straight line is divided at one point only. Q.E.D. (Forthcoming) Book X Introduction Proposition X.44 Proposition X.46. © 1996 D.E.Joyce Clark University Proposition 46 The side of a rational plus a medial area is divided at one point only. Let AB be the side of a rational plus a medial area divided at C, so that AC and CB are incommensurable in square and let the sum of the squares on AC and CB be medial, but twice the rectangle AC by CB rational. X.40 I say that AB is not so divided at another point. If possible, let it be divided at D also, so that AD and DB are also incommensurable in square and the sum of the squares on AD and DB is medial, but twice the rectangle AD by DB rational. Since, then, that by which twice the rectangle AC by CB differs from twice the rectangle AD by DB is also that by which the sum of the squares on AD and DB differs from the sum of the squares on AC and CB, while twice the rectangle AC by CB exceeds twice the rectangle AD by DB by a rational area, therefore the sum of the squares on AD and DB also exceeds the sum of the squares on AC and CB by a rational area, though they are medial, which is impossible. X.26 Therefore the side of a rational plus a medial area is not divided at different points, therefore it is divided at one point only. Therefore, the side of a rational plus a medial area is divided at one point only. Q.E.D. (Forthcoming) Book X Introduction Proposition X.45 Proposition X.47. © 1996 D.E.Joyce Clark University Proposition 47 The side of the sum of two medial areas is divided at one point only. Let AB be divided at C, so that AC and CB are incommensurable in square and let the sum of the squares on AC and CB be medial, and the rectangle AC by CB medial and also incommensurable with the sum of the squares on them. I say that AB is not divided at another point so as to fulfill the given conditions. If possible, let it be divided at D, so that again AC is of course not the same as BD, but AC is supposed greater. Set out a rational straight line EF, and apply to EF the rectangle EG equal to the sum of the squares on AC and CB, and the rectangle HK equal to twice the rectangle AC by CB. Then the whole EK equals the square on AB. II.4 Again, to EP apply EL, equal to the sum of the squares on AD and DB. Then the remainder, twice the rectangle AD by DB, equals the remainder MK. And since, by hypothesis, the sum of the squares on AC and CB is medial, therefore EG is also medial. And it is applied to the rational straight line EF, therefore HE is rational and incommensurable in length with EF. X.22 For the same reason HN is also rational and incommensurable in length with EF. And, since the sum of the squares on AC and CB is incommensurable with twice the rectangle AC by CB, therefore EG is also incommensurable with GN, so that EH is also incommensurable with HN. VI.1 X.11 And they are rational, therefore EH and HN are rational straight lines commensurable in square only. Therefore EN is a binomial straight line divided at H. X.36 Similarly we can prove that it is also divided at M. And EH is not the same with MN, therefore a binomial has been divided at different points, which is absurd. X.42 Therefore a side of the sum of two medial areas is not divided at different points. Therefore it is divided at one point only. Therefore, the side of the sum of two medial areas is divided at one point only. Q.E.D. (Forthcoming) Book X Introduction Proposition X.46 Definitions II of Book X. © 1996 D.E.Joyce Clark University Definitions II Definition 1. Given a rational straight line and a binomial, divided into its terms, such that the square on the greater term is greater than the square on the lesser by the square on a straight line commensurable in length with the greater, then, if the greater term is commensurable in length with the rational straight line set out, let the whole be called a first binomial straight line; Definition 2. But if the lesser term is commensurable in length with the rational straight line set out, let the whole be called a second binomial; Definition 3. And if neither of the terms is commensurable in length with the rational straight line set out, let the whole be called a third binomial. Definition 4. Again, if the square on the greater term is greater than the square on the lesser by the square on a straight line incommensurable in length with the greater, then, if the greater term is commensurable in length with the rational straight line set out, let the whole be called a fourth binomial; Definition 5. If the lesser, a fifth binomial; Definition 6. And, if neither, a sixth binomial. (Forthcoming) Book X Introduction Proposition X.47 Proposition X.48. © 1996 D.E.Joyce Clark University Proposition 48 To find the first binomial line. Set out two numbers AC and CB such that the sum of them AB has to BC the ratio which a square number has to a square number, but does not have to CA the ratio which a square number has to a square number. X.28,Lemma1 Set out any rational straight line D, and let EF be commensurable in length with D. Therefore EF is also rational. Let it be contrived that the number BA is to AC as the square on EF is to the square on FG. X.6,Cor. But AB has to AC the ratio which a number has to a number, therefore the square on EF also has to the square on FG the ratio which a number has to a number, so that the square on EF is commensurable with the square on FG. X.6 And EF is rational, therefore FG is also rational. And, since BA does not have to AC the ratio which a square number has to a square number, neither, therefore, has the square on EF to the square on FG the ratio which a square number has to a square number. Therefore EF is incommensurable in length with FG. X.9 Therefore EF and FG are rational straight lines commensurable in square only. Therefore EG is binomial. X.36 I say that it is also a first binomial straight line. Since the number BA is to AC as the square on EF is to the square on FG, while BA is greater than AC, therefore the square on EF is also greater than the square on FG. Let then the sum of the squares on FG and H equal the square on EF. Now since BA is to AC as the square on EF is to the square on FG, therefore, in conversion, AB is to BC as the square on EF is to the square on H. V.19,Cor. But AB has to BC the ratio which a square number has to a square number, therefore the square on EF also has to the square on H the ratio which a square number has to a square number. Therefore EF is commensurable in length with H. Therefore the square on EF is greater than the square on FG by the square on a straight line commensurable with EF. X.9 And EF and FG are rational, and EF is commensurable in length with D. Therefore EF is a first binomial straight line. Q.E.D. (Forthcoming) Book X Introduction Definitions II of Book X Proposition X.49. © 1996 D.E.Joyce Clark University Proposition 49 To find the second binomial line. Set out two numbers AC and CB such that the sum of them AB has to BC the ratio which a square number has to a square number, but does not have to AC the ratio which a square number has to a square number. Set out a rational straight line D, and let EF be commensurable in length with D, therefore EF is rational. Let it be contrived then that as the number CA is to AB, so is the square on EFto the square on FG, therefore the square on EF is commensurable with the square on FG. Therefore FG is also rational. X.6,Cor. X.6 Now, since the number CA does not have to AB the ratio which a square number has to a square number, neither does the square on EF have to the square on FG the ratio which a square number has to a square number. Therefore EF is incommensurable in length with FG. Therefore EF and FG are rational straight lines commensurable in square only. Therefore EG is binomial. X.9 X.36 It is next to be proved that it is also a second binomial straight line. Since, inversely, the number BA is to AC as the square on GF is to the square on FE, while BA is greater than AC, therefore the square on GF is greater than the square on FE. V.7.Cor Let the sum of the squares on EF and H equal the square on GF. Then, in conversion, AB is to BC as the square on FG is to the square on H. V.19,Cor. But AB has to BC the ratio which a square number has to a square number, therefore the square on FG also has to the square on H the ratio which a square number has to a square number. Therefore FG is commensurable in length with H, so that the square on FG is greater than the square on FE by the square on a straight line commensurable with FG. X.9 And FG and FE are rational straight lines commensurable in square only, and EF, the lesser term, is commensurable in length with the rational straight line D set out. Therefore EG is a second binomial straight line. Q.E.D. (Forthcoming) Book X Introduction Proposition X.48 Proposition X.50. © 1996 D.E.Joyce Clark University Proposition 50 To find the third binomial line. Set out two numbers AC and CB such that the sum of them AB has to BC the ratio which a square number has to a square number, but does not have to AC the ratio which a square number has to a square number. Also set out any other number D, not square, and let it not have to either of the numbers BA and AC the ratio which a square number has to a square number. Set out any rational straight line E, and let it be contrived that D is to AB as the square on E is to the square on FG. Then the square on E is commensurable with the square on FG. X.6,Cor. X.6 And E is rational, therefore FG is also rational. And, since D does not have to AB the ratio which a square number has to a square number, neither does the square on E have to the square on FG the ratio which a square number has to a square number, therefore E is incommensurable in length with FG. X.9 Next let it be contrived that the number BA is to AC as the square on FG is to the square on GH. Then the square on FG is commensurable with the square on GH. X.6,Cor. X.6 But FG is rational, therefore GH is also rational. And, since BA does not have to AC the ratio which a square number has to a square number, neither does the square on FG have to the square on HG the ratio which a square number has to a square number, therefore FG is incommensurable in length with GH. X.9 Therefore FG and GH are rational straight lines commensurable in square only. Therefore FH is binomial. X.36 I say next that it is also a third binomial straight line. Since D is to AB as the square on E is to the square on FG, and BA is to AC as the square on FG is to the square on GH, therefore, ex aequali, D is to AC as the square on E is to the square on GH. V.22 But D does not have to AC the ratio which a square number has to a square number, therefore neither does the square on E have to the square on GH the ratio which a square number has to a square number. Therefore E is incommensurable in length with GH. X.9 Since BA is to AC as the square on FG is to the square on GH, therefore the square on FG is greater than the square on GH. Let then the sum of the squares on GH and K equal the square on FG. Then, in conversion, AB is to BC as the square on FG is to the square on K. V.19,Cor. But AB has to BC the ratio which a square number has to a square number, therefore the square on FG also has to the square on K the ratio which a square number has to a square number. Therefore FG is commensurable in length with K. X.9 Therefore the square on FG is greater than the square on GH by the square on a straight line commensurable with FG. And FG and GH are rational straight lines commensurable in square only, and neither of them is commensurable in length with E. Therefore FH is a third binomial straight line. Q.E.D. (Forthcoming) Book X Introduction Proposition X.49 Proposition X.51. © 1996 D.E.Joyce Clark University Proposition 51 To find the fourth binomial straight line. Set out two numbers AC and CB such that AB has neither to BC nor to AC the ratio which a square number has to a square number. Set out a rational straight line D, and let EF be commensurable in length with D. Then EF is also rational. Let it be contrived that the number BA is to AC as the square on EF is to the square on FG. Then the square on EF is commensurable with the square on FG. Therefore FG is also rational. X.6,Cor. X.6 Now, since BA does not have to AC the ratio which a square number has to a square number, neither does the square on EF have to the square on FG the ratio which a square number has to a square number, therefore EF is incommensurable in length with FG. X.9 Therefore EF and FG are rational straight lines commensurable in square only, so that EG is binomial. I say next that it is also a fourth binomial straight line. Since BA is to AC as the square on EF is to the square on FG, therefore the square on EF is greater than the square on FG. Let then the sum of the squares on FG and H equal the square on EF. Then, in conversion, the number AB is to BC as the square on EF is to the square on H. V.19,Cor. But AB does not have to BC the ratio which a square number has to a square number, therefore neither does the square on EF have to the square on H the ratio which a square number has to a square number. Therefore EF is incommensurable in length with H. Therefore the square on EF is greater than the square on GF by the square on a straight line incommensurable with EF. X.9 And EF and FG are rational straight lines commensurable in square only, and EF is commensurable in length with D. Therefore EG is a fourth binomial straight line. Q.E.D. (Forthcoming) Book X Introduction Proposition X.50 Proposition X.52. © 1996 D.E.Joyce Clark University Proposition 52 To find the fifth binomial line. Set out two numbers AC and CB such that AB does not have to either of them the ratio which a square number has to a square number. Set out any rational straight line D, and let EF be commensurable with D. Then EF is rational. Let it be contrived that CA is to AB as the square on EF is to the square on FG. X.6,Cor. But CA does not have to AB the ratio which a square number has to a square number, therefore neither does the square on EF have to the square on FG the ratio which a square number has to a square number. Therefore EF and FG are rational straight lines commensurable in square only, therefore EG is binomial. X.36 X.9 I say next that it is also a fifth binomial straight line. Since CA is to AB as the square on EF is to the square on FG, therefore, inversely, BA is to AC as the square on FG is to the square on FE. Therefore the square on GF is greater than the square on FE. V.7.Cor Let then the sum of the squares on EF and H equal the square on GF. Then, in conversion, the number AB is to BC as the square on GF is to the square on H. V.19,Cor. But AB does not have to BC the ratio which a square number has to a square number, therefore neither does the square on FG have to the square on H the ratio which a square number has to a square number. Therefore FG is incommensurable in length with H, so that the square on FG is greater than the square on FE by the square on a straight line incommensurable with FG. X.9 And GF and FE are rational straight lines commensurable in square only, and the lesser term EF is commensurable in length with the rational straight line D set out. Therefore EG is a fifth binomial straight line. Q.E.D. (Forthcoming) Book X Introduction Proposition X.51 Proposition X.53. © 1996 D.E.Joyce Clark University Proposition 53 To find the sixth binomial line. Set out two numbers AC and CB such that AB does not have to either of them the ratio which a square number has to a square number, and let there also be another number D which is not square and which does not have to either of the numbers BA or AC the ratio which a square number has to a square number. Set out any rational straight line E, and let it be contrived that D is to AB as the square on E is to the square on FG. Then the square on E is commensurable with the square on FG. And E is rational, therefore FG is also rational. X.6,Cor. X.6 Now, since D does not have to AB the ratio which a square number has to a square number, neither does the square on E have to the square on FG the ratio which a square number has to a square number, therefore E is incommensurable in length with FG. X.9 Again, let it be contrived that BA is to AC as the square on FG is to the square on GH. Then the square on FG is commensurable with the square on HG. X.6,Cor. X.6 Therefore the square on HG is rational. Therefore HG is rational. And, since BA does not have to AC the ratio which a square number has to a square number, neither does the square on FG have to the square on GH the ratio which a square number has to a square number, therefore FG is incommensurable in length with GH. X.9 Therefore FG and GH are rational straight lines commensurable in square only. Therefore FH is binomial. X.36 It is next to be proved that it is also a sixth binomial straight line. Since D is to AB as the square on E is to the square on FG, and also BA is to AC as the square on FG is to the square on GH, therefore, ex aequali, D is to AC as the square on E is to the square on GH. V.22 But D does not have to AC the ratio which a square number has to a square number, therefore neither does the square on E have to the square on GH the ratio which a square number has to a square number, therefore E is incommensurable in length with GH. X.9 But it was also proved incommensurable with FG, therefore each of the straight lines FG and GH is incommensurable in length with E. And, since BA is to AC as the square on FG is to the square on GH, therefore the square on FG is greater than the square on GH. Let then the sum of the squares on GH and K equal the square on FG. Then, in conversion, AB is to BC as the square on FG is to the square on K. V.19,Cor. But AB does not have to BC the ratio which a square number has to a square number, so that neither does the square on FG have to the square on K the ratio which a square number has to a square number. Therefore FG is incommensurable in length with A. Therefore the square on FG is greater than the square on GH by the square on a straight line incommensurable with FG. X.9 And FG and GH are rational straight lines commensurable in square only, and neither of them is commensurable in length with the rational straight line E set out. Therefore FH is a sixth binomial straight line. Q.E.D. (Forthcoming) Book X Introduction Proposition X.52 Proposition X.54. © 1996 D.E.Joyce Clark University Proposition 54 Lemma. Let there be two squares AB and BC, and let them be placed so that DB is in a straight line with BE. Then FB is also in a straight line with BG. Complete the parallelogram AC. I say that AC is a square, that DG is a mean proportional between AB and BC, and further that DC is a mean proportional between AC and CB. Since DB equals BF, and BE is to BG, therefore the whole DE equals the whole FG. But DE equals each of the straight lines AH and KC, and FG equals each of the straight lines AK and HC, therefore each of the straight lines AH and KC also equals each of the straight lines AK and HC. I.34 Therefore the parallelogram AC is equilateral. And it is also rectangular, therefore AC is a square. Since FB is to BG as DB is to BE, while FB is to BG as AB is to DG, and DB is to BE as DG is to BC, therefore AB is to DG as DG is to BC. VI.1 VI.11 Therefore DG is a mean proportional between AB and BC. I say next that DC is also a mean proportional between AC and CB. Since AD is to DK as KG is to GC, for they are equal respectively, and, taken jointly, AK is to KD as KC is to CG, while AK is to KD as AC is to CD, and AC is to CG as DC is to CB, therefore AC is to DC as DC is to BC. V.18 VI.1 VI.11 Therefore DC is a mean proportional between AC and CB. Proposition 54 If an area is contained by a rational straight line and the first binomial, then the side of the area is the irrational straight line which is called binomial. Let the area AC be contained by the rational straight line AB and the first binomial AD. I say that the side of the area AC is the irrational straight line which is called binomial. Since AD is a first binomial straight line, divide it into its terms at E, and let AE be the greater term. It is then manifest that AE and ED are rational straight lines commensurable in square only, the square on AE is greater than the square on ED by the square on a straight line commensurable with AE, and AE is commensurable in length with the rational straight line AB set out. X.Def.II.1 Bisect ED at the point F. Then, since the square on AE is greater than the square on ED by the square on a straight line commensurable with AE, therefore, if there is applied to the greater AE a parallelogram equal to the fourth part of the square on the less, that is, to the square on EF, and deficient by a square figure, then it divides it into commensurable parts. X.17 Apply to AE the rectangle AG by GE equal to the square on EF. Then AG is commensurable in length with EG. Draw GH, EK, and FL from G, E, and F parallel to either of the straight lines AB and CD. Construct the square SN equal to the parallelogram AH, and the square NQ equal to GK, and place them so that MN is in a straight line with NO. Then RN is also in a straight line with NP. Complete the parallelogram SQ. Then SQ is a square. II.4 Lemma Now, since the rectangle AG by GE equals the square on EF, therefore AG is to EF as FE is to EG. VI.17 Therefore AH is to EL as EL is to KG. Therefore EL is a mean proportional between AH and GK. VI.1 But AH equals SN, and GK equals NQ, therefore EL is a mean proportional between SN and NQ. But MR is also a mean proportional between the same SN and NQ, therefore EL equals MR, so that it also equals PO. Lemma But AH and GK also equal SN and NQ, therefore the whole AC equals the whole SQ, that is, it equals the square on MO, Therefore MO is the side of AC. I say next that MO is binomial. Since AG is commensurable with GE, therefore AE is also commensurable with each of the straight lines AG and GE. X.15 But AE is also, by hypothesis, commensurable with AB, therefore AG and GE are also commensurable with AB. X.12 And AB is rational, therefore each of the straight lines AG and GE is also rational. Therefore each of the rectangles AH and GK is rational, and AH is commensurable with GK. X.19 But AH equals SN, and GK equals NQ, therefore the sum of SN and NQ, that is the squares on MN and NO, are rational and commensurable. Since AE is incommensurable in length with ED, while AE is commensurable with AG, and DE is commensurable with EF, therefore AG is also incommensurable with EF, so that AH is also incommensurable with EL. X.13 VI.1 X.11 But AH equals SN, and EL equals MR, therefore SN is also incommensurable with MR. But SN is to MR as PN is to NR, therefore PN is incommensurable with NR. VI.1 X.11 But PN equals MN, and NR equals NO, therefore MN is incommensurable with NO. And the square on MN is commensurable with the square on NO, and each is rational, therefore MN and NO are rational straight lines commensurable in square only. Therefore MO is binomial and the side of AC. X.36 Therefore, if an area is contained by a rational straight line and the first binomial, then the side of the area is the irrational straight line which is called binomial. Q.E.D. The lemma before the proposition is used in this proposition, X.60, and X.11. The proposition itself is used in X.71. Book X Introduction Proposition X.53 Proposition X.55. © 1996 D.E.Joyce Clark University Proposition 55 If an area is contained by a rational straight line and the second binomial, then the side of the area is the irrational straight line which is called a first bimedial. Let the area ABCD be contained by the rational straight line AB and the second binomial AD. I say that the side of the area AC is a first bimedial straight line. Since AD is a second binomial straight line, divide it into its terms at E, so that AE is the greater term. Then AE and ED are rational straight lines commensurable in square only, the square on AE is greater than the square on ED by the square on a straight line commensurable with AE, and the lesser term ED is commensurable in length with AB. X.Def.II.2 Bisect ED at F, and apply to AE the rectangle AG by GE equal to the square on EF and deficient by a square figure. Then AG is commensurable in length with GE. X.17 Draw GH, EK, and FL through G, E, and F parallel to AB and CD. Construct the square SN equal to the parallelogram AH, and the square NQ equal to GK, and place them so that MN is in a straight line with NO. Then RN is also in a straight line with NP. Complete the square SQ. It is then manifest from what was proved before that MR is a mean proportional between SN and NQ and equals EL, and that is the side of the area AC. It is now to be proved that MO is a first bimedial straight line. Since AE is commensurable in length with ED, while ED is commensurable with AB, therefore AE is incommensurable with AB. X.13 Since AG is commensurable with EG, therefore AE is also commensurable with each of the straight lines AG and GE. X.15 But AE is incommensurable in length with AB, therefore AG and GE are also incommensurable with AB. X.13 Therefore BA and AG, and BA and GE, are pairs of rational straight lines commensurable in square only, so that each of the rectangles AH and GK is medial. X.21 Hence, each of the squares SN and NQ is medial. Therefore MN and NO are also medial. Since AG is commensurable in length with GE, therefore AH is also commensurable with GK, that is, SN is commensurable with NQ, that is, the square on MN with the square on NO. VI.1 X.11 Since AE is incommensurable in length with ED, while AE is commensurable with AG, and ED is commensurable with EF, therefore AG is incommensurable with EF, so that AH is also incommensurable with EL, that is, SN is incommensurable with MR, that is, PN with NR, that is, MN is incommensurable in length with NO. X.13 VI.1 X.11 But MN and NO were proved to be both medial and commensurable in square, therefore MN and NO are medial straight lines commensurable in square only. I say next that they also contain a rational rectangle. Since DE is, by hypothesis, commensurable with each of the straight lines AB and EF, therefore EF is also commensurable with EK. X.12 And each of them is rational, therefore EL, that is, MR is rational, and MR is the rectangle MN by NO. X.19 But, if two medial straight lines commensurable in square only and containing a rational rectangle are added together, then the whole is irrational and is called a first bimedial straight line. Therefore MO is a first bimedial straight line. X.37 Therefore, if an area is contained by a rational straight line and the second binomial, then the side of the area is the irrational straight line which is called a first bimedial. Q.E.D. This proposition is used in X.71. Book X Introduction Proposition X.54 Proposition X.56. © 1996 D.E.Joyce Clark University Proposition 56 If an area is contained by a rational straight line and the third binomial, then the side of the area is the irrational straight line called a second bimedial. Let the area ABCD be contained by the rational straight line AB and the third binomial AD divided into its terms at E, of which terms AE is the greater. I say that the side of the area AC is the irrational straight line called a second bimedial. Make the same construction as before. Now, since AD is a third binomial straight line, therefore AE and ED are rational straight lines commensurable in square only, the square on AE is greater than the square on ED by the square on a straight line commensurable with AE, and neither of the terms AE and ED is commensurable in length with AB. X.Def.II.3 Then, in manner similar to the foregoing, we shall prove that MO is the side of the area AC, and MN and NO are medial straight lines commensurable in square only, so that MO is bimedial. It is next to be proved that it is also a second bimedial straight line. Since DE is incommensurable in length with AB, that is, with EK, and DE is commensurable with EF, therefore EF is incommensurable in length with EK. X.13 And they are rational, therefore FE and EK are rational straight lines commensurable in square only. Therefore EL, that is, MR, is medial. X.21 And it is contained by MN and NO, therefore the rectangle MN by NO is medial. Therefore MO is a second bimedial straight line. X.38 Therefore, if an area is contained by a rational straight line and the third binomial, then the side of the area is the irrational straight line called a second bimedial. Q.E.D. This proposition is used in X.72. Book X Introduction Proposition X.55 Proposition X.57. © 1996 D.E.Joyce Clark University Proposition 57 If an area is contained by a rational straight line and the fourth binomial, then the side of the area is the irrational straight line called major. Let the area AC be contained by the rational straight line AB and the fourth binomial AD divided into its terms at E, of which terms let AE be the greater. I say that the side of the area AC is the irrational straight line called major. Since AD is a fourth binomial straight line, therefore AE and ED are rational straight lines commensurable in square only, the square on AE is greater than the square on ED by the square on a straight line incommensurable with AE, and AE is commensurable in length with AB. X.Def.II.4 Bisect DE at F, and apply to AE a parallelogram, the rectangle AG by GE, equal to the square on EF. Then AG is incommensurable in length with GE. X.18 Draw GH, EK, and FL parallel to AB, and make the rest of the construction as before. It is then manifest that MO is the side of the area AC. It is next to be proved that MO is the irrational straight line called major. Since AG is incommensurable with EG, therefore AH is also incommensurable with GK, that is, SN with NQ. Therefore MN and NO are incommensurable in square. VI.1 X.11 Since AE is commensurable with AB, therefore AK is rational, and it equals the sum of the squares on MN and NO. Therefore the sum of the squares on MN and NO is also rational. X.19 Since DE is incommensurable in length with AB, that is, with EK, while DE is commensurable with EF, therefore EF is incommensurable in length with EK. X.13 Therefore EK and EF are rational straight lines commensurable in square only. Therefore LE, that is, MR, is medial. X.21 And it is contained by MN and NO, therefore the rectangle MN by NO is medial. And the sum of the squares on MN and NO is rational, and MN and NO are incommensurable in square. But, if two straight lines incommensurable in square and making the sum of the squares on them rational, but the rectangle contained by them medial, are added together, then the whole is irrational and is called major. Therefore MO is the irrational straight line called major and is the side of the area AC. X.39 Therefore, if an area is contained by a rational straight line and the fourth binomial, then the side of the area is the irrational straight line called major. Q.E.D. This proposition is used in X.70. Book X Introduction Proposition X.56 Proposition X.58. © 1996 D.E.Joyce Clark University Proposition 58 If an area is contained by a rational straight line and the fifth binomial, then the side of the area is the irrational straight line called the side of a rational plus a medial area. Let the area AC be contained by the rational straight line AB and the fifth binomial AD divided into its terms at E, so that AE is the greater term. I say that the side of the area AC is the irrational straight line called the side of a rational plus a medial area. Make the same construction shown before. It is then manifest that MO is the side of the area AC. It is then to be proved that MO is the side of a rational plus a medial area. Since AG is incommensurable with GE, therefore AH is also commensurable with HE, that is, the square on MN with the square on NO. Therefore MN and NO are incommensurable in square. X.18 VI.1 X.11 Since AD is a fifth binomial straight line, and ED the lesser segment, therefore ED is commensurable in length with AB. X.Def.II.5 But AE is incommensurable with ED, therefore AB is also incommensurable in length with AE. Therefore AK, that is, the sum of the squares on MN and NO, is medial. X.13 X.21 Since DE is commensurable in length with AB, that is, with EK, while DE is commensurable with EF, therefore EF is also commensurable with EK. X.12 And EK is rational, therefore EL, that is, MR, that is, the rectangle MN by NO, is also rational. X.19 Therefore MN and NO are straight lines incommensurable in square which make the sum of the squares on them medial, but the rectangle contained by them rational. Therefore MO is the side of a rational plus a medial area and is the side of the area AC. X.40 Therefore, if an area is contained by a rational straight line and the fifth binomial, then the side of the area is the irrational straight line called the side of a rational plus a medial area. Q.E.D. This proposition is used in X.71. Book X Introduction Proposition X.57 Proposition X.59. © 1996 D.E.Joyce Clark University Proposition 59 If an area is contained by a rational straight line and the sixth binomial, then the side of the area is the irrational straight line called the side of the sum of two medial areas. Let the area ABCD be contained by the rational straight line AB and the sixth binomial AD, divided into its terms at E, so that AE is the greater term. I say that the side of AC is the side of the sum of two medial areas. Make the same construction as shown before. It is then manifest that MO is the side of AC, and that l! IN is incommensurable in square with NO. Now, since EA is incommensurable in length with AB, therefore EA and AB are rational straight lines commensurable in square only, therefore AK, that is, the sum of the squares on MN and NO, is medial. X.21 Again, since ED is incommensurable in length with AB, therefore FE is also incommensurable with EK. Therefore FE and EK are rational straight lines commensurable in square only. Therefore EL, that is, MR, that is, the rectangle MN by NO, is medial. X.13 X.21 Since AE is incommensurable with EF, therefore AK is also incommensurable with EL. VI.1 X.11 But AK is the sum of the squares on MN and NO, and EL is the rectangle MN by NO, therefore the sum of the squares on MN and NO is incommensurable with the rectangle MN by NO. And each of them is medial, and MN and NO are incommensurable in square. Therefore MO is the side of the sum of two medial areas, and is the side of AC. X.41 Therefore, if an area is contained by a rational straight line and the sixth binomial, then the side of the area is the irrational straight line called the side of the sum of two medial areas. Q.E.D. This proposition is used in X.72. Book X Introduction Proposition X.58 Proposition X.60. © 1996 D.E.Joyce Clark University Proposition 60 Lemma. If a straight line is cut into unequal parts, then the sum of the squares on the unequal parts is greater than twice the rectangle contained by the unequal parts. Let AB be a straight line, and let it be cut into unequal parts at C, and let AC be the greater. I say that the sum of the squares on AC and CB is greater than twice the rectangle AC by CB. Bisect AB at D. Since a straight line is cut into equal parts at D and into unequal parts at C, therefore the rectangle AC by CB together with the square on CD equals the square on AD, so that the rectangle AC by CB is less than the square on AD. Therefore twice the rectangle AC by CB is less than double the square on AD. II.5 But the sum of the squares on AC and CB is double the sum of the squares on AD and DC, therefore the sum of the squares on AC and CB is greater than twice the rectangle AC by CB. II.9 Q.E.D. Proposition 60 The square on the binomial straight line applied to a rational straight line produces as breadth the first binomial. Let AB be a binomial straight line divided into its terms at C, so that AC is the greater term, let a rational straight line DE be set out, and let DEFG equal the square on AB be applied to DE producing DG as its breadth. I say that DG is a first binomial straight line. Apply to DE the rectangle DH equal to the square on AC, and KL equal to the square on BC. Then the remainder, twice the rectangle AC by CB, equals MF. Bisect MG at N, and draw NO parallel to ML or GF. Then each of the rectangles MO and NF equals once the rectangle AC by CB. Now, since AB is a binomial divided into its terms at C, therefore AC and CB are rational straight lines commensurable in square only. X.36 Therefore the squares on AC and CB are rational and commensurable with one an other, so that the sum of the squares on AC and CB is also rational. And it equals DL, therefore DL is rational. X.15 And it is applied to the rational straight line DE, therefore DM is rational and commensurable in length with DE. X.20 Again, since AC and CB are rational straight lines commensurable in square only, therefore twice the rectangle AC by CB, that is MF, is medial. X.21 And it is applied to the rational straight line ML, therefore MG is also rational and incommensurable in length with ML, that is, DE. X.22 But MD is also rational and is commensurable in length with DE, therefore DM is incommensurable in length with MG. X.13 And they are rational, therefore DM and MG are rational straight lines commensurable in square only. Therefore DG is binomial. X.36 It is next to be proved that it is also a first binomial straight line. Since the rectangle AC by CB is a mean proportional between the squares on AC and CB, therefore MO is also a mean proportional between DH and KL. X.54,Lemma Therefore DH is to MO as MO is to KL, that is DK is to MN as MN is to MK. Therefore the rectangle DK by KM equals the square on MN. VI.1 VI.17 Since the square on AC is commensurable with the square on CB, therefore DH is also commensurable with KL, so that DK is also commensurable with KM. VI.1 X.11 Since the sum of the squares on AC and CB is greater than twice the rectangle AC by CB, therefore DL is also greater than MF, so that DM is also greater than MG. VI.1 Lemma And the rectangle DK by KM equals the square on MN, that is, to the fourth part of the square on MG, and DK is commensurable with KM. But, if there are two unequal straight lines, and to the greater there is applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, and if it divides it into commensurable parts, then the square on the greater is greater than the square on the less by the square on a straight line commensurable with the greater. Therefore the square on DM is greater than the square on MG by the square on a straight line commensurable with DM. X.17 And DM and MG are rational, and DM, which is the greater term, is commensurable in length with the rational straight line DE set out. Therefore DG is a first binomial straight line. X.Def.II.1 Therefore, the square on the binomial straight line applied to a rational straight line produces as breadth the first binomial. Q.E.D. This proposition is used in X.72 and X.111. Book X Introduction Proposition X.59 Proposition X.61. © 1996 D.E.Joyce Clark University Proposition 61 The square on the first bimedial straight line applied to a rational straight line produces as breadth the second binomial. Let AB be a first bimedial straight line divided into its medials at C, of which medials AC is the greater. Let a rational straight line DE be set out, and let there be applied to DE the parallelogram DF equal to the square on AB, producing DG as its breadth. I say that DG is a second binomial straight line. Make the same construction as before. Then, since AB is a first bimedial divided at C, therefore AC and CB are medial straight lines commensurable in square only, and containing a rational rectangle, so that the squares on AC and CB are also medial. X.37 X.21 Therefore DL is medial. And it was applied to the rational straight line DE, therefore MD is rational and incommensurable in length with DE. X.15 X.23,Cor. X.22 Again, since twice the rectangle AC by CB is rational, therefore MF is also rational. And it is applied to the rational straight line ML, therefore MG is also rational and commensurable in length with ML, that is, DE. Therefore DM is incommensurable in length with MG. X.20 X.13 And they are rational, therefore DM and MG are rational straight lines commensurable in square only. Therefore DG is binomial. X.36 It is next to be proved that it is a second binomial straight line. Since the sum of the squares on AC and CB is greater than twice the rectangle AC by CB, therefore DL is also greater than MF, so that DM is also greater than MG. VI.1 Since the square on AC is commensurable with the square on CB, therefore DH is also commensurable with KL, so that DK is also commensurable with KM. VI.1 X.11 And the rectangle DK by KM equals the square on MN, therefore the square on DM is greater than the square on MG by the square on a straight line commensurable with DM. And MG is commensurable in length with DE. X.17 Therefore DG is a second binomial straight line. X.Def.II.2 Therefore, the square on the first bimedial straight line applied to a rational straight line produces as breadth the second binomial. Q.E.D. This proposition is used in X.72. Book X Introduction Proposition X.60 Proposition X.62. © 1996 D.E.Joyce Clark University Proposition 62 The square on the second bimedial straight line applied to a rational straight line produces as breadth the third binomial. Let AB be a second bimedial straight line divided into its medials at C, so that AC is the greater segment, let DE be any rational straight line, and to DE let there be applied the parallelogram DF equal to the square on AB and producing DG as its breadth. I say that DG is a third binomial straight line. Make the same construction as shown before. Then, since AB is a second bimedial divided at C, therefore AC and CB are medial straight lines commensurable in square only and containing a medial rectangle, so that the sum of the squares on AC and CB is also medial. X.38 X.15 X.23,Cor. And it equals DL, therefore DL is also medial. And it is applied to the rational straight line DE, therefore MD is also rational and incommensurable in length with DE. X.22 For the same reason, MG is also rational and incommensurable in length with ML, that is, with DE, therefore each of the straight lines DM and MG is rational and incommensurable in length with DE. Since AC is incommensurable in length with CB, and AC is to CB as the square on AC is to the rectangle AC by CB, therefore the square on AC is also incommensurable with the rectangle AC by CB. X.11 Hence the sum of the squares on AC and CB is incommensurable with twice the rectangle AC by CB, that is, DL is incommensurable with MF, so that DM is also incommensurable with MG. X.12 X.13 VI.1 X.11 And they are rational, therefore DG is binomial. X.36 It is to be proved that it is a third binomial straight line. In manner similar to the foregoing we may conclude that DM is greater than MG, and that DK is commensurable with KM. And the rectangle DK by KM equals the square on MN, therefore the square on DM is greater than the square on MG by the square on a straight line commensurable with DM. And neither of the straight lines DM nor MG is commensurable in length with DE. Therefore DG is a third binomial straight line. X.Def.II.3 Therefore, the square on the second bimedial straight line applied to a rational straight line produces as breadth the third binomial. Q.E.D. This proposition is used in X.72. Book X Introduction Proposition X.61 Proposition X.63. © 1996 D.E.Joyce Clark University Proposition 63 The square on the major straight line applied to a rational straight line produces as breadth the fourth binomial. Let AB be a major straight line divided at C, so that AC is greater than CB, let DE be a rational straight line, and to DE let there be applied the parallelogram DF equal to the square on AB and producing DG as its breadth. I say that DG is a fourth binomial straight line. Make the same construction as shown before. Since AB is a major straight line divided at C, therefore AC and CB are straight lines incommensurable in square which make the sum of the squares on them rational, but the rectangle contained by them medial. X.39 Since the sum of the squares on AC and CB is rational, therefore DL is rational. Therefore DM is also rational and commensurable in length with DE. X.20 Again, since twice the rectangle AC by CB, that is, MF, is medial, and it is applied to the rational straight line ML, therefore MG is also rational and incommensurable in length with DE. X.22 Therefore DM is also incommensurable in length with MG. Therefore DM and MG are rational straight lines commensurable in square only. Therefore DG is binomial. X.13 X.36 It is to be proved that it is a fourth binomial straight line. In manner similar to the foregoing we can prove that DM is greater than MG, and that the rectangle DK by KM equals the square on MN. Since the square on AC is incommensurable with the square on CB, therefore DH is also incommensurable with KL, so that DK is also incommensurable with KM. VI.1 X.11 But, if there are two unequal straight lines, and to the greater there is applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, and if it divides it into incommensurable parts, then the square on the greater is greater than the square on the less by the square on a straight line incommensurable in length with the greater, therefore the square on DM is greater than the square on MG by the square on a straight line incommensurable with DM. X.18 And DM and MG are rational straight lines commensurable in square only, and DM is commensurable with the rational straight line DE set out. Therefore DG is a fourth binomial straight line. X.Def.II.4 Therefore, the square on the major straight line applied to a rational straight line produces as breadth the fourth binomial. Q.E.D. This proposition is used in X.72. Book X Introduction Proposition X.62 Proposition X.64. © 1996 D.E.Joyce Clark University Proposition 64 The square on the side of a rational plus a medial area applied to a rational straight line produces as breadth the fifth binomial. Let AB be the side of a rational plus a medial area, divided into its straight lines at C, so that AC is the greater, let a rational straight line DE be set out, and let there be applied to DE the parallelogram DF equal to the square on AB, producing DG as its breadth. I say that DG is a fifth binomial straight line. Make the same construction as before. Since AB is the side of a rational plus a medial area, divided at C, therefore AC and CB are straight lines incommensurable in square which make the sum of the squares on them medial but the rectangle contained by them rational. X.40 Since, then, the sum of the squares on AC and CB is medial, therefore DL is medial, so that DM is rational and incommensurable in length with DE. X.22 Again, since twice the rectangle AC by CB, that is MF, is rational, therefore MG is rational and commensurable with DE. X.20 Therefore DM is incommensurable with MG. Therefore DM and MG are rational straight lines commensurable in square only. Therefore DG is binomial. X.13 X.36 I say next that it is also a fifth binomial straight line. For it can be proved similarly that the rectangle DK by KM equals the square on MN, and that DK is incommensurable in length with KM. Therefore the square on DM is greater than the square on MG by the square on a straight line incommensurable with DM. X.18 And DM and MG are commensurable in square only, and the less, MG, is commensurable in length with DE. Therefore DG is a fifth binomial. Therefore, the square on the side of a rational plus a medial area applied to a rational straight line produces as breadth the fifth binomial. Q.E.D. This proposition is used in X.72. Book X Introduction Proposition X.63 Proposition X.65. © 1996 D.E.Joyce Clark University Proposition 65 The square on the side of the sum of two medial areas applied to a rational straight line produces as breadth the sixth binomial. Let AB be the side of the sum of two medial areas, divided at C, let DE be a rational straight line, and let there be applied to DE the parallelogram DF equal to the square on AB, producing DG as its breadth. I say that DG is a sixth binomial straight line. Make the same construction as before. Since AB is the side of the sum of two medial areas divided at C, therefore AC and CB are straight lines incommensurable in square which make the sum of the squares on them medial, the rectangle contained by them medial, and moreover the sum of the squares on them incommensurable with the rectangle contained by them. X.41 So that, in accordance with what was before proved, each of the rectangles DL and MF is medial. And they are applied to the rational straight line DE, therefore each of the straight lines DM and MG is rational and incommensurable in length with DE. X.22 Since the sum of the squares on AC and CB is incommensurable with twice the rectangle AC by CB, therefore DL is incommensurable with MF. Therefore DM is also incommensurable with MG. VI.1 X.11 Therefore DM and MG are rational straight lines commensurable in square only. Therefore DG is binomial. X.36 I say next that it is a sixth binomial straight line. Similarly again we can prove that the rectangle DK by KM equals the square on MN, and that DK is incommensurable in length with KM, and, for the same reason, the square on DM is greater than the square MG by the square on a straight line incommensurable in length with DM. And neither of the straight lines DM nor MG is commensurable in length with the rational straight line DE set out. Therefore DG is a sixth binomial straight line. Therefore, the square on the side of the sum of two medial areas applied to a rational straight line produces as breadth the sixth binomial. Q.E.D. This proposition is used in X.72. Book X Introduction Proposition X.64 Proposition X.66. © 1996 D.E.Joyce Clark University Proposition 66 A straight line commensurable in length with a binomial straight line is itself also binomial and the same in order. Let AB be binomial, and let CD be commensurable in length with AB. I say that CD is binomial and the same in order with AB. Since AB is binomial, divide it into its terms at E, and let AE be the greater term, therefore AE and EB are rational straight lines commensurable in square only. X.36 Let it be contrived that AB is to CD as AE is to CF. Then the remainder EB is to the remainder FD as AB is to CD. VI.12 V.19 But AB is commensurable in length with CD, therefore AE is also commensurable with CF, and EB with FD. X.11 And AE and EB are rational, therefore CF and FD are also rational. And AE is to CF as EB is to FD. Therefore, alternately, AE is to EB as CF is to FD. V.11 V.16 But AE and EB are commensurable in square only, therefore CF and FD are also commensurable in square only. X.11 And they are rational, therefore CD is binomial. X.36 I say next that it is the same in order with AB. For the square on AE is greater than the square on EB either by the square on a straight line commensurable with AE or by the square on a straight line incommensurable with it. If then the square on AE is greater than the square on EB by the square on a straight line commensurable with AE, then the square on CF is also greater than the square on FD by the square on a straight line commensurable with CF. X.14 And, if AE is commensurable with the rational straight line set out, then CF is also commensurable with it, and for this reason each of the straight lines AB and CD is a first binomial, that is, the same in order. X.12 X.Def.II.1 But, if EB is commensurable with the rational straight line set out, then FD is also commensurable with it, and for this reason again CD is the same in order with AB, for each of them is a second binomial. X.12 X.Def.II.2 But, if neither of the straight lines AE nor EB is commensurable with the rational straight line set out, then neither of the straight lines CF nor FD is commensurable with it, and each of the straight lines AB and CD is a third binomial. X.13 X.Def.II.3 But, if the square on AE is greater than the square on EB by the square on a straight line incommensurable with AE, then the square on CF is also greater than the square on FD by the square on a straight line incommensurable with CF. X.14 And, if AE is commensurable with the rational straight line set out, then CF is also commensurable with it, and each of the straight lines AB and CD is a fourth binomial. X.Def.II.4 But, if EB is so commensurable, then FD is also, and each of the straight lines AB and CD is a fifth binomial. X.Def.II.5 But, if neither of the straight lines AE or EB is so commensurable, then neither of the straight lines CF or FD is commensurable with the rational straight line set out, and each of the straight lines AB and CD is a sixth binomial. X.Def.II.6 Hence a straight line commensurable in length with a binomial straight line is binomial and the same in order. Therefore, a straight line commensurable in length with a binomial straight line is itself also binomial and the same in order. Q.E.D. (Forthcoming) Book X Introduction Proposition X.65 Proposition X.67. © 1996 D.E.Joyce Clark University Proposition 67 A straight line commensurable with a bimedial straight line is itself also bimedial and the same in order. Let AB be bimedial, and let CD be commensurable in length with AB. I say that CD is bimedial and the same in order with AB. Since AB is bimedial, divide it into its medials at E. Then AE and EB are medial straight lines commensurable in square only. X.37 X.38 Let it be contrived that AB is to CD as AE is to CF. Then the remainder EB is to the remainder FD as AB is to CD. V.19 But AB is commensurable in length with CD, therefore AE and EB are commensurable with CF and FD respectively. X.11 But AE and EB are medial, therefore CF and FD are also medial. X.23 Since AE is to EB as CF is to FD, and AE and EB are commensurable in square only, therefore CF and FD are also commensurable in square only. V.11 X.11 But they were also proved medial, therefore CD is bimedial. I say next that it is also the same in order with AB. Since AE is to EB as CF is to FD, therefore the square on AE is to the rectangle AE by EB as the square on CF is to the rectangle CF by FD. Therefore, alternately, the square on AE is to the square on CF as the rectangle AE by EB is to the rectangle CF by FD. V.16 But the square on AE is commensurable with the square on CF, therefore the rectangle AE by EB is commensurable with the rectangle CF by FD. Therefore if the rectangle AE by EB is rational, then the rectangle CF by FD is also rational, and for this reason CD is a first bimedial, but if medial, medial, and each of the straight lines AB and CD is a second bimedial. And for this reason CD is the same in order with AB. X.37 X.23.Cor. X.38 Therefore, a straight line commensurable with a bimedial straight line is itself also bimedial and the same in order. Q.E.D. (Forthcoming) Book X Introduction Proposition X.66 Proposition X.68. © 1996 D.E.Joyce Clark University Proposition 68 A straight line commensurable with a major straight line is itself also major. Let AB be major, and let CD be commensurable with AB. I say that CD is major. Divide AB at E. Then AE and EB are straight lines incommensurable in square which make the sum of the squares on them rational but the rectangle contained by them medial. X.39 Make the same construction as before. Since AB is to CD as AE is to CF, and EB is to FD, therefore AE is to CF as EB is to FD. V.11 But AB is commensurable with CD, therefore AE and EB are commensurable with CF and FD respectively. X.11 Since AE is to CF as EB is to FD, alternately, also AE is to EB as CF is to FD, therefore, taken jointly, AB is to BE as CD is to DF. V.16 V.18 Therefore the square on AB is to the square on BE as the square on CD is to the square on DF. VI.20 Similarly we can prove that the square on AB is to the square on AE as the square on CD is to the square on CF. Therefore the square on AB is to the squares on AE and EB as the square on CD is to the squares on CF and FD, therefore, alternately, the square on AB is to the square on CD, so are the squares on AE and EB to the squares on CF and FD. V.16 But the square on AB is commensurable with the square on CD, therefore the squares on AE and EB are also commensurable with the squares on CF and FD. And the squares on AE and EB together are rational, therefore the squares on CF and FD together are rational. Similarly also twice the rectangle AE by EB is commensurable with twice the rectangle CF by FD. And twice the rectangle AE by EB is medial, therefore twice the rectangle CF by FD is also medial. X.23,Cor. Therefore CF and FD are straight lines incommensurable in square which make, at the same time, the sum of the squares on them rational, but the rectangle contained by them medial, therefore the whole CD is the irrational straight line called major. X.39 Therefore a straight line commensurable with the major straight line is major. Therefore, a straight line commensurable with a major straight line is itself also major. Q.E.D. (Forthcoming) Book X Introduction Proposition X.67 Proposition X.69. © 1996 D.E.Joyce Clark University Proposition 69 A straight line commensurable with the side of a rational plus a medial area is itself also the side of a rational plus a medial area. Let AB be the side of a rational plus a medial area, and let CD be commensurable with AB. It is to be proved that CD is also the side of a rational plus a medial area. Divide AB into its straight lines at E. Then AE and EB are straight lines incommensurable in square which make the sum of the squares on them medial but the rectangle contained by them rational. X.40 Make the same construction as before. We can then prove similarly that CF and FD are incommensurable in square, and the sum of the squares on AE and EB is commensurable with the sum of the squares on CF and FD, and the rectangle AE by EB with the rectangle CF by FD, so that the sum of the squares on CF and FD is also medial, and the rectangle CF by FD rational. Therefore CD is the side of a rational plus a medial area. Therefore, a straight line commensurable with the side of a rational plus a medial area is itself also the side of a rational plus a medial area. Q.E.D. (Forthcoming) Book X Introduction Proposition X.68 Proposition X.70. © 1996 D.E.Joyce Clark University Proposition 70 A straight line commensurable with the side of the sum of two medial areas is the side of the sum of two medial areas. Let AB be the side of the sum of two medial areas, and CD commensurable with AB. It is to be proved that CD is also the side of the sum of two medial areas. Since AB is the side of the sum of two medial areas, divide it into its straight lines at E, therefore AE and EB are straight lines incommensurable in square which make the sum of the squares on them medial, the rectangle contained by them medial, and furthermore the sum of the squares on AE and EB incommensurable with the rectangle AE by EB. X.41 Make the same construction as before. We can then prove similarly that CF and FD are also incommensurable in square, the sum of the squares on AE and EB is commensurable with the sum of the squares on CF and FD, and the rectangle AE by EB with the rectangle CF by FD, so that the sum of the squares on CF and FD is also medial, the rectangle CF by FD is medial, and moreover the sum of the squares on CF and FD is incommensurable with the rectangle CF by FD. Therefore CD is the side of the sum of two medial areas. Therefore, a straight line commensurable with the side of the sum of two medial areas is the side of the sum of two medial areas. Q.E.D. (Forthcoming) Book X Introduction Proposition X.69 Proposition X.71. © 1996 D.E.Joyce Clark University Proposition 71 If a rational and a medial are added together, then four irrational straight lines arise, namely a binomial or a first bimedial or a major or a side of a rational plus a medial area. Let AB be rational, and CD medial. I say that the side of the area AD is a binomial or a first bimedial or a major or a side of a rational plus a medial area. For AB is either greater or less than CD. First, let it be greater. Set out a rational straight line EF , apply to EF the rectangle EG equal to AB, producing EH as breadth, and apply to EF HI, equal to DC, producing HK as breadth. Then, since AB is rational and equals EG, therefore EG is also rational. And it is applied to EF, producing EH as breadth, therefore EH is rational and commensurable in length with EF. X.20 Again, since CD is medial and equals HI, therefore HI is also medial. And it is applied to the rational straight line EF, producing HK as breadth, therefore HK is rational and incommensurable in length with EF. X.22 Since CD is medial, while AB is rational, therefore AB is incommensurable with CD, so that EG is also incommensurable with HI. But EG is to HI as EH is to HK, therefore EH is also incommensurable in length with HK. VI.1 X.11 And both are rational, therefore EH and HK are rational straight lines commensurable in square only. Therefore EK is a binomial straight line, divided at H. X.36 Since AB is greater than CD, while AB equals EG and CD equals HI, therefore EG is also greater than HI. Therefore EH is also greater than HK. The square, then, on EH is greater than the square on HK either by the square on a straight line commensurable in length with EH or by the square on a straight line incommensurable with it. First, let the square on it be greater by the square on a straight line commensurable with itself. Now the greater straight line HE is commensurable in length with the rational straight line EF set out, therefore EK is a first binomial. X.Def.II.1 But EF is rational, and, if an area is contained by a rational straight line and the first binomial, then the side of the square equal to the area is binomial. Therefore the side of EI is binomial, so that the side of AD is also binomial. X.54 Next, let the square on EH be greater than the square on HK by the square on a straight line incommensurable with EH. Now the greater straight line EH is commensurable in length with the rational straight line EF set out, therefore EK is a fourth binomial. X.Def.II.4 But EF is rational, and, if an area be contained by a rational straight line and the fourth binomial, then the side of the area is the irrational straight line called major. Therefore the side of the area EI is major, so that the side of the area AD is also major. X.57 Next, let AB be less than CD. Then EG is also less than HI, so that EH is also less than HK. Now the square on HK is greater than the square on EH either by the square on a straight line commensurable with HK or by the square on a straight line incommensurable with it. First, let the square on it be greater by the square on a straight line commensurable in length with itself. Now the lesser straight line EH is commensurable in length with the rational straight line EF set out, therefore EK is a second binomial. X.Def.II.2 But EF is rational, and, if an area is contained by a rational straight line and the second binomial, then the side of the square it is a first bimedial, therefore the side of the area EI is a first bimedial, so that the side of AD is also a first bimedial. X.55 Next, let the square on HK be greater than the square on HE by the square on a straight line incommensurable with HK. Now the lesser straight line EH is commensurable with the rational straight line EF set out, therefore EK is a fifth binomial. X.Def.II.5 But EF is rational, and, if an area is contained by a rational straight line and the fifth binomial, then the side of the square equal to the area is a side of a rational plus a medial area. X.58 Therefore the side of the area EI is a side of a rational plus a medial area, so that the side of the area AD is also a side of a rational plus a medial area. Therefore, if a rational and a medial are added together, then four irrational straight lines arise, namely a binomial or a first bimedial or a major or a side of a rational plus a medial area. Q.E.D. (Forthcoming) Book X Introduction Proposition X.70 Proposition X.72. © 1996 D.E.Joyce Clark University Proposition 72 If two medial areas incommensurable with one another are added together, then the remaining two irrational straight lines arise, namely either a second bimedial or a side of the sum of two medial areas. Let two medial areas AB and CD incommensurable with one another be added together. I say that the side of the area AD is either a second bimedial or a side of the sum of two medial areas. For AB is either greater or less than CD. First, let AB be greater than CD. Set out the rational straight line EF, and apply to EF the rectangle EG equal to AB and producing EH as breadth, and the rectangle HI equal to CD and producing HK as breadth. Now, since each of the areas AB and CD is medial, therefore each of the areas EG and HI is also medial. And they are applied to the rational straight line FE producing EH and HK as breadth, therefore each of the straight lines EH and HK is rational and incommensurable in length with EF. X.22 Since AB is incommensurable with CD, and AB equals EG, and CD equals HI, therefore EG is also incommensurable with HI. But EG is to HI as EH is to HK, therefore EH is incommensurable in length with HK. VI.1 X.11 Therefore EH and HK are rational straight lines commensurable in square only, therefore EK is binomial. X.36 But the square on EH is greater than the square on HK either by the square on a straight line commensurable with EH or by the square on a straight line incommensurable with it. First, let the square on it be greater by the square on a straight line commensurable in length with itself. Now neither of the straight lines EH nor HK is commensurable in length with the rational straight line EF set out, therefore EK is a third binomial. X.Def.II.3 But EF is rational, and, if an area is contained by a rational straight line and the third binomial, then the side of the area is a second bimedial, therefore the side of EI, that is, of AD, is a second bimedial. X.56 Next, let the square on EH be greater than the square on HK by the square on a straight line incommensurable in length with EH. Now each of the straight lines EH and HK is incommensurable in length with EF, therefore EK is a sixth binomial. X.Def.II.6 But, if an area is contained by a rational straight line and the sixth binomial, then the side of the area is the side of the sum of two medial areas, so that the side of the area AD is also the side of the sum of two medial areas. X.59 Therefore, if two medial areas incommensurable with one another are added together, then the remaining two irrational straight lines arise, namely either a second bimedial or a side of the sum of two medial areas. Q.E.D. Proposition The binomial straight line and the irrational straight lines after it are neither the same with the medial nor with one another. For the square on a medial, if applied to a rational straight line, produces as breadth a straight line rational and incommensurable in length with that to which it is applied. But the square on the binomial, if applied to a rational straight line, produces as breadth the first binomial. X.22 X.60 The square on the first bimedial, if applied to a rational straight line, produces as breadth the second binomial. X.61 The square on the second bimedial, if applied to a rational straight line, produces as breadth the third binomial. X.62 The square on the major, if applied to a rational straight line, produces as breadth the fourth binomial. X.63 The square on the side of a rational plus a medial area, if applied to a rational straight line, produces as breadth the fifth binomial. X.64 The square on the side of the sum of two medial areas, if applied to a rational straight line, produces as breadth the sixth binomial. X.65 And the said breadths differ both from the first and from one another, from the first because it is rational, and from one another because they are not the same in order, so that the irrational straight lines themselves also differ from one another. (Forthcoming) Book X Introduction Proposition X.71 Proposition X.73. © 1996 D.E.Joyce Clark University Proposition 73 If from a rational straight line there is subtracted a rational straight line commensurable with the whole in square only, then the remainder is irrational; let it be called an apotome. From the rational straight line AB let the rational straight line BC, commensurable with the whole in square only, be subtracted. I say that the remainder AC is the irrational straight line called apotome. Since AB is incommensurable in length with BC, and AB is to BC as the square on AB is to the rectangle AB by BC, therefore the square on AB is incommensurable with the rectangle AB by BC. X.11 But the sum of the squares on AB and BC is commensurable with the square on AB, and twice the rectangle AB by BC is commensurable with the rectangle AB by BC. X.15 X.6 And, inasmuch as the sum of the squares on AB and BC equal twice the rectangle AB by BC together with the square on CA, therefore the sum of the squares on AB and BC is also incommensurable with the remainder, the square on AC. II.7 X.13 X.16 But the sum of the squares on AB and BC is rational, therefore AC is irrational. Let it be called an apotome. X.Def.4 Q.E.D. This proposition is used very frequently in the rest of Book X starting with X.75. It is also used in propositions XIII.6 and XIII.11. Book X Introduction Proposition X.72 Proposition X.74. © 1996 D.E.Joyce Clark University Proposition 74 If from a medial straight line there is subtracted a medial straight line which is commensurable with the whole in square only and which contains with the whole a rational rectangle, then the remainder is irrational; let it be called first apotome of a medial straight line. From the medial straight line AB let there be subtracted the medial straight line BC which is commensurable with AB in square only and with AB makes the rectangle AB by BC rational. I say that the remainder AC is irrational, and let it be called an apotome of a medial straight line. Since AB and BC are medial, the squares on AB and BC are also medial. But twice the rectangle AB by BC is rational, therefore the sum of the squares on AB and BC is incommensurable with twice the rectangle AB by BC. Therefore twice the rectangle AB by BC is also incommensurable with the remainder, the square on AC, since, if the whole is incommensurable with one of the magnitudes, then the original magnitudes are also incommensurable. cf. II.7 X.16 But twice the rectangle AB by BC is rational, therefore the square on AC is irrational, therefore AC is irrational. Let it be called a first apotome of a medial straight line. X.Def.4 Q.E.D. This proposition is used for a few later propositions in Book X starting with X.80. Book X Introduction Proposition X.73 Proposition X.75. © 1996 D.E.Joyce Clark University Proposition 75 If from a medial straight line there is subtracted a medial straight line which is commensurable with the whole in square only, and which contains with the whole a medial rectangle, then the remainder is irrational; let it be called second apotome of a medial straight line. From the medial straight line AB let there be subtracted the medial straight line CB which is commensurable with the whole AB in square only such that the rectangle AB by BC which it contains with the whole AB, is medial. X.28 I say that the remainder AC is irrational, and let it be called a second apotome of a medial straight line. Set out a rational straight line DI. Apply DE, equal to the sum of the squares on AB and BC, to DI producing DG as breadth. Apply DH, equal to twice the rectangle AB by BC, to DI producing DF as breadth. Then the remainder FE equals the square on AC. II.7 Now, since the squares on AB and BC are medial and commensurable, therefore DE is also medial. X.15 X.23,Cor. And it is applied to the rational straight line DI, producing DG as breadth, therefore DG is rational and incommensurable in length with DI. X.22 Again, since the rectangle AB by BC is medial, therefore twice the rectangle AB by BC is also medial. X.23,Cor. And it equals DH, therefore DH is also medial. And it is applied to the rational straight line DI, producing DF as breadth, therefore DF is rational and incommensurable in length with DI. X.22 Since AB and BC are commensurable in square only, therefore AB is incommensurable in length with BC. Therefore the square on AB is also incommensurable with the rectangle AB by BC. X.11 But the sum of the squares on AB and BC is commensurable with the square on AB, and twice the rectangle AB by BC is commensurable with the rectangle AB by BC, therefore twice the rectangle AB by BC is incommensurable with the sum of the squares on AB and BC. X.15 X.6 X.13 But DE equals the sum of the squares on AB and BC, and DH equals twice the rectangle AB by BC, therefore DE is incommensurable with DH. But DE is to DH as GD is to DF, therefore GD is incommensurable with DF. VI.1 X.11 And both are rational, therefore GD and DF are rational straight lines commensurable in square only. Therefore FG is an apotome. X.73 But DI is rational, and the rectangle contained by a rational and an irrational straight line is irrational, and its side is irrational. X.20 And AC is the side of FE, therefore AC is irrational. Let it be called a second apotome of a medial straight line. Q.E.D. This proposition is used for a few later propositions in Book X starting with X.81. Book X Introduction Proposition X.74 Proposition X.76. © 1996 D.E.Joyce Clark University Proposition 76 If from a straight line there is subtracted a straight line which is incommensurable in square with the whole and which with the whole makes the sum of the squares on them added together rational, but the rectangle contained by them medial, then the remainder is irrational; let it be called minor. From the straight line AB let there be subtracted the straight line BC which is incommensurable in square with the whole and fulfills the given conditions. X.33 I say that the remainder AC is the irrational straight line called minor. Since the sum of the squares on AB and BC is rational, while twice the rectangle AB by BC is medial, therefore the sum of the squares on AB and BC is incommensurable with twice the rectangle AB by BC, and, in conversion, the sum of the squares on AB and BC is incommensurable with the remainder, the square on AC. II.7 X.16 But the sum of the squares on AB and BC is rational, therefore the square on AC is irrational. Therefore AC is irrational. Let it be called minor. Q.E.D. This proposition is used for a few later propositions in Book X starting with X.82. Book X Introduction Proposition X.75 Proposition X.77. © 1996 D.E.Joyce Clark University Proposition 77 If from a straight line there is subtracted a straight line which is incommensurable in square with the whole, and which with the whole makes the sum of the squares on them medial but twice the rectangle contained by them rational, then the remainder is irrational; let it be called that which produces with a rational area a medial whole. From the straight line AB let there be subtracted the straight line BC which is incommensurable in square with AB and fulfills the given conditions. I say that the remainder AC is the irrational straight line aforesaid. Since the sum of the squares on AB and BC is medial, while twice the rectangle AB by BC is rational, therefore the sum of the squares on AB and BC is incommensurable with twice the rectangle AB by BC. Therefore the remainder, the square on AC, is also incommensurable with twice the rectangle AB by BC. II.7 X.16 And twice the rectangle AB by BC is rational, therefore the square on AC is irrational. Therefore AC is irrational. Let it be called that which produces with a rational area a medial whole. Q.E.D. This proposition is used for a few later propositions in Book X starting with X.83. Book X Introduction Proposition X.76 Proposition X.78. © 1996 D.E.Joyce Clark University Proposition 78 If from a straight line there is subtracted a straight line which is incommensurable in square with the whole and which with the whole makes the sum of the squares on them medial, twice the rectangle contained by them medial, and further the sum of the squares on them incommensurable with twice the rectangle contained by them, then the remainder is irrational; let it be called that which produces with a medial area a medial whole. From the straight line AB let there be subtracted the straight line BC incommensurable in square with AB and fulfilling the given conditions. X.35 I say that the remainder AC is the irrational straight line called that which produces with a medial area a medial whole. Set out a rational straight line DI. Apply DE, equal to the sum of the squares on AB and BC, to DI producing DG as breadth. Subtract DH equal twice the rectangle AB by BC. Then the remainder FE equals the square on AC, so that AC is the side of FE. II.7 Now, since the sum of the squares on AB and BC is medial and equals DE, therefore DE is medial. And it is applied to the rational straight line DI producing DG as breadth, therefore DG is rational and incommensurable in length with DI. X.22 Again, since twice the rectangle AB by BC is medial and equals DH, therefore DH is medial. And it is applied to the rational straight line DI producing DF as breadth, therefore DF is also rational and incommensurable in length with DI. X.22 Since the sum of the squares on AB and BC is incommensurable with twice the rectangle AB by BC, therefore DE is also incommensurable with DH. But DE is to DH as DG is to DF, therefore DG is incommensurable with DF. X.11 VI.1 And both are rational, therefore GD and DF are rational straight lines commensurable in square only. Therefore FG is an apotome. X.73 And FH is rational, but the rectangle contained by a rational straight line and an apotome is irrational, and its side is irrational. X.20 And AC is the side of FE, therefore AC is irrational. Let it be called that which produces with a medial area a medial whole. Q.E.D. This proposition is used for a few later propositions in Book X starting with X.84. Book X Introduction Proposition X.77 Proposition X.79. © 1996 D.E.Joyce Clark University Proposition 79 To an apotome only one rational straight line can be annexed which is commensurable with the whole in square only. Let AB be an apotome, and BC an annex to it. Then AC and CB are rational straight lines commensurable in square only. X.73 I say that no other rational straight line can be annexed to AB which is commensurable with the whole in square only. If possible, let BD be so annexed. Then AD and DB are also rational straight lines commensurable in square only. X.73 Now, since the excess of the sum of the squares on AD and DB over twice the rectangle AD by DB is also the excess of the sum of the squares on AC and CB over twice the rectangle AC by CB, for both exceed by the same, the square on AB, therefore, alternately, the excess of the sum of the squares on AD and DB over the sum of the squares on AC and CB is the excess of twice the rectangle AD by DB over twice the rectangle AC by CB. II.7 But the sum of the squares on AD and DB exceeds the sum of the squares on AC and CB by a rational area, for both are rational, therefore twice the rectangle AD by DB also exceeds twice the rectangle AC by CB by a rational area, which is impossible, for both are medial, and a medial area does not exceeded a medial by a rational area. X.21 X.26 Therefore no other rational straight line can be annexed to AB which is commensurable with the whole in square only. Therefore only one rational straight line can be annexed to an apotome which is commensurable with the whole in square only. Therefore, to an apotome only one rational straight line can be annexed which is commensurable with the whole in square only. Q.E.D. This proposition is used in X.81 and X.84. Book X Introduction Proposition X.78 Proposition X.80. © 1996 D.E.Joyce Clark University Proposition 80 To a first apotome of a medial straight line only one medial straight line can be annexed which is commensurable with the whole in square only and which contains with the whole a rational rectangle. Let AB be a first apotome of a medial straight line, and let KC be an annex to AB. Then AC and CB are medial straight lines commensurable in square only such that the rectangle AC by CB which they contain is rational. X.74 I say that no other medial straight line can be annexed to AB which is commensurable with the whole in square only and which contains with the whole a rational area. If possible, let DB also be so annexed. Then AD and DB are medial straight lines commensurable in square only such that the rectangle AD by DB which they contain is rational. X.74 Now, since the excess of the sum of the squares on AD and DB over twice the rectangle AD by DB is also the excess of the sum of the squares on AC and CB over twice the rectangle AC by CB, for they exceed by the same, the square on AB, therefore, alternately, the excess of the sum of the squares on AD and DB over the sum of the squares on AC and CB is also the excess of twice the rectangle AD by DB over twice the rectangle AC by CB. II.7 But twice the rectangle AD by DB exceeds twice the rectangle AC by CB by a rational area, for both are rational. Therefore the sum of the squares on AD and DB also exceeds the sum of the squares on AC and CB by a rational area, which is impossible, for both are medial, and a medial area does not exceed a medial by a rational area. X.15 X.23,Cor. X.26 Therefore, to a first apotome of a medial straight line only one medial straight line can be annexed which is commensurable with the whole in square only and which contains with the whole a rational rectangle. Q.E.D. (Forthcoming) Book X Introduction Proposition X.79 Proposition X.81. © 1996 D.E.Joyce Clark University Proposition 81 To a second apotome of a medial straight line only one medial straight line can be annexed which is commensurable with the whole in square only and which contains with the whole a medial rectangle. Let AB be a second apotome of a medial straight line and BC an annex to AB. Then AC and CB are medial straight lines commensurable in square only such that the rectangle AC by CB which they contain is medial. X.75 I say that no other medial straight line can be annexed to AB which is commensurable with the whole in square only and which contains with the whole a medial rectangle. If possible, let BD also be so annexed. Then AD and DB are also medial straight lines commensurable in square only such that the rectangle AD by DB which they contain is medial. X.75 Set out a rational straight line EF. Apply EG, equal to the sum of the squares on AC and CB, to EF producing EM as breadth. Subtract HG, equal to twice the rectangle AC by CB, producing HM as breadth. Then the remainder EL equals the square on AB, so that AB is the side of EL. II.7 Again, apply EI, equal to the sum of the squares on AD and DB, to EF producing EN as breadth. But EL also equals the square on AB, therefore the remainder HI equals twice the rectangle AD by DB. II.7 Now, since AC and CB are medial straight lines, therefore the squares on AC and CB are also medial. And they equal EG, therefore EG is also medial. X.15 X.23,Cor. And it is applied to the rational straight line EF, producing EM as breadth, therefore EM is rational and incommensurable in length with EF. X.22 Again, since the rectangle AC by CB is medial, twice the rectangle AC by CB is also medial. And it equals HG, therefore HG is also medial. X.23,Cor. And it is applied to the rational straight line EF, producing HM as breadth, therefore HM is also rational and incommensurable in length with EF. X.22 Since AC and CB are commensurable in square only, therefore AC is incommensurable in length with CB. But AC is to CB as the square on AC is to the rectangle AC by CB, therefore the square on AC is incommensurable with the rectangle AC by CB. X.11 But the sum of the squares on AC and CB is commensurable with the square on AC, while twice the rectangle AC by CB is commensurable with the rectangle AC by CB, therefore the sum of the squares on AC and CB is incommensurable with twice the rectangle AC by CB. X.6 X.13 And EG equals the sum of the squares on AC and CB, while GH equals twice the rectangle AC by CB, therefore EG is incommensurable with HG. But EG is to HG as EM is to HM, therefore EM is incommensurable in length with MH. VI.1 X.11 And both are rational, therefore EM and MH are rational straight lines commensurable in square only, therefore EH is an apotome, and HM an annex to it. X.73 Similarly we can prove that HN is also an annex to it. Therefore to an apotome different straight lines are annexed which are commensurable with the wholes in square only, which is impossible. X.79 Therefore, to a second apotome of a medial straight line only one medial straight line can be annexed which is commensurable with the whole in square only and which contains with the whole a medial rectangle. Q.E.D. (Forthcoming) Book X Introduction Proposition X.80 Proposition X.82. © 1996 D.E.Joyce Clark University Proposition 82 To a minor straight line only one straight line can be annexed which is incommensurable in square with the whole and which makes, with the whole, the sum of squares on them rational but twice the rectangle contained by them medial. Let AB be the minor straight line, and let BC be an annex to AB. Then AC and CB are straight lines incommensurable in square which make the sum of the squares on them rational, but twice the rectangle contained by them medial. X.76 I say that no other straight line can be annexed to AB fulfilling the same conditions. If possible, let BD be so annexed. Then AD and DB are both straight lines incommensurable in square which fulfill the aforesaid conditions. X.76 Now, since the excess of the sum of the squares on AD and DB over the sum of the squares on AC and CB is also the excess of twice the rectangle AD by DB over twice the rectangle AC by CB, while the sum of the squares on AD and DB exceed the sum of the squares on AC and CB by a rational area, for both are rational, therefore twice the rectangle AD by DB also exceeds twice the rectangle AC by CB by a rational area, which is impossible, for both are medial. X.26 Therefore, to a minor straight line only one straight line can be annexed which is incommensurable in square with the whole and which makes, with the whole, the sum of squares on them rational but twice the rectangle contained by them medial. Q.E.D. (Forthcoming) Book X Introduction Proposition X.81 Proposition X.83. © 1996 D.E.Joyce Clark University Proposition 83 To a straight line which produces with a rational area a medial whole only one straight line can be annexed which is incommensurable in square with the whole straight line and which with the whole straight line makes the sum of squares on them medial but twice the rectangle contained by them rational. Let AB be the straight line which produces with a rational area a medial whole, and let BC be an annex to AB. Then AC and CB are straight lines incommensurable in square which fulfill the given conditions. X.77 I say that no other straight line can be annexed to AB which fulfills the same conditions. If possible, let BD be so annexed. Then AD and DB are both straight lines incommensurable in square which fulfill the given conditions. X.77 As in the preceding cases, the excess of the sum of the squares on AD and DB over the sum of the squares on AC and CB is also the excess of twice the rectangle AD by DB over twice the rectangle AC by CB, while twice the rectangle AD by DB exceeds twice the rectangle AC by CB by a rational area, for both are rational, therefore the sum of the squares on AD and DB also exceeds the sum of the squares on AC and CB by a rational area, which is impossible, for both are medial. X.26 Therefore no other straight line can be annexed to AB which is incommensurable in square with the whole and which with the whole fulfills the aforesaid conditions, therefore only one straight line can be so annexed. Q.E.D. (Forthcoming) Book X Introduction Proposition X.82 Proposition X.84. © 1996 D.E.Joyce Clark University Proposition 84 To a straight line which produces with a medial area a medial whole only one straight line can be annexed which is incommensurable in square with the whole straight line and which with the whole straight line makes the sum of squares on them medial and twice the rectangle contained by them both medial and also incommensurable with the sum of the squares on them. Let AB be the straight line which produces with a medial area a medial whole, and BC an annex to it. Then AC and CB are straight lines incommensurable in square which fulfill the aforesaid conditions. X.78 I say that no other straight line can be annexed to AB which fulfills the aforesaid conditions. If possible, let BD be so annexed, so that AD and DB are also straight lines incommensurable in square which make the squares on AD and DB added together medial, twice the rectangle AD by DB medial, and also the sum of the squares on AD and DB incommensurable with twice the rectangle AD by DB. X.78 Set out a rational straight line EF. Apply EG, equal to the sum of the squares on AC and CB, to EF producing EM as breadth. Apply HG, equal to twice the rectangle AC by CB, to EF producing HM as breadth. Then the remainder, the square on AB, equals EL. Therefore AB is the side of EL. II.7 Again, apply EI, equal to the sum of the squares on AD and DB, to EF producing EN as breadth. But the square on AB also equals EL, therefore the remainder, twice the rectangle AD by DB, equals HI. II.7 Now, since the sum of the squares on AC and CB is medial and equals EG, therefore EG is also medial. And it is applied to the rational straight line EF producing EM as breadth, therefore EM is rational and incommensurable in length with EF. X.22 Again, since twice the rectangle AC by CB is medial and equals HG, therefore HG is also medial. And it is applied to the rational straight line EF producing HM as breadth, therefore HM is rational and incommensurable in length with EF. X.22 Since the sum of the squares on AC and CB is incommensurable with twice the rectangle AC by CB, therefore EG is also incommensurable with HG. Therefore EM is also incommensurable in length with MH. VI.1 X.11 And both are rational, therefore EM and MH are rational straight lines commensurable in square only. Therefore EH is an apotome, and HM an annex to it. X.73 Similarly we can prove that EH is again an apotome and HN an annex to it. Therefore to an apotome different rational straight lines are annexed which are commensurable with the wholes in square only, which was proved impossible. X.79 Therefore no other straight line can be so annexed to AB. Therefore to AB only one straight line can be annexed which is incommensurable in square with the whole and which with the whole makes the squares on them added together medial, twice the rectangle contained by them medial, and also the sum of the squares on them incommensurable with twice the rectangle contained by them. Q.E.D. (Forthcoming) Book X Introduction Definitions III of Book X Proposition X.85. © 1996 D.E.Joyce Clark University Definitions III Definition 1. Given a rational straight line and an apotome, if the square on the whole is greater than the square on the annex by the square on a straight line commensurable in length with the whole, and the whole is commensurable in length with the rational line set out, let the apotome be called a first apotome. Definition 2. But if the annex is commensurable with the rational straight line set out, and the square on the whole is greater than that on the annex by the square on a straight line commensurable with the whole, let the apotome be called a second apotome. Definition 3. But if neither is commensurable in length with the rational straight line set out, and the square on the whole is greater than the square on the annex by the square on a straight line commensurable with the whole, let the apotome be called a third apotome. Definition 4. Again, if the square on the whole is greater than the square on the annex by the square on a straight line incommensurable with the whole, then, if the whole is commensurable in length with the rational straight line set out, let the apotome be called a fourth apotome; Definition 5. If the annex be so commensurable, a fifth; Definition 6. And, if neither, a sixth. (Forthcoming) Book X Introduction Proposition X.84 Proposition X.85. © 1996 D.E.Joyce Clark University Proposition 85 To find the first apotome. Set out a rational straight line, and let BG be commensurable in length with A. Then BG is also rational. Set out two square numbers DE and EF, and let their difference FD not be square. Then ED does not have to DF the ratio which a square number has to a square number. Let it be contrived that ED is to DF as the square on BG is to the square on GC. Then the square on BG is commensurable with the square on GC. X.6,Cor. X.6 But the square on BG is rational, therefore the square on GC is also rational. Therefore GC is also rational. Since ED does not have to DF the ratio which a square number has to a square number, therefore neither has the square on BG to the square on GC the ratio which a square number has to a square number. Therefore BG is incommensurable in length with GC. X.9 And both are rational, therefore BG and GC are rational straight lines commensurable in square only. Therefore BC is an apotome. X.73 I say next that it is also a first apotome. Let the square on H be that by which the square on BG is greater than the square on GC. Now since ED is to FD as the square on BG is to the square on GC, therefore, in conversion, as DE is to EF as the square on GB is to the square on H. V.19,Cor. But DE has to EF the ratio which a square number has to a square number, for each is square, therefore the square on GB also has to the square on H the ratio which a square number has to a square number. Therefore BG is commensurable in length with H. X.9 And the square on BG is greater than the square on GC by the square on H, therefore the square on BG is greater than the square on GC by the square on a straight line commensurable in length with BG. And the whole BG is commensurable in length with the rational straight line A set out. Therefore BC is a first apotome. Therefore the first apotome BC has been found. X.Def.III.2 Q.E.F. (Forthcoming) Book X Introduction Definitions III of Book X Proposition X.86. © 1996 D.E.Joyce Clark University Proposition 86 To find the second apotome. Set out a rational straight line A, and let GC be commensurable in length with A. Then GC is rational. Set out two square numbers DE and EF, and let their difference DF not be square. Now let it be contrived that FD is to DE as the square on CG is to the square on GB. X.6,Cor. Then the square on CG is commensurable with the square on GB. X.6 But the square on CG is rational, therefore the square on GB is also rational. Therefore BG is rational. And, since the square on GC does not have to the square on GB the ratio which a square number has to a square number, therefore CG is incommensurable in length with GB. X.9 And both are rational, therefore CG and GB are rational straight lines commensurable in square only. Therefore BC is an apotome. X.73 I say next that it is also a second apotome. Let the square on H be that by which the square on BG is greater than the square on GC. Since the square on BG is to the square on GC as the number ED is to the number DF, therefore, in conversion, the square on BG is to the square on H as DE is to EF. V.19,Cor. And each of the numbers DE and EF is square, therefore the square on BG has to the square on H the ratio which a square number has to a square number. Therefore BG is commensurable in length with H. X.9 And the square on BG is greater than the square on GC by the square on H, therefore the square on BG is greater than the square on GC by the square on a straight line commensurable in length with BG. And CG, the annex, is commensurable with the rational straight line A set out, therefore BC is a second apotome. X.Def.III.2 Therefore the second apotome BC has been found. Q.E.F. (Forthcoming) Book X Introduction Proposition X.85 Proposition X.87. © 1996 D.E.Joyce Clark University Proposition 87 To find the third apotome. Set out a rational straight line A. Set out three numbers E, BC, and CD which do not have to one another the ratio which a square number has to a square number, but let CB have to BD the ratio which a square number has to a square number. Let it be contrived that E is to BC as the square on A is to the square on FG, and BC is to CD as the square on FG is to the square on GH. X.6,Cor. Since E is to BC as the square on A is to the square on FG, therefore the square on A is commensurable with the square on FG. X.6 But the square on A is rational, therefore the square on FG is also rational, therefore FG is rational. Since E does not have to BC the ratio which a square number has to a square number, therefore neither has the square on A to the square on FG the ratio which 3 square number has to a square number. Therefore A is incommensurable in length with FG. X.9 Again, since BC is to CD as the square on FG is to the square on GH, therefore the square on FG is commensurable with the square on GH. X.6 But the square on FG is rational, therefore the square on GH is also rational, therefore GH is rational. Since BC does not have to CD the ratio which a square number has to a square number, therefore neither has the square on FG to the square on GH the ratio which a square number has to a square number. Therefore FG is incommensurable in length with GH. X.9 And both are rational, therefore FG and GH are rational straight lines commensurable in square only. Therefore FH is an apotome. X.73 I say next that it is also a third apotome. Since E is to BC as the square on A is to the square on FG, and BC is to CD as the square on FG is to the square on HG, therefore, ex aequali, E is to CD as the square on A is to the square on HG. V.22 But E does not have to CD the ratio which a square number has to a square number, therefore neither has the square on A to the square on GH the ratio which a square number has to a square number. Therefore A is incommensurable in length with GH. X.9 Therefore neither of the straight lines FG nor GH is commensurable in length with the rational straight line A set out. Now let the square on K be that by which the square on FG is greater than the square on GH. Since BC is to CD as the square on FG is to the square on GH, therefore, in conversion, BC is to BD as the square on FG is to the square on K. V.19,Cor. But BC has to BD the ratio which a square number has to a square number, therefore the square on FG also has to the square on K the ratio which a square number has to a square number. Therefore FG is commensurable in length with K, and the square on FG is greater than the square on GH by the square on a straight line commensurable with FG. X.9 And neither of the straight lines FG nor GH is commensurable in length with the rational straight line A set out, therefore FH is a third apotome. X.Def.III.3 Therefore the third apotome FH has been found. Q.E.F. (Forthcoming) Book X Introduction Proposition X.86 Proposition X.88. © 1996 D.E.Joyce Clark University Proposition 88 To find the fourth apotome. Set out a rational straight line A, and let BG be commensurable in length with it. Set out two numbers DF and FE such that the whole DE has to neither of the numbers DE nor EF the ratio which a square number has to a square number. Let it be contrived that DE is to EF as the square on BG is to the square on GC. Then the square on BG is commensurable with the square on GC. X.6,Cor. X.6 But the square on BG is rational, therefore the square on GC is also rational. Therefore GC is rational. Now, since DE does not have to EF the ratio which a square number has to a square number, therefore neither has the square on BG to the square on GC the ratio which a square number has to a square number. Therefore BG is incommensurable in length with GC. X.9 And both are rational, therefore BG and GC are rational straight lines commensurable in square only. Therefore BC is an apotome. X.73 Now let the square on H be that by which the square on BG is greater than the square on GC. Since DE is to EF as the square on BG is to the square on GC, therefore, in conversion, ED is to DF as the square on GB is to the square on H. V.19,Cor. But ED does not have to DF the ratio which a square number has to a square number, therefore neither has the square on GB to the square on H the ratio which a square number has to a square number. Therefore BG is incommensurable in length with H. X.9 And the square on BG is greater than the square on GC by the square on H, therefore the square on BG is greater than the square on GC by the square on a straight line incommensurable with BG. And the whole BG is commensurable in length with the rational straight line A set out. Therefore BC is a fourth apotome. Therefore the fourth apotome has been found. X.Def.III.4 Q.E.F. (Forthcoming) Book X Introduction Proposition X.87 Proposition X.89. © 1996 D.E.Joyce Clark University Proposition 89 To find the fifth apotome. Set out a rational straight line A, and let CG be commensurable in length with A. Then CG is rational. Set out two numbers DF and FE such that DE again has to neither of the numbers DF nor FE the ratio which a square number has to a square number, and let it be contrived that FE is to ED as the square on CG is to the square on GB. Then the square on GB is also rational. Therefore BG is also rational. X.6 Now since DE is to EF as the square on BG is to the square on GC, while DE does not have to EF the ratio which a square number has to a square number, therefore neither does the square on BG have to the square on GC the ratio which a square number has to a square number. Therefore BG is incommensurable in length with GC. X.9 And both are rational, therefore BG and GC are rational straight lines commensurable in square only. Therefore BC is an apotome. X.73 I say next that it is also a fifth apotome. Let the square on H be that by which the square on BG is greater than the square on GC. Since the square on BG is to the square on GC as DE is to EF, therefore, in conversion, ED is to DF as the square on BG is to the square on H. V.19,Cor. But ED does not have to DF the ratio which a square number has to a square number, therefore neither has the square on BG to the square on H the ratio which a square number has to a square number. Therefore BG is incommensurable in length with H. X.9 And the square on BG is greater than the square on GC by the square on H, therefore the square on GB is greater than the square on GC by the square on a straight line incommensurable in length with GB. And the annex CG is commensurable in length with the rational straight line A set out, therefore BC is a fifth apotome. X.Def.III.5 Therefore the fifth apotome BC has been found. Q.E.F. (Forthcoming) Book X Introduction Proposition X.88 Proposition X.90. © 1996 D.E.Joyce Clark University Proposition 90 To find the sixth apotome. Set out a rational straight line A, and set out three numbers E, BC, and CD not having to one another the ratio which a square number has to a square number, and further let CB also not have to BD the ratio which a square number has to a square number. Let it be contrived that E is to BC as the square on A is to the square on FG, and BC is to CD as the square on FG is to the square on GH. X.6,Cor. Now since E is to BC as the square on A is to the square on FG, therefore the square on A is commensurable with the square on FG. X.6 But the square on A is rational, therefore the square on FG is also rational. Therefore FG is also rational. Since E does not have to BC the ratio which a square number has to a square number, therefore neither does the square on A have to the square on FG the ratio which a square number has to a square number, therefore A is incommensurable in length with FG. X.9 Again, since BC is to CD as the square on FG is to the square on GH, therefore the square on FG is commensurable with the square on GH. X.6 But the square on FG is rational, therefore the square on GH is also rational. Therefore GH is also rational. Since BC does not have to CD the ratio which a square number has to a square number, therefore neither does the square on FG have to the square on GH the ratio which a square number has to a square number. Therefore FG is incommensurable in length with GH X.9 And both are rational, therefore FG and GH are rational straight lines commensurable in square only. Therefore FH is an apotome. X.73 I say next that it is also a sixth apotome. Since E is to BC as the square on A is to the square on FG, and BC is to CD as the square on FG is to the square on GH, therefore, ex aequali, E is to CD as the square on A is to the square on GH. V.22 But E does not have to CD the ratio which a square number has to a square number, therefore neither does the square on A have to the square on GH the ratio which a square number has to a square number. Therefore A is incommensurable in length with GH. Therefore neither of the straight lines FG nor GH is commensurable in length with the rational straight line A. X.9 Now let the square on K be that by which the square on FG is greater than the square on GH. Since BC is to CD as the square on FG is to the square on GH, therefore, in conversion, CB is to BD as the square on FG is to the square on K. V.19,Cor. But CB does not have to BD the ratio which a square number has to a square number, therefore neither does the square on FG have to the square on K the ratio which a square number has to a square number, therefore FG is incommensurable in length with K. X.9 And the square on FG is greater than the square on GH by the square on K, therefore the square on FG is greater than the square on GH by the square on a straight line incommensurable in length with FG. And neither of the straight lines FG nor GH is commensurable with the rational straight line A set out. Therefore FH is a sixth apotome. X.Def.III.6 Therefore the sixth apotome FH has been found. Q.E.F. (Forthcoming) Book X Introduction Proposition X.89 Proposition X.91. © 1996 D.E.Joyce Clark University Proposition 91 If an area is contained by a rational straight line and a first apotome, then the side of the area is an apotome. Let the area AB be contained by the rational straight line AC and the first apotome AD. I say that the side of the area AB is an apotome. Since AD is a first apotome, let DG be its annex, therefore AG and GD are rational straight lines commensurable in square only. Also, the whole AG is commensurable with the rational straight line AC set out, and the square on AG is greater than the square on GD by the square on a straight line commensurable in length with AG. X.73 X.Def.III.2 Therefore if there is applied to AG a parallelogram equal to the fourth part of the square on DG and deficient by a square figure, then it divides it into commensurable parts. X.17 Bisect DG at E, apply to AG a parallelogram equal to the square on EG and deficient by a square figure, and let it be the rectangle AF by FG. Then AF is commensurable with FG. Draw EH, FI, and GK through the points E, F, and G parallel to AC. Now, since AF is commensurable in length with FG, therefore AG is also commensurable in length with each of the straight lines AF and FG. X.15 But AG is commensurable with AC, therefore each of the straight lines AF and FG is commensurable in length with AC. X.12 And AC is rational, therefore each of the straight lines AF and FG is also rational, so that each of the rectangles AI and FK is also rational. X.19 Now, since DE is commensurable in length with EG, therefore DG is also commensurable in length with each of the straight lines DE and EG. X.15 But DG is rational and incommensurable in length with AC, therefore each of the straight lines DE and EG is also rational and incommensurable in length with AC. Therefore each of the rectangles DH and EK is medial. X.13 X.21 Now make the square LM equal to AI, and subtract the square NO having a common angle with it, the angle LPM, and equal to FK. Then the squares LM and NO are about the same diameter. VI.26 Let PR be their diameter, and draw the figure. Since the rectangle AF by FG equals the square on EG, therefore AF is to EG as EG is to FG. VI.17 But AF is to EG as AI is to EK, and EG is to FG as EK is to KF, therefore EK is a mean proportional between AI and KF. VI.1 V.11 But it was proved before that MN is also a mean proportional between LM and NO, and AI equals the square LM, and KF equals NO, therefore MN also equals EK. X.54's Lemma But EK equals DH, and MN equals LO, therefore DK equals the gnomon UVW and NO. But AK also equals the sum of the squares LM and NO, therefore the remainder AB equals ST. But ST is the square on LN, therefore the square on LN equals AB. Therefore LN is the side of AB. I say next that LN is an apotome. Since each of the rectangles AI and FK is rational, and they equal LM and NO, therefore each of the squares LM and NO, that is, the squares on LP and PN respectively, is also rational. Therefore each of the straight lines LP and PN is also rational. Again, since DH is medial and equals LO, therefore LO is also medial. Since, then, LO is medial, while NO is rational, therefore LO is incommensurable with NO. But LO is to NO as LP is to PN, therefore LP is incommensurable in length with PN. VI.1 X.11 And both are rational, therefore LP and PN are rational straight lines commensurable in square only. Therefore LN is an apotome. X.73 And it is the side of the area AB, therefore the side of the area AB is an apotome. Therefore, if an area is contained by a rational straight line and a first apotome, then the side of the area is an apotome. Q.E.D. This proposition is used in X.108. Book X Introduction Proposition X.90 Proposition X.92. © 1996 D.E.Joyce Clark University Proposition 92 If an area is contained by a rational straight line and a second apotome, then the side of the area is a first apotome of a medial straight line. Let the area AB be contained by the rational straight line AC and the second apotome AD. I say that the side of the area AB is a first apotome of a medial straight line. Let DG be the annex to AD. Then AG and GD are rational straight lines commensurable in square only, and the annex DG is commensurable with the rational straight line AC set out, while the square on the whole AG is greater than the square on the annex GD by the square on a straight line commensurable in length with AG. X.73 X.Def.III.2 Since the square on AG is greater than the square on GD by the square on a straight line commensurable with AG, therefore, if there is applied to AG a parallelogram equal to the fourth part of the square on GD and deficient by a square figure, then it divides it into commensurable parts. X.17 Bisect, then, DG at E, apply to AG a parallelogram equal to the square on EG and deficient by a square figure, and let it be the rectangle AF by FG. Then AF is commensurable in length with FG. Therefore AG is also commensurable in length with each of the straight lines AF and FG. X.15 But AG is rational and incommensurable in length with AC, therefore each of the straight lines AF and FG is also rational and incommensurable in length with AC. Therefore each of the rectangle AI by FK is medial. X.13 X.21 Again, since DE is commensurable with EG, therefore DG is also commensurable with each of the straight lines DE and EG. X.15 But DG is commensurable in length with AC. Therefore each of the rectangles DH and EK is rational. X.19 Construct the square LM equal to AI, and subtract NO, equal to FK, about the same angle with LM, namely the angle LPM. Then the squares LM and NO are about the same diameter. VI.26 Let PR be their diameter, and draw the figure. Since AI and FK are medial and equal the squares on LP and PN, the squares on LP and PN are also medial, therefore LP and PN are also medial straight lines commensurable in square only. Since the rectangle AF by FG equals the square on EG, therefore AF is to EG as EG is to FG, while AF is to EG as AI is to EK, and EG is to FG as EK is to FK. Therefore EK is a mean proportional between AI and FK. VI.17 VI.1 V.11 But MN is also a mean proportional between the squares LM and NO, and AI equals LM while FK equals NO, therefore MN also equals EK. But DH equals EK, and LO equals MN, therefore the whole DK equals the gnomon UVW and NO. Since, then, the whole AK equals LM and NO, and, in these, DK equals the gnomon UVW and NO, therefore the remainder AB equals TS. But TS is the square on LN, therefore the square on LN equals the area AB. Therefore LN is the side of the area AB. I say that LN is a first apotome of a medial straight line. Since EK is rational and equals LO, therefore LO, that is, the rectangle LP by PN, is rational. But NO was proved medial, therefore LO is incommensurable with NO. But LO is to NO as LP is to PN, therefore LP and PN are incommensurable in length. VI.1 X.11 Therefore LP and PN are medial straight lines commensurable in square only, which contain a rational rectangle. Therefore LN is a first apotome of a medial straight line. X.74 And it is the side of the area AB. Therefore the side of the area AB is a first apotome of a medial straight line. Therefore, if an area is contained by a rational straight line and a second apotome, then the side of the area is a first apotome of a medial straight line. Q.E.D. This proposition is used in X.109. Book X Introduction Proposition X.91 Proposition X.93. © 1996 D.E.Joyce Clark University Proposition 93 If an area is contained by a rational straight line and a third apotome, then the side of the area is a second apotome of a medial straight line. Let the area AB be contained by the rational straight line AC and the third apotome AD. I say that the side of the area AB is a second apotome of a medial straight line. Let DG be the annex to AD. Then AG and GD are rational straight lines commensurable in square only, and neither of the straight lines AG and GD is commensurable in length with the rational straight line AC set out, while the square on the whole AG is greater than the square on the annex DG by the square on a straight line commensurable with AG. X.Def.III.3 Since, then, the square on AG is greater than the square on GD by the square on a straight line commensurable with AG, therefore, if there is applied to AG a parallelogram equal to the fourth part of the square on DG and deficient by a square figure, then it divides it into commensurable parts. X.17 Bisect DG at E, apply to AG a parallelogram equal to the square on EG and deficient by a square figure, and let it be the rectangle AF by FG. Draw EH, FI, and GK through the points E, F, and G parallel to AC. Then AF and FG are commensurable. Therefore AI is also commensurable with FK. VI.1 X.11 Since AF and FG are commensurable in length, therefore AG is also commensurable in length with each of the straight lines AF and FG. X.15 But AG is rational and incommensurable in length with AC, so that AF and FG are so also. X.13 Therefore each of the rectangles AI and FK is medial. X.21 Again, since DE is commensurable in length with EG, therefore DG is also commensurable in length with each of the straight lines DE and EG. X.15 But GD is rational and incommensurable in length with AC, therefore each of the straight lines DE and EG is also rational and incommensurable in length with AC. Therefore each of the rectangles DH and EK is medial. X.13 X.21 Since AG and GD are commensurable in square only, therefore AG is incommensurable in length with GD. But AG is commensurable in length with AF, and DG with EG, therefore AF is incommensurable in length with EG. X.13 But AF is to EG as AI is to EK, therefore AI is incommensurable with EK. VI.1 X.11 Now construct the square LM equal to AI, and subtract NO, equal to FK, about the same angle with LM. Then LM and NO are about the same diameter. VI.26 Let PR be their diameter, and draw the figure. Now, since the rectangle AF by FG equals the square on EG, therefore AF is to EG as EG is to FG. VI.17 But AF is to EG as AI is to EK, and EG is to FG as EK is to FK, therefore AI is to EK as EK is to FK. Therefore EK is a mean proportional between AI and FK. VI.1 VI.11 But MN is also a mean proportional between the squares LM and NO, and AI equals LM, and FK equals NO, therefore EK also equals MN. But MN equals LO, and EK equals DH, therefore the whole DK also equals the gnomon UVW and NO. But AK equals the sum of LM and NO, therefore the remainder AB equals ST, that is, to the square on LN. Therefore LN is the side of the area AB. I say that LN is a second apotome of a medial straight line. Since AI and FK were proved medial, and equal the squares on LP, therefore each of the squares on LP and PN is also medial. Therefore each of the straight lines LP and PN is medial. Since AI is commensurable with FK, therefore the square on LP is also commensurable with the square on PN. VI.1 X.11 Again, since AI was proved incommensurable with EK, therefore LM is also incommensurable with MN, that is, the square on LP with the rectangle LP by PN, so that LP is also incommensurable in length with PN. VI.1 X.11 Therefore LP and PN are medial straight lines commensurable in square only. I say next that they also contain a medial rectangle. Since EK was proved medial, and equals the rectangle LP by PN, therefore the rectangle LP by PN is also medial, so that LP and PN are medial straight lines commensurable in square only which contain a medial rectangle. Therefore LN is a second apotome of a medial straight line, and it is the side of the area AB. X.75 Therefore the side of the area AB is a second apotome of a medial straight line. Therefore, if an area is contained by a rational straight line and a third apotome, then the side of the area is a second apotome of a medial straight line. Q.E.D. This proposition is used in X.110. Book X Introduction Proposition X.92 Proposition X.94. © 1996 D.E.Joyce Clark University Proposition 94 If an area is contained by a rational straight line and a fourth apotome, then the side of the area is minor. Let the area be contained by the rational straight line AC and the fourth apotome AD. I say that the side of the area AB is minor. Let DG be the annex to AD, therefore AG and GD are rational straight lines commensurable in square only, AG is commensurable in length with the rational straight line AC set out, and the square on the whole AG is greater than the square on the annex DG by the square on a straight line incommensurable in length with AG. X.Def.III.4 Since the square on AG is greater than the square on GD by the square on a straight line incommensurable in length with AG, therefore, if there is applied to AG a parallelogram equal to the fourth part of the square on DG and deficient by a square figure, then it divides it into incommensurable parts. X.18 Bisect DG at E, apply to AG a parallelogram equal to the square on EG and deficient by a square figure, and let it be the rectangle AF by FG. Then AF is incommensurable in length with FG. Draw EH, FI, and GK through E, F, and G parallel to AC and BD. Since AG is rational and commensurable in length with AC, therefore the whole AK is rational. X.19 Again, since DG is incommensurable in length with AC, and both are rational, therefore DK is medial. X.21 Again, since AF is incommensurable in length with FG, therefore AI is incommensurable with FK. VI.1 X.11 Now construct the square LM equal to AI, and subtract NO, equal to FK, about the same angle, the angle LPM. Therefore the squares LM and NO are about the same diameter. Let PR be their diameter, and draw the figure. VI.26 Since the rectangle AF by FG equals the square on EG, therefore, AF is to EG as EG is to FG. VI.17 But AF is to EG as AI is to EK, and EG is to FG as EK is to FK, therefore EK is a mean proportional between AI and FK. VI.1 V.11 But MN is also a mean proportional between the squares LM and NO, and AI equals LM, and FK equals NO therefore EK also equals MN. But DH equals EK, and LO equals MN, therefore the whole DK equals the gnomon UVW and NO. Since, then, the whole AK equals the sum of the squares LM and NO, and, in these, DK equals the gnomon UVW and the square NO, therefore the remainder AB equals ST, that is, to the square on LN. Therefore LN is the side of the area AB. I say that LN is the irrational straight line called minor. Since AK is rational and equals the sum of the squares on LP and PN, therefore the sum of the squares on LP and PN is rational. Again, since DK is medial, and DK equals twice the rectangle LP by PN, therefore twice the rectangle LP by PN is medial. And, since AI was proved incommensurable with FK, therefore the square on LP is also incommensurable with the square on PN. Therefore LP and PN are straight lines incommensurable in square which make the sum of the squares on them rational, but twice the rectangle contained by them medial. Therefore LN is the irrational straight line called minor, and it is the side of the area AB. X.76 Therefore the side of the area AB is minor. Therefore, if an area is contained by a rational straight line and a fourth apotome, then the side of the area is minor. Q.E.D. This proposition is used in X.108. It is also used in proposition XIII.11. Book X Introduction Proposition X.93 Proposition X.95. © 1996 D.E.Joyce Clark University Proposition 95 If an area is contained by a rational straight line and a fifth apotome, then the side of the area is a straight line which produces with a rational area a medial whole. Let the area AB be contained by the rational straight line AC and the fifth apotome AD. I say that the side of the area AB is a straight line which produces with a rational area a medial whole. Let DG be the annex to AD. Then AG and GD are rational straight lines commensurable in square only, the annex GD is commensurable in length with the rational straight line AC set out, and the square on the whole AG is greater than the square on the annex DG by the square on a straight line incommensurable with AG. X.Def.III.5 Therefore, if there is applied to AG a parallelogram equal to the fourth part of the square on DG and deficient by a square figure, then it divides it into in commensurable parts. X.18 Bisect DG at the point E, apply to AG a parallelogram equal to the square on EG and deficient by a square figure, and let it be the rectangle AF by FG. Then AF is incommensurable in length with FG. Now, since AG is incommensurable in length with CA, and both are rational, therefore AK is medial. X.21 Again, since DG is rational and commensurable in length with AC, therefore DK is rational. X.19 Now construct the square LM equal to AI, and subtract the square NO, equal to FK and about the same angle, the angle LPM. Then the squares LM and NO are about the same diameter. Let PR be their diameter, and draw the figure. VI.26 Similarly then we can prove that LN is the side of the area AB. I say that LN is the straight line which produces with a rational area a medial whole. Since AK was proved medial and equals the sum of the squares on LP and PN, therefore the sum of the squares on LP and PN is medial. Again, since DK is rational and equals twice the rectangle LP by PN, therefore the latter is itself also rational. And, since AI is incommensurable with FK, therefore the square on LP is also incommensurable with the square on PN. Therefore LP and PN are straight lines incommensurable in square which make the sum of the squares on them medial but twice the rectangle contained by them rational. Therefore the remainder LN is the irrational straight line called that which produces with a rational area a medial whole, and it is the side of the area AB. X.77 Therefore the side of the area AB is a straight line which produces with a rational area a medial whole. Therefore, if an area is contained by a rational straight line and a fifth apotome, then the side of the area is a straight line which produces with a rational area a medial whole. Q.E.D. This proposition is used in X.109. Book X Introduction Proposition X.94 Proposition X.96. © 1996 D.E.Joyce Clark University Proposition 96 If an area is contained by a rational straight line and a sixth apotome, then the side of the area is a straight line which produces with a medial area a medial whole. Let the area AB be contained by the rational straight line AC and the sixth apotome AD. I say that the side of the area AB is a straight line which produces with a medial area a medial whole. Let DG be the annex to AD. Then AG and GD are rational straight lines commensurable in square only, neither of them is commensurable in length with the rational straight line AC set out, and the square on the whole AG is greater than the square on the annex DG by the square on a straight line incommensurable in length with AG. X.Def.III.6 Since the square on AG is greater than the square on GD by the square on a straight line incommensurable in length with AG, therefore, if there is applied to AG a parallelogram equal to the fourth part of the square on DG and deficient by a square figure, then it divides it into incommensurable parts. X.18 Bisect DG at E, apply to AG a parallelogram equal to the square on EG and deficient by a square figure, and let it be the rectangle AF by FG. Then AF is incommensurable in length with FG. But AF is to FG as AI is to FK, therefore AI is incommensurable with FK. X.11 Since AG and AC are rational straight lines commensurable in square only, therefore AK is medial. Again, since AC and DG are rational straight lines and incommensurable in length, DK is also medial. X.21 Now, since AG and GD are commensurable in square only, therefore AG is incommensurable in length with GD. But AG is to GD as AK is to KD, therefore AK is incommensurable with KD. VI.1 X.11 Now construct the square LM equal to AI, and subtract NO, equal to FK, about the same angle. Then the squares LM and NO are about the same diameter. VI.26 Let PR be their diameter, and draw the figure. Then in manner similar to the above we can prove that LN is the side of the area AB. I say that LN is a straight line which produces with a medial area a medial whole. Since AK was proved medial and equals the sum of the squares on LP and PN, therefore the sum of the squares on LP and PN is medial. Again, since DK was proved medial and equals twice the rectangle LP by PN, therefore twice the rectangle LP by PN is also medial. Since AK was proved incommensurable with DK, therefore the sum of the squares on LP and PN is also incommensurable with twice the rectangle LP by PN. And, since AI is incommensurable with FK, therefore the square on LP is also incommensurable with the square on PN. Therefore LP and PN are straight lines incommensurable in square which make the sum of the squares on them medial, twice the rectangle contained by them medial, and further, the sum of the squares on them incommensurable with twice the rectangle contained by them. Therefore LN is the irrational straight line called that which produces with a medial area a medial whole, and it is the side of the area AB. Therefore the side of the area is a straight line which produces with a medial area a medial whole. X.78 Therefore, if an area is contained by a rational straight line and a sixth apotome, then the side of the area is a straight line which produces with a medial area a medial whole. Q.E.D. This proposition is used in X.110. Book X Introduction Proposition X.95 Proposition X.97. © 1996 D.E.Joyce Clark University Proposition 97 The square on an apotome of a medial straight line applied to a rational straight line produces as breadth a first apotome. Let AB be an apotome, and CD rational, and to CD let there be applied CE equal to the square on AB and producing CF as breadth. I say that CF is a first apotome. Let BG be the annex to AB. Then AG and GB are rational straight lines commensurable in square only. X.73 To CD apply CH, equal to the square on AG, and KL, equal to the square on BG. Then the whole CL equals the sum of the squares on AG and GB, and, in these, CE equals the square on AB, therefore the remainder FL equals twice the rectangle AG by GB. II.7 Bisect FM at the point N, and draw NO through N parallel to CD. Then each of the rectangles FO and LN equals the rectangle AG by GB. Now, since the sum of the squares on AG and GB is rational, and DM equals the sum of the squares on AG and GB, therefore DM is rational. And it is applied to the rational straight line CD producing CM as breadth, therefore CM is rational and commensurable in length with CD. X.20 Again, since twice the rectangle AG by GB is medial, and FL equals twice the rectangle AG by GB, therefore FL is medial. And it is applied to the rational straight line CD producing FM as breadth, therefore FM is rational and incommensurable in length with CD. X.22 Since the squares on AG and GB are rational, while twice the rectangle AG by GB is medial, therefore the sum of the squares on AG and GB is incommensurable with twice the rectangle AG by GB. And CL equals the sum of the squares on AG and GB, and FL equals twice the rectangle AG by GB, therefore DM is incommensurable with FL. But DM is to FL as CM is to FM, therefore CM is incommensurable in length with FM. VI.1 X.11 And both are rational, therefore CM and MF are rational straight lines commensurable in square only. Therefore CF is an apotome. X.73 I say next that it is also a first apotome. Since the rectangle AG by GB is a mean proportional between the squares on AG and GB, CH equals the square on AG, KL equals the square on BG, and NL equals the rectangle AG by GB, therefore NL is also a mean proportional between CH and KL. Therefore CH is to NL as NL is to KL. But CH is to NL as CK is to NM, and NL is to KL as NM is to KM, therefore the rectangle CK by KM equals the square on NM, that is, the fourth part of the square on FM. VI.1 VI.17 Since the square on AG is commensurable with the square on GB, therefore CH is also commensurable with KL. But CH is to KL as CK is to KM, therefore CK is commensurable with KM. VI.1 X.11 Since CM and MF are two unequal straight lines, and to CM there has been applied the rectangle CK by KM equal to the fourth part of the square on FM and deficient by a square figure, while CK is commensurable with KM, therefore the square on CM is greater than the square on MF by the square on a straight line commensurable in length with CM. X.17 And CM is commensurable in length with the rational straight line CD set out, therefore CF is a first apotome. X.Def.III.2 Therefore, the square on an apotome of a medial straight line applied to a rational straight line produces as breadth a first apotome. Q.E.D. This proposition is used in X.111. It is also used in proposition XIII.6. Book X Introduction Proposition X.96 Proposition X.98. © 1996 D.E.Joyce Clark University Proposition 98 The square on a first apotome of a medial straight line applied to a rational straight line produces as breadth a second apotome. Let AB be a first apotome of a medial straight line and CD a rational straight line, and to CD let there be applied CE equal to the square on AB producing CF as breadth. I say that CF is a second apotome. Let BG be the annex to AB. Then AG and GB are medial straight lines commensurable in square only which contain a rational rectangle. X.74 To CD apply CH, equal to the square on AG, producing CK as breadth, and KL, equal to the square on GB, producing KM as breadth. Therefore the whole CL equals the sum of the squares on AG. Therefore CL is also medial. X.15 X.23,Cor. And it is applied to the rational straight line CD producing CM as breadth, therefore CM is rational and incommensurable in length with CD. X.22 Now, since CL equals the sum of the squares on AG and GB, and, in these, the square on AB equals CE, therefore the remainder, twice the rectangle AG by GB, equals FL. II.7 But twice the rectangle AG by GB is rational, therefore FL is rational. And it is applied to the rational straight line FE producing FM as breadth, therefore FM is also rational and commensurable in length with CD. X.20 Now, since the sum of the squares on AG and GB, that is, CL, is medial, while twice the rectangle AG by GB, that is, FL, is rational, therefore CL is incommensurable with FL. But CL is to FL as CM is to FM, therefore CM is incommensurable in length with FM. VI.1 X.11 And both are rational, therefore CM and MF are rational straight lines commensurable in square only. Therefore CF is an apotome. X.73 I say next that it is also a second apotome. Bisect FM at N, and draw NO through N parallel to CD. Then each of the rectangles FO and NL equals the rectangle AG by GB. Now, since the rectangle AG by GB is a mean proportional between the squares on AG and GB, the square on AG equals CH, the rectangle AG by GB equals NL, and the square on BG equals KL, therefore NL is also a mean proportional between CH and KL. Therefore CH is to NL as NL is to KL. But CH is to NL as CK is to NM, and NL is to KL as NM is to MK, therefore CK is to NM as NM is to KM. Therefore the rectangle CK by KM equals the square on NM, that is, the fourth part of the square on FM. VI.1 V.11 VI.17 Since CM and MF are two unequal straight lines, and the rectangle CK by KM, equal to the fourth part of the square on MF and deficient by a square figure, has been applied to the greater, CM, and divides it into commensurable parts, therefore the square on CM is greater than the square on MF by the square on a straight line commensurable in length with CM. X.17 And the annex FM is commensurable in length with the rational straight line CD set out, therefore CF is a second apotome. X.Def.III.2 Therefore, the square on a first apotome of a medial straight line applied to a rational straight line produces as breadth a second apotome. Q.E.D. This proposition is used in X.111. Book X Introduction Proposition X.97 Proposition X.99. © 1996 D.E.Joyce Clark University Proposition 99 The square on a second apotome of a medial straight line applied to a rational straight line produces as breadth a third apotome. Let AB be a second apotome of a medial straight line, and CD rational, and to CD let there be applied CE equal to the square on AB producing CF as breadth. I say that CF is a third apotome. Let BG be the annex to AB, therefore AG and GB are medial straight lines commensurable in square only which contains a medial rectangle. X.75 Apply CH, equal to the square on AG, to CD producing CK as breadth, and apply KL, equal to the square on BG, to KH producing KM as breadth. Then the whole CL equals the sum of the squares on AG and GB. Therefore CL is also medial. X.15 X.23,Cor. And it is applied to the rational straight line CD producing CM as breadth, therefore CM is rational and incommensurable in length with CD. X.22 Now, since the whole CL equals the sum of the squares on AG and GB, and, in these, CE equals the square on AB, therefore the remainder LF equals twice the rectangle AG by GB. II.7 Bisect FM at the point N, and draw NO parallel to CD. Then each of the rectangles FO and NL equals the rectangle AG by GB. But the rectangle AG by GB is medial, therefore FL is also medial. And it is applied to the rational straight line EF producing FM as breadth, therefore FM is also rational and incommensurable in length with CD. X.22 Since AG and GB are commensurable in square only, therefore AG is incommensurable in length with GB. Therefore the square on AG is also incommensurable with the rectangle AG by GB. VI.1 X.11 But the sum of the squares on AG and GB is commensurable with the square on AG, and twice the rectangle AG by GB with the rectangle AG by GB, therefore the sum of the squares on AG and GB is incommensurable with twice the rectangle AG by GB. X.13 But CL equals the sum of the squares on AG and GB, and FL equals twice the rectangle AG by GB, therefore CL is also incommensurable with FL. But CL is to FL as CM is to FM, therefore CM is incommensurable in length with FM. VI.1 And both are rational, therefore CM and MF are rational straight lines commensurable in square only, therefore CF is an apotome. X.73 I say next that it is also a third apotome. Since the square on AG is commensurable with the square on GB, therefore CH is also commensurable with KL, so that CK is also commensurable with KM. VI.1 X.11 Since the rectangle AG by GB is a mean proportional between the squares on AG and GB, CH equals the square on AG, KL equals the square on GB, and NL equals the rectangle AG by GB, therefore NL is also a mean proportional between CH and KL. Therefore CH is to NL as NL is to KL. But CH is to NL as CK is to NM, and NL is to KL as NM is to KM, therefore CK is to MN as MN is to KM. Therefore the rectangle CK by KM equals the square on MN, that is, to the fourth part of the square on FM. VI.1 V.11 Since, then, CM and MF are two unequal straight lines, and a parallelogram equal to the fourth part of the square on FM and deficient by a square figure has been applied to CM, and divides it into commensurable parts, therefore the square on CM is greater than the square on MF by the square on a straight line commensurable with CM. X.17 And neither of the straight lines CM nor MF is commensurable in length with the rational straight line CD set out, therefore CF is a third apotome. X.Def.III.3 Therefore, the square on a second apotome of a medial straight line applied to a rational straight line produces as breadth a third apotome. Q.E.D. This proposition is used in X.111. Book X Introduction Proposition X.98 Proposition X.100. © 1996 D.E.Joyce Clark University Proposition 100 The square on a minor straight line applied to a rational straight line produces as breadth a fourth apotome. Let AB be a minor and CD a rational straight line, and to the rational straight line CD let CE be applied equal to the square on AB and producing CF as breadth. I say that CF is a fourth apotome. Let BG be the annex to AB. Then AG and GB are straight lines incommensurable in square which make the sum of the squares on AG and GB rational, but twice the rectangle AG by GB medial. X.76 To CD apply CH, equal to the square on AG, producing CK as breadth, and KL, equal to the square on BG, producing KM as breadth. Then the whole CL equals the sum of the squares on AG and GB. And the sum of the squares on AG and GB is rational, therefore CL is also rational. And it is applied to the rational straight line CD producing CM as breadth, therefore CM is also rational and commensurable in length with CD. X.20 Since the whole CL equals the sum of the squares on AG and GB, and, in these, CE equals the square on AB, therefore the remainder FL equals twice the rectangle AG by GB. II.7 Bisect FM at the point N, and draw NO through N parallel to either of the straight lines CD or ML. Then each of the rectangles FO and NL equals the rectangle AG by GB. And, since twice the rectangle AG by GB is medial and equals FL, therefore FL is also medial. And it is applied to the rational straight line FE producing FM as breadth, therefore FM is rational and incommensurable in length with CD. X.22 Since the sum of the squares on AG and GB is rational, while twice the rectangle AG by GB is medial, therefore the sum of the squares on AG and GB is incommensurable with twice the rectangle AG by GB. But CL equals the sum of the squares on AG and GB, and FL equals twice the rectangle AG by GB, therefore CL is incommensurable with FL. But CL is to FL as CM is to MF, therefore CM is incommensurable in length with MF. VI.1 X.11 And both are rational, therefore CM and MF are rational straight lines commensurable in square only. Therefore CF is an apotome. X.73 I say that it is also a fourth apotome. Since AG and GB are incommensurable in square, therefore the square on AG is also incommensurable with the square on GB. And CH equals the square on AG, and KL equal to the square on GB, therefore CH is incommensurable with KL. But CH is to KL as CK is to KM, therefore CK is incommensurable in length with KM. VI.1 X.11 Since the rectangle AG by GB is a mean proportional between the squares on AG and GB the square on AG equals CH, the square on GB equals KL, and the rectangle AG by GB equals NL, therefore NL is a mean proportional between CH and KL. Therefore CH is to NL as NL is to KL. But CH is to NL as CK is to NM, and NL is to KL as NM is to KM, therefore CK is to MN as MN is to KM. VI.1 V.11 Therefore the rectangle CK by KM equals the square on MN, that is, to the fourth part of the square on FM. VI.17 Since CM and MF are two unequal straight lines, and the rectangle CK by KM, equal to the fourth part of the square on MF and deficient by a square figure, has been applied to CM and divides it into incommensurable parts, therefore the square on CM is greater than the square on MF by the square on a straight line incommensurable with CM. X.18 And the whole CM is commensurable in length with the rational straight line CD set out, therefore CF is a fourth apotome. X.Def.III.4 Therefore, the square on a minor straight line applied to a rational straight line produces as breadth a fourth apotome. Q.E.D. This proposition is used in X.111. Book X Introduction Proposition X.99 Proposition X.101. © 1996 D.E.Joyce Clark University Proposition 101 The square on the straight line which produces with a rational area a medial whole, if applied to a rational straight line, produces as breadth a fifth apotome. Let AB be the straight line which produces with a rational area a medial whole, and CD a rational straight line, and to CD let CE be applied equal to the square on AB and producing CF as breadth. I say that CF is a fifth apotome. Let BG be the annex to AB. Then AG and GB are straight lines incommensurable in square which make the sum of the squares on them medial but twice the rectangle contained by them rational. X.77 To CD apply CH equal to the square on AG, and KL equal to the square on GB. Then the whole CL equals the sum of the squares on AG and GB. But the sum of the squares on AG and GB together is medial, therefore CL is medial. And it is applied to the rational straight line CD producing CM as breadth, therefore CM is rational and incommensurable with CD. X.22 Since the whole CL equals the sum of the squares on AG and GB, and, in these, CE equals the square on AB, therefore the remainder FL equals twice the rectangle AG by GB. II.7 Bisect FM at N, and draw NO through N parallel to either of the straight lines CD or ML. Then each of the rectangles FO and NL equals the rectangle AG by GB. And, since twice the rectangle AG by GB is rational and equal to FL, therefore FL is rational. And it is applied to the rational straight line EF producing FM as breadth, therefore FM is rational and commensurable in length with CD. X.20 Now, since CL is medial, and FL rational, therefore CL is incommensurable with FL. But CL is to FL as CM is to MF, therefore CM is incommensurable in length with MF. VI.1 X.11 And both are rational, therefore CM and MF are rational straight lines commensurable in square only. Therefore CF is an apotome. X.73 I say next that it is also a fifth apotome. We can prove similarly that the rectangle CK by KM equals the square on NM, that is, the fourth part of the square on FM. And, since the square on AG is incommensurable with the square on GB, while the square on AG equals CH, and the square on GB equals KL, therefore CH is incommensurable with KL. But CH is to KL as CK is to KM, therefore CK is incommensurable in length with KM. VI.1 X.11 Since CM and MF are two unequal straight lines, and a parallelogram equal to the fourth part of the square on FM and deficient by a square figure has been applied to CM, and divides it into incommensurable parts, therefore the square on CM is greater than the square on MF by the square on a straight line incommensurable with CM. X.18 And the annex FM is commensurable with the rational straight line CD set out, therefore CF is a fifth apotome. X.Def.III.5 Therefore, the square on the straight line which produces with a rational area a medial whole, if applied to a rational straight line, produces as breadth a fifth apotome. Q.E.D. This proposition is used in X.111. Book X Introduction Proposition X.100 Proposition X.102. © 1996 D.E.Joyce Clark University Proposition 102 The square on the straight line which produces with a medial area a medial whole, if applied to a rational straight line, produces as breadth a sixth apotome. Let AB be the straight line which produces with a medial area a medial whole, and CD a rational straight line, and to CD let CE be applied equal to the square on AB and producing CF as breadth. I say that CF is a sixth apotome. Let BG be the annex to AB. Then AG and GB are straight lines incommensurable in square which make the sum of the squares on them medial, twice the rectangle AG by GB medial, and the sum of the squares on AG and GB incommensurable with twice the rectangle AG by GB. X.78 Now to CD apply CH equal to the square on AG and producing CK as breadth, and KL equal to the square on BG. Then the whole CL equals the sum of the squares on AG and GB. Therefore CL is also medial. And it is applied to the rational straight line CD producing CM as breadth, therefore CM is rational and incommensurable in length with CD. X.22 Since CL equals the sum of the squares on AG and GB, and, in these, CE equals the square on AB, therefore the remainder FL equals twice the rectangle AG by GB. And twice the rectangle AG by GB is medial, therefore FL is also medial. II.7 And it is applied to the rational straight line FE producing FM as breadth, therefore FM is rational and incommensurable in length with CD. X.22 Since the sum of the squares on AG and GB is incommensurable with twice the rectangle AG by GB, CL equals the sum of the squares on AG and GB, and FL equals twice the rectangle AG by GB, therefore CL is incommensurable with FL. But CL is to FL as CM is to MF, therefore CM is incommensurable in length with MF. And both are rational. VI.1 X.11 Therefore CM and MF are rational straight lines commensurable in square only, therefore CF is an apotome. X.73 I say next that it is also a sixth apotome. Since FL equals twice the rectangle AG by GB, bisect FM at N, and draw NO through N parallel to CD, therefore each of the rectangles FO and NL equals the rectangle AG by GB. And, since AG and GB are incommensurable in square, therefore the square on AG is incommensurable with the square on GB. But CH equals the square on AG, and KL equals the square on GB, therefore CH is incommensurable with KL. But CH is to KL as CK is to KM, therefore CK is incommensurable with KM. VI.1 X.11 Since the rectangle AG by GB is a mean proportional between the squares on AG and GB, CH equals the square on AG, KL equals the square on GB, and NL equals the rectangle AG by GB, therefore NL is also a mean proportional between CH and KL. Therefore CH is to NL as NL is to KL. And for the same reason as before the square on CM is greater than the square on MF by the square on a straight line incommensurable with CM. X.18 And neither of them is commensurable with the rational straight line CD set out, therefore CF is a sixth apotome. X.Def.III.6 Therefore, the square on the straight line which produces with a medial area a medial whole, if applied to a rational straight line, produces as breadth a sixth apotome. Q.E.D. This proposition is used in X.111. Book X Introduction Proposition X.101 Proposition X.103. © 1996 D.E.Joyce Clark University Proposition 103 A straight line commensurable in length with an apotome is an apotome and the same in order. Let AB be an apotome, B and let CD be commensurable in length with AB. I say that CD is also an apotome and the same in order with AB. Since AB is an apotome, let BE be the annex to it, therefore AE and EB are rational straight lines commensurable in square only. X.73 Let it be contrived that the ratio of BE to DF is the same as the ratio of AB to CD. Then one is to one as are all to all. Therefore the whole AE is to the whole CF as AB is to CD. VI.12 V.12 But AB is commensurable in length with CD, therefore AE is also commensurable with CF, and BE with DF. X.11 And AE and EB are rational straight lines commensurable in square only, therefore CF and FD are also rational straight lines commensurable in square only. X.13 Now since AE is to CF as BE is to DF, therefore, alternately, AE is to EB as CF is to FD. And the square on AE is greater than the square on EB either by the square on a straight line commensurable with AE or by the square on a straight line incommensurable with it. V.16 If then the square on AE is greater than the square on EB by the square on a straight line commensurable with AE, then the square on CF is also greater than the square on FD by the square on a straight line commensurable with CF. X.14 And, if AE is commensurable in length with the rational straight line set out, then CF is also; if BE, then DF also; and, if neither of the straight lines AE nor EB, then neither of the straight lines CF nor FD. X.12 X.13 But, if the square on AE is greater than the square on EB by the square on a straight line incommensurable with AE, then the square on CF is also greater than the square on FD by the square on a straight line incommensurable with CF. X.14 And, if AE is commensurable in length with the rational straight line set out, then CF is also; if BE, then DF also; and, if neither of the straight lines AE nor EB, then neither of the straight lines CF nor FD. Therefore CD is an apotome and the same in order with AB. X.12 X.13 Therefore, a straight line commensurable in length with an apotome is an apotome and the same in order. Q.E.D. (Forthcoming) Book X Introduction Proposition X.102 Proposition X.104. © 1996 D.E.Joyce Clark University Proposition 104 A straight line commensurable with an apotome of a medial straight line is an apotome of a medial straight line and the same in order. Let AB be an apotome of a medial straight line, and let CD be commensurable in length with AB. I say that CD is also an apotome of a medial straight line and the same in order with AB. Since AB is an apotome of a medial straight line, let EB be the annex to it. Then AE and EB are medial straight lines commensurable in square only. X.74 X.75 Let it be contrived that AB is to CD as BE is to DF. Then AE is also commensurable with CF, and BE with DF. VI.12 V.12 X.11 But AE and EB are medial straight lines commensurable in square only, therefore CF and FD are also medial straight lines commensurable in square only. X.23 X.13 Therefore CD is an apotome of a medial straight line. X.74 X.75 I say next that it is also the same in order with AB. Since AE is to EB as CF is to FD, therefore the square on AE is to the rectangle AE by EB as the square on CF is to the rectangle CF by FD. But the square on AE is commensurable with the square on CF, therefore the rectangle AE by EB is also commensurable with the rectangle CF by FD. V.16 X.11 Therefore, if the rectangle AE by EB is rational, then the rectangle CF by FD is also rational, and if the rectangle AE by EB is medial, the rectangle CF by FD is also medial. X.Def.4 X.23,Cor. Therefore CD is an apotome of a medial straight line and the same in order with AB. X.74 X.75 Therefore, a straight line commensurable with an apotome of a medial straight line is an apotome of a medial straight line and the same in order. Q.E.D. (Forthcoming) Book X Introduction Proposition X.103 Proposition X.105. © 1996 D.E.Joyce Clark University Proposition 105 A straight line commensurable with a minor straight line is minor. Let AB be a minor straight line, and CD commensurable with AB. I say that CD is also minor. Make the same construction as before. Then, since AE and EB are incommensurable in square, therefore CF and FD are also incommensurable in square. X.76 X.13 Now since AE is to EB as CF is to FD, therefore the square on AE is to the square on EB as the square on CF is to the square on FD. V.12 V.16 VI.22 Therefore, taken jointly, the sum of the squares on AE and EB is to the square on EB as the sum of the squares on CF and FD is to the square on FD. V.18 But the square on BE is commensurable with the square on DF, therefore the sum of the squares on AE and EB is also commensurable with the sum of the squares on CF and FD. V.16 X.11 But the sum of the squares on AE and EB is rational, therefore the sum of the squares on CF and FD is also rational. X.76 X.Def.4 Again, since the square on AE is to the rectangle AE by EB as the square on CF is to the rectangle CF by FD, while the square on AE is commensurable with the square on CF, therefore the rectangle AE by EB is also commensurable with the rectangle CF by FD. But the rectangle AE by EB is medial, therefore the rectangle CF by FD is also medial. X.76 X23,Cor. Therefore CF and FD are straight lines incommensurable in square which make the sum of the squares on them rational, but the rectangle contained by them medial. Therefore CD is minor. X.76 Therefore, a straight line commensurable with a minor straight line is minor. Q.E.D. (Forthcoming) Book X Introduction Proposition X.104 Proposition X.106. © 1996 D.E.Joyce Clark University Proposition 106 A straight line commensurable with that which produces with a rational area a medial whole is a straight line which produces with a rational area a medial whole. Let AB be a straight line which produces with a rational area a medial whole, and CD commensurable with AB. I say that CD is also a straight line which produces with a rational area a medial whole. Let BE be the annex to AB, therefore AE and EB are straight lines incommensurable in square which make the sum of the squares on AE and EB medial but the rectangle contained by them rational. X.77 Make the same construction. Then we can prove, in manner similar to the foregoing, that CF and FD are in the same ratio as AE and EB, the sum of the squares on AE and EB is commensurable with the sum of the squares on CF and FD, and the rectangle AE by EB is commensurable with the rectangle CF by FD, so that CF and FD are also straight lines incommensurable in square which make the sum of the squares on CF and FD medial but the rectangle contained by them rational. Therefore CD is a straight line which produces with a rational area a medial whole. X.77 Therefore, a straight line commensurable with that which produces with a rational area a medial whole is a straight line which produces with a rational area a medial whole. Q.E.D. (Forthcoming) Book X Introduction Proposition X.105 Proposition X.107. © 1996 D.E.Joyce Clark University Proposition 107 A straight line commensurable with that which produces a medial area and a medial whole is itself also a straight line which produces with a medial area a medial whole. Let AB be a straight line which produces with a medial area a medial whole, and let CD be commensurable with AB. I say that CD is also a straight line which produces with a medial area a medial whole. Let BE be the annex to AB, and make the same construction. Then AE and EB are straight lines incommensurable in square which make the sum of the squares on them medial, the rectangle contained by them medial, and further, the sum of the squares on them incommensurable with the rectangle contained by them. X.78 Now as was proved, AE and EB are commensurable with CF and FD, the sum of the squares on AE and EB with the sum of the squares on CF and FD, and the rectangle AE by EB with the rectangle CF by FD, therefore CF and FD are straight lines incommensurable in square which make the sum of the squares on them medial, the rectangle contained by them medial, and further, the sum of the squares on them incommensurable with the rectangle contained by them. Therefore CD is a straight line which produces with a medial area a medial whole. X.78 Therefore, a straight line commensurable with that which produces a medial area and a medial whole is itself also a straight line which produces with a medial area a medial whole. Q.E.D. (Forthcoming) Book X Introduction Proposition X.106 Proposition X.108. © 1996 D.E.Joyce Clark University Proposition 108 If a medial area is subtracted from a rational area, then the side of the remaining area becomes one of two irrational straight lines, either an apotome or a minor straight line. Let the medial area BD be subtracted from the rational area BC. I say that the side of the remainder EC becomes one of two irrational straight lines, either an apotome or a minor straight line. Set out a rational straight line FG, to FG apply the rectangular parallelogram GH equal to BC, and subtract GK equal to DB. Then the remainder EC equals LH. Since, then, BC is rational, and BD medial, while BC equals GH, and BD equals GK, therefore GH is rational, and GK is medial. And they are applied to the rational straight line FG, therefore FH is rational and commensurable in length with FG, while FK is rational and incommensurable in length with FG. Therefore FH is incommensurable in length with FK. X.20 X.22 X.13 Therefore FH and FK are rational straight lines commensurable in square only. Therefore KH is an apotome, and KF the annex to it. X.73 Now the square on HF is greater than the square on FK by the square on a straight line either commensurable with HF or not commensurable. First, let the square on it be greater by the square on a straight line commensurable with it. Now the whole HF is commensurable in length with the rational straight line FG set out, therefore KH is a first apotome. X.Def.III.2 But the side of the rectangle contained by a rational straight line and a first apotome is an apotome. Therefore the side of LH, that is, of EC, is an apotome. X.91 But, if the square on HF is greater than the square on FK by the square on a straight line incommensurable with HF, while the whole FH is commensurable in length with the rational straight line FG set out, then KH is a fourth apotome. X.Def.III.4 But the side of the rectangle contained by a rational straight line and a fourth apotome is minor. X.94 Therefore, if a medial area is subtracted from a rational area, then the side of the remaining area becomes one of two irrational straight lines, either an apotome or a minor straight line. Q.E.D. (Forthcoming) Book X Introduction Proposition X.107 Proposition X.109. © 1996 D.E.Joyce Clark University Proposition 109 If a rational area is subtracted from a medial area, then there arise two other irrational straight lines, either a first apotome of a medial straight line or a straight line which produces with a rational area a medial whole. Let the rational area BD be subtracted from the medial area BC. I say that the side of the remainder EC becomes one of two irrational straight lines, either a first apotome of a medial straight line or a straight line which produces with a rational area a medial whole. Set out a rational straight line FG, and apply the areas similarly. Then FH is rational and incommensurable in length with FG, while KF is rational and commensurable in length with FG, therefore FH and FK are rational straight lines commensurable in square only. X.13 Therefore KH is an apotome, and FK the annex to it. X.73 Now the square on HF is greater than the square on FK either by the square on a straight line commensurable with HF or by the square on a straight line incommensurable with it. If the square on HF is greater than the square on FK by the square on a straight line commensurable with HF, while the annex FK is commensurable in length with the rational straight line FG set out, then KH is a second apotome. X.Def.III.2 But FG is rational, so that the side of LH, that is, of EC, is a first apotome of a medial straight line. X.92 But, if the square on HF is greater than the square on FK by the square on a straight line incommensurable with HF, while the annex FK is commensurable in length with the rational straight line FG set out, then KH is a fifth apotome, so that the side of EC is a straight line which produces with a rational area a medial whole. X.Def.III.5 X.95 Therefore, if a rational area is subtracted from a medial area, then there arise two other irrational straight lines, either a first apotome of a medial straight line or a straight line which produces with a rational area a medial whole. Q.E.D. (Forthcoming) Book X Introduction Proposition X.108 Proposition X.110. © 1996 D.E.Joyce Clark University Proposition 110 If a medial area incommensurable with the whole is subtracted from a medial area, then two remaining irrational straight lines arise, either a second apotome of a medial straight line or a straight line which produces with a medial area a medial whole. As in the foregoing figures, let there be subtracted the medial area BD incommensurable with the whole from the medial area BC. I say that the side of EC is one of two irrational straight lines, either a second apotome of a medial straight line or a straight line which produces with a medial area a medial whole. Since each of the rectangles BC and BD is medial, and BC is incommensurable with BD, therefore each of the straight lines FH and FK is rational and incommensurable in length with FG. X.22 Since BC is incommensurable with BD, that is, GH with GK, therefore HF is also incommensurable with FK. VI.1 X.11 Therefore FH and FK are rational straight lines commensurable in square only. Therefore KH is an apotome. X.73 If then the square on FH is greater than the square on FK by the square on a straight line commensurable with FH, while neither of the straight lines FH nor FK is commensurable in length with the rational straight line FG set out, then KH is a third apotome. X.Def.III.3 But KL is rational, and the rectangle contained by a rational straight line and a third apotome is irrational, and the side of it is irrational, and is called a second apotome of a medial straight line, so that the side of LH, that is, of EC, is a second apotome of a medial straight line. X.93 But, if the square on FH is greater than the square on FK by the square on a straight line incommensurable with FH, while neither of the straight lines HF nor FK is commensurable in length with FG, then KH is a sixth apotome. X.Def.III.6 But the side of the rectangle contained by a rational straight line and a sixth apotome is a straight line which produces with a medial area a medial whole. X.96 Therefore the side of LH, that is, of EC, is a straight line which produces with a medial area a medial whole. Therefore, if a medial area incommensurable with the whole is subtracted from a medial area, then two remaining irrational straight lines arise, either a second apotome of a medial straight line or a straight line which produces with a medial area a medial whole. Q.E.D. (Forthcoming) Book X Introduction Proposition X.109 Proposition X.111. © 1996 D.E.Joyce Clark University Proposition 111 The apotome is not the same with the binomial straight line. Let AB be an apotome. I say that AB is not the same with the binomial straight line. If possible, let it be so. Set out a rational straight line DC, and to CD apply the rectangle CE equal to the square on AB and producing DE as breadth. Then, since AB is an apotome, DE is a first apotome. X.97 Let EF be the annex to it. Then DF and FE are rational straight lines commensurable in square only, the square on DF is greater than the square on FE by the square on a straight line commensurable with DF, and DF is commensurable in length with the rational straight line DC set out. X.Def.III.2 Again, since AB is binomial, therefore DE is a first binomial straight line. X.60 Divide it into its terms at G, and let DG be the greater term. Then DG and GE are rational straight lines commensurable in square only, the square on DG is greater than the square on GE by the square on a straight line commensurable with DG, and the greater term DG is commensurable in length with the rational straight line DC set out. X.Def.II.1 Therefore DF is also commensurable in length with DG. Therefore the remainder GF is also commensurable in length with DF. X.12 X.15 But DF is incommensurable in length with EF, therefore FG is also incommensurable in length with EF. X.13 Therefore GF and FE are rational straight lines commensurable in square only, therefore EG is an apotome. But it is also rational: which is impossible. X.73 Therefore, the apotome is not the same with the binomial straight line. Q.E.D. Remark The apotome and the irrational straight lines following it are neither the same with the medial straight line nor with one another. For the square on a medial straight line, if applied to a rational straight line, produces as breadth a straight line rational and incommensurable in length with that to which it is applied, X.22 while the square on an apotome, if applied to a rational straight line, produces as breadth a first apotome, X.97 the square on a first apotome of a medial straight line, if applied to a rational straight line, produces as breadth a second apotome, X.98 the square on a second apotome of a medial straight line, if applied to a rational straight line, produces as breadth a third apotome, X.99 the square on a minor straight line, if applied to a rational straight line, produces as breadth a fourth apotome, X.100 the square on the straight line which produces with a rational area a medial whole, if applied to a rational straight line, produces as breadth a fifth apotome, X.101 and the square on the straight line which produces with a medial area a medial whole, if applied to a rational straight line, produces as breadth a sixth apotome. X.102 Since the said breadths differ from the first and from one another, from the first because it is rational, and from one another since they are not the same in order, it is clear that the irrational straight lines themselves also differ from one an other. Since the apotome has been proved not to be the same as the binomial straight line, X.111 but, if applied to a rational straight line, the straight lines following the apotome produce breadths, each according to its own order, apotomes, and those following the binomial straight line themselves also, according to their order, produce the binomials as breadths, therefore those following the apotome are different, and those following the binomial straight line are different, so that there are, in order, thirteen irrational straight lines in all: Medial Binomial First bimedial Second bimedial Major Side of a rational plus a medial area Side of the sum of two medial areas Apotome First apotome of a medial straight line Second apotome of a medial straight line Minor Producing with a rational area a medial whole Producing with a medial area a medial whole (Forthcoming) Book X Introduction Proposition X.110 Proposition X.112. © 1996 D.E.Joyce Clark University Proposition 112 The square on a rational straight line applied to the binomial straight line produces as breadth an apotome the terms of which are commensurable with the terms of the binomial straight line and moreover in the same ratio; and further the apotome so arising has the same order as the binomial straight line. Let A be a rational straight line, let BC be a binomial, let DC be its greater term, and let the rectangle BC by EF equal the square on A. I say that EF is an apotome the terms of which are commensurable with CD and DB, and in the same ratio, and further, EF has the same order as BC. Again let the rectangle BD by G equal the square on A. Since, then, the rectangle BC by EF equals the rectangle BD by G, therefore CB is to BD as G is to EF. But CB is greater than BD, therefore G is also greater than EF. VI.16 (V.14) Let EH equal G. Then CB is to BD as HE is to EF, therefore, taken separately, CD is to BD as HF is to FE. V.17 Let it be contrived that HF is to FE as FK is to KE. Then the whole HK is to the whole KF as FK is to KE, for one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents. V.12 But FK is to KE as CD is to DB, therefore HK is to KF as CD is to DB. V.11 But the square on CD is commensurable with the square on DB, therefore the square on HK is commensurable with the square on KF. X.36 VI.22 X.11 And the square on HK is to the square on KF as HK is to KE, since the three straight lines HK, KF, and KE are proportional. Therefore HK is commensurable in length with KE, so that HE is also commensurable in length with EK. V.Def.9 X.15 Now, since the square on A equals the rectangle EH by BD, while the square on A is rational, therefore the rectangle EH by BD is also rational. And it is applied to the rational straight line BD, therefore EH is rational and commensurable in length with BD, so that EK, being commensurable with it, is also rational and commensurable in length with BD. X.20 Since, then CD is to DB as FK is to KE, while CD and DB are straight lines commensurable in square only, therefore FK and KE are also commensurable in square only. But KE is rational, therefore FK is also rational. X.11 Therefore FK and KE are rational straight lines commensurable in square only, therefore EF is an apotome. X.73 Now the square on CD is greater than the square on DB either by the square on a straight line commensurable with CD or by the square on a straight line incommensurable with it. If the square on CD is greater than the square on DB by the square on a straight line commensurable with CD, then the square on FK is also greater than the square on KE by the square on a straight line commensurable with FK. X.14 And, if CD is commensurable in length with the rational straight line set out, then FK is also; if BD is so commensurable, then KE is also; but, if neither of the straight lines CD nor DB is so commensurable, then neither of the straight lines FK nor KE is so. X.11 X.12 But, if the square on CD is greater than the square on DB by the square on a straight line incommensurable with CD, then the square on FK is also greater than the square on KE by the square on a straight line incommensurable with FK. X.14 And, if CD is commensurable with the rational straight line set out, then FK is also; if BD is so commensurable, then KE is also; but, if neither of the straight lines CD nor DB is so commensurable, then neither of the straight lines FK nor KE is so, so that FE is an apotome, the terms of which, FK and KE are commensurable with the terms CD and DB of the binomial straight line and in the same ratio, and it has the same order as BC. Therefore, the square on a rational straight line applied to the binomial straight line produces as breadth an apotome the terms of which are commensurable with the terms of the binomial straight line and moreover in the same ratio; and further the apotome so arising has the same order as the binomial straight line. Q.E.D. Note that it isn't proposition V.14 being invoked near the beginning of the proof, but an alternate form of it. See the Guide to V.14. Book X Introduction Proposition X.111 Proposition X.113. © 1996 D.E.Joyce Clark University Proposition 113 But BC is greater than BD , therefore KH is also greater than G . The square on a rational straight line, if applied to an apotome, produces as breadth the binomial straight line the terms of which are commensurable with the terms of the apotome and in the same ratio; and further the binomial so arising has the same order as the apotome. Let A be a rational straight line and BD an apotome, and let the rectangle BD by KH equal the square on A, so that the square on the rational straight line A when applied to the apotome BD produces KH as breadth. I say that KH is a binomial straight line the terms of which are commensurable with the terms of BD and in the same ratio, and further, KH has the same order as BD. Let DC be the annex to BD. Then BC and CD are rational straight lines commensurable in square only. Let the rectangle BC by G also equal the square on A. X.73 But the square on A is rational, therefore the rectangle BC by G is also rational. And it has been applied to the rational straight line BC, therefore G is rational and commensurable in length with BC. X.20 Since now the rectangle BC by G equals the rectangle BD by KH, therefore, CB is to BD as KH is to G. But BC is greater than BD, therefore KH is also greater than G. VI.16 (V.14) Make KE equal to G. Then KE is commensurable in length with BC. Since CB is to BD as HK is to KE, therefore, in conversion, BC is to CD as KH is to HE. V.19,Cor. Let it be contrived that KH is to HE as HF is to FE. Then the remainder KF is to FH as KH is to HE, that is BC is to CD. V.19 But BC and CD are commensurable in square only, therefore KF and FH are also commensurable in square only. V.11 Since KH is to HE as KF is to FH, while KH is to HE as HF is to FE, therefore KF is to FH as HF is to FE, so that also the first is to the third as the square on the first to the square on the second. Therefore KF is to FE as the square on KF is to the square on FH. V.11 V.Def.9 But the square on KF is commensurable with the square on FH, for KF and FH are commensurable in square, therefore KF is also commensurable in length with FE, so that KF is also commensurable in length with KE. X.11 X.15 But KE is rational and commensurable in length with BC, therefore KF is also rational and commensurable in length with BC. X.12 Since BC is to CD as KF is to FH, alternately, BC is to KF as DC is to FH. V.16 But BC is commensurable with KF, therefore FH is also commensurable in length with CD. X.11 But BC and CD are rational straight lines commensurable in square only, therefore KF and FH are also rational straight lines commensurable in square only. Therefore KH is binomial. X.Def.3 X.36 If now the square on BC is greater than the square on CD by the square on a straight line commensurable with BC, then the square on KF is also greater than the square on FH by the square on a straight line commensurable with KF. X.14 And, if BC is commensurable in length with the rational straight line set out, then KF is also; if CD is commensurable in length with the rational straight line set out, then FH is also; but, if neither of the straight lines BC nor CD, then neither of the straight lines KF nor FH. But, if the square on BC is greater than the square on CD by the square on a straight line incommensurable with BC, then the square on KF is also greater than the square on FH by the square on a straight line incommensurable with KF. X.14 And, if BC is commensurable with the rational straight line set out, then KF is also; if CD is so commensurable, then FH is also; but, if neither of the straight lines BC nor CD, then neither of the straight lines KF nor FH. Therefore KH is a binomial straight line, the terms of which KF and FH are commensurable with the terms BC and CD of the apotome and in the same ratio, and further, KH has the same order as BD. Therefore, the square on a rational straight line, if applied to an apotome, produces as breadth the binomial straight line the terms of which are commensurable with the terms of the apotome and in the same ratio; and further the binomial so arising has the same order as the apotome. Q.E.D. (Forthcoming) Book X Introduction Proposition X.112 Proposition X.114. © 1996 D.E.Joyce Clark University Proposition 114 If an area is contained by an apotome and the binomial straight line the terms of which are commensurable with the terms of the apotome and in the same ratio, then the side of the area is rational. Let an area, the rectangle AB by CD, be contained by the apotome AB and the binomial straight line CD, and let CE be the greater term of the latter, let the terms CE and ED of the binomial straight line be commensurable with the terms AF and FB of the apotome and in the same ratio, and let the side of the rectangle AB by CD be G. I say that G is rational. Set out a rational straight line H, and to CD apply a rectangle equal to the square on H and producing KL as breadth. Then KL is an apotome. Let its terms be KM and ML commensurable with the terms CE and ED of the binomial straight line and in the same ratio. X.112 But CE and ED are also commensurable with AF and FB and in the same ratio, therefore AF is to FB as KM is to ML. Therefore, alternately, AF is to KM as BF is to LM. Therefore the remainder AB is to the remainder KL as AF is to KM. V.19 But AF is commensurable with KM, therefore AB is also commensurable with KL. X.12 X.11 And AB is to KL as the rectangle CD by AB is to the rectangle CD by KL, therefore the rectangle CD by AB is also commensurable with the rectangle CD by KL. VI.1 X.11 But the rectangle CD by KL equals the square on H, therefore the rectangle CD by AB is commensurable with the square on H. But the square on G equals the rectangle CD by AB, therefore the square on G is commensurable with the square on H. But the square on H is rational, therefore the square on G is also rational. Therefore G is rational. And it is the side of the rectangle CD by AB. Therefore, if an area is contained by an apotome and the binomial straight line the terms of which are commensurable with the terms of the apotome and in the same ratio, then the side of the area is rational. Corollary. And it is made manifest to us by this also that it is possible for a rational area to be contained by irrational straight lines. Q.E.D. (Forthcoming) Book X Introduction Proposition X.113 Proposition X.115. © 1996 D.E.Joyce Clark University Proposition 115 From a medial straight line there arise irrational straight lines infinite in number, and none of them is the same as any preceding. Let A be a medial straight line. I say that from A there arise irrational straight lines infinite in number, and none of them is the same as any of the preceding. Set out a rational straight line B, and let the square on C equal the rectangle B by A. Then C is irrational, for that which is contained by an irrational and a rational straight line is irrational. X.Def.4 X.20 And it is not the same with any of the preceding, for the square on none of the preceding, if applied to a rational straight line will produce as breadth a medial straight line. Again, let the square on D equal the rectangle B by C. Then the square on D is irrational. X.20 Therefore D is irrational, and it is not the same with any of the preceding, for the square on none of the preceding, if applied to a rational straight line, will produce C as breadth. X.Def.4 Similarly, if this arrangement proceeds ad infinitum, it is manifest that from the medial straight line there arise irrational straight lines infinite in number, and none is the same with any of the preceding. Q.E.D. (Forthcoming) Book X Introduction Proposition X.114 Book XI Introduction. © 1996 D.E.Joyce Clark University Proposition 1 If two similar plane numbers multiplied by one another make some number, then the product is square. Let A and B be two similar plane numbers, and let A multiplied by B make C. I say that C is square. Multiply A by itself to make D. Then D is square. Since then A multiplied by itself makes D, and multiplied by B makes C, therefore A is to B as D is to C. VII.17 And, since A and B are similar plane numbers, therefore one mean proportional number falls between A and B. VIII.18 Since as many number fall in continued proportion between those which have the same ratio, therefore one mean proportional number falls between D and C also. VIII.8 And D is square, therefore C is also square. VIII.22 Therefore, if two similar plane numbers multiplied by one another make some number, then the product is square. Q.E.D. Although this is the first proposition in Book IX, it and the succeeding propositions continue those of Book VIII without break. To illustrate this proposition, consider the two similar plane numbers a = 18 and b = 8, as illustrated in the Guide to VII.Def.21. According to VIII.18, there is a mean proportional between them, namely, 12. And the square of the mean proportional is their product, ab = 144. Outline of the proof It is not clear why the proof of the proposition does not use the fact that the square of the mean proportional of two numbers equals their product, but instead uses slightly more complicated reasoning. Let a and b be the given similar plane numbers. Then there is a mean proportional between them (VIII.18). And, since a:b = a2:ab, therefore there is also a mean proportional between a2 and ab (VIII.1). But since a2 is a square, therefore ab is also a square (VIII.22). Thus, the product of the original similar plane numbers is a square. Use of this proposition The next proposition IX.2 is the converse of this one. This proposition is used in X.29. Next proposition: IX.2 Book IX introduction © 1996, 2002 D.E.Joyce Clark University Proposition 2 If two numbers multiplied by one another make a square number, then they are similar plane numbers. Let A and B be two numbers, and let A multiplied by B make the square number C. I say that A and B are similar plane numbers. Multiply A by itself to make D. Then D is square. Now, since A multiplied by itself makes D, and multiplied by B makes C, therefore A is to B as D is to C. VII.17 And, since D is square, and C is so also, therefore D and C are similar plane numbers. Therefore one mean proportional number falls between D and C. And D is to C as A is to B, therefore one mean proportional number falls between A and B also. VIII.18 VIII.8 But, if one mean proportional number falls between two numbers, then they are similar plane numbers, therefore A and B are similar plane numbers. VIII.20 Therefore, if two numbers multiplied by one another make a square number, then they are similar plane numbers. Q.E.D. This proposition is a converse of the previous one. As an example to illustrate this proposition, take any square number, such as 202 = 400. It can be factored as a product of two numbers in several ways. One such factorization is as a = 50 times b = 8. These two numbers have a mean proportional between them, namely, 20, so by VIII.20, they are similar plane numbers. (The actual shapes given by that proposition make 8 to be 2 by 4, and 50 to be 5 by 10.) Outline of the proof Let a and b be two numbers whose product ab is a square. Now, both a2 and ab are square numbers which means that they're similar plane numbers. By VIII.8, there's a mean proportional between them. But a2:ab = a:b, so there is also a mean proportional between a and b (VIII.8). Therefore, a and b are similar plane figures (VIII.20). As in the last proof, this one can be shortened. When the product ab is a square, say e2, then a mean proportional between a and b is e. Next proposition: IX.3 Previous: IX.1 Book IX introduction © 1996, 2002 D.E.Joyce Clark University Proposition 3 If a cubic number multiplied by itself makes some number, then the product is a cube. Let the cubic number A multiplied by itself make B. I say that B is cubic. Take C, the side of A. Multiply C by itself make D. It is then manifest that C multiplied by D makes A. Now, since C multiplied by itself makes D, therefore C measures D according to the units in itself. But further the unit also measures C according to the units in it, therefore the unit is to C as C is to D. VII.Def.20 Again, since C multiplied by D makes A, therefore D measures A according to the units in C. But the unit also measures C according to the units in it, therefore the unit is to C as D is to A. But the unit is to C as C is to D, therefore the unit is to C as C is to D, and as D is to A. Therefore between the unit and the number A two mean proportional numbers C and D have fallen in continued proportion. Again, since A multiplied by itself makes B, therefore A measures B according to the units in itself. But the unit also measures A according to the units in it, therefore the unit is to A as A is to B. VII.Def.20 But between the unit and A two mean proportional numbers have fallen, therefore two mean proportional numbers also fall between A and B. VIII.8 But, if two mean proportional numbers fall between two numbers, and the first is a cube, then the second is also a cube. And A is a cube, therefore B is also a cube. VIII.23 Therefore, if a cubic number multiplied by itself makes some number, then the product is a cube. Q.E.D. Modern algebra certainly makes short work of this proposition: (c3)2 = (c2)3. Use of this proposition This proposition is used in the next two propositions and IX.9. Next proposition: IX.4 Previous: IX.2 Book IX introduction © 1996 D.E.Joyce Clark University Proposition 4 If a cubic number multiplied by a cubic number makes some number, then the product is a cube. Let the cubic number A multiplied by the cubic number B make C. I say that C is cubic. Multiply A by itself to make D. Then D is a cube. IX.3 Since A multiplied by itself makes D, and multiplied by B makes C, therefore A is to B as D is to C. And, since A and B are cubic numbers, therefore A and B are similar solid numbers. Therefore two mean proportional numbers fall between A and B, so that two mean proportional numbers fall between D and C also. VII.17 VIII.19 VIII.8 And D is a cube, therefore C is also a cube. VIII.23 Therefore, if a cubic number multiplied by a cubic number makes some number, then the product is cubic. Q.E.D. Of course, m3n3 = (mn)3. Next proposition: IX.5 Previous: IX.3 Book IX introduction © 1996 D.E.Joyce Clark University Proposition 5 If a cubic number multiplied by any number makes a cubic number, then the multiplied number is also cubic. Let the cubic number A multiplied by any number B make the cubic number C. I say that B is cubic. Multiply A by itself to make D. Then D is a cube. IX.3 Now, since A multiplied by itself makes D, and multiplied by B makes C, therefore A is to B as D is to C. VII.17 And since D and C are cubes, therefore they are similar solid numbers. Therefore two mean proportional numbers fall between D and C. And D is to C as A is to B, therefore two mean proportional numbers fall between A and B, too. VIII.19 VIII.8 And A is a cube, therefore B is also a cube. VIII.23 Therefore, if a cubic number multiplied by any number makes a cubic number, then the multiplied number is also cubic. Q.E.D. This proposition is a converse of the previous one. When ab = c, and a is a cube, the previous propsition said that if b is a cube, then c is also, while this proposition says that if c is a cube, then b is also. Next proposition: IX.6 Previous: IX.4 Book IX introduction © 1996 D.E.Joyce Clark University Proposition 6 If a number multiplied by itself makes a cubic number, then it itself is also cubic. Let the number A multiplied by itself make the cubic number B. I say that A is also cubic. Multiply A by B to make C. Since, then, A multiplied by itself makes B, and multiplied by B makes C, therefore C is a cube. And, since A multiplied by itself makes B, therefore A measures B according to the units in itself. But the unit also measures A according to the units in it. Therefore the unit is to A as A is to B. And, since A multiplied by B makes C, therefore B measures C according to the units in A. VII.Def.20 But the unit also measures A according to the units in it. Therefore the unit is to A as B is to C. But the unit is to A as A is to B, therefore A is to B as B is to C. VII.Def.20 And, since B and C are cubes, therefore they are similar solid numbers. Therefore there are two mean proportional numbers between B and C. And B is to C as A is to B. Therefore there are two mean proportional numbers between A and B also. VIII.19 VIII.8 And B is a cube, therefore A is also a cube. cf. VIII.23 Therefore, if a number multiplied by itself makes a cubic number, then it itself is also cubic. Q.E.D. Outline of the proof Assume that a2 is a cube. Since a3 is also a cube, therefore there are two mean proportionals between them (VIII.19). But we have the proportion a:a2 = a2:a3, so there are also two mean proportionals between a and a2 (VIII.8). And since a2 is a cube, therefore a is also a cube (VIII.23). Use of this proposition This proposition is used in IX.10. Next proposition: IX.7 Previous: IX.5 Book IX introduction © 1996, 2002 D.E.Joyce Clark University Proposition 7 If a composite number multiplied by any number makes some number, then the product is solid. Let the composite number A multiplied by any number B make C. I say that C is solid. Since A is composite, it is measured by some number D. Let there be as many units in E as times that D measures A VII.Def.13 Since D measures A according to the units in E, therefore E multiplied by D makes A. And, since A multiplied by B makes C, and A is the product of D and E, therefore the product of D and E multiplied by B makes C. VII.Def.15 Therefore C is solid, and D, E, and B are its sides. Therefore, if a composite number multiplied by any number makes some number, then the product is solid. Q.E.D. Numbers with at least two factors are plain numbers; those with at least three are solid numbers. Perhaps Euclid takes extra steps that we would miss because he sees "d measures a a number e times" as saying something different from the product of d and e equals a." Next proposition: IX.8 Previous: IX.6 Book IX introduction © 1996 D.E.Joyce Clark University Proposition 8 If as many numbers as we please beginning from a unit are in continued proportion, then the third from the unit is square as are also those which successively leave out one, the fourth is cubic as are also all those which leave out two, and the seventh is at once cubic and square are also those which leave out five. Let there be as many numbers as we please, A, B, C, D, E, and F, beginning from a unit and in continued proportion. I say that B, the third from the unit, is square as are all those which leave out one; C, the fourth, is cubic as are all those which leave out two; and F, the seventh, is at once cubic and square as are all those which leave out five. Since the unit is to A as A is to B, therefore the unit measures the number A the same number of times that A measures B. But the unit measures the number A according to the units in it, therefore A also measures B according to the units in A. VII.Def.20 Therefore A multiplied by itself makes B, therefore B is square. And, since B, C, and D are in continued proportion, and B is square, therefore D is also square. For the same reason F is also square. VIII.22 Similarly we can prove that all those which leave out one are square. I say next that C, the fourth from the unit, is cubic are also all those which leave out two. Since the unit is to A as B is to C, therefore the unit measures the number A the same number of times that B measures C. But the unit measures the number A according to the units in A, therefore B also measures C according to the units in A. Therefore A multiplied by B makes C. Since then A multiplied by itself makes B, and multiplied by B makes C, therefore C is cubic. And, since C, D, E, and F are in continued proportion, and C is cubic, therefore F is also cubic. But it was also proved square, therefore the seventh from the unit is both cubic and square. Similarly we can prove that all the numbers which leave out five are also both cubic and square. VIII.23 Therefore, if as many numbers as we please beginning from a unit are in continued proportion, then the third from the unit is square as are also those which successively leave out one, the fourth is cubic as are also all those which leave out two, and the seventh is at once cubic and square are also those which leave out five. Q.E.D. In the continued proportion 1, a,a2, a3, a4, a5, a6, a7, etc., every second, a2, a4, a6, a8, etc., is a square; every third, a3, a6, a9, a12, etc., is a cube; and every sixth, a6, a12, a18, a24, etc., is both a square and a cube. Use of this proposition This proposition is used in four of the next five propositions. Next proposition: IX.9 Previous: IX.7 Book IX introduction © 1996 D.E.Joyce Clark University Proposition 9 If as many numbers as we please beginning from a unit are in continued proportion, and the number after the unit is square, then all the rest are also square; and if the number after the unit is cubic, then all the rest are also cubic. Let there be as many numbers as we please, A, B, C, D, E, and F, beginning from a unit and in continued proportion, and let A, the number after the unit, be square. I say that all the rest are also square. Now it was proved that B, the third from the unit, is square as are all those which leave out one. IX.8 I say that all the rest are also square. Since A, B, and C are in continued proportion, and A is square, therefore C is also square. Again, since B, C, and D are in continued proportion, and B is square, therefore D is also square. Similarly we can prove that all the rest are also square. VIII.22 Next, let A be a cube. I say that all the rest are also cubes. Now it was proved that C, the fourth from the unit, is a cube as are all those which leave out two. IX.8 I say that all the rest are also cubic. Since the unit is to A as A is to B, therefore the unit measures A the same number of times as A measures B. But the unit measures A according to the units in it, therefore A also measures B according to the units in itself, therefore A multiplied by itself makes B. And A is cubic. But, if cubic number multiplied by itself makes some number, then the product is also a cube, therefore B is also a cube. IX.3 And, since the four numbers A, B, C, and D are in continued proportion, and A is a cube, therefore D also is a cube. VIII.23 For the same reason E is also a cube, and similarly all the rest are cubes. Therefore, if as many numbers as we please beginning from a unit are in continued proportion, and the number after the unit is square, then all the rest are also square; and if the number after the unit is cubic, then all the rest are also cubic. Q.E.D. This proposition says that if a number is a square then all its powers are squares, too. Likewise for cubes. The following theorem is a converse of this one. Next proposition: IX.10 Previous: IX.8 Book IX introduction © 1996 D.E.Joyce Clark University Proposition 10 If as many numbers as we please beginning from a unit are in continued proportion, and the number after the unit is not square, then neither is any other square except the third from the unit and all those which leave out one; and, if the number after the unit is not cubic, then neither is any other cubic except the fourth from the unit and all those which leave out two. Let there be as many numbers as we please, A, B, C, D, E, and F, beginning from a unit and in continued proportion, and let A, the number after the unit, not be square. I say that neither are any other square except the third from the unit and those which leave out one. If possible, let C be square. But B is also square, therefore B and C have to one another the ratio which a square number has to a square number. IX.8 And B is to C as A is to B, therefore A and B have to one another the ratio which a square number has to a square number, so that A and B are similar plane numbers. VIII.26 converse And B is square, therefore A is also square, which is contrary to the hypothesis. Therefore C is not square. Similarly we can prove that neither is any other of the numbers square except the third from the unit and those which leave out one. Next, let A not be a cube. I say that neither are any other cubes except the fourth from the unit and those which leave out two. If possible, let D be a cube. Now C is also a cube, for it is fourth from the unit. And C is to D as B is to C, therefore B has to C the ratio which a cube has to a cube. And C is a cube, therefore B is also a cube. IX.8 VIII.25 And since the unit is to A as A is to B, and the unit measures A according to the units in it, therefore A also measures B according to the units in itself. Therefore A multiplied by itself makes the cubic number B. But, if a number multiplied by itself makes cubic number, then it is itself a cube. Therefore A is also a cube, contrary to the hypothesis. Therefore D is not a cube. Similarly we can prove that neither is any other of the numbers a cube except the fourth from the unit and those which leave out two. IX.6 Therefore, if as many numbers as we please beginning from a unit are in continued proportion, and the number after the unit is not square, then neither is any other square except the third from the unit and all those which leave out one; and, if the number after the unit is not cubic, then neither is any other cubic except the fourth from the unit and all those which leave out two. Q.E.D. This is a converse of the previous theorem. Next proposition: IX.11 Previous: IX.9 Book IX introduction © 1996 D.E.Joyce Clark University Proposition 11 If as many numbers as we please beginning from a unit are in continued proportion, then the less measures the greater according to some one of the numbers which appear among the proportional numbers. Let there be as many numbers as we please, B, C, D, and E, beginning from the unit A and in continued proportion. I say that B, the least of the numbers B, C, D, and E, measures E according to one of the numbers C or D. Since the unit A is to B as D is to E, therefore the unit A measures the number B the same number of times as D measures E. Therefore, alternately, the unit A measures D the same number of times as B measures E. VII.15 But the unit A measures D according to the units in it, therefore B also measures E according to the units in D, so that B the less measures E the greater according to some number of those which have place among the proportional numbers. Therefore, if as many numbers as we please beginning from a unit are in continued proportion, then the less measures the greater according to some one of the numbers which appear among the proportional numbers. Q.E.D. Corollary And it is manifest that, whatever place the measuring number has, reckoned from the unit, the same place also has the number according to which it measures, reckoned from the number measured, in the direction of the number before it. This proposition, along with the comment make in the corollary, says that ak divides an the number an-k times. Use of this proposition The corollary is used in the next proposition while the proposition itself is used in the one following that. Next proposition: IX.12 Previous: IX.10 Book IX introduction © 1996 D.E.Joyce Clark University Proposition 12 If as many numbers as we please beginning from a unit are in continued proportion, then by whatever prime numbers the last is measured, the next to the unit is also measured by the same. Let A, B, C, and D be as many numbers as we please beginning from a unit in continued proportion. I say that, by whatever prime numbers D is measured, A is also measured by the same. Let D be measured by any prime number E. I say that E measures A. Suppose it does not. Now E is prime, and any prime number is relatively prime to any which it does not measure, therefore E and A are relatively prime. And, since E measures D, let it measure it according to F, therefore E multiplied by F makes D. VII.29 Again, since A measures D according to the units in C, therefore A multiplied by C makes D. But, further, E multiplied by F makes D, therefore the product of A and C equals the product of E and F. IX.11 and Cor. Therefore A is to E as F is to C. But A and E are relatively prime, primes are also least, and the least measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent, therefore E measures C. Let it measure it according to G. VII.19 VII.21 VII.20 Therefore E multiplied by G makes C. But, further, by the previous theorem, A multiplied by B makes C. Therefore the product of A and B equals the product of E and G. IX.11 and Cor. Therefore A is to E as G is to B. But A and E are relatively prime, primes are also least, and the least numbers measure those which have the same ratio with them the same number of times, the antecedent the antecedent and the consequent the consequent, therefore E measures B. Let it measure it according to H. Then E multiplied by H makes B. VII.19 VII.21 VII.20 But, further, A multiplied by itself makes B, therefore the product of E and H equals the square on A. Therefore E is to A as A is to H. IX.8 VII.19 But A and E are relatively prime, primes are also least, and the least measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent, therefore E measures A antecedent antecedent. But, again, it also does not measure it, which is impossible. VII.21 VII.20 Therefore E and A are not relatively prime. Therefore they are relatively composite. But numbers relatively composite are measured by some number. VII.Def.14 And, since E is by hypothesis prime, and a prime is not measured by any number other than itself, therefore E measures A and E, so that E measures A. But it also measures D, therefore E measures A and D. Similarly we can prove that, by whatever prime numbers D is measured, A also is measured by the same. Therefore, if as many numbers as we please beginning from a unit are in continued proportion, then by whatever prime numbers the last is measured, the next to the unit is also measured by the same. Q.E.D. This proposition says that if a prime number p divides a power ak of a number a, then it divides the number a itself. Outline of the proof The proof is both elegant and inelegant. The elegant part is the reduction step from p dividing ak to p dividing ak-1. There are two inelegant parts. One is that the reduction step is applied three times starting with k equal to 4. The other is that three unnecessary statements are tacked on to the end of the proof after the goal is already reached. Assume that a prime number p divides a power ak of a number a. Suppose that p does not divide a. Then p is relatively prime to a (VII.29). From the proportion (ak/p):ak-1 = a:p, we see that the ratio (ak/p):ak-1 reduces to a:p in lowest terms (VII.21). Therefore, p divides ak-1 (VII.20). Apply this reduction step repeatedly until the conclusion p divides a is reached. Use of this proposition This proposition is used in the next one. Next proposition: IX.13 Previous: IX.11 Book IX introduction © 1996, 2002 D.E.Joyce Clark University Proposition 13 If as many numbers as we please beginning from a unit are in continued proportion, and the number after the unit is prime, then the greatest is not measured by any except those which have a place among the proportional numbers. Let there be as many numbers as we please, A, B, C, and D, beginning from a unit and in continued proportion, and let A, the number after the unit, be prime. I say that D, the greatest of them, is not measured by any other number except A, B, or C. If possible, let it be measured by E, and let E not be the same with any of the numbers A, B, or C. It is then manifest that E is not prime, for if E is prime and measures D, then it also measures A, which is prime, though it is not the same with it, which is impossible. Therefore E is not prime, so it is composite. IX.12 But any composite number is measured by some prime number, therefore E is measured by some prime number. VII.31 I say next that it is no measured by any other prime except A. If E is measured by another, and E measures D, then that other measures D, so that it also measures A, which is prime, though it is not the same with it, which is impossible. Therefore [only the prime] A measures E. IX.12 And, since E measures D, let it measure it according to F. I say that F is not the same with any of the numbers A, B, or C. If F is the same with one of the numbers A, B, or C, and measures D according to E, then one of the numbers A, B, or C also measures D according to E. But one of the numbers A, B, or C measures D according to some one of the numbers A, B, or C, therefore E is also the same with one of the numbers A, B or C, which is contrary to the hypothesis. IX.11 Therefore F is not the same as any one of the numbers A, B, or C. Similarly we can prove that F is measured by A, by proving again that F is not prime. If it is, and measures D, then it also measures A, which is prime, though it is not the same with it, which is impossible. Therefore F is not prime, so it is composite. IX.12 But any composite number is measured by some prime number, therefore F is measured by some prime number. VII.31 I say next that it is not measured by any other prime except A. If any other prime number measures F, and F measures D, then that other also measures D, so that it also measures A, which is prime, though it is not the same with it, which is impossible. Therefore [only the prime] A measures F. IX.12 And, since E measures D according to F, therefore E multiplied by F makes D. But, further, A multiplied by C makes D, therefore the product of A and C equals the product of E and F. IX.11 Therefore, proportionally A is to E as F is to C. VII.19 But A measures E, therefore F also measures C. Let it measure it according to G. Similarly, then, we can prove that G is not the same with any of the numbers A or B, and that it is measured by A. And, since F measures C according to G, therefore F multiplied by G makes C. But, further, A multiplied by B makes C, therefore the product of A and B equals the product of F and G. Therefore, proportionally A is to F as G is to B. IX.11 VII.19 But A measures F, therefore G also measures B. Let it measure it according to H. Similarly then we can prove that H is not the same with A. And, since G measures B according to H, therefore G multiplied by H makes B. But, further, A multiplied by itself makes B, therefore the product of H and G equals the square on A. IX.8 Therefore H is to A as A is to G. But A measures G, therefore H also measures A, which is prime, though it is not the same with it, which is absurd. VII.19 Therefore D the greatest is not measured by any other number except A, B, or C. Therefore, if as many numbers as we please beginning from a unit are in continued proportion, and the number after the unit is prime, then the greatest is not measured by any except those which have a place among the proportional numbers. Q.E.D. This proposition says that the only numbers that can divide a power of a prime are smaller powers of that prime. Outline of the proof The proof involves a reduction step like that in the proof of the previous proposition. Suppose a number e divides a power pk of a prime number p, but e does not equal any lower power of p. First note that e can't be prime itself, since then it would divide p (IX.12), which it doesn't. Then e is composite. Then some prime number q divides e (VII.31). Then q also divides pk, which it implies q divides p. Therefore, the only prime that can divide e is p. The rest of the proof is repeated reduction of the power k. Since e is not 1, it is divisible by p. Let g be e/p. Then g divides pk-1, but is not any lower power of p. Then the same argument can be applied. Continue in this manner until some number divides p but is not 1 or p, a contradiction. Thus, the only numbers that can divide a power of a prime are smaller powers of the prime. Use of this proposition This proposition is used in IX.32 and IX.36. Next proposition: IX.14 Previous: IX.12 Book IX introduction © 1996, 2002. D.E.Joyce Clark University Proposition 14 If a number is the least that is measured by prime numbers, then it is not measured by any other prime number except those originally measuring it. Let the number A be the least that is measured by the prime numbers B, C, and D. I say that A is not measured by any other prime number except B, C, or D. If possible, let it be measured by the prime number E, and let E not be the same as any one of the numbers B, C, or D. Now, since E measures A, let it measure it according to F, therefore E multiplied by F makes A. And A is measured by the prime numbers B, C, and D. But, if two numbers multiplied by one another make some number, and any prime number measures the product, then it also measures one of the original numbers, therefore each of B, C, and D measures one of the numbers E or F. VII.30 Now they do not measure E, for E is prime and not the same with any one of the numbers B, C, or D. Therefore they measure F, which is less than A, which is impossible, for A is by hypothesis the least number measured by B, C, and D. Therefore no prime number measures A except B, C, and D. Therefore, if a number is the least that is measured by prime numbers, then it is not measured by any other prime number except those originally measuring it. Q.E.D. This proposition states that the least common multiple of a set of prime numbers is not divisible by any other prime. The least common multiple is actually the product of those primes, but that isn't mentioned. The proof is clear, and it depends on VII.30, that if a prime divides a product, then it divides one of the factors. Next proposition: IX.15 Previous: IX.13 Book IX introduction © 1996 D.E.Joyce Clark University Proposition 15 If three numbers in continued proportion are the least of those which have the same ratio with them, then the sum of any two is relatively prime to the remaining number. Let A, B, and C, three numbers in continued proportion, be the least of those which have the same ratio with them. I say that the sum of any two of the numbers A, B, and C is relatively prime to the remaining number, that is, A plus B is relatively prime to C, B plus C is relatively prime to A, and A plus C is relatively prime to B. Take two numbers DE and EF to be the least of those which have the same ratio with A, B, and C. VIII.2 It is then manifest that DE multiplied by itself makes A, and multiplied by EF makes B, and that EF multiplied by itself makes C. Cor. to VIII.2 Now, since DE and EF are least, therefore they are relatively prime. But, if two numbers are relatively prime, then their sum is also relatively prime to each, therefore DF is relatively prime to each of the numbers DE and EF. VII.22 VII.28 But, further, DE is also relatively prime to EF, therefore DF and DE are relatively prime to EF. But, if two numbers are relatively prime to any number, then their product is also relatively prime to the other, so that the product of FD and DE is relatively prime to EF, hence the product of FD and DE is also relatively prime to the square on EF. VII.24 VII.25 But the product of FD and DE is the square on DE together with the product of DE and EF, therefore the sum of the square on DE and the product of DE and EF is relatively prime to the square on EF. II.3 And the square on DE is A, the product of DE and EF is B, and the square on EF is C, therefore the sum of A and B is prime to C. Similarly we can prove that the sum of B and C is relatively prime to A. I say next that the sum of A and C is also relatively prime to B. Since DF is relatively prime to each of the numbers DE and EF, therefore the square on DF is also relatively prime to the product of DE and EF. VII.24 VII.25 But the sum of the squares on DE and EF together with twice the product of DE and EF equals the square on DF, therefore the sum of the squares on DE and EF together with twice the product of DE and EF is relatively prime to the product of DE and EF. II.4 Taken separately, the sum of the squares on DE and EF together with the product of DE and EF is relatively prime to the product of DE and EF. Therefore, taken separately again, the sum of the squares on DE and EF is relatively prime to the product of DE and EF. And the square on DE is A, the product of DE and EF is B, and the square on EF is C. Therefore the sum of A and C is relatively prime to B. Therefore, if three numbers in continued proportion are the least of those which have the same ratio with them, then the sum of any two is relatively prime to the remaining number. Q.E.D. Outline of the proof Let a, b, c be three numbers in continued proportion. Then according to VIII.2, they are of the form a = d2, b = de, c = e2, where d and e are relatively prime. Then the sum, d + e, is relatively prime to both d and e (VII.28). Now, since both d and d + e are relatively prime to e, so is their product d2 + de relatively prime to e (VII.24), and therefore to e2 (VII.25). Thus, a + b is relatively prime to c. Likewise, b + c is relatively prime to a. Next, since d + e is relatively prime to both d and e, so is its square (d + e)2 relatively prime to the product de (VII.24 and VII.25). That is, d2 + e2 + 2de is relatively prime to de. Subtract 2de to conclude that d2 + e2 is relatively prime to de. Thus, b is relatively prime to a + c. Next proposition: IX.16 Previous: IX.14 Book IX introduction © 1996, 2002 D.E.Joyce Clark University Proposition 16 If two numbers are relatively prime, then the second is not to any other number as the first is to the second. Let the two numbers A and B be relatively prime. I say that B is not to any other number as A is to B. If possible as A is to B, let B be to C. Now A and B are relatively prime, numbers which are relatively prime are also least, and the least numbers measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent, therefore A measures B as antecedent antecedent. VII.21 VII.20 But it also measures itself, therefore A measures A and B which are relatively prime, which is absurd. Therefore B is not to C as A is to B. Therefore, if two numbers are relatively prime, then the second is not to any other number as the first is to the second. Q.E.D. Outline of the proof Let a and b be relatively prime. Then he ratio a:b is in lowest terms. Suppose that ratio is the same as the ratio b:c. Then the antecedent of the ratio a:b, namely a, divides the antecedent of the ratio b:c, namely b. But a cannot divide b since they're relatively prime. Use of this proposition This proposition is used in IX.18. Next proposition: IX.17 Previous: IX.15 Book IX introduction © 1996, 2002 D.E.Joyce Clark University Proposition 17 If there are as many numbers as we please in continued proportion, and the extremes of them are relatively prime, then the last is not to any other number as the first is to the second. Let there be as many numbers as we please, A, B, C, and D, in continued proportion, and let the extremes of them, A and D, be relatively prime. I say that D is not to any other number as A is to B. If possible A is to B, so let D be to E, therefore, alternately A is to D as B is to E. VII.13 But A and D are prime, primes are also least, and the least numbers measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent. Therefore A measures B. And A is to B as B is to C. Therefore B also measures C, so that A also measures C. VII.21 VII.20 And since B is to C as C is to D, and B measures C, therefore C also measures D. But A measures C, so that A also measures D. But it also measures itself, therefore A measures A and D which are relatively prime, which is impossible. Therefore D is not to any other number as A is to B. Therefore, if there are as many numbers as we please in continued proportion, and the extremes of them are relatively prime, then the last is not to any other number as the first is to the second. Q.E.D. This proposition generalizes the previous proposition from a ratio of two terms to a continued proportion of arbitrarily many. It says that a continued proportion in lowest terms cannot be extended. Outline of the proof Consider a continued proportion in lowest terms with the first term a relatively prime to the last term d, and having the ratio a:b. Suppose it can be extended to e so that a:b = d:e. Alternately, a:d = b:e. Since the first ratio a:d is in lowest terms, therefore a divides b. Then each term of the continued proportion divides the next, hence a divides d. But that's impossible since a and d are relatively prime. Therefore, the continued proportion cannot be extended. Next proposition: IX.18 Previous: IX.16 Book IX introduction © 1996, 2002 D.E.Joyce Clark University Proposition 18 Given two numbers, to investigate whether it is possible to find a third proportional to them. Let A and B be the given two numbers. It is required to investigate whether it is possible to find a third proportional to them. Now A and B are either relatively prime or not. And, if they are relatively prime, it was proved that it is impossible to find a third proportional to them. IX.16 Next, let A and B not be relatively prime, and let B multiplied by itself make C. Then A either measures C or does not measure it. First, let it measure it according to D, therefore A multiplied by D makes C. But, further, B multiplied by itself makes C, therefore the product of A and D equals the square on B. Therefore A is to B as B is to D, therefore a third proportional number D has been found to A and B. VII.19 Next, let A not measure C. I say that it is impossible to find a third proportional number to A and B. If possible, let D be such third proportional. Then the product of A and D equals the square on B. But the square on B is C, therefore the product of A and D equals C. Hence A multiplied by D makes C, therefore A measures C according to D. But, by hypothesis, it also does not measure it, which is absurd. Therefore it is not possible to find a third proportional number to A and B when A does not measure C. Q.E.D. Note that a third proportional d to a and b has to satisfy a:b = b:d, so d would have to be b2/a. So the third proportional exists when a divides b2. This conclusion is just what Euclid discovers in this proposition. Next proposition: IX.19 Previous: IX.17 Book IX introduction © 1996 D.E.Joyce Clark University Proposition 19 [The Greek text of this Proposition is corrupt. However, analogously to Proposition 18 the condition that a fourth proportional to A, B, and C exists is that A measure the product of B and C. ] Given three numbers, to investigate when it is possible to find a fourth proportional to them. Let A, B, and C be the given three numbers. It is required to investigate when it is possible to find a fourth proportional to them. Q.E.D. Note that a fourth proportional d to a, b and c has to satisfy a:b = c:d, so d would have to be bc/a. So the third proportional exists when a divides bc. No doubt that is what Euclid concludes in the missing part of this proposition. Next proposition: IX.20 Previous: IX.18 Book IX introduction © 1996, 2002 D.E.Joyce Clark University Proposition 20 Prime numbers are more than any assigned multitude of prime numbers. Let A, B, and C be the assigned prime numbers. I say that there are more prime numbers than A, B, and C. Take the least number DE measured by A, B, and C. Add the unit DF to DE. Then EF is either prime or not. First, let it be prime. Then the prime numbers A, B, C, and EF have been found which are more than A, B, and C. Next, let EF not be prime. Therefore it is measured by some prime number. Let it be measured by the prime number G. VII.31 I say that G is not the same with any of the numbers A, B, and C. If possible, let it be so. Now A, B, and C measure DE, therefore G also measures DE. But it also measures EF. Therefore G, being a number, measures the remainder, the unit DF, which is absurd. Therefore G is not the same with any one of the numbers A, B, and C. And by hypothesis it is prime. Therefore the prime numbers A, B, C, and G have been found which are more than the assigned multitude of A, B, and C. Therefore, prime numbers are more than any assigned multitude of prime numbers. Q.E.D. Outline of the proof Suppose that there are only a finite number of prime numbers. Let m be the least common multiple of all of them. (This least common multiple was also considered in proposition IX.14. It wasn't noted in the proof of that proposition that the least common multiple is the product of the primes, and it isn't noted in this proof, either.) Consider the number m + 1. It cannot be prime, since it's larger than all the primes. So it is not prime. Then according to VII.31, some prime g divides it. But g cannot be any of the primes, since they all divide m and do not divide m + 1. Thus, the assumption that there are a finite number of primes leads to a contradiction. Therefore, there are not a finite number of primes. Next proposition: IX.21 Previous: IX.19 Book IX introduction © 1996, 2002 D.E.Joyce Clark University Proposition 21 If as many even numbers as we please are added together, then the sum is even. Let as many even numbers as we please, AB, BC, CD, and DE, be added together. I say that the sum AE is even. Since each of the numbers AB, BC, CD, and DE is even, therefore each has a half part, so that the sum AE also has a half part. But an even number is that which is divisible into two equal parts, therefore AE is even. VII.Def.6 Therefore, if as many even numbers as we please are added together, then the sum is even. Q.E.D. With this proposition, Euclid commences the study of even and odd numbers. The study continues through proposition IX.34. The statements of these propositions probably constitute the oldest part of the Elements and date back to the Pythagoreans. Indeed, their proofs depend on no other propositions (except IX.31 which discusses prime numbers and may have been inserted among these propositions because it involves odd numbers), so that the statements together with the proofs may be the oldest part of the Elements. Commutativity and associativity of addition The proof of this proposition implicitly relies on a principle that the order that numbers are summed is irrelevant. For example, when showing that the sum of the two even numbers a and b is even, first a is divided into two equal parts, a = c + c, and b is divided into two equal parts, b = d + d, therefore a + b = (c + c) + (d + d). But to reach the conclusion that a + b is divisible into two equal parts, we need a + b = (c + d) + (c + d), which adds the terms in a different order. Of course the order that the terms are added has no effect on the sum. That is an implicit assumption made by Euclid and most everyone after him until the 19th century. The modern way to deal with this question is to recognize two properties of addition of numbers. One of them is commutativity. Addition is commutative since for any two numbers a and b, a + b = b + a. Commutativity says we can two numbers in any order and get the same result. The other property, associativity, is more subtle. When computing the sum a + b + c of three numbers, there is still a choice of which numbers to add first. You can either add a + b first to get d, then add d + c to get the sum, or you can add b + c first to get e, then add a + e to get the sum. The same sum should result. As an equation, associativity says you can move the parentheses around: (a + b) + c = a + (b + c). These two properties, commutativity and associativity, are enough to guarantee that when you add any number of terms together, the order that they're added is irrelvant. These properties should either be taken as postulates about numbers, or else proven from more basic assumptions. Besides these properties of addition, Euclid missed some other basic properties of the arithmetic operations. Use of this proposition This proposition is used in the next two and in IX.28. Next proposition: IX.22 Previous: IX.20 Book IX introduction © 1996, 2002 D.E.Joyce Clark University Proposition 22 If as many odd numbers as we please are added together, and their multitude is even, then the sum is even. Let as many odd numbers as we please, AB, BC, CD, and DE, even in multitude, be added together. I say that the sum AE is even. Since each of the numbers AB, BC, CD, and DE is odd, if a unit is subtracted from each, then each of the remainders is even, so that the sum of them is even. But the multitude of the units is also even. Therefore the sum AE is also even. (VII.Def.7) IX.21 Therefore, if as many odd numbers as we please are added together, and their multitude is even, then the sum is even. Q.E.D. A critical step in the proof is the claim that if 1 is subracted from an odd number, then the remainder is even. This was mentioned in VII.Def.7, but never proved. See the Guide for that defintion for details. Unless that gap is filled, this proposition, along with many that depend upon it, are unjustified. Use of this proposition This proposition is used in the next one. Next proposition: IX.23 Previous: IX.21 Book IX introduction © 1996, 2002. D.E.Joyce Clark University Proposition 23 If as many odd numbers as we please are added together, and their multitude is odd, then the sum is also odd. Let as many odd numbers as we please, AB, BC, and CD, the multitude of which is odd, be added together. I say that the sum AD is also odd. Subtract the unit DE from CD, therefore the remainder CE is even. VII.Def.7 But CA is also even, therefore the sum AE is also even. IX.22 IX.21 And DE is a unit. Therefore AD is odd. VII.Def.7 Therefore, if as many odd numbers as we please are added together, and their multitude is odd, then the sum is also odd. Q.E.D. This proposition is used in propositions IX.29 and IX.30. Next proposition: IX.24 Previous: IX.22 Book IX introduction © 1996 D.E.Joyce Clark University Proposition 24 If an even number is subtracted from an even number, then the remainder is even. Let the even number BC be subtracted from the even number AB. I say that the remainder CA is even. Since AB is even, therefore it has a half part. For the same reason BC also has a half part, so that the remainder CA also has a half part, and CA is therefore even. VII.Def.6 Therefore, if an even number is subtracted from an even number, then the remainder is even. Q.E.D. This proposition is used in four of the next five propositions. Next proposition: IX.25 Previous: IX.23 Book IX introduction © 1996 D.E.Joyce Clark University Proposition 25 If an odd number is subtracted from an even number, then the remainder is odd. Let the odd number BC be subtracted from the even number. I say that the remainder CA is odd. Subtract the unit CD from BC, therefore DB is even. VII.Def.7 But AB is also even, therefore the remainder AD is also even. And CD is a unit, therefore CA is odd. IX.24 VII.Def.7 Therefore, if an odd number is subtracted from an even number, then the remainder is odd. Q.E.D. This is the second of four propositions that examine the result of subtracting even and odd numbers from each other. Next proposition: IX.26 Previous: IX.24 Book IX introduction © 1996 D.E.Joyce Clark University Proposition 26 If an odd number is subtracted from an odd number, then the remainder is even. Let the odd number BC be subtracted from the odd number AB. I say that the remainder CA is even. Since AB is odd, subtract the unit BD, therefore the remainder AD is even. For the same reason CD is also even, so that the remainder CA is also even. VII.Def.7 IX.24 Therefore, if an odd number is subtracted from an odd number, then the remainder is even. Q.E.D. This proposition is used in IX.29. Next proposition: IX.27 Previous: IX.25 Book IX introduction © 1996 D.E.Joyce Clark University Proposition 27 If an even number is subtracted from an odd number, then the remainder is odd. Let the even number BC be subtracted from the odd number AB. I say that the remainder CA is odd. Subtract the unit AD, therefore DB is even. VII.Def.7 But BC is also even, therefore the remainder CD is even. Therefore CA is odd. IX.24 VII.Def.7 Therefore, if an even number is subtracted from an odd number, then the remainder is odd. Q.E.D. This is the last of four propositions that examine the result of subtracting even and odd numbers from each other. Next proposition: IX.28 Previous: IX.26 Book IX introduction © 1996 D.E.Joyce Clark University Proposition 28 If an odd number is multiplied by an even number, then the product is even. Let the odd number A multiplied by the even number B make C. I say that C is even. Since A multiplied by B makes C, therefore C is made up of as many numbers equal to B as there are units in A. And B is even, therefore C is made up of even numbers. VII.Def.15 But, if as many even numbers as we please be added together, the whole is even. Therefore C is even. IX.21 Therefore, if an odd number is multiplied by an even number, then the product is even. Q.E.D. This is one of two propositions that examine the result of multiplying even and odd numbers by each other. The third proposition, the product of two even numbers, is omitted. Note that the proof for this theorem makes no use of the assumption that A is an odd number. The statement of this theorem might just as well be "if any number is multiplied by an even number, then the product is even." Use of this proposition The proof of proposition IX.31 concludes at one point that since a number divides a odd number, it must be even. Such a statement follows from this proposition. Next proposition: IX.29 Previous: IX.27 Book IX introduction © 1996, 2002 D.E.Joyce Clark University Proposition 29 If an odd number is multiplied by an odd number, then the product is odd. Let the odd number A multiplied by the odd number B make C. I say that C is odd. Since A multiplied by B makes C, therefore C is made up of as many numbers equal to B as there are units in A. And each of the numbers A and B is odd, therefore C is made up of odd numbers, the multitude of which is odd. Thus C is odd. VII.Def.15 IX.23 Therefore, if an odd number is multiplied by an odd number, then the product is odd. Q.E.D. With the completion of this proposition, the study of addition, subtraction, and multiplication of even and odd numbers is also completed. There remain a few more propositions about even and odd numbers. Next proposition: IX.30 Previous: IX.28 Book IX introduction © 1996 D.E.Joyce Clark University Proposition 30 If an odd number measures an even number, then it also measures half of it. Let the odd number A measure the even number B. I say that it also measures the half of it. Since A measures B, let it measure it according to C. I say that C is not odd. If possible, let it be so. Then, since A measures B according to C, therefore A multiplied by C makes B. Therefore B is made up of odd numbers the multitude of which is odd. Therefore B is odd, which is absurd, for by hypothesis it is even. Therefore C is not odd, therefore C is even. IX.23 Thus A measures B an even number of times. For this reason then it also measures the half of it. Therefore, if an odd number measures an even number, then it also measures half of it. Q.E.D. This proposition is used in the next one. Next proposition: IX.31 Previous: IX.29 Book IX introduction © 1996 D.E.Joyce Clark University Proposition 31 If an odd number is relatively prime to any number, then it is also relatively prime to double it. Let the odd number A be relatively prime to any number B, and let C be double of B. I say that A is relatively prime to C. If they are not relatively prime, then some number will measure them. Let a number D measure them. Now A is odd, therefore D is also odd. And since D which is odd measures C, and C is even, therefore D measures the half of C also. (IX.28) IX.30 But B is half of C, therefore D measures B. But it also measures A, therefore D measures A and B which are relatively prime, which is impossible. Therefore A cannot but be relatively prime to C. Therefore A and C are relatively prime. Therefore, if an odd number is relatively prime to any number, then it is also relatively prime to double it. Q.E.D. A generalization of this proposition would be "If two numbers (2 and B in this proposition) are relatively prime to to any number (A), then their product (2B) is also relatively prime to it (A)." That is proposition VII.24. Next proposition: IX.32 Previous: IX.30 Book IX introduction © 1996 D.E.Joyce Clark University Proposition 32 Each of the numbers which are continually doubled beginning from a dyad is even-times even only. Let as many numbers as we please, B, C, and D, be continually doubled beginning from the dyad A. I say that B, C, and D are even-times even only. Now that each of the numbers B, C, and D is even-times even is manifest, for each is doubled from a dyad. I say that it is also even-times even only. Set out a unit. Since then as many numbers as we please beginning from a unit are in continued proportion, and the number A after the unit is prime, therefore D, the greatest of the numbers A, B, C, and D, is not measured by any other number except A, B, and C. And each of the numbers A, B, and C is even, therefore D is even-times even only. IX.13 VII.Def.8 Similarly we can prove that each of the numbers B and C is even-times even only. Therefore, each of the numbers which are continually doubled beginning from a dyad is eventimes even only. Q.E.D. Numbers which are even-times even only are just the powers of 2: 4, 8, 16, 32, etc. An alternate proof of this proposition uses IX.30 rather than IX.13. If an odd number could divide D, then by IX.30, it would divide half of it, namely C. Then it would divide B, then A, the diad, which is absurd. Note that the reduction step mentioned in this alternate proof is much simpler than the reduction step used in the proof of IX.30. Next proposition: IX.33 Previous: IX.31 Book IX introduction © 1996, 2002 D.E.Joyce Clark University Proposition 33 If a number has its half odd, then it is even-times odd only. Let the number A have its half odd. I say that A is even-times odd only. Now that it is even-times odd is manifest, for the half of it, being odd, measures it an even number of times. VII.Def.9 I say next that it is also even-times odd only. If A is even-times even also, then it is measured by an even number according to an even number, so that the half of it is also measured by an even number though it is odd, which is absurd. VII.Def.8 Therefore A is even-times odd only. Therefore, if a number has its half odd, then it is even-times odd only. Q.E.D. To say that a number is even-times odd only means that it is even-times odd, but it is not even-times even. As this proposition states, such numbers are the numbers which are twice odd numbers. Next proposition: IX.34 Previous: IX.32 Book IX introduction © 1996 D.E.Joyce Clark University Proposition 34 If an [even] number neither is one of those which is continually doubled from a dyad, nor has its half odd, then it is both even-times even and eventimes odd. Let the [even] number A neither be one of those doubled from a dyad, nor have its half odd. I say that A is both even-times even and even-times odd. Now that A is even-times even is manifest, for it has not its half odd. VII.Def.8 I say next that it is also even-times odd. If we bisect A, then bisect its half, and do this continually, we shall come upon some odd number which measures A according to an even number. If not, we shall come upon a dyad, and A will be among those which are doubled from a dyad, which is contrary to the hypothesis. Thus A is even-times odd. But it was also proved even-times even. Therefore A is both even-times even and eventimes odd. Therefore, if an [even] number neither is one of those which is continually doubled from a dyad, nor has its half odd, then it is both even-times even and even-times odd. Q.E.D. This completes the series of propostions on even and odd numbers that started with IX.21. Next proposition: IX.35 Previous: IX.33 Book IX introduction © 1996 D.E.Joyce Clark University Proposition 35 If as many numbers as we please are in continued proportion, and there is subtracted from the second and the last numbers equal to the first, then the excess of the second is to the first as the excess of the last is to the sum of all those before it. Let there be as many numbers as we please in continued proportion, A, BC, D, and EF, beginning from A as least, and let there be subtracted from BC and EF the numbers BG and FH, each equal to A. I say that GC is to A as EH is to the sum of A, BC, and D. Make FK equal to BC, and FL equal to D. Then, since FK equals BC, and of these the part FH equals the part BG, therefore the remainder HK equals the remainder GC. And since EF is to D as D is to BC, and as BC is to A, while D equals FL, BC equals FK, and A equals FH, therefore EF is to FL as LF is to FK, and as FK is to FH. Taken separately EL is to LF as LK is to FK, and as KH is to FH. VII.11 VII.13 Since one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents, therefore KH is to FH as the sum of EL, LK, and KH is to the sum of LF, FK, and HF. VII.12 But KH equals CG, FH equals A, and the sum of LF, FK, and HF equals the sum of D, BC, and A, therefore CG is to A as EH is to the sum of D, BC, and A. Therefore the excess of the second is to the first as the excess of the last is to the sum of those before it. Therefore, if as many numbers as we please are in continued proportion, and there is subtracted from the second and the last numbers equal to the first, then the excess of the second is to the first as the excess of the last is to the sum of all those before it. Q.E.D. This proposition says if a sequence of numbers a1, a2, a3, ..., an, an+1 is in continued proportion a1:a2 = a2:a3 = ... = an:an+1 then (a2 – a1) : a1 = (an+1 – a1) : (a1 + a2 + ... + an). This conclusion gives a way of computing the sum of the terms in the continued proportion as a1 + a2 + ... + an = a1 an+1 – a1 a2 – a1 . If we denote the first term by a and the ratio of the terms by r, then this gives the familiar formula a + ar + ar2 + ... + arn-1 = a rn – 1 r – 1 . Summary of the proof The proof is much more comprehisible when it's translated in to algebraic notation. The correspondence is as follows A = a1 BC = a2 ... D = an EF = an+1 BG = FH = a1 GC = a2 – a1 EH = an+1 – a1 a2:a1. For each proportion, say the first, an+1:an = an:an–1, take it separately according to VII.11 to get (an+1 – an):(an – an–1) = an:an–1, then alternately (an+1 – an):an = (an – an–1):an–1. Stringing the conclusions together, we have (an+1 – an):an = (an – an–1):an–1 = ... = (a2 – a1):a1. Next, using proposition VII.12, the sum of the antecedents is to the sum of the consequences equals this same ratio. Therefore (an+1–an + an–an–1 + ... + a2–a1) : (an + an–1 + ... + a2 + a1) = (a2 – a1) : a1. But an+1 – an + an – an–1 + ... + a2 – a1 equals an+1 – a1. That gives us the conclusion of the proposition (an+1 – a1) : (an + an–1 + ... + a2 + a1) = (a2 – a1) : a1. Use of this propostion This proposition is used in the next one. Next proposition: IX.36 Previous: IX.34 Book IX introduction © 1996, 2002 D.E.Joyce Clark University Proposition 36 If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if the sum multiplied into the last makes some number, then the product is perfect. Let as many numbers as we please, A, B, C, and D, beginning from a unit be set out in double proportion, until the sum of all becomes prime, let E equal the sum, and let E multiplied by D make FG. I say that FG is perfect. For, however many A, B, C, and D are in multitude, take so many E, HK, L, and M in double proportion beginning from E. Therefore, ex aequali A is to D as E is to M. Therefore the product of E and D equals the product of A and M. And the product of E and D is FG, therefore the product of A and M is also FG. VII.14 VII.19 Therefore A multiplied by M makes FG. Therefore M measures FG according to the units in A. And A is a dyad, therefore FG is double of M. But M, L, HK, and E are continuously double of each other, therefore E, HK, L, M, and FG are continuously proportional in double proportion. Subtract from the second HK and the last FG the numbers HN and FO, each equal to the first E. Therefore the excess of the second is to the first as the excess of the last is to the sum of those before it. Therefore NK is to E as OG is to the sum of M, L, KH, and E. IX.35 And NK equals E, therefore OG also equals M, L, HK, E. But FO also equals E, and E equals the sum of A, B, C, D and the unit. Therefore the whole FG equals the sum of E, HK, L, M, A, B, C, D, and the unit, and it is measured by them. I say also that FG is not measured by any other number except A, B, C, D, E, HK, L, M, and the unit. If possible, let some number P measure FG, and let P not be the same with any of the numbers A, B, C, D, E, HK, L, or M. And, as many times as P measures FG, so many units let there be in Q, therefore Q multiplied by P makes FG. But, further, E multiplied by D makes FG, therefore E is to Q as P is to D. VII.19 And, since A, B, C, and D are continuously proportional beginning from a unit, therefore D is not measured by any other number except A, B, or C. IX.13 And, by hypothesis, P is not the same with any of the numbers A, B, or C, therefore P does not measure D. But P is to D as E is to Q, therefore neither does E measure Q. VII.Def.20 And E is prime, and any prime number is prime to any number which it does not measure. Therefore E and Q are relatively prime. VII.29 But primes are also least, and the least numbers measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent, and E is to Q as P is to D, therefore E measures P the same number of times that Q measures D. VII.21 VII.20 But D is not measured by any other number except A, B, or C, therefore Q is the same with one of the numbers A, B, or C. Let it be the same with B. And, however many B, C, and D are in multitude, take so many E, HK, and L beginning from E. Now E, HK, and L are in the same ratio with B, C, and D, therefore, ex aequali B is to D as E is to L. VII.14 Therefore the product of B and L equals the product of D and E. But the product of D and E equals the product of Q and P, therefore the product of Q and P also equals the product of B and L. VII.19 Therefore Q is to B as L is to P. And Q is the same with B, therefore L is also the same with P, which is impossible, for by hypothesis P is not the same with any of the numbers set out. VII.19 Therefore no number measures FG except A, B, C, D, E, HK, L, M, and the unit. And FG was proved equal to the sum of A, B, C, D, E, HK, L, M, and the unit, and a perfect number is that which equals its own parts, therefore FG is perfect. VII.Def.22 Therefore, if as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if the sum multiplied into the last makes some number, then the product is perfect. Q.E.D. Summary of the proof Euclid begins by assuming that the sum of a number of powers of 2 (the sum beginning with 1) is a prime number. Let p be the number of powers of 2, and let s be their sum which is prime. s = 1 + 2 + 22 + ... + 2p-1 Note that the last power of 2 is 2p-1 since the sum starts with 1, which is 20. In Euclid's proof, A represents 2, B represents 22, C represents 23, and D is supposed to be the last power of 2, so it represents 2p-1. Also, E represents their sum s, and FG is the product of E and D, so it represents s2p-1. Let's denote that last by n. n = s2p-1 The goal is to show that n is a perfect number. In the first part of this proof, Euclid finds some proper divisors of n that sum to n. These come in two sequences: 1, 2, 22, ..., 2p-1 and s, 2s, 22s, ..., 2n-2s In his proof, the latter are represented by E, HK, L, and finally M. It is clear that each of these is a proper divisor of n, and later in the proof Euclid shows that they are the only proper divisors of n. Using the previous proposition, IX.35, Euclid finds the sum of the continued proportion, s + 2s + 22s + ... + 2n-2s, to be 2n-1s – s. But s was the sum 1 + 2 + 22 + ... + 2p-1, hence, n = 2n-1s = 1 + 2 + 22 + ... + 2p-1 + s + 2s + 22s + ... + 2n-2s Thus, n is a sum of these proper divisors. All that is left to do is to show that they are the only proper divisors of n, for then n will be the sum of all of its proper divisors, whence a perfect number. The remainder of the proof is detailed and difficult to follow. It hinges on IX.13 which implies that the only factors of 2p-1 are powers of 2, so all the factors of 2p-1 have been found. Here's a not-toofaithful version of Euclid's argument. Suppose n factors as ab where a is not a proper divisor of n in the list above. In Euclid's proof, P represents a and Q represents b. Since a divides s 2p-1, but is not a power of 2, and s is prime, therefore s divides a. Then b has to be a power of 2. But then a has to be a power of 2 times s. But all the powers of 2 times s are on the list of known proper divisors. Therefore, the list includes all the proper divisors. Mersenne primes and perfect numbers Note that the sum, s = 1 + 2 + 22 + ... + 2p-1, equals 2p – 1, by IX.35. As this fact is not needed in the proof, Euclid omits to mention it. Thus, we can restate the proposition as follows: If 2p – 1 is a prime number, then (2p – 1) 2p-1is a perfect number. Prime numbers of the form 2p – 1 have come to be called Mersenne primes named in honor of Marin Mersenne (1588-1648), one of many people who have studied these numbers. The four smallest perfect numbers, 6, 28, 496, and 8128, were known to the ancient Greek mathematicians. The Mersenne primes 2p – 1 corresponding to these four perfect numbers are 3, 7, 31, and 127, respectively, where the exponents p are 2, 3, 5, and 7, respectively. The observation that these four exponents are all prime suggests the following two questions: 1. In order for 2p – 1 to be prime, is it sufficient for p to be prime? 2. In order for 2p – 1 to be prime, is it necessary for p to be prime? Naturally, the next number to check for primality is 211 – 1, 2047, which, by a simple search for prime factors is found not to be prime. The number 2047 factors as 23 times 89. Therefore, primality of p is not sufficient. In 1640 Pierre de Fermat (1601-1665) wrote to Mersenne with his investigation of these primes. Fermat found three conditions on p that were necessary for 2p – 1 to be prime. One of these conditions answers the second question above- p does have to be prime. Here's a quick argument for that. If p did factor, say as ab, then 2p – 1, which is 2ab – 1, would also factor, namely as 2ab – 1 = (2a – 1) (2a(b-1) + 2a(b-2) + ... + 2a). Many mathematicians have studied Mersenne primes since then. A fairly practical testing algorithm was constructed by Lucas in 1876. He showed that the the number 2p – 1 is prime if and only if it divides the number S(p-1), where S(p-1) is defined recursively: S(1) = 4, and S(n+1) = S(n)2 – 2. The search for more Mersenne primes, and therefore more perfect numbers, continues. It is not known if there are infinitely many or finitely many even perfect numbers. Mersenne primes are scarce, but more continue to be found. There are at least 39 of them, the largest known (as of 2002) is 2213466917 – 1. There is a also a question about odd perfect numbers: Are there any? It has been shown that there are no small odd perfect numbers; it is known that odd numbers with fewer then 300 digits are not perfect. It may well be that there are no odd perfect numbers, but to date there is no proof. Next book: Book X introduction Previous proposition: IX.35 Book IX introduction © 1996 D.E.Joyce Clark University Proposition 1 If there are as many numbers as we please in continued proportion, and the extremes of them are relatively prime, then the numbers are the least of those which have the same ratio with them. Let there be as many numbers as we please, A, B, C, and D, in continued proportion, and let the extremes of them, A and D, be relatively prime. I say that A, B, C, and D are the least of those which have the same ratio with them. If not, let E, F, G, and H be less than A, B, C, and D, and in the same ratio with them. Now, since A, B, C, and D are in the same ratio with E, F, G, and H, and the multitude of the numbers A, B, C, and D equals the multitude of the numbers E, F, G, and H, therefore, ex aequali A is to D as E is to H. VII.14 But A and D are relatively prime, numbers which are relatively prime are also least, and the least numbers measure those which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent. Therefore A measures E, the greater the less, which is impossible. VII.21 VII.20 Therefore E, F, G, and H, which are less than A, B, C, and D, are not in the same ratio with them. Therefore A, B, C, and D are the least of those which have the same ratio with them. Therefore, if there are as many numbers as we please in continued proportion, and the extremes of them are relatively prime, then the numbers are the least of those which have the same ratio with them. Q.E.D. Continued proportions and geometric progressions Euclid doesn't define a continued proportion. In this proposition we consider only continued proportions with a constant ratio, proportions of the form a1:a2 = a2:a3 = a3:a4 = ... = an-1:an. In proposition VIII.4 continued proportions that don't have a constant ratio will be considered. An example of continued proportion with constant ratio is 1250:750 = 750:450 = 450:270 = 270:162 since each of the ratios is the same as the ratio 5:3. A modern expression for this situation is to say that the numbers a1, a2, a3, ..., an-1:an are in a geometric progression or a geometric sequence. The ratio of any consecutive pair in a geometric progression is constant. Many of the propositions in Books VIII and IX treat geometric progressions. The sum of a geometric progression is found in proposition IX.35. About this proposition This proposition says that if the end numbers are relatively prime in a continued proportion with constant ratio, then there is no continued proportion of the same length and same ratio having smaller numbers. We could say that the continued proportion is in lowest terms. The proposition generalizes VII.21 when there are only two numbers in continued proportion, that is, a ratio. VII.21 says if the two numbers are relatively prime, then the ratio is in lowest terms. Notice that the example of a continued proportion given above, 1250:750 = 750:450 = 450:270 = 270:162, is not in lowest terms, since all the numbers may be halved to get continued proportion of the same length and same ratio but with smaller numbers. Use of this proposition This proposition is used in the next one and in VIII.9. The converse of this proposition is VIII.3. Next proposition: VIII.2 Book VIII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 2 To find as many numbers as are prescribed in continued proportion, and the least that are in a given ratio. Let the ratio of A to B be the given ratio in least numbers. It is required to find numbers in continued proportion, as many as may be prescribed, the least that are in the ratio of A to B. Let four be prescribed. Multiply A by itself to make C, and by B to make D. Multiply B by itself to make E. Also multiply A by C, D, and E to make F, G, and H, and multiply B by E to make K. Now, since A multiplied by itself makes C, and multiplied by B makes D, therefore A is to B as C is to D. VII.17 Again, since A multiplied by B makes D, and B multiplied by itself makes E, therefore the numbers A and B multiplied by B make the numbers D and E respectively. Therefore A is to B as D is to E. But A is to B as C is to D, therefore C is to D as D is to E. VII.18 And, since A multiplied by C and D makes F and G, therefore C is to D as F is to G. VII.17 But C is to D as A is to B, therefore A is to B as F is to G. Again, since A multiplied by D and E makes G and H, therefore D is to E as G is to H. But D is to E as A is to B. Therefore A is to B as G is to H. VII.17 And, since A and B multiplied by E make H and K, therefore A is to B as H is to K. But A is to B as F is to G, and as G is to H. Therefore F is to G as G is to H, and as H is to K. VII.18 Therefore C, D, and E, and F, G, H, and K are proportional in the ratio of A to B. I say next that they are the least numbers that are so. Since A and B are the least of those which have the same ratio with them, and the least of those which have the same ratio are relatively prime, therefore A and B are relatively prime. VII.22 And the numbers A and B multiplied by themselves respectively make the numbers C and E, and multiplied by the numbers C and E respectively make the numbers F and K, therefore C and E and F and K are relatively prime respectively. VII.27 But, if there are as many numbers as we please in continued proportion, and the extremes of them are relatively prime, then they are the least of those which have the same ratio with them. Therefore C, D, and E and F, G, H and K are the least of those which have the same ratio with A and B. VIII.1 Q.E.D. Corollary. From this it is manifest that, if three numbers in continued proportion are the least of those which have the same ratio with them, then the extremes are squares, and, if four numbers, cubes. The problem is to construct n numbers in a continued proportion in lowest terms with a given constant ratio. If the ratio is a:b in lowest terms, then the numbers in the continued proportion are an-1, an-2b, an-3b2, ..., a1bn-2, bn-1 For instance, the five numbers in a continued proportion in lowest terms with a ratio of 2:3 form the sequence 24, 23 . 3, 22 . 32, 2 . 33, and 34, that is, the sequence 16, 24, 36, 54, and 81. The proof is not difficult. First, adjacent terms do have the correct ratio. Also, since a and b are relatively prime, proposition VII.27 implies that the end terms an-1 and bn-1 are relatively prime. The result then follows from the previous proposition VIII.1. Use of this proposition This proposition and its corollary are used in several propositions in Book VIII starting with the next and in proposition IX.15 in the next book. Next proposition: VIII.3 Previous: VIII.1 Book VIII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 3 If as many numbers as we please in continued proportion are the least of those which have the same ratio with them, then the extremes of them are relatively prime. Let as many numbers as we please, A, B, C, and D, in continued proportion be the least of those which have the same ratio with them. I say that the extremes of them, A and D, are relatively prime. Take two numbers E and F, the least that are in the ratio of A, B, C and D, then three others G, H and K with the same property, and others, more by one continually, until the multitude taken becomes equal to the multitude of the numbers A, B, C, and D. Let them be L, M, N, and O. VII.33 VIII.2 Since E and F are the least of those which have the same ratio with them, therefore they are relatively prime. And, since the numbers E and F multiplied by themselves respectively make the numbers G and K, and multiplied by the numbers G and K respectively make the numbers L and O, therefore both G and K and L and O are relatively prime. VII.22 VIII.2,Cor VII.27 And, since A, B, C, and D are the least of those which have the same ratio with them, while L, M, N, and O are the least that are in the same ratio with A, B, C, and D, and the multitude of the numbers A, B, C, and D equals the multitude of the numbers L, M, N, and O, therefore the numbers A, B, C, and D equal the numbers L, M, N, and O respectively. Therefore A equals L, and D equals O. And L and O are relatively prime. Therefore A and D are also relatively prime. Therefore, if as many numbers as we please in continued proportion are the least of those which have the same ratio with them, then the extremes of them are relatively prime. Q.E.D. This proposition, the converse of VIII.1, says if a continued proportion with constant ratio is in lowest terms, then its end numbers are relatively prime. The proof begins by using the previous proposition VIII.2 to construct the continued proportion in lowest terms, which must be the same as the given continued proportion, and it has relatively prime end numbers. Use of this proposition This proposition is used in propositions VIII.6, VIII.8, and VIII.21. Next proposition: VIII.4 Previous: VIII.2 Book VIII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 4 Given as many ratios as we please in least numbers, to find numbers in continued proportion which are the least in the given ratios. Let the given ratios in least numbers be that of A to B, that of C to D, and that of E to F. It is required to find numbers in continued proportion which are the least that are in the ratio of A to B, in the ratio of C to D, and in the ratio of E to F. Take G, the least number measured by B and C. VII.34 Let A measure H as many times as B measures G, and let D measure K as many times as C measures G. Now E either measures or does not measure K. First, let it measure it. Let K measure L as many times as E measures K. Now, since A measures H the same number of times that B measures G, therefore A is to B as H is to G. VII.Def.20 VII.13 For the same reason C is to D as G is to K, and E is to F as K is to L. Therefore H, G, K, and L are continuously proportional in the ratio of A to B, in the ratio of C to D, and in the ratio of E to F. I say next that they are also the least that have this property. If H, G, K, and L are not the least numbers continuously proportional in the ratios of A to B, of C to D, and of E to F, let them be N, O, M, and P. Then since A is to B as N is to O, while A and B are least, and the least numbers measure those which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent, therefore B measures O. VII.20 For the same reason C also measures O. Therefore B and C measure O. Therefore the least number measured by B and C also measures O. VII.35 But G is the least number measured by B and C, therefore G measures O, the greater the less, which is impossible. Therefore there are no numbers less than H, G, K, and L which are continuously in the ratio of A to B, of C to D, and of E to F. Next, let E not measure K. Take M, the least number measured by E and K. Let H and G measure N and O as many times as K measures M, respectively, and let F measure P as many times as E measures M. Since H measures N the same number of times that G measures O, therefore H is to G as N is to O. But H is to G as A is to B, therefore A is to B as is N is to O. For the same reason C is to D as is O is to M. VII.13 VII.Def.20 Again, since E measures M the same number of times that F measures P, therefore E is to F as M is to P. Therefore N, O, M, and P are continuously proportional in the ratios of A to B, of C to D, and of E to F. VII.13 VII.Def.20 I say next that they are also the least that are in the ratios A, B, C, D, E, and F. If not, there are numbers less than N, O, M, and P continuously proportional in the ratios A, B, C, D, E, and F. Let them be Q, R, S, and T. Now since Q is to R as A is to B, while A and B are least, and the least numbers measure those which have the same ratio with them the same number of times, the antecedent the antecedent and the consequent the consequent, therefore B measures R. For the same reason C also measures R, therefore B and C measure R. VII.20 Therefore the least number measured by B and C also measures R. But G is the least number measured by B and C, therefore G measures R. VII.35 And G is to R as K is to S, therefore K also measures S. VII.13 But E also measures S. Therefore E and K measure S. Therefore the least number measured by E and K also measures S. But M is the least number measured by E and K, therefore M measures S, the greater the less, which is impossible. VII.35 Therefore there are no numbers less than N, O, M, and P continuously proportional in the ratios of A to B, of C to D, and of E to F. Therefore N, O, M, and P are the least numbers continuously proportional in the ratios A, B, C, D, E, and F. Q.E.D. This is a generalization of VIII.2 to a more general concept of continued proportion. In this proposition we consider continued proportions having ratios that aren't necessarily constant. These, perhaps, should be called continued ratios. For example, the continued ratio 5:10:20 has a constant ratio of 1:2, but the continued ratio 5:10:30 does not; its first ratio is 1:2 while its second ratio is 1:3. Note that the continued ratio 5:10:30 is not the least with those given ratios since 1:2:6 is smaller with the same ratios of 1:2 and 1:3. The problem here is to construct the smallest continued ratio having the specified ratios. An examination of the proof shows that Euclid has a general process to attach two continued proportions into one long one with with the same ratios. Take, for example, the problem of placing the continued ratio 3:7:2:6 in front of the continued ratio 10:4:5 to make a seven-term continued ratio where the first four terms have the ratio 3:7:2:6 and the last three terms have the ratio 10:4:5. The resulting seven-term ratio should be least with the given ratios. The problem is that the last term of the first ratio, 6, does not equal the first term of the second ratio, 10. The solution is to increase the numbers in each ratio to match these numbers. Since 30 = LCM(6, 10), that can be done by multiplying each of the terms of the first ratio by 5 and each of the terms of the second ratio by 3. The resulting ratios, 15:35:10:30 and 30:12:15 can then be merged into the desired ratio 15:35:10:30:12:15. Outline of the proof We start with three given ratios a:b, c:d, and e:f, all in lowest terms. First, the two ratios a:b and c:d are merged into a three-term ratio h:g:k so that h:g = a:b and g:k = c:d. These are defined equationally as g = LCM(b, c), h = (g/b)a, and k = (g/c)d. Then h:g:k has the proper ratios. Next, the three-term ratio h:g:k is merged with the ratio e:f to get a four-term ratio n:o:m:p so that n:o = a:b, o:m = c:d, and m:p = c:d. These are defined equationally as m = LCM(e, k), n = (m/k)h, and o = (m/k)g, and p = (m/e)f. Again, n:o:m:p has the proper ratios. The bulk of the proof consists of showing that the resulting ratio n:o:m:p is the least with the given ratios. Use of this proposition This proposition is used in the next one and in proposition X.12. Next proposition: VIII.5 Previous: VIII.3 Book VIII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 5 Plane numbers have to one another the ratio compounded of the ratios of their sides. Let A and B be plane numbers, and let the numbers C and D be the sides of A, and E and F the sides of B. I say that A has to B the ratio compounded of the ratios of the sides. The ratios being given which C has to E and D to F, take the least numbers G, H, and K that are continuously in the ratios C, E, D, and F, so that C is to E as G is to H, and D is to F as H is to K. VIII.4 Multiply D by E to make L. Now, since D multiplied by C makes A, and multiplied by E makes L, therefore C is to E as A is to L. But C is to E as G is to H, therefore G is to H as A is to L. VII.17 Again, since E multiplied by D makes L, and further multiplied by F makes B, therefore D is to F as L is to B. But D is to F as H is to K, therefore H is to K as L is to B. VII.17 But it was also proved that, H as G is to H as A is to L, therefore, ex aequali, L as G is to K as A is to B. VII.14 But G has to K the ratio compounded of the ratios of the sides, therefore A also has to B the ratio compounded of the ratios of the sides. Therefore, plane numbers have to one another the ratio compounded of the ratios of their sides. Q.E.D. Compounded ratios Compound ratios as such only appear in a few places in the Elements. They appear here in this proposition, and in VI.23, an analogous proposition for rectangles and parallelograms. But duplicate and triplicate ratios are also special kinds of compound ratios, and they are used in Books VI, VIII, IX, X, XI, and XII. Duplicate and triplicate ratios were defined in general in V.9-10, where they are defined as the ratio of the ends of a continued proportion. That is, if a:b = b:c, then the duplicate ratio of a:b is a:c. For lines, constructing the third proportional c needed to duplicate the ratio a:b is done in proposition VI.11, but for numbers, the third proportional can be constructed by VIII.2. The ratio compounded from two given ratios a:b and b:c is just the ratio a:c. But if the middle term b is not shared by the two given ratios, then equal ratios must be found that do have a shared middle term. To find the ratio compounded from two given ratios a:b and c:d, first find e, f, and g so that e:f = a:b and f:g = c:d. Then, the ratio compounded from the ratios a:b and c:d will be the same as the ratio compounded from the ratios e:f and f:g, namely e:g. For numbers, this construction was done in the previous proposition VIII.4. Outline of the proof Let the plane number a be the product cd of its sides, and let the plane number b be the product ef of its sides. Use VIII.4 to construct a continued ratio g:h:k so that g:h = c:e and h:k = d:f so that g:k is the ratio compounded of the ratios c:e and d:f of the sides. Since a = cd, therefore c:e = a:de, and so g:h = a:de. Since b = ef, therefore d:f = de:b, and so h:k = de:b. From the two proportions g:h = a:de and h:k = de:b therefore, ex aequali, g:k = a:b. Thus, ratio the plane numbers is the ratio compounded of the ratios of their sides. The application of VIII.4 to find the least numbers continuously in the ratios c:d and e:f actually makes the proof more difficult. Here's a slightly shorter proof. Since c:e = cd:de = a:de, and d:f = de:ef = de:b, therefore, the ratio compounded from the ratios c:e and d:f of the sides is the ratio of the plane numbers a:b. Next proposition: VIII.6 Previous: VIII.4 Book VIII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 6 If there are as many numbers as we please in continued proportion, and the first does not measure the second, then neither does any other measure any other. Let there be as many numbers as we please, A, B, C, D, and E, in continued proportion, and let A not measure B. I say that neither does any other measure any [later] other. Now it is manifest that A, B, C, D, and E do not measure one another in order, for A does not even measure B. I say, then, that neither does any other measure any [later] other. If possible, let A measure C. And, however many A, B, and C are, take as many numbers F, G, and H, the least of those which have the same ratio with A, B, and C. VII.33 Now, since F, G, and H are in the same ratio with A, B, and C, and the multitude of the numbers A, B, and C equals the multitude of the numbers F, G, and H, therefore, ex aequali A is to C as F is to H. VII.14 And since A is to B as F is to G, while A does not measure B, therefore neither does F measure G. Therefore F is not a unit, for the unit measures any number. VII.Def.20 Now F and H are relatively prime. And F is to H as A is to C, therefore neither does A measure C. VIII.3 Similarly we can prove that neither does any other measure any other. Therefore, if there are as many numbers as we please in continued proportion, and the first does not measure the second, then neither does any other measure any other. Q.E.D. The proposition as stated isn't quite correct. For example, the numbers 24, 12, 6 and 3 are in continued proportion, and 24 does not divide 12, but each of the others does divide others, for instance, 3 divides 6. But none of the others divide others later in the sequence. Outline of the proof Consider a sequence of numbers in continued proportion where the first number does not divide the second. Since any number in that sequence has to the next number in the sequence the same ratio as the first has to the second, therefore no number divides the next. Suppose that some number in the sequence divides a later number. We may call that the former number a since it divides the next number in the sequence, and call the number it divides c. Take the continued proportion a, b, ..., c and, using VII.33, reduce it a continued proportion f, g, ..., h in lowest terms. Since that's in lowest terms, f and h are relatively prime. Since a:b f:g, and a does not divide b, therefore f does not divide g. Since f does not divide g, in particular f does not equal 1, but f and h are relatively prime by VIII.3, therefore f does not divide h. Finally, since a:c f:h, therefore a does not divide c either. Use of this proposition This proposition is used as a lemma for the following proposition. Next proposition: VIII.7 Previous: VIII.5 Book VIII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 7 If there are as many numbers as we please in continued proportion, and the first measures the last, then it also measures the second. Let there be as many numbers as we please, A, B, C, and D, in continued proportion, and let A measure D. I say that A also measures B. If A does not measure B, neither does any other of the numbers measure any other. But A measures D. Therefore A also measures B. VIII.6 Therefore, if there are as many numbers as we please in continued proportion, and the first measures the last, then it also measures the second. Q.E.D. This proposition is the contrapositive of the previous theorem. Use of this theorem This proposition is used in propositions VIII.14 and VIII.15. Next proposition: VIII.8 Previous: VIII.6 Book VIII introduction © 1996 D.E.Joyce Clark University Proposition 8 If between two numbers there fall numbers in continued proportion with them, then, however many numbers fall between them in continued proportion, so many also fall in continued proportion between the numbers which have the same ratios with the original numbers. Let the numbers C and D fall between the two numbers A and B in continued proportion with them, and make E in the same ratio to F as A is to B. I say that, as many numbers as have fallen between A and B in continued proportion, so many also fall between E and F in continued proportion. As many as A, B, C, and D are in multitude, take so many numbers G, H, K, and L, the least of those which have the same ratio with A, C, D, and B. Then the extremes of them G and L are relatively prime. VII.33 VIII.3 Now, since A, C, D, and B are in the same ratio with G, H, K, and L, and the multitude of the numbers A, C, D, and B equals the multitude of the numbers G, H, K, and L, therefore, ex aequali A is to B as G is to L. VII.14 But A is to B as E is to F, therefore G is to L as E is to F. (V.11) But G and L are relatively prime, numbers which are relatively prime are also least, and the least numbers measure those which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent. VII.21 VII.20 Therefore G measures E the same number of times as L measures F. Next, let H and K measure M and N, respectively, as many times as G measures E. Then G, H, K, and L measure E, M, N, and F the same number of times. Therefore G, H, K, and L are in the same ratio with E, M, N, and F. VII.Def.20 But G, H, K, and L are in the same ratio with A, C, D, and B, therefore A, C, D, and B are also in the same ratio with E, M, N, and F. But A, C, D, and B are in continued proportion, therefore E, M, N, and F are also in continued proportion. Therefore, as many numbers as have fallen between A and B in continued proportion with them, so many numbers have also fallen between E and F in continued proportion. Therefore, if between two numbers there fall numbers in continued proportion with them, then, however many numbers fall between them in continued proportion, so many also fall in continued proportion between the numbers which have the same ratios with the original numbers. Q.E.D. This proposition implies, among other things, that there is no number which forms a mean proportional between a number n and the number 2n, for if there were, there would be a number m so that 2, m, and 4 would form a continued proportion, but the only number between 2 and 4 is 3, and 2, 3, and 4 do not form a continued proportion. (If 1 is considered to be a number, the argument simplifies.) In modern terminology, this conclusion says the square root of 2 is not a rational number. See proposition X.9 for implications of this conclusion for imcommensurability of line segments. Outline of the proof Suppose that a:b = e:f and the sequence a, c, d, ..., b are in continued proportion. Use VII.33 to reduce that sequence to lowest terms g, h, k, ..., l. According to VIII.3, the ends of that sequence g and l are relatively prime. Since a:b = e:f, we also have g:l = e:f. But g is relatively prime to l, so the ratio g:l is in lowest terms (VII.21), and therefore g divides e the same number of times, say n, that l divides f (VII.20). Then the sequence ng, nh, nk, ..., nl is in continued proportion and starts with e and ends with f as required. Use of this proposition Although this proposition is not used in Book VIII, it is used in the first six propositions of Book IX. Next proposition: VIII.9 Previous: VIII.7 Book VIII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 9 If two numbers are relatively prime, and numbers fall between them in continued proportion, then, however many numbers fall between them in continued proportion, so many also fall between each of them and a unit in continued proportion. Let A and B be two numbers relatively prime, and let C and D fall between them in continued proportion, and let the unit E be set out. I say that, as many numbers fall between A and B in continued proportion as fall between either of the numbers A or B and the unit in continued proportion. Take two numbers F and G, the least that are in the ratio of A, C, D, and B, three numbers H, K, and L with the same property, and others more by one continually, until their multitude equals the multitude of A, C, D, and B. Let them be M, N, O, and P. VIII.2 It is now manifest that F multiplied by itself makes H and multiplied by H makes M, while G multiplied by itself makes L and multiplied by L makes P. VIII.2,Cor And, since M, N, O, and P are the least of those which have the same ratio with F and G, and A, C, D, and B are also the least of those which have the same ratio with F and G, while the multitude of the numbers M, N, O, and P equals the multitude of the numbers A, C, D, and B, therefore M, N, O, and P equal A, C, D, and B respectively. Therefore M equals A, and P equals B. VIII.1 Now, since F multiplied by itself makes H, therefore F measures H according to the units in F. But the unit E also measures F according to the units in it, therefore the unit E measures the number F the same number of times as F measures H. Therefore the unit E is to the number F as F is to H. VII.Def.20 Again, since F multiplied by H makes M, therefore H measures M according to the units in F. But the unit E also measures the number F according to the units in it, therefore the unit E measures the number F the same number of times as H measures M. Therefore the unit E is to the number F as H is to M. But it was also proved that the unit E is to the number F as F is to H, therefore the unit E is to the number F as F is to H, and as H is to M. But M equals A, therefore the unit E is to the number F as F is to H, and as H is to A. For the same reason also the unit E is to the number G as G is to L and as L is to B. Therefore as many numbers fall between A and B in continued proportion as fall between each of the numbers A and B and the unit E in continued proportion. Therefore, If two numbers are relatively prime, and numbers fall between them in continued proportion, then, however many numbers fall between them in continued proportion, so many also fall between each of them and a unit in continued proportion. Q.E.D. Suppose that relatively prime numbers a and b are the ends of a continued proportion with n terms, and that f / g is the ratio for the continued proportion in lowest terms. Then Euclid shows that a is the n–1st power of f, and a is the n–1st power of g. The argument is that the sequence fn-1, fn-2g, fn-3g,2, ..., fgn-2, gn-1 is in continued proportion with the correct ratio with relatively prime ends, so by VIII.1 they're the same sequence. Next proposition: VIII.10 Previous: VIII.8 Book VIII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 10 If numbers fall between two numbers and a unit in continued proportion, then however many numbers fall between each of them and a unit in continued proportion, so many also fall between the numbers themselves in continued proportion. Let the numbers D and E and the numbers F and G respectively fall between the two numbers A and B and the unit C in continued proportion. I say that, as many numbers have fallen between each of the numbers A and B and the unit C in continued proportion as fall between A and B in continued proportion. Multiply D by F to make H, and multiply the numbers D and F by H to make K and L respectively. Now, since the unit C is to the number D as D is to E, therefore the unit C measures the number D the same number of times as D measures E. But the unit C measures the number D according to the units in D, therefore the number D also measures E according to the units in D. Therefore D multiplied by itself makes E. VII.Def.20 Again, since C is to the number D as E is to A, therefore the unit C measures the number D the same number of times as E measures A. But the unit C measures the number D according to the units in D, therefore E also measures A according to the units in D. Therefore D multiplied by E makes A. For the same reason also F multiplied by itself makes G, and multiplied by G makes B. And, since D multiplied by itself makes E and multiplied by F makes H, therefore D is to F as E is to H. VII.17 For the same reason also D is to F as H is to G. Therefore E is to H as H is to G. VII.18 Again, since D multiplied by the numbers E and H makes A and K respectively, therefore E is to H as A is to K. But E is to H as D is to F, therefore D is to F as A is to K. VII.17 Again, since the numbers D and F multiplied by H make K and L respectively, therefore D is to F as K is to L. But D is to F as A is to K, therefore A is to K as K is to L. Further, since F multiplied by the numbers H and G makes L and B respectively, therefore H is to G as L is to B. VII.18 VII.17 But H is to G as D is to F, therefore D is to F as L is to B. But it was also proved that D is to F as A is to K and as K is to L, therefore A is to K as K is to L and as L is to B. Therefore A, K, L, and B are in continued proportion. Therefore, as many numbers as fall between each of the numbers A and B and the unit C in continued proportion, so many also fall between A and B in continued proportion. Therefore, if numbers fall between two numbers and a unit in continued proportion, then however many numbers fall between each of them and a unit in continued proportion, so many also fall between the numbers themselves in continued proportion. Q.E.D. This is a partial converse to the previous proposition; it doesn't require that the generating numbers d and f be relatively prime in order to conclude that the sequence dn-1, dn-2f, dn-3f,2, ..., dfn-2, fn-1 is in continued proportion. Next proposition: VIII.11 Previous: VIII.9 Book VIII introduction © 1996 D.E.Joyce Clark University Proposition 11 Between two square numbers there is one mean proportional number, and the square has to the square the duplicate ratio of that which the side has to the side. Let A and B be square numbers, and let C be the side of A, and D of B. I say that between A and B there is one mean proportional number, and A has to B the ratio duplicate of that which C has to D. Multiply C by D to make E. Now, since A is a square and C is its side, therefore C multiplied by itself makes A. For the same reason also, D multiplied by itself makes B. Since, then, C multiplied by the numbers C and D makes A and E respectively, therefore C is to D as A is to E. VII.17 For the same reason also C is to D as E is to B. Therefore A is to E as E is to B. Therefore between A and B there is one mean proportional number. VII.18 I say next that A also has to B the ratio duplicate of that which C has to D. Since A, E, and B are three numbers in proportion, therefore A has to B the ratio duplicate of that which A has to E. V.Def.9 But A is to E as C is to D, therefore A has to B the ratio duplicate of that which the side C has to D. Therefore, between two square numbers there is one mean proportional number, and the square has to the square the duplicate ratio of that which the side has to the side. Q.E.D. Between c2 and d2 is the mean proportional cd, and the ratio c2:d2 is the duplicate ratio of c:d. The argument for the latter statement is that c2:d2 is compounded of the two ratios c2:cd and cd:d2, but both of those are the same ratio as c:d. Use of this proposition This proposition is used in propositions VIII.14, VIII.15, and X.9. Next proposition: VIII.12 Previous: VIII.10 Book VIII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 12 Between two cubic numbers there are two mean proportional numbers, and the cube has to the cube the triplicate ratio of that which the side has to the side. Let A and B be cubic numbers, and let C be the side of A, and D of B. I say that between A and B there are two mean proportional numbers, and A has to K the ratio triplicate of that which C has to D. Multiply C by itself to make E, and by D to make F, multiply D by itself to make G, and multiply the numbers C and D by F to make H and K respectively. Now, since A is a cube, and C its side, and C multiplied by itself makes E, therefore C multiplied by itself makes E and multiplied by E makes A. For the same reason also D multiplied by itself makes G and multiplied by G makes B. And, since C multiplied by the numbers C and D makes E and F respectively, therefore C is to D as E is to F. For the same reason also C is to D as F is to G. Again, since C multiplied by the numbers E and F makes A and H respectively, therefore E is to F as A is to H. But E is to F as C is to D. Therefore C is to D as A is to H. VII.17 VII.18 Again, since the numbers C and D multiplied by F make H and K respectively, therefore C is to D as H is to K. Again, since D multiplied by each of the numbers F and G makes K and B respectively, therefore F is to G as K is to B. VII.18 VII.17 But F is to G as C is to D, therefore C is to D as A is to H, as H is to K, and as K is to B. Therefore H and K are two mean proportionals between A and B. I say next that A also has to B the ratio triplicate of that which C has to D. Since A, H, K, and B are four numbers in proportion, therefore A has to B the ratio triplicate of that which A has to H. V.Def.10 But A is to H as C is to D, therefore A also has to B the ratio triplicate of that which C has to D. Therefore, Between two cubic numbers there are two mean proportional numbers, and the cube has to the cube the triplicate ratio of that which the side has to the side. Q.E.D. This proposition is used in VIII.15. Next proposition: VIII.13 Previous: VIII.11 Book VIII introduction © 1996 D.E.Joyce Clark University Proposition 13 If there are as many numbers as we please in continued proportion, and each multiplied by itself makes some number, then the products are proportional; and, if the original numbers multiplied by the products make certain numbers, then the latter are also proportional. Let there be as many numbers as we please, A, B, and C, in continued proportion, so that A is to B as B is to C. Let A, B, and C multiplied by themselves make D, E, and F, and multiplied by D, E, and F let them make G, H, and K. I say that D, E, and F and G, H, and K are in continued proportion. 5 Multiply A by B to make L, and multiply the numbers A and B by L to make M and N respectively. Also multiply B by C to make O, and multiply the numbers B and C by O to make P and Q respectively. Then, in manner similar to the foregoing, we can prove that D, L, and E and G, M, N, and H are continuously proportional in the ratio of A to B, and further E, O, and F and H, P, Q, and K are continuously proportional in the ratio of B to C. Now A is to B as B is to C, therefore D, L, and E are also in the same ratio with E, O, and F, and further G, M, N, and H in the same ratio with H, P, Q, and K. And the multitude of D, L, and E equals the multitude of E, O, and F and that of G, M, N, and H to that of H, P, Q, and K, therefore, ex aequali D is to E as E is to F, and G is to H as H is to K. VII.14 Therefore, if there are as many numbers as we please in continued proportion, and each multiplied by itself makes some number, then the products are proportional; and, if the original numbers multiplied by the products make certain numbers, then the latter are also proportional. Q.E.D. The proposition says that if the terms of a continued proportion are squared or cubed, then the resulting sequences of numbers are also in continued proportion. Suppose that the original continued proportion has three terms: a, b, c. Then form two more sequences a2, ab, b2, bc, c2 and a3, a2b, a b2, b3, b2c, b c2, c3 Each of these are in continued proportion with the same ratio as the original sequence. The alternate terms in the second sequence form the continued proportion of the squares of the original sequence where the ratio is duplicate of the original ratio. Likewise, every third term of the third sequence make up a continued proportion of the cubes of the original sequence where the ratio is triplicate of the original ratio. Next proposition: VIII.14 Previous: VIII.12 Book VIII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 14 If a square measures a square, then the side also measures the side; and, if the side measures the side, then the square also measures the square. Let A and B be square numbers, let C and D be their sides, and let A measure B. I say that C also measures D. Multiply C by D to make E. Then A, E, and B are continuously proportional in the ratio of C to D. as in VIII.11 And, since A, E, and B are continuously proportional, and A measures B, therefore A also measures E. And A is to E as C is to D, therefore C measures D. VIII.7 VII.Def.20 Next, let C measure D. I say that A also measures B. With the same construction, we can in a similar manner prove that A, E, and B are continuously proportional in the ratio of C to D. And since C is to D as A is to E, and C measures D, therefore A also measures E. VII.Def.20 And A, E, and B are continuously proportional, therefore A also measures B. Therefore, if a square measures a square, then the side also measures the side; and, if the side measures the side, then the square also measures the square. Q.E.D. This proposition is to prove its contrapositive, VIII.16. Next proposition: VIII.15 Previous: VIII.13 Book VIII introduction © 1996 D.E.Joyce Clark University Proposition 15 If a cubic number measures a cubic number, then the side also measures the side; and, if the side measures the side, then the cube also measures the cube. Let the cubic number A measure the cube B, and let C be the side of A and D the side of B. I say that C measures D. Multiply C by itself to make E, multiply D by itself to make G, multiply C by D to make F, and multiply C and D by F to make H and K respectively. Now it is manifest that E, F, and G and A, H, K, and B are continuously proportional in the ratio of C to D. And, since A, H, K, and B are continuously proportional, and A measures B and G therefore it also measures H. VIII.11 VIII.12 VIII.7 And A is to H as C is to D, therefore C also measures D. VII.Def.20 Next, let C measure D. I say that A also measures B. With the same construction, we can prove in a similar manner that A, H, K, and B are continuously proportional in the ratio of C to D. And, since C measures D, and C is to D as A is to H, therefore A also measures H, so that A measures B also. VII.Def.20 Therefore, if a cubic number measures a cubic number, then the side also measures the side; and, if the side measures the side, then the cube also measures the cube. Q.E.D. This proposition is used to prove its contrapositive, VIII.17. Next proposition: VIII.16 Previous: VIII.14 Book VIII introduction © 1996 D.E.Joyce Clark University Proposition 16 If a square does not measure a square, then neither does the side measure the side; and, if the side does not measure the side, then neither does the square measure the square. Let A and B be square numbers, and let C and D be their sides, and let A not measure B. I say that neither does C measure D. If C measures D, A also measures B. But A does not measure B, therefore neither does C measure D. VIII.14 Next, let C not measure D. I say that neither does A measure B. If A measures B, then C also measures D. But C does not measure D, therefore neither does A measure B. VIII.14 Therefore, if a square does not measure a square, then neither does the side measure the side; and, if the side does not measure the side, then neither does the square measure the square. Q.E.D. This is simply the contrapositive of VIII.14. It is unclear why papyrus was wasted to state and prove it. Next proposition: VIII.17 Previous: VIII.15 Book VIII introduction © 1996 D.E.Joyce Clark University Proposition 17 If a cubic number does not measure a cubic number, then neither does the side measure the side; and, if the side does not measure the side, then neither does the cube measure the cube. Let the cubic number A not measure the cubic number B, and let C be the side of A, and D of B. I say that C does not measure D. For if C measures D, then A also measures B. But A does not measure B, therefore neither does C measure D. VIII.15 Next, let C not measure D. I say that neither does A measure B. If A measures B, then C also measures D. But C does not measure D, therefore neither does A measure B. VIII.15 Therefore, if a cubic number does not measure a cubic number, then neither does the side measure the side; and, if the side does not measure the side, then neither does the cube measure the cube. Q.E.D. This proposition is simply the contrapositive of VIII.15. "Contrariwise," continued Tweedledee, "if it was so, it would be; but as it isn't, it ain't. That's logic." Next proposition: VIII.18 Previous: VIII.16 Book VIII introduction © 1996 D.E.Joyce Clark University Proposition 18 Between two similar plane numbers there is one mean proportional number, and the plane number has to the plane number the ratio duplicate of that which the corresponding side has to the corresponding side. Let A and B be two similar plane numbers, and let the numbers C and D be the sides of A, and E and F of B. Now, since similar plane numbers are those which have their sides proportional, therefore C is to D as E is to F. VII.Def.21 I say then that between A and B there is one mean proportional number, and A has to B the ratio duplicate of that which C has to E, or as D has to F, that is, of that which the corresponding side has to the corresponding side. Now since C is to D as E is to F, therefore, alternately C is to E as D is to F. VII.13 And, since A is plane, and C and D are its sides, therefore D multiplied by C makes A. For the same reason also E multiplied by F makes B. Multiply D by E to make G. Then, since D multiplied by C makes A, and multiplied by E makes G, therefore C is to E as A is to G. VII.17 But C is to E as D is to F, therefore D is to F as A is to G. Again, since E multiplied by D makes G, and multiplied by F makes B, therefore D is to F as G is to B. VII.17 But it was also proved that D is to F as A is to G, therefore A is to G as G is to B. Therefore A, G, and B are in continued proportion. Therefore between A and B there is one mean proportional number. I say next that A also has to B the ratio duplicate of that which the corresponding side has to the corresponding side, that is, of that which C has to E or D has to F. Since A, G, and B are in continued proportion, A has to B the ratio duplicate of that which it has to G. And A is to G as C is to E, and as D is to F. Therefore A also has to B the ratio duplicate of that which C has to E or D has to F. V.Def.9 Therefore, between two similar plane numbers there is one mean proportional number, and the plane number has to the plane number the ratio duplicate of that which the corresponding side has to the corresponding side. Q.E.D. This proposition generalizes VIII.11 from squares to similar rectangles. Outline of the proof Assume that the similar plane numbers are cd and ef so that c:d = e:f, or, alternately, c:e = d:f. Then c:e = cd:de, and d:f = de:ef, therefore the ratio of the plane numbers cd:ef is compounded of the ratios of the corresponding sides c:e and d:f. Also, since the ratios of the corresponding sides are the same, the ratio of the plane numbers is the duplicate of the ratio of the sides. Furthermore, since cd:de = de:ef, the number de is a mean proportional between the two plane numbers cd and ef. Use of this proposition This proposition is used in several of the remaining propositions in this book beginning with the next. It is also used in the first two propositions of Book IX. Proposition VIII.20 is a partial converse of this one. Next proposition: VIII.19 Previous: VIII.17 Book VIII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 19 Between two similar solid numbers there fall two mean proportional numbers, and the solid number has to the solid number the ratio triplicate of that which the corresponding side has to the corresponding side. Let A and B be two similar solid numbers, and let C, D, and E be the sides of A, and let F, G, and H be the sides of B. Now, since similar solid numbers are those which have their sides proportional, therefore C is to D as F is to G, and D is to E as G is to H. VII.Def.21 I say that between A and B there fall two mean proportional numbers, and A has to B the ratio triplicate of that which C has to F, D has to G, and E has to H. Multiply C by D to make K, and multiply F by G to make L. Now, since C and D are in the same ratio with F and G, and K is the product of C and D, and L the product of F and G, K and L are similar plane numbers, therefore between K and L there is one mean proportional number M. VII.Def.21 VIII.18 Therefore M is the product of D and F was proved in the theorem preceding. VIII.18 Now, since D multiplied by C makes K, and multiplied by F makes M, therefore C is to F as K is to M. But K is to M as M is to L. Therefore K, M, and L are continuously proportional in the ratio of C to F. VII.17 And since C is to D as F is to G, alternately therefore C is to F as D is to G. For the same reason also D is to G as E is to H. VII.13 Therefore K, M, and L are continuously proportional in the ratio of C to F, in the ratio of D to G, and also in the ratio of E to H. Next, multiply E and H by M to make N and O respectively. Now, since A is a solid number, and C, D, and E are its sides, therefore E multiplied by the product of C and D makes A. But the product of C and D is K, therefore E multiplied by K makes A. For the same reason also H multiplied by L makes B. Now, since E multiplied by K makes A, and further also multiplied by M makes N, therefore K is to M as A is to N. VII.17 But K is to M as C is to F, as D is to G, and as E is to H, therefore C is to F as D is to G, as E is to H, and as A is to N. Again, since E and H multiplied by M make N and O respectively, therefore E is to H as N is to O. VII.18 But E is to H as C is to F and as D is to G, therefore C is to F as D is to G, as E is to H, as A is to N, and as N is to O. Again, since H multiplied by M makes O, and further also multiplied by L makes B, therefore M is to L as O is to B. But M is to L as C is to F, a D is to G, and as E is to H. Therefore C is to F as D is to G, and as E is to H, as are O to B, A to N, and N to O. VII.17 Therefore A, N, O, and B are continuously proportional in the aforesaid ratios of the sides. I say that A also has to B the ratio triplicate of that which the corresponding side has to the corresponding side, that is, of the ratio which the number C has to F, or D has to G, and also E has to H. Since A, N, O, and B are four numbers in continued proportion, therefore A has to B the ratio triplicate of that which A has to N. But it was proved that A is to N as C is to F, as D is to G, and as E is to H. V.Def.10 Therefore A also has to B the ratio triplicate of that which the corresponding side has to the corresponding side, that is, of the ratio which the number C has to F, D has to G, and also E has to H. Therefore, between two similar solid numbers there fall two mean proportional numbers, and the solid number has to the solid number the ratio triplicate of that which the corresponding side has to the corresponding side. Q.E.D. Assume cde and fgh are similar solid numbers so that c:d:e = f:g:h, or, expressed alternately, c:f = d:g = e:h. Then the numbers cde, fde, fge, and fgh are in continued proportion giving two mean proportionals between cde and fgh. Also, the ratio cde:fgh is compounded of the three equal ratios of the sides c:f, d:g, and e:h, so it is a triplicate ratio of each. Use of this proposition This proposition is used in a few propositions in Books VIII and IX starting with VIII.25. Proposition VIII.21 is a partial converse of this one. Next proposition: VIII.20 Previous: VIII.18 Book VIII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 20 If one mean proportional number falls between two numbers, then the numbers are similar plane numbers. Let one mean proportional number C fall between the two numbers A and B. I say that A and B are similar plane numbers. Take D and E, the least numbers of those which have the same ratio with A and C. Then D measures A the same number of times that E measures C. VII.33 VII.20 Let there be as many units in F as times that D measures A. Then F multiplied by D makes A, so that A is plane, and D and F are its sides. Again, since D and E are the least of the numbers which have the same ratio with C and B, therefore D measures C the same number of times that E measures B. VII.20 Let there be as many units in G as times that E measures B. Then E measures B according to the units in G. Therefore G multiplied by E makes B. Therefore B is plane, and E and G are its sides. Therefore A and B are plane numbers. I say next that they are also similar. Since F multiplied by D makes A, and multiplied by E makes C, therefore D is to E as A is to C, that is, C to B. VII.17 Again, since E multiplied by F and G makes C and B respectively, therefore F is to G as C is to B. But C is to B as D is to E, therefore D is to E as F is to G. And alternately D is to F as E is to G. VII.17 VII.13 Therefore A and B are similar plane numbers, for their sides are proportional. Therefore, if one mean proportional number falls between two numbers, then the numbers are similar plane numbers. Q.E.D. This is a partial converse of VIII.18. It says that if two numbers have a mean proportional, then they can be viewed as two similar plane numbers. An example might clarify the details. The variable refer to the outline of the proof below. The numbers a =18 and b =50 have a mean proportional c = 30. When a:c is to lowest terms, the result is d:e = 3:5. Then f is 6, and the number a = 18 is seen as the plane number d = 3 by f = 6. Also g is 10, and the number b = 50 is seen as the plane number e = 5 by g = 10. The sides of these plane numbers, 3 by 6 and 5 by 10, are proportional. Outline of the proof Suppose two numbers a and b have a mean proportional c. Reduce the ratio a:c to lowest terms d:e. Then d divides a the same number of times e divides c; call that number f. Then a is a plane number with sides d and f. Now since, c:b is the same ratio as a:c, it also reduces to the ratio d:e in lowest terms. Therefore, d divides c the same number of times that e divides b; call that number g. Then b is a plane number with sides e and g. Furthermore, the two plane numbers a and b are similar since we can show their sides are proportional as follows. From the three proportions d:e = a:c (which follows from a = fd and c = fe), a:c = c:b (since c is a mean proportional), and c:b = f:g (which follows from g = ef and b = ec), therefore, d:e = f:g, and alternately, d:f = e:g. Thus, the two plane numbers have proportional sides. Use of this proposition This proposition is used in the next two propositions and also IX.2. Next proposition: VIII.21 Previous: VIII.19 Book VIII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 21 If two mean proportional numbers fall between two numbers, then the numbers are similar solid numbers. Let two mean proportional numbers C and D fall between the two numbers A and B. I say that A and B are similar solid numbers. Take three numbers E, F, and G, the least of those which have the same ratio with A, C, and D. Then the extremes of them E and G are relatively prime. VII.33 or VIII.2 VIII.3 Now, since one mean proportional number F has fallen between E and G, therefore E and G are similar plane numbers. VIII.20 Let, then, H and K be the sides of E, and L and M the sides of G. Therefore it is manifest from the theorem before this that E, F, and G are continuously proportional in the ratio of H to L, and that of K to M. Now, since E, F, and G are the least of the numbers which have the same ratio with A, C, and D, and the multitude of the numbers E, F, and G equals the multitude of the numbers A, C, and D, therefore, ex aequali E is to G as A is to D. VII.14 But E and G are relatively prime, primes are also least, and the least measure those which have the same ratio with them the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent, therefore E measures A the same number of times that G measures D. VII.21 VII.20 Let there be as many units in N as times that E measures A. Then N multiplied by E makes A. But E is the product of H and K, therefore N multiplied by the product of H and K makes A. Therefore A is solid, and H, K, and N are its sides. Again, since E, F, and G are the least of the numbers which have the same ratio as C, D, and B, therefore E measures C the same number of times that G measures B. Let there be as many units in O as times that E measures C. Then G measures B according to the units in O, therefore O multiplied by G makes B. But G is the product of L and M, therefore O multiplied by the product of L and M makes B. Therefore B is solid, and L, M, and O are its sides. Therefore A and B are solid. I say that they are also similar. Since N and O multiplied by E make A and C, therefore N is to O as A is to C, that is, E to F. VII.18 But E is to F as H is to L, and as K is to M, therefore H is to L as K is to M, and as N is to O. And H, K, and N are the sides of A, and O, L, and M the sides of B. Therefore A and B are similar solid numbers. Therefore, if two mean proportional numbers fall between two numbers, then the numbers are similar solid numbers. Q.E.D. This is a partial converse of VIII.19. It says that if two numbers have two mean proportionals, then they can be viewed as two similar solid numbers. It's proof is analogous the previous proposition dealing with plane numbers, but naturally, it is longer and more involved. Use of this proposition This proposition is used in VIII.23. Next proposition: VIII.22 Previous: VIII.20 Book VIII introduction © 1996 D.E.Joyce Clark University Proposition 22 If three numbers are in continued proportion, and the first is square, then the third is also square. Let A, B, and C be three numbers in continued proportion, and let A the first be square. I say that C the third is also square. Since between A and C there is one mean proportional number, B, therefore A and C are similar plane numbers. But A is square, therefore C is also square. VIII.20 Therefore, if three numbers are in continued proportion, and the first is square, then the third is also square. Q.E.D. This proposition is used in a few propositions in this and the next book starting with VIII.24. Next proposition: VIII.23 Previous: VIII.21 Book VIII introduction © 1996 D.E.Joyce Clark University Proposition 23 If four numbers are in continued proportion, and the first is a cube, then the fourth is also a cube. Let A, B, C, and D be four numbers in continued proportion, and let A be a cube. I say that D is also a cube. Since between A and D there are two mean proportional numbers B and C, therefore A and D are similar solid numbers. But A is a cube, therefore D is also a cube. VIII.21 Therefore, if four numbers are in continued proportion, and the first is a cube, then the fourth is also a cube. Q.E.D. This proposition is used in several propositions in this and the next book starting with VIII.25. Next proposition: VIII.24 Previous: VIII.22 Book VIII introduction © 1996 D.E.Joyce Clark University Proposition 24 If two numbers have to one another the ratio which a square number has to a square number, and the first is square, then the second is also a square. Let the two numbers A and B have to one another the ratio which the square number C has to the square number D, and let A be square. I say that B is also square. Since C and D are square, C and D are similar plane numbers. Therefore one mean proportional number falls between C and D. VIII.18 And C is to D as A is to B, therefore one mean proportional number falls between A and B also. And A is square, therefore B is also square. VIII.18 VIII.22 Therefore, if two numbers have to one another the ratio which a square number has to a square number, and the first is square, then the second is also a square. Q.E.D. The proof of this proposition is straightforeward. Next proposition: VIII.25 Previous: VIII.23 Book VIII introduction © 1996 D.E.Joyce Clark University Proposition 25 If two numbers have to one another the ratio which a cubic number has to a cubic number, and the first is a cube, then the second is also a cube. Let the two numbers A and B have to one another the ratio which the cubic number C has to the cubic number D, and let A be a cube. I say that B is also a cube. Since C and D are cubes, C and D are similar solid numbers, therefore two mean proportional numbers fall between C and D. VIII.19 Since as many numbers fall in continued proportion between those which have the same ratio with C and D as fall between C and D, therefore two mean proportional numbers E and F fall between A and B. VIII.18 Since, then, the four numbers A, E, F, and B are in continued proportion, and A is a cube, therefore B is also a cube. VIII.23 Therefore, if two numbers have to one another the ratio which a cubic number has to a cubic number, and the first is a cube, then the second is also a cube. Q.E.D. This proposition is analogous to the previous one about squares. Its proof is straightforward. This proposition is used in IX.10. Next proposition: VIII.26 Previous: VIII.24 Book VIII introduction © 1996 D.E.Joyce Clark University Proposition 26 Similar plane numbers have to one another the ratio which a square number has to a square number. Let A and B be similar plane numbers. I say that A has to B the ratio which a square number has to a square number. Since A and B are similar plane numbers, therefore one mean proportional number C falls between A and B. VIII.18 Take D, E, and F, the least numbers of those which have the same ratio with A, C, and B VII.33 or VIII.2 Then the extremes of them D and F are square. And since D is to F as A is to B, and D and F are square, therefore A has to B the ratio which a square number has to a square number. VIII.2,Cor Therefore, similar plane numbers have to one another the ratio which a square number has to a square number. Q.E.D. This proposition is used in propositions IX.10 and X.9. Next proposition: VIII.27 Previous: VIII.25 Book VIII introduction © 1996 D.E.Joyce Clark University Proposition 27 Similar solid numbers have to one another the ratio which a cubic number has to a cubic number. Let A and B be similar solid numbers. I say that A has to B the ratio which cubic number has to cubic number. Since A and B are similar solid numbers, therefore two mean proportional numbers C and D fall between A and B. VIII.19 Take E, F, G, and H, the least numbers of those which have the same ratio with A, C, D, and B, and equal with them in multitude. VII.33 or VIII.2 Therefore the extremes of them, E and H, are cubes. And E is to H as A is to B, therefore A also has to B the ratio which a cubic number has to a cubic number. VIII.2,Cor. Therefore, similar solid numbers have to one another the ratio which a cubic number has to a cubic number. Q.E.D. This proposition is analogous to the previous proposition about similar plane numbers. Next book: Book IX Previous proposition: VIII.26 Book VIII introduction © 1996 D.E.Joyce Clark University Definitions 1 and 2 Def. 1. A unit is that by virtue of which each of the things that exist is called one. Def. 2. A number is a multitude composed of units. These 23 definitions at the beginning of Book VII are the definitions for all three books VII through IX on number theory. Some won't be used until Books VIII or IX. These first two definitions are not very constructive towards a theory of numbers. The numbers in definition 2 are meant to be whole positive numbers greater than 1, and definition 1 is meant to define the unit as 1. The word "monad," derived directly from the Greek, is sometimes used instead of "unit." Euclid treats the unit, 1, separately from numbers, 2, 3, and so forth. This makes his proofs awkward in some cases. Chrysippus (280–207), a Stoic philosopher, claimed that 1 is a number, but his pronouncement was not accepted for some time. Throughout these three books on number theory Euclid exhibits numbers as lines. In the diagram above, if A is the unit, then BE is the number 3. But, just because he draws them as lines does not mean they are lines, and he never calls them lines. It is not clear what the nature of these numbers is supposed to be. But their nature is irrelevant. Euclid could illustrate the unit as a line or as any other magnitude, and numbers would then be illustrated as multiples of that unit. There is a major distinction between lines and numbers. Lines are infinitely divisible, but numbers are not, in particular, the unit is not divisible into smaller numbers. Euclid has no postulates to elaborate the concept of number (other than the Common Notions which are meant to apply to numbers as well as magnitudes of various kinds). Indeed, mathematicians did not develop foundations for number theory until the late nineteenth century. Peano's axioms for numbers are the best known. The most important of Peano's axioms is the principle of mathematical induction which states that 1. if a property of numbers holds for 1, 2. and whenever property holds for n then it also holds for n + 1, 3. then the property holds for all numbers. Euclid does not use the principle of mathematical induction, but he does implicitly use a similar property of numbers, namely, that any decreasing sequence of numbers is finite. That property is known variously as the "well-ordering principle" for numbers and the "descending chain condition." We will discuss it later in more detail. Next definitions: VII.3-5 Book VII introduction © 1997, 2003 D.E.Joyce Clark University Definitions 3 through 5 Def. 3. A number is a part of a number, the less of the greater, when it measures the greater; Def. 4. But parts when it does not measure it. Def. 5. The greater number is a multiple of the less when it is measured by the less. These definitions are in preparation for the definition of proportion of numbers given in VII.Def.20. In the current definitions, the possible relations between a pair of numbers, m and n, are classified. Later in Book VII, the term "ratio" will be used for this relation. In all three of these definitions, the concept of "measures" is assumed to be understood. There is more to these definitions than meets the eye, though, at least part of the intent is evident. To illustrate VII.Def.3, take 2, which is a part of 6, namely, the one-third part of 6. If u is the unit, then 2 is represented as AB while 6 is represented by CF. As AB measures CF three times by CD, DE, and DF, therefore 2 is a part of 6, namely, the one-third part since it measures 6 three times. We can also use the same figure as an illustration of VII.Def.5 to see that 6 is a multiple of 2, in particular, the third multiple of 2. Definition VII.Def.4 is less clear, but its intent can be read from the use to which it's put in VII.Def.20 for proportions of numbers. For an example, consider the numbers 4 and 6. The number 4 does not measure the number 6, but it is parts of 6. Here, 4 is represented as AC while 6 is represented as DG. Clearly, AC does not measure DG. The way this definition is used in VII.Def.4, just the knowledge that "4 is parts of 6" is not enough, what is also needed is how many parts of 6 is 4. This will be needed to define a proportion such as 4:6 = 6:9. That proportion is supposed to hold since 4 is the same parts of 6 as 6 is of 9, namely 2 third parts. Thus, one number being parts of another also carries along with it how many of what parts. There is one more difficulty with this definition. It seems obvious that when one number m is less than another n, then in all cases m would be parts of n, namely m consists of m one-nth parts of n. Yet, the proposition VII.4 has a proof to show that m is either a part or parts of n. Divisors Where Euclid would say that m is a part of n, modern mathematicians would say that m is a proper divisor of n. A divisor of n is any whole number m (including 1) that divides n in the sense that there is another number k such that mk = n. A proper divisor of n is any divisor except n itself. For example, the proper divisors of the number 12 are 1, 2, 3, 4, and 6. Next definitions: VII.Def.6-10 Previous: VII.Def.1-2 Book VII introduction © 1997, 2002 D.E.Joyce Clark University Euclid's Elements Book VII Definitions 6 through 10 Def. 6. An even number is that which is divisible into two equal parts. Def. 7. An odd number is that which is not divisible into two equal parts, or that which differs by a unit from an even number. Def. 8. An even-times even number is that which is measured by an even number according to an even number. Def. 9. An even-times odd number is that which is measured by an even number according to an odd number. Def. 10. An odd-times odd number is that which is measured by an odd number according to an odd number. Definition 6 for "even number" is clear: the number n is even if it is of the form m + m. Definition 7 for "odd number" has two statements. The first can be taken as a definition of odd number, a number which is not divisible into two equal parts, that is to say not an even number. The other statement is not a definition for odd number, since one has already been given, but an unproved statement. It is easy to recognize that something has to be proved, since if we make the analogous definitions for another number, say 10, then analogous statement is false. Suppose we say a "decade number" is one divisible by 10, and and "undecade number' is one not divisible by 10. Then it is not the case that an undecade number differs by a unit from a decade number; the number 13, for instance, is not within 1 of a decade number. The unproved statement that a number differing from an even number by 1 is an odd number ought to be proved. That statment is used in proposition IX.22 and several propositions that follow it. It could be proved using, for instance, a principle that any decreasing sequence of numbers is finite. Definitions 8-10 are also clear. A product of two even numbers is an even-times even number; a product of an even and an odd number is an even-times odd number; and a product of of two odd numbers is an odd-times odd number. Note that a number like 12 is both even-times even and eventimes odd being at the same time 2 times 6 and 4 times 3. The numbers which are even-times even but not even-times odd are just the powers of 2: 4, 8, 16, 32, etc. These are the numbers which are even-times even only, and they occur in proposition IX.32. Next definitions: VII.Def.11-14 Previous: VII.Def.3-5 Book VII introduction © 1997, 2002. D.E.Joyce Clark University Definitions 11 through 14 Def. 11. A prime number is that which is measured by a unit alone. Def. 12. Numbers relatively prime are those which are measured by a unit alone as a common measure. Def. 13. A composite number is that which is measured by some number. Def. 14. Numbers relatively composite are those which are measured by some number as a common measure. Prime numbers form a very important class of numbers, and much of number theory is devoted to their analysis. The only proper divisor of a prime number is 1. The first few prime numbers are, of course, 2, 3, 5, 7, 11. Those numbers that aren't prime are composite, for instance, 4, 6, 8, 9, 10. The number 1 holds a special position. For Euclid, it was the unit rather than a number. For modern mathematicians 1 is also a unit, but in a different sense of the word, since it has a reciprocal, namely, itself. Numbers are relatively prime if their only common divisor is 1. For example, 6 and 35 are relatively prime (although neither is a prime number in itself). This situation is also phrased as "6 is prime to 35." For another example, the three numbers 6, 10, and 15 are relatively prime since no number (except 1) divides all three. If the numbers aren't relatively prime, then they're called "relatively composite," a term rarely used now. Next definitions: VII.Def.15-19 Previous: VII.Def.6-10 Book VII introduction © 1997 D.E.Joyce Clark University Definitions 15 through 19 Def. 15. A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other. Def. 16. And, when two numbers having multiplied one another make some number, the number so produced be called plane, and its sides are the numbers which have multiplied one another. Def. 17. And, when three numbers having multiplied one another make some number, the number so produced be called solid, and its sides are the numbers which have multiplied one another. Def. 18. A square number is equal multiplied by equal, or a number which is contained by two equal numbers. Def. 19. And a cube is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers. Notice that Euclid doesn't define addition and subtraction. Those operations are assumed to be understood. But multiplication and proportion are defined, and proportion is defined next in VII.Def.20. Definition 15 defines multiplication in terms of addition as a kind of composition. For instance, if 3 is multiplied by 6, then since 6 is 1+1+1+1+1+1, therefore, 3 multiplied by 6 is 3+3+3+3+3+3. The first proposition on multiplication is VII.16 which says multiplication is commutative. For our example, that would say 3 multiplied by 6 equals 6 multiplied by 3, which is 6+6+6. Figurate numbers Although Euclid never displays numbers except as lines, the Pythagoreans before him evidently did, that is, they displayed numbers as figures. The figures were in various shapes, such as triangles, squares, and so forth. Definitions 16 through 19 deal with figurate numbers, but without the figures. Euclid defines a plane number as a number which is the product of two numbers. Remember that for Euclid, 1 is the unit, not a number, so a prime number is not a plane number, even though it is a product of 1 and itself. Plane numbers are the composite numbers. Each composite number can be a plane number in at least one way, but most in more than one way. For instance, 16 can be viewed as a plane number either with sides 2 and 8 or with sides 4 and 4, that is, as a square number. Plane numbers can be displayed as rectangular configurations of dots. Alternatively, these "rectangular numbers" can be displayed as a configuration of squares. But most of the other figurate numbers, such as triangular numbers, could only easily be displayed by dots. Perhaps for the Pythagoreans, the most important figures were the triangular numbers: 3, 6, especially 10, 15, 21, etc. Each could be formed from the previous by adding a new row one unit longer. So, for instance, 10 = 1 + 2 + 3 + 4. For some reason, Euclid doesn't mention triangular numbers. Indeed, he doesn't address sums of arithmetic progressions at all, a subject of interest in many ancient cultures. Euclid does give the sum of a geometric progression, that is, a continued proportion, in proposition IX.35. Definition 18 defines solid numbers. For example, if 18 is presented as 3 times 3 times 2, then it is given as a solid number with three sides 3, 3, and 2. Solid numbers can be represented as a configuration of dots or cubes in three dimensions. Squares and cubes are are described as certain symmetric plane and solid numbers. Of course, some numbers, such as 64, can be simultaneously squares and cubes. Next definitions: VII.Def.20 Previous: VII.Def.11-14 Book VII introduction © 1997 D.E.Joyce Clark University Definition 20 Def. 20. Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth. Definition V.20 for proportionality of numbers is not the same as the definition of proportionality for magnitudes in Book V given in V.Def.5. This definition for numbers was probably the earlier one, but as not all magnitudes are commensurable, it cannot adequately define proportionality for magnitudes. This definition VII.20 is given by cases. The various cases correspond to defintions VII.Def.3 through VII.Def.6 for part, parts, and multiple. When four numbers, j, k, m, and n, are proportional, we'll write that symbolically as j:k = m:n, In the first case, j is the same multiple of k as m is of n. An example of this is the proportion 12:6 = 22:11, where 12 is twice 6 and 22 is twice 11. The second case is inverse to the first, j is the same part of k as m is of n. For an example take the proportion 6:12 = 22:11, where 6 is one half of 12, and 11 is one half of 22. For an example of the third case, consider 12:16 = 21:28. Since the first is the same parts of the second, namely 3 parts of 4, as the the third is of the fourth, the proportion holds. Actually, there should be a fourth case (inverse to the third case) when the second is the same parts of the first as the fourth is of the third, as 16:12 = 28:21. Of course, these cases could be merged into one by considering 1 to be a number and not distinguishing when the first is greater or less than the second. Ratios of numbers Although the word "ratio" doesn't appear in this definition, it appears frequently beginning in proposition VII.14. In book VII ratio is restricted to the use of saying when one ratio is the same as another, that is, there is a proportion as defined in this definition. In Book VIII, duplicate ratios, triplicate ratios, and other compounded ratios appear. Definitions for these concepts are not explicitly given, but once the concept of proportion has been defined, they have the same defintion given in Book V for duplicate and triplicate ratio in V.Def.9-10. Compound ratios aren't defined in Book V, but they can be understood by their use. See the Guide to V.Def.3. Various other definitions that go along with ratios and proportions were given in Book V, for instance, alternate ratios, inverse ratios, taken jointly, taken separately, and ex aequali. These definitions are also not repeated here in Book VII. Very soon in these books on number theory Euclid begins to rely on properties of proportion proved in Book V using the other definition of proportion. That these are valid for proportions of numbers could be verified individually or by showing that the two definitions of proportion are equivalent for numbers. Use of this defintion Proportions of numbers first appear in proposition VII.11, but all the propositions from VII.4 through VII.10 are in preparation for the study of numeric proportions. Next definition: VII.Def.21 Previous: VII.Def.15-19 Book VII introduction © 1997, 2002 D.E.Joyce Clark University Definition 21 Def. 21. Similar plane and solid numbers are those which have their sides proportional. The numbers 18 and 8 are similar plane numbers. When 18 is interpreted as a plane number with sides 6 and 3, and 8 has sides 4 and 2, then the sides are proportional. Proposition VIII.18 shows that the ratio of two similar plane numbers is the duplicate ratio of the corresponding sides. In this example, the ratio 18:8 is duplicate of the ratio 6:4. To illustrate similar solid numbers, consider the two numbers 240 and 810 when represented as 4 times 6 times 10 and 6 times 9 times 15, respectively. Proposition VIII.19 shows that the ratio of two similar solid numbers is the triplicate ratio of the corresponding sides. In this example, the ratio 240:810 is triplicate of the ratio 4:6. Next definition: VII.Def.22 Previous: VII.Def.20 Book VII introduction © 2002 D.E.Joyce Clark University Definition 22 A perfect number is that which is equal to the sum of its own parts. For example, the number 28 is perfect because its parts (that is, proper divisors) 1, 2, 4, 7, and 14 sum to 28. The four smallest perfect numbers were known to the ancient Greek mathematicians. They are 6, 28, 496, and 8128. In proposition IX.36 Euclid gives a construction of even perfect numbers. The divisors of these even perfect numbers can be listed in two columns, illustrated here for the divisors of 496. 1 2 4 8 16 31 62 124 248 (496) The first column lists powers of 2 from 20 up through 24. The sum of these powers of 2 is 31, which is one less than 25. That number 31 appears at the top of the second column, and its repeated doubles up through 496 appear on the second column. In such a tableau, the sum of all the numbers, except the last, will equal the last. The question of odd perfect numbers was not solved by Euclid. Probably the oldest open conjecture in mathematics is that there are no ood perfect numbers. There is no proof yet, but it is known that if there is an odd perfect number, then it has to be immensely huge. Next proposition: VII.1 Previous: VII.Def.21 Book VII introduction © 1997, 2002 D.E.Joyce Clark University Proposition 1 When two unequal numbers are set out, and the less is continually subtracted in turn from the greater, if the number which is left never measures the one before it until a unit is left, then the original numbers are relatively prime. The less of two unequal numbers AB and CD being continually subtracted from the greater, let the number which is left never measure the one before it until a unit is left. I say that AB and CD are relatively prime, that is, that a unit alone measures AB and CD. VII.Def.12 If AB and CD are not relatively prime, then some number E measures them. Let CD, measuring BF, leave FA less than itself, let AF, measuring DG, leave GC less than itself, and let GC, measuring FH, leave a unit HA. Since, then, E measures CD, and CD measures BF, therefore E also measures BF. But it also measures the whole BA, therefore it measures the remainder AF. But AF measures DG, therefore E also measures DG. But it also measures the whole DC, therefore it also measures the remainder CG. But CG measures FH, therefore E also measures FH. But it also measures the whole FA, therefore it measures the remainder, the unit AH, though it is a number, which is impossible. Therefore no number measures the numbers AB and CD. Therefore AB and CD are relatively prime. VII.Def.12 Therefore, when two unequal numbers are set out, and the less is continually subtracted in turn from the greater, if the number which is left never measures the one before it until a unit is left, then the original numbers are relatively prime. Q.E.D. Modern terminology uses the word "divides" rather than "measures," and the notation a | b is used to abbreviate the phrase "a divides b." This proposition assumes that 1 is the result of an antenaresis process. Antenaresis, also called the Euclidean algorithm, is a kind of reciprocal subtraction. Beginning with two numbers, the smaller, whichever it is, is repeatedly subtracted from the larger. If the initial two numbers are a1 (AB in the proof) and a2 (CD), with a1 greater than a2, then the first stage is to repeatedly subtract a2 from a1 until a remainder a3 (AF) less than a2 is found. That can be stated algebraically as a1 = m1 a2 + a3 where m1 is the number of times that a2 was subtracted from a1. The next stage repeatedly subtracts a3 from a2 leaving a remainder a4 (CG): a2 = m2 a3 + a4. For the hypotheses of this proposition, the algorithm stops when a remainder of 1 occurs: an-1 = mn-1 an + 1. (In Euclid's proof, an is a5 which is AH.) The conclusion is that a1 and a2 are relatively prime. The proof is not difficult. It depends on the observation that if b divides (that is, measures) both c and d, then b divides their difference c d. So, if some number b divides both a1 and a2, then it divides the remainder a3, too. And since it divides both a2 and a3, it divides the remainder a4. And so forth, with the final conclusion that b divides the last remainder 1. Since there is no number b (and by "number" is meant a number greater than 1) which divides 1, there is no number that divides both a1 and a2. Therefore a1 and a2 are relatively prime. Compare this proposition to X.2, a somewhat analogous statement about magnitudes. This proposition is used in the proof of the next one. Next proposition: VII.2 Previous: VII.Def.22 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 2 To find the greatest common measure of two given numbers not relatively prime. Let AB and CD be the two given numbers not relatively prime. It is required to find the greatest common measure of AB and CD. If now CD measures AB, since it also measures itself, then CD is a common measure of CD and AB. And it is manifest that it is also the greatest, for no greater number than CD measures CD. But, if CD does not measure AB, then, when the less of the numbers AB and CD being continually subtracted from the greater, some number is left which measures the one before it. For a unit is not left, otherwise AB and CD would be relatively prime, which is contrary to the hypothesis. VII.Def.12 VII.1 Therefore some number is left which measures the one before it. Now let CD, measuring BE, leave EA less than itself, let EA, measuring DF, leave FC less than itself, and let CF measure AE. Since then, CF measures AE, and AE measures DF, therefore CF also measures DF. But it measures itself, therefore it also measures the whole CD. But CD measures BE, therefore CF also measures BE. And it also measures EA, therefore it measures the whole BA. But it also measures CD, therefore CF measures AB and CD. Therefore CF is a common measure of AB and CD. I say next that it is also the greatest. If CF is not the greatest common measure of AB and CD, then some number G, which is greater than CF, measures the numbers AB and CD. Now, since G measures CD, and CD measures BE, therefore G also measures BE. But it also measures the whole BA, therefore it measures the remainder AE. But AE measures DF, therefore G also measures DF. And it measures the whole DC, therefore it also measures the remainder CF, that is, the greater measures the less, which is impossible. Therefore no number which is greater than CF measures the numbers AB and CD. Therefore CF is the greatest common measure of AB and CD. Corollary From this it is manifest that, if a number measures two numbers, then it also measures their greatest common measure. Euclid again uses antenaresis (the Euclidean algorithm) in this proposition, this time to find the greatest common divisor of two numbers that aren't relatively prime. Had Euclid considered the unit (1) to be a number, he could have merged these two propositions into one. The Euclidean algorithm, antenaresis The greatest common divisor of two numbers m and n is the largest number which divides both. It's usually denoted GCD(m, n). It can be found by antenaresis by repeatedly subtracting the smaller, whichever it happens to be at the time, from the larger until the smaller divides the larger. As an illustration consider the problem of computing the greatest common divisor of 884 and 3009. First, repeatedly subtract 884 from 3009 until the remainder is less than 884. An equivalent numerical operation is to divide 884 into 3009; you'll get the same remainder. In this case after subtracting 884 three times, the remainder is 357. The two numbers under our consideration are now 884 and 357. Repeatedly subtract 357 from 884 to get the remainder 170. Repeatedly subtract 170 from 357 to get the remainder 17. Finally, stop since 17 divides 170. We've found GCD(884,3009) equals 17. The stages of the algorithm are the same as in VII.1 except that the final remainder an+1, which divides the previous number an, is not 1. a1 = m1 a2 + a3 a2 = m2 a3 + a4 ... an-1 = mn-1 an + an+1. (In Euclid's proof a1 is AB, a2 is CD, a3 is AE, and a4 = an+1 is CF.) In the first part of the proof, Euclid shows that since an+1 divides an, it also divides an-1, ... , a2, and a1. Therefore an+1 is a common divisor of a2 and a1. In the last part of the proof, Euclid shows that if any number d divides both a2 and a1, then it also divides a3, ... , an, and an+1. Therefore an+1 is the greatest common divisor. The last part of the proof also shows that every common divisor divides the greatest common divisor as noted in the corollary. Foundations of number theory Euclid makes many implicit assumptions about numbers. For instance, he assumes that if m < n, then m can be repeatedly subtracted from n until there is eventually a remainder less than or equal to m. He seems to have recognized that magnitudes need not have this property since the property is used as a qualifier in the definition of ratios (V.Def.4), but he didn't recognize its importance for numbers. There are, in fact, nonstandard models of number theory which satisfy the usual properties of numbers, but do not have this property. In such models, there are numbers than can be decreased by 1 infinitely many times but not ever reach 1. The existance of such models implies an axiom is needed to exclude such behavior. There is a similar assumption that the process of antenaresis eventually reaches an end when applied to numbers. Euclid certainly knew it needn't halt for magnitudes since its halting is used as a criterion for incommensurability (X.2). There needs to be an explicit axiom to cover these situations. One such axiom is a descending chain condition which states that there is no infinite decreasing sequence of numbers a1 > a2 > ... > an > ... Use of this proposition This proposition and its corollary are used in the next two propositions. Note how similar this proposition is to X.3, even having the same diagram and the same corollary. The terminology is slightly different and X.3 deals with magnitudes rather than numbers. Next proposition: VII.3 Previous: VII.1 Book VII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 3 To find the greatest common measure of three given numbers not relatively prime. Let A, B, and C be the three given numbers not relatively prime. It is required to find the greatest common measure of A, B, and C. Take the greatest common measure, D, of the two numbers A and B. Then either D measures, or does not measure, C. VII.2 First, let it measure it. But it measures A and B also, therefore D measures A, B, and C. Therefore D is a common measure of A, B, and C. I say that it is also the greatest. If D is not the greatest common measure of A, B, and C, then some number E, greater than D, measures the numbers A, B, and C. Since then E measures A, B, and C, therefore it measures A and B. Therefore it also measures the greatest common measure of A and B. But the greatest common measure of A and B is D, therefore E measures D, the greater the less, which is impossible. VII.2,Cor. Therefore no number which is greater than D measures the numbers A, B, and C. Therefore D is the greatest common measure of A, B, and C. Next, let D not measure C. I say first that C and D are not relatively prime. Since A, B, and C are not relatively prime, therefore some number measures them. Now that which measures A, B, and C also measures A and B, and therefore measures D, the greatest common measure of A and B. But it measures C also, therefore some number measures the numbers D and C. Therefore D and C are not relatively prime. VII.2,Cor. Take their greatest common measure E. VII.2 Then, since E measures D, and D measures A and B, therefore E also measures A and B. But it measures C also, therefore E measures A, B, and C. Therefore E is a common measure of A, B, and C. I say next that it is also the greatest. If E is not the greatest common measure of A, B, and C, then some number F, greater than E, measures the numbers A, B, and C. Now, since F measures A, B, and C, it also measures A and B, therefore it measures the greatest common measure of A and B. But the greatest common measure of A and B is D, therefore F measures D. VII.2,Cor. And it measures C also, therefore F measures D and C. Therefore it also measures the greatest common measure of D and C. But the greatest common measure of D and C is E, therefore F measures E, the greater the less, which is impossible. VII.2,Cor. Therefore no number which is greater than E measures the numbers A, B, and C. Therefore E is the greatest common measure of A, B, and C. Q.E.D. A common modern notation for the greatest common divisor of two numbers a and b is GCD(a, b). Also, the notation a | b is typically used to indicate that a divides b. This proposition constructs the GCD(a, b, c) as GCD(GCD(a, b), c). The proof that this construction works is simplified if 1 is considered to be a number. Then, two numbers are relatively prime when their GCD is 1, and Euclid's first case in the proof is subsumed in the second. Let d = GCD(a, b), and let e = GCD(d, c). Since e | d, d | a, and d | b, it follows that e | a and e | b, so e, in fact, is a common divisor of a, b, and c. In order to show that e is the greatest common divisor, let f be any common divisor of a, b, and c. Then as f | a and f | b, therefore f | GCD(a, b), that is, f | d. Also, as f | d and f | c, therefore f | GCD(d, c), that is f | e. Therefore e is the greatest common divisor of a, b, and c. Q.E.D. This is the same proposition as X.4. This proposition is used in VII.33. Next proposition: VII.4 Previous: VII.2 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 4 Any number is either a part or parts of any number, the less of the greater. Let A and BC be two numbers, and let BC be the less. I say that BC is either a part, or parts, of A. Either A and BC are relatively prime or they are not. First, let A and BC be relatively prime. Then, if BC is divided into the units in it, then each unit of those in BC is some part of A, so that BC is parts of A. VII.Def.4 Next let A and BC not be relatively prime, then BC either measures, or does not measure, A. Now if BC measures A, then BC is a part of A. But, if not, take the greatest common measure D of A and BC, and divide BC into the numbers equal to D, namely BE, EF, and FC. VII.Def.3 VII.2 Now, since D measures A, therefore D is a part of A. But D equals each of the numbers BE, EF, and FC, therefore each of the numbers BE, EF, and FC is also a part of A, so that BC is parts of A. Therefore, any number is either a part or parts of any number, the less of the greater. Q.E.D. This proposition says that if b is a smaller number than a, then b is either a part of a, that is, b is a unit fraction of a, or b is parts of a, that is, a proper fraction, but not a unit fraction, of a. For instance, 2 is one part of 6, namely, one third part; but 4 is parts of 6, namely, 2 third parts of 6. It seems obvious that when one number b is less than another a, then in all cases b would be parts of a, namely b consists of b of the ath parts of a. For instance, 4 consists of 4 sixth parts of 6. Yet, the proof of this proposition ignores that possibility, except in the special case when b and a are relatively prime. In the case of 4 and 6, the proof will find that 4 is 2 third parts of 6. Thus, it appears that a satisfactory answer to the question "How mary parts of a is b?" requires finding the least number of parts. The proof has three cases. 1. If b and a are relatively prime, then b consists of b of the ath parts of a. 2. If b divides a, then b is one part of a. 3. Otherwise they're not relatively prime, and b does not divide a. Let d be their greatest common divisor. Then b consists of some number, say c parts, each part equal to d. But these parts also also parts of a. Therefore, b consists of c parts of a. Use of this proposition This proposition is used in VII.20. Next proposition: VII.5 Previous: VII.3 Book VII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 5 If a number is part of a number, and another is the same part of another, then the sum is also the same part of the sum that the one is of the one. Let the number A be a part of BC, and another number D be the same part of another number EF that A is of BC. I say that the sum of A and D is also the same part of the sum of BC and EF that A is of BC. Since, whatever part A is of BC, D is also the same part of EF, therefore, there are as many numbers equal to D in EF as there are in BC equal to A. Divide BC into the numbers equal to A, namely BG and GC, and EF into the numbers equal to D, namely EH and HF. Then the multitude of BG and GC equals the multitude of EH and HF. And, since BG equals A, and EH equals D, therefore the sum of BG and EH also equals the sum of A and D. For the same reason the sum of GC and HF also equals the sum of A and D. Therefore there are as many numbers in BC and EF equal to A and D as there are in BC equal to A. Therefore, the sum of BC and EF is the same multiple of the sum of A and D that BC is of A. Therefore, the sum of A and D is the same part of the sum of BC and EF that A is of BC. Therefore, if a number is part of a number, and another is the same part of another, then the sum is also the same part of the sum that the one is of the one. Q.E.D. This is the first of four propositions that deal with distributivity of division and multiplication over addition and subtraction. This one says division distributes over addition. Algebraically, if a = b/n and d = e/n, then a + d = (b + e)/n. As a single equation, this says b/n + e/n = (b + e)/n. Use of this proposition This proposition is used in the proofs of five of the next seven propositions. Next proposition: VII.6 Previous: VII.4 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 6 If a number is parts of a number, and another is the same parts of another, then the sum is also the same parts of the sum that the one is of the one. Let the number AB be parts of the number C, and another number DE be the same parts of another number F that AB is of C. I say that the sum of AB and DE is also the same parts of the sum of C and F that AB is of C. Since there are as many parts of DE in F as there are parts AB is of C, therefore there are as many parts of F in DE as there are parts of C in AB. Divide AB into the parts of C, namely AG and GB, and divide DE into the parts of F, namely DH, and HE. Then the multitude of AG and GB equals the multitude of DH and HE. And since DH is the same part of F that AG is of C, therefore the sum of AG and DH is the same part of the sum of C and F that AG is of C. For the same reason, the sum of GB and HE is the same parts of the sum of C and F that GB is of C. VII.5 Therefore the sum of AB and DE is the same parts of the sum of C and F that AB is of C. Therefore, if a number is parts of a number, and another is the same parts of another, then the sum is also the same parts of the sum that the one is of the one. Q.E.D. This proposition says multiplication by fractions distributes over addition. Algebraically, if a = (m/n)b and d = (m/n)e then a + d = (m/n)(b + e). As an equation, this says (m/n)b + (m/n)e = (m/n)(b + e). Use of this propositionVII.9. Next proposition: VII.7 Previous: VII.5 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 7 If a number is that part of a number which a subtracted number is of a subtracted number, then the remainder is also the same part of the remainder that the whole is of the whole. Let the number AB be that part of the number CD which AE subtracted is of CF subtracted. I say that the remainder EB is also the same part of the remainder FD that the whole AB is of the whole CD. Let EB be the same part of CG that AE is of CF. Now since EB is the same part of CG that AE is of CF, therefore AB is the same part of GF that AE is of CF. VII.5 But, by hypothesis, AB is the same part of CD that AE is of CF, therefore AB is the same part of CD that it is of GF. Therefore GF equals CD. Subtract CF from each. Then the remainder GC equals the remainder FD. Now since EB is the same part of GC that AE is of CF, and GC equals FD, therefore EB is the same part of FD that AE is of CF. But AB is the same part of CD that AE is of CF, therefore the remainder EB is the same part of the remainder FD that the whole AB is of the whole CD. Therefore, if a number is that part of a number which a subtracted number is of a subtracted number, then the remainder is also the same part of the remainder that the whole is of the whole. Q.E.D. This proposition is like VII.5 except it deals with subtraction instead of addition. It says division distributes over subtraction. Algebraically, if a = b/n and d = e/n, then a d = (b e)/n. This proposition is used in the next proposition and in VII.11. Next proposition: VII.8 Previous: VII.6 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 8 If a number is the same parts of a number that a subtracted number is of a subtracted number, then the remainder is also the same parts of the remainder that the whole is of the whole. Let the number AB be the same parts of the number CD that AE subtracted is of CF subtracted. I say that the remainder EB is also the same parts of the remainder FD that the whole AB is of the whole CD. Make GH equal to AB. Therefore AE is the same parts of CF that GH is of CD. Divide GH into the parts of CD, namely GK and KH, and divide AE into the parts of CF, namely AL and LE. Then the multitude of GK and KH equals the multitude of AL and LE. Now since AL is the same part of CF that GK is of CD, and CD is greater than CF, therefore GK is also greater than AL. Make GM equal to AL. Then GK is the same part of CD that GM is of CF. Therefore the remainder MK is the same part of the remainder FD that the whole GK is of the whole CD. VII.7 Again, since EL is the same part of CF that KH is of CD, and CD is greater than CF, therefore HK is also greater than EL. Make KN equal to EL. Therefore KN is the same part of CF that KH is of CD. Therefore the remainder NH is the same part of the remainder FD that the whole KH is of the whole CD. VII.7 But the remainder MK was proved to be the same part of the remainder FD that the whole GK is of the whole CD, therefore the sum of MK and NH is the same parts of DF that the whole HG is of the whole CD. But the sum of MK and NH equals EB, and HG equals BA, therefore the remainder EB is the same parts of the remainder FD that the whole AB is of the whole CD. Therefore, if a number is the same parts of a number that a subtracted number is of a subtracted number, then the remainder is also the same parts of the remainder that the whole is of the whole. Q.E.D. This proposition says multiplication by fractions distributes over subtraction. Algebraically, if a = (m/n)b and d = (m/n)e, then a + d = (m/n)(b + e). The sample value taken for m/n in the proof is 2/3. This proposition is used in VII.11. Next proposition: VII.9 Previous: VII.7 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 9 If a number is a part of a number, and another is the same part of another, then alternately, whatever part of parts the first is of the third, the same part, or the same parts, the second is of the fourth. Let the number A be a part of the number BC, and and another number D be the same part of another number EF that A is of BC. I say that, alternately, BC is the same part or parts of EF that A is of D. Since D is the same part of EF that A is of BC, therefore there are as many numbers BC equal to A as there are also in EF equal to D. Divide BC into the numbers equal to A, namely BG and GC, and divide EF into those equal to D, namely EH and HF. Then the multitude of BG and GC equals the multitude of EH and HF. Now, since the numbers BG and GC equal one another, and the numbers EH and HF also equal one another, while the multitude of BG and GC equals the multitude of EH and HF, therefore GC is the same part or parts of HF that BG is of EH, so that, in addition, the sum BC is the same part or parts of the sum EF that BG is of EH. VII.5 VII.6 But BG equals A, and EH equals D, therefore BC is the same part or parts of EF that A is of D. Therefore, if a number is a part of a number, and another is the same part of another, then alternately, whatever part of parts the first is of the third, the same part, or the same parts, the second is of the fourth. Q.E.D. In this proposition, Euclid shows that if a = b/n, and d = e/n, and if a = (m/n)d, then b = (m/n)e. The sample value taken for 1/n in the proof is 1/2. Proposition VII.15 can be construed as a special case of this one. This proposition is used in the proof of the next. Next proposition: VII.10 Previous: VII.8 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 10 If a number is parts of a number, and another is the same parts of another, then alternately, whatever part of parts the first is of the third, the same part, or the same parts, the second is of the fourth. Let the number AB be parts of the number C, and another number DE be the same parts of another number F. I say that, alternately, C is the same parts or part of F that AB is of DE. Since DE is the same parts of F as AB is of C, therefore F is the same parts of DE as C is of AB. Divide AB into the parts of C, namely AG and GB, and divide DE into the parts of F, namely DH and HE. Then the multitude of AG and GB equals the multitude of DH and HE. Now since DH is the same part of F as AG is of C, therefore, alternately, C is the same part or the same parts of F as AG is of DH. VII.9 For the same reason, C is the same part or the same parts of F as GB is of HE, so that, in addition, C is the same part or the same parts of F as AB is of DE. VII.9 VII.5 VII.6 Therefore, if a number is parts of a number, and another is the same parts of another, then alternately, whatever part of parts the first is of the third, the same part, or the same parts, the second is of the fourth. Q.E.D. In this proposition, Euclid shows that if a = (m/n)b, and d = (m/n)e, and if a = (p/q)d, then b = (p/q)e. The sample value taken for m/n in the proof is 2/3. Use of this proposition This proposition is used in VII.13. Next proposition: VII.11 Previous: VII.9 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 11 If a whole is to a whole as a subtracted number is to a subtracted number, then the remainder is to the remainder as the whole is to the whole. Let the whole AB be to the whole CD as AE subtracted is to CF subtracted. I say that the remainder EB is to the remainder FD as the whole AB is to the whole CD. Since AB is to CD as AE is to CF, therefore AE is the same part or parts of CF as AB is of CD. Therefore the remainder EB is the same part or parts of FD that AB is of CD. VII.Def.20 VII.7 VII.8 Therefore EB is to FD as AB is to CD. VII.Def.20 Therefore, if a whole is to a whole as a subtracted number is to a subtracted number, then the remainder is to the remainder as the whole is to the whole. Q.E.D. This proposition is the numerical analogue of proposition V.19 for general magnitudes. Algebraically, if a:c = e:f, then a – e:c – f = a:c. Note that Euclid only deals with two cases, when AB is a part or parts of CD, and leaves out the other two, when CD is a part or parts of AB. This proposition is used in IX.35. Next proposition: VII.12 Previous: VII.10 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 12 If any number of numbers are proportional, then one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents. Let A, B, C, and D be as many numbers as we please in proportion, so that A is to B as C is to D. I say that A is to B as the sum of A and C is to the sum of B and D. Since A is to B as C is to D, therefore A is the same part or parts of B as C is of D. Therefore the sum of A and C is the same part or parts of the sum of B and D that A is of B. VII.Def.20 VII.5 VII.6 Therefore A is to B as the sum of A and C is to the sum of B and D. VII.Def.20 Therefore, if any number of numbers are proportional, then one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents. Q.E.D. This proposition is the numerical analogue of V.12. Algebraically, If x1:y1 = x2:y2 = ... = xn:yn, then each of these ratios also equals the ratio (x1 + x2 + ... + xn) : (y1 + y2 + ... + yn). Euclid takes n to be 2 in his proof. This proposition is used in VII.15 , VII.20, and IX.35. Next proposition: VII.13 Previous: VII.11 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 13 If four numbers are proportional, then they are also proportional alternately. Let the four numbers A, B, C, and D be proportional, so that A is to B as C is to D. I say that they are also proportional alternately, so that A is to C as B is to D. Since A is to B as C is to D, therefore, A is the same part or parts of B as C is of D. VII.Def.20 Therefore, alternately, A is the same part or parts of C as B is of D. VII.10 Therefore A is to C as B is to D. VII.Def.20 Therefore, if four numbers are proportional, then they are also proportional alternately. Q.E.D. This is the numerical analogue of proposition V.16 for magnitudes. It says that if a : b = c : d, then a : c = b : d. This proposition is used frequently in Books VII through IX starting with the next proposition. Next proposition: VII.14 Previous: VII.12 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 14 If there are any number of numbers, and others equal to them in multitude, which taken two and two together are in the same ratio, then they are also in the same ratio ex aequali. Let there be as many numbers as we please A, B, and C, and others equal to them in multitude D, E, and F, which taken two and two are in the same ratio, so that A is to B as D is to E, and B is to C as E is to F. I say that, ex aequali A is to C as D is to F. Since A is to B as D is to E, therefore, alternately A is to D as B is to E. VII.13 Again, since B is to C as E is to F, therefore, alternately B is to E as C is to F. But B is to E as A is to D, therefore A is to D as C is to F. Therefore, alternately A is to C as D is to F. VII.13 (V.11) Therefore, if there are any number of numbers, and others equal to them in multitude, which taken two and two together are in the same ratio, then they are also in the same ratio ex aequali. Q.E.D. This is the numerical analogue of V.22 for magnitudes. It says that if x1:x2 = y1:y2, x2:x3 = y2:y3, ... , and xn-1:xn = yn-1:yn, then x1:xn = y1:yn. Euclid takes n to be 3 in his proof. The proof is straightforward, and a simpler proof than the one given in V.22 for magnitudes. Note that at one point, the missing analogue of proposition V.11 is used: from the two proportions B : E = C : F and B : E = A : D, the conclusion A : D = C : F is drawn. Similar missing analogues of propositions from Book V are used in other proofs in book VII. See, for instance, VII.19 where V.7 and V.9 are used. This proposition is used occasionally in Books VIII and IX starting with VIII.1. Next proposition: VII.15 Previous: VII.13 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 15 If a unit measures any number, and another number measures any other number the same number of times, then alternately, the unit measures the third number the same number of times that the second measures the fourth. Let the unit A measure any number BC, and let another number D measure any other number EF the same number of times. I say that, alternately also, the unit measures the number D the same number of times that BC measures EF. Since the unit A measures the number BC the same number of times that D measures EF, therefore there are as many numbers equal to D in EF as there are units in BC. Divide BC into the units in it, BG, GH, and HC, and divide EF into the numbers EK, KL, and LF equal to D. Then the multitude of BG, GH, and HC equals the multitude of EK, KL, and LF. And, since the units BG, GH, and HC equal one another, and the numbers EK, KL, and LF also equal one another, while the multitude of the units BG, GH, and HC equals the multitude of the numbers EK, KL, and LF, therefore the unit BG is to the number EK as the unit GH is to the number KL, and as the unit HC is to the number LF. Since one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents, therefore the unit BG is to the number EK as BC is to EF. VII.12 But the unit BG equals the unit A, and the number EK equals the number D. Therefore the unit A is to the number D as BC is to EF. Therefore the unit A measures the number D the same number of times that BC measures EF. Therefore, if a unit number measures any number, and another number measures any other number the same number of times, then alternately, the unit measures the third number the same number of times that the second measures the fourth. Q.E.D. This proposition expresses the commutativity of multiplication. If a number e is b times d, that is, 1 measures b the same number of times that b measures d, then e also is d times b. In other words, bd = db. The next proposition states this commutativity more explicitly. This proposition can be viewed as a special case of proposition VII.9. This proposition is used in the next proposition and a few others in Books VII and IX. Next proposition: VII.16 Previous: VII.14 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 16 If two numbers multiplied by one another make certain numbers, then the numbers so produced equal one another. Let A and B be two numbers, and let A multiplied by B make C, and B multiplied by A make D. I say that C equals D. Since A multiplied by B makes C, therefore B measures C according to the units in A. But the unit E also measures the number A according to the units in it, therefore the unit E measures A the same number of times that B measures C. Therefore, alternately, the unit E measures the number B the same number of times that A measures C. VII.15 Again, since B multiplied by A makes D, therefore A measures D according to the units in B. But the unit E also measures B according to the units in it, therefore the unit E measures the number B the same number of times that A measures D. But the unit E measures the number B the same number of times that A measures C, therefore A measures each of the numbers C and D the same number of times. Therefore C equals D. Therefore, if two numbers multiplied by one another make certain numbers, then the numbers so produced equal one another. Q.E.D. This proposition describes the commutativity mentioned in the last proposition more explicitly, ab = ba. It is used in VII.18 and a few others in Book VII. Next proposition: VII.17 Previous: VII.15 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 17 If a number multiplied by two numbers makes certain numbers, then the numbers so produced have the same ratio as the numbers multiplied. Let the number A multiplied by the two numbers B and C make D and E. I say that B is to C as D is to E. Since A multiplied by B makes D, therefore B measures D according to the units in A. But the unit F also measures the number A according to the units in it, therefore the unit F measures the number A the same number of times that B measures D. Therefore the unit F is to the number A as B is to D. VII.Def.20 For the same reason the unit F is to the number A as C is to E, therefore B is to D as C is to E. VII.Def.20 (V.11) Therefore, alternately B is to C as D is to E. VII.13 Therefore, if a number multiplied by two numbers makes certain numbers, then the numbers so produced have the same ratio as the numbers multiplied. Q.E.D. Algebraically, b : c = ab : ac. This proposition is used very frequently in Books VII through IX starting with the next proposition. Next proposition: VII.18 Previous: VII.16 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 18 If two numbers multiplied by any number make certain numbers, then the numbers so produced have the same ratio as the multipliers. Let two numbers A and B multiplied by any number C make D and E. I say that A is to B as D is to E. Since A multiplied by C makes D, therefore C multiplied by A makes D. For the same reason also C multiplied by B makes E. VII.16 Therefore the number C multiplied by the two numbers A and B makes D and E. Therefore A is to B as Dis to E. VII.17 Therefore, if two numbers multiplied by any number make certain numbers, then the numbers so produced have the same ratio as the multipliers. Q.E.D. Whereas the last proposition stated b : c = ab : ac, this one says b : c = ba : ca. The only difference is the order of multiplication, but VII.16 says multiplication is commutative, so that order is irrelevant. This proposition is used in the next one and occasionally in Book VIII. Next proposition: VII.19 Previous: VII.17 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 19 If four numbers are proportional, then the number produced from the first and fourth equals the number produced from the second and third; and, if the number produced from the first and fourth equals that produced from the second and third, then the four numbers are proportional. Let A, B, C, and D be four numbers in proportion, so that A is to B as C is to D, and let A multiplied by D make E, and let B multiplied by C make F. I say that E equals F. Multiply A by C to make G. Since, then, A multiplied by C makes G, and multiplied by D makes E, therefore the number A multiplied by the two numbers C and D makes G and E. Therefore C is to D as G is to E. But C is to D as A is to B, therefore A is to B as G is to E. VII.17 (V.11) Again, since A multiplied by C makes G, but, further, B multiplied by C makes F, therefore the two numbers A and B multiplied by a certain number C make G and F. Therefore A is to B as G is to F. VII.18 But further A is to B as G is to E, therefore G is to E as G is to F. Therefore G has to each of the numbers E and F the same ratio. Therefore E equals F. (V.11) (V.9) Again, let E equal F. I say that A is to B as C is to D. With the same construction, since E equals F, therefore G is to E as G is to F. (V.7) But G is to E as C is to D, and G is to F as A is to B, therefore A is to B as C is to D. VII.17 VII.18 (V.11) Therefore, if four numbers are proportional, then the number produced from the first and fourth equals the number produced from the second and third; and, if the number produced from the first and fourth equals that produced from the second and third, then the four numbers are proportional. Q.E.D. Algebraically, a : b = c : d if and only if ad = bc. These algebraic expressions are meaningful when the variables are all numbers, but not when they are magnitudes in general. They can be interpreted, however, when they are lines, and proposition VI.16 is the analogue in that case. Twice in this proof Euclid makes conclusions about proportions for numbers that he has neither stated nor proved. These places are indicated by (V.11), (V.9), and (V.7) in the margins, the analogous justifications for magnitudes. Some of the propositions in Book V for magnitudes are stated in proved in Book VII for numbers, in particular, V.16 and VII.13 correspond, and V.22 and VII.14 correspond. But many of the propositions in Book V have no analogue in Book VII, such as V.11, V.9, and V.7. Now it could be that Euclid considered the missing statements as being obvious, as Heath claims, but being obvious is usually not a reason for Euclid to omit a proposition. Furthermore, other propositions in the next three books assume properties about proportions of numbers without having proofs of those propositions. One explanation is that the books on number theory, including this one, are older, and when the material in Book V was developed by Eudoxus, or when it was incorporated into the Elements by Euclid, more careful attention was made to fundamental propositions like V.7, V.9, and V.11. This proposition is used frequently in Books VII and IX starting with VII.24. Next proposition: VII.20 Previous: VII.18 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 20 The least numbers of those which have the same ratio with them measure those which have the same ratio with them the same number of times; the greater the greater; and the less the less. Let CD and EF be the least numbers of those which have the same ratio with A and B. I say that CD measures A the same number of times that EF measures B. Now CD is not parts of A. If possible, let it be so. Therefore EF is also the same parts of B that CD is of A. VII.13 VII.Def.20 Therefore there are as many parts of B in EF are there are parts of A in CD. Divide CD into the parts of A, namely CG and GD, and divide EF into the parts of B, namely EH and HF. Thus the multitude of CG and GD equals the multitude of EH and HF. Now, since the numbers CG and GD equal one another, and the numbers EH and HF also equal one another, while the multitude of CG and GD equals the multitude of EH and HF, therefore CG is to EH as GD is to HF. Since one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents, therefore CG is to EH as CD is to EF. VII.12 Therefore CG and EH are in the same ratio with CD and EF, being less than they, which is impossible, for by hypothesis CD and EF are the least numbers of those which have the same ratio with them. Therefore CD is not parts of A, therefore it is a part of it. VII.4 And EF is the same part of B that CD is of A, therefore CD measures A the same number of times that EF measures B. VII.13 VII.Def.20 Therefore, the least numbers of those which have the same ratio with them measure those which have the same ratio with them the same number of times; the greater the greater; and the less the less. Q.E.D. This proposition says that given a ratio a:b, if c:d is the same ratio and the least among all those ratios with the same ratio, then, first of all, c divides a, and d divides b, but also, c divides a the same number of times that d divides b. For example, the ratio 91:132 is the same ratio as 7:11, which is least among all the ratios equal to 91:132, that is to say 91:132 reduces to 7:11 in lowest terms, therefore 7 divides 91 the same number of times that 11 divides 132, namely, 13 times. The proof goes along like this. Suppose a:b reduces to c:e in lowest terms. In order to show that c divides a, assume that it doesn't, assume that c = (m/n)a. Since a:b is the same ratio as c:e, therefore e = (m/n)d. But then c/m = (1/n)a, and e/m = (1/n)b. Therefore c/m:e/m is the same ratio as a:b, which shows that c:e is not in lowest terms, a contradiction. Therefore c does divide a, and e divides b the same number of times. Use of this proposition This proposition is used frequently in Books VII through IX starting with the next proposition. Next proposition: VII.21 Previous: VII.19 Book VII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 21 Numbers relatively prime are the least of those which have the same ratio with them. Let A and B be numbers relatively prime. I say that A and B are the least of those which have the same ratio with them. If not, there are some numbers less than A and B in the same ratio with A and B. Let them be C and D. Since, then, the least numbers of those which have the same ratio measure those which have the same ratio the same number of times, the greater the greater, and the less the less, that is, the antecedent the antecedent and the consequent the consequent, therefore C measures A the same number of times that D measures B. VII.20 Let there be as many units in E as the times that C measures A. Then D also measures B according to the units in E. And, since C measures A according to the units in E, therefore E also measures A according to the units in C. For the same reason E also measures B according to the units in D. VII.16 Therefore E measures A and B which are relatively prime, which is impossible. VII.Def.12 Therefore there are no numbers less than A and B which are in the same ratio with A and B. Therefore A and B are the least of those which have the same ratio with them. Therefore, numbers relatively prime are the least of those which have the same ratio with them. Q.E.D. The next proposition is the converse of this one. Together they say that a ratio a:b is reduced to lowest terms if and only if a is relatively prime to b. Although it appears that this proposition is pairs of numbers and their ratios, it is used in proposition VII.33 with any quantity of numbers. Stated in terms of three numbers a, b, and c, that proposition says that of all triples with the same ratio as a, b, and c, have, the triple of relatively prime numbers is least. Use of this proposition This proposition is used frequently in Books VII through IX starting with VII.24. Next proposition: VII.22 Previous: VII.20 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 22 The least numbers of those which have the same ratio with them are relatively prime. Let A and B be the least numbers of those which have the same ratio with them. I say that A and B are relatively prime. If they are not relatively prime, then some number C measures them. Let there be as many units in D as the times that C measures A, and as many units in E as the times that C measures B. Since C measures A according to the units in D, therefore C multiplied by D makes A. For the same reason C multiplied by E makes B. VII.Def.15 Thus the number C multiplied by the two numbers D and E makes A and B, therefore D is to E as A is to B. VII.17 Therefore D and E are in the same ratio with A and B, being less than they, which is impossible. Therefore no number measures the numbers A and B. Therefore A and B are relatively prime. Therefore, the least numbers of those which have the same ratio with them are relatively prime. Q.E.D. This proposition is the converse of the last one. Together they say that a ratio a:b is reduced to lowest terms if and only if a is relatively prime to b. Use of this proposition This proposition is used in propositions VIII.2, VIII.3, and IX.15. Next proposition: VII.23 Previous: VII.21 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 23 If two numbers are relatively prime, then any number which measures one of them is relatively prime to the remaining number. Let A and B be two numbers relatively prime, and let any number C measure A. I say that C and B are also relatively prime. If C and B are not relatively prime, then some number D measures C and B. Since D measures C, and C measures A, therefore D also measures A. But it also measures B, therefore D measures A and B which are relatively prime, which is impossible. VII.Def.12 Therefore no number measures the numbers C and B. Therefore C and B are relatively prime. Therefore, if two numbers are relatively prime, then any number which measures one of them is relatively prime to the remaining number. Q.E.D. The proof of this proposition is straightforward. Use of this proposition This proposition is used in the proof of the next one. Next proposition: VII.24 Previous: VII.22 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 24 If two numbers are relatively prime to any number, then their product is also relatively prime to the same. Let the two numbers A and B [each] be relatively prime to a number C, and let A multiplied by B make D. I say that C and D are relatively prime. If C and D are not relatively prime, then some number E measures C and D. Now, since C and A are relatively prime, and a certain number E measures C, therefore A and E are relatively prime. VII.23 Let there be as many units in F as the times that E measures D. Then F also measures D according to the units in E. VII.16 Therefore E multiplied by F makes D. Also, A multiplied by B makes D, therefore the product of E and F equals the product of A and B. VII.Def.15 But, if the product of the extremes equal that of the means, then the four numbers are proportional. Therefore E is to A as B is to F. VII.19 But A and E are relatively prime, numbers which are relatively prime are also the least of those which have the same ratio, and the least numbers of those which have the same ratio with them measure those which have the same ratio the same number of times, the greater the greater, and the less the less, that is, the antecedent the antecedent and the consequent the consequent, therefore E measures B. VII.21 VII.20 But it also measures C, therefore E measures B and C which are relatively prime, which is impossible. VII.Def.12 Therefore no number measures the numbers C and D. Therefore C and D are relatively prime. Therefore, if two numbers are relatively prime to any number, then their product is also relatively prime to the same. Q.E.D. Outline of the proof Assume that two numbers a and b are each relatively prime to a third number c. Suppose their product ab is not relatively prime to c. Then there is some number e (greater than 1) that divides both ab and c. Now, since e divides c, and c is relatively prime to a, therefore, by VII.23, e is also relatively prime to a. Let f be the number ab/e. Then e:a = b:f. Since e and a are relatively prime, then, by VII.21, e:a is in lowest terms. Therefore, by VII.20, e divides b. But then e divides both b and c contradicting the assumption that b and c are relatively prime. Therefore, the product ab is also relatively prime to c. Use of this proposition This proposition is used in the next two and in IX.15. Next proposition: VII.25 Previous: VII.23 Book VII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 25 If two numbers are relatively prime, then the product of one of them with itself is relatively prime to the remaining one. Let A and B be two numbers relatively prime, and let A multiplied by itself make C. I say that B and C are relatively prime. Make D equal to A. Since A and B are relatively prime, and A equals D, therefore D and B are also relatively prime. Therefore each of the two numbers D and A is relatively prime to B. Therefore the product of D and A is also relatively prime to B. VII.24 But the number which is the product of D and A is C. Therefore C and B are relatively prime. Therefore, if two numbers are relatively prime, then the product of one of them with itself is relatively prime to the remaining one. Q.E.D. This is a special case of the previous proposition. It is used in VII.27 and IX.15. Next proposition: VII.26 Previous: VII.24 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 26 If two numbers are relatively prime to two numbers, both to each, then their products are also relatively prime. Let the two numbers A and B be relatively prime to the two numbers C and D, both to each, and let A multiplied by B make E, and let C multiplied by D make F. I say that E and F are relatively prime. Since each of the numbers A and B is relatively prime to C, therefore the product of A and B is also relatively prime to C. But the product of A and B is E, therefore E and C are relatively prime. For the same reason E and D are also relatively prime. Therefore each of the numbers C and D is relatively prime to E. VII.24 Therefore the product of C and D is also relatively prime to E. But the product of C and D is F. Therefore E and F are relatively prime. VII.24 Therefore, if two numbers are relatively prime to two numbers, both to each, then their products are also relatively prime. Q.E.D. The proof of this proposition uses proposition VII.24 twice. If a and b are both relatively prime to both c and d, then so is their product ab. Now since c and d are both relatively prime to ab, therefore so is their product cd. This proposition is used in the proof of the next one. Next proposition: VII.27 Previous: VII.25 Book VII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 27 If two numbers are relatively prime, and each multiplied by itself makes a certain number, then the products are relatively prime; and, if the original numbers multiplied by the products make certain numbers, then the latter are also relatively prime. Let A and B be two relatively prime numbers, let A multiplied by itself make C, and multiplied by C make D, and let B multiplied by itself make E, and multiplied by E make F. I say that C and E are relatively prime, and that D and F are relatively prime. Since A and B are relatively prime, and A multiplied by itself makes C, therefore C and B are relatively prime. VII.25 Since, then, C and B are relatively prime, and B multiplied by itself makes E, therefore C and E are relatively prime. Again, since A and B are relatively prime, and B multiplied by itself makes E, therefore A and E are relatively prime. Since, then, the two numbers A and C are relatively prime to the two numbers B and E, both to each, therefore the product of A and C is relatively prime to the product of B and E. And the product of A and C is D, and the product of B and E is F. VII.26 Therefore D and F are relatively prime. Therefore, if two numbers are relatively prime, and each multiplied by itself makes a certain number, then the products are relatively prime; and, if the original numbers multiplied by the products make certain numbers, then the latter are also relatively prime. Q.E.D. The proposition states that if two numbers are relatively prime, then their powers are also relatively prime. Explicitly, it only says that their squares are relatively prime, and their cubes are relatively prime, but the way it is used in VIII.2, any powers need to be relatively prime. The proof of this proposition uses the last two propositions. Assume that a and b are relatively prime. Then applying VII.25 twice, we first get a2 and b relatively prime, then we get a2 and b2 relatively prime. Again, by VII.25, a and b2 are relatively prime. Now, a is relatively prime to b2, and b is relatively prime to a2, so by VII.26, a3 is relatively prime to b3. Likewise, higher powers of a and b can be shown to be relatively prime. Use of this proposition This proposition is used in VIII.2 and VIII.3. Next proposition: VII.28 Previous: VII.26 Book VII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 28 If two numbers are relatively prime, then their sum is also prime to each of them; and, if the sum of two numbers is relatively prime to either of them, then the original numbers are also relatively prime. Let two relatively prime numbers AB and BC be added. I say that their sum AC is also relatively prime to each of the numbers AB and BC. If CA and AB are not relatively prime, then some number D measures CA and AB. Since then D measures CA and AB, therefore it also measures the remainder BC. But it also measures BA, therefore D measures AB and BC which are relatively prime, which is impossible. VII.Def.12 Therefore no number measures the numbers CA and AB. Therefore CA and AB are relatively prime. For the same reason AC and CB are also relatively prime. Therefore CA is relatively prime to each of the numbers AB and BC. Next, let CA and AB be relatively prime. I say that AB and BC are also relatively prime. If AB and BC are not relatively prime, then some number D measures AB and BC. Now, since D measures each of the numbers AB and BC, therefore it also measures the whole CA. But it measures AB, therefore D measures CA and AB which are relatively prime, which is impossible. VII.Def.12 Therefore no number measures the numbers AB and BC. Therefore AB and BC are relatively prime. Therefore, if two numbers are relatively prime, then their sum is also prime to each of them; and, if the sum of two numbers is relatively prime to either of them, then the original numbers are also relatively prime. Q.E.D. This proposition is used in IX.15. Next proposition: VII.29 Previous: VII.27 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 29 Any prime number is relatively prime to any number which it does not measure. Let A be a prime number, and let it not measure B. I say that B and A are relatively prime. If B and A are not relatively prime, then some number C measures them. Since C measures B, and A does not measure B, therefore C is not the same as A. Now, since C measures B and A, therefore it also measures A which is prime, though it is not the same as it, which is impossible. Therefore no number measures B and A. Therefore A and B are relatively prime. Therefore, any prime number is relatively prime to any number which it does not measure. Q.E.D. This proposition is used in the next one and in propositions IX.12 and IX.36. Next proposition: VII.30 Previous: VII.28 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 30 If two numbers, multiplied by one another make some number, and any prime number measures the product, then it also measures one of the original numbers. Let the two numbers A and B multiplied by one another make C, and let any prime number D measure C. I say that D measures one of the numbers A or B. Let it not measure A. Now D is prime, therefore A and D are relatively prime. VII.29 Let as many units be in E as the times that D measures C. Since then D measures C according to the units in E, therefore D multiplied by E makes C. VII.Def.15 Further, A multiplied by B also makes C, therefore the product of D and E equals the product of A and B. Therefore D is to A as B is to E. VII.19 But D and A are relatively prime, relatively prime numbers are also least, and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent, therefore D measures B. VII.21 VII.20 Similarly we can also show that, if D does not measure B, then it measures A. Therefore D measures one of the numbers A or B. Therefore, if two numbers, multiplied by one another make some number, and any prime number measures the product, then it also measures one of the original numbers. Q.E.D. This proposition states that if p is a prime number, then whenever p divides a product of two numbers, then it divides at least one of them. This is actually a property that characterizes prime numbers, that is to say, no composite number has this property. (For if c is a composite number, c ab, so c divides the product but it doesn't divide either factor.) Outline of the proof Assume that a prime number d divides the product ab. The form of the proof is interesting. Euclid shows that if d doesn't divide a, then d does divide b, and similarly, if d doesn't divide b, then d does divide a. Therefore, it divides either one or the other. Suppose d does not divide a. Then, by VII.29, d is relatively prime to a. Let e be the number ab/d. Then d:a = b:e. By VII.21, the ratio d:a is in lowest terms, and so, by VII.20, d divides b. Use of this proposition This proposition is used in IX.14. Next proposition: VII.31 Previous: VII.29 Book VII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 31 Any composite number is measured by some prime number. Let A be a composite number. I say that A is measured by some prime number. Since A is composite, therefore some number B measures it. VII.Def.13 Now, if B is prime, then that which was proposed is done. But if it is composite, some number measures it. Let a number C measure it. VII.Def.11,13 Then, since C measures B, and B measures A, therefore C also measures A. And, if C is prime, then that which was proposed is done. But if it is composite, some number measures it. Thus, if the investigation is continued in this way, then some prime number will be found which measures the number before it, which also measures A. If it is not found, then an infinite sequence of numbers measures the number A, each of which is less than the other, which is impossible in numbers. Therefore some prime number will be found which measures the one before it, which also measures A. Therefore any composite number is measured by some prime number. Therefore, any composite number is measured by some prime number. Q.E.D. Euclid does not explain why there can't be an infinite sequence of numbers where each number divides the previous. He simply says that is impossible. Some justification is required such as the principle Euclid uses elsewhere that any decreasing sequence of numbers is finite. This proposition is used in the next one and in propositions IX.13 and IX.20. Next proposition: VII.32 Previous: VII.30 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 32 Any number is either prime or is measured by some prime number. Let A be a number. I say that A either is prime or is measured by some prime number. If now A is prime, then that which was proposed is done. But if it is composite, then some prime number measures it. VII.31 Therefore any number either is prime or is measured by some prime number. Therefore, any number is either prime or is measured by some prime number. Q.E.D. After the previous proposition, this one really doesn't need to be stated at all. Next proposition: VII.33 Previous: VII.31 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 33 Given as many numbers as we please, to find the least of those which have the same ratio with them. Let A, B, and C be the given numbers, as many as we please. It is required to find the least of those which have the same ratio with A, B, and C. Either A, B, and C are relatively prime or they are not. Now, if A, B, and C are relatively prime, then they are the least of those which have the same ratio with them. VII.21 But, if not, take D the greatest common measure of A, B, and C. Let there be as many units in the numbers E, F, and G as the times that D measures the numbers A, B, and C respectively. VII.3 Therefore the numbers E, F, and G measure the numbers A, B, and C respectively according to the units in D. Therefore E, F, and G measure A, B, and C the same number of times. Therefore E, F, and G are in the same ratio with A, B, and C. VII.16 VII.Def.20 I say next that they are the least that are in that ratio. If E, F, and G are not the least of those which have the same ratio with A, B, and C, then there are numbers less than E, F, and G in the same ratio with A, B, and C. Let them be H, K, and L. Therefore H measures A the same number of times that the numbers K and L measure the numbers B and C respectively. Let there be as many units in M as the times that H measures A. Then the numbers K and L also measure the numbers B and C respectively according to the units in M. And, since H measures A according to the units in M, therefore M also measures A according to the units in H. For the same reason M also measures the numbers B and C according to the units in the numbers K and L respectively. Therefore M measures A, B, and C. VII.16 Now, since H measures A according to the units in M, therefore H multiplied by M makes A. For the same reason also E multiplied by D makes A. VII.Def.15 Therefore the product of E and D equals the product of H and M. Therefore E is to H as M is to D. VII.19 But E is greater than H, therefore M is also greater than D. And it measures A, B, and C, which is impossible, for by hypothesis, D is the greatest common measure of A, B, and C. Therefore there cannot be any numbers less than E, F, and G which are in the same ratio with A, B, and C. Therefore E, F, and G are the least of those which have the same ratio with A, B, and C. Q.E.D. This proposition is unusual in that it discusses a ratio a:b:c of three (or more) numbers. It also has the proportion a:b:c = e:f:g. These multiterm ratios and proportions may have been left over from an earlier time. Euclid argues that the proportion holds because e, f, and g measure a, b, and c, respectively, the same number, d, times. By the definition of proportion, that observation directly implies a:e = b:f = c:g The desired proportion, a:b:c = e:f:g is an alternate form of that multiple proportion. See V.Def.13 for the definition of alternate ratios. Use of this proposition This proposition is used in the next one and several propositions in Book VIII starting with VIII.6. Next proposition: VII.34 Previous: VII.32 Book VII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 34 To find the least number which two given numbers measure. Let A and B be the two given numbers. It is required to find the least number which they measure. Now either A and B are relatively prime or they are not. First, let A and B be relatively prime. Multiply A by B to make C. Then B multiplied by A makes C. Therefore A and B measure C. I say next that it is also the least number they measure. If not, then A and B measure some number D less than C Let there be as many units in E as the times that A measures D, and as many units in F as the times that B measures D. Then A multiplied by E makes D, and B multiplied by F makes D. Therefore the product of A and E equals the product of B and F. Therefore A is to B as F is to E. VII.Def.15 VII.19 But A and B are relatively prime, primes are also least, and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less, therefore B measures E as the consequent the consequent. VII.21 VII.20 And, since A multiplied by B and by E makes C and D, therefore B is to E as C is to D. But B measures E, therefore C also measures D, the greater the less, which is impossible. VII.17 Therefore A and B do not measure any number less than C. Therefore C is the least that is measured by A and B. Next, let A and B not be relatively prime. Take F and E, the least numbers of those which have the same ratio with A and B. Therefore the product of A and E equals the product of B and F. VII.33 VII.19 Multiply A by E to make C. Then B multiplied by F makes C. Therefore A and B measure C. I say next that it is also the least number that they measure. If not, then A and B measure some number D less than C. Let there be as many units in G as the times that A measures D, and as many units in H as the times that B measures D. Then A multiplied by G makes D, and B multiplied by H makes D. Therefore the product of A and G equals the product of B and H. Therefore A is to B as H is to G. VII.19 But A is to B as F is to E. Therefore F is to E as H is to G. (V.11) But F and E are least, and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less, therefore E measures G. VII.20 And, since A multiplied by E and by G makes C and D, therefore E is to G as C is to D. VII.17 But E measures G, therefore C also measures D, the greater the less, which is impossible. Therefore A and B do not measure any number less than C. Therefore C is the least that is measured by A and B. Q.E.D. The least common multiple of two numbers a and b is the smallest number that they both divide. It is denoted LCM(a, b). This proposition construct it as the product divided by the greatest common divisor: LCM(a, b) = ab/LCM(a, b). Summary of the proof Let a and b be the two numbers. There are two cases depending on whether they are relatively prime or not. Case 1. Suppose a and b are relatively prime. An indirect proof shows that their least common multiple is their product ab. If not, then there is a smaller number d which both a and b divide. Since a:b = (d/b):(d/a), and a:b is in lowest terms (since a and b are relatively prime), therefore b divides d/a. Also, b:(d/a) = ab:d, so ab divides d, but d is smaller than ab, a contradiction. Thus, when a and b are relatively prime, their least common multiple is their product. Case 2. Suppose a and b are not relatively prime. Reduce the ratio a:b to its lowest terms f:e using the previous proposition VII.33. Then ae = bf. Let c denote this product. (Note that f = a/GCD(a, b), and e = b/GCD(a, b), so c = ab/GCD(a, b).) Both a and b divide c, therefore c is a common multiple of a and b. Suppose that it's not the least common multiple. Then there is a smaller number d which both a and b divide. Now f:e = a:b = (d/b):(d/a), and f:e is in lowest terms, therfore e divides d/a. But e:(d/a) = ae:d, therefore ae also divides d. But c = ae, and d is less than c, a contradiction. Thus, LCM(a, b) = ab/LCM(a, b). Use of this proposition This proposition is used in VII.36 and VIII.4. Next proposition: VII.35 Previous: VII.33 Book VII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 35 If two numbers measure any number, then the least number measured by them also measures the same. Let the two numbers A and B measure any number CD, and let E be the least that they measure. I say that E also measures CD. If E does not measure CD, let E, measuring DF, leave CF less than itself. Now, since A and B measure E, and E measures DF, therefore A and B also measure DF. But they also measure the whole CD, therefore they measure the remainder CF which is less than E, which is impossible. Therefore E cannot fail to measure CD. Therefore it measures it. Therefore, if two numbers measure any number, then the least number measured by them also measures the same. Q.E.D. Outline of the proof Assume both a and b divide c. Let e be their least common multiple. Suppose that e does not divide c. Then repeatedly subtract e from c to get c = ke + f, where the remainder f is less than e and k is some number. Since a and b both divide c and e, they also divide f making f a smaller common multiple than the least common multiple e, a contradiction. Thus the least common multiple also divides c. Use of this proposition This proposition is used in the next one and in VIII.4. Next proposition: VII.36 Previous: VII.34 Book VII introduction © 1996, 2002 D.E.Joyce Clark University Proposition 36 To find the least number which three given numbers measure. Let A, B, and C be the three given numbers. It is required to find the least number which they measure. Take D the least number measured by the two numbers A and B. VII.34 Then C either measures, or does not measure, D. First, let it measure it. But A and B also measure D, therefore A, B, and C measure D. I say next that it is also the least that they measure. If not, A, B, and C measure some number E less than D. Since A, B, and C measure E, therefore A and B measure E. Therefore the least number measured by A and B also measures E. VII.35 But D is the least number measured by A and B, therefore D measures E, the greater the less, which is impossible. Therefore A, B, and C do not measure any number less than D. Therefore D is the least that A, B, and C measure. Next, let C not measure D. Take E, the least number measured by C and D. VII.34 Since A and B measure D, and D measures E, therefore A and B also measure E. But C also measures E, therefore A, B, and C also measure E. I say next that it is also the least that they measure. If not, A, B, and C measure some number F less than E. Since A, B, and C measure F, therefore A and B measure F. Therefore the least number measured by A and B also measures F. But D is the least number measured by A and B, therefore D measures F. But C also measures F, therefore D and C measure F, so that the least number measured by D and C also measures F. VII.35 But E is the least number measured by C and D, therefore E measures F, the greater the less, which is impossible. Therefore A, B, and C do not measure any number which is less than E. Therefore E is the least that is measured by A, B, and C. Q.E.D. The least common multiple of three numbers a, b, and c can be found as LCM(a, b, c) = LCM(LCM(a, b), c)). This proposition is used in the proof of proposition VII.39. Next proposition: VII.37 Previous: VII.35 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 37 If a number is measured by any number, then the number which is measured has a part called by the same name as the measuring number. Let the number A be measured by any number B. I say that A has a part called by the same name as B. Let there be as many units in C as the times that B measures A. Since B measures A according to the units in C, and the unit D also measures the number C according to the units in it, therefore the unit D measures the number C the same number of times as B measures A. Therefore, alternately, the unit D measures the number B the same number of times as C measures A. Therefore, whatever part the unit D is of the number B, the same part is C of A also. But the unit D is a part of the number B called by the same name as it, therefore C is also a part of A called by the same name as B, so that A has a part C which is called by the same name as B. VII.15 Therefore, if a number is measured by any number, then the number which is measured has a part called by the same name as the measuring number. Q.E.D. This proposition says that if b divides a, then a has a one-bth part (namely, a/b). For example, 3 divides 12, therefore 12 has a one-third part. Use of this proposition. This proposition is used in the proof of proposition VII.39. Next proposition: VII.38 Previous: VII.36 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 38 If a number has any part whatever, then it is measured by a number called by the same name as the part. Let the number A have any part whatever, B, and let C be a number called by the same name as the part B. I say that C measures A. Since B is a part of A called by the same name as C, and the unit D is also a part of C called by the same name as it, therefore the part B of A is the same part of the unit D of the number C. Therefore the unit D measures the number C the same number of times that B measures A. Therefore, alternately, the unit D measures the number B the same number of times that C measures A. Therefore C measures A. VII.15 Therefore, if a number has any part whatever, then it is measured by a number called by the same name as the part. Q.E.D. This proposition says that if a has a one-cth part of a, then c divides a. For example, 12 has a onethird part, 3 divides 12. This is a converse of the last proposition. Use of this proposition. This proposition is used in the proof of the next proposition. Next proposition: VII.39 Previous: VII.37 Book VII introduction © 1996 D.E.Joyce Clark University Proposition 39 To find the number which is the least that has given parts. Let A, B, and C be the given parts. It is required to find the number which is the least that will have the parts A, B, and C. Let D, E, and F be numbers called by the same name as the parts A, B, and C. Take G, the least number measured by D, E, and F. VII.36 Therefore G has parts called by the same name as D, E, and F. VII.37 But A, B, and C are parts called by the same name as D, E, and F, therefore G has the parts A, B, and C. I say next that it is also the least number that has. If not, there is some number H less than G which has the parts A, B, and C. Since H has the parts A, B, and C, therefore H is measured by numbers called by the same name as the parts A, B, and C. But D, E, and F are numbers called by the same name as the parts A, B, and C, therefore H is measured by D, E, and F. VII.38 And it is less than G, which is impossible. Therefore there is no number less than G that has the parts A, B, and C. Q.E.D. The wording of the proposition is somewhat unclear, but an example will show its intent. Suppose you want to find the smallest number with given parts, say, a fourth part and a sixth part. Then take the LCM(4,6) which is 12. The number 12 has a 1/4 part, namely 3, and a 1/6 part, namely 2. Next book: Book VIII Previous proposition: VII.38 Book VII introduction © 1996, 2002 D.E.Joyce Clark University Definition 1 Similar rectilinear figures are such as have their angles severally equal and the sides about the equal angles proportional. The words in this definition do not quite express its entire intent. It is apparent from its use that the notion of similarity assumes a specific correspondence of consecutive vertices and sides. Consider, for instance, pentagons. In order for the pentagons ABCDE and FGHKL to be similar, it is required that 1. corresponding angles taken in order are equal, that is, A = F, B = G, C = H, D = K, and E = L, and 2. the sides about their equal angles are proportional in the same order: EA:AB = LF:FG, AB:BC = FG:GH, BC:CD = GH:HK, CD:EF = HK:KL, and EA:AB = KL:LF. It wouldn't be allowed, for instance, if the angles of one figure equalled the angles of the other, but in some haphazard order. And it wouldn't be allowed for the orders of the terms in the proportions to be permuted, or inverted, for instance, the second proportion could not be AB:BC = GH:FG. Use of this definition Propositions VI.4 and VI.5 give two criteria for two triangles to be similar. Proposition VI.4 says that condition 1 implies similarity, while VI.5 says condition 2 implies similarity. Proposition VI.6 is a side-angle-side similarity theorem, and VI.7 is a side-side-angle similarity theorem. Many of the other propositions in this and later books involve similarity in one way or another. Next definition: VI.Def.2 Book VI introduction © 1997, 2002 D.E.Joyce Clark University Definition 2 Two figures are reciprocally related when the sides about corresponding angles are reciprocally proportional. This isn't the actual definition that appears, but an approximation of its intent. A literal translation is incomplete, and this definition may have been added after Euclid. The intention can be seen in proposition VI.15 as illustrated here. The proposition states that if two triangles have one angle equal to one angle, then the triangles are equal if and only if the sides about the equal angles are reciprocally proportional. In the figure BAC and DAE are equal angles. So the two triangles are equal if and only if CA:AD = EA:AB. Euclid doesn't define the term "reciprocally proportional," but the meaning of the term is clear from its use. Although Euclid doesn't address the question, it would be interesting to characterize which triangles are reciprocally related, as shown to the left. The conditions are that AB:DE = DF:AC, BC:EF = DE:AB, and AC:DF = DF:BC. Multiplicatively, AB DE = BC EF = AC DF. Next definition: VI.Def.3 Previous: VI.Def.1 Book VI introduction © 1997 D.E.Joyce Clark University Definition 3 A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less. The line AB is cut in extreme and mean ratio at C since AB:AC = AC:CB. A construction to cut a line in this manner first appeared in Book II, proposition II.11. Of course that was before ratios were defined, and there an equivalent condition was stated in terms of rectangles, namely, that the square on AC equal the rectangle AB by BC. That construction was later used in Book IV in order to construct regular pentagons and 15-sided polygons (propositions IV.10 through 12 and 16). Now that the theory of ratios and proportions has been developed, it is time to define this section as a ratio, rather than using rectangles. An alternate construction is given in proposition VI.30. Next definition: VI.Def.4 Previous: VI.Def.3 Book VI introduction © 1997 D.E.Joyce Clark University Definition 4 The height of any figure is the perpendicular drawn from the vertex to the base. Evidently, what is meant by "vertex" is the highest point in the figure, or a highest point when many points are equally high. But to define "height" in terms of "highest point" would be a circular definition. Indeed, this definition only suggests what "height" might mean without defining it at all. Still, by the way the term is used in the Elements, we can determine its meaning. The only planar figures where heights are used in the Elements are triangles and parallelograms. If the figure is a triangle, and one side has been declared the base, then the height is the expected line, the line drawn from the opposite vertex perpendicular to the base. If the figure is a parallelogram, and one side has been declared the base, then the height may be taken to be a perpendicular from either of the two vertices not on the base. In the later books on solid geometry, other figures also can have bases and heights such as parallelepipeds, pyramids, prisms, cones, and cylinders. Different sides of a figure may be selected as the base depending on the application. In proposition XI.39 there are two triangular prisms. A triangle is chosen taken to be the base of one, while the base of the other is a parallelogram. The height of the first is a perpendicular drawn between two triangular opposite faces, but the height of the other is a perpendicular drawn between the parallelogram taken as the base and the opposite parallelogram. Next proposition: VI.1 Previous: VI.Def.3 Book VI introduction © 1997 D.E.Joyce Clark University Proposition 1 Triangles and parallelograms which are under the same height are to one another as their bases. Let ACB and ACD be triangles, and let CE and CF be parallelograms under the same height. I say that the base CB is to the base CD as the triangle ACB is to the triangle ACD, and as the parallelogram CE is to the parallelogram CF. Produce BD in both directions to the points H and L. Make any number of straight lines BG and GH equal to the base CB, and any number of straight lines DK and KL equal to the base CD. Join AG, AH, AK, and AL. I.3 Then, since CB, BG, and GH equal one another, the triangles ACB, ABG, and AGH also equal one another. I.38 Therefore, whatever multiple the base CH is of the base CB, the triangle ACH is also that multiple of the triangle ACB. For the same reason, whatever multiple the base CL is of the base CD, the triangle ACL is also that multiple of the triangle ACD. And, if the base CH equals the base CL, then the triangle ACH also equals the triangle ACL; if the base CH is in excess of the base CL, the triangle ACH is also in excess of the triangle ACL; and, if less, less. I.38 Thus, there being four magnitudes, namely two bases CB and CD, and two triangles ACB and ACD, equimultiples have been taken of the base CB and the triangle ACB, namely the base CH and the triangle ACH, and other, arbitrary, equimultiples of the base CD and the triangle ADC, namely the base CL and the triangle ACL, and it has been proved that, if the base CH is in excess of the base CL, the triangle ACH is also in excess of the triangle ACL; if equal, equal; and, if less, less. Therefore the base CB is to the base CD as the triangle ACB is to the triangle ACD. V.Def.5 Next, since the parallelogram CE is double the triangle ACB, and the parallelogram FC is double the triangle ACD, and parts have the same ratio as their equimultiples, therefore the triangle ACB is to the triangle ACD as the parallelogram CE is to the parallelogram FC. I.41 V.15 Since, then, it was proved that the base CB is to CD as the triangle ACB is to the triangle ACD, and the triangle ACB is to the triangle ACD as the parallelogram CE is to the parallelogram CF, therefore also the base CB is to the base CD as the parallelogram CE is to the parallelogram FC. V.11 Therefore, triangles and parallelograms which are under the same height are to one another as their bases. Q.E.D. In a more proper setting out of the proposition, the triangles under the same height would not have a common side, and the parallelograms would not have a common base and side with the triangles. Since triangles on equal bases and in the same parallels are equal (I.36), and parallelograms on equal bases and in the same parallels are equal (I.35), and equals may be substituted in proportions (V.7), Euclid's simplified setting out is sufficient. Nonetheless, a proper setting out does not require a more complicated proof. The goal of the proof is to show that three ratios, namely the ratio of the lines CB to CD, the ratio of the triangles ACB to ACD, and the ratio of the parallelograms CE to CF, are all the same ratio. That is CB:CD = ACB:ACD = CE:CF. The first stage of the proof shows that CB:CD = ACB:ACD. By the definition of proportion, V.Def.5, that means for any number m and any number n that m BC >=< n CD when m ABC >=< n ACD. Note that Euclid takes both m and n to be 3 in his proof. Now m BC equals the line CH, n CD equals the line CL, m ABC equals the triangle ACH, and n ACD equals the triangle ACL. So what has to be shown is that CH >=< CL when ACH >=< ACL. But that follows from proposition I.38. So the first stage of the proof is complete. The second stage is easier. Since the parallelograms are twice the triangles, they also have the same ratio. Other propositions that state fundamental proportions use the same outline for their proofs. Proposition VI.33: arcs of circles are proportional to angles on which they stand; XI.25: parallelepipeds are proportional to their bases; and XII.13: cylinders are proportional to their axes. On the method of modern analysis Heath remarked that "some American and German text-books adopt the less rigorous method of appealing to the theory of limits" for the foundation for the theory of proportion used here in geometry. Heath preferred Eudoxus' theory of proportion in Euclid's Book V as a foundation. It is remarkable how much mathematics has changed over the last century. In the beginning of the 20th century Heath could still gloat over the superiority of synthetic geometry, although he may have been one of the last to do so. Now, in the 21st century, synthetic geometry has receded into near oblivion while analysis, based on various concepts of limits, is preeminent. It took some time to find a foundation for mathematical analysis as solid, or more solid, than geometry. In the 17th century, the time of the creation of differential and integral calculus, geometry was seen as the most dependable justification for calculus. In the first half of the 19th century, the concept of limit was clarified and limits became the foundation of mathematical analysis. Heath's complaint would have been valid then since the theory of real numbers was still without any foundation except a geometric one, which, ultimately was based on Eudoxus' theory of proportion in Euclid's Book V. In the later 19th century Weierstrass, Cantor, and Dedekind succeeded in founding the theory of real numbers on that of natural numbers and a bit of set theory, so that by the beginning of the 20th century, there was a modern foundation for mathematical analysis. All the same, this new foundation could still be called Eudoxus' since the modern definition of real number is the same as his, but in a modern guise. Use of this proposition This is one of the most used propositions in the Elements. It is used frequently in Book VI starting with the next proposition, dozens of times in Book X, and and a few times in Books XI and XIII. Next proposition: VI.2 Previous: VI.Def.4 Book VI introduction © 1996, 2002 D.E.Joyce Clark University Proposition 2 If a straight line is drawn parallel to one of the sides of a triangle, then it cuts the sides of the triangle proportionally; and, if the sides of the triangle are cut proportionally, then the line joining the points of section is parallel to the remaining side of the triangle. Let DE be drawn parallel to BC, one of the sides of the triangle ABC. I say that BD is to AD as CE is to AE. Join BE and CD. Therefore the triangle BDE equals the triangle CDE, for they are on the same base DE and in the same parallels DE and BC. I.37 And ADE is another triangle. But equals have the same ratio to the same, therefore the triangle BDE is to the triangle ADE as the triangle CDE is to the triangle ADE. V.7 But the triangle BDE is to ADE as BD is to AD, for, being under the same height, the perpendicular drawn from E to AB, they are to one another as their bases. VI.1 For the same reason, the triangle CDE is to ADE as CE is to AE. Therefore BD is to AD also as CE is to AE. V.11 Next, let the sides AB and AC of the triangle ABC be cut proportionally, so that BD is to AD as CE is to AE. Join DE. I say that DE is parallel to BC. With the same construction, since BD is to AD as CE is to AE, but BD is to AD as the triangle BDE is to the triangle ADE, and CE is to AE as the triangle CDE is to the triangle ADE, therefore the triangle BDE is to the triangle ADE as the triangle CDE is to the triangle ADE. VI.1 V.11 Therefore each of the triangles BDE and CDE has the same ratio to ADE. Therefore the triangle BDE equals the triangle CDE, and they are on the same base DE. V.9 But equal triangles which are on the same base are also in the same parallels. I.39 Therefore DE is parallel to BC. Therefore, if a straight line is drawn parallel to one of the sides of a triangle, then it cuts the sides of the triangle proportionally; and, if the sides of the triangle are cut proportionally, then the line joining the points of section is parallel to the remaining side of the triangle. Q.E.D. Euclid prefers to prove a pair of converses in two stages, but in some propositions, as this one, the proofs in the two stages are almost inverses of each other, so both could be proved at once. In this proposition we have a given triangle ABC and a line DE joining a point D on the side BC to a point E on the side AC. The claim is that BD:AD = CE:AE if and only if DE || BC. By the previous proposition VI.1 we know in any case that BD:AD = triangle BDE : triangle ADE, and CE:AE = triangle CDE : triangle ADE. Hence, BD:AD = CE:AE if and only if BDE:ADE = CDE:ADE. By propositions V.7 and V.9 the latter condition is equivalent to BDE = CDE, and that, in turn, by propositions I.37 and I.39 is equivalent to DE || BC. Note It should be noted that a proportion such as BD:AD = AE:CE is not intended. In that case the sides are cut proportionally, but the correspondence is not the intended one. Use of this theorem This proposition is frequently used in the rest of Book VI starting with the next proposition. It is also used in Books XI and XII. Next proposition: VI.3 Previous: VI.1 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 3 If an angle of a triangle is bisected by a straight line cutting the base, then the segments of the base have the same ratio as the remaining sides of the triangle; and, if segments of the base have the same ratio as the remaining sides of the triangle, then the straight line joining the vertex to the point of section bisects the angle of the triangle. Let ABC be a triangle, and let the angle BAC be bisected by the straight line AD. I say that DB is to DC as AB is to AC. Draw CE through C parallel to DA, and carry AB through to meet it at E. I.31 Then, since the straight line AC falls upon the parallels AD and EC, the angle ACE equals the angle CAD. I.29 But the angle CAD equals the angle BAD by hypothesis, therefore the angle BAD also equals the angle ACE. Again, since the straight line BAE falls upon the parallels AD and EC, the exterior angle BAD equals the interior angle AEC. I.29 But the angle ACE was also proved equal to the angle BAD, therefore the angle ACE also equals the angle AEC, so that the side AE also equals the side AC. I.6 And, since AD is parallel to EC, one of the sides of the triangle BCE, therefore, proportionally DB is to DC as AB is to AE. VI.2 But AE equals AC, therefore DB is to DC as AB is to AC. V.7 Next, let DB be to DC as AB is to AC. Join AD. I say that the straight line AD bisects the angle BAC. With the same construction, since DB is to DC as AB is to AC, and also DB is to DC as AB is to AE, for AD is parallel to EC, one of the sides of the triangle BCE, therefore also AB is to AC as AB is to AE. VI.2 V.11 Therefore AC equals AE, so that the angle AEC also equals the angle ACE. V.9 I.5 But the angle AEC equals the exterior angle BAD, and the angle ACE equals the alternate angle CAD, therefore the angle BAD also equals the angle CAD. I.29 Therefore the straight line AD bisects the angle BAC. Therefore, if an angle of a triangle is bisected by a straight line cutting the base, then the segments of the base have the same ratio as the remaining sides of the triangle; and, if segments of the base have the same ratio as the remaining sides of the triangle, then the straight line joining the vertex to the point of section bisects the angle of the triangle. Q.E.D. This proposition characterizes an angle bisector of an angle in a triangle as the line that partitions the base into parts proportional to the adjacent sides. The second part of the statement of the proposition is the converse of the first part of the statement. The proof relies on basic properties of triangles and parallel lines developed in Book I along with the result of the previous proposition VI.2. Use of this proposition This proposition is not used in the remainder of the Elements. Next proposition: VI.4 Previous: VI.2 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 4 In equiangular triangles the sides about the equal angles are proportional where the corresponding sides are opposite the equal angles. Let ABC and DCE be equiangular triangles having the angle ABC equal to the angle DCE, the angle BAC equal to the angle CDE, and the angle ACB equal to the angle CED. I say that in the triangles ABC and DEC the sides about the equal angles are proportional where the corresponding sides are opposite the equal angles. Let BC be placed in a straight line with CE. Then, since the sum of the angles ABC and ACB is less than two right angles, and the angle ACB equals the angle DEC, therefore the sum of the angles ABC and DEC is less than two right angles. Therefore BA and ED, when produced, will meet. Let them be produced and meet at F. I.17 I.Post.5 Now, since the angle DCE equals the angle ABC, DC is parallel to FB. Again, since the angle ACB equals the angle DEC, AC is parallel to FE. I.28 Therefore FACD is a parallelogram, therefore FA equals DC, and AC equals FD. I.34 And, since AC is parallel to a side FE of the triangle FBE, therefore BA is to AF as BC is to CE. VI.2 But FD equals AC, therefore BC is to CE as AC is to DE, and alternately BC is to CA as CE is to ED. V.7 V.16 Since then it was proved that AB is to BC as DC is to CE, and BC is to CA as CE is to ED, therefore, ex aequali, BA is to AC as CD is to DE. V.22 Therefore, in equiangular triangles the sides about the equal angles are proportional where the corresponding sides are opposite the equal angles. Q.E.D. In the enunciation of this proposition the term "equiangular triangles" refers to two triangles whose corresponding angles are equal, not to two triangles each of which is equiangular (equilateral). Euclid has placed the triangles in particular positions in order to employ this particular proof. Such positioning is common in Book VI and is easily justified. This proposition implies that equiangular triangles are similar, a fact proved in detail in the proof of proposition VI.8. It also implies that triangles similar to the same triangle are similar to each other, also proved in detail in VI.8. The latter statement is generalized in VI.21 to rectilinear figures in general. This proposition is frequently used in the rest of Book VI starting with the next proposition, its converse. It is also used in Books X through XIII. Next proposition: VI.5 Previous: VI.3 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 5 If two triangles have their sides proportional, then the triangles are equiangular with the equal angles opposite the corresponding sides. Let ABC and DEF be two triangles having their sides proportional, so that AB is to BC as DE is to EF, BC is to CA as EF is to FD, and further BA is to AC as ED is to DF. I say that the triangle ABC is equiangular with the triangle DEF where the equal angles are opposite the corresponding sides, namely the angle ABC equals the angle DEF, the angle BCA equals the angle EFD, and the angle BAC equals the angle EDF. Construct the angle FEG equal to the angle CBA and the angle EFG equal to the angle BCA on the straight line EF and at the points E and F on it. Therefore the remaining angle at A equals the remaining angle at G. I.23 I.32 Therefore the triangle ABC is equiangular with the triangle GEF. Therefore in the triangles ABC and GEF the sides about the equal angles are proportional where the corresponding sides are opposite the equal angles, therefore AB is to BC as GE is to EF. VI.4 But, by hypothesis, AB is to BC as DE to EF, therefore DE is to EF as GE is to EF. V.11 Therefore each of the straight lines DE and GE has the same ratio to EF, therefore DE equals GE. V.9 For the same reason DF also equals GF. Then since DE equals GE, and EF is common, the two sides DE and EF equal the two sides GE and EF, and the base DF equals the base GF, therefore the angle DEF equals the angle GEF, and the triangle DEF equals the triangle GEF, and the remaining angles equal the remaining angles, namely those opposite the equal sides. I.8 I.4 Therefore the angle DFE also equals the angle GFE, and the angle EDF equals the angle EGF. And, since the angle DEF equals the angle GEF, and the angle GEF equals the angle ABC, therefore the angle ABC also equals the angle DEF. For the same reason the angle ACB also equals the angle DFE, and further, the angle at A equals the angle at D, therefore the triangle ABC is equiangular with the triangle DEF. Therefore, if two triangles have their sides proportional, then the triangles are equiangular with the equal angles opposite the corresponding sides. Q.E.D. Of course, this proposition is the converse of the previous. We now have two characterizations of similar triangles, either as equiangular triangles or as triangles with proportional sides. The next two propositions give two more characterizations corresponding to characterizations of congruent triangles. As in VI.2, a certain order is assumed for the proportionality. It is not intended, for instance, that AB:BC = DE:EF while BC:CA = FD:EF. See the remark about VI.Def.1. This proposition is used in the proof of proposition XII.12. Next proposition: VI.6 Previous: VI.4 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 6 If two triangles have one angle equal to one angle and the sides about the equal angles proportional, then the triangles are equiangular and have those angles equal opposite the corresponding sides. Let ABC and DEF be two triangles having one angle BAC equal to one angle EDF and the sides about the equal angles proportional, so that BA is to AC as ED is to DF. I say that the triangle ABC is equiangular with the triangle DEF, and has the angle ABC equal to the angle DEF, and the angle ACB equal to the angle DFE. On the straight line DF and at the points D and F on it, construct the angle FDG equal to either of the angles BAC or EDF, and the angle DFG equal to the angle ACB. I.23 Therefore the remaining angle at B equals the remaining angle at G. Therefore the triangle ABC is equiangular with the triangle DGF. I.32 Therefore, proportionally BA is to AC as GD is to DF. VI.4 But, by hypothesis, BA is to AC also as ED is to DF, therefore also ED is to DF as GD is to DF. V.11 Therefore ED equals GD. And DF is common, therefore the two sides ED and DF equal the two sides GD and DF, and the angle EDF equals the angle GDF, therefore the base EF equals the base GF, the triangle DEF equals the triangle DGF, and the remaining angles equal the remaining angles, namely those opposite the equal sides. V.9 I.4 Therefore the angle DFG equals the angle DFE, and the angle DGF equals the angle DEF. But the angle DFG equals the angle ACB, therefore the angle ACB also equals the angle DFE. And, by hypothesis, the angle BAC also equals the angle EDF, therefore the remaining angle at B also equals the remaining angle at E. Therefore the triangle ABC is equiangular with the triangle DEF. I.32 Therefore, if two triangles have one angle equal to one angle and the sides about the equal angles proportional, then the triangles are equiangular and have those angles equal opposite the corresponding sides. Q.E.D. This is a side-angle-side similarity theorem analogus to side-angle-side congruence theorem I.4. Here's a summary of the proof. Construct a triangle DGF equiangular with triangle ABC. Then triangle DGF is similar to triangle ABC ( VI.4), and that gives us the proportion BA:AC = GD:DF. But we have assumed the proportion BA:AC = ED:DF, and these two proportions together give us GD:DF = ED:DF (V.11), from which it follows that GD = ED (V.9). Therefore triangles DEF and DGF are congruent, and the rest follows easily. Use of this proposition This proposition is used in the proofs of propositions VI.20, VI.32, XII.1, and several times in XII.12. Next proposition: VI.7 Previous: VI.5 Book VI introduction © 1996, 2002 D.E.Joyce Clark University Proposition 7 If two triangles have one angle equal to one angle, the sides about other angles proportional, and the remaining angles either both less or both not less than a right angle, then the triangles are equiangular and have those angles equal the sides about which are proportional. Let ABC and DEF be two triangles having one angle equal to one angle, the angle BAC equal to the angle EDF, the sides about other angles ABC and DEF proportional, so that AB is to BC as DE is to EF. And, first, each of the remaining angles at C and F less than a right angle. I say that the triangle ABC is equiangular with the triangle DEF, the angle ABC equals the angle DEF, and the remaining angle, namely the angle at C, equals the remaining angle, the angle at F. If the angle ABC does not equal the angle DEF, then one of them is greater. Let the angle ABC be greater. Construct the angle ABG equal to the angle DEF on the straight line AB and at the point B on it. I.23 Then, since the angle A equals D, and the angle ABG equals the angle DEF, therefore the remaining angle AGB equals the remaining angle DFE. I.32 Therefore the triangle ABG is equiangular with the triangle DEF. Therefore AB is to BG as DE is to EF. VI.4 But, by hypothesis, DE is to EF as AB is to BC, therefore AB has the same ratio to each of the straight lines BC and BG. Therefore BC equals BG, so that the angle at C also equals the angle BGC. V.11 V.9 I.5 But, by hypothesis, the angle at C is less than a right angle, therefore the angle BGC is also less than a right angle, so that the angle AGB adjacent to it is greater than a right angle. I.13 And it was proved equal to the angle at F, therefore the angle at F is also greater than a right angle. But it is by hypothesis less than a right angle, which is absurd. Therefore the angle ABC is not unequal to the angle DEF. Therefore it equals it. But the angle at A also equals the angle at D, therefore the remaining angle at C equals the remaining angle at F. I.32 Therefore the triangle ABC is equiangular with the triangle DEF. Next let each of the angles at C and F be supposed not less than a right angle. I say again that, in this case too, the triangle ABC is equiangular with the triangle DEF. With the same construction, we can prove similarly that BC equals BG, so that the angle at C also equals the angle BGC. I.5 But the angle at C is not less than a right angle, therefore neither is the angle BGC less than a right angle. Thus in the triangle BGC the sum of two angles is not less than two right angles, which is impossible. I.17 Therefore, once more, the angle ABC is not unequal to the angle DEF. Therefore it equals it. But the angle at A also equals the angle at D, therefore the remaining angle at C equals the remaining angle at F. I.32 Therefore the triangle ABC is equiangular with the triangle DEF. Therefore, if two triangles have one angle equal to one angle, the sides about other angles proportional, and the remaining angles either both less or both not less than a right angle, then the triangles are equiangular and have those angles equal the sides about which are proportional. Q.E.D. This is a side-side-angle similarity proposition for triangles. The Elements does not have the analogous side-side-angle congruence proposition for triangles. See the note on congruence theorems after I.26 for more about congruence theorems. Next proposition: VI.8 Previous: VI.6 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 8 If in a right-angled triangle a perpendicular is drawn from the right angle to the base, then the triangles adjoining the perpendicular are similar both to the whole and to one another. Let ABC be a right-angled triangle having the angle BAC right, and let AD be drawn from A perpendicular to BC. I say that each of the triangles DBA and DAC is similar to the whole ABC, and, further, they are similar to one another. Since the angle BAC equals the angle BDA, for each is right, and the angle at B is common to the two triangles ABC and DBA, therefore the remaining angle ACB equals the remaining angle DAB. Therefore the triangle ABC is equiangular with the triangle DBA. I.32 Therefore BC, which is opposite the right angle in the triangle ABC, is to BA, which is opposite the right angle in the triangle DBA, as AB, which is opposite the angle at C in the triangle ABC, is to DB, which is opposite the equal angle BAD in the triangle DBA, and also as AC is to DA, which is opposite the angle at B common to the two triangles. VI.4 Therefore the triangle ABC is both equiangular to the triangle DBA and has the sides about the equal angles proportional. Therefore the triangle ABC is similar to the triangle DBA. VI.Def.1 In the same manner we can prove that the triangle DAC is also similar to the triangle ABC. Therefore each of the triangles DBA and DAC is similar to the whole ABC. I say next that the triangles DBA and DAC are also similar to one another. Since the right angle BDA equals the right angle ADC, and moreover the angle DAB was also proved equal to the angle at C, therefore the remaining angle at B also equals the remaining angle DAC. Therefore the triangle DBA is equiangular with the triangle ADC. I.32 Therefore BD, which is opposite the angle DAB in the triangle DBA, is to AD, which is opposite the angle at C in the triangle DAC equal to the angle DAB, as AD, itself which is opposite the angle at B in the triangle DBA, is to CD, which is opposite the angle DAC in the triangle DAC equal to the angle at B, and also as BA is to AC, these sides opposite the right angles. Therefore the triangle DBA is similar to the triangle DAC. VI.4 VI.Def.1 Therefore, if in a right-angled triangle a perpendicular is drawn from the right angle to the base, then the triangles adjoining the perpendicular are similar both to the whole and to one another. Q.E.D. Corollary From this it is clear that, if in a right-angled triangle a perpendicular is drawn from the right angle to the base, then the straight line so drawn is a mean proportional between the segments of the base. Essentially, triangles ABC and DBA are equiangular since they are right triangles with a common angle. Therefore, they are similar. Likewise, triangles ABC and DAC are similar. Note that Euclid verbosely draws from proposition VI.4 the conclusions that equiangular triangles are similar and that triangles similar to the same triangle are similar to each other. The general proposition that figures similar to the same figure are also similar to one another is proposition VI.21. There is no reason why that proposition could not have been placed before this one. This proposition may be used to give an alternate proof of proposition I.47. Indeed, Euclid presents such a proof in the lemma for X.33. That proof is probably older than Euclid's as given in I.47, but Euclid's proof has the advantage of not being dependent on Eudoxus' theory of proportion in Book V. This proposition and its corollary are used in propositions VI.13, VI.31, X.33, and often in Book XIII. Next proposition: VI.9 Previous: VI.7 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 9 To cut off a prescribed part from a given straight line. Let AB be the given straight line. It is required to cut off from AB a prescribed part. Let the third part be that prescribed. Draw a straight line AC through from A containing with AB any angle. Take a point D random on AC, and make DE and EC equal to AD. I.3 Join CB, and draw DF through D parallel to it. I.31 Then, since DF is parallel to a side CB of the triangle ABC, therefore, proportionally, AD is to DC as AF is to FB. VI.2 But DC is double AD, therefore FB is also double AF, therefore AB is triple of AF. Therefore from the given straight line AB the prescribed third part AF has been cut off. Q.E.F. The word "part" in this proposition means submultiple. The problem here is to divide a line AB into some given number n of equal parts, or actually, to to find just one of these parts. Euclid takes the case n = 3 in his proof. Simson complained that proving the general case by using a specific case, the one-third part, "is not at all like Euclid's manner." But it is very much Euclid's manner throughout books V and VI to prove a general numerical statement with a specific numerical value. Al-Nayrizi's construction Abu'l-Abbas al-Fadl ibn al-Nayrizi (fl. c. 897, d. c. 922) wrote a commentary on the first ten books of the Elements. He gives another construction to divide a line AB into n equal parts. First, construct equal perpendiculars at A and B in opposite directions, mark off n 1 equal parts on each of them, and connect the points as illustrated. The diagram shows AB divided into five equal parts. Al-Nayrizi's construction takes considerably less work than Euclid's. The proof that this construction is valid is about the same length as that for Euclid's construction. Use of this construction This construction is used in a few propositions in Book XIII to find a third or a fifth of a line. Next proposition: VI.10 Previous: VI.8 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 10 To cut a given uncut straight line similarly to a given cut straight line. Let AB be the given uncut straight line, and AC the straight line cut at the points D and E, and let them be so placed as to contain any angle. Join CB, and draw DF and EG through D and E parallel to CB, and draw DHK through D parallel to AB. I.31 Therefore each of the figures FH and HB is a parallelogram. Therefore DH equals FG and HK equals GB. I.34 Now, since the straight line EH is parallel to a side CK of the triangle DCK, therefore, proportionally, DE is to EC as DH is to HK. VI.2 But DH equals FG, and HK equals GB, therefore DE is to EC as FG is to GB. V.7 Again, since DF is parallel to a side EG of the triangle AEG, therefore, proportionally, AD is to DE as AF is to FG. VI.2 But it was also proved that DE is to EC as FG is to GB, therefore DE is to EC as FG is to GB, and AD is to DE as AF is to FG. Therefore the given uncut straight line AB has been cut similarly to the given cut straight line AC. Q.E.F. In a sense, this proposition is a generalization of the last one VI.9. Prop. VI.9 cut a line into two parts whose ratio was a given numerical ratio. This proposition cuts a line into two parts whose ratio is a given ratio of two other lines. Both propositions rely on VI.2 as a basis to make any conclusion about the ratio of two lines. Use of this construction This proposition is not used later in the Elements, but it is a basic construction of geometry. Next proposition: VI.11 Previous: VI.9 Book VI introduction © 1996, 2002 D.E.Joyce Clark University Proposition 11 To find a third proportional to two given straight lines. Let AB and AC be the two given straight lines, and let them be placed so as to contain any angle. It is required to find a third proportional to AB and AC. Produce them to the points D and E, and make BD equal to AC. Join BC, and draw DE through D parallel to it. I.3 I.31 Then since BC is parallel to a side DE of the triangle ADE, therefore, proportionally, AB is to BD as AC is to CE. VI.2 But BD equals AC, therefore AB is to AC as AC is to CE. V.7 Therefore a third proportional CE has been found to two given straight lines AB and AC. Q.E.F. If a and b are two magnitudes, then their third proportional is a magnitude c such that a:b = b:c. The third proportional is needed whenever a duplicate ratio is needed when the ratio itself is known. The duplicate ratio for a:b is a:c. Use of this proposition This construction is used in propositions VI.19, VI.22, and a few propositions in Book X. Next proposition: VI.12 Previous: VI.10 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 12 To find a fourth proportional to three given straight lines. Let A and B and C be the three given straight lines. It is required to find a fourth proportional to A, B, and C. Set out two straight lines DE and DF containing any angle EDF. Make DG equal to A, GE equal to B, and DH equal to C. Join GH, and draw EF through E parallel to it. I.3 I.31 Then since GH is parallel to a side EF of the triangle DEF, therefore DG is to GE as DH is to HF. VI.2 But DG equals A and GE to B, and DH to C, therefore A is to B as C is to HF. V.7 Therefore a fourth proportional HF has been found to the three given straight lines A, B, and C. Q.E.F. Of course, the previous proposition is a special case of this one. Descartes' geometric algebra Descartes (1591-1661) is well known for his coordinate geometry which he and Fermat developed in the 16th century. This subject, also called analytic geometry, places an x-y-coordinate system on a plane so that a curve in the plane corresponds to an equation in two variables x and y. The usual way this correspondence is used is to convert a problem in geometry into an algebraic problem about equations. Descartes was equally interested is using geometry to solve algebraic problems, but using a method quite distinct from that in the Elements which began in Book II. His idea was to take an equation in one variable and find a geometric figure which can be used to solve the equation. The idea wasn't particularly new as even Menaechmus (fl. about 350 B.C.E) had about 50 years before Euclid intersected two parabolas to find cube roots, which are solutions to particular cubic equations. Furthermore, about 1100, Omar Khayyam solved all cubic equations by means of parabolas and hyperbolas. But Descartes was systematic and was able to use the relatively recent invention of symbolic algebra to make more connections. Descartes began by interpreting the algebraic operations of addition, subtraction, multiplication, division, and extraction of square roots as geometric constructions on lines. He represented each (positive) magnitude by a line. Addition and subtraction were the same as Euclid's. To add two lines, just extend one by the length of the other. To subtract one line from another, just take the remainder after cutting it off the other. Multiplication and division, however, were different from Euclid's. Euclid represented the product of two lines by a rectangle, the product of three lines by a box in space, and Euclid didn't represent the product of four lines. But Descartes took the product of two lines to be another line. That required selecting a unit line, that is, a line of length 1. Then to find the product ab of two quantities a and b, he only needed to find the fourth proportional of 1, a, and b. This proposition VI.10 does that. In the diagram to the right, u is the unit line, a and b are to be multiplied, and ab is their product, the fourth proportional. This same proposition works to construct the quotient of two quantities. If b and c are two quantities, then the fourth proportional for b, c, and 1 is the quotient c/b. Descartes achieved the fifth operation, extraction of square roots, by means of the semicircle and right angle construction described in the next proposition VI.13. Use of this proposition This proposition is used in the proofs of VI.22, VI.23, and half a dozen propositions in Book X. Next proposition: VI.13 Previous: VI.11 Book VI introduction © 1996, 2002 D.E.Joyce Clark University Proposition 13 To find a mean proportional to two given straight lines. Let AB and BC be the two given straight lines. It is required to find a mean proportional to AB and BC. Place them in a straight line, and describe the semicircle ADC on AC. Draw BD from the point B at right angles to the straight line AC, and join AD and DC. I.11 Since the angle ADC is an angle in a semicircle, it is right. III.31 And, since, in the right-angled triangle ADC, BD has been drawn from the right angle perpendicular to the base, therefore BD is a mean proportional between the segments of the base, AB and BC. VI.8,Cor Therefore a mean proportional BD has been found to the two given straight lines AB and BC. Q.E.F. This construction of the mean proportional was used before in II.4 to find a square equal to a given rectangle. By proposition VI.17 coming up, the two constructions are equivalent. That is the mean proportional between two lines is the side of a square equal to the rectangle contained by the two lines. Algebraically, a : x = x : b if and only if ab = x2. Thus, x is the square root of ab. When b is taken to have unit length, this construction gives the construction for the square root of a. Use of this proposition This construction is used in the proofs of propositions VI.25, X.27, and X.28. Next proposition: VI.14 Previous: VI.12 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 14 In equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional; and equiangular parallelograms in which the sides about the equal angles are reciprocally proportional are equal. Let AB and BC be equal and equiangular parallelograms having the angles at B equal, and let DB and BE be placed in a straight line. Therefore FB and BG are also in a straight line. I.14 I say that, in AB and BC, the sides about the equal angles are reciprocally proportional, that is to say, DB is to BE as BG is to BF. Complete the parallelogram FE. I.31 Then since the parallelogram AB equals the parallelogram BC, and FE is another parallelogram, therefore AB is to FE as BC is to FE. V.7 But AB is to FE as DB is to BE, and BC is to FE as BG is to BF. Therefore DB is to BE as BG is to BF. VI.1 V.11 Therefore in the parallelograms AB and BC the sides about the equal angles are reciprocally proportional. Next, let DB be to BE as BG is to BF. I say that the parallelogram AB equals the parallelogram BC. Since DB is to BE as BG is to BF, while DB is to BE as the parallelogram AB is to the parallelogram FE, and, BG is to BF as the parallelogram BC is to the parallelogram FE, therefore also AB is to FE as BC is to FE. VI.1 V.11 Therefore the parallelogram AB equals the parallelogram BC. V.9 Therefore, in equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional; and equiangular parallelograms in which the sides about the equal angles are reciprocally proportional are equal. Q.E.D. This proposition is used in the proofs of propositions VI.16, VI.30, and X.22. Next proposition: VI.15 Previous: VI.13 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 15 In equal triangles which have one angle equal to one angle the sides about the equal angles are reciprocally proportional; and those triangles which have one angle equal to one angle, and in which the sides about the equal angles are reciprocally proportional, are equal. Let ABC and ADE be equal triangles having one angle equal to one angle, namely the angle BAC equal to the angle DAE. I say that in the triangles ABC and ADE the sides about the equal angles are reciprocally proportional, that is to say, that CA is to AD as EA is to AB. Place them so that CA is in a straight line with AD. Therefore EA is also in a straight line with AB. I.14 Join BD. Since then the triangle ABC equals the triangle ADE, and ABD is another triangle, therefore the triangle ABC is to the triangle ABD as the triangle ADE is to the triangle ABD. V.7 But ABC is to ABD as AC is to AD, and ADE is to ABD as AE is to AB. VI.1 Therefore also AC is to AD as AE is to AB. V.11 Therefore in the triangles ABC and ADE the sides about the equal angles are reciprocally proportional. Next, let the sides of the triangles ABC and ADE be reciprocally proportional, that is to say, let AE be to AB as CA is to AD. I say that the triangle ABC equals the triangle ADE. If BD is again joined, since AC is to AD as AE is to AB, while AC is to AD as the triangle ABC is to the triangle ABD, and AE is to AB as the triangle ADE is to the triangle ABD, therefore the triangle ABC is to the triangle ABD as the triangle ADE is to the triangle ABD. VI.1 V.11 Therefore each of the triangles ABC and ADE has the same ratio to ABD. Therefore the triangle ABC equals the triangle ADE. V.9 Therefore, in equal triangles which have one angle equal to one angle the sides about the equal angles are reciprocally proportional; and those triangles which have one angle equal to one angle, and in which the sides about the equal angles are reciprocally proportional, are equal. Q.E.D. This proposition is used in the proof of proposition VI.19. Next proposition: VI.16 Previous: VI.14 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 16 If four straight lines are proportional, then the rectangle contained by the extremes equals the rectangle contained by the means; and, if the rectangle contained by the extremes equals the rectangle contained by the means, then the four straight lines are proportional. Let the four straight lines AB, CD, E, and F be proportional, so that AB is to CD as E is to F. I say that the rectangle AB by F equals the rectangle CD by E. Draw AG and CH from the points A and C at right angles to the straight lines AB and CD, and make AG equal to F, and CH equal to E. I.11 I.3 Complete the parallelograms BG and DH. I.31 Then since AB is to CD as E is to F, while E equals CH, and F equals AG, therefore AB is to CD as CH is to AG. V.7 Therefore in the parallelograms BG and DH the sides about the equal angles are reciprocally proportional. But those equiangular parallelograms in which the sides about the equal angles are reciprocally proportional are equal, therefore the parallelogram BG equals the parallelogram DH. VI.14 And BG is the rectangle AB by F, for AG equals F, and DH is the rectangle CD by E, for E equals CH, therefore the rectangle AB by F equals the rectangle CD by E. Next, let the rectangle AB by F be equal to the rectangle CD by E. I say that the four straight lines are proportional, so that AB is to CD as E is to F. With the same construction, since the rectangle AB by F equals the rectangle CD by E, and the rectangle AB by F is BG, for AG equals F, and the rectangle CD by E is DH, for CH equals E, therefore BG equals DH. And they are equiangular. But in equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional. VI.14 Therefore AB is to CD as CH is to AG. V.7 But CH equals E, and AG to F, therefore AB is to CD as E is to F. Therefore, if four straight lines are proportional, then the rectangle contained by the extremes equals the rectangle contained by the means; and, if the rectangle contained by the extremes equals the rectangle contained by the means, then the four straight lines are proportional. Q.E.D. This proposition is a special case of VI.14. It hardly needs such a protracted proof. It is used occasionally in Book X, but the special case when the means are equal and the second figure is a square, as enunciated in the next proposition, is used throughout Book X and frequently in Book XIII. Next proposition: VI.17 Previous: VI.16 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 17 If three straight lines are proportional, then the rectangle contained by the extremes equals the square on the mean; and, if the rectangle contained by the extremes equals the square on the mean, then the three straight lines are proportional. Let the three straight lines A and B and C be proportional, so that A is to B as B is to C. I say that the rectangle A by C equals the square on B. Make D equal to B. I.3 Then, since A is to B as B is to C, and B equals D, therefore A is to B as D is to C. V.7 V.11 But, if four straight lines are proportional, then the rectangle contained by the extremes equals the rectangle contained by the means. VI.16 Therefore the rectangle A by C equals the rectangle B by D. But the rectangle B by D is the square on B, for B equals D, therefore the rectangle A by C equals the square on B. Next, let the rectangle A by C equal the square on B. I say that A is to B as B is to C. With the same construction, since the rectangle A by C equals the square on B, while the square on B is the rectangle B by D, for B equals D, therefore the rectangle A by C equals the rectangle B by D. But, if the rectangle contained by the extremes equals that contained by the means, then the four straight lines are proportional. VI.16 Therefore A is to B as D is to C. But B equals D, therefore A is to B as B is to C. Therefore, if three straight lines are proportional, then the rectangle contained by the extremes equals the square on the mean; and, if the rectangle contained by the extremes equals the square on the mean, then the three straight lines are proportional. Q.E.D. This is obviously a special case of the previous proposition. It is used very frequently in Books X and XIII. Next proposition: VI.18 Previous: VI.16 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 18 To describe a rectilinear figure similar and similarly situated to a given rectilinear figure on a given straight line. Let AB be the given straight line and CE the given rectilinear figure. It is required to describe on the straight line AB a rectilinear figure similar and similarly situated to the rectilinear figure CE. Join DF. Construct the angle GAB equal to the angle at C, and the angle ABG equal to the angle CDF, on the straight line AB at the points A and B on it. I.23 Therefore the remaining angle CFD equals the angle AGB. Therefore the triangle FCD is equiangular with the triangle GAB. I.32 Therefore, proportionally, FD is to GB as FC is to GA, and as CD is to AB. VI.4 V.16 Again, construct the angle BGH equal to the angle DFE, and the angle GBH equal to the angle FDE, on the straight line BG and at the points B and G on it. I.23 Therefore the remaining angle at E equals the remaining angle at H. Therefore the triangle FDE is equiangular with the triangle GBH. Therefore, proportionally, FD is to GB as FE is to GH, and as ED is to HB. I.32 VI.4 V.16 But it was also proved that FD is to GB as FC is to GA, and as CD is to AB. Therefore FC is to AG as CD is to AB, and as FE is to GH, and further as ED is to HB. V.11 And, since the angle CFD equals the angle AGB, and the angle DFE equals the angle BGH, therefore the whole angle CFE equals the whole angle AGH. For the same reason the angle CDE also equals the angle ABH. And the angle at C also equals the angle at A, and the angle at E equals the angle at H. Therefore AH is equiangular with CE, and they have the sides about their equal angles proportional. Therefore the rectilinear figure AH is similar to the rectilinear figure CE. VI.Def.1 Therefore the rectilinear figure AH has been described similar and similarly situated to the given rectilinear figure CE on the given straight line AB. Q.E.F. It is evident from the diagram, if not from the text, that AB should correspond to CD. Although the figure has only four sides, it is clear that the method applies to figures with more than four sides. This proposition is used in the proofs of propositions VI.22, VI.25, and VI.28, and the corollary is used in XII.17. Next proposition: VI.19 Previous: VI.17 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 19 Similar triangles are to one another in the duplicate ratio of the corresponding sides. Let ABC and DEF be similar triangles having the angle at B equal to the angle at E, and such that AB is to BC as DE is to EF, so that BC corresponds to EF. V.Def.11 I say that the triangle ABC has to the triangle DEF a ratio duplicate of that which BC has to EF. Take a third proportional BG to BC and EF so that BC is to EF as EF is to BG, and join AG. VI.11 Since AB is to BC as DE is to EF, therefore, alternately, AB is to DE as BC is to EF. V.16 But BC is to EF as EF is to BG, therefore also AB is to DE as EF is to BG. V.11 Therefore in the triangles ABG and DEF the sides about the equal angles are reciprocally proportional. But those triangles which have one angle equal to one angle, and in which the sides about the equal angles are reciprocally proportional, are equal. Therefore the triangle ABG equals the triangle DEF. VI.15 Now since BC is to EF as EF is to BG, and, if three straight lines are proportional, the first has to the third a ratio duplicate of that which it has to the second, therefore BC has to BG a ratio duplicate of that which BC has to EF. V.Def.9 But BC is to BG as the triangle ABC is to the triangle ABG, therefore the triangle ABC also has to the triangle ABG a ratio duplicate of that which BC has to EF. VI.1 V.11 But the triangle ABG equals the triangle DEF, therefore the triangle ABC also has to the triangle DEF a ratio duplicate of that which BC has to EF. V.7 Therefore, similar triangles are to one another in the duplicate ratio of the corresponding sides. Q.E.D. Corollary From this it is manifest that if three straight lines are proportional, then the first is to the third as the figure described on the first is to that which is similar and similarly described on the second. This proposition is used in the proof of the next one, and the corollary is used in the proofs of VI.22, VI.31, and X.6. Next proposition: VI.20 Previous: VI.18 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 20 Similar polygons are divided into similar triangles, and into triangles equal in multitude and in the same ratio as the wholes, and the polygon has to the polygon a ratio duplicate of that which the corresponding side has to the corresponding side. Let ABCDE and FGHKL be similar polygons, and let AB correspond to FG. I say that the polygons ABCDE and FGHKL are divided into similar triangles, and into triangles equal in multitude and in the same ratio as the wholes, and the polygon ABCDE has to the polygon FGHKL a ratio duplicate of that which AB has to FG. Join BE, CE, GL, and HL. Now, since the polygon ABCDE is similar to the polygon FGHKL, therefore the angle BAE equals the angle GFL, and AB is to AE as GF is to FL. VI.Def.1 Since then ABE and FGL are two triangles having one angle equal to one angle and the sides about the equal angles proportional, therefore the triangle ABE is equiangular with the triangle FGL, so that it is also similar, therefore the angle ABE equals the angle FGL. VI.6 VI.4 VI.Def.1 But the whole angle ABC also equals the whole angle FGH because of the similarity of the polygons, therefore the remaining angle EBC equals the angle LGH. And, since the triangles ABE and FGL are similar, BE is to AB as GL is to GF. Also, since the polygons are similar, AB is to BC as FG is to GH. Therefore, ex aequali, BE is to BC as GL is to GH, that is, the sides about the equal angles EBC and LGH are proportional. Therefore the triangle EBC is equiangular with the triangle LGH, so that the triangle EBC is also similar to the triangle LGH. V.22 VI.6 VI.4 VI.Def.1 For the same reason the triangle ECD is also similar to the triangle LHK. Therefore the similar polygons ABCDE and FGHKL have been divided into similar triangles, and into triangles equal in multitude. I say that they are also in the same ratio as the wholes, that is, in such manner that the triangles are proportional, and ABE, EBC, and ECD are antecedents, while FGL, LGH, and LHK are their consequents, and that the polygon ABCDE has to the polygon FGHKL a ratio duplicate of that which the corresponding side has to the corresponding side, that is AB to FG. Join AC and FH. Then since the polygons are similar, the angle ABC equals the angle FGH, and AB is to BC as FG is to GH, the triangle ABC is equiangular with the triangle FGH, therefore the angle BAC equals the angle GFH, and the angle BCA to the angle GHF. VI.6 And, since the angle BAM equals the angle GFN, and the angle ABM also equals the angle FGN, therefore the remaining angle AMB also equals the remaining angle FNG. Therefore the triangle ABM is equiangular with the triangle FGN. I.32 Similarly we can prove that the triangle BMC is also equiangular with the triangle GNH. Therefore, proportionally, AM is to MB as FN is to NG, and BM is to MC as GN is to NH. So that, in addition, ex aequali, AM is to MC as FN is to NH. V.22 But AM is to MC as the triangle ABM is to MBC, and as AME is to EMC, for they are to one another as their bases. VI.1 Therefore also one of the antecedents is to one of the consequents as are all the antecedents to all the consequents, therefore the triangle AMB is to BMC as ABE is to CBE. V.12 But AMB is to BMC as AM is to MC, therefore AM is to MC as the triangle ABE is to the triangle EBC. V.11 For the same reason also FN is to NH as the triangle FGL is to the triangle GLH. And AM is to MC as FN is to NH, therefore the triangle ABE is to the triangle BEC as the triangle FGL is to the triangle GLH, and, alternately, the triangle ABE is to the triangle FGL as the triangle BEC is to the triangle GLH. V.11 V.16 Similarly we can prove, if BD and GK are joined, that the triangle BEC is to the triangle LGH as the triangle ECD is to the triangle LHK. And since the triangle ABE is to the triangle FGL as EBC is to LGH, and further as ECD is to LHK, therefore also one of the antecedents is to one of the consequents as the sum of the antecedents to the sum of the consequents. Therefore the triangle ABE is to the triangle FGL as the polygon ABCDE is to the polygon FGHKL. V.12 But the triangle ABE has to the triangle FGL a ratio duplicate of that which the corresponding side AB has to the corresponding side FG, for similar triangles are in the duplicate ratio of the corresponding sides. VI.19 Therefore the polygon ABCDE also has to the polygon FGHKL a ratio duplicate of that which the corresponding side AB has to the corresponding side FG. V.11 Therefore, similar polygons are divided into similar triangles, and into triangles equal in multitude and in the same ratio as the wholes, and the polygon has to the polygon a ratio duplicate of that which the corresponding side has to the corresponding side. Q.E.D. Corollary Similarly also it can be proved in the case of quadrilaterals that they are in the duplicate ratio of the corresponding sides. And it was also proved in the case of triangles, therefore also, generally, similar rectilinear figures are to one another in the duplicate ratio of the corresponding sides. This proposition and its corollary are used occassionally in Books X, XII, and XIII, in particular, XII.1 and XII.8. Next proposition: VI.21 Previous: VI.19 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 21 Figures which are similar to the same rectilinear figure are also similar to one another. Let each of the rectilinear figures A and B be similar to C. I say that A is also similar to B. Since A is similar to C, it is equiangular with it and has the sides about the equal angles proportional. VI.Def.1 Again, since B is similar to C, it is equiangular with it and has the sides about the equal angles proportional. Therefore each of the figures A and B is equiangular with C and with C has the sides about the equal angles proportional, therefore A is similar to B. V.11 Therefore, figures which are similar to the same rectilinear figure are also similar to one another. Q.E.D. This proposition is used in the proofs of propositions VI.24, VI.28, and VI.29. It also would have been useful in the proof of VI.8. Next proposition: VI.22 Previous: VI.20 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 22 If four straight lines are proportional, then the rectilinear figures similar and similarly described upon them are also proportional; and, if the rectilinear figures similar and similarly described upon them are proportional, then the straight lines are themselves also proportional. Let the four straight lines AB, CD, EF, and GH be proportional, so that AB is to CD as EF is to GH. Let the similar and similarly situated rectilinear figures KAB and LCD be described on AB and CD, and the similar and similarly situated rectilinear figures MF and NH be described on EF and GH. I say that KAB is to LCD as MF is to NH. Take a third proportional O to AB and CD, and a third proportional P to EF and GH. VI.11 Then since AB is to CD as EF is to GH, therefore CD is to O as GH is to P. Therefore, ex aequali, AB is to O as EF is to P. V.11 V.22 But AB is to O as KAB is to LCD, and EF is to P as MF is to NH, therefore KAB is to LCD also as MF is to NH. VI.19,Cor V.11 Next, let KAB be to LCD as MF is to NH. I say also that AB is to CD as EF is to GH. For, if EF is not to GH as AB is to CD, let EF be to QR as AB is to CD. Describe the rectilinear figure SR similar and similarly situated to either of the two MF or NH on QR. VI.12 VI.18 Since then AB is to CD as EF is to QR, and there have been described on AB and CD the similar and similarly situated figures KAB and LCD, and on EF and QR the similar and similarly situated figures MF and SR, therefore KAB is to LCD as MF is to SR. Above But also, by hypothesis, KAB is to LCD as MF is to NH, therefore also MF is to SR as MF is to NH. V.11 Therefore MF has the same ratio to each of the figures NH and SR, therefore NH equals SR. V.9 But it is also similar and similarly situated to it, therefore GH equals QR. And, since AB is to CD as EF is to QR, while QR equals GH, therefore AB is to CD as EF is to GH. Therefore, if four straight lines are proportional, then the rectilinear figures similar and similarly described upon them are also proportional; and, if the rectilinear figures similar and similarly described upon them are proportional, then the straight lines are themselves also proportional. Q.E.D. There is a missing step near the end of the proof, namely, the justification of the statement that GH equals QR is missing. Just before it we have NH and SR are similar equal rectilinear figures, and we want to conclude the corresponding sides GH and QR are equal. The needed demonstration is not difficult to supply. Use of this proposition This proposition is used in the proofs of several propositions in Book X and in XII.4 in Book XII. Next proposition: VI.23 Previous: VI.21 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 23 Equiangular parallelograms have to one another the ratio compounded of the ratios of their sides. Let AC and CF be equiangular parallelograms having the angle BCD equal to the angle ECG. I say that the parallelogram AC has to the parallelogram CF the ratio compounded of the ratios of the sides. Let them be placed so that BC is in a straight line with CG. Then DC is also in a straight line with CE. I.14 Complete the parallelogram DG. Set out a straight line K, and make it so that BC is to CG as K is to L, and DC is to CE as L is to M. I.31 VI.12 Then the ratios of K to L and of L to M are the same as the ratios of the sides, namely of BC to CG and of DC to CE. But the ratio of K to M is compounded of the ratio of K to L and of that of L to M, so that K has also to M the ratio compounded of the ratios of the sides. Now since BC is to CG as the parallelogram AC is to the parallelogram CH, and BC is to CG as K is to L, therefore K is to L as AC is to CH. VI.1 V.11 Again, since DC is to CE as the parallelogram CH is to CF, and DC is to CE as L is to M, therefore L is to M as the parallelogram CH is to the parallelogram CF. VI.1 V.11 Since then it was proved that K is to L as the parallelogram AC is to the parallelogram CH, and L is to M as the parallelogram CH is to the parallelogram CF, therefore, ex aequali K is to M as AC is to the parallelogram CF. V.22 But K has to M the ratio compounded of the ratios of the sides, therefore AC also has to CF the ratio compounded of the ratios of the sides. Therefore, equiangular parallelograms have to one another the ratio compounded of the ratios of their sides. Q.E.D. This proposition is a generalization of the basic formula for the area of a rectangle, that is, the area of a rectangle is the product of its length and width. Such a formula depends on predetermined units of length and area so that the unit area is the area of a square whose sides have length equal to the unit length. Euclid and other Greek mathematicians did not use predetermined units of length or area, so they expressed this formula as a proportion. We would state that proportion as saying the ratio of the area of a given rectangle to the area of a given square is the product of the ratios of the lengths of the sides of the rectangle to the length of a side of the square. Of course, Euclid would say that without using the words 'area' and 'length' as follows: the ratio of the a given rectangle to a given square is the product of the ratios of the sides of the rectangle to a side of the square. Note that his terminology for a product of ratios involves "compounding the ratios." A natural generalization of the ratio of a rectangle to a square is the ratio of a rectangle to a rectangle. A broader generalization is the ratio of one parallelogram to another parallelogram having the same angles. That gives the generalization as stated in this proposition. Areas of rectangles and parallelograms These areas have been treated earlier in the Elements. Back in Book I and II the basic concept was "quadrature," that is, finding a square or other shaped figure of the same area as the given rectangle or parallelogram. That began with Proposition I.35 which said two triangles on the same base and with the same height are equal, and ended with I.14 which constructed a square equal to a given rectangle. Early in this book was the proposition VI.1 generalizing I.35 which said that parallelograms with the same height are proportional to their bases. Finally, in this proposition we have the full statement about areas of rectangles and parallelograms. Analogous statements in other books Proposition VIII.5 states that plane numbers have to one another the ratio compounded of the ratios of their sides. That proposition is probably a much older version that may go back to the Pythagoreans when "all was number." The discovery of incommensurable lines showed there were serious limitations to that version of the proposition. In Book XI there are analogous statements for volumes of parallelopipeds. For instance, Proposition XI.33 states that similar parallelepipeds are to one another in the triplicate ratio of their corresponding sides. That statement for parallelepipeds is analogous to this one for parallelograms. Use of this theorem Although this is a basic proposition on areas, it is actually not used in the rest of the Elements. Next proposition: VI.24 Previous: VI.22 Book VI introduction © 1996, 2002 D.E.Joyce Clark University Proposition 24 In any parallelogram the parallelograms about the diameter are similar both to the whole and to one another. Let ABCD be a parallelogram, and AC its diameter, and let EG and HK be parallelograms about AC. I say that each of the parallelograms EG and HK is similar both to the whole ABCD and to the other. For, since EF is parallel to a side BC of the triangle ABC, proportionally, BE is to EA as CF is to FA. VI.2 Again, since FG is parallel to a side CD of the triangle ACD, proportionally, CF is to FA as DG is to GA. VI.2 But it was proved that CF is to FA as BE is to EA, therefore BE is to EA as DG is to GA. Therefore, taken jointly, BA is to AE as DA is to AG, and, alternately, BA is to AD as EA is to AG. V.18 V.16 Therefore in the parallelograms ABCD and EG, the sides about the common angle BAD are proportional. And, since GF is parallel to DC, the angle AFG equals the angle ACD, and the angle DAC is common to the two triangles ADC and AGF, therefore the triangle ADC is equiangular with the triangle AGF. I.29 For the same reason the triangle ACB is also equiangular with the triangle AFE, and the whole parallelogram ABCD is equiangular with the parallelogram EG. Therefore, proportionally, AD is to DC as AG is to GF, DC is to CA as GF is to FA, AC is to CB as AF is to FE, and CB is to BA as FE is to EA. And, since it was proved that DC is to CA as GF is to FA, and AC is to CB as AF is to FE, therefore, ex aequali, DC is to CB as GF is to FE. V.22 Therefore in the parallelograms ABCD and EG the sides about the equal angles are proportional. Therefore the parallelogram ABCD is similar to the parallelogram EG. VI.Def.1 For the same reason the parallelogram ABCD is also similar to the parallelogram KH. Therefore each of the parallelograms EG and HK is similar to ABCD. But figures similar to the same rectilinear figure are also similar to one another, therefore the parallelogram EG is also similar to the parallelogram HK. VI.21 Therefore, in any parallelogram the parallelograms about the diameter are similar both to the whole and to one another. Q.E.D. With this proposition Euclid returns to applications of areas. Back in proposition I.45 rectilinear areas were applied to lines. In the upcoming propositions VI.28 and VI.29, rectilinear areas will be applied to lines but the areas will fall short of or extend beyond the end of the lines. Those propositions geometric solve two kinds of quadratic equations. This proposition is preporatory to them. It is used in the proofs of VI.26 (its converse) and VI.29. Next proposition: VI.25 Previous: VI.23 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 25 To construct a figure similar to one given rectilinear figure and equal to another. Let ABC be the given rectilinear figure to which the figure to be constructed must be similar, and D that to which it must be equal. It is required to construct one figure similar to ABC and equal to D. Let there be applied to BC the parallelogram BE equal to the triangle ABC, and to CE the parallelogram CM equal to D in the angle FCE which equals the angle CBL. I.44 I.45 Then BC is in a straight line with CF, and LE with EM. Take a mean proportional GH to BC and CF, and describe KGH similar and similarly situated to ABC on GH. VI.13 VI.18 Then, since BC is to GH as GH is to CF, and, if three straight lines are proportional, then the first is to the third as the figure on the first is to the similar and similarly situated figure described on the second, therefore BC is to CF as the triangle ABC is to the triangle KGH. V.19,Cor But BC is to CF as the parallelogram BE is to the parallelogram EF. VI.1 Therefore also the triangle ABC is to the triangle KGH as the parallelogram BE is to the parallelogram EF. Therefore, alternately, the triangle ABC is to the parallelogram BE as the triangle KGH is to the parallelogram EF. V.11 V.16 But the triangle ABC equals the parallelogram BE, therefore the triangle KGH also equals the parallelogram EF. And the parallelogram EF equals D, therefore KGH also equals D. (V.14) And KGH is also similar to ABC. Therefore this figure KGH has been constructed similar to the given rectilinear figure ABC and equal to the other given figure D. Q.E.F. Note that it isn't proposition V.14 being invoked near the end of the proof, but an alternate form of it. See the Guide to V.14. This proposition solves a similar problem, to find a figure with the size of one figure but the shape of another, a problem reputedly solved by Pythagoras. It is used in the proofs of propositions VI.28 and VI.29 Next proposition: VI.26 Previous: VI.24 Book VI introduction © 1996, 1997 D.E.Joyce Clark University Proposition 26 If from a parallelogram there is taken away a parallelogram similar and similarly situated to the whole and having a common angle with it, then it is about the same diameter with the whole. From the parallelogram ABCD let there be taken away the parallelogram AF similar and similarly situated to ABCD, and having the angle DAB common with it. I say that ABCD is about the same diameter with AF. For suppose it is not, but, if possible, let AHC be the diameter. Produce GF and carry it through to H. Draw HK through H parallel to either of the straight lines AD or BC. I.31 Since, then, ABCD is about the same diameter with KG, therefore DA is to AB as GA is to AK. VI.24 But also, since ABCD and EG are similar, therefore DA is to AB as GA is to AE. Therefore GA is to AK as GA is to AE. VI.Def.1 V.11 Therefore GA has the same ratio to each of the straight lines AK and AE. Therefore AE equals AK the less equals the greater, which is impossible. V.9 Therefore ABCD cannot fail to be about the same diameter with AF. Therefore the parallelogram ABCD is about the same diameter with the parallelogram AF. Therefore, if from a parallelogram there is taken away a parallelogram similar and similarly situated to the whole and having a common angle with it, then it is about the same diameter with the whole. Q.E.D. This proposition is the converse of VI.24. It is used in the proofs of the next three and for a few propositions in Book X. Next proposition: VI.27 Previous: VI.23 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 27 Of all the parallelograms applied to the same straight line falling short by parallelogrammic figures similar and similarly situated to that described on the half of the straight line, that parallelogram is greatest which is applied to the half of the straight line and is similar to the difference. Let AB be a straight line and let it be bisected at C. Let there be applied to the straight line AB the parallelogram AD falling short by the parallelogrammic figure DB described on the half of AB, that is, CB. I say that, of all the parallelograms applied to AB falling short by parallelogrammic figures similar and similarly situated to DB, AD is greatest. Let there be applied to the straight line AB the parallelogram AF falling short by the parallelogrammic figure FB similar and similarly situated to DB. I say that AD is greater than AF. Since the parallelogram DB is similar to the parallelogram FB, therefore they are about the same diameter. VI.26 Draw their diameter DB, and describe the figure. Then, since CF equals FE, and FB is common, therefore the whole CH equals the whole KE. I.43 But CH equals CG, since AC also equals CB. I.36 Therefore CG also equals KE. Add CF to each. Therefore the whole AF equals the gnomon LMN, so that the parallelogram DB, that is, AD, is greater than the parallelogram AF. Therefore, of all the parallelograms applied to the same straight line falling short by parallelogrammic figures similar and similarly situated to that described on the half of the straight line, that parallelogram is greatest which is applied to the half of the straight line and is similar to the difference. Q.E.D. This proposition clarifies the limitations of the next one, VI.28. In VI.28 a construction is made to apply a parallelogram equal to a given rectilinear figure to a line falling short by a parallelogrammic figure. This proposition implies that that construction cannot be made if the given rectilinear figure is too large. When that proposition is applied, the part which falls short is usually a square, not just any parallelogram, and this and the next proposition are much more easily understood in that case. In that case, the next proposition applies a rectangle equal to a given area to a line but falling short by a square. And this proposition implies that can only be done if the given area is at least the square on half the line, since that square is the greatest rectangle that can be so applied. Next proposition: VI.28 Previous: VI.26 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 28 To apply a parallelogram equal to a given rectilinear figure to a given straight line but falling short by a parallelogram similar to a given one; thus the given rectilinear figure must not be greater than the parallelogram described on the half of the straight line and similar to the given parallelogram. VI.27 Let C be the given rectilinear figure, AB the given straight line, and D the given parallelogram, and let C not be greater than the parallelogram described on the half of AB similar to the given parallelogram D. It is required to apply a parallelogram equal to the given rectilinear figure C to the given straight line AB but falling short by a parallelogram similar to D. Bisect AB at the point E. Describe EBFG similar and similarly situated to D on EB, and complete the parallelogram AG. I.9 VI.18 If then AG equals C, that which was proposed is done, for the parallelogram AG equal to the given rectilinear figure C has been applied to the given straight line AB but falling short by a parallelogram GB similar to D. But, if not, let HE be greater than C. Now HE equals GB, therefore GB is also greater than C. Construct KLMN equal to GB minus C and similar and similarly situated to D. VI.25 But D is similar to GB, therefore KM is also similar to GB. VI.21 Let, then, KL correspond to GE, and LM to GF. Now, since GB equals C and KM, therefore GB is greater than KM, therefore also GE is greater than KL, and GF than LM. Make GO equal to KL, and GP equal to LM, and let the parallelogram OGPQ be completed, therefore it is equal and similar to KM. Therefore GQ is also similar to GB, therefore GQ is about the same diameter with GB. VI.21 VI.26 Let GQB be their diameter, and describe the figure. Then, since BG equals C and KM, and in them GQ equals KM, therefore the remainder, the gnomon UWV, equals the remainder C. And, since PR equals OS, add QB to each, therefore the whole PB equals the whole OB. But OB equals TE, since the side AE also equals the side EB, therefore TE also equals PB. I.36 Add OS to each. Therefore the whole TS equals the whole, the gnomon VWU. But the gnomon VWU was proved equal to C, therefore TS also equals C. Therefore there the parallelogram ST equal to the given rectilinear figure C has been applied to the given straight line AB but falling short by a parallelogram QB similar to D. Q.E.F. When this proposition is used, the given parallelgram D usually is a square. Then the problem is to cut the line AB at a point S so that the rectangle AS by SB equals the given rectilinear figure C. This special case can be proved with the help of the propositions in Book II. See the Guide to proposition II.5 for more details. The outline of a simplified proof for rectangles The proof of the current propostion is difficult to follow. It is simplified when we take the special case mentioned above, namely, when the given parallelogram D is a square. The simplified proof is easier to follow since the rest of the parallelograms mentioned all become rectangles. The construction is as follows. Bisect AB at E, construct a square GFBE. The next stage is to construct a square GPQO equal to the square GFBE minus the figure C. For an alternate construction of GPQO, see the lemma for X.14 which applies I.47 to do that.) Complete the figure. We can understand the meaning of this construction more easily if we interpret it algebraically. Let a stand for the known quantity AB, and c the known quantity C. Then let x and y stand for the unknown quantities SB and SA. Then this construction finds x and y so that their sum is a and their product is c. In terms of the single variable x, the construction solves the quadratic equation ax – x2 = C. The next proposition solves a similar quadratic equation: ax + x2 = C. Use of this proposition This constuction in this proposition is used in propositions X.33 and X.34. Next proposition: VI.29 Previous: VI.27 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 29 To apply a parallelogram equal to a given rectilinear figure to a given straight line but exceeding it by a parallelogram similar to a given one. Let C be the given rectilinear figure, AB be the given straight line, and D the parallelogram to which the excess is required to be similar. It is required to apply a parallelogram equal to the the rectilinear figure C to the straight line AB but exceeding it by a parallelogram similar to D. Bisect AB at E. Describe the parallelogram BF on EB similar and similarly situated to D, and construct GH equal to the sum of BF and C and similar and similarly situated to D. VI.25 Let KH correspond to FL and KG to FE. Now, since GH is greater than FB, therefore KH is also greater than FL, and KG greater thanFE. Produce FL and FE. Make FLM equal to KH, and FEN equal to KG. Complete MN. Then MN is both equal and similar to GH. But GH is similar to EL, therefore MN is also similar to EL, therefore EL is about the same diameter with MN. VI.21 VI.26 Draw their diameter FO, and describe the figure. Since GH equals the sum of EL and C, while GH equals MN, therefore MN also equals the sum of EL and C. Subtract EL from each. Therefore the remainder, the gnomon XWV, equals C. Now, since AE equals EB, therefore AN equals NB, that is, LP. I.36 I.43 Add EO to each. Therefore the whole AO equals the gnomon VWX. But the gnomon VWX equals C, therefore AO also equals C. Therefore the parallelogram AO equal to the given rectilinear figure C has been applied to the given straight line AB but exceeding it by a parallelogram QP similar to D, since PQ is also similar to EL. VI.24 Q.E.F. The construction in this proposition is a generalization of that described in the Guide for II.6. In that proposition, the figure D is a square. The outline of a simplified proof for rectangles Like the last proposition, this one is more easily understood when the given parallelogram D is a square. In that case of this proposition a rectangle AO equal to a given rectilinear figure C is applied to a given straight line AB but exceeds it by a square (BQOP in the figure). So the rectangle being laid alongside the line extends past the end of the line AB, but the part that extends beyond the end is a square. For the construction, bisect AB at E, construct the square BEFL, then make the square FNOM equal to the square BEFL plus the figure C, and complete the figure. As in the last proposition we can understand the meaning of this construction more easily if we interpret it algebraically. Let a stand for the known quantity AB and x stand for the unknown quantity BP. Then this construction finds x so that (a + x) x = C. In other words it solves the quadratic equation ax + x2 = C. The construction of this proposition is used in the next proposition to cut a straight line in extreme and mean ratio. Next proposition: VI.30 Previous: VI.28 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 30 To cut a given finite straight line in extreme and mean ratio. Let AB be the given finite straight line. It is required to cut AB in extreme and mean ratio. Describe the square BC on AB. Apply the parallelogram CD to AC equal to the sum of BC and the figure AD similar to BC. I.46 VI.29 Now BC is a square, therefore AD is also a square. And, since BC equals CD, subtract CE from each, therefore the remainder BF equals the remainder AD. But it is also equiangular with it, therefore in BF and AD the sides about the equal angles are reciprocally proportional. Therefore FE is to ED as AE is to EB. VI.14 But FE equals AB, and ED equals AE. Therefore AB is to AE as AE is to EB. V.7 And AB is greater than AE, therefore AE is also greater than EB. Therefore the straight line AB has been cut in extreme and mean ratio at E, and the greater segment of it is AE. VI.Def.3 Q.E.F. The construction given here cuts a line into two parts A and B so that (A + B):B = B:A. By proposition VI.17, that condition is equivalent to making the rectangle A + B by A equal the square B by B, and that is the construction of proposition II.11. Use of this proposition This construction is used in XIII.17 to construct a pentagonal face of a regular dodecahedron. Next proposition: VI.31 Previous: VI.29 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 31 In right-angled triangles the figure on the side opposite the right angle equals the sum of the similar and similarly described figures on the sides containing the right angle. Let ABC be a right-angled triangle having the angle BAC right. I say that the figure on BC equals the sum of the similar and similarly described figures on BA and AC. Draw the perpendicular AD. I.12 Then, since in the right-angled triangle ABC, AD has been drawn from the right angle at A perpendicular to the base BC, therefore the triangles DBA and DAC adjoining the perpendicular are similar both to the whole ABC and to one another. VI.8 And, since ABC is similar to DBA, therefore BC is to BA as BA is to BD. VI.Def.1 And, since three straight lines are proportional, the first is to the third as the figure on the first is to the similar and similarly described figure on the second. VI.19,Cor Therefore BC is to BD as the figure on BC is to the similar and similarly described figure on BA. For the same reason also, BC is to CD as the figure on BC is to that on CA, so that, in addition, BC is to the sum of BD and DC as the figure on BC is to the sum of the similar and similarly described figures on BA and AC. V.24 But BC equals the sum of BD and DC, therefore the figure on BC equals the sum of the similar and similarly described figures on BA and AC. Therefore, in right-angled triangles the figure on the side opposite the right angle equals the sum of the similar and similarly described figures on the sides containing the right angle. Q.E.D. This proposition is a generalization of I.47 where the squares in I.47 are replaced by any similar rectilinear figures. Hippocrates' quadrature of lunes Proclus says that this proposition is Euclid's own, and the proof may be his, but the result, if not the proof, was known long before Euclid, at least in the time of Hippocrates a century before Euclid. The broad problem Hippocrates was investigating was that of quadrature of a circle, also called squaring a circle, which is to find a square (or any other polygon) with the same area as a given circle. Hippocrates did not solve that problem, but he did solve a related one involving lunes. A lune (also called a crescent) is a region of nonoverlap of two intersecting circles. Hippocrates did not succeed in squaring an arbitrary lune, but he did succeed in a couple special cases. Here is a summary of the simplist case. Draw a square ABCD with diameters AC and BD meeting at E. Circumscribe a semicircle AGBHC about the right isosceles triangle ABC. Draw the arc AFC from A to C of a circle with center D and radius DA. Hippocrates finds the area of the lune formed between the semicircle AGBHC and the arc AF as follows. Note that there are three segments of circles in the diagram; two of them are small, namely, AGB with base AB, and BHC with base BC, and one is large, namely, AFC with base AC. The first two are congruent, and the third is similar to them since all three are segments in quarters of circles. Hippocrates then uses a version of this proposition VI.31-generalized so the figures don't have to be rectilinear but may have curved sides- to conclude that the sum of the two small segments, AGB + BHC, equals the large segment AFC, since the bases of the small segments are sides of a right triangle while the base of the large segment is the triangle's hypotenuse. Therefore, the lune, which is the semicircle minus the large segment, equals the semicircle minus the sum of the small segments. But the semicircle minus the sum of the small segments is just the right triangle ABC. Thus, a rectilinear figure (the triangle) has been found equal to the lune, as required. Note that at Hippocrates' time, Eudoxus' theory of proportion had not been developed, so the understanding of the theory of similar figures (Book V) was not as complete as it was after Eudoxus. Also, Eudoxus' principle of exhaustion (see Book XI and proposition X.1) for finding areas of curved figures was still to come, so the concept of area of curved figures was on shaky ground, too. Such a situation is common in mathematics-mathematics advances into new territory long before the foundations of mathematics are developed to logically justify those advances. Next proposition: VI.32 Previous: VI.30 Book VI introduction © 1996, 2002 D.E.Joyce Clark University Proposition 32 If two triangles having two sides proportional to two sides are placed together at one angle so that their corresponding sides are also parallel, then the remaining sides of the triangles are in a straight line. Let ABC and DCE be two triangles having the two sides AB and AC proportional to the two sides DC and DE, so that AB is to AC as DC is to DE, and AB parallel to DC, and AC parallel to DE. I say that BC is in a straight line with CE. Since AB is parallel to DC, and the straight line AC falls upon them, therefore the alternate angles BAC and ACD equal one another. I.29 For the same reason the angle CDE also equals the angle ACD, so that the angle BAC equals the angle CDE. And, since ABC and DCE are two triangles having one angle, the angle at A, equal to one angle, the angle at D, and the sides about the equal angles proportional, so that AB is to AC as DC is to DE, therefore the triangle ABC is equiangular with the triangle DCE. Therefore the angle ABC equals the angle DCE. VI.6 But the angle ACD was also proved equal to the angle BAC, therefore the whole angle ACE equals the sum of the two angles ABC and BAC. Add the angle ACB to each. Therefore the sum of the angles ACE and ACB equals the sum of the angles BAC, ACB, and CBA. But the sum of the angles BAC, ABC, and ACB equals two right angles, therefore the sum of the angles ACE and ACB also equals two right angles. I.32 Therefore with a straight line AC, and at the point C on it, the two straight lines BC and CE not lying on the same side make the sum of the adjacent angles ACE and ACB equal to two right angles. Therefore BC is in a straight line with CE. I.14 Therefore, if two triangles having two sides proportional to two sides are placed together at one angle so that their corresponding sides are also parallel, then the remaining sides of the triangles are in a straight line. Q.E.D. The corresponding sides mentioned in the statement of the proposition are supposed to be directed in the same direction, even though that is not explicitly stated. This proposition is used in the proof of proposition XIII.17 which inscribes a regular dodecahedron in a sphere. Next proposition: VI.33 Previous: VI.31 Book VI introduction © 1996 D.E.Joyce Clark University Proposition 33 Angles in equal circles have the same ratio as the circumferences on which they stand whether they stand at the centers or at the circumferences. Let ABC and DEF be equal circles, and let the angles BGC and EHF be angles at their centers G and H, and the angles BAC and EDF angles at the circumferences. I say that the circumference BC is to the circumference EF as the angle BGC is to the angle EHF, and as the angle BAC is to the angle EDF. Make any number of consecutive circumferences CK and KL equal to the circumference BC, and any number of consecutive circumferences FM, MN equal to the circumference EF, and join GK and GL and HM and HN. Then, since the circumferences BC, CK, and KL equal one another, the angles BGC, CGK, and KGL also equal one another. Therefore, whatever multiple the circumference BL is of BC, the angle BGL is also that multiple of the angle BGC. III.27 For the same reason, whatever multiple the circumference NE is of EF, the angle NHE is also that multiple of the angle EHF. If the circumference BL equals the circumference EN, then the angle BGL also equals the angle EHN; if the circumference BL is greater than the circumference EN, then the angle BGL is also greater than the angle EHN; and, if less, less. III.27 There being then four magnitudes, two circumferences BC and EF, and two angles BGC and EHF, there have been taken, of the circumference BC and the angle BGC equimultiples, namely the circumference BL and the angle BGL, and of the circumference EF and the angle EHF equimultiples, namely the circumference EN and the angle EHN. And it has been proved that, if the circumference BL is in excess of the circumference EN, the angle BGL is also in excess of the angle EHN; if equal, equal; and if less, less. Therefore the circumference BC is to EF as the angle BGC is to the angle EHF. V.Def.5 But the angle BGC is to the angle EHF as the angle BAC is to the angle EDF, for they are doubles respectively. V.15 III.20 Therefore also the circumference BC is to the circumference EF as the angle BGC is to the angle EHF, and the angle BAC to the angle EDF. Therefore, angles in equal circles have the same ratio as the circumferences on which they stand whether they stand at the centers or at the circumferences. Q.E.D. This proposition stands apart from the rest in this book since it depends on none of them, but it is like the first proposition VI.1 which established a proportion between lines and plane figures since it establishes a proportion between angles and portions of circumferences cut off by those angles. This proposition is used in three consecutive propositions in Book XIII starting with XIII.8 to convert statements about arcs to statements about angles. Incidentally, all three have to do with regular pentagons inscribed in circles. Next book: Book VII Previous proposition: VI.32 Book VI introduction © 1996 D.E.Joyce Clark University Definitions 1 and 2 Def. 1. A magnitude is a part of a magnitude, the less of the greater, when it measures the greater. Def. 2. The greater is a multiple of the less when it is measured by the less. The two magnitudes mentioned in each definition are of the same kind. Following Euclid, they are illustrated here as lines, but they could both be planar figures, or solids, or angles, or any other kind of magnitude so long as they are of the same kind. The illustration shows two magnitudes, A and B, and A is one third of B since A measures B three times. Thus, A is a part of B, and B is a multiple of A. Next definition: V.Def.3-4 Book V introduction © 1997 D.E.Joyce Clark University Definition 3 A ratio is a sort of relation in respect of size between two magnitudes of the same kind. A convenient notation for a ratio of two magnitudes A and B of the same kind is A:B. No mixed ratios All of Euclid's ratios are pure ratios of two magnitudes of the same kind, in other words, there are no mixed ratios in the Elements. A familiar example of a mixed ratio is velocity, the ratio of a distance to a time, measured in units such as kilometers/hour. That isn't to say that ratios of different kinds of magnitudes aren't equated. In fact, that's one of the more important aspects of ratios. For example, the fundamental proposition of Book VI, proposition VI.1, says that given two triangles of the same height, the ratio of the triangles A:B is the same as the ratio of their heights Ah:Bh. That says that the ratio of two plane figures equals the ratio of two lines. Now, a common operation on proportions (equalities of ratios) is that of alternation (see V.Def.12 and V.16) which in its general form says that if A:B = C:D, then A:C = B:D. In the Elements alternation only applies when all four quantities are of the same kind. But if alternation is applied to the proportion of VI.1, then we get A:Ah = B:Bh, the equality of two mixed ratios, ratios of plane figures to lines. This step, and the acceptance of mixed ratios, which seems to us like a small thing, was not taken until centuries after Euclid. The nature of ratios A ratio is a pair of magnitudes of the same kind considered as a pair, but soon identified with other ratios. Definition 3 promises that ratios have sizes, that is, given two ratios A:B and C:D, either the first ratio is greater, equal, or less than the second ratio. That promise begins to be fulfilled in Definitions V.Def.5 and V.Def.7 where equality and order of ratios defined. Note that equality and order are defined for ratios, but they were assumed for numbers and magnitudes. Since equality and order are defined, their expected properties are proved in propositions, or at least some of the properties. For example, proposition V.11 states that two ratios that are the same as a third are the same as each other, a statement analogous to C.N.1 for magnitudes. Equivalence relations (Equivalence relations were mentioned before in the guide for the Common Notions. It was mentioned there that equality of magnitudes of the same kinds is an equivalence relation.) The process used for defining ratios of magnitudes was something new for Eudoxus and Euclid, but that process is now commonplace in mathematics to construct new kinds of things. The process starts with entities x, y, z, etc., that are well understood, such as pairs of magnitudes of the same kind. Then a relation E on these entities is found which is intended to be equality for them. For ratios, that is given in V.Def.5. Right now, let xEy denote that x is related to y by the relation E. Next, it is verified that the relation E is an equivalence relation, that is, a reflexive, symmetric, and transitive relation. A relation E is reflexive if for any x it is the case that xEx, that is, anything is related to itself by E. A relation is E is symmetric if whenever xEy, then yEx. And it is transitive if whenever xEy and yEz, then xEz. Once E is known to be an equivalence relation, new entities are conceived which are named by the old entities x, y, z, etc., but the new entities are taken to be equal, x = y, when their names are equivalent under the relation E, that is, xEy. Proportion as an equivalence relation is discussed in the Guide to definition V.Def.5. Operations on ratios and proportions, compounded ratios There are several operations on ratios and proportions defined soon. For instance, V.Def.9 defines duplicate ratios, under certain assumptions, which may be thought of as the squares of ratios. See also definitions V.Def.12 through V.18. But ratios are neither numbers nor magnitudes, and the usual operations of addition, subtraction, multiplication, and division that apply to numbers don't apply to ratios. Numbers can be added and subtracted, and so can magnitudes of the same kind, but ratios cannot. Take for example a ratio A:B of plane figures and a ratio C:D of angles. What could be meant by their sum (A:B) + (C:D)? One obvious approach is to treat ratios as quotients. That suggests A/B + C/D = (AD + BC)/BD, but a product of a plane figure and an angle, such as AD, has no meaning, so the obvious approach has obvious difficulties. Multiplication and division are not automatic for ratios. Ratios A:B and B:C are compounded to form A:C, which may be thought of as the product of the two ratios, and the duplicate ratio mentioned above is a special case of a compound ratio. But the compound of two ratios A:B and C:D depends on the middle terms B and C being the same. The proof of proposition V.18 assumes that fourth proportionals exist, a property unjustified by any postulate, but if fourth proportionals do exist, then the ratio C:D is equal to some ratio B:E, and then the compound of A:B and C:D is the compound of A:B and B:E, and that compound is A:E. Thus, multiplication is an operation when fourth proportionals exist. Division is also an operation when fourth proportionals exist since D:C may be thought of as the reciprocal of C:D. Ratios of various kinds Several kinds of ratios appear in the Elements. There are ratios of numbers, ratios of lines constructable in plane geometry, ratios of rectilinear angles, ratios of plane figures constructable in plane geometry, and ratios of solids. Numeric ratios, that is, ratios of numbers, are treated in the books on number theory, Books VII through VIII. In modern terminology these numeric ratios are called "positive rational numbers." Numeric ratios and proportions have a separate, simpler definition in VII.Def.20. That definition is compatible with the definitions here in Book V, but that compatibility is not demonstrated in the Elements. The problem with numeric ratios is that there are not enough of them. That is ratios of magnitudes are not always equal to ratios of numbers. The illustration to the right shows a square with side A and diameter B. The ratio B to A does exist according to the next definition V.Def.4 since some multiple of each is greater than the other. In modern terms this ratio would be identified with the square root of 2 and is known to be an irrational number, that is, it is not equal to a numeric ratio. It is, nonetheless, a ratio in Euclid's terminology. The ratio B:A is a ratio of lines, but it is not a ratio of numbers. Since this and other ratios of lines are not ratios of numbers, a more general definition of ratio is required. That more general definition is the one given here and continuing through V.Def.6. In modern terminology, the numeric ratios are positive rational numbers. The field of all rational numbers including 0 and the negative rational numbers is commonly denoted Q. The ratios of lines constructable in plane geometry form the field extended from Q by closure under square roots. A convenient notation for that field is Q . It is a much larger field, but does not include all real numbers. For instance, the cube root of 2, needed for doubling a cube, the sine of 20°, needed for trisecting angles, and pi, needed for squaring the circle, all are missing from Q . The conic sections are part of solid geometry but they are not treated in the Elements. Cones are discussed in Book XII, but their sections (intersections with planes) which include ellipses, parabolas, and hyperbolas are not even defined in the Elements. Euclid's work on the Conics was superceded by Apollonius' and no longer exists. Intersections of conics lead to lines of new lengths that can be used to solve problems such as doubling a cube and trisecting an angle, but they don't help in squaring the circle. Thus, there are more ratios of lines constructable in solid geometry than ratios of lines constructable in plane geometry. The ratios of rectilinear figures form the same field Q as the ratios of lines. This follows from the theory of application of areas developed in Book I, see proposition I.44. But there are other plane figures besides rectilinear ones: circles. The ratio of a circle to the square on its radius is pi. Thus, pi is a ratio of plane figures even though it is not a ratio of lines. Ratios of more than two terms Throughout Book V the only ratios that are considered are those with two terms in accordance with V.Def.3, but in Book VII ratios of three or more terms are used in proposition VI.33. In that proposition, a ratio A:B:C of three numbers is considered, and a certain proportion A:B:C = E:F:G is shown. These multiterm ratios and proportions are probably left over from an earlier time. In any case, the multiterm proportion may be interpreted as an abbreviation for two proportions A:B = E:F and B:C = F:G. Then it follows ex aequali that A:C = E:G. Next defintion: V.Def.4 Previous: V.Def.1-2 Book V introduction © 1997 D.E.Joyce Clark University Definition 4 Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another. This definition limits the existence of ratios to comparable magnitudes of the same kind where comparable means each, when multiplied, can exceed the other. The ratio doesn't exist when one magnitude is so small or the other so large that no multiple of the one can exceed the other. This definition excludes the ratio of a finite straight line to an infinite straight line and the ratio of an infinitesimal straight line, should any exist, to a finite straight line. The result on horn angles in proposition III.16 excludes ratios between horn angles and rectilinear angles. That proposition states that a horn angle is less than any rectilinear angle, hence no multiple of a horn angle is greater than a rectilinear angle. The situation of horn angles is much worse than that, however, since horn angles of different sizes aren't even comparable. Definition 4 as an axiom of comparability This definition is used repeatedly as a axiom for magnitudes rather than a definition. It is frequently invoked in this book, starting with proposition V.8 but also required for more fundamental properties, and elsewhere, such as the important proposition X.1. In the proofs of these propositions one magnitude is less than another, and it is asserted that some multiple of the smaller is greater than the larger. Euclid implicitly assumes that the magnitudes he discusses, except horn angles, are all comparable. Straight lines, rectilinear angles, plane figures, and solids are all comparable to any other of the same type. This principle of comparability should be explicit in order to justify the principle of comparability for magnitudes of these kinds. One solution is to make it a postulate that straight lines are comparable. From that postulate comparability of each of the other kinds of magnitudes could be proved. Several of the propositions, stated and unstated, depend on this principle. Without it, some are simply false for kinds of magnitude that have infinitesimals. If x and y are two magnitudes of the same kind, then x is infinitesimal with respect to y, or y is infinite with respect to x, if no multiple of x is greater than y. For example, horn angles are infinitesimal with respect to rectilinear angles. Although this definition excludes ratios between horn angles and rectilinear angles, it allows a ratio between a rectilinear angle B and the sum of a horn angle A and the rectilinear angle, and, according to the next three definitions, the two ratios B:(A + B) and B:B so not satisfy the law of trichotomy, that is, they aren't the same ratio but neither is greater than the other either. Examples involving infinitesimals can be useful to show which propositions require treating this definition as an axiom. Next devinition: V.Def.5-6 Previous: Definition V.Def.3 Book V introduction © 1997 D.E.Joyce Clark University Definitions 5 and 6 Def. 5. Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever are taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order. Def. 6. Let magnitudes which have the same ratio be called proportional. Definition 5 defines two ratios w:x and y:z to be the same, written w:x = y:z, when for all numbers n and m it is the case that if nw is greater, equal, or less than mx, then ny is greater, equal, or less than mz, respectively, that is, if nw > mx, then ny > mz, if nw = mx, then ny = mz, and if nw < mx, then ny < mz. It is very convenient to use the shorter notation if nw >=< mx, then ny >=< mz. Note that whenever the symbol >=< is used there are three parallel statements being made. The four magnitudes do not all have to be of the same kind, but the first pair w and x need to be of the one kind, and the second pair y and z of one kind, either the same kind as that of w and x or a different kind. Perhaps the best illustration of these definitions comes from proposition VI.1 in which Euclid first uses them to construct a proportion. The goal in this proposition is to show that the lines are proportional to the triangles. More precisely, the line BC is to the line CD as the triangle ABC is to the triangle ACD, that is, the ratio BC:CD of lines is the same as the ratio ABC:ACD of triangles. Even though the ratios derive from different kinds of magnitudes, they are to be compared and shown equal. According to Definition 5, in order to show the ratios are the same, Euclid takes any one multiple of BC and ABC (which he illustrates by taking three times each), and any one multiple of CD and ACD (which he also illustrates by taking three times each). Then he proceeds to show that the former equimultiples, namely HC and CL, alike exceed, are alike equal to, or alike fall short of, the latter equimultiples, namely, AHC and ACL. Symbolically, in order to prove BC:CD = ABC:ACD, Euclid proves for any numbers n and m that the line n BC is greater, equal, or less than the line m CD when the triangle n ABC is greater, equal, or less than the triangle m ACD. We will abbreviate this condition symbolically as if n BC >=< m CD, then n ABC >=< m ACD. Note that in order to check this condition, it is only necessary to compare lines to lines and planar figures to planar figures. To see how Euclid does this, refer to VI.1. Numerical ratios and commensurability As it sometimes happens, a ratio of two magnitudes A:B is the same as a ratio of numbers m:n. Take for instance the case when A is a line that is twice a line U while B is a line that is three times the line U. Then, we could show that the ratio of magnitudes A:B is the same as the numrical ratio 2:3. Such ratios are studied in detail in Book X. That book begins by defining in X.Def.1. what it means for two quantities to be "commensurable." For instance, the two lines A and B are commensurable since there is a unit U that measures both. Later in Book X (propositions X.5 and X.6) it is explicitly shown that two magnitudes are commensurable if and only if their ratio is a numeric ratio. Using modern concepts and notations, we can more easily see what the general definition of equality of two magnitudes means. If we treat ratios as real numbers, the a proportion such as the one described above, BC:CD = ABC:ACD, means that the ratio BC:CD compares to all numerical ratios (that is, rational numbers) m/n the same way that ABC:ACD does. Another way of saying this is that equality of two real numbers is determined by their relation to all rational numbers. This is often expressed by saying that the set of rational numbers is dense in the set of real numbers. Of course, Euclid did not have what modern mathematicians call real numbers. Indeed, there is an ontological difference between real numbers and Euclid's ratios. Some real numbers are not ratios of the magnitudes of any kind mentioned in the Elements. Proportions as equivalence relations Equivalence relations were defined in the Guide for V.Def.3. Three things need to be checked to see if proportion is an equivalence relation: reflexivity, symmetry, and transitivity. First, reflexivity. Is it the case for any pair of magnitudes of the same type A and B that A and B are in the same ratio as A and B? That means for any numbers m and n, if nA >=< mB, then nA >=< mB. That is trivial. Second, symmetry. Is it the case that if A and B are in the same ratio as C and D, then C and D are in the same ratio as A and B? The first says if nA >=< mB, then nC >=< mD, while the second says if nC >=< mD, then nA >=< mB. This can be shown using the law of trichotomy for magnitudes. (Suppose nC > mD. If nA is not greater than mB, then it is less or equal, but then nC is less or equal to mD, contradicting nC > mD. etc.) Euclid missed symmetry, but he uses it very frequently. Third, transitivity. Euclid states this explicitly in proposition V.11. The proof relies only on the definition. Thus, proportion is an equivalence relation. Are proportions equalities of ratios? When A and B are in the same ratio as C and D, then the four magnitudes are said to be proportional, or in proportion, according to definition 6. Is that the same as saying the ratios A:B and C:D are equal? A more fundamental question is "do ratios exist?" Are they some kind of mathematical object like numbers and magnitudes? The Elements do not require it. Instead, proportion is a relation held between one pair of magnitudes and another pair of magnitudes. Yet it is very easy to read Book V as though ratios are mathematical objects of some abstract variety. And it's easy to read "A and B have the same ratio as C and D" as saying that the ratio A:B is the same ratio as C:D. Not every relation allows that reading, but equivalence relations do, and proportion is an equivalence relation. The philosophical questions "do ratios exist?" and "is a proportion equality of ratios?" can be converted to the question "why do equivalence relations create entities?" or a little more conservatively, "why do equivalence relations allow us to think and act as if the entities exist?" It is hard to imagine that Euclid did not think of ratios as things and proportions as equalities, especially since the next definition defines when one ratio is larger than another. Perhaps he did but continued to write noncommittally. Proportions are written as equalities in the Guide. Next definition: V.Def.7 Previous: V.Def.4 Book V introduction © 1997, 2002 D.E.Joyce Clark University Definition 7 When, of the equimultiples, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then the first is said to have a greater ratio to the second than the third has to the fourth. Definition 5 explained when two ratios were equal, namely, w:x = y:z when for all numbers n and m, if nw >=< mx, then ny >=< mz. Definition 7 now says w:x > y:z when there are numbers n and m such that nw > mx but ny is not greater than mz. Of course, y:z is called the lesser ratio. When defining greater and lesser, there are a number of properties that should be verified. These are various transitivities and the law of trichotomy, some of the same properities of greater and lesser that magnitudes have. (See the Guide for the Common Notions.) If u:v < w:x, and w:x = y:z, then u:v < y:z. If u:v = w:x, and w:x < y:z, then u:v < y:z. If u:v < w:x, and w:x < y:z, then u:v < y:z. Euclid only has the first property, which is is proposition V.13. Its proof depends only on the definitions. The second is so much like it that it isn't mentioned but it is used in the same way as the third. The third one is quite easy to prove. The law of trichotomy for ratios Stated for ratios, the law of trichotomy says that for any two ratios w:x and y:z, exactly one of the following three cases holds: w:x < y:z, or w:x = y:z, orw:x > y:z. Euclid missed the law of trichotomy for magnitudes in his list of Common Notions, and he missed it for ratios, too. The side of the law which says at most one of the three cases can occur is first used in proposition V.9, while the side which says at least one occurs is first used in V.10. The side of the law which says at most one of the three cases can occur is fairly easy to prove. From the definitions themselves it is clear that w:x > y:z contradicts w:x = y:z. The first says there are n and m such that nw > mx but ny is not greater than mz, while the second concludes from nw > mx that ny > mz yielding a contradiction based on the law of trichotomy for magnitudes. Similarly w:x < y:z contradicts w:x = y:z. Once transitivity has been shown, it can be shown that w:x > y:z contradicts w:x < y:z, for then w:x > w:x, and that contradicts w:x = w:x. (There are also proofs that don't depend on transitivity.) The other side of the law of trichotomy, the one that says at least one of the three cases holds, is a bit harder to prove, and it depends on treating V.Def.4 as an axiom of comparability. In fact, it is false without it. First, a proof using V.Def.4 as an axiom, then a counterexample to show that's necessary. A proof of trichotomy. Let w:x and y:z be any two ratios. We need to show that one of the three cases holds. We'll assume the ratios aren't the same and show one of them is greater than the other. When they're not the same, then there are numbers m and n such that nw >=< mx but not ny >=< mz. We have three cases to consider, and two of them are easy. In one case, nw > mx but not ny > mz, so for that case w:x > y:z. In another case, nw < mx but not ny < mz, so for that case w:x < y:z. Consider now the last case: nw = mx but ny does not equal mz. Then one of ny and mz is greater, say ny > mz. Now using V.Def.4 as an axiom, there is some some number k such that k(ny mz) > z. Since k(ny mz) = kny kmz, therefore kny > kmz + z, that is, kny > (km + 1)z. But knw = kmx, and kmx < (km + 1)x. Therefore, (kn)w < (km + 1)x. But (kn)y > (km + 1)z. Therefore w:x < y:z. Q.E.D. A counterexample to trichotomy. This counterexample has infinitesimals, so it doesn't satisfy the axiom of comparability, that is, V.Def.4 treated as an axiom. Let y be an infinitesimal with respect to x, that is, for any number n, ny < x. We'll show that the two ratios x:x and x:(x + y) do not satisfy the law of trichotomy. First, though, note that the second ratio does satisfy V.Def.4 as a definition since twice each of x and x + y is greater than the other. Now, the ratios x:x and x:(x + y) are not equal since 2x = 2x but 2x < 2(x + y). Next, x:(x + y) is not greater than x:x, since nx > m(x + y) implies nx > mx. Finally, x:x is not greater than x:(x + y), for if nx > mx, then n > m, so nx is not less than mx > x, and since y is an infinitesimal with respect to x, x > my, therefore nx mx > my, that is, nx > m(x + y). In summary, since we have a proof of trichotomy that uses V.Def.4 as an axiom of comparability, and a counterexample of trichotomy that violates the axiom of comparability, we can conclude that any proof trichotomy requires the axiom of comparability. Next definition: V.Def.8-10 Previous: V.Def.5-6 Book V introduction © 1997 D.E.Joyce Clark University Definitions 8 through 10 Def. 8. A proportion in three terms is the least possible. Def. 9. When three magnitudes are proportional, the first is said to have to the third the duplicate ratio of that which it has to the second. Def. 10. When four magnitudes are continuously proportional, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on continually, whatever be the proportion. In the illustration A, B, and C form three terms for the proportion A:B = B:C, therefore the ratio A:C is the duplicate ratio of A:B. For a numerical example, 9:4 is the duplicate ratio of 3:2. The illustration also shows a continued proportion of four magnitudes, A, B, C, and D, since A:B = B:C = C:D. Also, A:D is the triplicate ratio of A:B. For a numerical example, 27:8 is the triplicate ratio of 3:2. Next definition: V.Def.11-13 Previous: V.Def.7 Book V introduction © 1997 D.E.Joyce Clark University Definitions 11 through 13 Def. 11. Antecedents are said to correspond to antecedents, and consequents to consequents. Def. 12. Alternate ratio means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent. Def. 13. Inverse ratio means taking the consequent as antecedent in relation to the antecedent as consequent. The figure illustrates the proportion A:B = C:D. Thus, A and C are corresponding terms since they're the antecedents. Also, B and D are corresponding terms since they're the consequents. Note that for alternate ratios to exist, all four magnitudes must be of the same kind. The alternate ratios in this proportion are A:C and B:D. Euclid proves these are the same ratio in proposition V.16. Thereafter, given one proportion A:B = C:D, he concludes alternately the alternate proportion A:C = B:D. The ratio inverse to A:B is B:A. It is evident from the definition V.Def.5 that A:B = C:D and B:A = D:C reduce to the same conditions on A, B, C, and D. Therefore, if two ratios are the same, then their two inverse ratios are also the same. For some reason, this statement is misplaced as the corollary after proposition V.7. Several of the propositions are stated using the antecedent terms but they apply as well for the consequent terms by inversion. For example, proposition V.24 says that if u:v = w:x and y:v = z:x, then (u + y):v = (w + z):x. But the statement using consequents is valid, too: if v:u = x:w and v:y = x:z, then v:(u + y) = x(w + z). The symmetry of the antecedent and consequent terms of a ratio a:b is not, however, one of perfect parallelism. They're opposite in regard to order. Proposition V.8 shows that the ratio is a:b increasing in a since if a increases then the ratio increases, but the ratio is decreasing in b since the if b increases then the ratio decreases. But that's still a kind of symmetry. Next proposition: V.Def.14-16 Previous: V.Def.8-10 Book V introduction © 1997 D.E.Joyce Clark University Definitions 14 through 16 Def. 14. A ratio taken jointly means taking the antecedent together with the consequent as one in relation to the consequent by itself. Def. 15. A ratio taken separately means taking the excess by which the antecedent exceeds the consequent in relation to the consequent by itself. Def. 16. Conversion of a ratio means taking the antecedent in relation to the excess by which the antecedent exceeds the consequent. Taking jointly the ratio u:v yields the ratio (u + v):v. Taking separately the ratio (u + v):v returns the ratio u:v. Taking the ratio (u + v):v in conversion yields the ratio (u + v):u. These conversions are only important when the ratios are in proportions. The following three proportions are shown to be equivalent in propositions V.17 and V.18. 1. (u + v):v = (x + y):y. 2. (u + v):u = (x + y):x. 3. u:v = x:y. Proposition V.17 and V.18 show proportions 1 and 3 are equivalent. That means proportion 2 and the inverse of 3, v:u = y:x, are also equivalent. And of course, 3 and its inverse are equivalent, so all three proportions are equivalent. Furthermore, when all the magnitudes are of the same kind, then the alternate proportions are also equivalent by V.16 making six equivalent statements. 4. (u + v):(x + y) = v:y. 5. (u + v):(x + y) = u:x 6. u:x = v:y Proposition V.19 goes on to say that 4 implies 5, and its corollary says 1 implies 2. Heath translates "taken jointly," "taken separately," and "in conversion" by the Latin words componendo, separando, and convertendo, respectively. Next proposition: V.Def.17-18 Previous: V.Def.11-13 Book V introduction © 1997 D.E.Joyce Clark University Definitions 17 and 18 Def. 17. A ratio ex aequali arises when, there being several magnitudes and another set equal to them in multitude which taken two and two are in the same proportion, the first is to the last among the first magnitudes as the first is to the last among the second magnitudes. Or, in other words, it means taking the extreme terms by virtue of the removal of the intermediate terms. Def. 18. A perturbed proportion arises when, there being three magnitudes and another set equal to them in multitude, antecedent is to consequent among the first magnitudes as antecedent is to consequent among the second magnitudes, while, the consequent is to a third among the first magnitudes as a third is to the antecedent among the second magnitudes. If A:B = D:E, and B:C = E:F, then as shown in proposition V.22, ex aequali, A:C = D:F. However, if G:H = M:N, and H:K = L:M, then a perturbed proportion holds as shown in proposition V.23, namely, G:K = L:N. Next proposition: V.1 Previous: V.Def.14-16 Book V introduction © 1997 D.E.Joyce Clark University Proposition 1 If any number of magnitudes are each the same multiple of the same number of other magnitudes, then the sum is that multiple of the sum. Let any number of magnitudes AB and CD each be the same multiple of magnitudes E and F respectively. V.Def.2 I say that the sum of AB and CD is the same multiple of the sum of E and F that AB is of E. Since AB is the same multiple of E that CD is of F, therefore there are as many magnitudes in AB equal to E as there are in CD equal to F. Divide AB into magnitudes AG and GB equal to E, and divide CD into CH and HD equal to F. Then the number of the magnitudes AG and GB equals the number of the magnitudes CH and HD. Now, since AG equals E, and CH equals F, therefore the sum of AG and CH equals the sum of E and F. For the same reason GB equals E, and the sum of GB and HD equals the sum of E and F. Therefore, there are as many magnitudes in AB equal to E as there are in the sum of AB and CD equal to the sum of E and F. Therefore, the sum of AB and CD is the same multiple of the sum of E and F that AB is of E. Therefore, if any number of magnitudes are each the same multiple of the same number of other magnitudes, then the sum is that multiple of the sum. Q.E.D. In modern terminology, this proposition states that multiplication by numbers distributes over addition of magnitudes, that is, m (x1 + x2 + ... + xn) = m x1 + m x2 + ... + m xn. Here, the m is a number, and all the xi's are magnitudes of the same kind. Euclid always displays his magnitudes as lines, but they could be magnitudes of other kinds, like plane regions, for instance. In this proposition, all the magnitudes are of the same kind. Euclid's proof is only for the simplest nontrivial case. He takes the number n of magnitudes to be 2, and the multiple m also to be 2, so he proves that if x1 = m y1 and x2 = m y2, then x1 + x2 = m (y1 + y2). Throughout Book V, Euclid proves the general numerical case by a particular case. The numbers he chooses are usually 2 and 3. Use of this proposition Proposition V.1 is used in the proofs of four other propositions, namely, V.5, V.8, V.12, and V.17. Next proposition: V.2 Previous: V.Def.17-18 Book V introduction © 1996 D.E.Joyce Clark University Proposition 2 If a first magnitude is the same multiple of a second that a third is of a fourth, and a fifth also is the same multiple of the second that a sixth is of the fourth, then the sum of the first and fifth also is the same multiple of the second that the sum of the third and sixth is of the fourth. Let a first magnitude AB be the same multiple of a second C that a third DE is of a fourth F, and let a fifth BG be the same multiple of the second C that a sixth EH is of the fourth F. V.Def.2 I say that the sum AG of the first and fifth is the same multiple of the second, C, that the sum DH of the third and sixth is of the fourth, F. Since AB is the same multiple of C that DE is of F, therefore there are as many magnitudes in AB equal to C as there are in DE equal to F. For the same reason there are as many magnitudes in BG equal to C as there are in EH equal to F. Therefore, there are as many magnitudes in the whole AG equal to C as there are in the whole DH equal to F. Therefore, AG is the same multiple of C that DH is of F. Therefore the sum AG of the first and fifth is the same multiple of the second, C, that the sum DH of the third and sixth is of the fourth, F. Therefore, if a first magnitude is the same multiple of a second that a third is of a fourth, and a fifth also is the same multiple of the second that a sixth is of the fourth, then the sum of the first and fifth also is the same multiple of the second that the sum of the third and sixth is of the fourth. Q.E.D. This proposition simply states that sums of equimultiples are equimultiples, that is, if mc and mf are equimultiples of c and f, and nc and nf are also equimultiples of c and f, then the sums mc + nc and mf + nf are also equimultiples of c and f. The proof depends on a form of distributivity, namely, that multiplication by magnitudes distributes over addition of numbers. (m + n)c = mc + nc. Note that the magnitudes don't all have to be of the same kind. Different colors are used in the figures here to indicate different kinds of magnitudes. Use of this proposition Proposition V.2 is used in the proofs of three other propositions, namely, V.3, V.6, and V.17. Next proposition: V.3 Previous: V.1 Book V introduction © 1996 D.E.Joyce Clark University Proposition 3 If a first magnitude is the same multiple of a second that a third is of a fourth, and if equimultiples are taken of the first and third, then the magnitudes taken also are equimultiples respectively, the one of the second and the other of the fourth. Let a first magnitude A be the same multiple of a second B that a third C is of a fourth D, and let equimultiples EF and GH be taken of A and C. V.Def.2 I say that EF is the same multiple of B that GH is of D. Since EF is the same multiple of A that GH is of C, therefore there are as many magnitudes as in EF equal to A as there are in GH equal to C. Divide EF into the magnitudes EK and KF equal to A, and divide GH into the magnitudes GL and LH equal to C. Then the number of the magnitudes EK and KF equals the number of the magnitudes GL and LH. And, since A is the same multiple of B that C is of D, while EK equals A, and GL equals C, therefore EK is the same multiple of B that GL is of D. For the same reason KF is the same multiple of B that LH is of D. Since a first magnitude EK is the same multiple of a second B that a third GL is of a fourth D, and a fifth KF is the same multiple of the second B that a sixth LH is of the fourth D, therefore the sum EF of the first and fifth is the same multiple of the second B that the sum GH of the third and sixth is of the fourth D. V.2 Therefore, if a first magnitude is the same multiple of a second that a third is of a fourth, and if equimultiples are taken of the first and third, then the magnitudes taken also are equimultiples respectively, the one of the second and the other of the fourth. Q.E.D. This proposition says that equimultiples of equimultiples are equimultiples, that is, if w and x are equimultiples of y and z, and u and v are equimultiples of w and x, then u and v are equimultiples of y and z. The proof depends on an associativity of multiplication: m (ny) = (mn) y. In Euclid's proof, n is 3 and m is 2. As in the last proposition, the magnitudes need not all be of the same kind. Although this proposition is not actually a statement about ratios, it can be interpreted as one. The hypotheses that A and C are equimultiples of B and D can be interpreted as a proportion A:B = C:D, and the conclusion that mA and mC are equimultiples of B and D can be interpreted as a proportion mA:B = mC:D. Under these interpretations this proposition becomes a special case of the next, and it is the special case that is used to prove the general case in the next proposition. Next proposition: V.4 Previous: V.2 Book V introduction © 1996 D.E.Joyce Clark University Proposition 4 If a first magnitude has to a second the same ratio as a third to a fourth, then any equimultiples whatever of the first and third also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order. Let a first magnitude A have to a second B the same ratio as a third C to a fourth D, and let equimultiples E and F be taken of A and C, and G and H other, arbitrary, equimultiples of B and D. I say that E is to G as F is to H. Take equimultiples K and L of E and F, and other, arbitrary, equimultiples M and N of G and H. Since E is the same multiple of A that F is of C, and equimultiples K and L of E and F have been taken, therefore K is the same multiple of A that L is of C. For the same reason M is the same multiple of B that N is of D. V.3 And, since A is to B as C is to D, and equimultiples K and L have been taken of A and C, and other, arbitrary, equimultiples M and N of B and D, therefore, if K is in excess of M, then L is in excess of N; if it is equal, equal; and if less, less. V.Def.5 And K and L are equimultiples of E and F, and M and N are other, arbitrary, equimultiples of G and H, therefore E is to G as F is to H. V.Def.5 Therefore, if a first magnitude has to a second the same ratio as a third to a fourth, then any equimultiples whatever of the first and third also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order. Q.E.D. Note how Euclid uses the definition to prove that the two ratios pa:qb and pc:qd are the same. (Here, a and b are magnitudes of one kind, and c and d are magnitudes of another kind, but p and q are numbers.) We are given a:b = c:d. That means for any numbers m and n that if ma >=< nb, then mc >=< nd. We have to prove that pa:qb = pc:qd for any numbers p and q. That means, we have to prove that for any m and n, if mpa >=< nqb, then mpc >=< nqd. But that's just a special case of the given relation if ma >=< nb, then mc >=< nd. Use of this proposition Proposition V.4 is used in the proof of one other proposition, namely, V.22. Next proposition: V.5 Previous: V.3 Book V introduction © 1996 D.E.Joyce Clark University Proposition 5 If a magnitude is the same multiple of a magnitude that a subtracted part is of a subtracted part, then the remainder also is the same multiple of the remainder that the whole is of the whole. Let the magnitude AB be the same multiple of the magnitude CD that the subtracted part AE is of the subtracted part CF. I say that the remainder EB is also the same multiple of the remainder FD that the whole AB is of the whole CD. Make CG so that EB is the same multiple of CG that AE is of CF. Then, since AE is the same multiple of CF that EB is of GC, therefore AE is the same multiple of CF that AB is of GF. V.1 But, by the assumption, AE is the same multiple of CF that AB is of CD. Therefore AB is the same multiple of each of the magnitudes GF and CD. Therefore GF equals CD. Subtract CF from each. Then the remainder GC equals the remainder FD. And, since AE is the same multiple of CF that EB is of GC, and GC equals DF, therefore AE is the same multiple of CF that EB is of FD. But, by hypothesis, AE is the same multiple of CF that AB is of CD, therefore EB is the same multiple of FD that AB is of CD. That is, the remainder EB is the same multiple of the remainder FD that the whole AB is of the whole CD. Therefore, If a magnitude is the same multiple of a magnitude that a subtracted part is of a subtracted part, then the remainder also is the same multiple of the remainder that the whole is of the whole. Q.E.D. This proposition is analogous to V.1 but involves differences rather than sums. It states that multiplication by numbers distributes over subtraction of magnitudes: m (x y) = mx my. Note that all the magnitudes in this proposition are of the same kind. The problem of parts There's a construction at the beginning of the proof to make a part of a magnitude: Make CGso that EBis the same multiple of CGthat AEis of CF. so that, if for example, AE is a third of CF, then CG is to be made a third of EB. Such constructions cannot be made for all kinds of magnitudes, in particular, angles and arcs. Alternative proofs that don't require constructions of parts are relatively easy to find. A more interesting problem of general constructions for magnitudes is discussed in the Guide for proposition V.18. This proposition is not used in the rest of the Elements. Next proposition: V.6 Previous: V.4 Book V introduction © 1996 D.E.Joyce Clark University Proposition 6 If two magnitudes are equimultiples of two magnitudes, and any magnitudes subtracted from them are equimultiples of the same, then the remainders either equal the same or are equimultiples of them. Let two magnitudes AB and CD be equimultiples of two magnitudes E and F, and let AG and CH subtracted from them be equimultiples of the same two E and F. I say that the remainders GB and HD either equal E and F or are equimultiples of them. First, let GB equal E. I say that HD also equals F. Make CK equal to F. Since AG is the same multiple of E that CH is of F, while GB equals E, and KC equals F, therefore AB is the same multiple of E that KH is of F. V.2 But, by hypothesis, AB is the same multiple of E that CD is of F, therefore KH is the same multiple of F that CD is of F. Since then each of the magnitudes KH and CD is the same multiple of F, therefore KH equals CD. Subtract CH from each. Then the remainder KC equals the remainder HD. But F equals KC, therefore HD also equals F. Hence, if GB equals E, HD also equals F. Similarly we can prove that, even if GB is a multiple of E, HD is also the same multiple of F. Therefore, if two magnitudes are equimultiples of two magnitudes, and any magnitudes subtracted from them are equimultiples of the same, then the remainders either equal the same or are equimultiples of them. Q.E.D. The proposition states that if ma and mb are equimultiples of a and b, and na and nb are also equimultiples, then the differences, ma na and mb nb are more equimultiples. It's analogous to proposition V.2 which was for addition. Its proof depends on a distributivity, namely that multiplication by magnitudes distributes over subtraction of numbers: (m n)a = ma na. Euclid takes 4 as m and 3 as n. He has two cases since since he doesn't take 1 to be a number. This proposition is not used in the rest of the Elements. Next proposition: V.7 Previous: V.5 Book V introduction © 1996 D.E.Joyce Clark University Proposition 7 Equal magnitudes have to the same the same ratio; and the same has to equal magnitudes the same ratio. Let A and B be equal magnitudes and C an arbitrary magnitude. I say that each of the magnitudes A and B has the same ratio to C, and C has the same ratio to each of the magnitudes A and B. Take equimultiples D and E of A and B, and take an arbitrary multiple F of C. Then, since D is the same multiple of A that E is of B, and A equals B, therefore D equals E. But F is another, arbitrary, magnitude. If therefore D is in excess of F, then E is also in excess of F; if equal, equal; and, if less, less. And D and E are equimultiples of A and B, while F is another, arbitrary, multiple of C, therefore A is to C as B is to C. V.Def.5 I say next that C also has the same ratio to each of the magnitudes A and B. With the same construction, we can prove similarly that D equals E, and F is some other magnitude. If therefore F is in excess of D, it is also in excess of E; if equal, equal; and, if less, less. And F is a multiple of C, while D and E are other, arbitrary, equimultiples of A and B, therefore C is to A as C is to B. V.Def.5 Therefore, equal magnitudes have to the same the same ratio; and the same has to equal magnitudes the same ratio. Q.E.D. Corollary From this it is manifest that, if any magnitudes are proportional, then they are also proportional inversely. This proposition says that if a = b, then a:c = b:c, and c:a = c:b. The proposition is evident. Its converse is given in V.9. The corollary is misplaced. There is nothing relevant in the proposition. There's no way it could yield the corollary since the proposition requires all the magnitudes to be of the same kind and the corollary doesn't. But the corollary is valid, and it follows easily from definition V.Def.5. Use of this proposition Such a basic property of ratios as this is used frequently when ratios are mentioned. It is used in a few times in Book V starting with V.10, frequently in Book VI, and ocassionally in later books. Next proposition: V.8 Previous: V.6 Book V introduction © 1996, 1997 D.E.Joyce Clark University Proposition 8 Of unequal magnitudes, the greater has to the same a greater ratio than the less has; and the same has to the less a greater ratio than it has to the greater. Let AB and C be unequal magnitudes, and let AB be greater, and let D be another, arbitrary, magnitude. I say that AB has to D a greater ratio than C has to D, and D has to C a greater ratio than it has to AB. Since AB is greater than C, make EB equal to C. Then the less of the magnitudes AE and EB, if multiplied, will eventually be greater than D. (V.Def.4) First, let AE be less than EB. Let AE be multiplied, and let FG be a multiple of it which is greater than D. Make GH the same multiple of EB and K the same multiple of C that FG is of AE. Take L double of D and M triple of it, and successive multiples increasing by one, until what is taken is the first multiple of D that is greater than K. Let it be taken, and let it be N which is quadruple of D and the first multiple of it greater than K. (V.Def.4) Since K is less than N first, therefore K is not less than M. And, since FG is the same multiple of AE that GH is of EB, therefore FG is the same multiple of AE that FH is of AB. V.1 But FG is the same multiple of AE that K is of C, therefore FH is the same multiple of AB that K is of C. Therefore FH and K are equimultiples of AB and C. Again, since GH is the same multiple of EB that K is of C, and EB equals C, therefore GH equals K. But K is not less than M, therefore neither is GH less than M. And FG is greater than D, therefore the whole FH is greater than the sum of D and M. But the sum of D and M equals N, inasmuch as M is triple D, and the sum of M and D is quadruple D, while N is also quadruple D, therefore the sum of M and D equals N. But FH is greater than the sum of M and D, therefore FH is in excess of N, while K is not in excess of N. And FH and K are equimultiples of AB and C, while N is another, arbitrary, multiple of D, therefore AB has to D a greater ratio than C has to D. V.Def.7 I say next, that D has to C a greater ratio than D has to AB. With the same construction, we can prove similarly that N is in excess of K, while N is not in excess of FH. And N is a multiple of D, while FH and K are other, arbitrary, equimultiples of AB and C, therefore D has to C a greater ratio than D has to AB. V.Def.7 Next, let AE be greater than EB. Then the less, EB, if multiplied, will eventually be greater than D. (V.Def.4) Let it be multiplied, and let GH be a multiple of EB and greater than D. Make FG the same multiple of AE, and K the same multiple of C that GH is of EB. Then we can prove similarly that FH and K are equimultiples of AB and C, and, similarly, take N the first multiple of D that is greater than FG, so that FG is again not less than M. (V.Def.4) But GH is greater than D, therefore the whole FH is in excess of the sum of D and M, that is, of N. Now K is not in excess of N, inasmuch as FG also, which is greater than GH, that is, than K, is not in excess of N. And in the same manner, by following the above argument, we complete the demonstration. Therefore, of unequal magnitudes, the greater has to the same a greater ratio than the less has; and the same has to the less a greater ratio than it has to the greater. Q.E.D. Although the statement of this proposition is easy to comprehend, its proof is difficult. It says that if x > y, then x:z > y:z but z:x < z:y. Its converse is proposition V.10. At four points in the proof V.Def.4 is used as an axiom of comparabililty rather than a definition. The first instance: Then the less of the magnitudes AEand EB,if multiplied, will eventually be greater than D. In fact the axiom of comparability is required for this proposition since it is false when infinitesimals are allowed. When y is infinitesimal with respect to x, then the first statement of the proposition doesn't hold since x > x y but it is not the case that x:x > (x y):x, and the second statement doesn't hold since x + y > x, but not x:x > x:(x + y). Explanation of the proof The proof is slightly more comprehensible when modern algebraic notation is used since that clarifies its overall structure. Every magnitude in Euclid's proof is represented by a name and illustrated by a line. With an algebraic notation, we can refer to a magnitude by a formula. For instance, if we let a be AB and c be C, then we can use a c for AE, thus reducing the number of variables and easing comprehension. We can also have variables for numbers, instead of having to choose a specific number as Euclid does when he takes N to be 4D. But algebra obscures much, too. Euclid carefully proved distributivity of multiplication by numbers over addition of magnitudes in V.1, which is used in this proof. We manipulate algebraic expressions almost automatically. In order to be as correct as Euclid, we should verify the rules of algebra and be aware when we use them. With these preliminary qualifications, let's look at a translation of the proof into symbolic algebra. To prove: if a > c, then a:d > c:d, but d:c > d:a. a = AB c = C = EB d = D Let a > c. Either a c < c, or a c > c, [or a c = c]. a c = AE Case 1: Suppose a c < c. Let m be a number such that m(a c) > d. m(a c) = FG mc = GH = K Let n be the smallest number such that nd > mc. [What happens when n = 1 to Euclid's proof?] nd = N (n 1)d = M Since mc is not less than (n 1)d, and m(a c) > d, therefore, by adding, ma > nd. But mc is not greater than nd. Therefore a:d > c:d. ma = FH Also nd > mc but nd is not greater than ma. Therefore d:c > d:a. Case 2: Suppose c < a c. Let Let m be a number such that mc > d. (Same as above) Let n be the smallest number such that nd > m(a c). Since m(a c) is not less than (n 1)d, and mc > d, therefore, by adding, ma > nd. [Euclid says to do the rest in the same manner: Since a c > c, therefore m(a c) > mc. But nd > m(a c), therefore nd > mc. But ma > nd, therefore a:d > c:d. Also nd > mc but nd is not greater than ma. Therefore d:c > d:a.] [Case 3 when a c = c is left to the reader.] Thus, the conclusion is reached in any case. Q.E.D. Use of this proposition Proposition V.8 is used a few times in Book V starting with the next proposition. Next proposition: V.9 Previous: V.7 Book V introduction © 1996, 1997 D.E.Joyce Clark University Proposition 9 Magnitudes which have the same ratio to the same equal one another; and magnitudes to which the same has the same ratio are equal. Let each of the magnitudes A and B have the same ratio to C. I say that A equals B. Otherwise, each of the magnitudes A and B would not have the same ratio to C, but they do, therefore A equals B. V.8 Next, let C have the same ratio to each of the magnitudes A and B. I say that A equals B. Otherwise, C would not have the same ratio to each of the magnitudes A and B, but it does, therefore A equals B. V.8 Therefore, magnitudes which have the same ratio to the same equal one another; and magnitudes to which the same has the same ratio are equal. Q.E.D. This converse to proposition V.7 has two statements: If a:c = b:c, then a = b. If c:a = c:b, then a = b. Besides the previous proposition, the proof relies on the law of trichotomy for ratios, the part which says that a:b < a:c and a:b = a:c cannot both occur. Although Euclid didn't prove that, it follows easily from the definitions in V.Def.5 and V.Def.7. This proposition relies on using V.Def.4 as an axiom of comparability through its use of the previous proposition. The axiom is required since the statement of the proposition is false when a is c + y, and b is c + 2y where y is an infinitesimal with respect to c. This proposition is used occasionally in Books VI, VII, X, XI, and XII to conclude equality of geometric magnitudes. Next proposition: V.10 Previous: V.8 Book V introduction © 1996, 1997 D.E.Joyce Clark University Proposition 10 Of magnitudes which have a ratio to the same, that which has a greater ratio is greater; and that to which the same has a greater ratio is less. Let A have to C a greater ratio than B has to C. I say that A is greater than B. If not, then A either equals B or is less than it. Now A does not equal B, for in that case each of the magnitudes A and B would have the same ratio to C, but they do not, therefore A does not equal B. V.7 Nor is A less than B, for in that case A would have to C a less ratio than B has to C, but it does not, therefore A is not less than B. V.8 But it was proved not to be equal either, therefore A is greater than B. Next, let C have to B a greater ratio than C has to A. I say that B is less than A. If not, it is either equal or greater. Now B does not equal A, for in that case C would have the same ratio to each of the magnitudes A and B, but it does not, therefore A does not equal B. V.7 Nor is B greater than A, for in that case C would have to B a less ratio than it has to A, but it does not, therefore B is not greater than A. V.8 But it was proved that it is not equal either, therefore B is less than A. Therefore, of magnitudes which have a ratio to the same, that which has a greater ratio is greater; and that to which the same has a greater ratio is less. Q.E.D. This converse to proposition V.8 has two statements. If a:c > b:c, then a > b. If c:b > c:a, then b < a. Part of the law of trichotomy for ratios is used in this proof, the part which says at most one of the three cases a:c < b:c, a:c = b:c, or a:c > b:c, can occur. Euclid's proof relies on using V.Def.4 as an axiom of comparability since it uses proposition V.8 and the law of trichotomy for ratios. But the proposition can also be proved without the axiom. Suppose a:c > b:c.Then there are numbers mand nsuch that na > mcbut nbis not greater than mc.Therefore na > nb.Therefore a > b.Thus a:c > b:cimplies a > b. The other implication of the proposition can be proved similarly. This proposition is used a few times in book V starting with V.14. Next proposition: V.11 Previous: V.9 Book V introduction © 1996 D.E.Joyce Clark University Proposition 11 Ratios which are the same with the same ratio are also the same with one another. Let A be to B as C is to D, and let C be to D as E is to F. I say that A is to B as E is to F. Take equimultiples G, H, and K of A, C, and E, and take other, arbitrary, equimultiples L, M, and N of B, D, and F. Then since A is to B as C is to D, and of A and C equimultiples G and H have been taken, and of B and D other, arbitrary, equimultiples L and M, therefore, if G is in excess of L, H is also in excess of M; if equal, equal; and if less, less. V.Def.5 Again, since C is to D as E is to F, and of C and E equimultiples H and K have been taken, and of D and F other, arbitrary, equimultiples M and N, therefore, if H is in excess of M, K is also in excess of N; if equal, equal; and if less, less. But we saw that, if H was in excess of M, G was also in excess of L; if equal, equal; and if less, less, so that, in addition, if G is in excess of L, K is also in excess of N; if equal, equal; and if less, less. V.Def.5 And G and K are equimultiples of A and E, while L and N are other, arbitrary, equimultiples of B and F, therefore A is to B as E is to F. V.Def.5 Therefore, ratios which are the same with the same ratio are also the same with one another. Q.E.D. This proposition expresses the transitivity of the relation of being the same when applied to ratios. After this proposition (and the easily proved properties of reflexivity and symmetry, see the Guide to definition V.Def.6), the expression "two ratios are the same," or the equivalent expression "two ratios are equal," is justified. The proof follows directly from the definition. What is remarkable is that Eudoxus, or Euclid, recognized that this proposition needed to be proved. The magnitudes may be of three different kinds with A and B of one kind, C and D of a second kind, and E and F of a third kind. This proposition is used very frequently whenever ratios are used. Next proposition: V.12 Previous: V.10 Book V introduction © 1996 D.E.Joyce Clark University Proposition 12 If any number of magnitudes are proportional, then one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents. Let any number of magnitudes A, B, C, D, E, and F be proportional, so that A is to B as C is to D, and as E is to F. I say that A is to B as the sum of A, C, and E is to the sum of B, D, and F. Take equimultiples G, H, and K of A, C, and E, and take other, arbitrary, equimultiples L, M, and N of B, D, and F. Then since A is to B as C is to D, and as E is to F, and of A, C, and E equimultiples G, H, and K have been taken, and of B, D, and F other, arbitrary, equimultiples L, M, and N, therefore, if G is in excess of L, then H is also in excess of M, and K of N; if equal, equal; and if less, less. So that, in addition, if G is in excess of L, then the sum of G, H, and K is in excess of the sum of L, M, and N; if equal, equal; and if less, less. V.Def.5 Now G and the sum of G, H, and K are equimultiples of A and the sum of A, C, and E, since, if any number of magnitudes are each the same multiple the same number of other magnitudes, then the sum is that multiple of the sum. V.1 For the same reason L and the sum of L, M, and N are also equimultiples of B and the sum of B, D, and F, therefore A is to B as the sum of A, C, and E is to the sum of B, D, and F. V.Def.5 Therefore, if any number of magnitudes are proportional, then one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents. Q.E.D. The general form for this proposition is that if x1:y1 = x2:y2 = ... = xn:yn, then each of these ratios also equals the ratio (x1 + x2 + ... + xn) : (y1 + y2 + ... + yn). This proposition is used in V.15 and a few other propositions in books VI, X, and XII. Next proposition: V.13 Previous: V.11 Book V introduction © 1996 D.E.Joyce Clark University Proposition 13 If a first magnitude has to a second the same ratio as a third to a fourth, and the third has to the fourth a greater ratio than a fifth has to a sixth, then the first also has to the second a greater ratio than the fifth to the sixth. Let a first magnitude A have to a second B the same ratio as a third C has to a fourth D, and let the third C have to the fourth D a greater ratio than a fifth E has to a sixth F. I say that the first A also has to the second B a greater ratio than the fifth E to the sixth F. Since there are some equimultiples of C and E, and of D and F other equimultiples, such that the multiple of C is in excess of the multiple of D, while the multiple of E is not in excess of the multiple of F, let them be taken. Let G and H be equimultiples of C and E, and K and L other, arbitrary, equimultiples of D and F, so that G is in excess of K, but H is not in excess of L. Whatever multiple G is of C, let M also be that multiple of A, and, whatever multiple K is of D, let N also be that multiple of B. V.Def.7 Now, since A is to B as C is to D, and of A and C equimultiples M and G have been taken, and of B and D other, arbitrary, equimultiples N and K, therefore, if M is in excess of N, G is also in excess of K; if equal, equal; and if less, less. V.Def.5 But G is in excess of K, therefore M is also in excess of N. But H is not in excess of L, and M and H are equimultiples of A and E, and N and L other, arbitrary, equimultiples of B and F, therefore A has to B a greater ratio than E has to F. V.Def.7 Therefore, if a first magnitude has to a second the same ratio as a third to a fourth, and the third has to the fourth a greater ratio than a fifth has to a sixth, then the first also has to the second a greater ratio than the fifth to the sixth. Q.E.D. This proposition states that if two ratios are equal, and one is greater than a third, then so is the other. That is, if a:b = c:d and c:d > e:f, then a:b > e:f. The magnitudes may be of three different kinds with a and b of one kind, c and d of a second kind, and e and f of a third kind. The analogous statement for lesser ratios isn't stated, but, of course, it holds as well. Euclid uses it as well as this proposition, in V.20. So does transitivity: if a:b > c:d and c:d > e:f, then a:b > e:f. The proof isn't difficult, but without symbolic algebra it becomes unwieldly. Euclid would have required 20 lines in his diagram. This proposition is used in the next one as well as V.20 and V.21. Next proposition: V.14 Previous: V.12 Book V introduction © 1996 D.E.Joyce Clark University Proposition 14 If a first magnitude has to a second the same ratio as a third has to a fourth, and the first is greater than the third, then the second is also greater than the fourth; if equal, equal; and if less, less. Let a first magnitude A have the same ratio to a second B as a third C has to a fourth D, and let A be greater than C. I say that B is also greater than D. Since A is greater than C, and B is another, arbitrary, magnitude, therefore A has to B a greater ratio than C has to B. V.8 But A is to B as C is to D, therefore C has to D a greater ratio than C has to B. V.13 But that to which the same has a greater ratio is less, therefore D is less than B, so that B is greater than D. V.10 Similarly we can prove that, if A equals C, then B equals D, and, if A is less than C, then B is less than D. Therefore, if a first magnitude has to a second the same ratio as a third has to a fourth, and the first is greater than the third, then the second is also greater than the fourth; if equal, equal; and if less, less. Q.E.D. The statement is that if a:b = c:dand a >=< c,then b >=< d. In this form all four magnitudes need to be of the same kind. The alternate form of the proposition Curiously, sometimes the alternate form if a:b = c:dand a >=< b,then c >=< d is used. This other form is more general since a and b may be of one kind while c and d can be of a different kind. (See definition V.Def.12 and proposition V.16 for alternate proportions.) For example, in proposition VI.25 there are the statements: ...the triangle ABCis to the triangle KGHas the parallelogram BEis to the parallelogram EF.Therefore, alternately, the triangle ABCis to the parallelogram BEas the triangle KGHis to the parallelogram EF.But the triangle ABCequals the parallelogram BE,therefore the triangle KGHalso equals the parallelogram EF. First, the proportion is converted to its alternate form by V.16. Then, it is claimed that since the first equals the second, therefore the third equals the fourth. Clearly, V.14 is not being invoked otherwise the alternate form of the proportion would not be mentioned. Another example comes from X.112. ... the rectangle BCby EFequals the rectangle BDby G,therefore CBis to BDas Gis to EF.But CBis greater than BD,therefore Gis also greater than EF. Here the first is greater than the second, so the third is greater than the fourth. Proposition VI.16 (if the rectangle contained by the extremes equals the rectangle contained by the means, then the four straight lines are proportional) was used to derive the proportion CB:BD = G:EF from the equality of the rectangles, but it would have been just as easy to conclude CB:G = BD:EF, and then V.14 could be used. Clearly, this proposition V.14 is not being invoked in either of these propositions, but the alternate form is used instead. That suggests that the proofs of VI.25 and X.112 were written when V.14 wasn't available. The proof of the statement if a:b = c:dand a >=< b,then c >=< d is not difficult using the definition V.Def.5. Since a >=< b, therefore 2a >=< 2b. From the proportion a:b = c:d it follows that 2c >=< 2d. Therefore c >=< d. Q.E.D. The proof is even easier when 1 is considered to be a number. Although this alternate form does not rely on using V.Def.4 as an axiom of comparability, the original form does. The statement of the proposition is false when infinitesimals are allowed. For a particular example, take the proportion x:(x + y) = x: (x + 2y) or its inverse. Use of this proposition This proposition is used in V.16 and a few other propositions in Books V, VI, X, XII, and XIII. Next proposition: V.15 Previous: V.13 Book V introduction © 1996, 1997 D.E.Joyce Clark University Proposition 15 Parts have the same ratio as their equimultiples. Let AB be the same multiple of C that DE is of F. I say that C is to F as AB is to DE. Since AB is the same multiple of C that DE is of F, as many magnitudes as there are in AB equal to C, there are also in DE equal to F. Divide AB into the magnitudes AG, GH, and HB equal to C, and divide DE into the magnitudes DK, KL, and LE equal to F. Then the number of the magnitudes AG, GH, and HB equals the number of the magnitudes DK, KL, and LE. And, since AG, GH, and HB equal one another, and DK, KL, and LE also equal one another, therefore AG is to DK as GH is to KL, and as HB is to LE. V.7 Therefore one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents. Therefore AG is to DK as AB is to DE. V.12 But AG equals C and DK equals F, therefore C is to F as AB is to DE. Therefore, parts have the same ratio as their equimultiples. Q.E.D. This proposition states that if n is any number and c and f any magnitudes of the same kind, then c:f = nc:nf. This proposition is used in the next one and a few others in Books V, VI, and XIII. Next proposition: V.16 Previous: V.14 Book V introduction © 1996 D.E.Joyce Clark University Proposition 16 If four magnitudes are proportional, then they are also proportional alternately. Let A, B, C, and D be four proportional magnitudes, so that A is to B as C is to D. I say that they are also so alternately, that is A is to C as B is to D. V.Def.12 Take equimultiples E and F of A and B, and take other, arbitrary, equimultiples G and H of C and D. Then, since E is the same multiple of A that F is of B, and parts have the same ratio as their equimultiples, therefore A is to B as E is to F. V.15 But A is to B as C is to D, therefore C is to D also as E is to F. V.11 Again, since G and H are equimultiples of C and D, therefore C is to D as G is to H. V.15 But C is to D as E is to F, therefore as E is to F also as G is to H. V.11 But, if four magnitudes are proportional, and the first is greater than the third, then the second is also greater than the fourth; if equal, equal; and if less, less. V.14 Therefore, if E is in excess of G, F is also in excess of H; if equal, equal; and if less, less. Now E and F are equimultiples of A and B, and G and H other, arbitrary, equimultiples of C and D, therefore A is to C as B is to D. V.Def.5 Therefore, if four magnitudes are proportional, then they are also proportional alternately. Q.E.D. The four magnitudes A, B, C, and D need to be of the same kind. If A and B are of a different kind than C and D, then the alternate ratios A:C and B:D would be "mixed." The Greek geometers did not accept mixed ratios, but modern physicists and engineers routinely use them, as do we all since such a common measurement as velocity is made out of a ratio of a distance to a time. This proposition requires using V.Def.4 as an axiom of comparability. Use of this proposition This proposition is used in V.19 and a couple others in Book V, and frequently in Books VI, X, XI, and XII. Occasionally it is used when the magnitudes need not be all of the same kind, as it ought not. Next proposition: V.17 Previous: V.15 Book V introduction © 1996 D.E.Joyce Clark University Proposition 17 If magnitudes are proportional taken jointly, then they are also proportional taken separately. Let AB, BE, CD, and DF be magnitudes proportional taken jointly, so that AB is to BE as CD is to DF. V.Def.14 I say that they are also proportional taken separately, that is, AE is to EB as CF is to DF. V.Def.15 Take equimultiples GH, HK, LM, and MN of AE, EB, CF, and FD, and take other, arbitrary, equimultiples, KO and NP of EB and FD. Then, since GH is the same multiple of AE that HK is of EB, therefore GH is the same multiple of AE that GK is of AB. V.1 But GH is the same multiple of AE that LM is of CF, therefore GK is the same multiple of AB that LM is of CF. Again, since LM is the same multiple of CF that MN is of FD, therefore LM is the same multiple of CF that LN is of CD. V.1 But LM was the same multiple of CF that GK is of AB, therefore GK is the same multiple of AB that LN is of CD. Therefore GK and LN are equimultiples of AB and CD. Again, since HK is the same multiple of EB that MN is of FD, and KO is also the same multiple of EB that NP is of FD, therefore the sum HO is also the same multiple of EB that MP is of FD. V.2 And, since AB is to BE as CD is to DF, and of AB and CD equimultiples GK and LN have been taken, and of EB and FD equimultiples HO and MP, therefore, if GK is in excess of HO, and LN is also in excess of MP; if equal, equal; and if less, less. Let GK be in excess of HO. Subtract HK from each. Therefore GH is also in excess of KO. But we saw that, if GK was in excess of HO, then LN was also in excess of MP, therefore LN is also in excess of MP, and, if MN is subtracted from each, then LM is also in excess of NP, so that, if GH is in excess of KO, then LM is also in excess of NP. Similarly we can prove that, if GH equals KO, then LM also equals NP; and if less, less. And GH and LM are equimultiples of AE and CF, while KO and NP are other, arbitrary, equimultiples of EB and FD, therefore AE is to EB as CF is to FD. V.Def.5 Therefore, if magnitudes are proportional taken jointly, then they are also proportional taken separately. Q.E.D. The proposition says that if (w + x):x = (y + z):z, then w:x = y:z. Two of the magnitudes w and x can be of one kind while the other two y and z are of another kind. The converse is given in the next proposition. This proposition is used in the next two propositions and a couple in Book X. Next proposition: V.18 Previous: V.16 Book V introduction © 1996, 1997 D.E.Joyce Clark University Proposition 18 If magnitudes are proportional taken separately, then they are also proportional taken jointly. Let AE, EB, CF, and FD be magnitudes proportional taken separately, so that AE is to EB as CF is to FD. V.Def.15 I say that they are also proportional taken jointly, that is, AB is to BE as CD is to FD. V.Def.14 For, if CD is not to DF as AB is to BE, then AB is to BE as CD is either to some magnitude less than DF or to a greater. First, let it be in that ratio to a less magnitude DG. Then, since AB is to BE as CD is to DG, they are magnitudes proportional taken jointly, so that they are also proportional taken separately. Therefore AE is to EB as CG is to GD. V.17 But also, by hypothesis, AE is to EB as CF is to FD. Therefore CG is to GD as CF is to FD. V.11 But the first CG is greater than the third CF, therefore the second GD is also greater than the fourth FD. V.14 But it is also less, which is impossible. Therefore AB is to BE as CD is not to a less magnitude than FD. Similarly we can prove that neither is it in that ratio to a greater, it is therefore in that ratio to FD itself. Therefore, if magnitudes are proportional taken separately, then they are also proportional taken jointly. Q.E.D. This proposition is the converse of the last one. It says that if w:x = y:z, then (w + x):x = (y + z):z. As in the last proposition, two of the magnitudes w and x can be of one kind while the other two y and z are of another. On the existence of fourth proportionals At the beginning of this proof we have, paraphrased: If CD:DFdoes not equal AB:BE,then AB:BE = CD:DGwhere DGis some magnitude greater or less than DF. Given the other three magnitudes, a fourth proportional DG is assumed. It is not asserted that the fourth proportional can be constructed; it is only hypothetical. This is the beginning of a proof by contradiction. This technique of assuming the existence of a fourth proportional to derive a contradiction is also used in Book XII to prove various proportionalities of areas and volumes, for example, in proposition XII.2 which shows circles are proportional to the squares on their diameters. Eudoxus, who developed the techniques of both Books V and XII, or Euclid, or both of them, accepted this technique as valid. The problem is: do fourth proportionals exist? They certainly can't be constructed in all cases. The problems of doubling a cube, squaring a circle, and trisecting an angle cannot be solved by plane Euclidean methods, and they all involve inconstructable fourth proportionals. Take doubling a cube for example. If C is a cube with an edge A, then the inconstructable edge B of a cube with double the volume of C is the fourth proportional in C:(C+C) = A:B. Is there a difference between existence and constructibility? Constructibility is a fairly clear concept since there are postulates for what can be constructed. There are no postulates for things that exist but aren't constructed, but the existence of a fourth proportional is a good candidate for a such a postulate. There is a similar situation in modern mathematics with the axiom of choice for set theory. That axiom says that in certain situations there is at least one set satisfying certain criteria. It does not construct anything in the usual sense of "construct," and it doesn't even specify a particular set. Although it is useful in many situations, mathematicians prefer not to use it unless it's necessary. For this proposition, the assumption of the existence of fourth proportionals is unnecessary. An alternate proof Proposition: If w:x = y:z, then (w + x):x = (y + z):z. Proof: Suppose w:x = y:z. Let n and m be any numbers. Either n < m or not. Case 1: n < m. Suppose n(w + x) >=< mx.Subtract nxto get nw >=< (m n)x.But w:x = y:z,so ny >=< (m n)z.Add nzto get n(y + z) >=< mz. Case 2: n is not less than m. Then both n(w + x) > mx,and n(y + z) > mz. In any case n(w + x) >=< mx implies n(y + z) >=< mz. Therefore (w + x):x = (y + z):z. Q.E.D. Use of this proposition This proposition is used in proposition V.24 and a few in Books VI, X, XII, and XIII. Next proposition: V.`9 Previous: V.17 Book V introduction © 1996, 1997 D.E.Joyce Clark University Proposition 19 If a whole is to a whole as a part subtracted is to a part subtracted, then the remainder is also to the remainder as the whole is to the whole. Let the whole AB be to the whole CD as the part AE subtracted is to the part CF subtracted. I say that the remainder EB is also to the remainder FD as the whole AB is to the whole CD. Since AB is to CD as AE is to CF, therefore alternately, BA is to AE as DC is to CF. V.16 And, since the magnitudes are proportional taken jointly, they are also proportional taken separately, that is, BE is to EA as DF is to CF, and, alternately, BE is to DF as EA is to FC. V.17 V.16 But, by hypothesis, AE is to CF as is the whole AB to the whole CD. Therefore the remainder EB is also to the remainder FD as the whole AB is to the whole CD. V.11 Therefore if a whole is to a whole as a part subtracted is to a part subtracted, then the remainder is also to the remainder as the whole is to the whole. Q.E.D. Corollary From this it is manifest that, if magnitudes are proportional taken jointly, then they are also proportional in conversion. V.Def.16 This proposition says that if (u + v):(x + y) equals v:y, then it also equals u:x. The transformations of proportions taken jointly, taken separately, and in conversion are summarized in the Guide for V.Def.14. The magnitudes in this proposition must all be of the same kind, but those in the corollary can be of two different kinds. Thus, the corollary is out of place. It should probably be after the last proposition since it follows from the previous two propositions by inversion. As Heiberg and Heath agree, the corollary was probably interpolated before Theon's time. This proposition relies on using V.Def.4 as an axiom of comparability. (Infinitesimal counterexample: when y is infinitesimal with respect to x, then 2x:(2x + 2y) equals x:(x + 2y) but does not equal x:x.) The corollary, however, does not rely on an axiom of comparability. Use of the proposition and the corollary This proposition is used in V.25 and a few propositions in Book X. The corollary is used once in each of Books VI and XIII and fairly often in Book X. Next proposition: V.20 Previous: V.18 Book V introduction © 1996, 1997 D.E.Joyce Clark University Proposition 20 If there are three magnitudes, and others equal to them in multitude, which taken two and two are in the same ratio, and if ex aequali the first is greater than the third, then the fourth is also greater than the sixth; if equal, equal, and; if less, less. Let there be three magnitudes A, B, and C, and others D, E, and F equal to them in multitude, which taken two and two are in the same ratio, so that A is to B as D is to E, and B is to C as E is to F. Let A be greater than C ex aequali. I say that D is also greater than F; if A equals C, equal; and, if less, less. Since A is greater than C, and B is some other magnitude, and the greater has to the same a greater ratio than the less has, therefore A has to B a greater ratio than C has to B. V.8 But A is to B as D is to E, and, C is to B, inversely, as F is to E, therefore D has to E a greater ratio than F has to E. V.7.Cor V.13 But, of magnitudes which have a ratio to the same, that which has a greater ratio is greater, therefore D is greater than F. V.10 Similarly we can prove that, if A equals C, then D also equals F, and if less, less. Therefore, if there are three magnitudes, and others equal to them in multitude, which taken two and two are in the same ratio, and if ex aequali the first is greater than the third, then the fourth is also greater than the sixth; if equal, equal, and; if less, less. Q.E.D. This proposition is in preparation for V.22, and its proof is clear. Next proposition: V.21 Previous: V.19 Book V introduction © 1996 D.E.Joyce Clark University Proposition 21 If there are three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them is perturbed, then, if ex aequali the first magnitude is greater than the third, then the fourth is also greater than the sixth; if equal, equal; and if less, less. Let there be three magnitudes A, B, and C, and others D, E, and F equal to them in multitude, which taken two and two are in the same ratio, and let the proportion of them be perturbed, so that A is to B as E is to F, and B is to C as D is to E. V.Def.18 Let A be greater than C ex aequali. I say that D is also greater than F; if A equals C, equal; and if less, less. Since A is greater than C, and B is some other magnitude, therefore A has to B a greater ratio than C has to B. V.8 But A is to B as E is to F, and, inversely, C is to B as E is to D. Therefore also E has to F a greater ratio than E has to D. V.7.Cor V.13 But that to which the same has a greater ratio is less, therefore F is less than D, therefore D is greater than F. V.10 Similarly we can prove that, if A equals C, then D also equals F; and if less, less. Therefore, if there are three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them is perturbed, then, if ex aequali the first magnitude is greater than the third, then the fourth is also greater than the sixth; if equal, equal; and if less, less. Q.E.D. This proposition is in preparation for V.23. Both this and V.23 rely on treating V.Def.4 as an axiom of comparability. Next proposition: V.22 Previous: V.20 Book V introduction © 1996, 1997 D.E.Joyce Clark University Proposition 22 If there are any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, then they are also in the same ratio ex aequali. Let there be any number of magnitudes A, B, and C, and others D, E, and F equal to them in multitude, which taken two and two together are in the same ratio, so that A is to B as D is to E, and B is to C as E is to F. I say that they are also in the same ratio ex aequali, that is, A is to C as D is to F. V.Def.17 Take equimultiples G and H of A and D, and take other, arbitrary, equimultiples K and L of B and E, and, further, take other, arbitrary, equimultiples M and N of C and F. Then, since A is to B as D is to E, and of A and D equimultiples G and H have been taken, and of B and E other, arbitrary, equimultiples K and L, therefore G is to K as H is to L. V.4 For the same reason also K is to M as L is to N. Since, then, there are three magnitudes G, K, and M, and others H, L, and N equal to them in multitude, which taken two and two together are in the same ratio, therefore, ex aequali, if G is in excess of M, H is also in excess of N; if equal, equal; and if less, less. V.20 And G and H are equimultiples of A and D, and M and N other, arbitrary, equimultiples of C and F. Therefore A is to C as D is to F. V.Def.5 Therefore, if there are any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, then they are also in the same ratio ex aequali. Q.E.D. The general statement for this proposition is that for magnitudes x1, x2, ... , and xn of one kind, and magnitudes y1, y2, ... , and yn of the same or another kind, if x1:x2 = y1:y2, x2:x3 = y2:y3, ... , and xn1:xn = yn-1:yn, then x1:xn = y1:yn. The proof builds on proposition V.20. Assume A:B = D:E, and B:C = E:F. To show A:C = D:F. Let n, m, and k be three numbers. By V.4, nA:mB = nD:mE, and mA:kC = mD:kF. By V.20, nA >=< kCimplies nD >=< kF. Therefore, A:C = D:F. Q.E.D. This proposition can also be proved directly from the definition Def.V.5 very easily. The analogous proposition for ratios of numbers is given in proposition VII.14. The proof given there works for magnitudes as well, but they all have to be of the same kind. This proposition is used in V.24 and several propositions in Books VI, X, and XII. Next proposition: V.23 Previous: V.21 Book V introduction © 1996, 1997 D.E.Joyce Clark University Proposition 23 If there are three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, then they are also in the same ratio ex aequali. Let there be three magnitudes A, B, and C, and others D, E, and F, equal to them in multitude, which, taken two and two together, are in the same proportion, and let the proportion of them be perturbed, so that A is to B as E is to F, and B is to C as D is to E. V.Def.18 I say that A is to C as D is to F. Take equimultiples G, H, and K of A, B, and D, and take other, arbitrary, equimultiples L, M, and N of C, E, and F. Then, since G and H are equimultiples of A and B, and parts have the same ratio as their multiples, therefore A is to B as G is to H. V.15 For the same reason E is to F as M is to N. And A is to B as E is to F, therefore G is to H as M is to N. V.11 Next, since B is to C as D is to E, alternately, also, B is to D as C is to E. (V.16) And, since H and K are equimultiples of B and D, and parts have the same ratio as their equimultiples, therefore B is to D as H is to K. V.15 But B is to D as C is to E, therefore also, H is to K as C is to E. V.11 Again, since L and M are equimultiples of C and E, therefore C is to E as L is to M. V.15 But C is to E as H is to K, therefore also, H is to K as L is to M, and, alternately, H is to L as K is to M. V.11 (V.16) But it was also proved that G is to H as M is to N. Since, then, there are three magnitudes G, H, and L, and others equal to them in multitude K, M, and N, which taken two and two together are in the same ratio, and the proportion of them is perturbed, therefore, ex aequali, if G is in excess of L, K is also in excess of N; if equal, equal; and if less, less. V.21 And G and K are equimultiples of A and D, and L and N of C and F. Therefore A is to C as D is to F. V.Def.5 Therefore, if there are three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, then they are also in the same ratio ex aequali. Q.E.D. This proposition says that when a, b, and c are of one kind, and d, e, and f are of the same or another kind, if a:b = e:f and b:c = d:e, then a:c = d:f. The proof given here uses proposition V.16 and alternate ratios, and that means it only applies when all six magnitudes are of the same kind. It is also rather clumsy, since it uses V.15 and V.11 instead of V.4 as the previous proposition V.22 did. It doesn't seem likely that this proof would be written when the better proof of V.22 could serve as a guide, so it seems likely that V.4 was inserted later and an older proof of V.22 was cleaned up, but that of V.23 wasn't for some reason such as its relative unimportance. After all, it is not used in the rest of the Elements. Here's a summary of the proof as given. Assume A:B = E:F,and B:C = D:E.To show A:C = D:F. Let n and m be two arbitrary numbers. By V.15, both A:B = nA:nB, and E:F = mE:mF. Therefore, by V.11, nA:nB = mE:mF. (The last two sentences would reduce to one with V.4.) Using alternation V.16 on the other proportion B:C = D:E yields B:D = C:E (but that requires that all the magnitudes are of the same kind). For similar reasons nB:nD = B:D = C:E = mC:mE. Therefore, alternately, nB:mC = nD:mE. Now use V.21 on the two proportions nB:mC = nD:mE and nA:nB = mE:mF, to conclude nA >=< mC implies nD >=< mF Therefore, A:C = D:F.Q.E.D. Although the last proposition on proportions ex aequali did not depend on treating V.Def.4 as an axiom of comparability, this proposition on perturbed proportions ex aequali does. For a counterexample involving infinitesimals, take a = c = e, b = e = (a + y), and f = (a + 2y), where y is infinitesimal with respect to a. Next proposition: V.24 Previous: V.22 Book V introduction © 1996, 1997 D.E.Joyce Clark University Proposition 24 If a first magnitude has to a second the same ratio as a third has to a fourth, and also a fifth has to the second the same ratio as a sixth to the fourth, then the sum of the first and fifth has to the second the same ratio as the sum of the third and sixth has to the fourth. Let a first magnitude AB have to a second C the same ratio as a third DE has to a fourth F, and let also a fifth BG have to the second C the same ratio as a sixth EH has to the fourth F. I say that the sum of the first and fifth, AG, has to the second C the same ratio as the sum of the third and sixth, DH, has to the fourth F. Since BG is to C as EH is to F, inversely, C is to BG as F is to EH. V.7.Cor Then, since AB is to C as DE is to F, and C is to BG as F is to EH, therefore, ex aequali, AB is to BG as DE is to EH. V.22 And, since the magnitudes are proportional taken separately, they are also proportional taken jointly, therefore AG is to GB as DH is to HE. V.18 But also BG is to C as EH is to F, therefore, ex aequali, AG is to C as DH is to F. V.22 Therefore, if a first magnitude has to a second the same ratio as a third has to a fourth, and also a fifth has to the second the same ratio as a sixth to the fourth, then the sum of the first and fifth has to the second the same ratio as the sum of the third and sixth has to the fourth. Q.E.D. This proposition says that if u:v = w:x and y:v = z:x, then (u + y):v = (w + z):x. Although the proposition is stated using the antecedent terms of the proportions, by inversion it applies to the consequent terms as well. This proposition is used in proposition VI.31. Next proposition: V.25 Previous: V.23 Book V introduction © 1996, 1997 D.E.Joyce Clark University Proposition 25 If four magnitudes are proportional, then the sum of the greatest and the least is greater than the sum of the remaining two. Let the four magnitudes AB, CD, E, and F be proportional so that AB is to CD as E is to F, and let AB be the greatest of them and F the least. I say that the sum of AB and F is greater than the sum of CD and E. Make AG equal to E, and CH equal to F. Since AB is to CD as E is to F, and E equals AG, and F equals CH, therefore AB is to CD as AG is to CH. V.7 V.11 And since the whole AB is to the whole CD as the part AG subtracted is to the part CH subtracted, therefore the remainder GB is also to the remainder HD as the whole AB is to the whole CD. V.19 But AB is greater than CD, therefore GB is also greater than HD. (V.14) And, since AG equals E, and CH equals F, therefore the sum of AG and F equals the sum of CH and E. And if, GB and HD being unequal, and GB greater, the sum of AG and F is added to GB, and the sum of CH and E is added to HD, it follows that the sum of AB and F is greater than the sum of CD and E. Therefore, if four magnitudes are proportional, then the sum of the greatest and the least is greater than the sum of the remaining two. Q.E.D. This proposition says that if w:x = y:z and w is the greatest of the four magnitudes while z is the least, then w + z > x + y. All four magnitudes must be of the same kind. This proposition is not used in the rest of the Elements but is an end in itself. Arithmetic and geometric means A special case of it is when the middle terms are the same: x:y = y:z. In that case y is the mean proportional, equivalent to the geometric mean for real numbers and described as the square root of the product xz. The conclusion of the proposition, after dividing by 2, says (x + z)/2 > y. The arithmetic mean, or average, of two magnitudes is half their sum. Thus, this proposition gives as a corollary The arithmetic mean of two magnitudes is less than their geometric mean. This proposition relies on treating V.Def.4 as an axiom of comparability. Infinitesimal counterexample: when y is infinitesimal with respect to x, consider the proportion (x + 5y):(x + 2y) = (x + 4y):x. Next book: Book VI Introduction Previous: V.24 Book V introduction © 1996, 1997 D.E.Joyce Clark University Common Notions C.N.1. Things which equal the same thing also equal one another. C.N.2. If equals are added to equals, then the wholes are equal. C.N.3. If equals are subtracted from equals, then the remainders are equal. C.N.4. Things which coincide with one another equal one another. C.N.5. The whole is greater than the part. These common notions, sometimes called axioms, refer to magnitudes of one kind. The various kinds of magnitudes that occur in the Elements include lines, angles, plane figures, and solid figures. The first Common Notion could be applied to plane figures to say, for instance, that if a triangle equals a rectangle, and the rectangle equals a square, then the triangle also equals the square. Magnitudes of the same kind can be compared and added, but magnitudes of different kinds cannot. For instance, a line cannot be added to a rectangle, nor can an angle be compared to a pentagon. C.N.4 requires interpretation. On the face of it, it seems to say that if two things are identical (that is, they are the same one), then they are equal, in other words, anything equals itself. But the way it traditionally is interpreted is as a justification of a principle of superposition, which is used, for instance, in proposition I.4. Using this principle, if one thing can be moved to coincide with another, then they are equal. See the notes on I.4 for more discussion on this point. C.N.5, the whole is greater than the part, could be interpreted as a definition of "greater than." To say one magnitude B is a part of another A could be taken as saying that A is the sum of B and C for some third magnitude C, the remainder. Symbolically, A > B means that there is some C such that A = B + C. At any rate, Euclid frequently treats these two conditions as being equivalent. There are a number of properties of magnitudes used in Book I besides the listed Common Notions. Here are a few of them and locations where they are used. 1. If not x = y, then x > y or x < y. I.6 2. Not both x < y and x = y. I.6 3. If not not x = y, then x = y. I.6 4. If x < y and y = z, then x < z. I.7 5. If x < y and y < z, then x < z. I.7 6. If x = y and y < z, then x < z. I.16 7. If x < y, then x + z < y + z. I.17 8. If not x > y, then x = y or x < y. I.19 9. If not x < y and not x = y, then x > y. I.19 10. If 2x = 2y, then x = y. I.37 11. If x = y, then 2x = 2y. I.42 Number 3 is an instance of the logical principle of double negation, rather than a common notion. Number 11 is a special case of C.N.2 since doubling is a special case of addition, that is, 2x is just x + x. Some of the others are logical variants of each other, for instance, numbers 1, 8, and 9 are all equivalent to the statement that at least one of the three cases x < y, x = y, or x > y holds. Statement 2 says that two of those cases cannot simultaneously hold. The statement that for two magnitudes xand yof the same kind, exactly one of the three cases x<y, x = y,or x> yholds is called the law of trichotomy for magnitudes. This law, in particular, really ought to have been made an explicit common notion. A modern presentation In modern mathematics, axioms such as these would form the basis of an abstract algebra. Typically a presentation is given symbolically and in terms of set theory, although the set theory isn't necessary. Here is an outline for a presentation for magnitudes. This outline doesn't have many of the details that would normally be included. First, assume there is a binary relation on a set of magnitudes of the same kind called equality, denoted as usual with an equal sign as in x = y. (This equality is not identity as we want different magnitudes, such as two different triangles, to be equal. Alternatively, we could identify equal magnitudes so that equality is identity.) Assume that equality is what is called an equivalence relation, that is, it satisfies three axioms: Reflexivity: For each x, x = x. Symmetry: If x = y, then y = x. Transitivity: If x = y and y = z, then x = z. Next, assume a binary operation called addition and written the usual way, x + y. Furthermore, assume addition satisfies the axioms Substitution of equals: If x = y, then x + z = y + z, and z + x = z + y. Associativity: For each x, y, and z, (x + y) + z = x + (y + z). Commutativity: For each x and y, x + y = y + x. Associativity and commutativity together imply that the order that addition is performed is irrelevant. An algebra satisfying only associativity is called a semigroup, while a semigroup that also satisfies commutativity is called a commutative semigroup or an Abelian semigroup. When other axioms are added for zero and negation, then the algebra is called a group, and when commutative, an Abelian group. Groups are some of the most important algebraic structures in modern mathematics. We can now define order in terms of addition. Define a binary relation less than by taking x < y to mean that there is some z such that x + z = y. And let greater than just have the opposite order, that is, x > y means y < x. A number of properties of order can be easily proved. If x<yand y = z,then x<z. If x = y and y < z, then x < z. If x < y and y < z, then x < z. If x < y, then x + z < y + z, and z + x < z + y. Next, assume an axiom for cancellation: If x + z = y + z, then x = y. With this axiom, subtraction can be defined, at least up to equality. If x < y, that is to say, there is some z such that x + z = y, then we may define y x as that z, since, under the axiom of cancelation, any other magnitude w such that x + w = y would equal z. Subtraction is characterized by the property that x + z = yif and only if z = yx. The expected properties of subtraction, listed below, can be easily proved. Whenever a difference is indicated, such as x y, it is implicitly assumed that x > y. Only a few of these properties are used in Book I. If x = y,then xz = yz,and wx = wy If x = y and w = z, then x w = y z. (x + y) y = x. (x y) + y = x. (x y) - (w z) = (x w) - (y z). If x < y, then z x > z y. If x < y and w = z, then x w < y z. If x = y and w < z, then x w > y z. If x < y and w > z, then x w < y z. The law of trichotomy still isn't covered. It can be split into two parts: at most one of the three cases can occur, and at least one of the three cases occurs. The first can be stated as an axiom of addition as It is not the case that x = x + y. And that says it is not the case that x > x. The other half requires the axiom For each x and y, either x = y or there is some z such that x + z = y, or there is some z such that x = y + z. With these axioms, all the properties of magnitudes needed in the first few books of the Elements can be proved. For instance, we can prove If 2x = 2y,then x = y. using the same outline that Euclid used to prove proposition I.6. Let twice xequal twice y.I say that xequals y. If x does not equal y, then one of them is greater. Let x be greater. Then x + x > y + y, that is, twice x is greater than twice y. But twice x was assumed to equal twice y, the less equals the greater, which is absurd. Therefore x and y are not unequal. Therefore they are equal. Q.E.D. Book V will require more properties of magnitudes, and in that book, pairs of magnitudes of different kinds will be compared by using ratios. Next proposition: I.1 Previous: Postulate 5 Book I introduction © 1996 D.E.Joyce Clark University Definitions Def. 1. A rectilinear figure is said to be inscribed in a rectilinear figure when the respective angles of the inscribed figure lie on the respective sides of that in which it is inscribed. Def. 2. Similarly a figure is said to be circumscribed about a figure when the respective sides of the circumscribed figure pass through the respective angles of that about which it is circumscribed. Def. 3. A rectilinear figure is said to be inscribed in a circle when each angle of the inscribed figure lies on the circumference of the circle. Def. 4. A rectilinear figure is said to be circumscribed about a circle when each side of the circumscribed figure touches the circumference of the circle. Def. 5. Similarly a circle is said to be inscribed in a figure when the circumference of the circle touches each side of the figure in which it is inscribed. Def. 6. A circle is said to be circumscribed about a figure when the circumference of the circle passes through each angle of the figure about which it is circumscribed. Def. 7. A straight line is said to be fitted into a circle when its ends are on the circumference of the circle. The first figure shows a smaller quadrilateral inscribed in a larger quadrilateral, and the larger one is circumscribed about the smaller one. The second figure shows a quadrilateral inscribed in a circle, and the circle is circumscribed about the quadrilateral. The third figure shows a circle inscribed in a quadrilateral, and the quadrilateral is circumscribed about the circle. Note also that in the second figure, each side of the quadrilateral is fitted into the circle. First proposition: IV.1 Book IV introduction © 1996 D.E.Joyce Clark University Proposition 1 To fit a straight line into a given circle equal to a given straight line which is not greater than the diameter of the circle. Let ABC be the given circle, and D the given straight line not greater than the diameter of the circle. It is required to fit a straight line into the circle ABC equal to the straight line D. Draw a diameter BC of the circle ABC. If BC equals D, then that which was proposed is done, for BC has been fitted into the circle ABC equal to the straight line D. But, if BC is greater than D, make CE equal to D, describe the circle EAF with center C and radius CE, and join CA. I.3 Then, since the point C is the center of the circle EAF, CA equals CE. But CE equals D, therefore D also equals CA. Therefore CA has been fitted into the given circle ABC equal to the given straight line D. IV.Def.7 Q.E.F. The hypothesis that the line to be fitted into the circle is no longer than the diameter of the circle is certainly necessary, but Euclid did not show it was sufficient. That is sufficient to conclude the two circles actually meet at a point A is never demonstrated. This logical gap has appeared before in the Elements, for instance in Propositions I.1 and I.22. This proposition is used in the proofs of IV.10, IV.16, and occasionally in Books X, XI, and XII. Next proposition: IV.2 Previous: Definitions Book IV introduction © 1996 D.E.Joyce Clark University Proposition 2 To inscribe a triangle equiangular with a given triangle in a given circle. Let ABC be the given circle, and DEF the given triangle. It is required to inscribe a triangle equiangular with the triangle DEF in the circle ABC. Draw GH touching the circle ABC at A. Construct the angle HAC equal to the angle DEF on the straight line AH and at the point A on it, and construct the angle GAB equal to the angle DFE on the straight line AG and at the point A on it. Join BC. III.16,Cor I.23 Then, since a straight line AH touches the circle ABC, and from the point of contact at A the straight line AC is drawn across in the circle, therefore the angle HAC equals the angle ABC in the alternate segment of the circle. III.32 But the angle HAC equals the angle DEF, therefore the angle ABC also equals the angle DEF. For the same reason the angle ACB also equals the angle DFE, therefore the remaining angle BAC also equals the remaining angle EDF. I.32 Therefore a triangle equiangular with the given triangle has been inscribed in the given circle. IV.Def.2 Q.E.F. This construction is used in propositions IV.11, IV.16, and XIII.13. Next proposition: IV.3 Previous: IV.1 Book IV introduction © 1996 D.E.Joyce Clark University Proposition 3 To circumscribe a triangle equiangular with a given triangle about a given circle. Let ABC be the given circle, and DEF the given triangle. It is required to circumscribe a triangle equiangular with the triangle DEF about the circle ABC. Produce EF in both directions to the points G and H. Take the center K of the circle ABC, and draw a radius KB at random. On the straight line KB and at the point K on it, construct the angle BKA equal to the angle DEG, and the angle BKC equal to the angle DFH. Through the points A, B, and C draw LAM, MBN, and NCL touching the circle ABC. III.1 I.23 III.16,Cor Now, since LM, MN, and NL touch the circle ABC at the points A, B, and C, and KA, KB, and KC have been joined from the center K to the points A, B, and C, therefore the angles at the points A, B, and C are right. III.18 And, since the four angles of the quadrilateral AMBK equal four right angles, inasmuch as AMBK is in fact divisible into two triangles, and the angles KAM and KBM are right, therefore the sum of the remaining angles AKB and AMB equals two right angles. But the sum of the angles DEG and DEF also equals two right angles, therefore the sum of the angles AKB and AMB equals the sum of the angles DEG and DEF, of which the angle AKB equals the angle DEG, therefore the remaining angle AMB equals the remaining angle DEF. I.13 Similarly it can be proved that the angle LNB also equals the angle DFE, therefore the remaining angle MLN equals the angle EDF. I.32 Therefore the triangle LMN is equiangular with the triangle DEF, and it has been circumscribed about the circle ABC. IV.Def.4 Therefore a triangle equiangular with the given triangle has been circumscribed about a given circle. Q.E.F. This proposition is not used elsewhere in the Elements, but is included as a mate to the previous proposition in which a triangle is inscribed inside rather than circumscribed outside a given circle. Next proposition: IV.4 Previous: IV.2 Book IV introduction © 1996 D.E.Joyce Clark University Proposition 4 To inscribe a circle in a given triangle. Let ABC be the given triangle. It is required to inscribe a circle in the triangle ABC. Bisect the angles ABC and ACB by the straight lines BD and CD, and let these meet one another at the point D. Draw DE, DF, and DG from D perpendicular to the straight lines AB, BC, and CA. I.9 I.12 Now, since the angle ABD equals the angle CBD, and the right angle BED also equals the right angle BFD, EBD and FBD are two triangles having two angles equal to two angles and one side equal to one side, namely that opposite one of the equal angles, which is BD common to the triangles, therefore they will also have the remaining sides equal to the remaining sides, therefore DE equals DF. I.26 For the same reason DG also equals DF. Therefore the three straight lines DE, DF, and DG equal one another. Therefore the circle described with center D and radius one of the straight lines DE, DF, or DG also passes through the remaining points and touches the straight lines AB, BC, and CA, because the angles at the points E, F, and G are right. For, if it cuts them, the straight line drawn at right angles to the diameter of the circle from its end will be found to fall within the circle, which was proved absurd, therefore the circle described with center D and radius one of the straight lines DE, DF, or DG does not cut the straight lines AB, BC, and CA Therefore it touches them, and is the circle inscribed in the triangle ABC. III.16 IV.Def.5 Let it be inscribed as FGE. Therefore the circle EFG has been inscribed in the given triangle ABC. Q.E.F. It is easy to supply the missing argument that the angle bisectors BD and CD do meet. Incircles and excircles This circle inscribed in a triangle has come to be known as the incircle of the triangle, its center the incenter of the triangle, and its radius the inradius of the triangle. The incircle is a circle tangent to the three lines AB, BC, and AC. If these three lines are extended, then there are three other circles also tangent to them, but outside the triangle. They are called the excircles. The points on the internal angle bisector AD are equidistant from the two sides of the triangle AB and AC. The line KL is perpendicular to AD at A, and the points on it are also equidistant from the extended sides AB and AC. The line B'C' is called the external angle bisector at A. Whereas the incenter D lies at the confluence of the three internal angle bisectors, the excenter B' lies at the confluence of two external angle bisectors AB' and CB' and one internal angle bisector BB'. Likewise for the other two excenters A' and C'. Heron's formula Heron of Alexandria (first century C.E.) was an important Greek mathematician who wrote, among other things, a commentary on the Elements which is lost now but was known to Proclus and anNairizi. In Heron's Metrica, which was rediscovered in 1896, there appears a proof of what is called Heron's formula. It states that the area of a triangle is the square root of s(s-a)(s-b)(s-c) where a = BC, b = AC, and c = AB, the sides of the triangle, and s is the semiperimeter (a + b + c)/2. Archimedes may have known this formula, but but we don't have his proof. Heath gives Heron's complete proof, but here we'll just look at the first part that involves the incircle. Let D be the incenter of the triangle ABC, and let DE, DF, and DG be perpendicular lines drawn to the sides as in Euclid's proof. These three lines are radii of the incircle, and therefore have length r, the inradius. The triangle ABD has base AB and height r, so its area is r AB/2. Likewise, the area of triangle BCD is r BC/2, and the area of triangle CAD is r CA/2. Adding these together we find the area of triangle ABC is r (AB + BC + CA)/2. Therefore we have Area(ABC) = rs an interesting result in itself. We'll leave Heron's proof now and consider the corresponding statement for excircles. Now let A' be the excenter on the bisector of the internal angle at A. Let A'E', A'F', and A'G' be the perpendiculars drawn from A' to the sides of the triangle. They are radii of the excircle of length rA. Triangle ABA' has base AB and height A'E', so its area is rA AB/2. Likewise, the area of triangle BCA' is rA BC/2, and the area of triangle CAA' is rA CA/2. Triangle ABC is the sum of triangles ABA' and ACA' minus triangle BCA', so its area is rA (AB + AC – CA)/2 which equals rA(s – A). From the other excircles we get two more equations. We then have Area(ABC) = rs = rA(s – A) = rB(s – B) = rC(s – C). There are other relationships among these radii, for instance, 1/r = 1/rA + 1/rB + 1/rC, but let's stop here to go on to the circumcircle of a triangle. Next proposition: IV.5 Previous: IV.3 Book IV introduction © 1996 D.E.Joyce Clark University Proposition 5 To circumscribe a circle about a given triangle. Let ABC be the given triangle. It is required to circumscribe a circle about the given triangle ABC. Bisect the straight lines AB and AC at the points D and E. Draw DF and EF from the points D and E at right angles to AB and AC. They will then meet within the triangle ABC, or on the straight line BC, or outside BC. I.10 I.11 First let them meet within at F. Join FB, FC, and FA. Then, since AD equals DB, and DF is common and at right angles, therefore the base AF equals the base FB. I.4 Similarly we can prove that CF also equals AF, so that FB also equals FC, therefore the three straight lines FA and FB and FC equal one another. Therefore the circle described with center F and radius one of the straight lines FA, FB, or FC also passes through the remaining points, and the circle is circumscribed about the triangle ABC. IV.Def.6 Let it be circumscribed as ABC. Next, let DF and EF meet on the straight line BC at F, as is the case in the second figure. Join AF. Then, similarly, we can prove that the point F is the center of the circle circumscribed about the triangle ABC. Next, let DF and EF meet outside the triangle ABC at F, as is the case in the third figure. Join AF, BF, and CF. Then again, since AD equals DB, and DF is common and at right angles, therefore the base AF equals the base BF. I.4 Similarly we can prove that CF also equals AF, so that BF also equals FC. Therefore the circle described with center F and radius one of the straight lines FA, FB, or FC also passes through the remaining points, and is circumscribed about the triangle ABC. IV.Def.6 Therefore a circle has been circumscribed about the given triangle. Q.E.F. [Corollary] And it is manifest that when the center of the circle falls within the triangle, the angle BAC, being in a segment greater than the semicircle, is less than a right angle, when the center falls on the straight line BC, the angle BAC, being in a semicircle, is right, and when the center of the circle falls outside the triangle, the angle BAC, being in a segment less than the semicircle, is greater than a right angle. III.31 As noted by Simson and others, Euclid does not justify the intersection of the perpendicular bisectors DF and EF. Such a justification is necessary but easy to supply. The note following the proposition is not actually called a corollary in the Greek text. It is just a remark following the proposition. Circumcircles This circle drawn about a triangle is called, naturally enough, the circumcircle of the triangle, its center the circumcenter of the triangle, and its radius the circumradius. Much has been discovered about the theory of incircles and circumcircles since Euclid. The ratio that appears in the law of sines in trigonometry is the diameter of the circumcircle: 2R = BC / sin A = CA / sin B = AB / sin C where R is the circumradius. This relation is easy to derive from the figure. Angle AFC is twice the angle at B [III.20], but it is also twice angle AFE since the triangles AFE and CFE are congruent. Therefore angle AFE equals the angle at B. Then the sine of B can be found in the right triangle AFE as the ratio of the side AE opposite angle AFE to the hypotenuse AF. Since AE is half of AC, it follows that sin B = AC/(2R) which yields one of the three equations for the law of sines. There is also an equation relating the circumradius R, the inradius r, and the three exradii rA, rB, and rC: 4R = rA + rB + rC – r, and a number of other interesting results about circumcircles, incircles, and other constructions based on an arbitrary triangle. This construction is used in propositions IV.10 and XI.23. Next proposition: IV.6 Previous: IV.4 Book IV introduction © 1996 D.E.Joyce Clark University Proposition 6 To inscribe a square in a given circle. Let ABCD be the given circle. It is required to inscribe a square in the circle ABCD. Draw two diameters AC and BD of the circle ABCD at right angles to one another, and join AB, BC, CD, and DA. III.1 I.11 Then, since BE equals ED, for E is the center, and EA is common and at right angles, therefore the base AB equals the base AD. I.4 For the same reason each of the straight lines BC and CD also equals each of the straight lines AB and AD. Therefore the quadrilateral ABCD is equilateral. I say next that it is also right-angled. For, since the straight line BD is a diameter of the circle ABCD, therefore BAD is a semicircle, therefore the angle BAD is right. III.31 For the same reason each of the angles ABC, BCD, and CDA is also right. Therefore the quadrilateral ABCD is right-angled. But it was also proved equilateral, therefore it is a square, and it has been inscribed in the circle ABCD. Therefore the square ABCD has been inscribed in the given circle. Q.E.F. This construction is used in a few propositions of Book XII, the first being XII.2. Next proposition: IV.7 Previous: IV.5 Book IV introduction © 1996 D.E.Joyce Clark University Proposition 7 To circumscribe a square about a given circle. Let ABCD be the given circle. It is required to circumscribe a square about the circle ABCD. Draw two diameters AC and BD of the circle ABCD at right angles to one another. Draw FG, GH, HK, and KF through the points A, B, C, and D touching the circle ABCD. III.1 I.11 III.16,Cor Then, since FG touches the circle ABCD, and EA has been joined from the center E to the point of contact at A, therefore the angles at A are right. III.18 For the same reason the angles at the points B, C, and D are also right. Now, since the angle AEB is right, and the angle EBG is also right, therefore GH is parallel to AC. I.28 For the same reason AC is also parallel to FK, so that GH is also parallel to FK. I.30 Similarly we can prove that each of the straight lines GF and HK is parallel to BED. Therefore GK, GC, AK, FB, and BK are parallelograms, therefore GF equals HK, and GH equals FK. I.34 And, since AC equals BD, and AC also equals each of the straight lines GH and FK, and BD equals each of the straight lines GF and HK, therefore the quadrilateral FGHK is equilateral. I.34 I say next that it is also right-angled. For, since GBEA is a parallelogram, and the angle AEB is right, therefore the angle AGB is also right. I.34 Similarly we can prove that the angles at H, K, and F are also right. Therefore FGHK is right-angled. But it was also proved equilateral, therefore it is a square, and it has been circumscribed about the circle ABCD. Therefore a square has been circumscribed about the given circle. Q.E.F. This proposition is used in XII.10. Next proposition: IV.8 Previous: IV.6 Book IV introduction © 1996 D.E.Joyce Clark University Proposition 8 To inscribe a circle in a given square. Let ABCD be the given square. It is required to inscribe a circle in the given square ABCD. Bisect the straight lines AD and AB at the points E and F respectively. Draw EH through E parallel to either AB or CD, and draw FK through F parallel to either AD or BC. Therefore each of the figures AK, KB, AH, HD, AG, GC, BG, and GD is a parallelogram, and their opposite sides are evidently equal. I.10 I.31 I.34 Now, since AD equals AB, and AE is half of AD, and AF half of AB, therefore AE equals AF, so that the opposite sides are also equal, therefore FG equals GE. Similarly we can prove that each of the straight lines GH and GK equals each of the straight lines FG and GE. Therefore the four straight lines GE, GF, GH, and GK equal one another. Therefore the circle described with center G and radius one of the straight lines GE, GF, GH, or GK also passes through the remaining points. And it touches the straight lines AB, BC, CD, and DA, because the angles at E, F, H, and K are right. For, if the circle cuts AB, BC, CD, or DA, the straight line drawn at right angles to the diameter of the circle from its end will fall within the circle, which was proved absurd. Therefore the circle described with center G and radius one of the straight lines GE, GF, GH, or GK does not cut the straight lines AB, BC, CD, and DA. III.16 Therefore it touches them, and has been inscribed in the square ABCD. Therefore a circle has been inscribed in the given square. Q.E.F. This is a straightforward proposition, one of four in the sequence IV.6 through IV.9 about circles and squares. Next proposition: IV.9 Previous: IV.7 Book IV introduction © 1996 D.E.Joyce Clark University Proposition 9 To circumscribe a circle about a given square. Let ABCD be the given square. It is required to circumscribe a circle about the square ABCD. Join AC and BD, and let them cut one another at E. Then, since DA equals AB, and AC is common, therefore the two sides DA and AC equal the two sides BA and AC, and the base DC equals the base BC, therefore the angle DAC equals the angle BAC. I.8 Therefore the angle DAB is bisected by AC. Similarly we can prove that each of the angles ABC, BCD, and CDA is bisected by the straight lines AC and DB. Now, since the angle DAB equals the angle ABC, and the angle EAB is half of the angle DAB, and the angle EBA half of the angle ABC, therefore the angle EAB also equals the angle EBA, so that the side EA also equals EB. I.6 Similarly we can prove that each of the straight lines EA and EB equals each of the straight lines EC and ED. Therefore the four straight lines EA, EB, EC, and ED equal one another. Therefore the circle described with center E and radius one of the straight lines EA, EB, EC, or ED also passes through the remaining points, and it is circumscribed about the square ABCD. Let it be circumscribed, as ABCD. Therefore a circle has been circumscribed about the given square. Q.E.F. This is a straightforward proposition, one of four in the sequence IV.6 through IV.9 about circles and squares. Next proposition: IV.10 Previous: IV.8 Book IV introduction © 1996 D.E.Joyce Clark University Proposition 10 To construct an isosceles triangle having each of the angles at the base double the remaining one. Set out any straight line AB, and cut it at the point C so that the rectangle AB by BC equals the square on CA. Describe the circle BDE with center A and radius AB. Fit in the circle BDE the straight line BD equal to the straight line AC which is not greater than the diameter of the circle BDE. II.11 IV.1 Join AD and DC, and circumscribe the circle ACD about the triangle ACD. IV.5 Then, since the rectangle AB by BC equals the square on AC, and AC equals BD, therefore the rectangle AB by BC equals the square on BD. And, since a point B was taken outside the circle ACD, and from B the two straight lines BA and BD fall on the circle ACD, and one of them cuts it while the other falls on it, and the rectangle AB by BC equals the square on BD, therefore BD touches the circle ACD. III.37 Since, then, BD touches it, and DC is drawn across from the point of contact at D, therefore the angle BDC equals the angle DAC in the alternate segment of the circle. III.32 Since, then, the angle BDC equals the angle DAC, add the angle CDA to each, therefore the whole angle BDA equals the sum of the two angles CDA and DAC. But the exterior angle BCD equals the sum of the angles CDA and DAC, therefore the angle BDA also equals the angle BCD. I.32 But the angle BDA equals the angle CBD, since the side AD also equals AB, so that the angle DBA also equals the angle BCD. I.5 Therefore the three angles BDA, DBA, and BCD equal one another. And, since the angle DBC equals the angle BCD, the side BD also equals the side DC. I.6 But BD equals CA by hypothesis, therefore CA also equals CD, so that the angle CDA also equals the angle DAC. Therefore the sum of the angles CDA and DAC is double the angle DAC. I.5 And the angle BCD equals the sum of the angles CDA and DAC, therefore the angle BCD is also double the angle CAD. But the angle BCD equals each of the angles BDA and DBA, therefore each of the angles BDA and DBA is also double the angle DAB. Therefore the isosceles triangle ABD has been constructed having each of the angles at the base DB double the remaining one. Q.E.F. The goal of the proposition is to construct a 36°-72°-72° isosceles triangle ABD. It's actually constructed on a given side AB. The base will equal the larger part of AB when AB is cut at a point C so that AB BC = AC2. The constuction for that cut was given in proposition II.11. The difficulty of the proof is showing that this construction results in the desired triangle. Cutting AB at that point is also called cutting the line "in extreme and mean ratio," see VI.Def.3 for the definition of "extreme and mean ratio," and see proposition VI.30 for details. Euclid uses a surprizing amount of the theory of circles from Book III. After drawing circle ACD, he uses III.37 to conclude from AB BC = DB2 that the line DB is tangent to the circle. Next, he uses III.32 to conclude that the angle BDC between the tangent DB and the chord DC equals the angle CAD which cuts off that chord. At this point Euclid has shown that one of the two angles at D, namely angle BDC, equals the angle A. When he shows that the other, namely angle CDA also equals angle A, then since the triangle ABD is isosceles, he will have shown each of the base angles of triangle ABD is twice the vertex angle A, and the proof will be complete. The rest is relatively easy. First, the small triangle BCD is isosceles, a fact that can be seen from the following equation about angles: B = BDA = BDC + CDA = CAD + CDA = BCD. Therefore, the sides CD and BD are equal, but from the original construction, BD = CA. Hence, the triangle ADC is also isosceles, so the two angles CDA and A are equal, as needed. Comments Euclid could have split the statement and the proof of this proposition into two. The first part would state that if an isosceles triangle has its base equal to a segment of its side so that square on the base equals the rectangle contained by the side and the remaining segment of the side, then each base angle of the triangle is twice the vertex angle. Most of the proof of this proposition IV.10 is actually a proof of this first part. The other part would be the constuction. There is a converse of this proposition, one the Euclid did not state. Namely, if an isosceles triangle has each base angle equal to twice the vertex angle, then the base is equal to a segment of its side so that square on the base equals the rectangle contained by the side and the remaining segment of the side. In other words, 36°-72°-72° isosceles triangles are characterized by this property. The triangle ABD constructed in this proposition is one of ten sectors of a regular decagon (10-gon). Thus, it is one short step from this proposition to the construction of a regular decagon inscribed in a circle. If alternate vertices of a regular decagon are connected, then a regular pentagon is formed which is inscribed in the circle. It is unclear why Euclid did not use such a construction rather than the one he chose in the next proposition An alternate proof involving similar triangles It was probably Euclid who made a concerted effort to include as many propositions that he could in the first four books that did not rely on proportions. The theory of similar triangles is not broached until Book VI which depends on the theory of proportion in Book V. The clever proof that Euclid gave to this proposition does not depend on similar triangles, and so it could be placed here in Book IV. There is, however, a simpler proof that does depend on similar triangles. As Euclid does, begin by cutting a straight line AB at the point C so that the rectangle AB by BC equals the square on CA (II.11). Otherwise said, the straight line AB has been cut in extreme and mean ratio at C so that the proportion AB:AC = AC:BC holds. (See VI.Def.3, VI.17, and VI.30.) Next, construct an isosceles triangle with one side AB, a second side AD equal to side AB, and the base equal BD equal to AC (I.22). Then we have the proportion AD:BD = BD:BC. Therefore, two triangles ADB and DBC have one angle equal to one angle (angle D of the first triangle equals angle B of the second) and the sides about the equal angles proportional. Therefore, by VI.6, the triangles are equiangular. It easily follows that both triangles have their base angles each equal to twice their vertex angles. Use of this proposition This construction was designed to be used in the next proposition which inscribes a regular pentagon in a circle. Next proposition: IV.11 Previous: IV.9 Book IV introduction © 1996, 2002 D.E.Joyce Clark University Proposition 11 To inscribe an equilateral and equiangular pentagon in a given circle. Let ABCDE be the given circle. It is required to inscribe an equilateral and equiangular pentagon in the circle ABCDE. Set out the isosceles triangle FGH having each of the angles at G and H double the angle at F. Inscribe in the circle ABCDE the triangle ACD equiangular with the triangle FGH, so that the angles CAD, ACD, and CDA equal the angles at F, G, and H respectively. Therefore each of the angles ACD and CDA is also double the angle CAD. IV.10 IV.2 Now bisect the angles ACD and CDA respectively by the straight lines CE and DB, and join AB, BC, DE, and EA. I.9 Then, since each of the angles ACD and CDA is double the angle CAD, and they are bisected by the straight lines CE and DB, therefore the five angles DAC, ACE, ECD, CDB, and BDA equal one another. But equal angles stand on equal circumferences, therefore the five circumferences AB, BC, CD, DE, and EA equal one another. III.26 But straight lines that cut off equal circumferences are equal, therefore the five straight lines AB, BC, CD, DE, and EA equal one another. Therefore the pentagon ABCDE is equilateral. III.29 I say next that it is also equiangular. For, since the circumference AB equals the circumference DE, add BCD to each, therefore the whole circumference ABCD equals the whole circumference EDCB. And the angle AED stands on the circumference ABCD, and the angle BAE on the circumference EDCB, therefore the angle BAE also equals the angle AED. III.27 For the same reason each of the angles ABC, BCD, and CDE also equals each of the angles BAE and AED, therefore the pentagon ABCDE is equiangular. But it was also proved equilateral, therefore an equilateral and equiangular pentagon has been inscribed in the given circle. Q.E.F. Richmond's construction The construction of this proposition is rather tedious to carry out. First, a line has to be cut according to the construction in II.11. Next, that is used in IV.10 for the construction of a 36°-72°-72° isosceles triangle. Next, that triangle is fit into the given circle using the construction IV.2. Finally, a couple more lines are drawn to finish the pentagon. Various alternatives have have been given by others, such as Ptolemy. One of the nicest was given in 1893 by H. W. Richmond. To inscribe a regular pentagon in a circle, first draw perpendicular radii OA and OB from the center O of a circle. Let C be the midpoint of OB and draw AC. Bisect angle ACO to meet OA at D. Draw a perpendicular DE to OA to the circle. Then AE is one side of the pentagon. The remaining sides can then be constructed. java applet or image The easiest way to verify that Richmond's construction works is by means of trigonometry. The angles and sections in extreme and mean ratio in a pentagram Consider the regular pentagon ABCDE constructed in this proposition. Draw the diagonals of the pentagram to create a regular star pentagram ACEBD inside the pentagon. These diagonals meet forming a smaller regular pentagon in the center of the original pentagon. The diagonals of that pentagon can be drawn to make an inscribed pentagram which, in turn, bound a yet smaller regular pentagon. And so forth. For purposes of analysis, let d1 and s1 denote the diagonal and side of the first regular pentagon ABCDE. Also let d2 and s2 denote the diagonal and side of the second regular pentagon FGHKL. And so forth. It is evident that there are many lines parallel to the base CD of the triangle, namely BE and LG, as well as innumerable ones in the smaller pentagrams. That means that there will be many 36°-72°-72° triangles besides the large one ACD. The next smaller one is ALG, then FKH, then many smaller ones. Also, each of these various sized 36°-72°-72° triangles is congruent many others in the diagram. For instance, triangles ALG and EAK are congruent. There are also a series of obtuse 36°-36°-108° isosceles triangles of varying sizes. All these parallel lines and similar triangles yield numerous relationships among the various diagonals and sides of the pentagons. Some of these relationships are additive equations: d1= s1+ d2 s1= d2+ s2 d2= s2+ d3 s2= d3+ s3 and so forth. Other relationships are based on the property of 36°-72°-72° triangles used in their construction in IV.10, namely that the square of the base of such a triangle equals the product of a side and the difference between the side and the base. In terms of the diagonals and sides of the pentagons, this gives the equations: d1d2= s12 s1s2= d22 d2d3= s22 s2s3= d32 and so forth. After ratios are proportions are developed in Book V and Book VI, we can add the following continued proportion to the list of relationships: d1:s1= s1:d2= d2:s2= s2:d3= ... See the Guide to proposition X.2 which shows that diagonal and side of a regular pentagon are incommensurable. In more modern terms we would say that their ratio, which is called the "golden ratio," is an irrational number. Use of this proposition This construction is used in the next proposition to circumscribe a regular pentagon around a circle and later in IV.16 to construct a regular 15-gon. It is also used in XIII.16 for the construction of a regular icosahedron (a 20-sided polyhedron each of whose faces is an equilateral triangle). Surprizingly, it is not used in XIII.17 for construct a regular dodecahedron (a 12-sided polyhedron each of whose faces is a regular pentagon); the regular pentagons needed for it are constructed in space directly without the help of this proposition. Next proposition: IV.12 Previous: IV.10 Book IV introduction © 1996, 1998, 2002 D.E.Joyce Clark University Proposition 12 To circumscribe an equilateral and equiangular pentagon about a given circle. Let ABCDE be the given circle. It is required to circumscribe an equilateral and equiangular pentagon about the circle ABCDE. Let A, B, C, D, and E be conceived to be the angular points of the inscribed pentagon, so that the circumferences AB, BC, CD, DE, and EA are equal. Draw GH, HK, KL, LM, and MG through A, B, C, D, and E touching the circle. Take the center F of the circle ABCDE, and join FB, FK, FC, FL, and FD. IV.11 III.16,Cor III.1 Then, since the straight line KL touches the circle ABCDE at C, and FC has been joined from the center F to the point of contact at C, therefore FC is perpendicular to KL. Therefore each of the angles at C is right. III.18 For the same reason the angles at the points B and D are also right. And, since the angle FCK is right, therefore the square on FK equals the sum of the squares on FC and CK. I.47 For the same reason the square on FK also equals the sum of the squares on FB and BK, so that the sum of the squares on FC and CK equals the sum of the squares on FB and BK, of which the square on FC equals the square on FB, therefore the remaining square on CK equals the square on BK. I.47 Therefore BK equals CK. And, since FB equals FC, and FK is common, the two sides BF and FK equal the two sides CF and FK, and the base BK equals the base CK, therefore the angle BFK equals the angle KFC, and the angle BKF equals the angle FKC. Therefore the angle BFC is double the angle KFC, and the angle BKC double the angle FKC. I.8 For the same reason the angle CFD is also double the angle CFL, and the angle DLC double the angle FLC. Now, since the circumference BC equals CD, the angle BFC also equals the angle CFD. III.27 And the angle BFC is double the angle KFC, and the angle DFC double the angle LFC, therefore the angle KFC also equals the angle LFC. But the angle FCK also equals the angle FCL, therefore FKC and FLC are two triangles having two angles equal to two angles and one side equal to one side, namely FC which is common to them, therefore they will also have the remaining sides equal to the remaining sides, and the remaining angle to the remaining angle, therefore the straight line KC equals CL, and the angle FKC equals the angle FLC. I.26 And, since KC equals CL, therefore KL is double KC. For the same reason it can be proved that HK is also double BK. And BK equals KC, therefore HK also equals KL. Similarly each of the straight lines HG, GM, and ML can also be proved equal to each of the straight lines HK and KL, therefore the pentagon GHKLM is equilateral. I say next that it is also equiangular. For, since the angle FKC equals the angle FLC, and the angle HKL was proved double the angle FKC, and the angle KLM double the angle FLC, therefore the angle HKL also equals the angle KLM. Similarly each of the angles KHG, HGM, and GML can also be proved equal to each of the angles HKL and KLM. Therefore the five angles GHK, HKL, KLM, LMG, and MGH equal one another. Therefore the pentagon GHKLM is equiangular. And it was also proved equilateral, and it has been circumscribed about the circle ABCDE. Q.E.F. This construction depends on the last. First, inscribe a regular pentagon in the circle, then take tangents to the circle at the five vertices of the inscribed pentagon. The result will be a circumscribed pentagon. This method generally works to create a regular circumscribed n-gon given a regular inscribed n-gon. Conversely, if you have a regular circumscribed n-gon, then you can connect the points of tangency in sequence to get a regular inscribed n-gon. Next proposition: IV.13 Previous: IV.11 Book IV introduction © 1996 D.E.Joyce Clark University Proposition 13 To inscribe a circle in a given equilateral and equiangular pentagon. Let ABCDE be the given equilateral and equiangular pentagon. It is required to inscribe a circle in the pentagon ABCDE. Bisect the angles BCD and CDE by the straight lines CF and DF respectively. Join the straight lines FB, FA, and FE from the point F at which the straight lines CF and DF meet one another. I.9 Then, since BC equals CD, and CF common, the two sides BC and CF equal the two sides DC and CF, and the angle BCF equals the angle DCF, therefore the base BF equals the base DF, and the triangle BCF equals the triangle DCF, and the remaining angles equal the remaining angles, namely those opposite the equal sides. I.4 Therefore the angle CBF equals the angle CDF. And, since the angle CDE is double the angle CDF, and the angle CDE equals the angle ABC, while the angle CDF equals the angle CBF, therefore the angle CBA is also double the angle CBF. Therefore the angle ABF equals the angle FBC. Therefore the angle ABC is bisected by the straight line BF. Similarly it can be proved that the angles BAE and AED are also bisected by the straight lines FA and FE respectively. Now draw FG, FH, FK, FL, and FM from the point F perpendicular to the straight lines AB, BC, CD, DE, and EA. I.12 Then, since the angle HCF equals the angle KCF, and the right angle FHC also equals the angle FKC, FHC and FKC are two triangles having two angles equal to two angles and one side equal to one side, namely FC which is common to them and opposite one of the equal angles, therefore they also have the remaining sides equal to the remaining sides. Therefore the perpendicular FH equals the perpendicular FK. I.26 Similarly it can be proved that each of the straight lines FL, FM, and FG also equals each of the straight lines FH and FK, therefore the five straight lines FG, FH, FK, FL, and FM equal one another. Therefore the circle described with center F and radius one of the straight lines FG, FH, FK, FL, or FM also passes through the remaining points, and it touches the straight lines AB, BC, CD, DE, and EA, because the angles at the points G, H, K, L, and M are right. For, if it does not touch them. but cuts them, it will result that the straight line drawn at right angles to the diameter of the circle from its end falls within the circle, which was proved absurd. III.16 Therefore the circle described with center F and radius one of the straight lines FG, FH, FK, FL, or FM does not cut the straight lines AB, BC, CD, DE, and EA. Therefore it touches them. Let it be described, as GHKLM. Therefore a circle has been inscribed in the given equilateral and equiangular pentagon. Q.E.F. The method given here to inscribe a circle in a regular pentagon works in general to inscribe a circle in a regular n-gon. Simply bisect two of the angles of the n-gon to find the center of the circle. Then draw a perpendicular to one of the sides. The foot of the perpendicular gives a point on the circumference of the circle. Next proposition: IV.14 Previous: IV.12 Book IV introduction © 1996 D.E.Joyce Clark University Proposition 14 To circumscribe a circle about a given equilateral and equiangular pentagon. Let ABCDE be the given pentagon, which is equilateral and equiangular. It is required to circumscribe a circle about the pentagon ABCDE. Bisect the angles BCD and CDE by the straight lines CF and DF respectively. Join the straight lines FB, FA, and FE from the point F at which the straight lines meet to the points B, A, and E. I.9 Then in manner similar to the preceding it can be proved that the angles CBA, BAE, and AED are also bisected by the straight lines FB, FA, and FE respectively. Now, since the angle BCD equals the angle CDE, and the angle FCD is half of the angle BCD, and the angle CDF half of the angle CDE, therefore the angle FCD also equals the angle CDF, so that the side FC also equals the side FD. I.6 Similarly it can be proved that each of the straight lines FB, FA, and FE also equals each of the straight lines FC and FD. Therefore the five straight lines FA, FB, FC, FD, and FE equal one another. Therefore the circle described with center F and radius one of the straight lines FA, FB, FC, FD, or FE also passes through the remaining points, and is circumscribed. Let it be circumscribed, and let it be ABCDE. Therefore a circle has been circumscribed about the given equilateral and equiangular pentagon. Q.E.F. This construction is used in propositions XIII.8 and XIII.18. Next proposition: IV.15 Previous: IV.13 Book IV introduction © 1996 D.E.Joyce Clark University Proposition 15 To inscribe an equilateral and equiangular hexagon in a given circle. Let ABCDEF be the given circle. It is required to inscribe an equilateral and equiangular hexagon in the circle ABCDEF. Draw the diameter AD of the circle ABCDEF. Take the center G of the circle. Describe the circle EGCH with center D and radius DG. Join EG and CG and carry them through to the points B and F. Join AB, BC, CD, DE, EF, and FA. III.1 I say that the hexagon ABCDEF is equilateral and equiangular. For, since the point G is the center of the circle ABCDEF, GE equals GD. Again, since the point D is the center of the circle GCH, DE equals DG. But GE was proved equal to GD, therefore GE also equals ED. Therefore the triangle EGD is equilateral, and therefore its three angles EGD, GDE, and DEG equal one another, inasmuch as, in isosceles triangles, the angles at the base equal one another. I.5 And the sum of the three angles of the triangle equals two right angles, therefore the angle EGD is one-third of two right angles. I.32 Similarly, the angle DGC can also be proved to be one third of two right angles. And, since the straight line CG standing on EB makes the sum of the adjacent angles EGC and CGB equal to two right angles, therefore the remaining angle CGB is also one-third of two right angles. I.13 Therefore the angles EGD, DGC, and CGB equal one another, so that the angles vertical to them, the angles BGA, AGF, and FGE, are equal. I.15 Therefore the six angles EGD, DGC, CGB, BGA, AGF, and FGE equal one another. But equal angles stand on equal circumferences, therefore the six circumferences AB, BC, CD, DE, EF, and FA equal one another. III.26 And straight lines that cut off equal circumferences are equal, therefore the six straight lines equal one another. Therefore the hexagon ABCDEF is equilateral. III.29 I say next that it is also equiangular. For, since the circumference FA equals the circumference ED, add the circumference ABCD to each, therefore the whole FABCD equals the whole EDCBA. And the angle FED stands on the circumference FABCD, and the angle AFE on the circumference EDCBA, therefore the angle AFE equals the angle DEF. III.27 Similarly it can be proved that the remaining angles of the hexagon ABCDEF are also severally equal to each of the angles AFE and FED, therefore the hexagon ABCDEF is equiangular. But it was also proved equilateral, and it has been inscribed in the circle ABCDEF. Therefore an equilateral and equiangular hexagon has been inscribed in the given circle. Q.E.F. Corollary From this it is manifest that the side of the hexagon equals the radius of the circle. And, in like manner as in the case of the pentagon, if through the points of division on the circle we draw tangents to the circle, there will be circumscribed about the circle an equilateral and equiangular hexagon in conformity with what was explained in the case of the pentagon. And further by means similar to those explained in the case of the pentagon we can both inscribe a circle in a given hexagon and circumscribe one about it. The corollary is used in several propositions in Book XIII starting with XIII.9. Next proposition: IV.16 Previous: IV.14 Book IV introduction © 1996 D.E.Joyce Clark University Proposition 16 To inscribe an equilateral and equiangular fifteen-angled figure in a given circle. Let ABCD be the given circle. It is required to inscribe in the circle ABCD a fifteen-angled figure which shall be both equilateral and equiangular. Inscribe a side AC of an equilateral triangle and a side AB of an equilateral pentagon in in the circle ABCD. Therefore, of the equal segments of which there are fifteen in the circle ABCD, there will be five in the circumference ABC which is one-third of the circle, and there will be three in the circumference AB which is one-fifth of the circle. Therefore in the remainder BC there will be two of the equal segments. IV.2 IV.11 Inscribe a side AC of an equilateral triangle and a side AB of an equilateral pentagon in in the circle ABCD. Therefore, of the equal segments of which there are fifteen in the circle ABCD, there will be five in the circumference ABC which is one-third of the circle, and there will be three in the circumference AB which is onefifth of the circle. Therefore in the remainder BC there will be two of the equal segments. IV.2 IV.11 Bisect BC at E. Therefore each of the circumferences BE and EC is a fifteenth of the circle ABCD. III.30 If therefore we join BE and EC and continually fit into the circle ABCD straight lines equal to them, a fifteen-angled figure which is both equilateral and equiangular will be inscribed in it. IV.1 Q.E.F. Corollary And, in like manner as in the case of the pentagon, if through the points of division on the circle we draw tangents to the circle, there will be circumscribed about the circle a fifteen-angled figure which is equilateral and equiangular. And further, by proofs similar to those in the case of the pentagon, we can both inscribe a circle in the given fifteen-angled figure and circumscribe one about it. The arc AC is 1/3 of the circle, since A and B are two of the three equally spaced vertices of a regular triangle. Likewise, the arc AC is 1/5 of the circle, since A and C are two adjacent points of a regular pentagon. Therefore, the difference of these two arcs, AC – AB, which is the arc BC is 1/3 –1/5 of the circle, that is 2/15 of the circle. Since E bisects that arc BC, therefore BE and EC are each 1/15 of the circle. The rest of the regular 15-gon can then easily be constructed. Constructable regular polygons Now, by the end of Book IV, Euclid has described how to construct many regular polygons. The regular 3-gon, known as the equilateral triangle, was constructed in I.1, while the regular 4-gon, known as the square, was constructed in I.46. In book IV, regular 5-gons and regular 6-gons have been constructed. An application of III.30 (which was used in this proposition) can double the number of sides of a regular polygon, and therefore regular polygons with 8, 10, 12, 16, 20, 24, etc., sides can be constructed. This proposition shows how to use a regular m-gon and a regular n-gon to produce a regular mn-gon, provided that m and n are relatively prime numbers. That produced a 15gon, and from that we can produce regular polygons with 30, 60, 120, etc., sides. Thus, a regular ngon can be constructed if the only prime numbers that divide n are 2, 3, and 5, where 2 can be a repeated factor, but 3 and 5 are not repeated. But are there any others? What about regular polygons with 7, 9, 11, 13, 17, 18, 19, etc., sides? Euclid said nothing about them, but the ancient Greek mathematicians expected that they couldn't be constructed with only the Euclidean tools of straightedge and compass. There were constructions involving conic sections (hyperbolas, parabolas, ellipses) to trisect an angle. With such a construction a 9-gon can be made. But methods involving conic sections go beyond Euclidean tools. With the help of non-algebraic curves, like Archimedes' spiral, an angle can be divided into any number of equal parts, and with the aid of those curves any n-gon can be constructed. But, again, they go beyond Euclidean tools. The problem of constructing other regular polygons with Euclidean tools remained just that, a problem, for over 2000 years. Finally, Carl Friedrich Gauss (1777-1855) made progress. He described in his Disquitiones Arithmeticae, a major work on number theory, how to construct a regular 17-gon with Euclidean tools. Thus, 17 can be added to 3 and 5 as prime numbers that can divide n, but at most once. Furthermore, he showed that any prime number which is of the form 22k + 1 can be included. Such prime numbers are called Fermat primes. The known Fermat primes are 3 (which is 220 + 1), 5 (which is 221 + 1), 17 (which is 222 + 1), 257 (which is 223 + 1), and 65537 (which is 224 + 1). Thus, 257 and 65537 can be appended to the list 3, 5, 17. It is not known whether there are any more Fermat primes. Gauss was convinced that the only constructable n-gons were those where n was only divisible by 2 and the Fermat primes, where the Fermat primes were not repeated. But he had no proof of that, but in 1837 Wantzel did. Next book: Book V Introduction Previous proposition: IV.15 Book IV introduction © 1996, 2002 D.E.Joyce Clark University Proposition 11 To cut a given straight line so that the rectangle contained by the whole and one of the segments equals the square on the remaining segment. Let AB be the given straight line. It is required to cut AB so that the rectangle contained by the whole and one of the segments equals the square on the remaining segment. Describe the square ABDC on AB. Bisect AC at the point E, and join BE. Draw CA through to F, and make EF equal to BE. Describe the square FH on AF, and draw GH through to K. I.46 I.10 I.3 I.46 I say that AB has been cut at H so that the rectangle AB by BH equals the square on AH. Since the straight line AC has been bisected at E, and FA is added to it, the rectangle CF by FA together with the square on AE equals the square on EF. II.6 But EF equals EB, therefore the rectangle CF by FA together with the square on AE equals the square on EB. But the sum of the squares on BA and AE equals the square on EB, for the angle at A is right, therefore the rectangle CF by FA together with the square on AE equals the sum of the squares on BA and AE. I.47 Subtract the square on AE from each. Therefore the remaining rectangle CF by FA equals the square on AB. Now the rectangle CF by FA is FK, for AF equals FG, and the square on AB is AD, therefore FK equals AD. Subtract AK from each. Therefore FH, which remains, equals HD. And HD is the rectangle AB by BH, for AB equals BD, and FH is the square on AH, therefore the rectangle AB by BH equals the square on HA. Therefore the given straight line AB has been cut at H so that the rectangle AB by BH equals the square on HA. Q.E.F. This construction cuts a line into two parts to solve the equation a (a – x) = x2 geometrically. This construction is used in the proof of IV.10, which is later used to construct a regular pentagon. It accomplishes the same thing as the construction of proposition VI.30, which cuts a line into extreme and mean ratio, defined in VI.Def.3, and that construction is used later in XIII.17. The difference between this proposition and VI.30 is a matter of terminology. Propositions dealing with ratios of lines are postponed until Book VI, but any ratio concerning lines can be converted into a statement about areas of rectangles. Proposition VI.16 states that the line A is to the line B as the line C is to the line D is equivalent to the statement that the rectangle A by D equals the rectangle B by C. The construction of this proposition cuts a line into two parts A and B so that the rectangle A + B by A equals the square B by B. The construction in VI.30 cuts a line so that A + B : B = B : A, which by VI.16, or by its special case VI.17, is the same thing. Construction steps For the purposes of cutting the line AB, the entire diagram does not have to be constructed. The points D, G, and K are unnecessary. In the diagram to the right, only those lines and circles necessary for the construction are shown, and only those parts of them that are relevant. Altogether, there are six circles to be drawn, two lines connected, and one line extended. In order, they are as follows. Extend BA and draw a circle centered at A with radius AB to determine the point L. Draw circles centered at B and L with radius BL to determine points M and N. Draw the straight line MN. Then C is where it meets the circle centered at A. Draw a circle centered at C with radius CA, and connect the two points where it meets the circle centered at A. Then E is where that line meets AC. Draw the circle centered at E with radius EB to determine F. Draw the circle centered at A with radius F to determine H, the desired point to cut AB. The golden ratio, the 36°-72°-72° triangle, and regular pentagons This is the first of several propositions in the Elements that treats these concepts. At this point, ratios have not been introduced, so Euclid describes it in basic terms, that a given straight line is cut so that "the rectangle contained by the whole and one of the segments equals the square on the remaining segment." Next proposition: II.12 Previous: II.10 Book II introduction © 1996 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Definition 1 Equal circles are those whose diameters are equal, or whose radii are equal. This should not be a definition but a postulate or a theorem. The subject of the area of circles is developed in Proposition XII.2 and this definition is not used there. Instead the concept of equality, or rather, inequality is the same as it is in the rest of the Elements. For instance, if one figure is the contained in another, then the first is less than the other. Thus, there is a prior defintion for the equality of two figures. Two circles are illustrated, namely circle BCD and circle FGH. The center of circle BCD is A, while the center of circle FGH is E. They are equal by Euclid's definition since their diameters BC and FG are equal, or since their radii AB and EF are equal. Next definition: III.Def.2-3 Book III introduction © 1996, 1997 D.E.Joyce Clark University Definitions 2 and 3 Def. 2. A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle. Def. 3. Circles are said to touch one another which meet one another but do not cut one another. Consider circle BCD in the figure. The line EF touches this circle at the point C. Another expression for the same thing is that EF is tangent to the circle at C. Two circles can touch each other either internally or externally. Circle HKL touches circle BCD externally, while circle DMN touches circle BCD internally. Next definitions: III.Def.4-5 Previous: III.Def.1 Book III introduction © 1996 D.E.Joyce Clark University Definitions 4 and 5 Def. 4. Straight lines in a circle are said to be equally distant from the center when the perpendiculars drawn to them from the center are equal. Def. 5. And that straight line is said to be at a greater distance on which the greater perpendicular falls. These definitions could have been broadened to distances from a line to a point, but Euclid's needs are for this situation. The perpendiculars AE and AH drawn from the center A to the lines CD and FG respectively are equal, so the lines CD and FG are equally distant from the center. As the perpendicular AM is greater, the line KL is at a greater distance from the center. Next definitions: III.Def.6-9 Previous: III.Def.2-3 Book III introduction © 1996 D.E.Joyce Clark University Definitions 6 through 9 Def. 6. A segment of a circle is the figure contained by a straight line and a circumference of a circle. Def. 7. An angle of a segment is that contained by a straight line and a circumference of a circle. Def. 8. An angle in a segment is the angle which, when a point is taken on the circumference of the segment and straight lines are joined from it to the ends of the straight line which is the base of the segment, is contained by the straight lines so joined. Def. 9. And, when the straight lines containing the angle cut off a circumference, the angle is said to stand upon that circumference. A line in a circle, such as the line BC, divides the circle into two segments, the small blue segment BDC, and the large yellow segment BEC. An angle of the segment BDC is not a rectilinear angle, since only one of its sides, BC, is a straight line. The other side is curved, namely, an arc of a circle. These angles of segments only appear in proposition III.16, and are not important in Euclid's development of geometry. An example of an angle in a segment is the angle BFC in the yellow segment BEC. This angle BFC stands upon the circumference (arc) BDC. Angles in segments are rectilinear, and they are important. In proposition III.21, Euclid proves that all the angles in a given segment are equal. Next definition: III.Def.10 Previous: III.Def.4-5 Book III introduction © 1996 D.E.Joyce Clark University Definition 10 A sector of a circle is the figure which, when an angle is constructed at the center of the circle, is contained by the straight lines containing the angle and the circumference cut off by them. Here a sector BAC is illustrated. The angle BAC encloses the sector. Note that the remainder of the circle would not be considered a sector by Euclid since the angle at the center would be greater than 180°. Next definition: III.Def.11 Previous: III.Def.6-9 Book III introduction © 1996 D.E.Joyce Clark University Definition 11 Similar segments of circles are those which admit equal angles, or in which the angles equal one another. Since the segments BGC and EHF have in them equal angles BGC and EHF, this definition declares them to be similar segments. This is hardly a proper definition considering that proposition III.21 has yet to be proved in which it is shown that all the angles in one segment are equal. Next: Proposition III.1 Previous: Definition III.10 Book III introduction © 1996 D.E.Joyce Clark University Proposition 1 To find the center of a given circle. Let ABC be the given circle. It is required to find the center of the circle ABC. Draw a straight line AB through it at random, and bisect it at the point D. Draw DC from D at right angles to AB, and draw it through to E. Bisect CE at F. I.10 I.11 I.10 I say that F is the center of the circle ABC. For suppose it is not, but, if possible, let G be the center. Join GA, GD, and GB. Then, since AD equals DB, and DG is common, the two sides AD and DG equal the two sides BD and DG respectively. And the base GA equals the base GB, for they are radii, therefore the angle ADG equals the angle GDB. I.Def.15 I.8 But, when a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, therefore the angle GDB is right. I.Def.10 But the angle FDB is also right, therefore the angle FDB equals the angle GDB, the greater equals the less, which is impossible. Therefore G is not the center of the circle ABC. Similarly we can prove that neither is any other point except F. Therefore the point F is the center of the circle ABC. Q.E.F. Corollary From this it is manifest that if in a circle a straight line cuts a straight line into two equal parts and at right angles, then the center of the circle lies on the cutting straight line. Since the definition of a circle, I.Def.15, includes the existence of a center, Euclid is justified in taking a point G as the center. In this proof G is shown to lie on the perpendicular bisector of the line AB. He leaves to the reader to show that G actually is the point F on the perpendicular bisector, but that's clear since only the midpoint F is equidistant from the two points C and E on the circle. From that observation it also follows that the center of a circle is unique, although the uniqueness can easily be proved in other ways. As Todhunter remarked, Euclid implicitly assumes that the perpendicular bisector of AB actually intersects the circle in points C and E. Use of this proposition and its corollary About half the proofs in Book III and several of those in Book IV begin with taking the center of a circle, but in plane geometry, it isn't necessary to invoke this proposition III.1 since the only way that circles can occur is if they are constructed around a center to begin with. Even in solid geometry, the center of a circle is usually known so that III.1 isn't necessary. Indeed, that is the case whenever the center is needed in Euclid's books on solid geometry (see XI.23, XIII.9 through XIII.13, and XIII.16). Sections of spheres cut by planes are also circles as are certain sections of cylinders and cones, but in these cases too, the centers can easily be found without recourse to III.1. Thus, III.1 redundant, although it is an interesting construction. The corollary is used in propositions III.9 and III.10. Next proposition: III.2 Previous: III.Def.11 Book III introduction © 1996, 1997 D.E.Joyce Clark University Proposition 2 If two points are taken at random on the circumference of a circle, then the straight line joining the points falls within the circle. Let ABC be a circle, and let two points A and B be taken at random on its circumference. I say that the straight line joined from A to B falls within the circle. For suppose it does not, but, if possible, let it fall outside, as AEB. Take the center D of the circle ABC. Join DA and DB, and draw DFE through. III.1 Then, since DA equals DB, the angle DAE also equals the angle DBE. I.Def.15 I.5 And, since one side AEB of the triangle DAE is produced, the angle DEB is greater than the angle DAE. I.16 And the angle DAE equals the angle DBE, therefore the angle DEB is greater than the angle DBE. And the side opposite the greater angle is greater, therefore DB is greater than DE. But DB equals DF, therefore DF is greater than DE, the less greater than the greater, which is impossible. I.19 I.Def.15 Therefore the straight line joined from A to B does not fall outside the circle. Similarly we can prove that neither does it fall on the circumference itself, therefore it falls within. Therefore if two points are taken at random on the circumference of a circle, then the straight line joining the points falls within the circle. Q.E.D. The figure for this proposition is rather strange, but that is necessary since it refers to a hypothetical situation which is shown to be impossible. In this figure AEB is supposed to be a straight line that lies on outside the circle. There are other impossible figures in later propositions in this Book. That Euclid even has this proposition is remarkable. Of course, it should be included, but there are equally obvious statements (but difficult to prove) left out in earlier books. For instance, that the two circles constructed in a plane on a line AB intersect is not proved although it is used in I.1. This indicates that more care has been given to the foundations for this book than for the previous books. Euclid leaves to the reader to prove that AB cannot lie on the circumference, and that is not particularly difficult to prove. This propostion is used in the next one. Next proposition: III.3 Previous: III.1 Book III introduction © 1996 D.E.Joyce Clark University Proposition 3 If a straight line passing through the center of a circle bisects a straight line not passing through the center, then it also cuts it at right angles; and if it cuts it at right angles, then it also bisects it. Let a straight line CD passing through the center of a circle ABC bisect a straight line AB not passing through the center at the point F. I say that it also cuts it at right angles. Take the center E of the circle ABC, and join EA and EB. III.1 Then, since AF equals FB, and FE is common, two sides equal two sides, and the base EA equals the base EB, therefore the angle AFE equals the angle BFE. I.Def.15 I.8 But, when a straight line standing on another straight line makes the adjacent angles equal to one another, each of the equal angles is right, therefore each of the angles AFE and BFE is right. I.Def.10 Therefore CD, which passes through the center and bisects AB which does not pass through the center, also cuts it at right angles. Next, let CD cut AB at right angles. I say that it also bisects it, that is, that AF equals FB. For, with the same construction, since EA equals EB, the angle EAF also equals the angle EBF. I.5 But the right angle AFE equals the right angle BFE, therefore EAF and EBF are two triangles having two angles equal to two angles and one side equal to one side, namely EF, which is common to them, and opposite one of the equal angles. Therefore they also have the remaining sides equal to the remaining sides I.26 Therefore AF equals FB. Therefore if a straight line passing through the center of a circle bisects a straight line not passing through the center, then it also cuts it at right angles; and if it cuts it at right angles, then it also bisects it. Q.E.D. Compare this statement to the corollary of proposition III.1. This proposition is used in the next one, a few others in Book III, and XII.16. Next proposition: III.4 Previous: III.2 Book III introduction © 1996 D.E.Joyce Clark University Proposition 4 If in a circle two straight lines which do not pass through the center cut one another, then they do not bisect one another. Let ABCD be a circle, and in it let the two straight lines AC and BD, which do not pass through the center, cut one another at E. I say that they do not bisect one another. For, if so, let them bisect one another, so that AE equals EC, and BE equals ED. Take the center F of the circle ABCD. Join FE. III.1 Then, since a straight line FE passing through the center bisects a straight line AC not passing through the center, it also cuts it at right angles, therefore the angle FEA is right. III.3 Again, since a straight line FE bisects a straight line BD, it also cuts it at right angles. Therefore the angle FEB is right. III.3 But the angle FEA was also proved right, therefore the angle FEA equals the angle FEB, the less equals the greater, which is impossible. Therefore AC and BD do not bisect one another. Therefore if in a circle two straight lines which do not pass through the center cut one another, then they do not bisect one another. Q.E.D. This proposition is not used in the rest of the Elements. The contrapositive of this statement is more positive: if two straight lines in a circle bisect each other, then they meet at the center. This proposition is not used in the rest of the Elements. Next proposition: III.5 Previous: III.3 Book III introduction © 1996 D.E.Joyce Clark University Proposition 5 If two circles cut one another, then they do not have the same center. Let the circles ABC and CDG cut one another at the points B and C. I say that they do not have the same center. For, if possible, let it be E. Join EC, and draw EFG through at random. Then, since the point E is the center of the circle ABC, EC equals EF. Again, since the point E is the center of the circle CDG, EC equals EG. I.Def.15 But EC was proved equal to EF also, therefore EF also equals EG, the less equals the greater which is impossible. Therefore the point E is not the center of the circles ABC and CDG. Therefore if two circles cut one another, then they do not have the same center. Q.E.D. Note that no use was made in the proof of the point B. That means the proof actually shows that if two circles meet, then they do not have the same center, and that covers not only this proposition but the next, too, where the two touch each other. This proposition is used in III.10 which states that circles cannot intersect at more than two points. Next proposition: III.6 Previous: III.4 Book III introduction © 1996 D.E.Joyce Clark University Proposition 6 If two circles touch one another, then they do not have the same center. Let the two circles ABC and CDE touch one another at the point C. I say that they do not have the same center. For, if possible, let it be F. Join FC, and draw FEB through at random. Then, since the point F is the center of the circle ABC, FC equals FB. Again, since the point F is the center of the circle CDE, FC equals FE. I.Def.15 But FC was proved equal to FB, therefore FE also equals FB, the less equals the greater, which is impossible. Therefore F is not the center of the circles ABC and CDE. Therefore if two circles touch one another, then they do not have the same center. Q.E.D. As mentioned before, this proposition is almost the same as the previous. Both could be included in one statement: circles that meet don't have the same center, or the contrapositive: concentric circles don't meet. This propostion is not used in the rest of the Elements. Next proposition: III.7 Previous: III.5 Book III introduction © 1996 D.E.Joyce Clark University Proposition 7 If on the diameter of a circle a point is taken which is not the center of the circle, and from the point straight lines fall upon the circle, then that is greatest on which passes through the center, the remainder of the same diameter is least, and of the rest the nearer to the straight line through the center is always greater than the more remote; and only two equal straight lines fall from the point on the circle, one on each side of the least straight line. Let ABCD be a circle, and let AD be a diameter of it. Let F be a point F on AD which is not the center of the circle. Let E be the center of the circle. Let straight lines FB, FC, and FG fall upon the circle ABCD from F. I say that FA is greatest, FD is least, and of the rest FB is greater than FC, and FC greater than FG. Join BE, CE, and GE. Then, since in any triangle the sum of any two sides is greater than the remaining one, the sum of EB and EF is greater than BF. I.20 But AE equals BE, therefore AF is greater than BF. Again, since BE equals CE, and FE is common, the two sides BE and EF equal the two sides CE and EF. But the angle BEF is also greater than the angle CEF, therefore the base BF is greater than the base CF. I.24 For the same reason CF is also greater than GF. Again, since the sum of GF and FE is greater than EG, and EG equals ED, the sum of GF and FE is greater than ED. I.20 Subtract EF from each. Therefore the remainder GF is greater than the remainder FD. Therefore FA is greatest, FD is least, FB is greater than FC, and FC greater than FG. I say also that from the point F only two equal straight lines fall on the circle ABCD, one on each side of the least FD. Construct the angle FEH equal to the angle GEF on the straight line EF and at the point E on it. Join FH. I.23 Then, since GE equals EH, and EF is common, the two sides GE and EF equal the two sides HE and EF, and the angle GEF equals the angle HEF, therefore the base FG equals the base FH. I.4 I say again that another straight line equal to FG does not fall on the circle from the point F. For, if possible, let FK so fall. Then, since FK equals FG, and FH equals FG, FK also equals FH, the nearer to the straight line through the center being thus equal to the more remote, which is impossible. Above Therefore another straight line equal to GF does not fall from the point F upon the circle. Therefore only one straight line so falls. Therefore if on the diameter of a circle a point is taken which is not the center of the circle, and from the point straight lines fall upon the circle, then that is greatest on which passes through the center, the remainder of the same diameter is least, and of the rest the nearer to the straight line through the center is always greater than the more remote; and only two equal straight lines fall from the point on the circle, one on each side of the least straight line. Q.E.D. The statement of this proposition is daunting. It concerns the distances from a point F inside a circle to the points on the circumference. The point F is assumed not to be the center. If a diameter AD is passed through F, then one of the points A is the point on the circumference furthest from F and the other D is the closest. As a point travels the circumference from A to D it gets closer to F. The final part of the statement is that if G is one point on the circumference, then there is exactly one other point H on the circumference the same distance from F (assuming, of course, that G is neither A nor D). Note There is some ambiguity in the statement of this proposition. It is not clear what the phrase "the nearer to the straight line through the center" means. It may well refer to the angle, so that FB is considered nearer to FA than FC since the angle BFA is less than the angle CFA. If so, there there is a missing detail in the proof, as De Morgan pointed out. It is declared that the angle BEF is greater than the angle CEF, but that hasn't been proved. DeMorgan and others have described various ways to fill this logical gap. This propostion is not used in the rest of the Elements. Next proposition: III.8 Previous: III.6 Book III introduction © 1996 D.E.Joyce Clark University Proposition 8 If a point is taken outside a circle and from the point straight lines are drawn through to the circle, one of which is through the center and the others are drawn at random, then, of the straight lines which fall on the concave circumference, that through the center is greatest, while of the rest the nearer to that through the center is always greater than the more remote, but, of the straight lines falling on the convex circumference, that between the point and the diameter is least, while of the rest the nearer to the least is always less than the more remote; and only two equal straight lines fall on the circle from the point, one on each side of the least. Let ABC be a circle, and let a point D be taken outside ABC. Let straight lines DA, DE, DF, and DC be drawn through from D, and let DA be drawn through the center. I say that, of the straight lines falling on the concave circumference AEFC, the straight line DA through the center is greatest, while DE is greater than DF, and DF greater than DC. But, of the straight lines falling on the convex circumference HLKG, the straight line DG between the point and the diameter AG is least, and the nearer to the least DG is always less than the more remote, namely DK is less than DL, and DL is less than DH. Take the center M of the circle ABC. Join ME, MF, MC, MK, ML, and MH. III.1 Then, since AM equals EM, add MD to each, therefore AD equals the sum of EM and MD. But the sum of EM and MD is greater than ED, therefore AD is also greater than ED. I.20 Again, since ME equals MF, and MD is common, therefore EM and MD equal FM and MD, and the angle EMD is greater than the angle FMD, therefore the base ED is greater than the base FD. I.24 Similarly we can prove that FD is greater than CD. Therefore DA is greatest, while DE is greater than DF, and DF is greater than DC. Next, since the sum of MK and KD is greater than MD, and MG equals MK, therefore the remainder KD is greater than the remainder GD, so that GD is less than KD. I.20 And, since on MD, one of the sides of the triangle MLD, two straight lines MK and KD are constructed meeting within the triangle, therefore the sum of MK and KD is less than the sum of ML and LD. And MK equals ML, therefore the remainder DK is less than the remainder DL. I.21 Similarly we can prove that DL is also less than DH. Therefore DG is least, while DK is less than DL, and DL is less than DH. I say also that only two equal straight lines will fall from the point D on the circle, one on each side of the least DG. Construct the angle DMB equal to the angle KMD on the straight line MD and at the point M on it. Join DB. I.23 Then, since MK equals MB, and MD is common, the two sides KM and MD equal the two sides BM and MD respectively, and the angle KMD equals the angle BMD, therefore the base DK equals the base DB. I.4 I say that no other straight line equal to the straight line DK falls on the circle from the point D. For, if possible, let a straight line so fall, and let it be DN. Then, since DK equals DN, and DK equals DB, DB also equals DN, that is, the nearer to the least DG equal to the more remote, which was proved impossible. Above Therefore no more than two equal straight lines fall on the circle ABC from the point D, one on each side of DG the least. Therefore if a point is taken outside a circle and from the point straight lines are drawn through to the circle, one of which is through the center and the others are drawn at random, then, of the straight lines which fall on the concave circumference, that through the center is greatest, while of the rest the nearer to that through the center is always greater than the more remote, but, of the straight lines falling on the convex circumference, that between the point and the diameter is least, while of the rest the nearer to the least is always less than the more remote; and only two equal straight lines fall on the circle from the point, one on each side of the least. Q.E.D. This proposition has a statement even more complicated than the previous proposition. This one deals with the distances from a point D outside a circle to the points on the circumference. If the diameter AG extended passes through D, then one of its endpoints G is the point on the circumference closest to D and the other A is furthest. As a point moves along the circumference from A to D it gets closer to D. Euclid considers two parts of the circumference, the convex part is the near part exposed to the point D, while the concave part is the part on the far side of the circle. The final part of the statement is that if K is one point on the circumference, then there is exactly one other point B on the circumference the same distance from D (assuming, of course, that K is neither G nor A). Note There is a logical gap in the proof of this proposition similar to that in the previous proposition. Again, various ways have been proposed to fill it. This propostion is not used in the rest of the Elements. Next proposition: III.9 Previous: III.7 Book III introduction © 1996 D.E.Joyce Clark University Proposition 9 If a point is taken within a circle, and more than two equal straight lines fall from the point on the circle, then the point taken is the center of the circle. Let D a point within a circle ABC, and from D let more than two equal straight lines, namely DA and DB and DC, fall on the circle ABC. I say that the point D is the center of the circle ABC. Join AB and BC, and bisect them at the points E and F. Join ED and FD, and draw them through to the points G, K, H, and L. I.10 Then, since AE equals EB, and ED is common, the two sides AE and ED equal the two sides BE and ED, and the base DA equals the base DB, therefore the angle AED equals the angle BED. I.8 Therefore the angles AED and BED are each right. Therefore GK cuts AB into two equal parts and at right angles. And since, if in a circle a straight line cuts a straight line into two equal parts and at right angles, the center of the circle is on the cutting straight line, therefore the center of the circle is on GK. III.1,Cor For the same reason the center of the circle ABC is also on HL. And the straight lines GK and HL have no other point common but the point D, therefore the point D is the center of the circle ABC. Therefore if a point is taken within a circle, and more than two equal straight lines fall from the point on the circle, then the point taken is the center of the circle. Q.E.D. The statement of this proposition is already covered by the last part of Proposition III.7 which says for a point in a circle that is not the center at most two points lie on the circumference at any distance from that point. This proposition is used in III.25. Next proposition: III.10 Previous: III.8 Book III introduction © 1996 D.E.Joyce Clark University Proposition 10 A circle does not cut a circle at more than two points. For, if possible, let the circle ABC cut the circle DEF at more points than two, namely B, G, F, and H. Join BH and BG, and bisect them at the points K and L. Draw KC and LM from K and L at right angles to BH and BG, and carry them through to the points A and E. I.10 I.11 Then, since in the circle ABC a straight line AC cuts a straight line BH into two equal parts and at right angles, the center of the circle ABC lies on AC. Again, since in the same circle ABC a straight line NO cuts a straight line BG into two equal parts and at right angles, the center of the circle ABC lies on NO. III.1,Cor But it was also proved to lie on AC, and the straight lines AC and NO meet at no point except at P, therefore the point P is the center of the circle ABC. Similarly we can prove that P is also the center of the circle DEF, therefore the two circles ABC and DEF which cut one another have the same center P, which is impossible. III.5 Therefore a circle does not cut a circle at more than two points. Q.E.D. The figure is another impossible figure. Both curves are supposed to be circumferences of circles, but, of course, they cannot both be drawn as circles since the situation is proved not to occur. Although Euclid names four points where the circles meet, only three, B, G, and H, are used in the proof. The proof actually shows that the two circles cannot meet in more than two points, where "meet" could be either cut or touch. Heath remarks that the lines bisecting BG and BH have not been shown to meet. In fact, they have, since the center of the circle ABC has been shown to be on both. This proposition is used in III.24. Next proposition: III.11 Previous: III.9 Book III introduction © 1996 D.E.Joyce Clark University Proposition 11 If two circles touch one another internally, and their centers are taken, then the straight line joining their centers, being produced, falls on the point of contact of the circles. Let the two circles ABC and ADE touch one another internally at the point A, and let the centers F and G of the circles ABC and ADE be taken. III.1 I say that the straight line joined from G to F and produced falls on A. For suppose it does not, but, if possible, let it fall as FGH. Join AF and AG. Then, since the sum of AG and GF is greater than FA, that is, than FH, subtract FG from each, therefore the remainder AG is greater than the remainder GH. I.20 But AG equals GD, therefore GD is also greater than GH, the less greater than the greater, which is impossible. Therefore the straight line joined from F to G does not fall outside. Therefore it falls on A, the point of contact. Therefore if two circles touch one another internally, and their centers are taken, then the straight line joining their centers, being produced, falls on the point of contact of the circles. Q.E.D. In order to carry through the proof, in particular so that FA = FH, the circle ABC needs to be the larger circle. Various conclusions in the proof are based on the figure rather than rigorous deductive reasoning. Camerer and others have suggested ways of filling the gaps. This proposition is used in III.13. Next proposition: III.12 Previous: III.10 Book III introduction © 1996 D.E.Joyce Clark University Proposition 12 If two circles touch one another externally, then the straight line joining their centers passes through the point of contact. Let the two circles ABC and ADE touch one another externally at the point A. Take the center F of ABC, and the center G of ADE. III.1 I say that the straight line joined from F to G passes through the point of contact at A. For suppose it does not, but, if possible, let it pass as FCDG. Join AF and AG. Then, since the point F is the center of the circle ABC, FA equals FC. Again, since the point G is the center of the circle ADE, GA equals GD. But FA was also proved equal to FC, therefore FA and AG equal FC and GD, so that the whole FG is greater than FA and AG, but it is also less, which is impossible. I.20 Therefore the straight line joined from F to G does not fail to pass through the point of contact at A, therefore it passes through it. Therefore if two circles touch one another externally, then the straight line joining their centers passes through the point of contact. Q.E.D. This proposition was certainly added to the Elements after Euclid, perhaps by Heron or a later commentator. This proposition is not used in the rest of the Elements. Next proposition: III.13 Previous: III.11 Book III introduction © 1996 D.E.Joyce Clark University Proposition 13 A circle does not touch another circle at more than one point whether it touches it internally or externally. For, if possible, let the circle ABDC touch the circle EBFD, first internally, at more points than one, namely D and B. Take the center G of the circle ABDC and the center H of EBFD. III.1 Therefore the straight line joined from G to H falls on B and D. III.11 Let it so fall, as BGHD. Then, since the point G is the center of the circle ABCD and BG equals GD, therefore BG is greater than HD. Therefore BH is much greater than HD. Again, since the point H is the center of the circle EBFD, BH equals HD, but it was also proved much greater than it, which is impossible. Therefore a circle does not touch a circle internally at more points than one. I say further that neither does it so touch it externally. For, if possible, let the circle ACK touch the circle ABDC at more points than one, namely A and C. Join AC. Then, since on the circumference of each of the circles ABDC and ACK two points A and C have been taken at random, the straight line joining the points falls within each circle, but it fell within the circle ABCD and outside ACK, which is absurd. III.2 III.Def.3 Therefore a circle does not touch a circle externally at more points than one. And it was proved that neither does it so touch it internally. Therefore a circle does not touch another circle at more than one point whether it touches it internally or externally. Q.E.D. In the second impossible figure there are three curves connecting A to C. The two circles are not supposed to cut each other, but just to touch each other at the two points A and C, and the straight line AC should lie between the two circles and not within either one. There are logical flaws in this proof similar to those in the last two proofs. This proposition is not used in the rest of the Elements. Next proposition: III.14 Previous: III.12 Book III introduction © 1996 D.E.Joyce Clark University Proposition 14 Equal straight lines in a circle are equally distant from the center, and those which are equally distant from the center equal one another. Let AB and CD be equal straight lines in a circle ABDC. I say that AB and CD are equally distant from the center. Take the center E of the circle ABDC. Draw EF and EG from E perpendicular to AB and CD, and join AE and EC. III.1 I.12 Then, since a straight line EF passing through the center cuts a straight line AB not passing through the center at right angles, it also bisects it. Therefore AF equals FB. Therefore AB is double AF. III.3 For the same reason CD is also double CG. But AB equals CD, therefore AF also equals CG. Also, since AE equals EC, the square on AE also equals the square on EC. But the sum of the squares on AF and EF equals the square on AE, for the angle at F is right, and the sum of the squares on EG and GC equals the square on EC, for the angle at G is right. Therefore the sum of the squares on AF and FE equals the sum of the squares on CG and GE, of which the square on AF equals the square on CG, for AF equals CG. Therefore the remaining square on FE equals the square on EG. Therefore EF equals EG. I.47 But straight lines in a circle are said to be equally distant from the center when the perpendiculars drawn to them from the center are equal. Therefore AB and CD are equally distant from the center. III.Def.4 Next, let the straight lines AB and CD be equally distant from the center, that is, let EF equal EG. I say that AB also equals CD. For, with the same construction, we can prove, as before, that AB is double AF, and CD double CG. And, since AE equals CE, the square on AE equals the square on CE. But the sum of the squares on EF and FA equals the square on AE, and the sum of the squares on EG and GC equals the square on CE. I.47 Therefore the sum of the squares on EF and FA equals the sum of the squares on EG and GC, of which the square on EF equals the square on EG, for EF equals EG. Therefore the remaining square on AF equals the square on CG. Therefore AF equals CG. And AB is double AF, and CD double CG, therefore AB equals CD. Therefore equal straight lines in a circle are equally distant from the center, and those which are equally distant from the center equal one another. Q.E.D. Note how Euclid has proved twice in the course of this proof the side-side-right angle congruence theorem. See the note after I.26 about congruence theorems for triangles. This proposition is used in the next one. Next proposition: III.15 Previous: III.13 Book III introduction © 1996 D.E.Joyce Clark University Proposition 15 Of straight lines in a circle the diameter is greatest, and of the rest the nearer to the center is always greater than the more remote. Let ABCD be a circle, AD its diameter, and E its center. Let BC be nearer to the center AD, and FG more remote. I say that AD is greatest and BC greater than FG. Draw EH and EK from the center E perpendicular to BC and FG. I.12 Then, since BC is nearer to the center and FG more remote, EK is greater than EH. III.Def.5 Make EL equal to EH. Draw LM through L at right angles to EK, and carry it through to N. Join ME, EN, FE, and EG. I.3 I.11 Then, since EH equals EL, BC also equals MN. III.14 Again, since AE equals EM, and ED equals EN, AD equals the sum of ME and EN. But the sum of ME and EN is greater than MN, and MN equals BC, therefore AD is greater than BC. I.20 And, since the two sides ME and EN equal the two sides FE and EG, and the angle MEN greater than the angle FEG, therefore the base MN is greater than the base FG. I.24 But MN was proved equal to BC. Therefore the diameter AD is greatest and BC greater than FG. Therefore of straight lines in a circle the diameter is greatest, and of the rest the nearer to the center is always greater than the more remote. Q.E.D. This proposition is not used in the rest of the Elements. Next proposition: III.16 Previous: III.14 Book III introduction © 1996 D.E.Joyce Clark University Proposition 16 The straight line drawn at right angles to the diameter of a circle from its end will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed, further the angle of the semicircle is greater, and the remaining angle less, than any acute rectilinear angle. Let ABC be a circle about D as center and AB as diameter. I say that the straight line drawn from A at right angles to AB from its end will fall outside the circle. For suppose it does not, but, if possible, let it fall within as CA, and join DC. Since DA equals DC, the angle DAC also equals the angle ACD. I.5 But the angle DAC is right, therefore the angle ACD is also right. Thus, in the triangle ACD, the two angles DAC and ACD equal two right angles, which is impossible. I.17 Therefore the straight line drawn from the point A at right angles to BA will not fall within the circle. Similarly we can prove that neither will it fall on the circumference, therefore it will fall outside. Let it fall as AE. I say next that into the space between the straight line AE and the circumference CHA another straight line cannot be interposed. For, if possible, let another straight line be so interposed, as FA. Draw DG from the point D perpendicular to FA. I.12 Then, since the angle AGD is right, and the angle DAG is less than a right angle, AD is greater than DG. I.17 I.19 But DA equals DH, therefore DH is greater than DG, the less greater than the greater, which is impossible. Therefore another straight line cannot be interposed into the space between the straight line and the circumference. I say further that the angle of the semicircle contained by the straight line BA and the circumference CHA is greater than any acute rectilinear angle, and the remaining angle contained by the circumference CHA and the straight line AE is less than any acute rectilinear angle. For, if there is any rectilinear angle greater than the angle contained by the straight line BA and the circumference CHA, and any rectilinear angle less than the angle contained by the circumference CHA and the straight line AE, then into the space between the circumference and the straight line AE a straight line will be interposed such as will make an angle contained by straight lines which is greater than the angle contained by the straight line BA and the circumference CHA, and another angle contained by straight lines which is less than the angle contained by the circumference CHA and the straight line AE. But such a straight line cannot be interposed, therefore there will not be any acute angle contained by straight lines which is greater than the angle contained by the straight line BA and the circumference CHA, nor yet any acute angle contained by straight lines which is less than the angle contained by the circumference CHA and the straight line AE. Above Therefore the straight line drawn at right angles to the diameter of a circle from its end will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed, further the angle of the semicircle is greater, and the remaining angle less, than any acute rectilinear angle. Q.E.D. Corollary From this it is manifest that the straight line drawn at right angles to the diameter of a circle from its end touches the circle. This proposition is used in the proof of proposition IV.4 and two others in Book IV. The corollary is used in III.33, III.37, a few propositions in Book IV, and XII.16. Next proposition: III.17 Previous: III.15 Book III introduction © 1996 D.E.Joyce Clark University Proposition 17 From a given point to draw a straight line touching a given circle. Let A be the given point, and BCD the given circle. It is required to draw from the point A a straight line touching the circle BCD. Take the center E of the circle, and join AE. Describe the circle AFG with center E and radius EA. Draw DF from D at right angles to EA. Join EF and AB. III.1 I.11 I say that AB has been drawn from the point A touching the circle BCD. For, since E is the center of the circles BCD and AFG, EA equals EF, and ED equals EB. Therefore the two sides AE and EB equal the two sides FE and ED, and they contain a common angle, the angle at E, therefore the base DF equals the base AB, and the triangle DEF equals the triangle BEA, and the remaining angles to the remaining angles, therefore the angle EDF equals the angle EBA. I.4 But the angle EDF is right, therefore the angle EBA is also right. Now EB is a radius, and the straight line drawn at right angles to the diameter of a circle, from its end, touches the circle, therefore AB touches the circle BCD. III.16,Cor Therefore from the given point A the straight line AB has been drawn touching the circle BCD. Q.E.F. The construction in this proposition is used in propositions III.34 and XII.2. Next proposition: III.18 Previous: III.16 Book III introduction © 1996 D.E.Joyce Clark University Proposition 18 If a straight line touches a circle, and a straight line is joined from the center to the point of contact, the straight line so joined will be perpendicular to the tangent. For let a straight line DE touch the circle ABC at the point C. Take the center F of the circle ABC, and join FC from F to C. III.1 I say that FC is perpendicular to DE. For, if not, draw FG from F perpendicular to DE. I.12 Then, since the angle FGC is right, the angle FCG is acute, and the side opposite the greater angle is greater, therefore FC is greater than FG. I.17 I.19 But FC equals FB, therefore FB is also greater than FG, the less greater than the greater, which is impossible. Therefore FG is not perpendicular to DE. Similarly we can prove that neither is any other straight line except FC. Therefore FC is perpendicular to DE. Therefore if a straight line touches a circle, and a straight line is joined from the center to the point of contact, the straight line so joined will be perpendicular to the tangent. Q.E.D. This proposition is used in a few propositions in Books III and IV beginning with III.36. Next proposition: III.19 Previous: III.17 Book III introduction © 1996 D.E.Joyce Clark University Proposition 19 If a straight line touches a circle, and from the point of contact a straight line is drawn at right angles to the tangent, the center of the circle will be on the straight line so drawn. For let a straight line DE touch the circle ABC at the point C. Draw CA from C at right angles to DE. I.11 I say that the center of the circle is on AC. For suppose it is not, but, if possible, let F be the center, and join CF. Since a straight line DE touches the circle ABC, and FC has been joined from the center to the point of contact, FC is perpendicular to DE. Therefore the angle FCE is right. III.18 But the angle ACE is also right, therefore the angle FCE equals the angle ACE, the less equals the greater, which is impossible. Therefore F is not the center of the circle ABC. Similarly we can prove that neither is any other point except a point on AC. Therefore if a straight line touches a circle, and from the point of contact a straight line is drawn at right angles to the tangent, the center of the circle will be on the straight line so drawn. Q.E.D. This proposition is used in proposition III.32. Next proposition: III.20 Previous: III.18 Book III introduction © 1996 D.E.Joyce Clark University Proposition 20 In a circle the angle at the center is double the angle at the circumference when the angles have the same circumference as base. Let ABC be a circle, let the angle BEC be an angle at its center, and the angle BAC an angle at the circumference, and let them have the same circumference BC as base. I say that the angle BEC is double the angle BAC. Join AE, and draw it through to F. Then, since EA equals EB, the angle EAB also equals the angle EBA. Therefore the sum of the angles the angles EAB and EBA is double the angle EAB. I.5 But the angle BEF equals the sum of the angles EAB and EBA, therefore the angle BEF, is also double the angle EAB. I.32 For the same reason the angle FEC is also double the angle EAC. Therefore the whole angle BEC is double the whole angle BAC. Again let another straight line be inflected, and let there be another angle BDC. Join DE and produced it to G. Similarly then we can prove that the angle GEC is double the angle EDC, of which the angle GEB is double the angle EDB. Therefore the remaining angle BEC is double the angle BDC. Therefore in a circle the angle at the center is double the angle at the circumference when the angles have the same circumference as base. Q.E.D. This proposition is used in the next one, III.27, and VI.33. Next proposition: III.21 Previous: III.19 Book III introduction © 1996 D.E.Joyce Clark University Proposition 21 In a circle the angles in the same segment equal one another. Let ABCD be a circle, and let the angles BAD and BED be angles in the same segment BAED. I say that the angles BAD and BED equal one another. Take the center F of the circle ABCD, and join BF and FD. III.1 Now, since the angle BFD is at the center, and the angle BAD at the circumference, and they have the same circumference BCD as base, therefore the angle BFD is double the angle BAD. III.20 For the same reason the angle BFD is also double the angle BED. Therefore the angle BAD equals the angle BED. Therefore in a circle the angles in the same segment equal one another. Q.E.D. This proposition is used in the next one. Next proposition: III.22 Previous: III.20 Book III introduction © 1996 D.E.Joyce Clark University Proposition 22 The sum of the opposite angles of quadrilaterals in circles equals two right angles. Let ABCD be a circle, and let ABCD be a quadrilateral in it. I say that the sum of the opposite angles equals two right angles. Join AC and BD. Then, since in any triangle the sum of the three angles equals two right angles, the sum of the three angles CAB, ABC, and BCA of the triangle ABC equals two right angles. I.32 But the angle CAB equals the angle BDC, for they are in the same segment BADC, and the angle ACB equals the angle ADB, for they are in the same segment ADCB, therefore the whole angle ADC equals the sum of the angles BAC and ACB. III.21 Add the angle ABC to each. Therefore the sum of the angles ABC, BAC, and ACB equals the sum of the angles ABC and ADC. But the sum of the angles ABC, BAC, and ACB equals two right angles, therefore the sum of the angles ABC and ADC also equal two right angles. Similarly we can prove that the sum of the angles BAD and DCB also equals two right angles. Therefore the sum of the opposite angles of quadrilaterals in circles equals two right angles. Q.E.D. This proposition is used in III.32. Next proposition: III.23 Previous: III.21 Book III introduction © 1996 D.E.Joyce Clark University Proposition 23 On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side. For, if possible, on the same straight line AB let two similar and unequal segments of circles ACB and ADB be constructed on the same side. Draw ACD through, and join CB and DB. Then, since the segment ACB is similar to the segment ADB, and similar segments of circles are those which admit equal angles, the angle ACB equals the angle ADB, the exterior to the interior, which is impossible. III.Def.11 I.16 Therefore on the same straight line there cannot be constructed two similar and unequal segments of circles on the same side. Q.E.D. This proposition is used in the next one. Next proposition: III.24 Previous: III.22 Book III introduction © 1996 D.E.Joyce Clark University Proposition 24 Similar segments of circles on equal straight lines equal one another. Let AEB and CFD be similar segments of circles on equal straight lines AB and CD. I say that the segment AEB equals the segment CFD. For, if the segment AEB is superposed on CFD, and if the point A is placed on C and the straight line AB on CD, then the point B coincides with the point D, because AB equals CD, and, AB coinciding with CD, the segment AEB also coincides with CFD. For, if the straight line AB coincides with CD but the segment AEB does not coincide with CFD, then it either falls within it, or outside it, or it falls awry, as CGD, and a circle cuts a circle at more points than two, which is impossible. III.23 III.10 Therefore, if the straight line AB is superposed on CD, then the segment AEB does not fail to coincide with CFD also, therefore it coincides with it and equals it. C.N.4 Therefore similar segments of circles on equal straight lines equal one another. Q.E.D. The proof here uses the method of superposition which was also used for I.4 and I.8. Next proposition: III.25 Previous: III.23 Book III introduction © 1996 D.E.Joyce Clark University Proposition 25 Given a segment of a circle, to describe the complete circle of which it is a segment. Let ABC be the given segment of a circle. It is required to describe the complete circle belonging to the segment ABC, that is, of which it is a segment. Bisect AC at D, draw DB from the point D at right angles to AC, and join AB. I.10 I.11 The angle ABD is then greater than, equal to, or less than the angle BAD. First let it be greater. Construct the angle BAE on the straight line BA, and at the point A on it, equal to the angle ABD. Draw DB through to E, and join EC. I.23 Then, since the angle ABE equals the angle BAE, the straight line EB also equals EA. I.6 And, since AD equals DC, and DE is common, the two sides AD and DE equal the two sides CD and DE respectively, and the angle ADE equals the angle CDE, for each is right, therefore the base AE equals the base CE. I.4 But AE was proved equal to BE, therefore be also equals CE. Therefore the three straight lines AE, EB, and EC equal one another. Therefore the circle drawn with center E and radius one of the straight lines AE, EB, or EC also passes through the remaining points and has been completed. III.9 Therefore, given a segment of a circle, the complete circle has been described. And it is manifest that the segment ABC is less than a semicircle, because the center E happens to be outside it. Similarly, even if the angle ABD equals the angle BAD and AD being equal to each of the two BD and DC, the three straight lines DA, DB, and DC will equal one another, D will be the center of the completed circle, and ABC will clearly be a semicircle. But, if the angle ABD is less than the angle BAD, and if we construct, on the straight line BA and at the point A on it, an angle equal to the angle ABD, the center will fall on DB within the segment ABC, and the segment ABC will clearly be greater than a semicircle. I.23 Therefore, given a segment of a circle, the complete circle has been described. Q.E.F. The construction in this proposition is not used in the rest of the Elements. Next proposition: III.26 Previous: III.24 Book III introduction © 1996 D.E.Joyce Clark University Proposition 26 In equal circles equal angles stand on equal circumferences whether they stand at the centers or at the circumferences. Let ABC and DEF be equal circles, and in them let there be equal angles, namely at the centers the angles BGC and EHF, and at the circumferences the angles BAC and EDF. I say that the circumference BKC equals the circumference ELF. Join BC and EF. Now, since the circles ABC and DEF are equal, the radii are equal. Thus the two straight lines BG and GC equal the two straight lines EH and HF, and the angle at G equals the angle at H, therefore the base BC equals the base EF. I.4 And, since the angle at A equals the angle at D, the segment BAC is similar to the segment EDF, and they are upon equal straight lines. III.Def.11 But similar segments of circles on equal straight lines equal one another, therefore the segment BAC equals EDF. But the whole circle ABC also equals the whole circle DEF, therefore the remaining circumference BKC equals the circumference ELF. III.24 Therefore in equal circles equal angles stand on equal circumferences whether they stand at the centers or at the circumferences. Q.E.D. This proposition is used in III.28, IV.11, IV,15, and XIII.10. Next proposition: III.27 Previous: III.25 Book III introduction © 1996 D.E.Joyce Clark University Proposition 27 In equal circles angles standing on equal circumferences equal one another whether they stand at the centers or at the circumferences. For in equal circles ABC and DEF, on equal circumferences BC and EF, let the angles BGC and EHF stand at the centers G and H, and the angles BAC and EDF at the circumferences. I say that the angle BGC equals the angle EHF, and the angle BAC equals the angle EDF. For, if the angle BGC does not equal the angle EHF, one of them is greater. Let the angle BGC be greater. Construct the angle BGK equal to the angle EHF on the straight line BG and at the point G on it. I.23 Now equal angles stand on equal circumferences when they are at the centers, therefore the circumference BK equals the circumference EF. I.26 But EF equals BC, therefore BK also equals BC, the less equals the greater, which is impossible. Therefore the angle BGC is not unequal to the angle EHF, therefore it equals it. And the angle at A is half of the angle BGC, and the angle at D half of the angle EHF, therefore the angle at A also equals the angle at D. III.20 Therefore in equal circles angles standing on equal circumferences equal one another whether they stand at the centers or at the circumferences. Q.E.D. This proposition is used in a few propositions in Books III, IV, VI, and XII starting with III.29. Next proposition: III.28 Previous: III.26 Book III introduction Proposition 28 In equal circles equal straight lines cut off equal circumferences, the greater circumference equals the greater and the less equals the less. Let ABC and DEF be equal circles, and in the circles let AB and DE be equal straight lines cutting off ACB and DFE as greater circumferences and AGB and DHE as lesser. I say that the greater circumference ACB equals the greater circumference DFE, and the less circumference AGB equals DHE. Take the centers K and L of the circles, and join AK, KB, DL, and LE. III.1 Now, since the circles are equal, the radii are also equal, therefore the two sides AK and KB equal the two sides DL and LE, and the base AB equals the base DE, therefore the angle AKB equals the angle DLE. I.8 But equal angles stand on equal circumferences when they are at the centers, therefore the circumference AGB equals DHE. III.26 And the whole circle ABC also equals the whole circle DEF, therefore the remaining circumference ACB also equals the remaining circumference DFE. Therefore in equal circles equal straight lines cut off equal circumferences, the greater circumference equals the greater and the less equals the less. Q.E.D. This proposition is used in III.30 and XIII.18. Next proposition: III.29 Previous: III.27 Book III introduction © 1996 D.E.Joyce Clark University Proposition 29 In equal circles straight lines that cut off equal circumferences are equal. Let ABC and DEF be equal circles, and in them let equal circumferences BGC and EHF be cut off. Join the straight lines BC and EF. I say that BC equals EF. Take the centers K and L of the circles. Join BK, KC, EL, and LF. III.1 Now, since the circumference BGC equals the circumference EHF, the angle BKC also equals the angle ELF. III.27 And, since the circles ABC and DEF are equal, the radii are also equal, therefore the two sides BK and KC equal the two sides EL and LF, and they contain equal angles, therefore the base BC equals the base EF. I.4 Therefore in equal circles straight lines that cut off equal circumferences are equal. Q.E.D. This proposition is used in IV.11 and IV.15. Next proposition: III.30 Previous: III.28 Book III introduction © 1996 D.E.Joyce Clark University Proposition 30 To bisect a given circumference. Let ADB be the given circumference. It is required to bisect the circumference ADB. Join AB, and bisect it at C. Draw CD from the point C at right angles to the straight line AB. Join AD and DB. I.10 I.11 Then, since AC equals CB, and CD is common, the two sides AC and CD equal the two sides BC and CD, and the angle ACD equals the angle BCD, for each is right, therefore the base AD equals the base DB. I.4 But equal straight lines cut off equal circumferences, the greater equal to the greater, and the less to the less, and each of the circumferences AD and DB is less than a semicircle, therefore the circumference AD equals the circumference DB. III.28 Therefore the given circumference has been bisected at the point D. Q.E.F. The construction in this proposition is used in IV.16. Next proposition: III.31 Previous: III.29 Book III introduction © 1996 D.E.Joyce Clark University Proposition 31 In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle; further the angle of the greater segment is greater than a right angle, and the angle of the less segment is less than a right angle. Let ABCD be a circle, let BC be its diameter, and E its center. Join BA, AC, AD, and DC. I say that the angle BAC in the semicircle BAC is right, the angle ABC in the segment ABC greater than the semicircle is less than a right angle, and the angle ADC in the segment ADC less than the semicircle is greater than a right angle. Join AE, and carry BA through to F. Then, since BE equals EA, the angle ABE also equals the angle BAE. Again, since CE equals EA, the angle ACE also equals the angle CAE. Therefore the whole angle BAC equals the sum of the two angles ABC and ACB. I.5 But the angle FAC exterior to the triangle ABC also equals the sum of the two angles ABC and ACB. Therefore the angle BAC also equals the angle FAC. Therefore each is right. Therefore the angle BAC in the semicircle BAC is right. I.32 Next, since in the triangle ABC the sum of the two angles ABC and BAC is less than two right angles, and the angle BAC is a right angle, the angle ABC is less than a right angle. And it is the angle in the segment ABC greater than the semicircle. I.17 Next, since ABCD is a quadrilateral in a circle, and the sum of the opposite angles of quadrilaterals in circles equals two right angles, while the angle ABC is less than a right angle, therefore the remaining angle ADC is greater than a right angle. And it is the angle in the segment ADC less than the semicircle. III.22 I say further that the angle of the greater segment, namely that contained by the circumference ABC and the straight line AC, is greater than a right angle, and the angle of the less segment, namely that contained by the circumference ADC and the straight line AC, is less than a right angle. This is at once manifest. For, since the angle contained by the straight lines BA and AC is right, the angle contained by the circumference ABC and the straight line AC is greater than a right angle. Again, since the angle contained by the straight lines AC and AF is right, the angle contained by the straight line CA and the circumference ADC is less than a right angle. Therefore in a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle; further the angle of the greater segment is greater than a right angle, and the angle of the less segment is less than a right angle. Q.E.D. This proposition is used in III.32 and in each of the rest of the geometry books, namely, Books IV, VI, XI, XII, XIII. It is also used in Book X. Next proposition: III.32 Previous: III.30 Book III introduction © 1996 D.E.Joyce Clark University Proposition 32 If a straight line touches a circle, and from the point of contact there is drawn across, in the circle, a straight line cutting the circle, then the angles which it makes with the tangent equal the angles in the alternate segments of the circle. For let a straight line EF touch the circle ABCD at the point B, and from the point B let there be drawn across, in the circle ABCD, a straight line BD cutting it. I say that the angles which BD makes with the tangent EF equal the angles in the alternate segments of the circle, that is, that the angle FBD equals the angle constructed in the segment BAD, and the angle EBD equals the angle constructed in the segment DCB. Draw BA from B at right angles to EF, take a point C at random on the circumference BD, and join AD, DC, and CB. I.11 Then, since a straight line EF touches the circle ABCD at B, and BA has been drawn from the point of contact at right angles to the tangent, the center of the circle ABCD is on BA. III.19 Therefore BA is a diameter of the circle ABCD. Therefore the angle ADB, being an angle in a semicircle, is right. III.31 Therefore the sum of the remaining angles BAD and ABD equals one right angle. I.32 But the angle ABF is also right, therefore the angle ABF equals the sum of the angles BAD and ABD. Subtract the angle ABD from each. Therefore the remaining angle DBF equals the angle BAD in the alternate segment of the circle. Next, since ABCD is a quadrilateral in a circle, the sum of its opposite angles equals two right angles. III.22 But the sum of the angles DBF and DBE also equals two right angles, therefore the sum of the angles DBF and DBE equals the sum of the angles BAD and BCD, of which the angle BAD was proved equal to the angle DBF, therefore the remaining angle DBE equals the angle DCB in the alternate segment DCB of the circle. Therefore if a straight line touches a circle, and from the point of contact there is drawn across, in the circle, a straight line cutting the circle, then the angles which it makes with the tangent equal the angles in the alternate segments of the circle. Q.E.D. This proposition is used in the next two propositions and a couple of the propositions in Book IV. Next proposition: III.33 Previous: III.31 Book III introduction © 1996 D.E.Joyce Clark University Proposition 33 On a given straight line to describe a segment of a circle admitting an angle equal to a given rectilinear angle. Let AB be the given straight line, and the angle at C the given rectilinear angle. It is required to describe on the given straight line AB a segment of a circle admitting an angle equal to the angle at C. The angle at C is then acute, or right, or obtuse. First let it be acute as in the first figure. Construct the angle BAD equal to the angle at C on the straight line AB and at the point A. Therefore the angle BAD is also acute. I.23 Draw AE at right angles to DA. Bisect AB at F. Draw FG from the point F at right angles to AB, and join GB. I.10 I.12 Then, since AF equals FB, and FG is common, the two sides AF and FG equal the two sides BF and FG, and the angle AFG equals the angle BFG, therefore the base AG equals the base BG. I.4 Therefore the circle described with center G and radius GA passes through B also. Draw it as ABE, and join EB. Now, since AD is drawn from A, the end of the diameter AE, at right angles to AE, therefore AD touches the circle ABE. III.16,Cor. Since then a straight line AD touches the circle ABE, and from the point of contact at A a straight line AB has been drawn across in the circle ABE, the angle DAB equals the angle AEB in the alternate segment of the circle. III.32 But the angle DAB equals the angle at C, therefore the angle at C also equals the angle AEB. Therefore on the given straight line AB the segment AEB of a circle has been described admitting the angle AEB equal to the given angle, the angle at C. Next let the angle at C be right, and let it be again required to describe on AB a segment of a circle admitting an angle equal to the right angle at C. Let the angle BAD be constructed equal to the right angle at C, as is the case in the second figure. Bisect AB at F. Describe the circle AEB with center F and radius either FA or FB. I.23 I.10 Therefore the straight line AD touches the circle ABE, because the angle at A is right. III.16 Cor. And the angle BAD equals the angle in the segment AEB, for the latter too is itself a right angle, being an angle in a semicircle. III.31 But the angle BAD also equals the angle at C, therefore the angle AEB also equals the angle at C. Therefore again the segment AEB of a circle has been described on AB admitting an angle equal to the angle at C. Next, let the angle at C be obtuse. Construct the angle BAD equal to C on the straight line AB and at the point A as is the case in the third figure. Draw AE at right angles to AD. Bisect AB again at F. Draw FG at right angles to AB, and join GB. I.23 I.11 I.12 Then, since AF again equals FB, and FG is common, the two sides AF and FG equal the two sides BF and FG, and the angle AFG equals the angle BFG, therefore the base AG equals the base BG. I.4 Therefore the circle described with center G and radius GA also passes through B. Let it so pass, as AEB. Now, since AD is drawn at right angles to the diameter AE from its end, AD touches the circle AEB. III.16 Cor. And AB has been drawn across from the point of contact at A, therefore the angle BAD equals the angle constructed in the alternate segment AHB of the circle. III.32 But the angle BAD equals the angle at C. Therefore the angle in the segment AHB also equals the angle at C. Therefore on the given straight line AB the segment AHB of a circle has been described admitting an angle equal to the angle at C. Q.E.F. This proposition is not used in the rest of the Elements. Next proposition: III.34 Previous: III.32 Book III introduction © 1996 D.E.Joyce Clark University Proposition 34 From a given circle to cut off a segment admitting an angle equal to a given rectilinear angle. Let ABC be the given circle, and the angle at D the given rectilinear angle. It is required to cut off from the circle ABC a segment admitting an angle equal to the given rectilinear angle, the angle at D. Draw EF touching ABC at the point B. Construct the angle FBC equal to the angle at D on the straight line FB and at the point B on it. III.17 I.23 Then, since a straight line EF touches the circle ABC, and BC has been drawn across from the point of contact at B, the angle FBC equals the angle constructed in the alternate segment BAC. III.32 But the angle FBC equals the angle at D, therefore the angle in the segment BAC equals the angle at D. Therefore from the given circle ABC the segment BAC has been cut off admitting an angle equal to the given rectilinear angle, the angle at D. Q.E.F. This proposition is not used in the rest of the Elements. Next proposition: III.35 Previous: III.33 Book III introduction © 1996 D.E.Joyce Clark University Proposition 35 If in a circle two straight lines cut one another, then the rectangle contained by the segments of the one equals the rectangle contained by the segments of the other. For in the circle ABCD let the two straight lines AC and BD cut one another at the point E. I say that the rectangle AE by EC equals the rectangle DE by EB. If now AC and BD are through the center, so that E is the center of the circle ABCD, it is manifest that, AE, EC, DE, and EB being equal, the rectangle AE by EC also equals the rectangle DE by EB. Next let AC and DB not be through the center. Take the center F let the center of ABCD. Draw FG and FH from F perpendicular to the straight lines AC and DB. Join FB, FC, and FE. III.1 I.12 Then, since a straight line GF through the center cuts a straight line AC not through the center at right angles, it also bisects it, therefore AG equals GC. III.3 Since, then, the straight line AC has been cut into equal parts at G and into unequal parts at E, the rectangle AE by EC together with the square on EG equals the square on GC. II.5 Add the square on GF. Therefore the rectangle AE by EC plus the sum of the squares on GE and GF equals the sum of the squares on CG and GF. But the square on FE equals the sum of the squares on EG and GF, and the square on FC equals the sum of the squares on CG and GF. Therefore the rectangle AE by EC plus the square on FE equals the square on FC. I.47 And FC equals FB, therefore the rectangle AE by EC plus the square on EF equals the square on FB. For the same reason, also, the rectangle DE by EB plus the square on FE equals the square on FB. But the rectangle AE by EC plus the square on FE was also proved equal to the square on FB, therefore the rectangle AE by EC plus the square on FE equals the rectangle DE by EB plus the square on FE. Subtract the square on FE from each. Therefore the remaining rectangle AE by EC equals the rectangle DE by EB. Therefore if in a circle two straight lines cut one another, then the rectangle contained by the segments of the one equals the rectangle contained by the segments of the other. Q.E.D. By means of proposition VI.16, the statement, "the rectangle AE by EC equals the rectangle DE by EB," may be converted into one about ratios, namely, "AE : EB = DE : EC." This proposition is not used in the rest of the Elements. Next proposition: III.36 Previous: III.34 Book III introduction © 1996 D.E.Joyce Clark University Proposition 36 If a point is taken outside a circle and two straight lines fall from it on the circle, and if one of them cuts the circle and the other touches it, then the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference equals the square on the tangent. Let a point D be taken outside the circle ABC, and from D let the two straight lines DCA and DB fall on the circle ABC. Let DCA cut the circle ABC, and let BD touch it. I say that the rectangle AD by DC equals the square on DB. Then DCA is either through the center or not through the center. First let it be through the center, and let F be the center of the circle ABC. Join FB. Therefore the angle FBD is right. III.18 And, since AC has been bisected at F, and CD is added to it, the rectangle AD by DC plus the square on FC equals the square on FD. II.6 But FC equals FB, therefore the rectangle AD by DC plus the square on FB equals the square on FD. And the sum of the squares on FB and BD equals the square on FD, therefore the rectangle AD by DC plus the square on FB equals the sum of the squares on FB and BD. I.47 Subtract the square on FB from each. Therefore the remaining rectangle AD by DC equals the square on the tangent DB. Again, let DCA not be through the center of the circle ABC. Take the center E, and draw EF from E perpendicular to AC. Join EB, EC, and ED. III.1 Then the angle EBD is right. III.18 And, since a straight line EF through the center cuts a straight line AC not through the center at right angles, it also bisects it, therefore AF equals FC. III.3 Now, since the straight line AC has been bisected at the point F, and CD is added to it, the rectangle AD by DC plus the square on FC equals the square on FD. II.6 Add the square on FE to each. Therefore the rectangle AD by DC plus the sum of the squares on CF and FE equals the sum of the squares on FD and FE. But the square on EC equals the sum of the squares on CF and FE, for the angle EFC is right, and the square on ED equals the sum of the squares on DF and FE, therefore the rectangle AD by DC plus the square on EC equals the square on ED. I.47 And EC equals EB, therefore the rectangle AD by DC plus the square on EB equals the square on ED. But the sum of the squares on EB and BD equals the square on ED, for the angle EBD is right, therefore the rectangle AD by DC plus the square on EB equals the sum of the squares on EB and BD. I.47 Subtract the square on EB from each. Therefore the remaining rectangle AD by DC equals the square on DB. Therefore if a point is taken outside a circle and two straight lines fall from it on the circle, and if one of them cuts the circle and the other touches it, then the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference equals the square on the tangent. Q.E.D. This proposition is used in the next one. Next proposition: III.37 Previous: III.35 Book III introduction © 1996 D.E.Joyce Clark University Proposition 37 If a point is taken outside a circle and from the point there fall on the circle two straight lines, if one of them cuts the circle, and the other falls on it, and if further the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference equals the square on the straight line which falls on the circle, then the straight line which falls on it touches the circle. Let a point D be taken outside the circle ABC, from D let the two straight lines DCA and DB fall on the circle ACB, let DCA cut the circle and DB fall on it, and let the rectangle AD by DC equal the square on DB. I say that DB touches the circle ABC. Draw DE touching ABC. Take the center F of the circle ABC, and join FE, FB, and FD. III.17 III.1 Thus the angle FED is right. III.18 Now, since DE touches the circle ABC, and DCA cuts it, the rectangle AD by DC equals the square on DE. III.36 But the rectangle AD by DC was also equal to the square on DB, therefore the square on DE equals the square on DB. Therefore DE equals DB. And FE equals FB, therefore the two sides DE and EF equal the two sides DB and BF, and FD is the common base of the triangles, therefore the angle DEF equals the angle DBF. I.8 But the angle DEF is right, therefore the angle DBF is also right. And FB produced is a diameter, and the straight line drawn at right angles to the diameter of a circle, from its end, touches the circle, therefore DB touches the circle. III.16 Cor Similarly this can be proved to be the case even if the center is on AC. Therefore if a point is taken outside a circle and from the point there fall on the circle two straight lines, if one of them cuts the circle, and the other falls on it, and if further the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference equals the square on the straight line which falls on the circle, then the straight line which falls on it touches the circle. Q.E.D. This proposition is used in IV.10. Next: Book IV Introduction Previous proposition: Proposition III.36 Book III introduction © 1996 D.E.Joyce Clark University Definitions Def. 1. Any rectangular parallelogram is said to be contained by the two straight lines containing the right angle. Def. 2. And in any parallelogrammic area let any one whatever of the parallelograms about its diameter with the two complements be called a gnomon. Guide According to the first definition, the rectangle ABCD illustrated on the left is contained by the lines AB and BC, and this rectangle can be called the rectangle AB by BC. Of course, it could also be called the rectangle BC by CD, or two other names. On the right, in the parallelogram EFGH, there is a diameter EG with a parallelogram LNGO about it and the two complements KLOF and MHNL, and these three parallelograms together make up the gnomon. In other words a gnomon is an L-shaped figure made by removing a parallelogram from a larger similar parallelogram. (The "g" in "gnomon" is silent.) Euclid illustrated gnomons by arcs of circles around the inner vertex. In this example, the gnomon is called the gnomon PQR. First proposition: II.1 Book II introduction © 1996 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 1 If there are two straight lines, and one of them is cut into any number of segments whatever, then the rectangle contained by the two straight lines equals the sum of the rectangles contained by the uncut straight line and each of the segments. Let A and BC be two straight lines, and let BC be cut at random at the points D and E. I say that the rectangle A by BC equals the sum of the rectangle A by BD, the rectangle A by DE, and the rectangle A by EC. Draw BF from B at right angles to BC. Make BG equal to A. Draw GH through G parallel to BC. Through D, E, and C draw DK, EL, and CH parallel to BG. I.11 I.3 I.31 Then BH equals the sum of BK, DL, and EH. Now BH is the rectangle A by BC, for it is contained by GB and BC, and BG equals A; BK is the rectangle A by BD, for it is contained by GB and BD, and BG equals A; and DL is the rectangle A by DE, for DK, that is BG, equals A. Similarly also EH is the rectangle A by EC. II.Def.1 I.34 Therefore the rectangle A by BC equals the sum of the rectangle A by BD, the rectangle A by DE, and the rectangle A by EC. Therefore if there are two straight lines, and one of them is cut into any number of segments whatever, then the rectangle contained by the two straight lines equals the sum of the rectangles contained by the uncut straight line and each of the segments. Q.E.D. The phrase "the rectangle contained by the two straight lines" means any rectangle constructed with two sides equal to the two given sides. In some sense this is the product of the two lines. When the sides have names, such as A and BC, we will refer to that rectangle by "the rectangle A by BC" since that is a little clearer than Euclid's terse "the rectangle A, BC." In this proposition Euclid proves that if BC = BD + DE + EC, then (A by BC) = (A by BD) + (A by DE) + (A by EC). In modern algebraic notation this could be stated as follows: If y = y1 + y2 + ... + yn, then xy = x y1 + x y2 + ... + x yn. This can also be stated in a single equation as x (y1 + y2 + ... + yn) = x y1 + x y2 + ... + x yn. Here x and the various yi's are all lines, and n is an arbitrary number. In modern terminology this identity is called the distributive law for multiplication over addition. Use of this proposition This proposition is not specifically invoked in the rest of the Elements. The next two propositions, however, are special cases of it, and they are each explicitly used once. Next proposition: II.2 Previous: II.Def.1-2 Book II introduction © 1996 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 2 If a straight line is cut at random, then the sum of the rectangles contained by the whole and each of the segments equals the square on the whole. Let the straight line AB be cut at random at the point C. I say that the sum of the rectangle BA by AC and the rectangle AB by BC equals the square on AB Describe the square ADEB on AB, and draw CF through C parallel to either AD or BE. Then AE equals AF plus CE. I.46 I.31 Now AE is the square on AB; AF is the rectangle BA by AC, for it is contained by DA and AC, and AD equals AB; and CE is the rectangle AB by BC, for BE equals AB. II.Def.1 Therefore the sum of the rectangle BA by AC and the rectangle AB by BC equals the square on AB. Therefore if a straight line is cut at random, then the sum of the rectangles contained by the whole and each of the segments equals the square on the whole. Q.E.D. This proposition is actually a special case of II.1. In II.1 Euclid shows that the product of one line by a sum of any number of lines is the sum of the products of that line by each of the lines. In this proposition, there are just two of those lines and their sum equals the one line. Rather than using II.1 to prove II.2, Euclid proves II.2 directly. This suggests that II.1 may have been inserted into the Elements after II.2 was included, either by Euclid or someone else. In modern algebraic notation this proposition says that if y = y1 + y2, then xy = x y1 + x y2. This can also be stated in a single equation as x (y1 + y2) = x y1 + x y2. Use of this proposition This proposition is used in the proof of proposition XIII.10 which shows that a certain relationship holds for the sides of a regular pentagon, regular hexagon, and regular decagon that are all inscribed in the same circle, namely, the square on the side of the pentagon equals the sum of the squares on the side of a hexagon and on the side of a decagon. Next proposition: II.3 Previous: II.1 Book II introduction © 1996 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 3 If a straight line is cut at random, then the rectangle contained by the whole and one of the segments equals the sum of the rectangle contained by the segments and the square on the aforesaid segment. Let the straight line AB be cut at random at C. I say that the rectangle AB by BC equals the sum of the rectangle AC by CB and the square on BC. Describe the square CDEB on CB. Draw ED through to F, and draw AF through A parallel to either CD or BE. I.46 I.31 Then AE equals AD plus CE. Now AE is the rectangle AB by BC, for it is contained by AB and BE, and BE equals BC; AD is the rectangle AC by CB, for DC equals CB; and DB is the square on CB. Therefore the rectangle AB by BC equals the sum of the rectangle AC by CB and the square on BC. Therefore if a straight line is cut at random, then the rectangle contained by the whole and one of the segments equals the sum of the rectangle contained by the segments and the square on the aforesaid segment. Q.E.D. This proposition is another special case of II.1. In modern algebraic notation it says that if x = y + z, then xy = y2 + yz. Identities that are logically equivalent to this implication can be found by eliminating one of the three variables x, y, or z. Here are two of them. (y + z) y = y2 + yz, and xy = y2 + y (x – y). Use of this proposition This proposition refers to lines and rectangles, but the analogous statement for numbers is used in a proposition in one of the Euclid's books on number theory, namely, of proposition IX.15. Next proposition: II.4 Previous: II.2 Book II introduction © 1996 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 4 If a straight line is cut at random, then the square on the whole equals the sum of the squares on the segments plus twice the rectangle contained by the segments. Let the straight line AB be cut at random at C. I say that the square on AB equals the sum of the squares on AC and CB plus twice the rectangle AC by CB. Describe the square ADEB on AB. Join BD. Draw CF through C parallel to either AD or EB, and draw HK through G parallel to either AB or DE. I.46 I.31 Then, since CF is parallel to AD, and BD falls on them, the exterior angle CGB equals the interior and opposite angle ADB. I.29 But the angle ADB equals the angle ABD, since the side BA also equals AD. Therefore the angle CGB also equals the angle GBC, so that the side BC also equals the side CG. I.5 I.6 But CB equals GK, and CG to KB. Therefore GK also equals KB. Therefore CGKB is equilateral. I.34 I say next that it is also right-angled. Since CG is parallel to BK, the sum of the angles KBC and GCB equals two right angles. I.29 But the angle KBC is right. Therefore the angle BCG is also right, so that the opposite angles CGK and GKB are also right. I.34 Therefore CGKB is right-angled, and it was also proved equilateral, therefore it is a square, and it is described on CB. For the same reason HF is also a square, and it is described on HG, that is AC. Therefore the squares HF and KC are the squares on AC and CB. I.34 Now, since AG equals GE, and AG is the rectangle AC by CB, for GC equals CB, therefore GE also equals the rectangle AC by CB. Therefore the sum of AG and GE equals twice the rectangle AC by CB. I.43 But the squares HF and CK are also the squares on AC and CB, therefore the sum of the four figures HF, CK, AG, and GE equals the sum of the squares on AC and CB plus twice the rectangle AC by CB. But HF, CK, AG, and GE are the whole ADEB, which is the square on AB. Therefore the square on AB equals the the sum of the squares on AC and CB plus twice the rectangle AC by CB. Therefore if a straight line is cut at random, the square on the whole equals the squares on the segments plus twice the rectangle contained by the segments. Q.E.D. The statement of the proposition can be interpreted in modern notation as saying that if x = y + z, then x2 = y2 + z2 + 2yz. More simply, as an identity, it says (y + z)2 = y2 + z2 + 2yz. Use of this proposition This is one of the more frequently used propositions of Book II. It is used in II.12 later in this book, frequently in BookX, and in XIII.2. Also, the analogous statement for numbers is used in IX15, Next proposition: II.5 Previous: II.3 Book II introduction © 1996 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 5 If a straight line is cut into equal and unequal segments, then the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section equals the square on the half. Let a straight line AB be cut into equal segments at C and into unequal segments at D. I say that the rectangle AD by DB together with the square on CD equals the square on CB. Describe the square CEFB on CB, and join BE. Draw DG through D parallel to either CE or BF, again draw KM through H parallel to either AB or EF, and again draw AK through A parallel to either CL or BM. I.46 I.31 Then, since the complement CH equals the complement HF, add DM to each. Therefore the whole CM equals the whole DF. I.43 But CM equals AL, since AC is also equal to CB. Therefore AL also equals DF. Add CH to each. Therefore the whole AH equals the gnomon NOP. I.36 II.Def.2 But AH is the rectangle AD by DB, for DH equals DB, therefore the gnomon NOP also equals the rectangle AD by DB. Add LG, which equals the square on CD, to each. Therefore the sum of the gnomon NOP and LG equals the sum of the rectangle AD by DB and the square on CD. But the gnomon NOP together with LG is the whole square CEFB, which is described on CB. Therefore the rectangle AD by DB together with the square on CD equals the square on CB. Therefore if a straight line is cut into equal and unequal segments, then the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section equals the square on the half. Q.E.D. In the figure there is a part of a circle denoted with the points NOP. This is supposed to indicate the gnomon which is three parts of the square BCEF, the only part left out for the gnomon being the square EGHL. Later versions of the Elements did not have this curve in the figure. Instead, they named the same gnomon as LBG. Either way is sufficient to specify the gnomon. Explanation of the proof We can represent a rectangle algebraically as xy where the sides are x and y. In the diagram above, take x as the line AD and y as the line DH, so that xy is the rectangle AH. According to this proposition, this product, or rectangle, is the difference of two squares, the large one being the square of (x + y)/2, which is the square on the line BC in the diagram, and the small one being the square of (x – y)/2, which is the square on the line LH (which equals the square on the line CD). Using symbolic algebra, we can easily verify the identity xy = ( x + y 2 ) 2 – ( y – x 2 ) 2 But Euclid was resticted to geometric arguments. The argument isn't difficult. The original rectangle AH is the sum of the rectangles AL and CH. By proposition I.43, the rectangles CH and HF are equal. And, of course, the rectangles AL and CM are equal. Therefore, AH = AL + CH = CM + HF = CB2 – LH2, as required. Solution to a quadratic problem This proposition is set up to help in the solution of a quadratic problem of the following form. Find two numbers xand yso that their sum is a known value band their product is a known value c2. In terms of the single variable x, this is equivalent to solving the quadratic equation, x(b – x) = c2. This equation can be written in a standard form as x2 + c2 = bx. If b is represented as the line AB in the diagram, and with x = AD and y = BD, the first condition x + y = b is satisfied. This proposition says that the product xy equals the square on BC (which is b/2) minus the square on CD. Thus, the remaining condition reduces to finding CD so that (b/2)2 – CD2 = c2. By I.47, if a right triangle is constructed with one side equal to b/2 and another equal to c, then the hypotenuse will equal the required value for CD. Algebraically, the solutions AD for x and BD for y have the values This analysis yields a construction to solve the quadratic problem stated above. To cut a given straight line so that the rectangle contained by the unequal segments equals a given square. Thus the given square must not be greater than the square described on the half of the given straight line. Let AB be the given straight line. Bisect it at C. Draw the semicircle AQB with center C and radius BC. Construct a perpendicular CR to AB at C equal to the side of the given square. Draw RS parallel to AB intersecting the semicircle at S, and draw SD perpendicular to AB. Then, as described above, AB has been cut at G so that AG times GB equals the given square. This proposition is not found in the Elements, but a generalization is. After II.14 the given square could be replaced by any given rectilinear figure, since II.14 constructs a square equal to a given rectilinear figure. But the full generalization is not given until proposition VI.28. Not only has the given square become a general rectilinear figure, but all the rectangles and squares have been replaced by parallelograms. That requires generalizing II.4 to parallelograms. That's done in VI.25 which constructs a parallelgram similar to a given parallelogram and equal to a given rectilinear figure. It also requires a few technical propositions to carry out the proof. Use of this proposition This proposition is used in II.14, III.35, and occasionally in Book X. Next proposition: II.6 Previous: II.4 Book II introduction © 1996, 2002 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 6 If a straight line is bisected and a straight line is added to it in a straight line, then the rectangle contained by the whole with the added straight line and the added straight line together with the square on the half equals the square on the straight line made up of the half and the added straight line. Let a straight line AB be bisected at the point C, and let a straight line BD be added to it in a straight line. I say that the rectangle AD by DB together with the square on CB equals the square on CD. Describe the square CEFD on CD, and join DE. Draw BG through the point B parallel to either EC or DF, draw KM through the point H parallel to either AB or EF, and further draw AK through A parallel to either CL or DM. I.46 I.31 Then, since AC equals CB, AL also equals CH. But CH equals HF. Therefore AL also equals HF. I.36 I.43 Add CM to each. Therefore the whole AM equals the gnomon NOP. II.Def.2 But AM is the rectangle AD by DB, for DM equals DB. Therefore the gnomon NOP also equals the rectangle AD by DB. Add LG, which equals the square on BC, to each. Therefore the rectangle AD by DB together with the square on CB equals the gnomon NOP plus LG. But the gnomon NOP and LG are the whole square CEFD, which is described on CD. Therefore the rectangle AD by DB together with the square on CB equals the square on CD. Therefore if a straight line is bisected and a straight line is added to it in a straight line, then the rectangle contained by the whole with the added straight line and the added straight line together with the square on the half equals the square on the straight line made up of the half and the added straight line. Q.E.D. Explanation of the proof This proposition is remarkably similar to the last one, II.5, except the point D does not lie on the line AB but on that line extended. Let b denote the line AB, x denote AD, and y denote BD as in II.5. Then x – y = b (as opposed to x + y = b as in II.5). According to this proposition the rectangle AD by DB, which is the product xy, is the difference of two squares, the large one being the square on the line CD, that is the square of x – b/2, and the small one being the square on the line CB, that is, the square of b/2. Algebraically, x(x – b) = (x – b/2)2 – (b/2)2. This equation is easily verified with modern algebra, but it's also easily verified in geometry, as done here in the proof. The geometric proof is primarily an exercise in cutting and pasting. The rectangle AB by DB is the rectangle AM, which is the sum of the rectangles AL and CM. But the rectangles AL, CH, and HF are all equal. Therefore, the rectangle AB by DB equals the gnomon formed by the rectangles CM and HF. That gnomon is the square CF minus the square LG, but the latter equals the square on BC. Thus, the rectangle AB by DB equals the square on DB minus the ssquare on CB. Solution to a quadratic problem As was II.5, this proposition is set up to help in the solution of a quadratic problem: Find two numbers xand yso that their difference x – yis a known value band their product is a known value c2. In terms of x alone, this is equivalent to solving the quadratic equation x(x – b) = c2. Since this proposition says that x(x – b) = (x – b/2)2 – (b/2)2, the problem reduces to solving the equation c2 = (x – b/2)2 – (b/2)2, that is, finding CD so that CD2 = (b/2)2 + c2. By I.47, if a right triangle is constructed with one side equal to b/2 and another equal to c, then the hypotenuse will equal the required value for CD. Algebraically, the solutions AD for x and BD for y have the values This analysis yields a construction to solve the quadratic problem stated above. To apply a rectangle equal to a given square to a given straight line but exceeding it by a square. Let AB be the given straight line. Bisect it at C. Construct a perpendicular BQ to AB at B equal to the side of the given square. Draw CQ. Extend AB to D so that BD equals CQ. Then, as described above, AB has been extended to D so that AD times BD equals the given square. This construction is not found in the Elements, but a generalization of it to parallelograms is proposition VI.29. Use of this proposition This proposition is used in II.11, III.36, and a lemma for X.29. Next proposition: II.7 Previous: II.5 Book II introduction © 1996, 2002 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 7 If a straight line is cut at random, then the sum of the square on the whole and that on one of the segments equals twice the rectangle contained by the whole and the said segment plus the square on the remaining segment. Let a straight line AB be cut at random at the point C. I say that the sum of the squares on AB and BC equals twice the rectangle AB by BC plus the square on CA. Describe the square ADEB on AB, and let the figure be drawn. I.46 I.31 Then, since AG equals GE, add CF to each, therefore the whole AF equals the whole CE. I.43 Therefore the sum of AF and CE is double AF. But the sum of AF and CE equals the gnomon KLM plus the square CF, therefore the gnomon KLM plus the square CF is double AF. But twice the rectangle AB by BC is also double AF, for BF equals BC, therefore the gnomon KLM plus the square CF equal twice the rectangle AB by BC. Add DG, which is the square on AC, to each. Therefore the gnomon KLM plus the sum of the squares BG and GD equals twice the rectangle AB by BC plus the square on AC. But the gnomon KLM plus the sum of the squares BG and GD equals the whole ADEB plus CF, which are squares described on AB and BC. Therefore the sum of the squares on AB and BC equals twice the rectangle AB by BC plus the square on CA. Therefore if a straight line is cut at random, then the sum of the square on the whole and that on one of the segments equals twice the rectangle contained by the whole and the said segment plus the square on the remaining segment. Q.E.D. We can interpret this algebraically with x for AB, y for AC, and z for CB. Then the proposition says that if x = y + z, then x2 + z2 = 2xz + y2. This can be rewritten as various identities depending on which variable is eliminated, the simplest being x2 + z2 = 2xz + (x – z)2. This proposition is used later in Book II to prove proposition II.13, and it is used repeatedly in Book X. Next proposition: II.8 Previous: II.6 Book II introduction © 1996 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 8 If a straight line is cut at random, then four times the rectangle contained by the whole and one of the segments plus the square on the remaining segment equals the square described on the whole and the aforesaid segment as on one straight line. Let a straight line AB be cut at random at the point C. I say that four times the rectangle AB by BC plus the square on AC equals the square described on AB and BC as on one straight line. Produce the straight line BD in a straight line with AB, and make BD equal to CB. Describe the square AEFD on AD, and let the figure be drawn. I.3 I.46 I.31 Then, since CB equals BD, while CB equals GK, and BD equals KN, therefore GK also equals KN. I.34 For the same reason QR also equals RP. And, since BC equals BD, and GK equals KN, therefore CK also equals KD, and GR equals RN. I.36 But CK equals RN, for they are complements of the parallelogram CP. Therefore KD also equals GR. Therefore the four areas DK, CK, GR, RN equal one another. Therefore the four are quadruple of CK. I.43 Again, since CB equals BD, while BD equals BK, that is CG, and CB equals GK, that is GQ, therefore CG also equals GQ. I.34 And, since CG equals GQ, and QR equals RP, AG also equals MQ, and QL equals RF. I.36 But MQ equals QL, for they are complements of the parallelogram ML, therefore AG also equals RF. Therefore the four areas AG, MQ, QL, RF equal one another. Therefore the four are quadruple of AG. But the four areas CK, KD, GR, RN were proved to be quadruple of CK, therefore the eight areas, which contain the gnomon STU, are quadruple of AK. I.43 Now, since AK is the rectangle AB by BD, for BK equals BD, therefore four times the rectangle AB by BD is quadruple of AK. But the gnomon STU was also proved to be quadruple of AK, therefore four times the rectangle AB by BD equals the gnomon STU. Add OH, which equals the square on AC, to each. Therefore four times the rectangle AB by BD plus the square on AC equals the gnomon STU plus OH. But the gnomon STU and OH are the whole square AEFD, which is described on AD. Therefore four times the rectangle AB by BD plus the square on AC equals the square on AD. But BD equals BC. Therefore four times the rectangle AB by BC together with the square on AC equals the square on AD, that is to the square described on AB and BC as on one straight line. Therefore if a straight line is cut at random, then four times the rectangle contained by the whole and one of the segments plus the square on the remaining segment equals the square described on the whole and the aforesaid segment as on one straight line. Q.E.D. Algebraically, if x = y + z, then 4xy + z2 = (x + y)2. As an identity, 4xy + (x – y)2 = (x + y)2. This proposition is not used in the rest of the Elements. Next proposition: II.9 Previous: II.7 Book II introduction © 1996 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 9 If a straight line is cut into equal and unequal segments, then the sum of the squares on the unequal segments of the whole is double the sum of the square on the half and the square on the straight line between the points of section. Let a straight line AB be cut into equal segments at C, and into unequal segments at D. I say that the sum of the squares on AD and DB is double the sum of the squares on AC and CD. Draw CE from C at right angles to AB, and make it equal to either AC or CB. Join EA and EB. Draw DF through D parallel to EC and FG through F parallel to AB. Join AF. I.11 I.3 I.31 Then, since AC equals CE, the angle EAC also equals the angle AEC. I.5 And, since the angle at C is right, the sum of the remaining angles EAC and AEC equals one right angle. I.32 And they are equal, therefore each of the angles CEA and CAE is half of a right angle. For the same reason each of the angles CEB and EBC is also half of a right angle, therefore the whole angle AEB is right. And, since the angle GEF is half of a right angle, and the angle EGF is right, for it equals the interior and opposite angle ECB, the remaining angle EFG is half of a right angle. Therefore the angle GEF equals the angle EFG, so that the side EG also equals GF. I.29 I.32 I.6 Again, since the angle at B is half of a right angle, and the angle FDB is right, for it is again equal to the interior and opposite angle ECB, the remaining angle BFD is half of a right angle. Therefore the angle at B equals the angle DFB, so that the side FD also equals the side DB. I.29 I.32 I.6 Now, since AC equals CE, the square on AC also equals the square on CE, therefore the sum of the squares on AC and CE is double the square on AC. But the square on EA equals the sum of the squares on AC and CE, for the angle ACE is right, therefore the square on EA is double the square on AC. I.47 Again, since EG equals GF, the square on EG also equals the square on GF. Therefore the sum of the squares on EG and GF is double the square on GF. But the square on EF equals the sum of the squares on EG and GF, therefore the square on EF is double the square on GF. I.47 But GF equals CD, therefore the square on EF is double the square on CD. I.34 But the square on EA is also double of the square on AC, therefore the sum of the squares on AE and EF is double the sum of the squares on AC and CD. And the square on AF equals sum of the squares on AE and EF, for the angle AEF is right. Therefore the square on AF is double the sum of the squares on AC and CD. I.47 But the sum of the squares on AD and DF equals the square on AF, for the angle at D is right, therefore the sum of the squares on AD and DF is double the sum the squares on AC and CD. I.47 And DF equals DB. Therefore the sum of the squares on AD and DB is double the sum of the squares on AC and CD. Therefore if a straight line is cut into equal and unequal segments, then the sum of the squares on the unequal segments of the whole is double the sum of the square on the half and the square on the straight line between the points of section. Q.E.D. Outline of the proof Start with the given line AB bisected at C and cut at another point D. The construction will result in a line AF whose square is simultaneously equal to both AD2 + DB2 and 2(AC2+CD2). Draw CE at right angles to AB and equal to half of it. Finish the diagram by drawing parallel lines and connecting points. That results in four isosceles right triangles, ACE, ECB, EGF, and FDB, as well as two other right triangles, AEB and AEF. Then, AF2 = AE2 + EF2 = (AC2 + CE2) + (EG2 + GF2) = 2(AC2+CD2) But also, AF2 = AD2 + DF2 = AD2 + DB2 Thus, AD2 + DB2 = 2(AC2+CD2), as desired. An alternate cut-and-paste proof Up until this proposition, Euclid has only used cut-and-paste proofs, and such a proof can easily be made for this proposition as well. It would start with the same line AB bisected at C and also cut at D. Then lines at right angles and parallel to line AB would be constructed to make squares and rectangles of various sizes. The goal is to show that AD2 + DB2 = 2(AC2 + CD2). The left hand side of the equation is displayed to the right as the two squares AR plus RF. Cut out the rectangle KQ and move it to the right to cover the rectangle MP. Also move the rectangle CN up to the rectangle MH. Now the upper right square MF is completely covered once, with two extra coverings of the small square MR. That is, two squares of size AC2 are covered as well as two squares of size CD2. The two sides of the equation are, therefore, equal. Interpretations of the proposition The equation, AD2 + DB2 = 2(AC2+CD2), can be interpreted in various ways depending on which parts of the given information are taken as basic. These interpretations are more of an aid to the modern reader than as intrinsic aspects of the proposition, since they are interpretations in modern symbolic algebra. For instance, when AC and CB are set to y while CD is set to z, then the algebraic identity (y + z)2 + (y – z)2 = 2 (y2 + z2) results. Alternatively, when AC and CB are set to y while BC is set to x,, then we get the identity (2y – x)2 + x2 = 2 (y2 + (y – x)2) There are yet other interpretiations. This proposition is used in Book X to prove a lemma for X.60, and in that lemma, w = AD and x = DB are given first, so that w2 + x2 = 2 (((w + x)/2)2 + ((w – x)/2)2) Next proposition: II.10 Previous: II.8 Book II introduction © 1996, 2002 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 10 If a straight line is bisected, and a straight line is added to it in a straight line, then the square on the whole with the added straight line and the square on the added straight line both together are double the sum of the square on the half and the square described on the straight line made up of the half and the added straight line as on one straight line. Let a straight line AB be bisected at C, and let a straight line BD be added to it in a straight line. I say that the sum of the squares on AD and DB is double the sum of the squares on AC and CD. Draw CE from the point C at right angles to AB and equal to either AC or CB. Join EA and EB. Draw EF through E parallel to AD, and draw FD through D parallel to CE. I.11 I.3 I.31 Then, since a straight line EF falls on the parallel straight lines EC and FD, the sum of the angles CEF and EFD equals two right angles. Therefore the sum of the angles FEB and EFD is less than two right angles. I.29 But straight lines produced from angles less than two right angles meet. Therefore EB and FD, if produced in the direction B and D, will meet. Post.5 Let them be produced and meet at G, and join AG. Then, since AC equals CE, the angle EAC also equals the angle AEC. The angle at C is right, therefore each of the angles EAC and AEC is half of a right angle. I.5 I.32 For the same reason each of the angles CEB and EBC is also half of a right angle, therefore the angle AEB is right. And, since the angle EBC is half of a right angle, the angle DBG is also half of a right angle. But the angle BDG is also right, for it equals the angle DCE, since they are alternate. Therefore the remaining angle DGB is half of a right angle. Therefore the angle DGB equals the angle DBG, so that the side BD also equals the side GD. I.15 I.29 I.32 I.6 Again, since the angle EGF is half of a right angle, and the angle at F is right, for it equals the opposite angle, the angle at C, the remaining angle FEG is half of a right angle. Therefore the angle EGF equals the angle FEG, so that the side GF also equals the side EF. I.34 I.32 I.6 Now, since the square on EC equals the square on CA, the sum of the squares on EC and CA is double the square on CA. But the square on EA equals the sum of the squares on EC and CA, therefore the square on EA is double the square on AC. I.47 Again, since FG equals EF, the square on FG also equals the square on FE. Therefore the sum of the squares on GF and FE is double the square on EF. But the square on EG equals the sum of the squares on GF and FE, therefore the square on EG is double the square on EF. I.47 And EF equals CD, therefore the square on EG is double the square on CD. But the square on EA was also proved to be double the square on AC, therefore the sum of the squares on AE and EG is double the sum of the squares on AC and CD. I.34 And the square on AG equals the sum of the squares on AE and EG, therefore the square on AG is double the sum of the squares on AC and CD. But the sum of the squares on AD and DG equals the square on AG, therefore the sum of the squares on AD and DG is double the sum of the squares on AC and CD. I.47 And DG equals DB, therefore the sum of the squares on AD and DB is double the sum of the squares on AC and CD. Therefore if a straight line is bisected, and a straight line is added to it in a straight line, then the square on the whole with the added straight line and the square on the added straight line both together are double the sum of the square on the half and the square described on the straight line made up of the half and the added straight line as on one straight line. Q.E.D. Outline of the proof The proof is almost the same as the previous proposition. The entire situation is the same, except the point D is moved past the end B of the line AB. As before draw CE at right angles to AB and equal to half of it. Finish the diagram. That results in isosceles right triangles, ACE, ECB, and EFG, as well as two other right triangles, AEG and ADG. Then, AG2 = AE2 + EG2 = (AC2 + CE2) + (EF2 + FG2) = 2(AC2+CD2) But also, AG2 = AD2 + DG2 = AD2 + DB2 Thus, AD2 + DB2 = 2(AC2+CD2), as desired. Interpretations of this proposition This proposition can be interpreted to derive the same algebraic identities as the previous proposition. For example, when y is identified with CD and z is identified with CB, then the proposition states that (y + z)2 + (y – z)2 = 2 (y2 + z2). Side and diagonal numbers The diagonal of a square is incommensurable with the side of a square, as noted elsewhere in the Elements. Another way of saying that is that the ratio of the diagonal to a side, what is generally known as the square root of 2, is not the ratio of two whole numbers, in other words 2 is an irrational number. Nonetheless, finding a close rational approximation for 2 is an important goal. For practical reasons, close aprroximations are useful, and for theoretical reasons, close approximations are interesting. Some time before Euclid, Plato refers in his Republic the "rational diameter" of 5. A square of side length s = 5 has a diameter (diagonal) of d = 50, which is irrational. But 49 is close to 50, so 7 is close to this irrational length. There are a number of side lengths s whose squares s2 differ from a square number d2 by only 1. The first few are displayed in the table below. The column labelled "angle" gives the angle opposite the diameter d. By the law of cosines, the cosine of the angle equals 1 – d2/(2s2). side s diagonal d relation angle ratio d/s 1 1 2 . 12 – 1 = 12 60° 1.0 2 3 2 . 22 + 1 = 32 97.1808° 1.5 5 7 2 . 52 – 1 = 72 88.854° 1.4 12 17 2 . 122 + 1 = 172 90.1989° 1.416667 29 41 2 . 292 – 1 = 412 89.9659° 1.413793 70 99 2 . 702 + 1 = 992 90.0058° 1.414286 Note how the squares of the diagonals alternately are less and greater by 1 than twice the squares of the sides, and the angle correspondingly alternates between acute and obtuse, but quickly approaches a right angle. Likewise, the ratio d/s is alternately less and greater then the irrational number 2. To the right are displayed triangles for the first four lines in the table. The first triangle has side s1 = OA = 1 and diagonal d1 = AB = 1. For the others s2 = OC = 2 and d2 = CD = 3; s3 = OE = 5 and d3 = EF = 7; also s4 = OG = 12 and d4 = GH = 17. The pattern of the lengths of the sides and diagonals is evident. The next side s' is the sum of the current side and diagonal, while the next diagonal d' is the sum of twice the current side and diagonal: s' = s + d d' = 2s + d. A pattern requires a verification, and this proposition supplies that. What needs to be verified is that if 2s2 differs from d2 by exactly 1, then so does 2s'2 differ from d'2 by exactly 1. Let CD equal s, and let BC equal d. Then s' = AC = CB, and d' = AD. By this proposition d'2 + d2 = 2(s'2 + s2), so that d'2 – 2s'2 = 2s2 – d2. Thus, if 2s2 differs from d2 by 1, then 2s'2 also differs from d'2 by 1. More precisely, if one difference is +1, then the other difference is –1. Next proposition: II.11 Previous: II.9 Book II introduction © 1996, 2002 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 12 In obtuse-angled triangles the square on the side opposite the obtuse angle is greater than the sum of the squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides about the obtuse angle, namely that on which the perpendicular falls, and the straight line cut off outside by the perpendicular towards the obtuse angle. Let ABC be an obtuse-angled triangle having the angle BAC obtuse, and draw BD from the point B perpendicular to CA produced. I.12 I say that the square on BC is greater than the sum of the squares on BA and AC by twice the rectangle CA by AD. Since the straight line CD has been cut at random at the point A, the square on DC equals the sum of the squares on CA and AD and twice the rectangle CA by AD. II.4 Add the square on DB be added to each. Therefore the sum of the squares on CD and DB equals the sum of the squares on CA, AD, and DB plus twice the rectangle CA by AD. But the square on CB equals the sum of the squares on CD and DB, for the angle at D is right, and the square on AB equals the sum of the squares on AD and DB, therefore the square on CB equals the sum of the squares on CA and AB plus twice the rectangle CA by AD, so that the square on CB is greater than the sum of the squares on CA and AB by twice the rectangle CA by AD. I.47 Therefore in obtuse-angled triangles the square on the side opposite the obtuse angle is greater than the sum of the squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides about the obtuse angle, namely that on which the perpendicular falls, and the straight line cut off outside by the perpendicular towards the obtuse angle. Q.E.D. This proposition for obtuse angles, together with the next one for acute triangles, complement the Pythagorean theorem, Proposition I.47, for right triangles. Prop.I.47 says that if triangle ABC has a right angle at A, then a2 = b2 + c2 where a, b, and c are the sides opposite the angles A, B, and C, respectively. In this proposition, II.12, the angle A is obtuse rather than right, and the conclusion is that a2 = b2 + c2 2ch where h is the height of the triangle when c is taken as the base of the triangle. The next proposition, II.13 has the same conclusion, but the hypothesis is that the angle at A is acute rather than obtuse. This conclusion is very close to the law of cosines for oblique triangles. a2 = b2 + c2 – 2bc cos A, since the height h equals b cos A. Trigonometry was developed some time after the Elements was written, and the negative numbers needed here (for the cosine of an obtuse angle) were not accepted until long after most of trigonometry was developed. Nonetheless, this proposition and the next may be considered geometric versions of the law of cosines. Neither this nor the next is used in the rest of the Elements. Next proposition: II.13 Previous: II.11 Book II introduction © 1996 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 13 In acute-angled triangles the square on the side opposite the acute angle is less than the sum of the squares on the sides containing the acute angle by twice the rectangle contained by one of the sides about the acute angle, namely that on which the perpendicular falls, and the straight line cut off within by the perpendicular towards the acute angle. Let ABC be a triangle having the angle at B acute, and draw AD from the point A perpendicular to BC. I.12 I say that the square on AC is less than the sum of the squares on CB and BA by twice the rectangle CB by BD. Since the straight line CB has been cut at random at D, the sum of the squares on CB and BD equals twice the rectangle CB by BD plus the square on DC. II.7 Add the square on DA to each. Therefore the sum of the squares on CB, BD, and DA equals twice the rectangle CB by BD plus the sum of the squares on AD and DC. But the square on AB equals the sum of the squares on BD and DA, for the angle at D is right, and the square on AC equals the sum of the squares on AD and DC, therefore the sum of the squares on CB and BA equals the square on AC plus twice the rectangle CB by BD, so that the square on AC alone is less than the sum of the squares on CB and BA by twice the rectangle CB by BD. I.47 Therefore in acute-angled triangles the square on the side opposite the acute angle is less than the sum of the squares on the sides containing the acute angle by twice the rectangle contained by one of the sides about the acute angle, namely that on which the perpendicular falls, and the straight line cut off within by the perpendicular towards the acute angle. Q.E.D. See the guide for the previous proposition II.12. Next proposition: II.14 Previous: II.12 Book II introduction © 1996 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 14 To construct a square equal to a given rectilinear figure. Let A be the given rectilinear figure. It is required to construct a square equal to the rectilinear figure A. Construct the rectangular parallelogram BD equal to the rectilinear figure A. I.45 Then, if BE equals ED, then that which was proposed is done, for a square BD has been constructed equal to the rectilinear figure A. But, if not, one of the straight lines BE or ED is greater. Let BE be greater, and produce it to F. Make EF equal to ED, and bisect BF at G. I.3 I.10 Describe the semicircle BHF with center G and radius one of the straight lines GB or GF. Produce DE to H, and join GH. I.Def.18 Then, since the straight line BF has been cut into equal segments at G and into unequal segments at E, the rectangle BE by EF together with the square on EG equals the square on GF. II.5 But GF equals GH, therefore the rectangle BE by EF together with the square on GE equals the square on GH. But the sum of the squares on HE and EG equals the square on GH, therefore the rectangle BE by EF together with the square on GE equals the sum of the squares on HE and EG. I.47 Subtract the square on GE from each. Therefore the remaining rectangle BE by EF equals the square on EH. But the rectangle BE by EF is BD, for EF equals ED, therefore the parallelogram BD equals the square on HE. And BD equals the rectilinear figure A. Therefore the rectilinear figure A also equals the square which can be described on EH. Therefore a square, namely that which can be described on EH, has been constructed equal to the given rectilinear figure A. Q.E.F. The construction of a square equal to a given is short as described in the proof. The verification that this construction works is also short with the help of Proposition II.5 and Proposition I.47, the Pythagorean theorem. First, Prop. II.5 allows us to convert the rectangle, BE by ED, into the difference of two squares, GF2 – GE2. Note that GF equals GH, the hypotenuse of a right triangle GHE. Using I.47 we can replace the difference of two squares, GH2 – GE2, by the single square, EH2. Thus, the original rectangle equals the squre EH2. Quadrature of rectilinear figures This proposition finishes the for quadrature of rectilinear figures. The narrow meaning of the word "quadrature" is to find a square with the same area of a given figure, also called "squaring" the figure. In a broader sense, "quadrature" means finding the area of a given figure. Proposition I.45 on application of areas of rectilinear figures allows us to replace the figure under question with a rectangle of the same area. Now, the semicircle construction in this proposition finds what is called the "mean proportional" between the sides of the rectangle. If the sides of the rectangle are denoted a and b, then the mean proportional x between them satisfies the proportion a:x = x:b, and that's equivalent to an equality of areas ab = x2, that is to say, the square on this mean proportional has the same area as the rectangle. Thus, any rectilinear figure can be squared. This result is an end in itself. It is not used in the rest of the Elements. There is another proof of this proposition that is based on similar triangles. Referring to the figure in the proposition, draw lines BH and BF, and you'll see three similar right triangles: BFH, BHE, and HGE. From their similarity it follows that BE:EH = EH:EF. That says EH is the required mean proportional. Proportions aren't developed until Book V, and similar triangles aren't mentioned until Book VI. So in order to complete the theory of quadrature of rectilinear figures early in the Elements, Euclid chose a different proof that doesn't depend on similar triangles. Note that this same result appears in the garb of proportions in Proposition VI.13. Also in Book VI, Proposition VI.17 shows that the square on the mean proportional equals the rectangle on the two straight lines. Squaring the circle What about circles and other shapes? The general theory of circles is treated in Book III, but there are no propositions about the areas of circles until book XII. Proposition XII.2 says the areas of circles are proportional to the squares on their diameters. That allows the area of two circles to be compared, but it doesn't answer the question "what's the area of this circle?" in the same way that this proposition does for rectilinear figures. That would require finding a square equal to a given circle. This problem of "quadrature of the circle" was one of three famous problems that goes back at least to the time of Anaxagoras, about 150 years before Euclid. It is equivalent to constructing a line segment of length pi (relative to a unit length). This problem was solved by ancient Greek geometers but not by means of the Euclidean tools of straightedge and compass; higher curves were required. In fact, by the time of Pappus it was believed that the circle could not be squared using only straightedge, compass, and, furthermore, couldn't be squared even with the help of the conic sections (parabola, hyperbola, and ellipse). But the ancient Greeks had no mathematical proof that it could not be squared. That the circle could not be squared with Euclidean tools was not shown until 1882 when Lindemann proved that pi is a transcendental number. Next book: Book III Previous: II.13 Book II introduction © 1996,2002 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University Proposition 47 In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle. Let ABC be a right-angled triangle having the angle BAC right. I say that the square on BC equals the sum of the squares on BA and AC. Describe the square BDEC on BC, and the squares GB and HC on BA and AC. Draw AL through A parallel to either BD or CE, and join AD and FC. I.46 I.31 Post.1 Since each of the angles BAC and BAG is right, it follows that with a straight line BA, and at the point A on it, the two straight lines AC and AG not lying on the same side make the adjacent angles equal to two right angles, therefore CA is in a straight line with AG. I.Def.22 I.14 For the same reason BA is also in a straight line with AH. Since the angle DBC equals the angle FBA, for each is right, add the angle ABC to each, therefore the whole angle DBA equals the whole angle FBC. I.Def.22 Post.4 C.N.2 Since DB equals BC, and FB equals BA, the two sides AB and BD equal the two sides FB and BC respectively, and the angle ABD equals the angle FBC, therefore the base AD equals the base FC, and the triangle ABD equals the triangle FBC. I.Def.22 I.4 Now the parallelogram BL is double the triangle ABD, for they have the same base BD and are in the same parallels BD and AL. And the square GB is double the triangle FBC, for they again have the same base FB and are in the same parallels FB and GC. I.41 Therefore the parallelogram BL also equals the square GB. Similarly, if AE and BK are joined, the parallelogram CL can also be proved equal to the square HC. Therefore the whole square BDEC equals the sum of the two squares GB and HC. C.N.2 And the square BDEC is described on BC, and the squares GB and HC on BA and AC. Therefore the square on BC equals the sum of the squares on BA and AC. Therefore in right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle.. Q.E.D. This proposition is generalized in VI.31 to arbitrary similar figures placed on the sides of the triangle ABC. If the rectilinear figures on the sides of the triangle are similar, then that on the hypotenuse is the sum of the other two figures. A bit of history This proposition, I.47, is often called the "Pythagorean theorem," called so by Proclus and others centuries after Pythagoras and even centuries after Euclid. The statement of the proposition was very likely known to the Pythagoreans if not to Pythagoras himself. The Pythagoreans and perhaps Pythagoras even knew a proof of it. But the knowledge of this relation was far older than Pythagoras. More than a millennium before Pythagoras, the Old Babylonians (ca. 1900-1600 B.C.E) used this relation to solve geometric problems involving right triangles. Moreover, the tablet known as Plimpton 322 shows that the Old Babylonians could construct all the so-called Pythagorean triples, those triples of numbers a, b, and c such that a2 + b2 = c2 which describe triangles with integral sides. (The smallest of these is 3, 4, 5.) The hypotenuse diagram in the Zhou bi suan jing The rule for computing the hypotenuse of a right triangle was well known in ancient China. It is used in the Zhou bi suan jing, a work on astronomy and mathematics compiled during the Han period, and in the later important mathematical work Jiu zhang suan shu [Nine Chapters] to solve right triangles. The Zhou bi includes a very interesting diagram known as the "hypotenuse diagram." This diagram may not have been in the original text but added by its primary commentator Zhao Shuang sometime in the third century C.E. A particular case of this proposition is illustrated by this diagram, namely, the 3-4-5 right triangle. Place four 3 by 4 rectangles around a 1 by 1 square. A 7 by 7 square results. The four diagonals of the rectangles bound a tilted square as illustrated. The area of tilted square is 49 minus 4 times 6 (the 6 is the area of one right triangle with legs 3 and 4), which is 25. Therefore the tilted square is 5 by 5, and the diagonal of the original 3 by 4 rectangles is 5. Zhao Shuang referred to the hypotenuse figure in a general way. He described all the ways the sides, the hypotenuse, and their squares can be found in terms of each other. The Zhou bi has recently been translated into English with an excellent commentary. See Astronomy and mathematics in ancient China: the Zhou bi suan jing, by Christopher Cullin, Cambridge University Press, 1996. Alternate methods of proof According to Proclus, the specific proof of this proposition given in the Elements is Euclid's own. It is likely that older proofs depended on the theories of proportion and similarity, and as such this proposition would have to wait until after Books V and VI where those theories are developed. It appears that Euclid devised this proof so that the proposition could be placed in Book I. Euclid presents a proof based on proportion and similarity in the lemma for proposition X.33. Compare it, summarized here, to the proof in I.47. Let ABC be a right-angled triangle with a right angle at A. Draw AM perpendicular to BC. According to VI.8, the triangles ABM and AMC are similar both to the whole ABC and to one another. (VI.8 concludes the triangles are similar after showing they have the same angles, see VI.4.) Since triangle ABC is similar to triangle ABD, therefore, by the definition of similarity VI.Def.1, CB is to BA as BA is to BM. Next VI.17 converts this proportion to a statement about areas, namely, the rectangle CB by BM (which is the parallelogram BL in the proof of I.47) equals the square on AB. For the same reason the rectangle BC by CM (which is the parallelogram CL in the proof of I.47) also equals the square on AC. Therefore the sum of the two rectangles CB by BM and BC by CM, which is the square on BC, equals the sum of the squares on AC and BC. Q.E.D. (Actually, the final sentence is not part of the lemma, probably because Euclid moved that statement to the first Book as I.47.) So, although Euclid's proof in I.47 may be more complicated than some others, we can see how it well it corresponds to a simpler proof that depends on the theories of proportion and similarity. Generalizations of I.47 Propositions II.12 and II.13 consider triangles other than right triangles. In II.12 the right angle is replaced by an obtuse angle, while in II.13 the right angle is replaced by an acute angle. The resulting statements are actually geometric forms of the law of cosines. Proposition VI.31 generalizes the figures that can be placed on the sides of the right triangle to any three similar figures instead of the three squares here in I.47. Use of this proposition This proposition is used in the next one, which its converse, in propositions II.9 through II.14 in Book II, and several propositions in the rest of the books on geometry. Next proposition: I.48 Previous: I.46 Book I introduction © 1996 D.E.Joyce Clark University Proposition 4 If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides. Let ABC and DEF be two triangles having the two sides AB and AC equal to the two sides DE and DF respectively, namely AB equal to DE and AC equal to DF, and the angle BAC equal to the angle EDF. I say that the base BC also equals the base EF, the triangle ABC equals the triangle DEF, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides, that is, the angle ABC equals the angle DEF, and the angle ACB equals the angle DFE. If the triangle ABC is superposed on the triangle DEF, and if the point A is placed on the point D and the straight line AB on DE, then the point B also coincides with E, because AB equals DE. Again, AB coinciding with DE, the straight line AC also coincides with DF, because the angle BAC equals the angle EDF. Hence the point C also coincides with the point F, because AC again equals DF. But B also coincides with E, hence the base BC coincides with the base EF and equals it. C.N.4 Thus the whole triangle ABC coincides with the whole triangle DEF and equals it. C.N.4 And the remaining angles also coincide with the remaining angles and equal them, the angle ABC equals the angle DEF, and the angle ACB equals the angle DFE. Therefore if two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides. Q.E.D. This is the first of the congruence propositions for triangles. Euclid did not explicitly use the concept of congruence, although it would have simplified his exposition a bit. The definition of congruence would include the hypotheses and conclusions of this proposition, that is, two triangles ABC and DEF are congruent if angles A, B, and C are equal to angles D, E, and F respectively, and sides AB, BC, and AC are equal to sides DE, EF, and DF respectively, and the triangle ABC equals the triangle DEF (by which is meant that they have the same area). In the books on solid geometry, Euclid uses the phrase "similar and equal" for congruence, but similarity is not defined until Book VI, so that phrase would be out of place in the first part of the Elements. For more discussion of congruence theorems see the note after proposition I.26, the last of the congruence propositions. Euclid frequently refers to one side of a triangle as its "base," leaving the other two named "sides." Any one of the sides might be chosen as the base, but once chosen, it remains the base for the rest of the discussion. This is simply a linguistic device to save words. The method of superposition The method of proof used in this proposition is sometimes called "superposition." It apparently is not a method that Euclid prefers since he so rarely uses it, only here in I.4 and in I.8 and III.24, but not in many other propositions in which he could have used it. It is not entirely clear what is meant by "superposing a triangle on a triangle" means. It has been variously interpreted as actually moving one triangle to cover the other or as simply associating parts of one triangle with parts of the other. For the two triangles illustrated in the figure, you can actually slide one over the other in a continuous motion within the plane. Note, however, that if one triangle is the mirror image of the other, then any continuous motion would require moving one triangle outside of the plane. But the triangles don't have to be same plane to begin with, and they often are not in the same plane when this proposition is invoked in the books on solid geometry. Whatever the intended meaning of superposition may be, there are no postulates to allow any conclusions based on superposition. One possibility is to add postulates based on a group of transformations of space, or if restricted to plane geometry, on a group of transformations of the plane. Charles Dodgson (a.k.a. Lewis Carroll) would have said that using group theory is not appropriate to an elementary exposition of Euclidean geometry. Heath has described a more elementary conservative basis in his commentary on this proposition. Yet another alternative is to simply take this proposition as a postulate, or part of it as a postulate. For instance, Hilbert in his Foundations of Geometry takes as given that under the hypotheses of this proposition that the remaining angles equal the remaining angles. Then, Hilbert proves that the base equals the base. Use of Proposition 4 Of the various congruence theorems, this one is the most used. This proposition is used frequently in Book I starting with the next two propositions, and it is often used in the rest of the books on geometry, namely, Books II, III, IV, VI, XI, XII, and XIII. Although the two triangles in this proposition appear to be in the same plane, that is not necessary. In Proposition XI.4 and many others in Book XI this proposition is applied to pairs of triangles in different planes. Next proposition: I.5 Previous: I.3 Book I introduction © 1996, 1997 D.E.Joyce Clark University Proposition 8 If two triangles have the two sides equal to two sides respectively, and also have the base equal to the base, then they also have the angles equal which are contained by the equal straight lines. Let ABC and DEF be two triangles having the two sides AB and AC equal to the two sides DE and DF respectively, namely AB equal to DE and AC equal to DF, and let them have the base BC equal to the base EF. I say that the angle BAC also equals the angle EDF. If the triangle ABC is applied to the triangle DEF, and if the point B is placed on the point E and the straight line BC on EF, then the point C also coincides with F, because BC equals EF. Then, BC coinciding with EF, therefore BA and AC also coincide with ED and DF, for, if the base BC coincides with the base EF, and the sides BA and AC do not coincide with ED and DF but fall beside them as EG and GF, then given two straight lines constructed on a straight line and meeting in a point, there will have been constructed on the same straight line and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each to that which has the same end with it. But they cannot be so constructed. I.7 Therefore it is not possible that, if the base BC is applied to the base EF, the sides BA and AC do not coincide with ED and DF. Therefore they coincide, so that the angle BAC coincides with the angle EDF, and equals it. C.N.4 Therefore if two triangles have the two sides equal to two sides respectively, and also have the base equal to the base, then they also have the angles equal which are contained by the equal straight lines. Q.E.D. This, the "side-side-side" congruence theorem, is the second of Euclid's three congruence theorems for triangles. See the note on congruence theorems after proposition I.26. As in the proof of I.4, this proof employs the hazy method of superposition. Use of Proposition 8 This proposition is used for the a few of the propositions in Book I starting with the next one. It is also used several times in the Books III, IV, XI, and XIII. As in I.4 the two triangles need not lie in one plane. Propositions such as XI.4 in Book XI apply this theorem to the case when the two triangles are not coplanar. Next proposition: I.9 Previous: I.7 Book I introduction © 1996 D.E.Joyce Clark University Proposition 26 If two triangles have two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that opposite one of the equal angles, then the remaining sides equal the remaining sides and the remaining angle equals the remaining angle. Let ABC and DEF be two triangles having the two angles ABC and BCA equal to the two angles DEF and EFD respectively, namely the angle ABC to the angle DEF, and the angle BCA to the angle EFD, and let them also have one side equal to one side, first that adjoining the equal angles, namely BC equal to EF. I say that the remaining sides equal the remaining sides respectively, namely AB equals DE and AC equals DF, and the remaining angle equals the remaining angle, namely the angle BAC equals the angle EDF. If AB does not equal DE, then one of them is greater. Let AB be greater. Make BG equal to DE, and join GC. I.3 Post.1 Since BG equals DE, and BC equals EF, the two sides GB and BC equal the two sides DE and EF respectively, and the angle GBC equals the angle DEF, therefore the base GC equals the base DF, the triangle GBC equals the triangle DEF, and the remaining angles equal the remaining angles, namely those opposite the equal sides. Therefore the angle GCB equals the angle DFE. But the angle DFE equals the angle ACB by hypothesis. Therefore the angle BCG equals the angle BCA, the less equals the greater, which is impossible. I.4 C.N.1 Therefore AB is not unequal to DE, and therefore equals it. But BC also equals EF. Therefore the two sides AB and BC equal the two sides DE and EF respectively, and the angle ABC equals the angle DEF. Therefore the base AC equals the base DF, and the remaining angle BAC equals the remaining angle EDF. I.4 Next, let sides opposite equal angles be equal, as AB equals DE. I say again that the remaining sides equal the remaining sides, namely AC equals DF and BC equals EF, and further the remaining angle BAC equals the remaining angle EDF. If BC is unequal to EF, then one of them is greater. Let BC be greater, if possible. Make BH equal to EF, and join AH. I.3 Post.1 Since BH equals EF, and AB equals DE, the two sides AB and BH equal the two sides DE and EF respectively, and they contain equal angles, therefore the base AH equals the base DF, the triangle ABH equals the triangle DEF, and the remaining angles equal the remaining angles, namely those opposite the equal sides. Therefore the angle BHA equals the angle EFD. I.4 But the angle EFD equals the angle BCA, therefore, in the triangle AHC, the exterior angle BHA equals the interior and opposite angle BCA, which is impossible. C.N.1 I.16 Therefore BC is not unequal to EF, and therefore equals it. But AB also equals DE. Therefore the two sides AB and BC equal the two sides DE and EF respectively, and they contain equal angles. Therefore the base AC equals the base DF, the triangle ABC equals the triangle DEF, and the remaining angle BAC equals the remaining angle EDF. I.4 Therefore if two triangles have two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that opposite one of the equal angles, then the remaining sides equal the remaining sides and the remaining angle equals the remaining angle. Q.E.D. There are two statements in this theorem which are different only in their hypotheses. In one, the known side lies between the two angles, in the other, the known side lies opposite one of the angles. If this proposition had come after proposition I.32 which states the sum of the angles in a triangle equals two right angles, then these two hypotheses could have been merged into one, since then if two angles are known, then is the third. But proposition I.32 depends on the parallel postulate Post.5, which, it is apparent, Euclid did not want to use unless necessary. Thus, this proposition, I.26, appears where it is with two distinct hypotheses. On congruence theorems This is the last of Euclid's congruence theorems for triangles. Euclid's congruence theorems are I.4 (side-angle-side), I.8 (side-side-side), and this one, I.26 (side and two angles). Calling them congruence theorems is anachronistic, since Euclid did not explicitly use the concept of congruence. We would say that two triangles ABC and DEF are congruent if the angles A, B, and C equal the angles D, E, and F respectively, and the sides AB, BC, and AC equal the sides DE, EF, and DF respectively, and the triangle ABC equals the triangle DEF (by which is meant that they have the same area). The remaining congruence theorem, side-side-angle, includes some ambiguous cases. Suppose triangles ABC and DEF are such that sides AB and BC are equal to sides DE and EF respectively, and angle A equals angle D. If it is also known that AB is less than or equal to BC, then it follows that the two triangles are congruent. If, however, AB is greater than BC, then the two triangles need not be congruent. Euclid does not include any form of a side-side-angle congruence theorem, but he does prove one special case, side-side-right angle, in the course of the proof of proposition III.14. Although Euclid does not include a side-side-angle congruence theorem, he does have a side-sideangle similarity theorem, namely proposition VI.7. The analogous congruence theorem could be stated as follows: If two triangles have one angle equal to one angle, two sides adjoining the equal angles equal, namely, one side adjoining the equal angles, and one opposite the equal angles, and the remaining angles either both less or both not less than a right angle, then the remaining side equals the remaining side and the remaining angles equal the remaining angles. Use of Proposition 26 This proposition is used in the proofs of proposition I.34 and several propositions in Books III, IV, XI, XII, and XIII. As in propositions I.4 and I.8, it appears that the triangles are in the same plane, but, again, that is not necessary. Indeed, this proposition is invoked in proposition XI.35 when two triangles do not lie in the same plane. Next proposition: I.27 Previous: I.25 Book I introduction © 1996 D.E.Joyce Clark University Definition 1 A point is that which has no part. The Elements is the prime example of an axiomatic system from the ancient world. Its form has shaped centuries of mathematics. An axiomatic system should begin with a list of the terms that it will use. This definition says that one term that will be used is that of point. The next few definitions give some more terms that will be used. Although there is some description to go along with the terms, that description is actually never used in the exposition of the axiomatic system. It can, at most, be used to orient the reader. The description of a point, "that which has no part," indicates that Euclid will be treating a point as having no width, length, or breadth, but as an indivisible location. Later definitions will define terms by means of terms defined before them, but the first few terms in the Elements are not defined by means of other terms; they're "primitive" terms. Their meaning comes from properties about them that are assumed later in axioms. In the Elements, the axioms come in two kinds: postulates and common notions. The first postulate, I.Post.1, for instance, gives some meaning to the term "point." It states that a straight line may be drawn between any two points. Other postulates add more meaning to the term "point." Actually, Euclid failed to notice that he made a number of conclusions without complete justification at a number of places in the Elements. This usually means that a postulate, that is, a explicit assumption, is missing. Next definition: I.Def.2 Book I introduction © 1996 D.E.Joyce Clark University Definition 2 A line is breadthless length. "Line" is the second primitive term in the Elements. The description, "breadthless length," says that a line will have one dimension, length, but it won't have breadth or depth. In I.Def.5 a surface is defined with the two dimensions length and breadth, and in XI.Def.1 a solid is defined with the three dimensions length, breadth, and depth. One cannot tell from this definition what kind of line is meant by "line," but later a "straight" line defined to be a special kind of line. One can conclude, then, that "lines" need not be straight. Perhaps "curve" would be a better translation than "line" since Euclid meant what is commonly called a curve in modern English, where a curve may or may not be straight. Also, from the next definition, it is apparent that Euclid's lines may have ends, so they are "line segments" or "curve segments." But they need not have ends in all cases since the entire circumference of a circle is an example of a line. Indeed, lines need not be finite in all cases; there are a few instances in the Elements where a line is not bounded, and that is usually indicated by the language. See, for example, proposition I.12. One piece of terminology that Euclid did not mention explicitly in a definition is a phrase to indicate when a line passes through a point. That would be a "primitive" relation that could hold between a line and a point. Postulates would be included as well to give meaning to the phrase as they are in modern treatments of elementary geometry. Next definition: I.Def.3 Previous: I.Def.1 Book I introduction © 1996 D.E.Joyce Clark University Definition 3 The ends of a line are points. This statement can be taken as indicating that between certain lines and points a relation holds, that a point can be an end of a line. It doesn't say what ends are. It also doesn't indicate how many ends a line can have. For instance, the circumference of a circle has no ends, but a finite line has its two end points. Next definition: I.Def.4 Previous: I.Def.2 Book I introduction © 1996 D.E.Joyce Clark University Definition 4 A straight line is a line which lies evenly with the points on itself. This statement indicates, at least, that the term "straight line" refers to a kind of line. It is hard to tell what else it means, if anything. Various commentators have interpreted in a variety of ways. The definition of plane surface in I.Def.7 uses a similar language that is equally opaque. There are a some postulates that come a little later in Book I and give meaning to straight lines. I.Post.1 says that a straight line can be drawn between any two points, I.Post.2 says that a straight line can be extended, and the remaining postulates use the concept of straight line in one way or another. Next definition: I.Def.5 Previous: I.Def.3 Book I introduction © 1996 D.E.Joyce Clark University Definition 5 A surface is that which has length and breadth only. This statement suggests that a surface has two dimensions, but says very little, if anything, since neither length nor breadth have been defined yet, nor will they be. From the next definition, it is clear that a surface does not have to be a plane. Other examples of surfaces that appear in the Elements are surfaces of cones, cylinders, and spheres. Next proposition: I.Def.6 Previous: I.Def.4 Book I introduction © 1996 D.E.Joyce Clark University Definition 6 The edges of a surface are lines. As in I.Def.3, this statement describes a certain relationship, but this time between surfaces and lines. For example, a hemisphere is a surface, and its edge is the circumference of a circle, a kind of line. This definition cannot actually be used since there are no postulates to go along with it to connect the edges of a surface in any way to the surface. Euclid uses the same term for the end of a line in I.Def.3, the edge of a surface in this definition, and the surface of a solid in XI.Def.2. That term could be translated as "that which is around," "the limits of," or "the extremities of," but in English the terms "the ends of" a line, "the edges of" a surface, and either "the surfaces of" or "the faces of" a solid are fairly standard for different dimensions. Next definition: I.Def.7 Previous: I.Def.5 Book I introduction © 1996 D.E.Joyce Clark University Definition 7 A plane surface is a surface which lies evenly with the straight lines on itself. We see now that a plane surface, usually abbreviated to the single word "plane," is a kind of surface. Perhaps the remainder of the statement is a definition of content, but, if so, some words are missing. One interpretation often given is that if a plane surface contains two points, then it contains the line connecting the two points. If that were the meaning, then it would be just as well to make that the explicit definition or to make it a postulate. But that does not seem to be Euclid's intent. His proposition XI.7 has a detailed proof that the line joining two points on two parallel lines lies in the plane of the two parallel lines. No proof at all would be necessary if that line were by definition or by postulate contained in a plane that contained its ends. Note that a plane surface may be infinite, but needn't be infinite. It can be a square, a circle, or any other plane figure (Def.I.19). There are no postulates in the Elements for the existance of plane surfaces, either finite or infinite. Post.3 says circles can be drawn, but a ambient plane is implicitly required there. Rectilinear figures are assumed to exist once the bounding lines have been constructed, but again, a plane is presumed to exist first. Throughout Books I through IV and Book VI, the books on plane geometry, there is the implicit assumption of one plane in which all the points, lines, and circles lie. In the books on solid geometry, Books XI through XIII, there is sometimes mentioned a "plane of reference," and proposition XI.2 claims that two intersecting lines determine a plane as does any triangle (but its proof fails completely). Next definition: I.Def.8 Previous: I.Def.6 Book I introduction © 1996 D.E.Joyce Clark University Definition 8 A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line. The concept of angle is a very important concept for all of Greek geometry. Many of the propositions require angles even for their statements. The two lines are meant to emanate from the same point; two intersecting lines will actually make four angles. The concept is also a difficult one, and, surprisingly, broader than our modern concept of angle. As can be seen from the next definition of rectilinear angle, angles do not have to have straight sides; they can have curves as sides. The size of the angle does not depend on the length of the sides, but is determined only by how the two sides meet. In the Elements nearly all the angles are rectilinear, but angles with curved sides appear in proposition III.16. In that proposition, a so-called horn angle CAE is described as the angle between a circle and a straight tangent line and is shown to be smaller than any rectilinear angle FAE. Even though the curved side of the horn angle extends beyond any rectilinear angle, it is considered to be smaller since near the vertex A of the angle, the curvilinear angle CAE is entirely contained in the rectilinear angle FAE. Thus, an angle doesn't have an extent. Next definition: I.Def.9 Previous: I.Def.7 Book I introduction © 1996 D.E.Joyce Clark University Definition 9 And when the lines containing the angle are straight, the angle is called rectilinear. This continues the previous definition of angle. Nearly all the angles that appear in the Elements are rectilinear as is the illustrated angle BAC. Angles usually are named by three points, the middle point the vertex of the angle. When there is no ambiguity it is sufficient to name the angle by its vertex, in this example, A. Angles as magnitudes As treated by Euclid, rectilinear angles are magnitudes that can be added together. When the sum of angles happens greater than two right angles, it is continued to be treated as a sum of angles rather than an individual angle. For instance, in proposition I.32 it is proved that the sum of the interior angles of a triangle equals two right angles. Treating angles as magnitudes should not be confused with measuring angles. The angles themselves are the magnitudes. The only measurement of angles in the Elements is in terms of right angles (defined in the next definition). Degree measurement and radian measurement were not used until later. In terms of degrees a right angle is 90°, while in terms of radians a right angle is pi/2 radians. Throughout ancient Greek mathematics, only positive magnitudes were considered. Zero and negative magnitudes were not conceived. For the most part, a lack of zero and negative magnitudes complicates mathematics, but occasionally simplifies it. In any case, the power of a mathematics without zero and negative magnitudes is no less in the sense that any statement made using the language of zero or negative magnitudes can be translated into a statement that doesn't use them, although the translated statement may be longer and less understandable. Although in modern mathematics, angles can be positive, negative, or zero, and can be greater than a full circle (360° or 2 pi radians), in the Elements angles are always greater than zero and less than two right angles (180° or pi radians), except perhaps in one interpretation of proposition III.20 where the central angle of a circle could be greater than two right angles. Next definition: I.Def.10 Previous: I.Def.8 Book I introduction © 1996 D.E.Joyce Clark University Definition 10 When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands. In the figure, the two angles DBA and DBC are equal, so they are right angles by definition, and so the line BD set up on the line AC is perpendicular to it. Later there will be a postulate (Post.4) which states that all right angles are equal, and after a few propositions, it can be shown that AC is also perpendicular to BD. There are no postulates that explicitly state perpendiculars exist. Instead a construction for them is given and proved in proposition I.11. The word "orthogonal" is frequently used in mathematics as a synonym for "perpendicular." This is the first mention in the Elements of magnitudes being equal. There are several different kinds of magnitudes in the Elements besides angles. Lines, plane figures, and solids are also kinds of magnitudes. Some of the assumptions about magnitudes are stated later as "common notions" C.N., which are often called "axioms." One thing that magnitudes of the same kind can be is "equal," as the angles in this definition can be. Nowhere does Euclid explicitly state what it means for angles to be equal, or for that matter, for lines, plane figures, or solids to be equal, although much can be determined by the way he uses equality. Next definition: I.Def.11-12 Previous: I.Def.9 Book I introduction © 1996 D.E.Joyce Clark University Definitions 11 and 12 Def. 11. An obtuse angle is an angle greater than a right angle. Def. 12. An acute angle is an angle less than a right angle. The angle ABD in the figure is an obtuse angle. It is greater than a right angle, but less than two right angles. Recall that Euclid required that any angle be less than two right angles. The angle CBD is an acute angle. It is less than a right angle. Note that there is no requirement that the angle be rectilinear, indeed, the horn angles mentioned before are not rectilinear, but they are less than right angles, and so are acute (notwithstanding Proclus' remarks to the contrary). With these definition, we see another aspect of magnitudes, namely, two magnitudes of the same kind, such as two angles, can be compared for size. Euclid frequently uses what is known as the law of trichotomy: given two magnitudes F and G of the same kind, exactly one of the following three situations hold, F is less than G, F equals G, or F is greater than G. See the comments after the Common Notions for more discussion on magnitudes and the law of trichotomy. Next definition: I.Def.13-14 Previous: I.Def.10 Book I introduction © 1996 D.E.Joyce Clark University Definitions 13 and 14 Def. 13. A boundary is that which is an extremity of anything. Def. 14. A figure is that which is contained by any boundary or boundaries. These are rather nebulous definitions since they are based on the undefined terms "extremity" and "contained by." Euclid deals with two kinds of figures in the Elements: plane figures and solid figures. Plane figures are defined in the upcoming definitions: circles and semicircles in I.Def.15 and I.Def.18, rectilinear figures in I.Def.19 and particular kinds of rectilinear figures such as triangles and quadrilaterals following that. Specific solid figures such as spheres, cones, pyramids, and various polyhedra are defined in Book XI. Plane figures are not solid figures since they are not contained by any boundaries in space. Thus, implicit to the concept of figure is the ambient plane or space of the figure. Extremities, boundaries, and topology Euclid deals with three kinds of extremities, or boundaries. There are the ends of lines (I.Def.3), the edges of surfaces (I.Def.3), and the surfaces of solids (XI.Def.2). A finite line has two points as its boundaries. A circle is defined in I.Def.15 as is a plane figure and has its circumference as its boundary. A sphere is defined in XI.Def.14 as a solid figure and has a spherical surface as its boundary. The modern subject of topology studies space in a different way than geometry does. The geometric concepts of straightness, distance, and angle are excluded from topology, but the concept of boundary is central to topology. In topology, a sphere remains a sphere even when it's squeezed or stretched. Not everything has a boundary. For instance, the circumference of a circle has no boundary. Also a spherical surface has no boundary. In topology, a finite region with no boundary is called a cycle. Circles and spherical surfaces are cycles. In general, if something is a boundary, it has no boundary itself. So boundaries are cycles. But not all cycles are boundaries. Topology uses observation to distinguish various spaces. For instance, on a spherical surface, every circle is the boundary of a region on that surface. But on a toroidal surface (rotate a circle around a line in the plane of the circle that doesn't meet the circle), there are circles (for instance, that circle mentioned parenthetically) that don't bound any region on the surface. Thus, spherical surfaces are topologically different from toroidal surfaces. Figures and their boundaries The definition of figure needs to be fleshed out. In order to be a figure, a region must be bounded, that is, held in by a boundary. For instance, an infinite plane is unbounded, so it is not intended to be a figure. Neither is the region between two parallel lines even though that region has the two parallel lines as its extremities. Other figures may be considered if other ambient spaces are allowed, although Euclid only uses plane and solid figures. For a one-dimensional example, a line segment could be considered to be a figure in an infinite line with its endpoints as its boundary. Also, a hemisphere could be considered to be a figure on the surface of a sphere with the equator as its boundary. Next definition: I.Def.15-18 Previous: I.Def.11-12 Book I introduction © 1996 D.E.Joyce Clark University Definitions 15, 16, 17, and 18 Def. 15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another. Def. 16. And the point is called the center of the circle. Def. 17. A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle. Def. 18. A semicircle is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle. A definition such as this describes what circles are. Definitions do not guarantee the existence of the things they define. The existence of circles follows from a postulate, namely, Post.3. Note that a circle for Euclid is a two-dimensional figure. But in modern mathematics, usually the word "circle" refers to what Euclid calls the circumference of a circle. The center of the circle in the diagram is the point C. It's interesting that the English word "center" derives from the Greek word which also means a prod or a poker, and it refers to the fixed leg of a compass used to draw a circle. There is no assumption in the definition that there is only one center of a circle. Proposition III.1 gives a construction for the center of a circle, and the proof of that proposition shows that the center is unique. The (curved) line ABD that contains the circle is its circumference. Euclid typically names a circle by three points on its circumference. Perhaps a better translation than "circumference" would be "periphery" since that is the Greek word while "circumference" derives from the Latin. Euclid doesn't have a term for "radius" other than "that from the center," but "radius" is such a useful word that it is used here to translate "that from the center," such as the radius CD. An example diameter is the line AB which passes through the center. Of course, a diameter is twice a radius, and since the radii are all equal to each other by definition, the diameters also all equal to each other. That diameters "also bisects the circle" should not be part of the definition, but either assumed as a postulate or proved as a proposition. It depends on the fact that circles are drawn on planes, and planes have constant curvature. The analogous figure on a surface of nonconstant curvature does not have this property. For such figures the two "semicircles" on either side of a "diameter" need not be equal. Although circles are used throughout Book I, the proper theory of circles doesn't begin until Book III. That book begins with more definitions relating to circles including the equality of circles, when circles touch (are tangent to) lines and other circles, and so forth. Next definition: I.Def.19 Previous: I.Def.13-14 Book I introduction © 1996, 1997, 2003 D.E.Joyce Clark University Definition 19 Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines. Euclid classifies rectilinear figures by their number of sides in this definition. Classifying them by their number of angles could lead to complications since an angle has to be less than two right angles, and a non-convex figure would have an internal angle greater than two right angles. The modern English names, however, are based an the number of angles (except quadrilateral): triangle, pentagon, hexagon, heptagon, octagon, etc. From pentagon on up these names derive from the Greek, but they're rarely used past octagon. Next proposition: I.Def.20-21 Previous: I.Def.15-18 Book I introduction © 1996 D.E.Joyce Clark University Definitions 20 and 21 Def. 20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal. Def. 21. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute. Definition 20 classifies triangles by their symmetries, while definition 21 classifies them by the kinds of angles they contain. The scalene triangle C has no symmetries, but the isosceles triangle B has a bilateral symmetry. The equilateral triangle A not only has three bilateral symmetries, but also has 120°-rotational symmetries. As defined by Euclid, an equilateral triangle is not to be considered as an isosceles triangle, but in modern terminology, it is usually the case that equilateral triangles are included among the isosceles triangles, that is, it is only required that at least two sides be equal in order for a triangle to be isosceles. Generally speaking, modern definitions are inclusive whereas Euclid's definitions are usually exclusive. Equilateral triangles are constructed in the very first proposition of the Elements, I.1. An alternate characterization of isosceles triangles, namely that their base angles are equal, is demonstrated in propositions I.5 and I.6. Since triangle D has a right angle, it is a right triangle. Proposition I.17 states that the sum of any two angles in a triangle is less than two right angles, therefore, no triangle can contain more than one right angle. Furthermore, there can be at most one obtuse angle, and a right angle and an obtuse angle cannot occur in the same triangle. Triangle E is an obtuse triangle since it has an obtuse angle, while triangle F is an acute triangle since all its angles are acute. Next definition: I.Def.22 Previous: I.Def.19 Book I introduction © 1996 D.E.Joyce Clark University Definition 22 Of quadrilateral figures, a square is that which is both equilateral and rightangled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia. The figure A is, of course, a square. Figure B is an oblong, or a rectangle. Figure C is a rhombus. Figure D is a trapezium (sometimes called a trapeze or trapezoid). And figure E is a parallelogram. The only figure defined here that Euclid actually uses is the square. The other names of figures may have been common at the time of Euclid's writing, or they may have been left over from earlier authors' versions of the Elements. Euclid makes much use of parallelogram, or parallelogrammic area, which he does not define, but clearly means quadrilateral with parallel opposite sides. Parallelograms include rhombi and rhomboids as special cases. And rather than oblong, he uses rectangle, or rectangular parallelogram, which includes both squares and oblongs. Squares and oblongs are defined to be "right-angled." Of course, that is intended to mean that all four angles are right angles. Sometimes Euclid's definitions are too brief, but the intended meaning can easily be determined from the way the definitions are used. In particular, proposition I.46 constructs a square, and all four angles are constructed to be right, not just one of them. Next definition: I.Def.23 Previous: I.Def.20-21 Book I introduction © 1996 D.E.Joyce Clark University Definition 23 Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction. This definition only defines what it means for straight lines to be parallel; it does not say that there are any parallel lines. Proposition I.31 gives a construction for a line parallel to a given line through a given point. Next proposition: I.Post.1 Previous: I.Def.22 Book I introduction © 1996 D.E.Joyce Clark University Postulate 1 To draw a straight line from any point to any point. This first postulate says that given any two points such as A and B, there is a line AB which has them as endpoints. This is one of the constructions that may be done with a straightedge (the other being described in the next postulate). Although it doesn't explicitly say so, there is a unique line between the two points. Since Euclid uses this postulate as if it includes the uniqueness as part of it, he really ought to have stated the uniqueness explicitly. The last three books of the Elements cover solid geometry, and for those, the two points mentioned in the postulate may be any two points in space. Proposition XI.1 claims that if part of a line is contained in a plane, then the whole line is. In the books on plane geometry, it is implicitly assumed that the line AB joining A to B lies in the plane of discussion. Next postulate: I.Post.2 Previous: I.Def.23 Book I introduction © 1996 D.E.Joyce Clark University Postulate 2 To produce a finite straight line continuously in a straight line. Here we have the second ability of a straightedge, namely, to extend a given line AB to CD. This postulate does not say how far a line can be extended. Sometimes it is used so that the extension equals some other line. Other times it is extended arbitrarily far. As with the first postulate, it is implicitly assumed in the books on plane geometry that when a line is extended, it remains in the plane of discussion. The first proposition on solid geometry, proposition XI.1, claims that line can't be only partly in a plane. The central step in the proof of that proposition is to show that a line cannot be extended in two ways, that is, there is only one continuation of a line. The proof is hardly convincing. Rather, this postulate should include a clause to that effect. Neusis: fitting a line into a diagram Other uses of a straightedge can be imagined. For instance, it might be marked at two points on it, then fit into a diagram so that the two points fall on two lines, perhaps curved. This operation is an example of "neusis" or "verging" where lines are adjusted to fit the diagram. For instance, Archimedes, who lived in the century after Euclid, used neusis in several constructions in his work On Spirals. In the Book of Lemmas, attributed by Thabit ibn-Qurra to Archimedes, neusis is used to trisect an angle. Suppose the angle ABC is to be trisected. Draw a circle DEF with center B and any radius. Extend CB through D and beyond. Fit in a line GHE passing through E and a point G on the line CB extended so that a segment (colored green) equal to the radius BD of the circle starts at G and ends at a point H on the circle. (You'll have to move G around until H lands on the circle.) Draw BH. With the help of Euclid's propositions here in Book I, we can show that angle EGC is one-third of angle ABC. Since the lines GH, HB, and BE are equal, therefore the triangles GHB and HBE are isosceles. Therefore, by I.5, angle HGB equals HBG, and angle BHE equals angle BEH. By I.32, the exterior angle BHE of triangle GHB equals the sum of the equal angles HGB and HBG, therefore angle BHE is double angle HGB. And angle BEH equals BHE, so it is also double angle HGB. Again by I.32, the exterior angle ABC of triangle BEG is the sum of angles HGB and BEH. But angle BEH is double angle HGB, therefore angle ABC is triple angle GHB. Therefore, angle GHB is one-third of angle ABC. Q.E.D. The ancient Greek geometers believed that angle trisection required tools beyond those given in Euclid's postulates. They were right, but it wasn't proved until Wantzel in 1837 proved that a 60°angle cannot be so trisected using only Euclidean tools. Euclid has no postulate for neusis constructions, and since neusis constructions can trisect angles, we conclude from Wantzel's theorem that another postulate is required to justify neusis constructions. Next postulate: I.Post.3 Previous: I.Post.1 Book I introduction © 1996 D.E.Joyce Clark University Postulate 3 To describe a circle with any center and radius. This is the third assumed construction in the Elements. It corresponds to drawing a circle with a compass. Circles were defined in Def.I.15 and Def.I.16 as plane figures with the property that there is a certain point, called the center of the circle, such that all straight lines from the center to the boundary are equal. That is, all the radii are equal. The given data are (1) a point A to be the center of the circle, (2) another point B to be on the circumference of the circle, and (3) a plane in which the two points lie. In the first few books of the Elements, there is but one plane under consideration and needn't be mentioned, but in the last three books which develop solid geometry, the plane has to be specified. Note that this postulate does not allow for the compass to be moved. The usual way that a compass is used is that is is opened to a given width, then the pivot is placed on the drawing surface, then a circle is drawn as the compass is rotated around the pivot. But this postulate does not allow for transferring distances. It is as if the compass collapses as soon as it's removed from the plane. Proposition I.3, however, gives a construction for transferring distances. Therefore, the same constructions that can be made with a regular compass can also be made with Euclid's collapsing compass. Next postulate: I.Post.4 Previous: I.Post.2 Book I introduction © 1996 D.E.Joyce Clark University Postulate 4 That all right angles equal one another. In the definition of right angle, it is clear that the two angles at the foot of a perpendicular, such as angles ACD and BCD, are equal. This postulate says that an angle at the foot of one perpendicular, such as angle ACD, equals an angle at the foot of any other perpendicular, such as angle EGH. Next postulate: I.Post.5 Previous: I.Post.3 Book I introduction © 1996 D.E.Joyce Clark University Postulate 5 That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. Of course, this is a postulate for plane geometry. It should include the condition that the two straight lines lie in a plane, otherwise, skew lines in space would satisfy the hypotheses. Also, without an ambient plane, the term "that side [of the straight line]" has no meaning. In the diagram, if angle ABE plus angle BED is less than two right angles (180°), then lines AC and DF will meet when extended in the direction of A and D. This postulate is usually called the "parallel postulate" since it can be used to prove properties of parallel lines. Euclid develops the theory of parallel lines in propositions through I.31. The parallel postulate is historically the most interesting postulate. Geometers throughout the ages have tried to show that it could be proved from the remaining postulates so that it wasn't necessary to assume it. The process tried was to assume its falsehood, then derive a contradiction. Many strange conclusions follow from denying the parallel postulate, and several geometers found such great absurdities that they concluded that the parallel postulate did follow from the rest. Nevertheless, these apparent absurdities are not contradictions. In the early nineteenth century, Bolyai, Lobachevsky, and Gauss found ways of dealing with this "non-Euclidean" geometry by means of analysis and accepted it as a valid kind of geometry, although very different from Euclidean geometry. This hyperbolic geometry, as it is called, is just as consistent as Euclidean geometry and has many uses. Thus, we know now that we must include the parallel postulate to derive Euclidean geometry. For more on noneuclidean geometries, see the notes on hyperbolic geometry after I.29 and elliptic geometry after I.16. Euclid does not use this parallel postulate until Proposition I.29, but nearly all of the rest of Book I depends on it. For more commentary about the postulate see the Guides to I.29 and I.30. Next proposition: I.1 Previous: I.Post.4 Book I introduction © 1996, 2003 D.E.Joyce Clark University Proposition 1 To construct an equilateral triangle on a given finite straight line. Let AB be the given finite straight line. It is required to construct an equilateral triangle on the straight line AB. Describe the circle BCD with center A and radius AB. Again describe the circle ACE with center B and radius BA. Join the straight lines CA and CB from the point C at which the circles cut one another to the points A and B. Post.3 Post.1 Now, since the point A is the center of the circle CDB, therefore AC equals AB. Again, since the point B is the center of the circle CAE, therefore BC equals BA. I.Def.15 But AC was proved equal to AB, therefore each of the straight lines AC and BC equals AB. And things which equal the same thing also equal one another, therefore AC also equals BC. C.N.1 Therefore the three straight lines AC, AB, and BC equal one another. Therefore the triangle ABC is equilateral, and it has been constructed on the given finite straight line AB. I.Def.20 Q.E.F. This proposition is a very pleasant choice for the first proposition in the Elements. The construction of the triangle is clear, and the proof that it is an equilateral triangle is evident. Of course, there are two choices for the point C, but either one will do. Euclid could have chosen proposition I.4 to come first, since it doesn't logically depend on the previous three, but there are some good reasons for putting I.1 first. For one thing, the Elements ends with constructions of the five regular solids in Book XIII, so it is a nice aesthetic touch to begin with the construction of a regular triangle. More important, though, is I.1 is needed in I.2, and that in I.3. Propositions I.2 and I.3 give constructions for moving lines, and I.4, although not logically dependent on I.2 or I.3, does use the concept of superposition which involves, in some sense, moving points and lines. Marginal references to postulates, definitions, etc. The abbreviations in the right column refer to postulates, definitions, common notions, and previously proved propositions. Each indicates a justification of a construction or conclusion in a sentence to its left. They are not part of Euclid's Elements, but it is a tradition to include them as a guide to the reader. Sometimes the justification is quoted as C.N.1 is quoted here, but usually it is left to the reader to determine the justification. Q.E.F. and Q.E.D. at the ends of proofs The Q.E.F. at the end of the proof is an abbreviation for the Latin words "quod erat faciendum" which means "which was to be done." A few of the propositions, as this one and the next two, solve problems by constructions. These are the ones that end with Q.E.F. (They're also printed in red here in the listings of propositions for each book.) The rest of the proofs end with Q.E.D. instead, an abbreviation for "quod erat demonstrandum" which means "which was to be demonstrated." It's convenient to have a standard way to indicate the end of a proof. These Latin abbreviations are a bit of an anachronism. It would be less of an anachronism to use abbreviations for the original Greek phrase, or abbreviations for a modern English phrase since the rest of this version of the Elements is in English. But by now, Q.E.F. and Q.E.D. are traditional. In recent decades a small square has become common as a symbol to indicate the end of a proof. Critiques of the proof It is surprising that such a short, clear, and understandable proof can be so full of holes. These are logical gaps where statements are made with insufficient justification. Having the first proof in the Elements this proposition has probably received more criticism over the centuries than any other. Why does the point C exist? Near the beginning of the proof, the point C is mentioned where the circles are supposed to intersect, but there is no justification for its existence. The only one of Euclid's postulate that says a point exists the parallel postulate, and that postulate is not relevant here. Thus, there is no assurance that the point C actually exists. Indeed, there are models of geometry in which the circles do not intersect. Thus, other postulates not mentioned by Euclid are required. In Book III, Euclid takes some care in analyzing the possible ways that circles can meet, but even with more care, there are missing postulates. Why is ABC a plane figure? After concluding the three straight lines AC, AB, and BC are equal, what is the justification that they contain a plane figure ABC? Recall that a triangle is a plane figure bounded by contained by three lines. These lines have not been shown to lie in a plane and that the entire figure lies in a plane. It is proposition XI.1 that claims that all parts of a line lie in a plane, and XI.2 that claims that the entire triangle lie in a plane. Logically, they should precede I.1. The reason they don't, of course, is that those propositions belong to solid geometry, and plane geometry is developed first in the Elements, also, no doubt, plane geometry developed first historically. Why does ABC contain an equilateral triangle? Proclus relates that early on there were critiques of the proof and describes that of Zeno of Sidon, an Epicurean philosopher of the early first century B.C.E. (not to be confused with Zeno of Elea famous of the paradoxes who lived long before Euclid), and whose criticisms, Proclus says, were refuted in a book by Posidonius. The critique is sound, however, and the refutation faulty. Zeno of Sidon criticized the proof because it was not shown that the sides do not meet before they reach the vertices. Suppose AC and BC meet at E before they reach C, that is, the straight lines AEC and BEC have a common segment EC. Then they would contain a triangle ABE which is not equilateral, but isosceles. Zeno recognized that in order to destroy his counterexample it was necessary to assume that straight lines cannot have a common segment. Proclus relates a supposed proof of that statement, the same one found in proposition XI.1, but it is faulty. Proclus and Posidonius quoted properties of lines and circles that were never proven and never explicitly assumed as postulates. The possibilities that haven't been excluded are much more numerous than Zeno's example. The sides could meet numerous times and the region they contain could look like a necklace of bubbles. What needs to be shown (or assumed as a postulate) is that two infinitely extended straight lines can meet in at most one point. Use of Proposition 1 The construction in this proposition is directly used in propositions I.2, I.9, I.10, I.11, XI.11, and XI.22. Next proposition: I.2 Previous: I.Post.5 Book I introduction © 1996, 1997 D.E.Joyce Clark University Proposition 2 To place a straight line equal to a given straight line with one end at a given point. Let A be the given point, and BC the given straight line. It is required to place a straight line equal to the given straight line BC with one end at the point A. Join the straight line AB from the point A to the point B, and construct the equilateral triangle DAB on it. Post. 1 I.1. Produce the straight lines AE and BF in a straight line with DA and DB. Describe the circle CGH with center B and radius BC, and again, describe the circle GKL with center D and radius DG. Post.2 Post.3 Since the point B is the center of the circle CGH, therefore BC equals BG. Again, since the point D is the center of the circle GKL, therefore DL equals DG. I.Def.15 And in these DA equals DB, therefore the remainder AL equals the remainder BG. C.N.3 But BC was also proved equal to BG, therefore each of the straight lines AL and BC equals BG. And things which equal the same thing also equal one another, therefore AL also equals BC. C.N.1 Therefore the straight line AL equal to the given straight line BC has been placed with one end at the given point A. Q.E.F. This is a very clever construction to solve what seems to be a simple problem. One would like simply to slide the line BC along so that one end coincides with the point A. But there is no motion in the geometry of Euclid. There is something like motion used in proposition I.4, but nothing is actually moved there. The only basic constructions that Euclid allows are those described in Postulates 1, 2, and 3. Euclid then builds new constructions (such as the one in this proposition) out of previously described constructions. So at this point, the only constructions available are those of the three postulates and the construction in proposition I.1, and Euclid uses all four here. Another, different, expectation is that one might use a compass to transfer the distance BC over to the point A. It is clear from Euclid's use of postulate 3 that the point to be used for the center and a point that will be on the circumference must be constructed before applying the postulate; postulate 3 is not used to transfer distance. Sometimes postulate 3 is likened to a collapsing compass, that is, when the compass is lifted off the drawing surface, it collapses. It could well be that in some earlier Greek geometric theory abstracted compasses that could transfer distances. If that speculation is correct, then this proposition would be a late addition to the theory. The construction of the proposition allows a weaker postulate (namely postulate 3) to be assumed. Construction steps When using a compass and a straightedge to perform this construction there are more circles drawn than shown in the diagram that accompanies the proposition. These are the two circles needed to construct the equilateral triangle ABD. One side, AB, of that triangle isn't necessary for the construction. Altogether, four circles and two lines are required for this construction. Use of Proposition 2 The construction in this proposition is only used in Proposition I.3. Note that this constuction assumes that all the point A and the line BC lie in a plane. It may also be used in space, however, since Proposition XI.2 implies that A and BC do lie in a plane. Next proposition: I.3 Previous: I.1 Book I introduction © 1996 D.E.Joyce Clark University Proposition 3 To cut off from the greater of two given unequal straight lines a straight line equal to the less. Let AB and C be the two given unequal straight lines, and let AB be the greater of them. It is required to cut off from AB the greater a straight line equal to C the less. Place AD at the point A equal to the straight line C, and describe the circle DEF with center A and radius AD. I.2 Post. 3 Now, since the point A is the center of the circle DEF, therefore AE equals AD. I.Def.15 But C also equals AD, therefore each of the straight lines AE and C equals AD, so that AE also equals C. C.N.1 Therefore, given the two straight lines AB and C, AE has been cut off from AB the greater equal to C the less. Q.E.F. Now it is clear that the purpose of Proposition 2 is to effect the construction in this proposition. According to Proclus (410-485 C.E.) in his Commentary on Book I, Hippocrates of Chios (fl. ca. 430 B.C.E.) was the first to write an Elements. Leon and Theudius also wrote versions before Euclid (fl. ca. 295 B.C.E.). These other Elements have all been lost since Euclid's replaced them. It is conceivable that in some of these earlier versions the construction in proposition I.2 was not known, so this proposition would instead have been a postulate (a stronger version of Post.3). Once the construction in I.2 was discovered, the current weaker Post.3 would do. Then again, I.2 might go back to the time of Hippocrates. Construction steps This construction takes one more step beyond that of I.2, and that is the final circle, the circle shown in the diagram accompanying this proposition. Altogether, therefore, five circles and two lines are required for this construction. Frequently, though, one end of the line C is already placed at A, and then the construction of I.2 isn't required. In that case, only one circle needs to be drawn. Use of Proposition 3 This proposition begins the geometric arithmetic of lines. Explicitly, it allows lines to be subtracted, but it can also be used to compare lines for equality and to add lines, that is, one line can be placed alongside another to determine if they are equal, or if not, which is greater. In other words, this construction justifies the law of trichotomy for lines. The construction is use more often in the Elements than any other starting with proposition I.5. It is used in all the books on geometry, that is in Books I through IV, VI, and XI through XIII. Naturally a construction of this sort is needed in the solid geometry of Books XI through XIII. Surprisingly, the construction given here also works in solid geometry, even the lines AB and C don't lie in the same plane. Since the point A and the line C lie in one plane, the construction of I.2 produces a line AD equal to C in that plane. Now AD and AB also lie in one plane, but not the same one, and the circle AEF can be drawn there. Next proposition: I.4 Previous: I.2 Book I introduction © 1996 D.E.Joyce Clark University Proposition 5 In isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another. Let ABC be an isosceles triangle having the side AB equal to the side AC, and let the straight lines BD and CE be produced further in a straight line with AB and AC. I.Def.20 Post.2 I say that the angle ABC equals the angle ACB, and the angle CBD equals the angle BCE. Take an arbitrary point F on BD. Cut off AG from AE the greater equal to AF the less, and join the straight lines FC and GB. I.3. Post.1 Since AF equals AG, and AB equals AC, therefore the two sides FA and AC equal the two sides GA and AB, respectively, and they contain a common angle, the angle FAG. Therefore the base FC equals the base GB, the triangle AFC equals the triangle AGB, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides, that is, the angle ACF equals the angle ABG, and the angle AFC equals the angle AGB. I.4 Since the whole AF equals the whole AG, and in these AB equals AC, therefore the remainder BF equals the remainder CG. C.N.3 But FC was also proved equal to GB, therefore the two sides BF and FC equal the two sides CG and GB respectively, and the angle BFC equals the angle CGB, while the base BC is common to them. Therefore the triangle BFC also equals the triangle CGB, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides. Therefore the angle FBC equals the angle GCB, and the angle BCF equals the angle CBG. I.4 Accordingly, since the whole angle ABG was proved equal to the angle ACF, and in these the angle CBG equals the angle BCF, the remaining angle ABC equals the remaining angle ACB, and they are at the base of the triangle ABC. But the angle FBC was also proved equal to the angle GCB, and they are under the base. C.N.3 Therefore in isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another. Q.E.D. There are two conclusions for this proposition, first that the internal base angles ABC and ACB are equal, second that the external base angles FBC and GCB are equal. From the diagram it looks like it would be easy to prove the second conclusion from the first by simply subtracting the equal angles ABC and ACB the straight angles ABF and ACG, respectively. But Euclid doesn't accept straight angles, and even if he did, he hasn't proved that all straight angles are equal. Proposition I.13 would be enough, since it implies the sum of angles ABC and FBC equals two right angles, and the sum of angles ACB and GCB also equals two right angles, and so the two sums are equal effectively saying all straight angles are equal. Unfortunately, such an argument would be circular. I.13 depends on I.11, I.11 on I.8, I.8 on I.7, and I.7 on I.5. Thus, I.13 cannot be used in the proof of I.5. It may appear that I.7 only depends on the first conclusion of I.5, but a case of I.7 that Euclid does not discuss relies on the second conclusion of I.5. This proposition has been called the Pons Asinorum, or Asses' Bridge. Whether this name is due to its difficulty (which it isn't) or the resemblance of its figure to a bridge is not clear. Very few of the propositions in the Elements are known by names. Pappus' proof Pappus (fl. ca. 320 C.E.) gave a much shorter proof of the first conclusion, but it is also conceptually more difficult. The two triangles BAC and CAB have two sides equal to two sides, namely side BA of the first triangle equals side CA of the second triangle, and side AC of the first triangle equal to side AB of the second, and the contained angles are equal, namely angle BAC of the first triangle equals angle CAB of the second, therefore, by I.4, the corresponding parts of the two triangles are equal, in particular, the angle B in the first triangle equals the angle C of the second. The difficulty lies in treating one triangle as two, or in making a correspondence between a triangle and itself, but not the correspondence of identity. There is nothing wrong with this proof formally, but it might be more difficult for a student just learning geometry. Use of Proposition 5 This proposition is used in Book I for the proofs of several propositions starting with I.7 It is also used frequently in Books II, III, IV, VI, and XIII. Next proposition: I.6 Previous: I.4 Book I introduction © 1996, 1997 D.E.Joyce Clark University Proposition 6 If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another. Let ABC be a triangle having the angle ABC equal to the angle ACB. I say that the side AB also equals the side AC. If AB does not equal AC, then one of them is greater. C.N Let AB be greater. Cut off DB from AB the greater equal to AC the less, and join DC. I.3 Post.1 Since DB equals AC, and BC is common, therefore the two sides DB and BC equal the two sides AC and CB respectively, and the angle DBC equals the angle ACB. Therefore the base DC equals the base AB, and the triangle DBC equals the triangle ACB, the less equals the greater, which is absurd. Therefore AB is not unequal to AC, it therefore equals it. I.4 C.N.5 Therefore if in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another. Q.E.D. Converses of propositions This is the converse of (part of) the previous proposition I.5. Proposition I.6 says that if angle B equals angle C, then side AB equals side AC. Proposition I.5 says that if side AB equals side AC, then angle B equals angle C. In general, the converse of a proposition of the form "If P, then Q" is the proposition "If Q, then P." When both a proposition and its converse are valid, Euclid tends to prove the converse soon after the proposition, a practice that has continued to this day. A proposition and its converse are not logically equivalent. There are examples where "If P, then Q" is valid, but "If Q, then P" is not valid. An example from the Elements is proposition III.5 which states "If two circles cut one another, then they do not have the same center." The converse would be "If two circles do not have the same center, then they cut one another" which is certainly not valid since if one circle lies entirely outside the other, then they don't have the same center. Proofs by contradiction This is the first "proof by contradiction," also called "reductio ad absurdum," in the Elements. In this proof, in order to prove AB equals AC, Euclid assumes they are unequal and derives a contradiction, namely, that the triangle ACB equals a part of itself, triangle DBC, which contradicts C.N.5, the whole is greater than the part. The contradiction is that triangle ACB both equals and does not equal triangle DBC. In general, to prove a statement of the form "P" with a proof by contradiction, begin with an assumption "not P" and derive some contradiction "Q and not Q," and finally conclude "P." Euclid often uses proofs by contradiction, but he does not use them to conclude the existence of geometric objects. That is, he does not use them in constructions. But he does use them to show what has been constructed is correct. In modern mathematics nonconstructive proofs by contradiction do occur. Famous examples are Brouwer's fixed point theorems published in 1912. One of these states that any continuous transformation f of a circle (circular disk) to itself has a fixed point x, that is, a point such that f(x) = x. In his proof, he assumed that such a point did not exist and derived a contradiction. Although his proof is logically correct, he was not satisfied since the proof does not help in constructing a fixed point. Brouwer was an adherent of a philosophy of mathematics called "intuitionism" that holds, among other things, that mathematical objects have not been shown to exist until constructions have been given for them. The law of trichotomy in practice The proof uses the law of trichotomy for lines. "If AB does not equal AC, then one of them is greater." There are three cases: AB < AC, AB = AC, or AB > AC. If the middle possibility is excluded, then only the two others remain, so one of the lines is greater. The law of trichotomy is not explicitly stated as a Common Notion, but it is the sort of property of magnitudes listed as Common Notions. Proposition I.3 can be read as a construction to determine whether one line is less than, equal to, or greater than another. Using I.3, one line is laid along another, and it will fall short, fall equal, or extend beyond the other. For this proposition I.6, the construction simplifies since the two lines AB and AC already have one end in common. The other part of the law of trichotomy is also used in the proof, the part that says only one of the three cases can occur. "... the triangle DBC equals the triangle ACB, the less equals the greater, which is absurd." C.N.5, the whole is greater than the part, allows the conclusion that triangle DBC (the part) is less than triangle ACB (the whole). But the contradiction arises because only one of the two cases DBC = ACB and DBC < ACB can occur. Use of Proposition 6 This proposition is not used in the proofs of any of the later propositions in Book I, but it is used in Books II, III, IV, VI, and XIII. Next proposition: I.7 Previous: I.5 Book I introduction © 1996, 1997 D.E.Joyce Clark University Proposition 7 Given two straight lines constructed from the ends of a straight line and meeting in a point, there cannot be constructed from the ends of the same straight line, and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each equal to that from the same end. If possible, given two straight lines AC and CB constructed on the straight line AB and meeting at the point C, let two other straight lines AD and DB be constructed on the same straight line AB, on the same side of it, meeting in another point D and equal to the former two respectively, namely each equal to that from the same end, so that AC equals AD which has the same end A, and CB equals DB which has the same end B. Join CD. Post.1 Since AC equals AD, therefore the angle ACD equals the angle ADC. Therefore the angle ADC is greater than the angle DCB. Therefore the angle CDB is much greater than the angle DCB. I.5 C.N.5 C.N. Again, since CB equals DB, therefore the angle CDB also equals the angle DCB. But it was also proved much greater than it, which is impossible. I.5 C.N. Therefore given two straight lines constructed from the ends of a straight line and meeting in a point, there cannot be constructed from the ends of the same straight line, and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each equal to that from the same end. Q.E.D. In order to conclude "the angle ADC is greater than the angle DCB" it is necessary for angle ADC to be greater than angle DCB, but that won't happen unless the point D lies outside the triangle ABC. Euclid hasn't considered the case when D lies inside triangle ABC as well as other special cases. This is not unusual as Euclid frequently treats only one case. Commentators over the centuries have inserted other cases in this and other propositions. It is usually easy to modify Euclid's proof for the remaining cases. In this proposition for the case when D lies inside triangle ABC, the second conclusion of I.5 may be used to justify the proof. Hidden justifications The sentences ... the angle ACDequals the angle ADC.Therefore the angle ADCis greater than the angle DCB.Therefore the angle CDBis much greater than the angle DCB. use several properties of magnitudes. C.N.5 justifies the unstated angle ACD > DCB since DCB is part of ACD. The statement that ADC is greater than the angle DCB is justified by the property of magnitudes If x < y and y = z, then x < z. This property is not among the listed Common Notions. Next, transitivity of "less than" If x < y and y < z, then x < z. justifies the last statement "CDB is much greater than the angle DCB." Transitivity is another property not listed as a Common Notion. As in the proof of the last proposition and many to come, the law of trichotomy is also used. Here it's used to reach the final contradiction. Use of Proposition 7 This proposition is used in the proof of the next proposition. Next proposition: I.8 Previous: I.6 Book I introduction © 1996, 1997 D.E.Joyce Clark University Proposition 9 To bisect a given rectilinear angle. Let the angle BAC be the given rectilinear angle. It is required to bisect it. Take an arbitrary point D on AB. Cut off AE from AC equal to AD, and join DE. Construct the equilateral triangle DEF on DE, and join AF. I.3 Post.1 I.1 I say that the angle BAC is bisected by the straight line AF. Since AD equals AE, and AF is common, therefore the two sides AD and AF equal the two sides EA and AF respectively. And the base DF equals the base EF, therefore the angle DAF equals the angle EAF. I.Def.20 I.8 Therefore the given rectilinear angle BAC is bisected by the straight line AF. Q.E.F. Construction steps When using a compass and a straightedge to perform this construction, three circles and the final bisecting line need to be drawn. One circle with center A and radius AD is needed to determine the point E. The other two circles with centers at D and E and common radius DE intersect to give the point F. The sides of the equilateral triangle aren't needed for the construction. There is an alternate construction where the circles centered at D and E have a different radius, namely, AD, which equals AE. A different proof is required to show that this alternate construction works. On angle trisection Angle bisection is an easy construction to make using Euclidean tools of straightedge and compass. Also, line bisection is quite easy (see the next proposition I.10), and division of a line into any number of equal parts is not especially difficult (see proposition VI.9). Dividing an angle into an odd number of equal parts is not so easy, in fact, it is impossible to trisect a 60°-angle using Euclidean tools (the Postulates 1 through 3). Euclid's predecessors employed a variety higher curves for this purpose. Archimedes, after Euclid, created two constructions: his spiral could divide an angle into any number of parts, and his neusis construction could trisect angles (see the note on Post.2). By Pappus' time it was believed that angle trisection was not possible using Euclidean tools, but that wasn't proven until 1837 when Wantzel published his proof. Nevertheless, amateur geometers continue to search in vain for such a construction and frequently bother mathematicians with their purported solutions. Their solutions are of two forms. Sometimes they simply construct approximate trisections. Other times they use neusis or some other other tool that goes beyond Euclid's tools. Students of geometry are cautioned not to waste their time on this problem and, if they do, not to bother others with their purported solutions. Much better would be to study Galois theory, the mathematics that proves the impossibility of angle trisection. Use of Proposition 9 The construction of this proposition is used in the next one and a few propositions in Books IV, VI, and XIII. Next proposition: I.10 Previous: I.8 Book I introduction © 1996 D.E.Joyce Clark University Proposition 10 To bisect a given finite straight line. Let AB be the given finite straight line. It is required to bisect the finite straight line AB. Construct the equilateral triangle ABC on it, and bisect the angle ACB by the straight line CD. I.1 I.9 I say that the straight line AB is bisected at the point D. Since CA equals CB, and CD is common, therefore the two sides CA and CD equal the two sides CB and CD respectively, and the angle ACD equals the angle BCD, therefore the base AD equals the base BD. I.Def.20 I.4 Therefore the given straight line AB is bisected at D. Q.E.F. While this construction divides a line into two equal parts, the construction in proposition VI.9 divides a line into any given number of equal parts. Construction steps This method for bisecting lines takes less actual work than it appears to. It is really no more than the double-equilateral-triangle. First, the equilateral triangle ABC needs to be constructed. According to I.1 two circles need to be drawn: one with center A and radius AB, the other with center B and radius BA. One of the points of intersection of the two circles is C. Then to bisect angle ACB, according to I.9, an arbitrary point is chosen on one side of the angle, and it might as well be the point A on the side AC, and a point equally far from C on the side BC, which is, of course, B. Then an equilateral triangle is constructed on the line AB. There are two such equilateral triangles, the one already constructed ACB, and another one, call it AEB. The point E is the other intersection of the two circles already drawn. Then, by I.9, the line CE bisects the angle ACB, and according to this proposition, the point D bisects the line AB. Actually, only two circles and the straight line CE need to be drawn. The straight lines AC, CB, AE, and EB, aren't necessary for the construction; they are only used to show that the construction is correct. Use of Proposition 10 The construction of this proposition in Book I is used in propositions I.12, I.16, and I.42. It is also used in several propositions in the Books II, III, IV, X, and XIII. Next proposition: I.11 Previous: I.9 Book I introduction © 1996 D.E.Joyce Clark University Proposition 11 To draw a straight line at right angles to a given straight line from a given point on it. Let AB be the given straight line, and C the given point on it. It is required to draw a straight line at right angles to the straight line AB from the point C. Take an arbitrary point D on AC. Make CE equal to CD. Construct the equilateral triangle FDE on DE, and join CF. I.3 I.1 Post.1 I say that the straight line CF has been drawn at right angles to the given straight line AB from C the given point on it. Since CD equals CE, and CF is common, therefore the two sides CD and CF equal the two sides CE and CF respectively, and the base DF equals the base EF. Therefore the angle DCF equals the angle ECF, and they are adjacent angles. I.Def.20 I.8 But, when a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, therefore each of the angles DCF and FCE is right. I.Def.10 Therefore the straight line CF has been drawn at right angles to the given straight line AB from the given point C on it. Q.E.F. This and the next proposition both construct a perpendicular to a line through a given point. The difference is that the given point lies on the line in this proposition but doesn't in the next. Construction steps The actual construction here is the same doubleequilateral-triangle construction of the previous proposition that is used to bisect the line DE, except that it is preceded by the selection of points D and E on AB equidistant from C. This construction actually only requires drawing three circles and the one line FG. Use of Proposition 11 Thia construction is used in propositions I.13, I.46, I.48, and numerous propositions in Books II, III, VI, VI, XI, XII, and XIII. Next proposition: I.12 Previous: I.10 Book I introduction © 1996 D.E.Joyce Clark University Proposition 12 To draw a straight line perpendicular to a given infinite straight line from a given point not on it. Let AB be the given infinite straight line, and C the given point which is not on it. It is required to draw a straight line perpendicular to the given infinite straight line AB from the given point C which is not on it. Take an arbitrary point D on the other side of the straight line AB, and describe the circle EFG with center C and radius CD. Bisect the straight line EG at H, and join the straight lines CG, CH, and CE. Post.3 I.10 Post.1 I say that CH has been drawn perpendicular to the given infinite straight line AB from the given point C which is not on it. Since GH equals HE, and HC is common, therefore the two sides GH and HC equal the two sides EH and HC respectively, and the base CG equals the base CE. Therefore the angle CHG equals the angle EHC, and they are adjacent angles. I.Def.15 I.8 But, when a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands. I.Def.10 Therefore CH has been drawn perpendicular to the given infinite straight line AB from the given point C which is not on it. Q.E.F. Again, the double-equilateral-triangle construction is used, but this time the preparation of the starting line EG is different. The point D is taken on the other side of the line AB to insure that circle meets the line AB in at least two points, E and G. If D is taken on the line AB, it might be taken at H, and the resulting circle would touch the line only at H; and if D is taken on the same side of AB, then the circle could miss the line entirely. Euclid does not precede this proposition with propositions investigating how lines meet circles. He is much more careful in Book III on circles in which the first dozen or so propositions lay foundations. For instance, Proposition III.10 states that a circle does not cut a circle at more than two points. Even so, some propositions are missing. One is needed for this proposition to justify the existence of the two points C and E where the line AB meets circle with center C and radius CD. Such a proposition would state "A circle whose center is on one side of a line and on whose circumference lies a point on the other side of the line meets the line at two points." Incidentally, Proclus explains in his commentary on Book I that the problem of constructing the perpendicular was investigated by Oenopides of Chios who lived sometime in the middle of the fifth century B.C.E., a century and a half before Euclid. Use of Proposition 12 The construction of this proposition is not used in Book I, but it is used on occasion in the remaining geometric books, namely, Books II through IV, VI, and XI through XIII. Next proposition: I.13 Previous: I.11 Book I introduction © 1996 D.E.Joyce Clark University Proposition 13 If a straight line stands on a straight line, then it makes either two right angles or angles whose sum equals two right angles. Let any straight line AB standing on the straight line CD make the angles CBA and ABD. I say that either the angles CBA and ABD are two right angles or their sum equals two right angles. Now, if the angle CBA equals the angle ABD, then they are two right angles. I.Def.10 But, if not, draw BE from the point B at right angles to CD. Therefore the angles CBE and EBD are two right angles. I.11 Since the angle CBE equals the sum of the two angles CBA and ABE, add the angle EBD to each, therefore the sum of the angles CBE and EBD equals the sum of the three angles CBA, ABE, and EBD. C.N.2 Again, since the angle DBA equals the sum of the two angles DBE and EBA, add the angle ABC to each, therefore the sum of the angles DBA and ABC equals the sum of the three angles DBE, EBA, and ABC. C.N.2 But the sum of the angles CBE and EBD was also proved equal to the sum of the same three angles, and things which equal the same thing also equal one another, therefore the sum of the angles CBE and EBD also equals the sum of the angles DBA and ABC. But the angles CBE and EBD are two right angles, therefore the sum of the angles DBA and ABC also equals two right angles. C.N.1 Therefore if a straight line stands on a straight line, then it makes either two right angles or angles whose sum equals two right angles. Q.E.D. With this proposition, we begin to see what the arithmetic of magnitudes means to Euclid, in particular, how to add angles. Euclid says that the angle CBE equals the sum of the two angles CBA and ABE. So, one way a sum of angles occurs is when the two angles have a common vertex (B in this case) and a common side (BA in this case), and the angles lie on opposite sides of their common side. Thus, addition of angles can be performed by joining adjacent angles. But that's not the only addition that occurs here. Euclid also says that the sum of the angles CBE and EBD equals the sum of the three angles CBA, ABE, and EBD. That sum being mentioned is a straight angle, which is not to be considered as an angle according to Euclid. It is a formal sum equal to two right angles. In other propositions formal sums of four right angles occur. These and larger formal sums are not angles themselves, merely sums of angles. Only if an angle sum is less than two right angles can it be identified with a single angle. Use of Proposition 13 This proposition is used in the proofs of the next two propositions and several others in this book as well as a few propositions in Books IV and VI. Next proposition: I.14 Previous: I.12 Book I introduction © 1996 D.E.Joyce Clark University Proposition 14 If with any straight line, and at a point on it, two straight lines not lying on the same side make the sum of the adjacent angles equal to two right angles, then the two straight lines are in a straight line with one another. With any straight line AB, and at the point B on it, let the two straight lines BC and BD not lying on the same side make the sum of the adjacent angles ABC and ABD equal to two right angles. I say that BD is in a straight line with CB. If BD is not in a straight line with BC, then produce BE in a straight line with CB. Post.2 Since the straight line AB stands on the straight line CBE, therefore the sum of the angles ABC and ABE equals two right angles. But the sum of the angles ABC and ABD also equals two right angles, therefore the sum of the angles CBA and ABE equals the sum of the angles CBA and ABD. I.13 Post.4 C.N.1 Subtract the angle CBA from each. Then the remaining angle ABE equals the remaining angle ABD, the less equals the greater, which is impossible. Therefore BE is not in a straight line with CB. C.N.3 Similarly we can prove that neither is any other straight line except BD. Therefore CB is in a straight line with BD. Therefore if with any straight line, and at a point on it, two straight lines not lying on the same side make the sum of the adjacent angles equal to two right angles, then the two straight lines are in a straight line with one another. Q.E.D. This is a converse of the last proposition. This is a proposition in plane geometry. If A, B, C, and D do not lie in a plane, then CBD cannot be a straight line. An ambient plane is necessary to talk about the sides of the line AB The qualifying sentence, "Similarly we can prove that neither is any other straight line except BD," is meant to take care of the cases when E does not lie inside the angle ABD. Use of Proposition 14 This proposition is used in propositions I.45, I.47, and a few in Books VI and XI. Next proposition: I.15 Previous: I.13 Book I introduction © 1996 D.E.Joyce Clark University Proposition 15 If two straight lines cut one another, then they make the vertical angles equal to one another. Let the straight lines AB and CD cut one another at the point E. I say that the angle CEA equals the angle DEB, and the angle BEC equals the angle AED. Since the straight line AE stands on the straight line CD making the angles CEA and AED, therefore the sum of the angles CEA and AED equals two right angles. I.13 Again, since the straight line DE stands on the straight line AB making the angles AED and DEB, therefore the sum of the angles AED and DEB equals two right angles. I.13 But the sum of the angles CEA and AED was also proved equal to two right angles, therefore the sum of the angles CEA and AED equals the sum of the angles AED and DEB. Subtract the angle AED from each. Then the remaining angle CEA equals the remaining angle DEB. Post.4 C.N.1 C.N.3 Similarly it can be proved that the angles BEC and AED are also equal. Therefore if two straight lines cut one another, then they make the vertical angles equal to one another. Q.E.D. Corollary From this it is manifest that, if two straight lines cut one another, then they make the angles at the point of section equal to four right angles. Although the term "vertical angles" is not defined in the list of definitions at the beginning of Book I, its meaning is clear form its use in this proposition. A corollary that follows a proposition is a statement that immediately follows from the proposition or the proof in the proposition. It is possible that this and the other corollaries in the Elements are interpolations inserted after Euclid wrote the Elements. During the writing, he could have either bundled the corollary into the proposition or made it a separate proposition. Notes Proclus includes another corollary: If any number of straight lines intersect one another at one point, then the sum of all the angles so formed equals four right angles. Use of Proposition 15 This proposition is used in the next one, a few others in this book, II.10, IV.15 Next proposition: I.16 Previous: I.14 Book I introduction © 1996 D.E.Joyce Clark University Proposition 16 In any triangle, if one of the sides is produced, then the exterior angle is greater than either of the interior and opposite angles. Let ABC be a triangle, and let one side of it BC be produced to D. I say that the exterior angle ACD is greater than either of the interior and opposite angles CBA and BAC. Bisect AC at E. Join BE, and produce it in a straight line to F. I.10 Post.1 Post.2 Make EF equal to BE, join FC, and draw AC through to G. I.3 Post.1 Post.2 Since AE equals EC, and BE equals EF, therefore the two sides AE and EB equal the two sides CE and EF respectively, and the angle AEB equals the angle FEC, for they are vertical angles. Therefore the base AB equals the base FC, the triangle ABE equals the triangle CFE, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides. Therefore the angle BAE equals the angle ECF. I.15 I.4 But the angle ECD is greater than the angle ECF, therefore the angle ACD is greater than the angle BAE. C.N.5 Similarly, if BC is bisected, then the angle BCG, that is, the angle ACD, can also be proved to be greater than the angle ABC. I.15 Therefore in any triangle, if one of the sides is produced, then the exterior angle is greater than either of the interior and opposite angles. Q.E.D. In the later proposition I.32, after he invokes the parallel postulate Post.5, Euclid shows the stronger result that the exterior angle of a triangle equals the sum of the interior, opposite angles. Elliptic geometry There are geometries besides Euclidean geometry. Two of the more important geometries are elliptic geometry and hyperbolic geometry, which were developed in the nineteenth century. The first 15 propositions in Book I hold in elliptic geometry, but not this one. (For more on hyperbolic geometry, see the note after Proposition I.29.) Plane elliptic geometry is closely related to spherical geometry, but it differs in that antipodal points on the sphere are identified. Thus, a "point" in an elliptic plane is a pair of antipodal points on the sphere. A "straight line" in an elliptic plane is an arc of great circle on the sphere. When a "straight line" is extended, its ends eventually meet so that, topologically, it becomes a circle. This is very different from Euclidean geometry since here the ends of a line never meet when extended. The illustration on the right shows the stereographic projection of one hemisphere. Since only one hemisphere is displayed, each "point" is represented by one point except those "points" such as D, E, and F on the blue bounding great circle which appear twice. A "triangle" in elliptic geometry, such as ABC, is a spherical triangle (or, more precisely, a pair of antipodal spherical triangles). The internal angle sum of a spherical triangle is always greater than 180°, but less than 540°, whereas in Euclidean geometry, the internal angle sum of a triangle is 180° as shown in Proposition I.32. Elliptic geometry satisfies some of the postulates of Euclidean geometry, but not all of them under all interpretations. Usually, Post.1, to draw a straight line from any point to any point, is interpreted to include the uniqueness of that line. But in elliptic geometry a completed "straight line" is topologically a circle so that any pair of points on it divide it into two arcs. Therefore, in elliptic geometry exactly two "straight lines" join any two given "points." Also, Post.2, to produce a finite straight line continuously in a straight line, is sometimes interpreted to include the condition that its ends don't meet when extended. Under that interpretation, elliptic geometry fails Postulate 2. Elliptic geometry fails Post.5, the parallel postulate, as well, since any two "straight lines" in an elliptic plane meet. That is, any two great circles on the sphere meet at a pair of antipodal points. Finally, a completed "straight line" in the elliptic plane does not divide the plane into two parts as infinite straight lines do in the Euclidean plane. A completed "straight line" in the elliptic plane is a great circle on the sphere. Any two "points" not on that "straight line" include two points in the same hemisphere, and they can be joined by an arc that doesn't meet the great circle. Therefore two "points" lie on the same side of the completed "straight line." The proof of this particular proposition fails for elliptic geometry, and the statement of the proposition is false for elliptic geometry. In particular, the statement "the angle ECD is greater than the angle ECF" is not true of all triangles in elliptic geometry. The line CF need not be contained in the angle ACD. All the previous propositions do hold in elliptic geometry and some of the later propositions, too, but some need different proofs. Use of Proposition 16 This proposition is used in the proofs of the next two propositions, a few others in this book, and a couple in Book III. Next proposition: I.17 Previous: I.15 Book I introduction © 1996, 1997 D.E.Joyce Clark University Proposition 17 In any triangle the sum of any two angles is less than two right angles. Let ABC be a triangle. I say that the sum of any two angles of the triangle ABC is less than two right angles. Produce BC to D. Post.2 Since the angle ACD is an exterior angle of the triangle ABC, therefore it is greater than the interior and opposite angle ABC. Add the angle ACB to each. Then the sum of the angles ACD and ACB is greater than the sum of the angles ABC and BCA. I.16 C.N. But the sum of the angles ACD and ACB is equal to two right angles. Therefore the sum of the angles ABC and BCA is less than two right angles. I.13 Similarly we can prove that the sum of the angles BAC and ACB is also less than two right angles, and so the sum of the angles CAB and ABC as well. Therefore in any triangle the sum of any two angles is less than two right angles. Q.E.D. The statements ... the angle ACD... is greater than the interior and opposite angle ABC.Add the angle ACBto each. Then the sum of the angles ACDand ACBis greater than the sum of the angles ABCand BCA. use the property of magnitudes If x> y,then x + z> y + z. This property is not listed among the Common Notions. This proposition is strengthened in Proposition I.32 to say the sum of all three angles in a triangle equals two right angles. Use of Proposition 17 This proposition is used in III.16 and a couple other propositions of Books III, and a few in Books VI and XI. Next proposition: I.18 Previous: I.16 Book I introduction © 1996 D.E.Joyce Clark University Proposition 18 In any triangle the angle opposite the greater side is greater. Let ABC be a triangle having the side AC greater than AB. I say that the angle ABC is also greater than the angle BCA. Since AC is greater than AB, make AD equal to AB, and join BD. I.3 Post.1 Since the angle ADB is an exterior angle of the triangle BCD, therefore it is greater than the interior and opposite angle DCB. I.16 But the angle ADB equals the angle ABD, since the side AB equals AD, therefore the angle ABD is also greater than the angle ACB. Therefore the angle ABC is much greater than the angle ACB. I.5 Therefore in any triangle the angle opposite the greater side is greater. Q.E.D. On word order In this translation of Euclid's Elements the order of the words differs from the original Greek. In each of Euclid's Greek sentences, the data, that is the geometric objects given or already constructed, appear first, and the remaining geometric objects appear later. This is possible in Greek since it is an inflected language and the word order is very flexible. On the other hand, the word order in English is intrinsic to the syntax and semantics of the sentence and is not very flexible. Take, for instance, the statements of this and the next proposition. Very literal translations of these are (I.18) "In any triangle, the greater side [as subject] the greater angle [as object] subtends," and (I.19) "In any triangle, the greater angle [as object] the greater side [as subject] subtends." Heath keeps the word order in his translation but makes the second statement passive: (I.18) "In any triangle the greater side subtends the greater angle," and (I.19) "In any triangle the greater angle is subtended by the greater side." Without the understanding that the data come first, these two sentences are logically equivalent. In this translation the original word order is abandoned in order to make for more readable sentences and to clarify the meaning. Thus, (I.18) "In any triangle the angle opposite the greater side is greater," and (I.19) "In any triangle the side opposite the greater angle is greater." It may sound like these two propositions really do say the same thing, but they don't. They're actually disguised converses of each other. I.18 says "if side AC > side AB, then angle ABC > angle BCA" (but it hasn't yet been shown that there is no other way for angle ABC to be greater), while I.19 says "if angle ABC > angle BCA, then side AC > side AB." Use of Proposition 18 This proposition is used in the proof of proposition I.19. Next proposition: I.19 Previous: I.17 Book I introduction © 1996 D.E.Joyce Clark University Proposition 19 In any triangle the side opposite the greater angle is greater. Let ABC be a triangle having the angle ABC greater than the angle BCA. I say that the side AC is greater than the side AB. If not, either AC equals AB or it is less than it. Now AC does not equal AB, for then the angle ABC would equal the angle ACB, but it does not. Therefore AC does not equal AB. I.5 Neither is AC less than AB, for then the angle ABC would be less than the angle ACB, but it is not. Therefore AC is not less than AB. I.18 And it was proved that it is not equal either. Therefore AC is greater than AB. Therefore in any triangle the side opposite the greater angle is greater. Q.E.D. As mentioned before, this proposition is a disguised converse of the previous one. As Euclid often does, he uses a proof by contradiction involving the already proved converse to prove this proposition. It is not that there is a logical connection between this statement and its converse that makes this tactic work, but some kind of symmetry involved. In this case, if one side is less than another, then the other is greater than the one, and the previous proposition applies. So the relevant symmetry is between "less" and "greater." The law of sines Although some of the geometric underpinnings of trigonometry appear in the Elements, trigonometry itself does not. Trigonometry makes its appearance among later Greek mathematics where the the basic trigonometric function is the chord, which is related to the sine. Without going into details, the law of sines contains more precise information about the relation between angles and sides of a triangle than this and the last proposition did. The law of sines states that (sin A)/BC = (sin B)/AC = (sin C)/AB. Alternately, the first equation may be read a proportion sin A is to sin B as BC is to AC. In other words, the sine of an angle in a triangle is proportional to the opposite side. (Proportions aren't defined in the Elements until Book V.) Use of Proposition 19 This proposition is used in the proofs of propositions I.20, I.24, and some others in Book III. Next proposition: I.20 Previous: I.18 Book I introduction © 1996 D.E.Joyce Clark University Proposition 20 In any triangle the sum of any two sides is greater than the remaining one. Let ABC be a triangle. I say that in the triangle ABC the sum of any two sides is greater than the remaining one, that is, the sum of BA and AC is greater than BC, the sum of AB and BC is greater than AC, and the sum of BC and CA is greater than AB. Draw BA through to the point D, and make DA equal to CA. Join DC. Post.2 I.3 Post.1 Since DA equals AC, therefore the angle ADC also equals the angle ACD. Therefore the angle BCD is greater than the angle ADC. I.5 C.N.5 Since DCB is a triangle having the angle BCD greater than the angle BDC, and the side opposite the greater angle is greater, therefore DB is greater than BC. I.19 But DA equals AC, therefore the sum of BA and AC is greater than BC. Similarly we can prove that the sum of AB and BC is also greater than CA, and the sum of BC and CA is greater than AB. Therefore in any triangle the sum of any two sides is greater than the remaining one. Q.E.D. This proposition is known as "the triangle inequality." It is part of the statement that the shortest path between two points is a straight line, but there are many other conceivable paths besides broken lines. A minimum distance This proposition on the triangle inequality, along with I.15 on vertical angles, allows us to solve a problem on minimum distance, described and solved by Heron of Alexandria. Suppose there are two points A and B on the same side of a line CD. The problem is to find the shortest path which goes first from the point A to some point P on the line CD, then from P to the point B. We will only consider paths that are made out of straight lines; call such a path a bent line. But that still leaves us the question of which point P to choose on the line CD to minimize the sum of the distances AP plus PB. The solution is that the shortest path will be the path AEB where angle of incidence, namely, angle AEC, equals the angle of reflection, namely, angle BED. First, we should show how to construct the bent line where the angle of incidence equals the angle of reflection. Draw a perpendicular BF from the point B to the line CD (I.12), and extend it to B' so that FB' = BF (Post.2, I.3). Draw AB' and let E be the point where AB' intersects CD. (There will be a point of intersection since A and B' are on opposite sides of CD.) Draw BE. Now, triangles BFE and B'FE are congruent since they have two sides and the included angle equal (I.4), the included angles being right angles. Therefore, angles BFE and BF'E are equal. The vertical angle AEC across from angle B'ED also equals these angles (I.15). Thus, the angle of incidence AEC equals the angle of reflection BED. We still have to show that the distance AE + EB is less than any distance AP + EP for any point P other than E that lies on the line CD. Let P be such a point and draw lines AP, BP, and B'P. Then by proposition I.20, above, AP + EP is less than AB'. But AB' = AE + EB', and EB' = EB, therefore AP + EP is less than AE + EB. Thus, the shortest bent line between two points on the same side of a line that meets that line is the one where the angle of incidence equals the angle of reflection. Q.E.D. Use of Proposition 20 This proposition is used in the next two propositions, several in Book III, and XI.20. Next proposition: I.21 Previous: I.19 Book I introduction © 1996, 2002 D.E.Joyce Clark University Proposition 21 If from the ends of one of the sides of a triangle two straight lines are constructed meeting within the triangle, then the sum of the straight lines so constructed is less than the sum of the remaining two sides of the triangle, but the constructed straight lines contain a greater angle than the angle contained by the remaining two sides. From the ends B and C of one of the sides BC of the triangle ABC, let the two straight lines BD and DC be constructed meeting within the triangle. I say that the sum of BD and DC is less than the sum of the remaining two sides of the triangle BA and AC, but BD and DC contain an angle BDC greater than the angle BAC. Draw BD through to E. Post.2 Since in any triangle the sum of two sides is greater than the remaining one, therefore, in the triangle ABE, the sum of the two sides AB and AE is greater than BE. I.20 Add EC to each. Then the sum of BA and AC is greater than the sum of BE and EC. C.N. Again, since, in the triangle CED, the sum of the two sides CE and ED is greater than CD, add DB to each, therefore the sum of CE and EB is greater than the sum of CD and DB. I.20 C.N. But the sum of BA and AC was proved greater than the sum of BE and EC, therefore the sum of BA and AC is much greater than the sum of BD and DC. C.N. Again, since in any triangle the exterior angle is greater than the interior and opposite angle, therefore, in the triangle CDE, the exterior angle BDC is greater than the angle CED. I.16 For the same reason, moreover, in the triangle ABE the exterior angle CEB is greater than the angle BAC. But the angle BDC was proved greater than the angle CEB, therefore the angle BDC is much greater than the angle BAC. I.16 C.N. Therefore if from the ends of one of the sides of a triangle two straight lines are constructed meeting within the triangle, then the sum of the straight lines so constructed is less than the sum of the remaining two sides of the triangle, but the constructed straight lines contain a greater angle than the angle contained by the remaining two sides. Q.E.D. Pappus and others before him noticed that if the lines are not drawn from the ends of the side, then the sum of the the constructed straight lines can be greater than the sum of the remaining two sides of the triangle. In fact that sum can be made almost as large as twice the longest side of the triangle. Use of Proposition 21 This proposition is used in proposition III.8. Next proposition: I.22 Previous: I.20 Book I introduction © 1996 D.E.Joyce Clark University Proposition 22 To construct a triangle out of three straight lines which equal three given straight lines: thus it is necessary that the sum of any two of the straight lines should be greater than the remaining one. I.20 Let the three given straight lines be A, B, and C, and let the sum of any two of these be greater than the remaining one, namely, A plus B greater than C, A plus C greater than B, and B plus C greater than B. It is required to construct a triangle out of straight lines equal to A, B, and C. Set out a straight line DE, terminated at D but of infinite length in the direction of E. Make DF equal to A, FG equal to B, and GH equal to C. Post.2 I.3 Describe the circle DKL with center F and radius FD. Again, describe the circle KLH with center G and radius GH. Join KF and KG. Post.3 Post.1 I say that the triangle KFG has been constructed out of three straight lines equal to A, B, and C. Since the point F is the center of the circle DKL, therefore FD equals FK. But FD equals A, therefore KF also equals A. I.Def.16 C.N.1 Again, since the point G is the center of the circle LKH, therefore GH equals GK. But GH equals C, therefore KG also equals C. I.Def.16 C.N.1 And FG also equals B, therefore the three straight lines KF, FG, and GK equal the three straight lines A, B, and C. Therefore out of the three straight lines KF, FG, and GK, which equal the three given straight lines A, B, and C, the triangle KFG has been constructed. Q.E.F. The qualifier in the statement of the proposition, "thus it is necessary that the sum of any two of the straight lines should be greater than the remaining one," refers to the triangle inequality, Proposition I.20. This condition is, indeed, necessary. It is also sufficient, but Euclid failed to show that sufficiency. This construction is actually a generalization of the very first proposition I.1 in which the three lines are all equal. There too, as was noted, Euclid failed to prove that the two circles intersected. Use of Proposition 22 The construction in this proposition is used for the construction in proposition I.23. It is also used in XI.22 Next proposition: I.23 Previous: I.21 Book I introduction © 1996 D.E.Joyce Clark University Proposition 23 To construct a rectilinear angle equal to a given rectilinear angle on a given straight line and at a point on it. Let the angle DCE be the given rectilinear angle, AB the given straight line, and A the point on it. It is required to construct a rectilinear angle equal to the given rectilinear angle DCE on the given straight line AB and at the point A on it. Take the points D and E at random on the straight lines CD and CE respectively, and join DE. Out of three straight lines which equal the three straight lines CD, DE, and CE construct the triangle AFG in such a way that CD equals AF, CE equals AG, and DE equals FG. Post.1 I.22 Since the two sides DC and CE equal the two sides FA and AG respectively, and the base DE equals the base FG, therefore the angle DCE equals the angle FAG. I.8 Therefore on the given straight line AB, and at the point A on it, the rectilinear angle FAG has been constructed equal to the given rectilinear angle DCE. Q.E.F. As Proclus and Heath point out, a very minor variant of the construction in I.22 is needed to make the triangle AFG. The problem is that in I.22 the triangle is placed not at the end of line, but somewhere beyond that, and in I.23, the triangle needs to be placed right at the end A of the line. Construction steps This construction that moves an angle requires a number of steps involving a straightedge and compass. Unless there is some special reason for selecting particular points D and E on the sides of the angle C, they might as well be taken equidistant from C as Apollonius suggested. Then only two distances need to be transferred instead of three. In order to make CE equal to CD, one circle is required. Next, in order to transfer the distance CD to A, four circles (not shown) are required as per Propositions I.2 and I.1, and four more circles (also not shown) to transfer ED to G. Finally, two more circles are required, one with center A and radius CD (which has been transferred), the other with center G and radius ED. These last two circles meet at the point F, and the line AF is the other side of the required angle. In all there are ten circles and one line that must be drawn. The lines in the intermediate stages may be suppressed as usual since they're only needed to verify the construction is correct. Use of Proposition 23 The construction in this proposition is used in the next one and a couple others in Book I. It is also used frequently in the later books. It is also used frequently in BookS III and VI and occasionally in Books IV and XI. Although it may appear that the triangles are to be in the same plane, that is not necessary. Indeed, the construction in this proposition is used to construct an angle in a different plane in proposition XI.31. Next proposition: I.24 Previous: I.22 Book I introduction © 1996 D.E.Joyce Clark University Proposition 24 If two triangles have two sides equal to two sides respectively, but have one of the angles contained by the equal straight lines greater than the other, then they also have the base greater than the base. Let ABC and DEF be two triangles having the two sides AB and AC equal to the two sides DE and DF respectively, so that AB equals DE, and AC equals DF, and let the angle at A be greater than the angle at D. I say that the base BC is greater than the base EF. Since the angle BAC is greater than the angle EDF, construct the angle EDG equal to the angle BAC at the point D on the straight line DE. Make DG equal to either of the two straight lines AC or DF. Join EG and FG. I.23 I.3 Post.1 Since AB equals DE, and AC equals DG, the two sides BA and AC equal the two sides ED and DG, respectively, and the angle BAC equals the angle EDG, therefore the base BC equals the base EG. I.4 Again, since DF equals DG, therefore the angle DGF equals the angle DFG. Therefore the angle DFG is greater than the angle EGF. I.5 Therefore the angle EFG is much greater than the angle EGF. Since EFG is a triangle having the angle EFG greater than the angle EGF, and side opposite the greater angle is greater, therefore the side EG is also greater than EF. I.19 But EG equals BC, therefore BC is also greater than EF. Therefore if two triangles have two sides equal to two sides respectively, but have one of the angles contained by the equal straight lines greater than the other, then they also have the base greater than the base. Q.E.D. Use of Proposition 24 This proposition is used in the next proposition as well as a few in Book III and XI.22. Next proposition: I.25 Previous: I.23 Book I introduction © 1996 D.E.Joyce Clark University Proposition 25 If two triangles have two sides equal to two sides respectively, but have the base greater than the base, then they also have the one of the angles contained by the equal straight lines greater than the other. Let ABC and DEF be two triangles having two sides AB and AC equal to two sides DE and DF respectively, namely AB to DE, and AC to DF, and let the base BC be greater than the base EF. I say that the angle BAC is also greater than the angle EDF. If not, it either equals it or is less. Now the angle BAC does not equal the angle EDF, for then the base BC would equal the base EF, but it is not. Therefore the angle BAC does not equal the angle EDF. I.4 Neither is the angle BAC less than the angle EDF, for then the base BC would be less than the base EF, but it is not. Therefore the angle BAC is not less than the angle EDF. I.24 But it was proved that it is not equal either. Therefore the angle BAC is greater than the angle EDF. Therefore if two triangles have two sides equal to two sides respectively, but have the base greater than the base, then they also have the one of the angles contained by the equal straight lines greater than the other. Q.E.D. The conclusions of this proposition and the previous are partial converses of each other. Together they say that if two triangles have two sides equal to two sides respectively, then the base greater than the base if and only if the one of the angles contained by the equal straight lines greater than the other. Use of Proposition 25 This proposition is not used in the rest of Book I, but it is used in XI.20 and XI.23. Next proposition: I.26 Previous: I.24 Book I introduction © 1996 D.E.Joyce Clark University Proposition 27 If a straight line falling on two straight lines makes the alternate angles equal to one another, then the straight lines are parallel to one another. Let the straight line EF falling on the two straight lines AB and CD make the alternate angles AEF and EFD equal to one another. I say that AB is parallel to CD. If not, AB and CD when produced meet either in the direction of B and D or towards A and C. Let them be produced and meet, in the direction of B and D, at G. Then, in the triangle GEF, the exterior angle AEF equals the interior and opposite angle EFG, which is impossible. I.16 Therefore AB and CD when produced do not meet in the direction of B and D. Similarly it can be proved that neither do they meet towards A and C. But straight lines which do not meet in either direction are parallel. Therefore AB is parallel to CD. I.Def.23 Therefore if a straight line falling on two straight lines makes the alternate angles equal to one another, then the straight lines are parallel to one another. Q.E.D. There is implicitly assumed an ambient plane. The term "alternate angles" doesn't have a meaning unless the lines all lie in a plane. Note that Euclid does not consider two other possible ways that the two lines could meet, namely, in the directions A and D or toward B and C. About logical inverses Although this is the first proposition about parallel lines, it does not require the parallel postulate Post.5 as an assumption. This proposition I.27 and the parallel postulate can be made to look more similar if they are reworded (with the help of I.13). Proposition 1.27. If a straight line falls on two straight lines, then if the alternate angles are equal, then the straight lines do not meet. Post.5. If a straight line falls on two straight lines, then if the alternate angles are not equal, then the straight lines meet [on a certain side of the line]. If the remark about the side is dropped, then the conclusions are logical inverses of each other, and the logical inverse of a statement is logically equivalent to the converse. This little table summarizes the logical relations between similarly looking statements. Statement If P then Q. Converse If Q then P. Not logically equivalent to the statement. Contrapositive If not Q then not P. Logically equivalent to the statement. Inverse If not P then not Q. Logically equivalent to the converse. Although the contrapositive is logically equivalent to the statement, Euclid always proves the contrapositive separately using a proof by contradiction and the original statement. Similarly, the inverse is proved using the converse. Sometimes all four statements appear in separate propositions as in propositions X.5 through X.8. Other times the four appear as four statements in one proposition as in X.9. More often than not, however, the contrapositive and inverse make no appearance, and, of course, the converse only appears when it can be proved. Use of Proposition 27 At this point, parallel lines have yet to be constructed. That occurs in proposition I.31 which uses this proposition to verify that lines constructed there are parallel. This proposition is also used in the next one and in I.33. Next proposition: I.28 Previous: I.26 Book I introduction © 1996, 1997 D.E.Joyce Clark University Proposition 28 If a straight line falling on two straight lines makes the exterior angle equal to the interior and opposite angle on the same side, or the sum of the interior angles on the same side equal to two right angles, then the straight lines are parallel to one another. Let the straight line EF falling on the two straight lines AB and CD make the exterior angle EGB equal to the interior and opposite angle GHD, or the sum of the interior angles on the same side, namely BGH and GHD, equal to two right angles. I say that AB is parallel to CD. Since the angle EGB equals the angle GHD, and the angle EGB equals the angle AGH, therefore the angle AGH equals the angle GHD. And they are alternate, therefore AB is parallel to CD. I.15 C.N.1 I.27 Next, since the sum of the angles BGH and GHD equals two right angles, and the sum of the angles AGH and BGH also equals two right angles, therefore the sum of the angles AGH and BGH equals the sum of the angles BGH and GHD. I.13 C.N.1 Post.4 Subtract the angle BGH from each. Therefore the remaining angle AGH equals the remaining angle GHD. And they are alternate, therefore AB is parallel to CD. C.N.3 I.27 Therefore if a straight line falling on two straight lines makes the exterior angle equal to the interior and opposite angle on the same side, or the sum of the interior angles on the same side equal to two right angles, then the straight lines are parallel to one another. Q.E.D. This proposition states two useful minor variants of the previous proposition. The three statements differ only in their hypotheses which are easily seen to be equivalent with the help of proposition I.13. Use of Proposition 28 This proposition is used in IV.7, VI.4, and a couple times in Book XI. Next proposition: I.29 Previous: I.27 Book I introduction © 1996 D.E.Joyce Clark University Proposition 29 A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the sum of the interior angles on the same side equal to two right angles. Let the straight line EF fall on the parallel straight lines AB and CD. I say that it makes the alternate angles AGH and GHD equal, the exterior angle EGB equal to the interior and opposite angle GHD, and the sum of the interior angles on the same side, namely BGH and GHD, equal to two right angles. If the angle AGH does not equal the angle GHD, then one of them is greater. Let the angle AGH be greater. Add the angle BGH to each. Therefore the sum of the angles AGH and BGH is greater than the sum of the angles BGH and GHD. But sum of the angles AGH and BGH equals two right angles. Therefore the sum of the angles BGH and GHD is less than two right angles. I.13 But straight lines produced indefinitely from angles less than two right angles meet. Therefore AB and CD, if produced indefinitely, will meet. But they do not meet, because they are by hypothesis parallel. Post.5 Therefore the angle AGH is not unequal to the angle GHD, and therefore equals it. Again, the angle AGH equals the angle EGB. Therefore the angle EGB also equals the angle GHD. I.15 C.N.1 Add the angle BGH to each. Therefore the sum of the angles EGB and BGH equals the sum of the angles BGH and GHD. C.N.2 But the sum of the angles EGB and BGH equals two right angles. Therefore the sum of the angles BGH and GHD also equals two right angles. I.13 C.N.1 Therefore a straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the sum of the interior angles on the same side equal to two right angles. Q.E.D. The statement of this proposition includes three parts, one the converse of I.27, the other two the converse of I.28. Like those propositions, this one assumes an ambient plane containing all the three lines. This is the first proposition which depends on the parallel postulate. As such it does not hold in hyperbolic geometry. Hyperbolic geometry Two important geometries alternative to Euclidean geometry are elliptic geometry and hyperbolic geometry. Elliptic geometry was discussed in the note after Proposition I.16, that being the first proposition which doesn't hold in elliptic geometry. This, I.29, is the first which doesn't hold in hyperbolic geometry. These three geometries can be distinguished by the number of lines parallel to a given line passing through a given point. For elliptic geometry, there is no such parallel line; for Euclidean geometry (which may be called parabolic geometry), there is exactly one; and for hyperbolic geometry, there are infinitely many. It is not possible to illustrate hyperbolic geometry with correct distances on a flat surface since a flat surface is Euclidean. Poincaré, however, described a useful model of hyperbolic geometry where the "points" in a hyperbolic plane are taken to be points inside a fixed circle (but not the points on the circumference). The "lines" in the hyperbolic plane are the parts of circles orthogonal, that is, at right angles to the fixed circle. And in this model, "angles" in the hyperbolic plane are angles between these arcs, or, more precisely, angles between the tangents to the arcs at the point of intersection. Since "angles" are just angles, this model is called a conformal model. Distances in the hyperbolic plane, however, are not measured by distances along the arcs. There is a more complicated relation between distances so that near the edge of the fixed circle a very short arc models a very long "line." Once this model is accepted, it is easy to see why there are infinitely many "lines" parallel to a given "line" through a given "point." That is just that there are infinitely many circles orthogonal to the fixed circle which don't intersect the given circle orthogonal to the fixed circle but do pass through the given point. In the diagram, AB is a "line" in the hyperbolic plane, that is, a circle orthogonal to the circumference of the shaded disk which represents the hyperbolic plane. A "point" C lies in that plane. Two "lines" are shown passing through C, one gets close to the line AB in the direction of A, the other gets close in the direction of B. But these two "lines" don't intersect AB since the arcs representing them only intersect on the circumference of the disk, and points on the circumference don't represent "points" in the hyperbolic plane. These two parallel "lines" are called the asymptotic parallels of AB since they approach AB at one end or the other. There are infinitely many parallels between them. (In much of the literature on hyperbolic geometry, the word "parallels" is used for what are called "asymptotic parallels" here, while "nonintersecting lines" is used for what are called "parallels" here.) Use of Proposition 29 This proposition is used in very frequently in Book I starting with the next proposition. It is also used frequently in Book II, VI, and XI, and once in Book XII. Next proposition: I.30 Previous: I.28 Book I introduction © 1996 D.E.Joyce Clark University Proposition 30 Straight lines parallel to the same straight line are also parallel to one another. Let each of the straight lines AB and CD be parallel to EF. I say that AB is also parallel to CD. Let the straight line GK fall upon them. Since the straight line GK falls on the parallel straight lines AB and EF, therefore the angle AGK equals the angle GHF. I.29 Again, since the straight line GK falls on the parallel straight lines EF and CD, therefore the angle GHF equals the angle GKD. I.29 But the angle AGK was also proved equal to the angle GHF. Therefore the angle AGK also equals the angle GKD, and they are alternate. C.N.1 Therefore AB is parallel to CD. Therefore straight lines parallel to the same straight line are also parallel to one another. Q.E.D. For this proposition it is supposed that the three lines lie in one plane. Proposition XI.9 applies to the case where the three lines do not lie in a plane. Playfair's axiom A number of the propositions in the Elements are equivalent to the parallel postulate Post.5 in the sense that if the rest of the postulates are assumed and any one of these propositions is assumed, then the parallel postulate can be proved as a proposition. This one I.30, the last I.29, either part of I.32, and almost any later one. Thus, Euclid had many statements to choose from to take as a postulate. In many modern expositions of synthetic geometry, Playfair's axiom (John Playfair, 1748-1819) is chosen as that postulate instead of Euclid's parallel postulate Post.5. Playfair's axiom states that there is at most one line parallel to a given line passing through a given point. (That there is at least one follows from the next proposition I.31 which doesn't depend on the parallel postulate.) Two advantages of Playfair's axiom over Euclid's parallel postulate are that it is a simpler statement, and it emphasizes the distinction between Euclidean and hyperbolic geometry. Two disadvantages are that it does not have the historical importance of Euclid's parallel postulate, and the proof of the parallel postulate from Playfair's axiom is nonconstructive. That proof is a proof by contradiction that begins assuming that a point does not exist, deriving a contradiction, and concluding that the point must exist, but does not construct it. It may well be that Euclid chose to make the construction an assumption of his parallel postulate rather rather than choosing some other equivalent statement for his postulate. Elegance in mathematics Euclid's Elements form one of the most beautiful works of science in the history of humankind. This beauty lies more in the logical development of geometry rather than in geometry itself. It is not the diagrams that excite our interest; rather it is the concepts, the way the concepts interconnect, and the way Euclid selected and presented these concepts and their interconnections. The Elements are elegant. Elegance in mathematics is characterized by simplicity and clarity. An elegant presentation is easy for the reader to follow. But elegance is not only in the presentation, it is in the selection of definitions and proofs. The elegant definition is the one that makes the rest of the theory easy. The elegant proof is the one that is easiest to follow, one that is designed just right to fit the goal. Extraneous concepts should not be involved. Even the goals need to be adjusted to the right level of generality to cover the concepts, but not so abstract that the abstraction itself obscures the goal. One of the criticisms of Euclid's parallel postulate was that it isn't simple. The statement of this proposition, I.30, is much simpler, and Playfair's axiom is much simpler. As they're each logically equivalent to Euclid's parallel postulate, if elegance were the primary goal, then Euclid would have chosen one of them in place of his postulate. Perhaps the reasons mentioned above explain why Euclid used Post.5 instead. Use of Proposition 30 This proposition is used in I.45 and IV.7. Next proposition: I.31 Previous: I.29 Book I introduction © 1996, 1997, 2003 D.E.Joyce Clark University Proposition 31 To draw a straight line through a given point parallel to a given straight line. Let A be the given point, and BC the given straight line. It is required to draw a straight line through the point A parallel to the straight line BC. Take a point D at random on BC. Join AD. Construct the angle DAE equal to the angle ADC on the straight line DA and at the point A on it. Produce the straight line AF in a straight line with EA. Post.1 I.23 Post.2 Since the straight line AD falling on the two straight lines BC and EF makes the alternate angles EAD and ADC equal to one another, therefore EAF is parallel to BC. I.27 Therefore the straight line EAF has been drawn through the given point A parallel to the given straight line BC. Q.E.F. The parallel line EF constructed in this proposition is the only one passing through the point A. If there were another, then the interior angles on one side or the other of AD it makes with BC would be less than two right angles, and therefore by the parallel postulate Post.5, it would meet BC, a contradiction. Incidentally, this construction also works in hyperbolic geometry, although different parallel lines through A are constructed for different points D. Construction steps The construction needed is that of I.23 to construct an angle. That construction required ten circles and one line in general. In the specific case needed here, however, one of the distances does not have to be transferred, and that eliminates the need to construct four of the circles. Therefore this construction actually only requires six circles and a line. Use of Proposition 31 This construction is frequently used in the remainder of Book I starting with the next proposition. It is also frequently used in Books II, IV, VI, XI, XII, and XIII. Next proposition: I.32 Previous: I.30 Book I introduction © 1996, 1997 D.E.Joyce Clark University Proposition 32 In any triangle, if one of the sides is produced, then the exterior angle equals the sum of the two interior and opposite angles, and the sum of the three interior angles of the triangle equals two right angles. Let ABC be a triangle, and let one side of it BC be produced to D. I say that the exterior angle ACD equals the sum of the two interior and opposite angles CAB and ABC, and the sum of the three interior angles of the triangle ABC, BCA, and CAB equals two right angles. Draw CE through the point C parallel to the straight line AB. I.31 Since AB is parallel to CE, and AC falls upon them, therefore the alternate angles BAC and ACE equal one another. I.29 Again, since AB is parallel to CE, and the straight line BD falls upon them, therefore the exterior angle ECD equals the interior and opposite angle ABC. I.29 But the angle ACE was also proved equal to the angle BAC. Therefore the whole angle ACD equals the sum of the two interior and opposite angles BAC and ABC. Add the angle ACB to each. Then the sum of the angles ACD and ACB equals the sum of the three angles ABC, BCA, and CAB. C.N.2 But the sum of the angles ACD and ACB equals two right angles. Therefore the sum of the angles ABC, BCA, and CAB also equals two right angles. I.13 C.N.1 Therefore in any triangle, if one of the sides is produced, then the exterior angle equals the sum of the two interior and opposite angles, and the sum of the three interior angles of the triangle equals two right angles. Q.E.D. Corollaries of Proclus There are two corollaries of this proposition given by Proclus. Corollary 1. The sum of the interior angles of a convex rectilinear figure equals twice as many angles as the figure has sides, less four. Corollary 2. The sum of the exterior angles of any convex rectilinear figure together equal four right angles. Use of Proposition 32 Although this proposition isn't used in the rest of Book I, it is frequently used in the rest of the books on geometry, namely Books II, III, IV, VI, XI, XII, and XIII. The corollaries, however, are not used in the Elements. Next proposition: I.33 Previous: I.31 Book I introduction © 1996 D.E.Joyce Clark University Proposition 33 Straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel. Let AB and CD be equal and parallel, and let the straight lines AC and BD join them at their ends in the same directions. I say that AC and BD are also equal and parallel. Join BC. Post.1 Since AB is parallel to CD, and BC falls upon them, therefore the alternate angles ABC and BCD equal one another. I.29 Since AB equals CD, and BC is common, the two sides AB and BC equal the two sides DC and CB, and the angle ABC equals the angle BCD, therefore the base AC equals the base BD, the triangle ABC equals the triangle DCB, and the remaining angles equals the remaining angles respectively, namely those opposite the equal sides. Therefore the angle ACB equals the angle CBD. I.4 Since the straight line BC falling on the two straight lines AC and BD makes the alternate angles equal to one another, therefore AC is parallel to BD. I.27 And it was also proved equal to it. Therefore straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel. Q.E.D. The qualifier "in the same directions" in the statement of this proposition is necessary since without it the lines AD and BC could join the endpoints of the parallel lines, and AD and BC are not parallel but intersect. But these words of Euclid words are informal, and it would take some work to determine geometrically which end of AD corresponds to which end of a parallel line BC. In general, given four points A, B, C, and D, exactly one of the three pairs of lines, AB and CD, AC and BD, and AD and BC, intersects. (If extended to infinite lines, all three pairs of lines might intersect, but as line segments only one pair does.) This statement belongs to the fundamental part of plane geometry that includes betweenness and sides of lines that wasn't developed until the late nineteenth century. Use of Proposition 33 This proposition is used in I.36, I.45, and a few prpositions in Books XI through XIII. Next proposition: I.34 Previous: I.32 Book I introduction © 1996, 1997 D.E.Joyce Clark University Proposition 34 In parallelogrammic areas the opposite sides and angles equal one another, and the diameter bisects the areas. Let ACDB be a parallelogrammic area, and BC its diameter. I say that the opposite sides and angles of the parallelogram ACDB equal one another, and the diameter BC bisects it. Since AB is parallel to CD, and the straight line BC falls upon them, therefore the alternate angles ABC and BCD equal one another. I.29 Again, since AC is parallel to BD, and BC falls upon them, therefore the alternate angles ACB and CBD equal one another. I.29 Therefore ABC and DCB are two triangles having the two angles ABC and BCA equal to the two angles DCB and CBD respectively, and one side equal to one side, namely that adjoining the equal angles and common to both of them, BC. Therefore they also have the remaining sides equal to the remaining sides respectively, and the remaining angle to the remaining angle. Therefore the side AB equals CD, and AC equals BD, and further the angle BAC equals the angle CDB. I.26 Since the angle ABC equals the angle BCD, and the angle CBD equals the angle ACB, therefore the whole angle ABD equals the whole angle ACD. C.N.2 And the angle BAC was also proved equal to the angle CDB. Therefore in parallelogrammic areas the opposite sides and angles equal one another. I say, next, that the diameter also bisects the areas. Since AB equals CD, and BC is common, the two sides AB and BC equal the two sides DC and CB respectively, and the angle ABC equals the angle BCD. Therefore the base AC also equals DB, and the triangle ABC equals the triangle DCB. I.4 Therefore the diameter BC bisects the parallelogram ACDB. Therefore in parallelogrammic areas the opposite sides and angles equal one another, and the diameter bisects the areas. Q.E.D. In this proposition Euclid uses the term "parallelogrammic area" rather than the word "parallelogram" which first occurs in the next proposition. Proclus indicated that the word "parallelogram" was created by Euclid. This proposition begins the study of areas of rectilinear figures. It is a modest beginning, but it allows the comparison of triangles and parallelograms so that problems and results concerning one can be converted to problems and results concerning the other. Use of Proposition 34 This proposition is used in the next four propositions and some others in Book I, several in Book II, a few in Books IV, VI, X, XI, and XII. Next proposition: I.35 Previous: I.33 Book I introduction © 1996 D.E.Joyce Clark University Proposition 35 Parallelograms which are on the same base and in the same parallels equal one another. Let ABCD and EBCF be parallelograms on the same base BC and in the same parallels AF and BC. I say that ABCD equals the parallelogram EBCF. Since ABCD is a parallelogram, therefore AD equals BC. I.34 For the same reason EF equals BC, so that AD also equals EF. And DE is common, therefore the whole AE equals the whole DF. C.N.1 C.N.2 But AB also equals DC. Therefore the two sides EA and AB equal the two sides FD and DC respectively, and the angle FDC equals the angle EAB, the exterior equals the interior. Therefore the base EB equals the base FC, and the triangle EAB equals the triangle FDC. I.34 I.29 I.4 Subtract DGE from each. Then the trapezium ABGD which remains equals the trapezium EGCF which remains. C.N.3 Add the triangle GBC to each. Then the whole parallelogram ABCD equals the whole parallelogram EBCF. C.N.2 Therefore parallelograms which are on the same base and in the same parallels equal one another. Q.E.D. Euclid's proof specifically treats the case when the point D lies between A and E in which case subtraction of a triangle is necessary. There are other cases to consider, for instance, when E lies between A and D. In that case the point G is irrelevant and the trapezium BCED may be added to the congruent triangles ABE and DCF to derive the conclusion. Euclid often supplies a proof for only one case, although occasionally he gives proofs for two or three cases. Rectilinear figures as magnitudes We see how Euclid treats figures as magnitudes by adding as subtracting them. The triangles EAB and FDC are shown directly to be equal. Then the triangle DGE, which is contained in each, is subtracted from each, and Euclid concludes that the remaining trapezia ABGD and EGCF are therefore equal. These trapezia are not congruent, but they do have the same area. Next, the triangle GBC is added to each trapezium to conclude the two parallelograms ABCD and EBCF are equal. These are the same kinds of cut-and-paste operations that Euclid used on lines and angles earlier in Book I, but these are applied to rectilinear figures. In later books cut-and-paste operations will be applied to other kinds of magnitudes such as solid figures and parts of circumferences of circles. Use of Proposition 35 This proposition is used in the next two propositions and in XI.31. Next proposition: I.36 Previous: I.34 Book I introduction © 1996 D.E.Joyce Clark University Proposition 36 Parallelograms which are on equal bases and in the same parallels equal one another. Let ABCD and EFGH be parallelograms which are on the equal bases BC and FG and in the same parallels AH and BG. I say that the parallelogram ABCD equals EFGH. Join BE and CH. Post.1 Since BC equals FG and FG equals EH, therefore BC equals EH. I.34 C.N.1 But they are also parallel, and EB and HC join them. But straight lines joining equal and parallel straight lines in the same directions are equal and parallel, therefore EBCH is a parallelogram. I.33 And it equals ABCD, for it has the same base BC with it and is in the same parallels BC and AH with it. I.35 For the same reason also EFGH equals the same EBCH, so that the parallelogram ABCD also equals EFGH. C.N.1 Therefore parallelograms which are on equal bases and in the same parallels equal one another. Q.E.D. This proposition is a generalization of the previous proposition I.35, and its proof depends directly on it. Euclid could have bundled the two propositions into one. Then the special case of I.35 would have been proven first and then used to prove the general case of I.36. In an introductory book like Book I this separation makes it easier to follow the logic, but in later books special cases are often bundled into the general proposition. Use of Proposition 36 This proposition is used in I.38, a few propositions in Books II and VI, and XI.29 Next proposition: I.37 Previous: I.35 Book I introduction © 1996 D.E.Joyce Clark University Proposition 37 Triangles which are on the same base and in the same parallels equal one another. Let ABC and DBC be triangles on the same base BC and in the same parallels AD and BC. I say that the triangle ABC equals the triangle DBC. Produce AD in both directions to E and F. Draw BE through B parallel to CA, and draw CF through C parallel to BD. Post.2 I.31 Then each of the figures EBCA and DBCF is a parallelogram, and they are equal, for they are on the same base BC and in the same parallels BC and EF. I.35 Moreover the triangle ABC is half of the parallelogram EBCA, for the diameter AB bisects it. And the triangle DBC is half of the parallelogram DBCF, for the diameter DC bisects it. I.34 Therefore the triangle ABC equals the triangle DBC. C.N Therefore triangles which are on the same base and in the same parallels equal one another. Q.E.D. In this proposition the triangles have the same base while in the next one the triangles have equal bases. Since the proofs are the same except that this depends on I.35 while the next depends on I.36, and the next is more general, there is no purpose to include this proposition. The justification of the last conclusion is missing. From the statement that the doubles of two magnitudes are equal, we want to conclude that the magnitudes themselves are equal. Although Euclid included no such common notion, others inserted it later. See the commentary on Common Notions for a proof of this halving principle based on other properties of magnitudes. Use of Proposition 37 This proposition is used in I.39, I.41, and VI.2. Next proposition: I.38 Previous: I.36 Book I introduction © 1996, 1997 D.E.Joyce Clark University Proposition 38 Triangles which are on equal bases and in the same parallels equal one another. Let ABC and DEF be triangles on equal bases BC and EF and in the same parallels BF and AD. I say that the triangle ABC equals the triangle DEF. Produce AD in both directions to G and H. Draw BG through B parallel to CA, and draw FH through F parallel to DE. Post.2 I.31 Then each of the figures GBCA and DEFH is a parallelogram, and GBCA equals DEFH, for they are on equal bases BC and EF and in the same parallels BF and GH. I.36 Moreover the triangle ABC is half of the parallelogram GBCA, for the diameter AB bisects it. And the triangle FED is half of the parallelogram DEFH, for the diameter DF bisects it. I.34 Therefore the triangle ABC equals the triangle DEF. C.N. Therefore triangles which are on equal bases and in the same parallels equal one another. Q.E.D. The idea of the argument is clear: since parallelograms on equal bases and in the same parallels are equal by I.36, and the triangles are half the parallelograms by I.34, therefore the triangles are also equal. Use of Proposition 38 This proposition is used in I.40, I.42, and VI.1. Next proposition: I.39 Previous: I.37 Book I introduction © 1996 D.E.Joyce Clark University Proposition 39 Equal triangles which are on the same base and on the same side are also in the same parallels. Let ABC and DBC be equal triangles which are on the same base BC and on the same side of it. Join AD. I say that AD is parallel to BC. Post.1 If not, draw AE through the point A parallel to the straight line BC, and join EC. I.31 Post.1 Therefore the triangle ABC equals the triangle EBC, for it is on the same base BC with it and in the same parallels. I.37 But ABC equals DBC, therefore DBC also equals EBC, the greater equals the less, which is impossible. C.N.1 Therefore AE is not parallel to BC. Similarly we can prove that neither is any other straight line except AD, therefore AD is parallel to BC. Therefore equal triangles which are on the same base and on the same side are also in the same parallels. Q.E.D. This is a partial converse to proposition I.37, only partial since the two triangles ABC and DBC have to be on the same side of the line BC. If they weren't, then of course AD would not be parallel to BC but instead cross it at the midpoint Use of Proposition 39 This proposition is used in VI.2. Next proposition: I.40 Previous: I.38 Book I introduction © 1996 D.E.Joyce Clark University Proposition 40 Equal triangles which are on equal bases and on the same side are also in the same parallels. Let ABC and CDE be equal triangles on equal bases BC and CE and on the same side. I say that they are also in the same parallels. Join AD. I say that AD is parallel to BE. Post.1 If not, draw AF through A parallel to BE, and join FE. I.31 Post.1 Therefore the triangle ABC equals the triangle FCE, for they are on equal bases BC and CE and in the same parallels BE and AF. I.38 But the triangle ABC equals the triangle DCE, therefore the triangle DCE also equals the triangle FCE, the greater equals the less, which is impossible. Therefore AF is not parallel to BE. C.N.1 Similarly we can prove that neither is any other straight line except AD, therefore AD is parallel to BE. Therefore equal triangles which are on equal bases and on the same side are also in the same parallels. Q.E.D. The setting out of this proposition is not up to Euclid's standards. There is no justification for assuming that the point C is a common vertex of the two triangles. Fortunately, the proof works just as well if C is split into two points. For some of the propositions and many of the lemmas and corollaries in the Elements, there is evidence that Euclid did not write them, but they were added later. The process of incorporating new material in textbooks was almost automatic when the books were copied by hand instead of printed. Scholars wrote comments (called "scholia") in the margins of the texts, and copyists (some of whom were later scholars) would include those comments as part of the text in their new copies. Heiberg could show by means of an early papyrus fragment that this proposition was an early interpolation. For others, such as I.37 there is no direct evidence, only a doubt that a mathematician of Euclid's caliber would have included them. Unlike the other propositions in Book I, this one is not used later in the Elements. Next proposition: I.41 Previous: I.39 Book I introduction © 1996 D.E.Joyce Clark University Proposition 41 If a parallelogram has the same base with a triangle and is in the same parallels, then the parallelogram is double the triangle. Let the parallelogram ABCD have the same base BC with the triangle EBC, and let it be in the same parallels BC and AE. I say that the parallelogram ABCD is double the triangle BEC. Join AC. Post.1 Then the triangle ABC equals the triangle EBC, for it is on the same base BC with it and in the same parallels BC and AE. I.37 But the parallelogram ABCD is double the triangle ABC, for the diameter AC bisects it, so that the parallelogram ABCD is also double the triangle EBC. I.34 Therefore if a parallelogram has the same base with a triangle and is in the same parallels, then the parallelogram is double the triangle. Q.E.D. This partially generalizes I.34, that a parallelogram is twice the triangle by its diameter and two of its sides. A slightly more general statement would be that If a parallelogram has an equal base with a triangle and is in the same parallels, then the parallelogram is double the triangle. Use of Proposition 41 This proposition is used in the next one, I.47, VI.1, and X.38. Next proposition: I.42 Previous: I.40 Book I introduction © 1996 D.E.Joyce Clark University Proposition 42 To construct a parallelogram equal to a given triangle in a given rectilinear angle. Let ABC be the given triangle, and D the given rectilinear angle. It is required to construct D a parallelogram equal to the triangle ABC in the rectilinear angle. Bisect BC at E, and join AE. Construct the angle CEF on the straight line EC at the point E on it equal to the angle D. Draw AG through A parallel to EC, and draw CG through C parallel to EF. I.10 Post.1 I.23 I.31 Then FECG is a parallelogram. Since BE equals EC, therefore the triangle ABE also equals the triangle AEC, for they are on equal bases BE and EC and in the same parallels BC and AG. Therefore the triangle ABC is double the triangle AEC. I.38 But the parallelogram FECG is also double the triangle AEC, for it has the same base with it and is in the same parallels with it, therefore the parallelogram FECG equals the triangle ABC. I.41 C.N.1 And it has the angle CEF equal to the given angle D. Therefore the parallelogram FECG has been constructed equal to the given triangle ABC, in the angle CEF which equals D. Q.E.F. The idea of the construction is as follows. First make a triangle half the size of the given triangle. Next skew the half-size triangle to make one of its angles the desired angle without changing its area. Complete the resulting half-size triangle to a parallelogram. That's the desired parallelogram equal to the original triangle in the desired angle. Application of areas With this proposition Euclid moves to the next phase in his study of areas, the application of areas. Before this, he has exhibited various situations when triangles or parallelograms have equal areas, or when a triangle has half the area of a parallelogram. But now he's interested in constructing another figure with the same area as a given figure. In this proposition, he constructs a parallelogram that has a given angle and has the same area as a given triangle. But his goals are coming up, application of areas in I.45 and quadrature in II.14. In proposition I.45, given a rectilinear figure an equal parallelogram is constructed on a given side within a given angle. This kind of construction is called "applying" an area to a side. The area is sort of laid along the line. It may be that before Euclid the area was always applied to a rectangle along the line, but Euclid generalized the construction to parallelograms. This extra generalization is not often used. Later, in proposition II.14 a square is constructed equal to a given rectilinear figure, a process called "quadrature" (making into a square) of the figure. This square is a canonical measure of the area. Use of Proposition 42 This construction is used as part of the constructions in the two propositions following the next one. Next proposition: I.43 Previous: I.41 Book I introduction © 1996, 1997 D.E.Joyce Clark University Proposition 43 In any parallelogram the complements of the parallelograms about the diameter equal one another. Let ABCD be a parallelogram, and AC its diameter, and about AC let EH and FG be parallelograms, and BK and KD the so-called complements. I say that the complement BK equals the complement KD. Since ABCD is a parallelogram, and AC its diameter, therefore the triangle ABC equals the triangle ACD. I.34 Again, since EH is a parallelogram, and AK is its diameter, therefore the triangle AEK equals the triangle AHK. For the same reason the triangle KFC also equals KGC. I.34 Now, since the triangle AEK equals the triangle AHK, and KFC equals KGC, therefore the triangle AEK together with KGC equals the triangle AHK together with KFC. C.N.2 And the whole triangle ABC also equals the whole ADC, therefore the remaining complement BK equals the remaining complement KD. C.N.3 Therefore in any parallelogram the complements of the parallelograms about the diameter equal one another. Q.E.D. The meaning of the statement has to be found in its use. The term "the parallelograms about the diameter" refers to the two parallelograms having the same angles as the original parallelogram and with diameters AK and KC which are two parts of a diameter AC of the original parallelogram. The "complements" are the two parallelograms left over after removing those two parallelograms from the original parallelogram. Use of Proposition 43 The immediate purpose of this proposition is to change the shape of a parallelogram (one of the complements) into an equal parallelogram with the same angles (the other complement). That's how it is used in the next proposition. It is also used in several propositions in Book II, and a couple in Book VI. Next proposition: I.44 Previous: I.42 Book I introduction © 1996 D.E.Joyce Clark University Proposition 44 To a given straight line in a given rectilinear angle, to apply a parallelogram equal to a given triangle. Let AB be the given straight line, D the given rectilinear angle, and C the given triangle. It is required to apply a parallelogram equal to the given triangle C to the given straight line AB in an angle equal to D. Construct the parallelogram BEFG equal to the triangle C in the angle EBG which equals D, and let it be placed so that BE is in a straight line with AB. I.42 Draw FG through to H, and draw AH through A parallel to either BG or EF. Join HB. Post.2 I.31 Post.1 Since the straight line HF falls upon the parallels AH and EF, therefore the sum of the angles AHF and HFE equals two right angles. Therefore the sum of the angles BHG and GFE is less than two right angles. And straight lines produced indefinitely from angles less than two right angles meet, therefore HB and FE, when produced, will meet. I.29 Post.5 Let them be produced and meet at K. Draw KL through the point K parallel to either EA or FH. Produce HA and GB to the points L and M. I.31 Then HLKF is a parallelogram, HK is its diameter, and AG and ME are parallelograms, and LB and BF are the so-called complements about HK. Therefore LB equals BF. I.43 But BF equals the triangle C, therefore LB also equals C. C.N.1 Since the angle GBE equals the angle ABM, while the angle GBE equals D, therefore the angle ABM also equals the angle D. I.15 C.N.1 Therefore the parallelogram LB equal to the given triangle C has been applied to the given straight line AB, in the angle ABM which equals D. Q.E.F. There are two steps in this construction. The first uses proposition I.42 to construct some parallelogram with the correct angle equal to the given triangle. The second uses I.43 to change its length to the proper length. To "apply an area to a line in an angle" means just what this construction accomplishes, namely, to construct a parallelogram equal to that area with one side as the given line and one angle equal to the given angle. In practice the angle is often a right angle. The given line may be thought of as a "unit" line. Then the length of the resulting rectangle represents the the area. Use of Proposition 44 Besides being used in the next proposition, this construction is used in VI.25 to make a figure similar to one rectilinear figure but equal to another. Next proposition: I.45 Previous: I.43 Book I introduction © 1996, 1997 D.E.Joyce Clark University Proposition 45 To construct a parallelogram equal to a given rectilinear figure in a given rectilinear angle. Let ABCD be the given rectilinear figure and E the given rectilinear angle. It is required to construct a parallelogram equal to the rectilinear figure ABCD in the given angle E. Join DB. Construct the parallelogram FH equal to the triangle ABD in the angle HKF which equals E. Apply the parallelogram GM equal to the triangle DBC to the straight line GH in the angle GHM which equals E. Post.1 I.42 I.44 Since the angle E equals each of the angles HKF and GHM, therefore the angle HKF also equals the angle GHM. C.N.1 Add the angle KHG to each. Therefore the sum of the angles FKH and KHG equals the sum of the angles KHG and GHM. C.N.2 But the sum of the angles FKH and KHG equals two right angles, therefore the sum of the angles KHG and GHM also equals two right angles. I.29 C.N.1 Thus, with a straight line GH, and at the point H on it, two straight lines KH and HM not lying on the same side make the adjacent angles together equal to two right angles, therefore KH is in a straight line with HM. I.14 Since the straight line HG falls upon the parallels KM and FG, therefore the alternate angles MHG and HGF equal one another. I.29 Add the angle HGL to each. Then the sum of the angles MHG and HGL equals the sum of the angles HGF and HGL. C.N.2 But the sum of the angles MHG and HGL equals two right angles, therefore the sum of the angles HGF and HGL also equals two right angles. Therefore FG is in a straight line with GL. I.29 C.N.1 I.14 Since FK is equal and parallel to HG, and HG equal and parallel to ML also, therefore KF is also equal and parallel to ML, and the straight lines KM and FL join them at their ends. Therefore KM and FL are also equal and parallel. Therefore KFLM is a parallelogram. I.34 I.30 C.N.1 I.33 Since the triangle ABD equals the parallelogram FH, and DBC equals GM, therefore the whole rectilinear figure ABCD equals the whole parallelogram KFLM. C.N.2 Therefore the parallelogram KFLM has been constructed equal to the given rectilinear figure ABCD in the angle FKM which equals the given angle E. Q.E.F. With this construction any rectilinear area can be applied to a line in an angle, that is, it can be transformed into a parallelogram with whatever angle you want and with one side whatever you want. That is a satisfactory solution to the question "what's the area of this figure?" But the question "what's the area of a circle?" is not answered in the Elements. See the note on squaring the circle after proposition II.14 for more discussion of this question. Use of Proposition 45 This construction is used in propositions II.14, VI.25, and XI.32. Like many of the other constructions in Book I, it is used to make constructions in different planes as is done in XI.32. Next proposition: I.46 Previous: I.44 Book I introduction © 1996 D.E.Joyce Clark University Proposition 46 To describe a square on a given straight line. Let AB be the given straight line. It is required to describe a square on the straight line AB. Draw AC at right angles to the straight line AB from the point A on it. Make AD equal to AB. Draw DE through the point D parallel to AB, and draw BE through the point B parallel to AD. I.11 I.3 I.31 Then ADEB is a parallelogram. Therefore AB equals DE, and AD equals BE. I.34 But AB equals AD, therefore the four straight lines BA, AD, DE, and EB equal one another. Therefore the parallelogram ADEB is equilateral. I say next that it is also right-angled. Since the straight line AD falls upon the parallels AB and DE, therefore the sum of the angles BAD and ADE equals two right angles. I.29 But the angle BAD is right, therefore the angle ADE is also right. And in parallelogrammic areas the opposite sides and angles equal one another, therefore each of the opposite angles ABE and BED is also right. Therefore ADEB is right-angled. I.34 And it was also proved equilateral. Therefore it is a square, and it is described on the straight line AB. I.Def.22 Q.E.F. We now have the second regular polygon, the first being the equilateral triangle of proposition I.1. Regular polygons with 5, 6, and 15 sides are constructed in Book IV. Consruction steps There are quite a few steps needed to construct a square on AB. In order to construct the perpendicular AC, first AB has to be extended in the direction of A and a point F on the far side the same distance from A as B is, then two more circles centered at B and F to get a perpendicular line, and then it needs to be cut off at length C, but fortunately, the needed circle has already been drawn. Next, EB is to be drawn through B parallel to AD. In general that construction given in I.31 takes six circles, but in this case if EB is drawn perpendicular to AB, then it will be parallel to AD, too, and that construction only takes three circles with radii BA, AG, and GA. This abbreviation of Euclid's construction requires six circles and four lines. There are alternate constructions that are a bit shorter. For instance, E may be found as the other intersection of the circles of radii BA and DA. Use of Proposition 46 The construction of a square given in this proposition is used in the next proposition, numerous propositions in Book II, and others in Books VI, XII, and XIII. Next proposition: I.47 Previous: I.45 Book I introduction © 1996 D.E.Joyce Clark University Proposition 48 If in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right. In the triangle ABC let the square on one side BC equal the sum of the squares on the sides BA and AC I say that the angle BAC is right. Draw AD from the point A at right angles to the straight line AC. Make AD equal to BA, and join DC. I.11 I.3 Post.1 Since DA equals AB, therefore the square on DA also equals the square on AB. Add the square on AC to each. Then the sum of the squares on DA and AC equals the sum of the squares on BA and AC. C.N.2 But the square on DC equals the sum of the squares on DA and AC, for the angle DAC is right, and the square on BC equals the sum of the squares on BA and AC, for this is the hypothesis, therefore the square on DC equals the square on BC, so that the side DC also equals BC. I.47 C.N.1 Since DA equals AB, and AC is common, the two sides DA and AC equal the two sides BA and AC, and the base DC equals the base BC, therefore the angle DAC equals the angle BAC. But the angle DAC is right, therefore the angle BAC is also right. I.8 Therefore if in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right. Q.E.D. This proposition is the converse of the previous. It is used in proposition XI.35. Next book: Book II introduction Previous proposition: I.47 Book I introduction © 1996 D.E.Joyce Clark University Using the Geometry Applet The Geometry Applet is used to illustrate the figures in the Elements. With the help of this applet, you can manipulate the figures by dragging points. In order to take advantage of this applet, be sure that you have enabled Java on your browser. If you disable Java, or if your browser is not Java-capable, then the illustrations in the elements will still appear, but as plain images. If you click on a point in the figure, you can usually move it in some way. The free points, usually colored red, can be freely dragged about, and as they move, the rest of the diagram (except the other free points) will adjust appropriately. Sliding points, usually colored orange, can be dragged about like the free points, except their motion is limited to either a straight line, a circle, a plane, or a sphere, depending on the point. Other points can be dragged to translate the entire diagram. But if a pivot point appears, usually colored green, then the diagram will be rotated and scaled around that pivot point. Take, for example, the figure below showing the relation between a tetrahedron and a cube inscribed in a sphere. The diameter of the sphere has length AB, and you can drag the endpoints A and B to change the size of the sphere. The side of the cube has length BD, and the side of the tetrahedron has length AD. The cube is drawn with red edges while the tetrahedron is shaded light blue and drawn with blue edges. The center of the sphere is the red dot, and you can drag it to move the sphere around. The point E can be dragged anywhere on the surface of the sphere. The point F has to be at length BD from E on the surface of the sphere, and so it drags along a certain circle on the sphere. The rest of the cube and tetrahedron are then determined. (See proposition XIII.15 for background on the mathematics.) ***If your browser doesn't deal with java applets, then the illustrations in the Elements will still appear but only as plain images and can't be manipulated. Those images were captured from the running Geometry Applet.*** Note that you can't drag a point off the diagram, but frequently parts of the diagram will be moved off as you drag other points around. But if you type r or the space key while the cursor is over the diagram, then the diagram will be reset to its original configuration. You can also lift the figure off the page into a separate window. When you type u or return the figure is moved to its own window. Typing d or return while the cursor is over the original window will return the diagram to the page. Note that you can resize the floating window to make the diagram larger. Table of Contents Copyright © 1996, 1997. http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University These pages are located at http://aleph0.clarku.edu/~djoyce/java/elements/elements.html. About the Text This text of this version of Euclid's Elements is similar to Heath's edition which he translated from Heiberg's definitive edition in Greek, but it is slightly less literal to make it more readable. ● Heath, Sir Thomas Little (1861-1940) The thirteen books of Euclid's Elements translated from the text of Heiberg with introduction and commentary. Three volumes. University Press, Cambridge, 1908. Second edition: University Press, Cambridge, 1925. Reprint: Dover Publ., New York, 1956. Reviewed: Isis 10 (1928), 60-62. ● Heiberg, J. L. (Johan Ludwig) (1854-1928), and H. Menge. Euclidis opera omnia. 8 vol. & supplement, in Greek. Teubner, Leipzig, 1883-1916. Edited by J. L. Heiberg and H. Menge. Heath's excellent critical commentary is as important as the text itself, and since Heath's edition is in publication (Dover), a purchase of that edition is recommended. The text of Heath's translation of Euclid's Elements is also available on-line at the Perseus Project at Tuft's University starting with the first definition of book I. Not just Heath's translation, but his commentary as well as the Greek text is available at the Perseus Project. (The figures are not included.) Other versions of Euclid's Elements I have also used other versions of the Elements for this translation including ● Peyrard, F. Les Oeuvres d'Euclide, en Grec, en Latin et en Français. Three volumes. M. Patris, Paris, 1814. ● Todhunter's edition (1862) of Simson's translation (various editions from 1756-1830) which was published in 1933 by Dent, London, with an introduction by Heath. Copyright © 1996, 1997. http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University These pages are located at http://aleph0.clarku.edu/~djoyce/java/elements/elements.html. Euclid Little is known about Euclid's actual life. He was living in Alexandria about 300 B.C.E. based on a passage in Proclus' Commentary on the First Book of Euclid's Elements. Indeed, much of what is known or conjectured is based on what Proclus says. After mentioning two students of Plato, Proclus writes All those who have written histories bring to this point their account of the development of this science. Not long after these men came Euclid, who brought together the Elements, systematizing many of the theorems of Eudoxus, perfecting many of those of Theatetus, and putting in irrefutable demonstrable form propositions that had been rather loosely established by his predecessors. He lived in the time of Ptolemy the First, for Archimedes, who lived after the time of the first Ptolemy, mentions Euclid. It is also reported that Ptolemy once asked Euclid if there was not a shorter road to geometry that through the Elements, and Euclid replied that there was no royal road to geometry. He was therefore later than Plato's group but earlier than Eratosthenes and Archimedes, for these two men were contemporaries, as Eratosthenes somewhere says. Euclid belonged to the persuasion of Plato and was at home in this philosophy; and this is why he thought the goal of the Elements as a whole to be the construction of the so-called Platonic figures. (Proclus, ed. Friedlein, p. 68, tr. Morrow) It is apparent that Proclus had no direct evidence for when Euclid lived, but managed to place him between Plato's students and Archimedes, putting him, very roughly, about 300 B.C.E. Proclus lived about 800 years later, in the fifth century C.E. There are a few other historical comments about Euclid. The most important being Pappus' (fourth century C.E.) comment that Apollonius (third century B.C.E.) studied "with the students of Euclid at Alexandria." Thus, we know almost nothing about Euclid's life. But we have more of his writings than any other ancient mathematician. Besides the Elements, there are the Data, On Divisions of Figures, the Phaenomena, and the Optics. All are included in the Euclidis opera omnia of Heiberg and Menge (see below) in Greek and translated into Latin. Other translations are listed below. Euclid also wrote other books which no longer exist but were mentioned by later writers. They include Surface Loci, Porisms, Conics, and the Pseudaria (that is, the Book of Fallacies). ● Archibald, Raymond Clare (1875-1957). Euclid's book on division of figures. Cambridge University Press, Cambridge, 1915. ● Berggren, J. L. Euclid's Phaenomena: a translation and study of a Hellenistic treatise in spherical astronomy. Garland, 1996? ● Bretschneider, Karl Anton. Die Geometrie und die Geometer vor Eukleides; ein historischer Versuch. Teubner, Leipzig, 1870. ● Busard, H.L.L. First Latin translation of Euclid's "Elements" commonly ascribe to Adelard of Bath. Pontifical Institute. ● Chasles, M. (Michel) (1793-1880) Les trois livres de porismes d'Euclide, rétablis ... d'aprés la notice ... de Pappus. MalletBachelier, Paris, 1860. ● Frankland, William Barrett. The first book of Euclid's Elements with a commentary based principally upon that of Proclus Diadochus. Cambridge Univ Press, New York, 1905. ● Heath, Sir Thomas Little (1861-1940) The thirteen books of Euclid's Elements translated from the text of Heiberg with introduction and commentary. Three volumes. University Press, Cambridge, 1908. Second edition: University Press, Cambridge, 1925. Reprint: Dover Publ., New York, 1956. Reviewed: Isis 10 (1928),60-62. The text of Heath's translation of Euclid's Elements is also available on-line at the Perseus Project at Tuft's University starting with the first definition of book I. ● Heiberg, J. L. (Johan Ludwig) (1854-1928) Euclidis opera omnia. 8 vol. & supplement. 1883-1916. Edited by J. L. Heiberg and H. Menge. ● Kayas, G. J. Les Eléments. CNRS, 1978. ● Knorr, Wilbur Richard The evolution of the Euclidean elements. A study of the theory of incommensurable magnitudes and its significance for Greek geometry. Synthese Historical Library, vol. 15. Reidel, Dordrecht-Boston, 1975. Reviewed: MR 57#12003. ● Morrow, Glenn R. Proclus: A commentary on the first book of Euclid's elements. Translated by G. R. Morrow. Princeton Univ Press, Princeton, 1970. ● Mueller, Ian. Philosophy of mathematics and deductive structure in Euclid's Elements. MIT Press, Cambridge, Mass., 1981. ● Schmidt, Robert. Euclid's Recipients, commonly called the Data. Golden Hind Press, 1988. ● Taisbak, C. M. Colored quadrangles. A guide to the tenth book of Euclid's Elements. Opuscula Graecolatina, 24. Museum Tusculanum Press, Copenhagen, 1982. Reviewed: MR 84i:01022. ● Thomas-Stanford, Charles Early editions of Euclid's Elements. Bibliographical Society, London, 1926. Reviewed: Isis 10 (1928), 59-60. ● Thomson, William. Pappus' commentary on Euclid's Elements. Cambridge, 1930. Review: Isis 16 (1931), 132-136. Table of Contents Copyright © 1998. http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University These pages are located at http://aleph0.clarku.edu/~djoyce/java/elements/elements.html. A Quick Trip through the Elements To get an idea of what's in the Elements, here are a few highlights in the order that they appear: Book I on basic plane geometry ● Def. I.23, definition of parallel lines, one of many definitions in Book I ● Post. I.5, the parallel postulate ● Common notions, the axioms for magnitudes ● Prop. I.1, the first proposition which shows how to construct an equilateral triangle ● The congruence theorems for triangles: Prop. I.4, side-angle-side, Prop. I.8, side-side-side, and Prop. I.26, angle-side-angle ● Propositions on isosceles triangles: Prop. I.5, equal angles imply equal sides, and the converse, Prop. I.6, equal sides imply equal angles ● Prop. I.9 and Prop. I.10, constructions to bisect angles and line segments ● Prop. I.11 and Prop. I.12, constructions to draw perpendicular lines ● Prop. I.16, an exterior angle of a triangle is greater than either of the opposite interior angles (compare I.32 below) ● Prop. I.29, about angles made when a line crosses two parallel lines ● Prop. I.20, the triangle inequality (the sum of two sides is greater than the third) ● Prop. I.22, to construct a triangle with given sides ● Prop. I.32, an exterior angle of a triangle is the sum of the two opposite interior angles; the sum of the three interior angles equals two right angles. ● On application of areas: Prop. I.42 to find a parallelogram equal in area to any given triangle, and Prop. I.45 to find a parallelogram equal in area to any given polygon ● Prop. I.47, the Pythagorean theorem and its converse Prop. I.48 Book II on geometric algebra ● Prop. II.4, a geometric version of the algebraic identity (x + y)2 = x2 + 2xy + y< /i>2 ● Prop. II.5, a sample proposition showing how to factor the difference of two squares ● Prop. II.6, a geometric version to solve the quadratic equation (b – x)x = c ● Prop. II.11, construction to cut a line in the golden ratio ● Prop. II.12 and Prop. II.13, a pre-trigonometry version of the the law of cosines ● Prop. II.14, a final proposition on application of areas-to find a square equal in area to any given polygon Book III on circles and angles ● Prop. III.1, how to find the center of a circle ● Prop. III.17, how to draw a line tangent to a circle ● Propositions on angles in circles: Prop. III.20, Prop. III.21, and Prop. III.22 ● Prop. III.31, Thales' theorem that an angle inscribed in a semicircle is right, and similar statements giving acute and obtuse angles ● Prop. III.35, when two chords are drawn through a point inside a circle, then the product of the two segments of one chord equals the product of the two segments of the other chord ● Prop. III.36, if from a point outside a circle both a tangent and a secant are drawn, then the square of the tangent is the product of the whole secant and the external segment of the secant, and the converse in Prop. III.37 Book IV on constructions of regular polygons ● Inscribing and circumscribing circles and arbitrary triangles Prop. IV.2, Prop. IV.3, Prop. IV.4, and Prop. IV.5, ● Prop. IV.10, how to construct a particular triangle needed for regular pentagons ● Prop. IV.11, how to construct a regular pentagon ● Prop. IV.15, how to construct a regular hexagon ● Prop. IV.16, how to construct a regular 15-gon Book V on Eudoxus' abstract theory of ratio and proportion, abstract algebra ● Def. V.3, the definition and nature of ratio ● Def. V.5 and V.6, the definition of proportion (equality of ratios) ● Def. V.9, the definition of duplicate proportion (the square of a ratio) ● Prop. V.2, distributivity of multiplication over addition ● Prop. V.3, associativity of multiplication of whole numbers ● Prop. V.11, transitivity of equality of ratios ● Prop. V.16, alternate proportions ● Prop. V.22, ratios ex aequali Book VI on similar figures and geometric proportions ● Def. VI.1, definition of similar figures ● Prop. VI.1, areas of triangles (also parallelograms) of the same height are proportional to their bases ● Prop. VI.2, a line parallel to the base of a triangle cuts the sides proportionally ● Propositions on similar triangles: Prop. VI.4, Prop. VI.5, ● Prop. VI.6, side-angle-side similarity theorem ● Prop. VI.9, to cut a line into a given number of equal segments ● Prop. VI.10, to cut a line into a specified ratio ● Constructions of fourth proportionals Prop. VI.12, and mean proportionals Prop. VI.13, ● Prop. VI.16, if four lines are proportional, w:x = y:z, then the rectangle contained by the extremes, w by z, has the same area as the rectangle contained by the means, x by y ● Prop. VI.19, on areas of similar triangles ● Prop. VI.25, on application of areas ● Prop. VI.31, a generalization of the Pythagorean theorem to figures other than squares Book VII on basic number theory ● Def. VII.11, definition of prime number ● Prop. VII.12, the Euclidean algorithm for finding greatest common divisors ● Several basic properties of numbers, such as Prop. VII.16, commutativity of multiplication of numbers, mn = nm. ● Prop. VII.29, if a prime number doesn't divide a number, then it's relatively prime to it ● Prop. VII.30, if a prime number divides a product of two numbers, then it divides one of them ● Prop. VII.34, on constructing least common multiples Book VIII on continued proportions (geometric progressions) in number theory ● Prop. VIII.2 and Prop. VIII.4, on finding continued proportions of numbers ● Many propositions on squares and cubes, such as Prop. VIII.22, if three numbers are in continued proportion, and the first is square, then the third is also square Book IX on number theory ● Prop. IX.14, a partial version of the fundamental theorem of arithmetic that says no prime number can divide a product of other prime numbers ● Prop. IX.20, there are infinitely many prime numbers ● Several propositions on even and odd numbers, such as Prop. IX.23 which says that if you add an odd number of odd numbers together, then the sum is odd ● Prop. IX.35, how to get the sum of a geometric progression ● Prop. IX.36, on perfect numbers Book X on classification of irrational magnitudes ● Def. X.1, definition of commensurable magnitudes ● Prop. X.1, a principle of exhaustion ● Prop. X.2, a characterization of incommensurable magnitudes ● Prop. X.9, commensurability in square as opposed to commensurability in length ● Prop. X.12, transitivity of commensurability ● Lemma 1 for Prop. X.29, to find two square numbers whose sum is also a square Book XI on basic solid geometry ● Def. XI.14, definition of a sphere ● Def. XI.25 through 28, definitions of regular polygons ● Prop. XI.3, the intersection of two planes is a straight line ● Prop. XI.6, two lines perpendicular to a plane are parallel ● Constructions to draw lines perpendicular to planes: Prop. XI.11 and Prop. XI.12 ● Prop. XI.14, two planes perpendicular to the same line are parallel ● Prop. XI.23, how to construct solid angles ● Several propositions on volumes of parallelopipeds, such as Prop. XI.32 ● Prop. XI.39 on volumes of prisms Book XII on measurement of solids ● Prop. XII.2, areas of circle are proportional to the squares on their diameters ● Prop. XII.6 and Prop. XII.7, a triangular prism can be divided into three pyramids of equal volume, hence, the volume of a pyramid is one third of that of the prism with the same base and same height ● Prop. XII.10, the volume of a cone is one third of that of the cylindar with the same base and same height ● Prop. XII.11, volumes of cones and cylinders are proportional to their heights ● Prop. XII.18, on volumes of spheres Book XIII on constructing regular polyhedra ● Prop. XIII.9, on hexagons and decagons inscribed in a circle, and the golden ratio ● Prop. XIII.10, on hexagons and decagons inscribed in a circle, and the golden ratio ● Prop. XIII.11, when a pentagon, hexagon, and decagon are inscribed in a circle, the square on the side of the pentagon equals the sum of the squares on the sides of the hexagon and the decagon ● Constructions of regular polyhedra XIII.13, XIII.15, XIII.14, XIII.16, and XIII.17 ● These five are shown to be the only regular solids in proposition XIII.18. Table of Contents Copyright © 2002. http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University These pages are located at http://aleph0.clarku.edu/~djoyce/java/elements/elements.html. References on the web This version of Euclid's Elements is located at http://aleph0.clarku.edu/~djoyce/java/elements/elements.html. The site http://www.euclides.org/ in Catalan has links for Euclid and the Elements. It includes a translation into Catalan of the statments of the definitions, postulates, and axioms of the Elements. The text of Heath's translation of Euclid's Elements on-line at the Perseus Project, http://www.perseus.tufts.edu/, at Tuft's University. Not just Heath's translation, but his commentary as well as the Greek text is available at the Perseus Project. In 1847 Oliver Byrne designed a wonderful version of the first six books of the Elements with an imaginative use of color to illustrate geometry. Bill Casselman has made this available at http://www.math.ubc.ca/people/faculty/cass/Euclid/byrne.html, at the University of British Columbia. Copyright © 1996, 1997, 2003. http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E.Joyce Dept. Math. & Comp. Sci. Clark University These pages are located at http://aleph0.clarku.edu/~djoyce/java/elements/elements.html. Subject index for the Elements Headings A B C D E F G H I J K L M N O P Q R S T U V W Y Z Numbers & symbols A acute angle. See angle, acute. algorithm, Euclidean See Euclidean algorithm. alternate angles I.27 alternate proportions and ratios definition V.Def.12 for magnitudes V.16 for numbers VII.13 amicable numbers VII.Def.22 angle (plane) See also solid angle. obtuse angle I.Def.12 alternate angles I.27 bisection I.9 construction I.23 definition I.Def.8, I.Def.9 exterior angle I.16, I.32 horn angle I.Def.8, III.16, V.Def.4 angles as magnitudes I.Def.9 proportional to arc VI.33 in a segment III.Def.8 obtuse angle I.Def.11 of a segment III.Def.7 on a circumference III.Def.9, III.26, III.27 rectilinear angle I.Def.9 right angle I.Def.10 right angles are equal Post.4 angles about a transversal I.27, I.28, I.29, trisection Post.2 two right angles are straight I.13, I.14 vertical angles I.15 antecedents in proportions V.Def.11 antenaresis See Euclidean algorithm. application of areas in an angle I.42, I.44, I.45 exceeding by a parallelogram VI.29 exceeding by a square II.6 falling short by a parallelogram VI.27 VI.28 falling short by a square II.5 approximation of circles by polygons XII.2, Apollonius of Perge (ca. 250-175 B.C.E.) terms for conic sections XI.Def.18 arc proportional to angle VI.33 Archimedes of Syracuse (ca. 287-212 B.C.E.) angle trisection Post.2 neusis Post.2 property of magnitudes X.1 area of a triangle Heron's formula IV.4 arithmetic, fundamental theorem of VII.31 arithmetic mean or average V.25 associativity of addition for magnitudes C.N. associativity of multiplication for magnitudes V.3 average, arithmetic and geometric V.25 axiom axiom of comparability V.Def.4 for magnitudes C.N. axis of a cone XI.Def.19 of a cylinder XI.Def.22 of a sphere XI.Def.15 B base of a cone XI.Def.19 of a cylinder XI.Def.23 of a triangle I.4 bisect an angle I.9 a circumference (arc) III.30 a line I.10 boundary I.Def.13 Brouwer nonconstructive fixed point theorem I.5 C cancellation for addition C.N. in proportions V.9 center of a circle characterization III.9 construction III.1 definition I.Def.16 intersecting circles have distinct centers III.5 tangent circles have distinct centers III.6 Chrysippus (280–207) 1 as a number VII.Def.1-2 circumference a circumference (arc) III.30 circle area of XII.2, central angle double angle at circumference III.20 chord inside circle III.2 center of. See center of a circle. construct circle from segment III.25 construction Post.3 definition I.Def.15 diameter of. See diameter. equal angles in segments III.21 equal chords at equal distances III.14 equal circles III.Def.1 intersection of circles III.10 product of secants III.37 product of secants equals tangent2 III.36 products of chord sections III.35 proportional to diameter2 XII.2, radius of. See radius of a circle. right angle in semicircle III.31 sector of. See sector of a circle. segment of. See segment of a circle. tangent to. See tangent. circumcircle of a triangle IV.5 circumference proportional to angle VI.33 circumscribed figures circle circumscribed about a pentagon IV.14 circle circumscribed about a rectilinear figure IV.Def.6 circle circumscribed about a square IV.9 circle circumscribed about a triangle IV.5 pentagon circumscribed about a circle IV.12 rectilinear figure circumscribed about a circle IV.Def.4 rectilinear figure circumscribed about a rectilinear figure IV.Def.2 square circumscribed about a circle IV.7 triangle circumscribed about a circle IV.3 commensurable definition X.Def.1 and numerical ratios V.Def.5 in square X.Def.2 magnitudes and numerical ratios X.5,, X.6, X.7, X.8 common notions C.N. commutativity for addition of magnitudes C.N. of multiplication VII.15 VII.18 compass construction Post.3 componendo V.Def.14 composite numbers definition VII.Def.13 divisible by a prime VII.31 cone axis XI.Def.18 base XI.Def.19 cone one third of cylinder XII.10 definition XI.Def.20 proportional to base XII.11 proportional to height XII.14 reciprocally proportional XII.15 right-angled, acute-angled, obtuse angled XI.Def.18 similar cones XI.Def.24, XII.12 congruent figures I.4 solids XI.Def.10 congruence propositions for triangles. See triangle. consequents in proportions V.Def.11 constructions, 2and 3-dimensional XI.20 continued proportion VIII.1 in lowest terms VIII.1, VIII.2, VIII.3, VIII.4 sum of a IX.35 contradiction, proof by I.5 contrapositive proposition I.27 converse of a proposition I.5 conversion of a proportion or ratio definition V.Def.16 proposition for magnitudes V.19 convertendo V.Def.16 cosines, law of II.12, II.13 cross multiplication of proportions for lines VI.16 for numbers VII.19 cube construction XIII.15 definition XI.Def.25 relation to dodecahedron XIII.17 relation to tetrahedron XIII.15 cubic numbers VII.Def.19, IX.3, IX.4, IX.5, IX.6 cut into extreme and mean ratio. See extreme and mean ratio. cylinder axis of XI.Def.22 bases of XI.Def.23 cone one third of cylinder XII.10 definition XI.Def.21 proportional to base XII.11 proportional to height XII.13, XII.14 reciprocally proportional XII.15 similar cylinders XI.Def.24, XII.12 D decagon, regular (10-gon) side of hexagon to side of decagon XIII.9 sides of pentagon, hexagon, & decagon XIII.10 Descartes (1591-1661) geometric algebra VI.12 diameter of a circle bisecting chord III.3 definition I.Def.17 diameter is greatest chord III.15 distance, line to point III.Def.4 distributivity of division over addition VII.5 of division over subtraction VII.7 of multiplication over addition for lines II.1, II.2 for magnitudes V.1, V.2 for numbers VII.6, VII.8 of multiplication over subtraction for magnitudes V.5, V.6 divisor of a number VII.Def.3 dodecahedron construction XIII.17 definition XI.Def.28 relation to cube XIII.17 dual of a polyhedron XIII.14 duplicate ratio V.Def.9 E elegance in mathematics I.30 ellipse XI.Def.18 elliptic geometry I.16 equal circles III.Def.11 equal and similar solids XI.Def.10 equilateral triangle (60°-60°-60° triangle) construction I.1 definition I.Def.20 side of XIII.12 equivalence relation V.Def.3 equality as an equivalence relation C.N. proportion as an equivalence relation V.Def.5 Euclid (fl. ca. 300 B.C.E.). Euclidean algorithm VII.2, VII.3, X.3 characterization of incommensurability of magnitudes X.2 test for relatively prime numbers VII.1 Eudoxus (ca. 408-355 B.C.E) definition or proportion V.Def.6 principle of exhaustion XII.2 property of magnitudes X.1 even even number VII.Def.6, IX.21, IX.24, IX.27, IX.28, IX.30 even-times even number VII.Def.8, IX.32, IX.34 even-times odd number VII.Def.9, IX.33, IX.34 ex aequali ratios and proportions definition V.Def.17 for magnitudes V.22 for numbers VII.14 excircle of a triangle IV.4 exhaustion, principle of XII.2 exterior angle greater than opposite interior angle of triangle I.16 sum of opposite interior angles of triangle I.32 extreme and mean ratio algebra on segments XIII.1, XIII.2, XIII.3, XIII.4, XIII.5 construction II.11, VI.30 definition VI.Def.3 is irrational called apotome XIII.6, in a 36°-72°-72° triangle IV.10 in a pentagram IV.11, XIII.8 side of hexagon to side of decagon XIII.9 F face of a solid XI.Def.2 figure I.Def.14 rectilinear I.Def.19 fit a straight line into a circle, construction IV.1 into a circle, definition IV.Def.7 into a diagram Post.2 Fermat, Pierre de (1601-1665). Fermat primes IV.16 Mersenne primes and perfect numbers IX.36 fourth proportionals V.18 friendly numbers VII.Def.22 fundamental theorem of arithmetic VII.31 G Gauss, Carl Friedrich (1777-1855). regular polygons IV.16 GCD. See greatest common divisor. geometric mean or average V.25 geometric progression or sequence. See continued proportion. geometry elliptic I.16 hyperbolic I.29 nonEuclidean Post.5 gnomon II.Def.2 golden ratio. See extreme and mean ratio. greatest common divisor Euclidean algorthim for VII.3, VII.2 for several numbers VII.4 greatest common measure of several commensurable magnitudes X.4 of two commensurable magnitudes X.3 group C.N. H height of a figure VI.Def.4 Heron of Alexandria (ca. 1st century C.E.) defintion of equal and similar solids XI.Def.10 Heron's formula for area of a triangle IV.4 minimum distance problem I.20 hexagon, regular inscribed in a circle IV.15 side of hexagon to side of decagon XIII.9 sides of pentagon, hexagon, & decagon XIII.10 hexahedron, regular. See cube. Hilbert, David (1862-1943) Foundations of Geometry I.4 Hippocrates of Chios (fl. ca. 430 B.C.E.). his Elements I.3 quadrature of lunes VI.31 horn angle. See angle, horn. hyperbola XI.Def.18 hyperbolic geometry I.29 I icosahedron construction XIII.16 definition XI.Def.27 incircle of a triangle IV.4 inclination line to a line. See angle. line to a plane XI.Def.5 plane to a plane XI.Def.6 similar XI.Def.7 incommensurable. See commensurable. infinitude of prime numbers IX.20 inscribed figures 15-gon inscribed in a circle IV.16 circle in a pentagon IV.13 circle in a rectilinear figure IV.Def.5 circle inscribed in a square IV.8 circle inscribed in a triangle IV.4 hexagon inscribed in a circle IV.15 pentagon inscribed in a circle IV.11 rectilinear figure in a circle IV.Def.3 rectilinear figure in a rectilinear figure IV.Def.1 square inscribed in a circle IV.6 triangle inscribed in a circle IV.2 inverse proportions and ratios definition V.Def.13 proposition V.7 inverse proposition I.27 irrational. See rational. irrationality of surds VIII.8 isosceles triangle definition I.Def.20 has equal base angles I.5, I.5 larger vertex angle & larger base I.24, I.24 J jointly ratios and magnitudes taken jointly V.Def.14, V.17, V.18 K L law of cosines II.12, II.13 law of sines I.19 law of trichotomy. See trichotomy. LCM. See least common multiple. least common multiple VII.33, VII.34, VII.35 of several numbers VII.36 Lindemann, Ferdinand (1852-1939) transcendence of pi II.14 line See also straight line. definition I.Def.2 ends of a line I.Def.3 lowest terms VII.20 are are relatively prime VII.21, VII.22, VIII.1 reduce to VII.33 lunes, quadrature VI.31 M magnitude V.Def.1 commensurable. See commensurable infinite and infinitesimal magnitudes V.Def.4 multiple of a magnitude V.Def.2 part of a magnitude V.Def.1 proportional magnitudes V.Def.5 ratio of magnitudes V.Def.3, V.Def.4 magnitudes in the same ratio V.Def.5 marginal references I.1 mean and extreme ratio. See extreme and mean ratio. mean, arithmetic and geometric V.25 Mersenne, Marin (1588-1648). Mersenne primes IX.36 monad, definition VII.Def.1 modern analysis, method of VI.1 multilateral figure I.Def.19. See polygon. multiple a magnitude V.Def.2 of a number VII.Def.5 multiplication of numbers VII.Def.15 N neusis Post.2 nonEuclidean geometry Post.5 number amicable numbers VII.Def.22 composite number VII.Def.13 cubic number VII.Def.19 definition VII.Def.2 divisible by a prime VII.32 divisor of a number VII.Def.3 even. See even number. even-times even. See even number. even-times odd. See even number. friendly numbers VII.Def.22 multiple of a number VII.Def.5 odd. See odd number. odd-times odd. See odd number. part of a number VII.Def.3 parts of a number VII.Def.4 perfect number VII.Def.22 plane number VII.Def.16 prime number VII.Def.11 relatively composite numbers VII.Def.14 relatively prime numbers VII.Def.12 sides of a plane number VII.Def.16 sides of a solid number VII.Def.17 similar plane and solid numbers VII.Def.21 solid number VII.Def.17 square number VII.Def.18 triangular number VII.Def.16 number theory foundations of VII.1 Peano's axioms VII.Def.1 O oblong I.Def.22 obtuse angle. See angle, obtuse. octahedron, regular construction XIII.14 definition XI.Def.26 odd odd number VII.Def.7, IX.22, IX.23, IX.25, IX.26, IX.27, IX.29, IX.30, IX.31 odd-times odd number VII.Def.10 P Pappus of Alexandria (fl. ca. 320 C.E.) proof of I.5 parabola XI.Def.18 parallel lines I.Def.23, I.31 planes XI.Def.8 postulate Post.5 transitivity of parallelism I.30, XI.9 parallelogram area of I.35, I.36 basic properties I.34 definition I.34 about the diameter I.43 equiangular parallelograms proportional to sides VI.23 proportional to base VI.1 reciprocally proportional parallelograms VI.14 similar parallelograms about the diameter VI.24 VI.26 parallelepiped (parallelepipedal solid) bisected by diagonal XI.28 construct similar one XI.27 definition XI.24 equal XI.29, XI.30, XI.31 proportional to base XI.25, XI.32 proportional to sides XI.33, XI.36, XI.37 reciprocally proportional parallelepipeds XI.34 part of a magnitude definition V.Def.1 problem of parts V.5 part of a number definition VII.Def.3 parts of a number definition VII.Def.4 Peano, Giuseppe (1858-1932). Peano's axioms for number theory VII.Def.1 pentagon, regular circumscribed about a circle IV.12 criterion of regularity XIII.7 diagonals cut in extreme and mean ratio XIII.8 inscribed in a circle IV.11 Richmond's construction IV.11 sides of pentagon, hexagon, & decagon XIII.10 side of pentagon is irrational called minor XIII.11 perfect number definition VII.Def.22 construction IX.36 perpendicular, line to a line construction given a point I.11, I.12 definition I.Def.10, perpendicular, line to a plane definition XI.Def.3 propositions XI.4, XI.6, XI.8, XI.11, XI.12, XI.13 perturbed proportion definition V.Def.18 proposition V.22 plane definition I.Def.7 determined by intersecting lines XI.2 determined by triangle XI.2 inclination to a line XI.Def.5 inclination to a plane XI.Def.6 intersection of two planes XI.3 parallel planes XI.Def.8, XI.14, XI.15, XI.16, XI.17 perpendicular to a line XI.Def.3, XI.14 perpendicular to a plane XI.Def.4, XI.18, XI.19 plane angle. See angle. plane number definition VII.Def.16 similar plane numbers VII.Def.21, VIII.26, IX.1, IX.2 proportional to sides VIII.5 Playfair axiom of parallels I.30, point definition I.Def.1 polygons approximating circles XII.2, areas of similar polygons VI.20, XII.1 constructable regular polygons IV.16 polyhedra, regular See tetrahedron, cube, octahedron, icosahedron, and dodecahedron. classification XIII.18 duals of XIII.14 Pons Asinorum I.5 postulates Post.1-5 powers of 2 IX.32 prime numbers definition VII.Def.11 dividing products VII.30 Fermat primes IV.16 infinitude of IX.20 Mersenne primes IX.36 powers of IX.13 products of IX.14 relatively prime VII.Def.12 principle of exhaustion XII.2, prism See also parallelepiped. defintion XI.Def.13 equal prisms XI.39 triangular prism partitioned into three equal pyramids XII.5, Proclus (410-485 C.E.) Commentary on Book I I.3 proof by contradiction I.5 nonconstructive I.5 progression, geometric. See continued proportion. proportion alternate proportions V.Def.12, V.16 VII.13 antecedents in proportions V.Def.11 consequents in proportions V.Def.11 continued. See continued proportion. conversion of a proportion V.Def.16, VII.19 cross multiplication VII.19 definition V.Def.6 proportions as equivalence relations V.Def.5 proportions ex aequali V.Def.17, V.22 VII.14 inverse proportions V.Def.13 V.7 magnitudes V.Def.6 numbers VII.Def.20 proportions taken jointly V.Def.14, V.17, V.18 perturbed proportion V.Def.18, V.22 proportions taken separately V.Def.15, V.17, V.18 operations on proportions V.Def.3 proportion in three terms V.Def.8 reciprocal. See reciprocal proportion transitivity V.11 proportional construct third proportional VI.11 construct fourth proportional VI.12 construct mean proportional VI.13 fourth proportionals V.18 fourth proportional of numbers IX.19 magnitudes V.Def.6 mean proportionals between cubic numbers VIII.12 mean proportional between similar plane numbers VIII.18, VIII.20 mean proportionals between similar solid numbers VIII.19, VIII.21 mean proportional between square numbers VIII.11 numbers VII.Def.20 third proportional of numbers IX.18 proposition contrapositive I.27 converse of I.5 inverse of I.27 pyramid See also tetrahedron, regular defintion XI.Def.12 pyramids proportional to their sides XII.8 pyramids proportional to their bases XII.5, XII.6 pyramid third of prism with same base XII.5 reciprocally proportional pyramids XII.9 Pythagorean theorem I.47 converse I.48 generalized to similar figures VI.31 Q Q.E.D. and Q.E.F. I.1 quadratic equation, solution by application of areas II.5, II.6, VI.28, VI.29 quadrilateral figure I.Def.19 quadrature of circles II.14, XII.2, of lunes VI.31 of rectilinear figures II.14 quadrilateral Varignon parallelogram of a XI.9 R radius of a circle definition I.Def.15 perpendicular to tangent III.18, III.19 ratio alternate ratio V.Def.12, V.16, VII.13 compounded ratio V.Def.3, VIII.5 conversion of a ratio V.Def.16 VII.19 definition V.Def.3 duplicate ratio V.Def.9 extreme and mean. See extreme and mean ratio. ratios ex aequali V.Def.17, V.22 VII.14 greater ratio V.Def.7 inverse ratio V.Def.13 ratios taken jointly V.Def.14, V.17, V.18 in lowest terms VII.20 ratios of magnitudes V.Def.4 magnitudes in the same ratio V.Def.5 mixed ratio V.Def.3 nature of ratios V.Def.3 numerical ratio VII.Def.20, V.Def.5 operations on ratios V.Def.3 ratios taken separately V.Def.15, V.17, V.18 ratios of more than two terms V.Def.3 ratios of various kinds V.Def.3 triplicate ratio V.Def.9 rational line X.Def.3 number V.Def.3 numbers and commensurable magnitudes X.5, X.6, X.7, X.8 squares and areas X.Def.4 reciprocally proportional figures definition VI.Def.2 parallelograms VI.14 pyramids XII.9 triangles VI.15 rectangle (rectangular parallelogram) contained by sides II.Def.1 rectilinear figure definition I.Def.19 reflexive relation. See equivalence relation. regular polygons, constructable IV.16 relation equivalence relation V.Def.3 reflexive relation V.Def.3 symmetric relation V.Def.3 transitive relation V.Def.3 relatively composite numbers VII.Def.14 relatively prime numbers definition VII.Def.12 are in lowest terms VII.21, VII.22 numbers dividing them are VII.23 primes are VII.29 products of VII.24, VII.25, VII.26, VII.27 sums of VII.28 revolution, solid of XI.Def.14 rhombus & rhomboid I.Def.22 right triangles. See triangles, right. S scalene triangle definition I.Def.20 section into extreme and mean ratio. See extreme and mean ratio. sector of a circle definition III.Def.10 segment of a circle definition III.Def.6 angle in III.Def.8, III.31 angle of III.Def.7 construct circle from segment III.25 equal angles in segments III.21 equal segments III.24 similar segments III.Def.11 separately ratios taken separately V.Def.15, V.17, V.18 VII.11 separando V.Def.15 sequence, geometric. See continued proportion. series (sum), geometric, IX.35 sides of a plane number VII.Def.16 of a solid number VII.Def.17 semicircle definition I.Def.18 semigroup C.N. similar areas of similar polygons VI.20 figures on proportional lines VI.22 equal and similar solids XI.Def.10 plane and solid numbers VII.Def.21 rectilinear figures construction VI.18 similar cylinders and cones XI.Def.24 definition VI.Def.1 construction of given area VI.25 segments of circles III.Def.11 solids XI.Def.9 transitivity of similarity VI.21 triangles, See triangles, similar. sines, law of I.19 solid congruent solids XI.Def.10 definition XI.Def.1 equal and similar solids XI.Def.10 face of XI.Def.1 of revolution XI.Def.14 similar solids XI.Def.9 solid angle definition XI.Def.11, propositions XI.20, XI.21, XI.23, XI.26 solid number definition VII.Def.17 proposition IX.7 similar solid numbers VII.Def.21 VIII.27 sphere axis of XI.Def.15 center of XI.Def.16 defintion XI.Def.14 diameter of XI.Def.17 proportional to diameter3 XII.18 volume XII.18 square construction I.46, definition I.Def.22 of the hypotenuse I.47 square number VII.Def.18 squaring (finding areas). See quadrature. straight line bisection I.10 construct third proportional VI.11 construct fourth proportional VI.12 construct mean proportional VI.13 cut off line I.3 cut off a part VI.9 cut proportionally VI.10 definition I.Def.4 distance to a point III.Def.4 draw between two points Post.1 equidistant lines I.Def.23 extend a line Post.2 fit in a circle IV.Def.7, IV.1 inclination to a plane XI.Def.5 parallel lines I.Def.23, I.31 planarity of XI.1, XI.5 perpendicular lines. See perpendicular, line to a line. perpendicular to a plane. See perpendicular, line to a plane. place a line I.2 tangent. See tangent. substitution of equals C.N. superposition, method of I.4 surface See also plane. definition I.Def.5 edges of a surface I.Def.6 surds, irrationality of VIII.8 symmetric relation. See equivalence relation. T tangent circles definition III.Def.3 have distinct centers III.6 meet at common diameter III.11, III.12 meet at one point III.13 tangent line to a circle definition III.Def.2 construction III.17 perpendicular to radius III.18, III.19 tetrahedron, regular called a pyramid XI.Def.25 construction XIII.13 relation to cube XIII.15 Thales of Miletus (ca. 624-547 B.C.E.) right angle in semicircle III.31 topology I.Def.13 touch. See tangent. transitivity See also equivalence relation. of equality of ratios V.11 of "less than" I.7 of parallel lines I.30, XI.9 of similarity VI.21 transversal, angles about a I.27, I.28, I.29, trapezium I.Def.22 triangle 36°-72°-72° triangle IV.10 acute triangle I.Def.21 angle bisector cuts base proportionally VI.3 area of a triangle I.37, I.38 proportional to base VI.1 similar triangles VI.19 circumcircle of a triangle IV.5 congruence proposition angle-angle-side I.26 angle-side-angle I.26 side-angle-side I.4 side-side-angle I.26 side-side-side I.8 construction given 3 sides I.22 equilateral I.Def.20. See equilateral triangle. excircle of a triangle IV.4 exterior angle sum of opposite interior angles I.32 greater side opposite greater angle I.18, I.19 Heron's formula for area IV.4 incircle of a triangle IV.4 inscribed in a circle IV.2 isosceles triangle I.Def.20 obtuse triangle I.Def.21 parallel cuts sides proportionally VI.2 reciprocally proportional triangles VI.15 right triangle I.Def.21 perpendicular creates similar right triangles VI.8 scalene triangle I.Def.20 similar areas in duplicate ratio VI.19 equiangular triangles are VI.4 proportional triangles are VI.5 side-angle-side proposition VI.6 side-side-angle proposition VI.7 triangle inequality I.20 triangular number VII.Def.16 trichotomy, law of for magnitudes C.N., V.Def.5 in practice I.5 for ratios V.Def.7 trilateral figure I.Def.19. See triangle. triplicate ratio V.Def.9 trisection of an angle Post.2, I.9 U unit, definition VII.Def.1 V Varignon (1654-1722) Varignon parallelogram of a quadrilateral XI.9 vertical angles I.15 W word order I.18 X Y Z Zeno of Sidon (1st century B.C.E) criticism of proposition I.1 Zhou bi suan jing Pythagorean theorem I.47 Symbols 36°-72°-72° triangle IV.10 >=< V.Def.5 Table of Contents

<UN> © koninklijke brill nv, leiden, 2018 | doi 10.1163/17455243-01503004 Manuel Vargas and Gideon Yaffe (eds.) Rational and Social Agency: The Philosophy of Michael Bratman (Oxford: Oxford University Press, 2014), 368 pp. isbn 9780199794515. Hardback: $90.00. I have it on good authority that publishers hate Festschrifts. The reason for their aversion is not difficult to imagine: the typical Festschrift is seen as lacking in novel ideas and arguments, and filled with flattering displays of praise and an absence of critical depth. When viewed in another light, though, the Festschrift presents a unique opportunity to reflect on the work of an eminent scholar, and to raise questions about where a particular field of scholarship has been and is going. The present Festschrift honouring the work of Michael Bratman is nearly perfect in this regard. Without flattery or superficiality, this impressive collection of essays both broadens and refines the extraordinary work that has been devoted to attaining a critical understanding of Bratman's corpus. The collection contains ten essays bookended by a useful introduction by the editors and incisive replies by Bratman. The first four essays, by Richard Holton, Al Mele, Kieran Setiya, and David Velleman, discuss Bratman's planning theory of intention. The next four, by Jay Wallace, Geoffrey Sayre-McCord and Michael Smith, Elijah Millgram, and Christine Korsgaard, examine Bratman's accounts of rational self-governance, autonomy, and identification. The final two, by Margaret Gilbert and Scott Shapiro, focus on Bratman's account of shared agency. In my opinion, none of the essays directly challenges the foundational assumptions that provide the reductive framework within which Bratman operates, but one essay in particular, by Elijah Millgram, comes very close. Hence, in what follows, I begin by sketching two important presuppositions of Bratman's work and then I discuss Millgram's essay and Bratman's response, which struck me as among the most fruitful combination of the bunch. Like many Anglophone philosophers of action working over the past four decades or so, Bratman is interested in differentiating between those attitudes and actions with which you identify as agent and those that you legitimately disown. For Bratman, intentions play a central role in discriminating between the two. On his influential model, intentions are sui generis mental states that function as persisting and stable plans whose normative content specifies which other attitudes you should treat as reasons when deliberating about what to do. Crucially, for Bratman, when you act on the basis of deliberation guided by the relevant intentions, and when the structure of the relevant intentions is consistent and coherent, the intentions are rationally self-governing. That is, your rational self-governance consists in the proper functioning of this system of mental states: when this system causes your behaviour in the appropriate manner, you can be said to govern what is taking place. book reviews 371 Downloaded from Brill.com01/05/2019 09:25:40PM via Columbia University Libraries book reviews journal of moral philosophy 15 (2018) 363-382 <UN> 372 Equally as important, Bratman's account of rational self-governance rejects homuncular models of the agent. Typical homuncular models assume that as a conscious agent you exist in a manner that differs from your system of mental states, stepping back from and reflecting upon that system much like the notorious Cartesian spectre existing as a thinking substance in addition to the parts of which that system is composed. In rejecting homuncular models of the agent, Bratman makes a controversial metaphysical assumption of his own. He assumes that you are identical with a system of mental states unified across time by relations of psychological continuity, which he describes in terms of a broadly Lockean approach to personal identity. That is, you are not a persisting physical object of any kind, e.g., a particular living animal, but a temporally fragmented collection of mental states stitched together by psychologically continuous relations. Together, these assumptions form the core of Bratman's reductive model of rational self-governed action. On the one hand, Bratman assumes that when your action is appropriately caused by the correctly structured intentions, this system of mental states governs what is taking place. On the other hand, Bratman rejects homuncular models of the agent and assumes that you are identical with the relevant system of mental states standing in relations of psychological continuity. Together, these assumptions depict you not as a physical object of any kind, but as a collection of mental states held together by relations of psychological continuity. In one of the most challenging essays of the collection, Elijah Millgram's "Segmented Agency" (pp. 152–89) targets the reductive model of rational selfgoverned agency endorsed by Bratman. Using an evocative example of a Jewish academic living in Germany during the 1930s, Millgram argues that there are times in your life when you experience genuinely unanticipated circumstances that require immediate action, where you must abandon your environment and act in novel ways that cannot be guided by the kinds of persisting and stable intentions that are of interest to Bratman. According to Millgram, intentions make sense only for agents who reside in static environments that are largely predictable, but you are not such an agent. Rather, because you inhabit and experience a world that is deeply surprising, there are many circumstances in which you must move on, exiting one niche while searching for another. In this way, you are a segmented agent whose life is normally divided into parts and who often moves from place to place. If Millgram is correct and we are segmented agents, an important question arises. When you exit one niche and move to another, can this be a rational self-governed action? According to the model defended by Bratman, when you act on the basis of deliberation guided by the relevant intentions, and when the structure of the intentions is consistent and coherent, your intentions Downloaded from Brill.com01/05/2019 09:25:40PM via Columbia University Libraries 373book reviews journal of moral philosophy 15 (2018) 363-382 <UN> rationally guide you and you thereby govern what is taking place. However, according to Millgram, your stable and persisting intentions are niche-specific, so they cannot rationally guide you when leaving a known environment and transitioning to unknown circumstances. But, if that is so, then leaving your niche and heading into the unfamiliar cannot be a rational self-governed action, at least not according to the model defended by Bratman. This, says Millgram, is a problem, for in such cases praiseworthy forms of rational self-governed action occur, where you rely on forms of rationality that are appropriate to segmented agents, such as feelings of boredom or frustration that prompt you to move to a new niche. Thus, for Millgram, the model of rational self-governed agency that Bratman endorses is incomplete, and as a segmented agent you are not identical with a collection of mental states fused together across time by psychologically continuous relations. Rather, you are not a single agent at all, but "the substrate of a series of them" (p. 175), i.e., the substance underlying a series of distinct agents that "you conjure up to meet the needs of the moment" (ibid.), so the broadly Lockean approach to personal identity endorsed by Bratman does not apply. In his response to Millgram, Bratman insists that for a radical shift to be a rationally self-governed action there would have to be some stable, plan-like intention operative on your part, or else when faced with such a fundamental change in the world you would move from niche to niche on the basis of good luck, rather than as a rational self-governed action. Granted, Millgram does not (here) specify how to identify those movements between niches that are the result of good luck and those that are rationally self-governed actions, but Bratman's response does not address the deeper challenge raised by examples of segmented agency. Although Millgram does not quite frame it this way, his example targets the reductive ambition inherent within Bratman's model of rational self-governance. Examples of segmented agency challenge the reductive claim that you are identical with a collection of mental states, by suggesting that there is a real distinction between you and those mental states. Crucially, acknowledging the existence of this distinction does not require that we follow Bratman and assume that we must choose between homuncular and reductive models of the agent. Plausibly, we can recognize this distinction by claiming that, e.g., you are a particular physical object that has conscious mental states, is subject to the occurrence of conscious mental events, stands in relations to other physical objects, undergoes change without annihilation, persists through time while occupying space, and has the ability to perform various intentional actions. If such an alternative is plausible, then, contrary to what Bratman has long assumed to be the case, avoiding the sort of homuncular model of the agent that would make Descartes proud does not require the Downloaded from Brill.com01/05/2019 09:25:40PM via Columbia University Libraries book reviews journal of moral philosophy 15 (2018) 363-382 <UN> 374 reductive assumption that you are identical with a psychologically continuous collection of mental states. Of course, exploring the plausibility of such alternatives is work for future scholars who need not share the controversial, reductive assumptions upon which Bratman has built his model of rational self-governed agency. Perhaps had this Festschrift included additional work that directly challenges such assumptions, the collection would have achieved perfection. This, however, is but a minor concern. The editors are to be commended for assembling an outstanding collection of essays, which, like Bratman's own corpus, are essential reading for anyone working in Anglophone philosophy of action. Michael Brent University of Denver [email protected] Downloaded from Brill.com01/05/2019 09:25:40PM via Columbia University Libraries

Times Literary Supplement, No. 5444 (August 3, 2007): 26. Zero-horrors game Trenton Merricks Peter van Inwagen The Problem of Evil 183 pp. Oxford: Clarendon Press. £ 19.99. 0 19 924560 6 A "proof" is an argument that just about any intellectually honest person would find convincing, unless that person were irrational or stupid or otherwise cognitively defective. Peter van Inwagen, a leading metaphysician and philosopher of religion, considers whether there is a proof for the non-existence of God that starts with instances of evil, such as pain and suffering. Van Inwagen claims that, so far, there have been no proofs in philosophy, at least not of major philosophical theses. Though initially jarring, perhaps this claim should not be surprising. After all, any argument-including any argument aspiring to be a proof- must begin somewhere. It must have premises. And if its conclusion is striking enough, there will almost certainly be some smart people of good will who will find denying one of that argument's premises, even if that denial is itself striking, more reasonable than accepting its conclusion. Given philosophy's track record, van Inwagen doubts that there will be any argument with the conclusion 'God does not exist' that virtually every intellectually honest (etc.) person would find convincing. Indeed, he thinks philosophy's track record 2 suggests that no argument against God's existence would convince even virtually every intellectually honest agnostic, never mind the intellectually honest who approach such an argument believing firmly in God. But there is a first time for everything. So he takes a careful look at arguments that purport to prove, from evil, that God does not exist. Van Inwagen's central objections to these arguments turn on the idea that free will is a great good, even though free creatures might choose to cause pain and suffering and other evils. Of course, this idea is familiar. Van Inwagen's contribution is not this idea itself, but rather the ways in which he motivates it, responds to objections, and brings new clarity to various points. Along the way, he makes many interesting and novel moves. For example, he asks us to suppose the following for the sake of argument: a man commits assault; sending that man to prison brings about the good of deterring further assaults; this good could be had by sending the man to prison for ten years; this same good could be had by sending him to prison for ten years minus one day; in general, sending him to prison for n-1 days deters assault no less than does sending him to prison for n days; finally, each day spent in prison is an evil. Given these suppositions, sending the man to prison for ten years brings about no good not also brought about by sending him to prison for ten years minus one day. So it seems that we ought to send him to prison for ten years minus one day, rather than for the full ten years. At least, this is implied by the following plausible moral principle: if one is able to prevent an evil, one should not allow that evil, unless allowing it thereby brings about a greater good (or prevents a greater evil). 3 But the very same reasoning then tells us that, rather than send the man to prison for ten years minus one day, we ought to send him to prison for ten years minus two days. It then tells us that we ought to send him to prison for ten years minus three days. Eventually, this sort of reasoning will tell us that we should send the man to prison for zero days, that is, not send him to prison at all. Something has gone wrong. Van Inwagen thinks the culprit is the plausible moral principle itself. He takes this reasoning to show that that principle leads to a false result, and so is itself false. Van Inwagen also argues that, given certain conditions, it is good that the world is risky, even to the extent that sometimes this or that 'horror' occurs, such as a brutal murder or a cancerous tumor. And he thinks that there are certain horrors that help to make the world risky, but serve no other good purpose. He adds that no particular such horror is necessary for a risky world. Now consider one such horror. God could have left everything pretty much just as it is, except for preventing that horror, and nothing good would have been lost, not even the riskiness of the world. The most natural reaction is that God should have prevented that horror. But if the aforementioned plausible moral principle is false, this reaction is not obviously right. Moreover, van Inwagen thinks that those who endorse this reaction must add that God should prevent every horror that is, first, not absolutely necessary for the world's riskiness and, second, whose only good is contributing to that riskiness. This addition presupposes, says van Inwagen, that there is some minimum number of horrors necessary for that riskiness. He rejects this presupposition, saying: 'For any n, if the existence of at most n horrors is consistent with God's plan, the existence of at most n-1 horrors will be equally consistent with God's plan' (p. 106). 4 I think that van Inwagen should not reject this presupposition. For suppose God prevents one horror, whose only good would have been contributing to the world's riskiness. That riskiness remains, since van Inwagen thinks that 'God's plan' does not turn on any single horror. Suppose God prevents another such horror. The riskiness remains. You see where this is going. Eventually, we get the result that zero horrors are necessary for riskiness. Something has gone wrong. I conclude that the culprit is van Inwagen's thesis that there is no minimum number of horrors necessary for riskiness. Van Inwagen will reject my reasoning, which is meant to show that his 'no minimum number' principle is false. But that reasoning mimics his own argument against the above plausible moral principle. So either his argument against the plausible moral principle fails or his 'no minimum number' principle is false. Something has to give. Even these critical remarks point to the virtues of van Inwagen's book. For example, consider van Inwagen's idea that there is no minimum number of horrors necessary to achieve a particular good. This idea, van Inwagen argues, implies that God, even if all-powerful and morally perfect, can achieve that good only by arbitrarily 'drawing a line' between the horrors allowed and the horrors prevented. Van Inwagen's conclusion that such arbitrariness is part and parcel of providence is not only bold and provocative, but is also made more believable than one would have expected. The same goes for van Inwagen's attack on the plausible moral principle. And for his idea that riskiness is a good. And for much else in this fine book. Trenton Merricks is Professor of Philosophy at the University of Virginia and the author of Objects and Persons and Truth and Ontology.

LE FONCTIONNALISME FACE AU PROBLÈME DES QUALIA Author(s): Ned Block Source: Les Études philosophiques, No. 3, LA THÉORIE COMPUTATIONNELLE DE L'ESPRIT (JUILLET-SEPTEMBRE 1992), pp. 337-369 Published by: Presses Universitaires de France Stable URL: http://www.jstor.org/stable/20848652 . Accessed: 08/06/2014 16:55 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Presses Universitaires de France is collaborating with JSTOR to digitize, preserve and extend access to Les Études philosophiques. http://www.jstor.org This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions LE FONCTIONNALISME FACE AU PROBL?ME DES QU ALI A1 1.0. Fonctionnalisme, b?haviorisme et physicalisme L'approche fonctionnaliste de l'esprit fait aujourd'hui l'objet d'un large consensus2. Comme le b?haviorisme et le physicalisme, le fonction nalisme cherche ? r?pondre ? la question : ? Que sont les ?tats men taux ? ? Je ne m'occuperai ici que des versions du fonctionnalisme qui prennent la forme d'une th?orie de l'identit?. Elles posent, par exemple, que la douleur est un ?tat fonctionnel, de m?me que, en tant qu'il se pr? sente comme une th?orie de l'identit?, le physicalisme pose que la dou leur est un ?tat physique. Je commencerai par fournir une pr?sentation du fonctionnalisme et un rapide aper?u de sa critique du b?haviorisme et du physicalisme. Puis je tenterai de montrer que le fonctionnalisme succombe en fait aux m?mes reproches qu'il adresse ? ces deux doctrines. Voici une fa?on de caract?riser le fonctionnalisme suffisamment vague pour ?tre recevable par la plupart de ceux qui s'en r?clament : cha que type d'?tat mental est un ?tat qui, ?tant donn? certains inputs senso riels et certains ?tats mentaux, consiste dans une disposition ? agir d'une certaine fa?on et ? avoir certains ?tats mentaux. Ainsi caract?ris?, le fonc tionnalisme peut appara?tre comme une nouvelle incarnation du b?ha viorisme. Le b?haviorisme identifie les ?tats mentaux ? des dispositions ? 1. ? Le fonctionnalisme face au probl?me des qualia ? est la traduction d'une version ?court?e de ? Troubles with functionalism ? (1978) parue en 1990 sous le titre Qualia-ba sed objections to Functionalism dans Mind and Cognition, William Lycan (ed.), Basil Blackwell. Reproduit avec Faimable autorisation de l'auteur. 2. Voir Fodor, 1965 ; Lewis, 1972 ; Putnam, 1966, 1967, 1970, 1975* ; Armstrong, 1968 ; Locke, 1968 ; peut-?tre Sellars, 1968 ; peut-?tre Dennett, 1969, 1978? ; Nelson, 1969, 1975 (voir cependant Nelson, 1976) ; Pitcher, 1971 ; Smart, 1971 ; Block et Fodor, 1972 ; H?rman, 1973 ; Grice, 1975 ; Shoemaker, 1975 ; Wiggins, 1975. Les Etudes philosophiques, n? 3/1992 This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions 338 Ned Block agir d'une certame fa?on quand certains inputs sont donn?s. Mais comme l'ont fait valoir plusieurs de ses critiques (Chisolm, 1957 ; Geach, 1957 ; Putnam, 1963), le d?sir d'atteindre le but ne peut ?tre identifi? ? la disposition, par exemple, ? faire A dans les circonstances o? les inputs font que A conduit ? parce que, apr?s tout, il se peut fort bien que l'agent ne sache pas que A conduit ? B, et donc ne soit pas dispos? ? faire A. Le fonctionnalisme remplace les ? inputs sensoriels ? du b?ha viorisme par ? des inputs sensoriels et des ?tats mentaux ? ; il remplace en outre les ? dispositions .? agir ? du b?haviorisme par des ? disposi tions ? agir et ? avoir certains ?tats mentaux ?. Les fonctionnalistes veu lent individuer les ?tats mentaux causalement, et puisque les ?tats men taux ont des causes et des effets mentaux aussi bien que des causes sensorielles et des effets comportementaux, les fonctionnalistes indivi duent les ?tats mentaux en partie en termes de relations causales avec d'autres ?tats mentaux. L'une des cons?quences de cette diff?rence entre le fonctionnalisme et le b?haviorisme est qu'il peut y avoir des orga nismes qui pour le second sont dot?s d'?tats mentaux, tandis qu'ils en sont d?pourvus pour le premier. Les conditions n?cessaires pour que quelque chose ait des propri?t?s mentales qui sont postul?es par le fonctionnalisme sont donc d'une cer taine fa?on plus contraignantes que celles qui sont postul?es par le b?ha viorisme. Selon le b?haviorisme, il est n?cessaire et suffisant pour d?si rer qu'un syst?me soit caract?ris? par un certain ensemble (peut-?tre infini) de relations input-output ; c'est-?-dire que, selon le b?haviorisme, un syst?me ne d?sire que dans le cas o? un certain ensemble de condi tionnels de la forme ? si I, il ?mettra O ? sont vrais de lui. Mais selon le fonctionnalisme, un syst?me pourrait avoir ces relations input-output et cependant ne pas d?sirer ; car selon lui, le fait qu'un syst?me d?sire ou non d?pend du fait qu'il ait ou non des ?tats internes ayant certaines relations causales avec d'autres ?tats internes (ainsi qu'avec les inputs et les outputs). Puisque le b?haviorisme n'exige pas d'?tats internes, il peut y avoir des syst?mes dont le b?haviorisme affirme qu'ils ont des ?tats mentaux, tandis que le fonctionnalisme le nie1. En d'autres termes, aux yeux du fonctionnalisme, le b?haviorisme est coupable de lib?ralisme ? c'est-?-dire d'assigner des propri?t?s mentales ? des choses qui en sont en fait d?pourvues. Malgr? cette divergence, l'esprit du fonctionnalisme n'est pas n?ces sairement radicalement diff?rent de celui du b?haviorisme2. Shoemaker (1975) ?crit par exemple que ? selon l'une de ses interpr?tations, le fonc tionnalisme est, en philosophie de l'esprit, la doctrine que les termes 1. L'inverse est ?galement vrai. 2. En fait, si Ton d?finit le b?haviorisme comme la th?orie que les ?tats mentaux peuvent ?tre d?finis en termes non mentaux, le fonctionnalisme est une variante du b?ha viorisme. This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions Le fonctionnalisme face au probl?me des qualia 339 mentaux et psychologiques sont, en principe, ?liminables d'une certaine fa?on ? (p. 306-307). Les fonctionnalistes ont en effet g?n?ralement accord?, dans leur caract?risation fonctionnelle d'un ?tat mental, un trai tement diff?rent aux termes qui d?signent des ?tats mentaux et ? ceux qui d?signent les inputs et les outputs. Ainsi, dans la plus simple version que re?oit la th?orie en termes de machine de Turing (Putnam, 1967 ; Block et Fodor, 1972), les ?tats mentaux sont identifi?s avec l'ensemble des ?tats d'une machine de Turing, qui eux-m?mes sont implicitement d?finis par une table de machine qui mentionne explicitement des inputs et des outputs d?crits d'une fa?on non mentaliste. Dans le fonctionnalisme de Lewis, les termes d?signant des ?tats mentaux sont d?finis au moyen d'une version de la m?thode de Ramsey qui permet d'?liminer le recours essentiel ? toute terminologie mentale dans les d?finitions, mais non pas la terminologie input-output. Ainsi, ? douleur ? est consid?r? comme synonyme d'une description d?finie qui contient des termes d'inputs et d'outputs mais aucune terminologie men tale (cf. Lewis, 1972). De plus, le fonctionnalisme, tant dans sa version machinique que non machinique, a toujours insist? sur le fait que la caract?risation des ?tats mentaux doit contenir des descriptions d'inputs et d'outputs formul?es en langage physique. Armstrong (1968) ?crit par exemple que Nous devons distinguer entre le ? comportement physique ?, qui d?signe n'importe quelle action ou passion du corps, et le ? comportement proprement dit ?, qui implique une relation avec l'esprit... Or, si dans notre formule ? l'?tat d'une personne apte ? adopter une certaine forme de comportement ? ? compor tement ? est consid?r? comme l'?quivalent de ? comportement proprement dit ?, alors nous expliquons les concepts mentaux au moyen d'un concept qui fait d?j? appel au mental, ce qui est cir culaire. Il est donc clair que dans notre formule, ? comporte ment ? doit vouloir dire ? comportement physique ? (p. 84). Par cons?quent, le fonctionnalisme peut ?tre consid?r? comme ? n'?pinglant ? les ?tats mentaux qu'? la p?riph?rie ? c'est-?-dire au moyen d'une caract?risation physique, ou ? tout le moins non mentale, des inputs et des outputs. L'une des th?ses principales de cet article est que le fonctionnalisme ne parvient pas, pour cette raison, ? ?viter le pro bl?me au nom duquel il condamne ? juste titre le b?haviorisme. Le fonc tionnalisme est lui aussi coupable de lib?ralisme, pour les m?mes raisons que le b?haviorisme. A la diff?rence du b?haviorisme toutefois, le fonc tionnalisme peut ?tre modifi? sans artifice de fa?on ? ?viter le lib?ralisme ? mais sans parvenir alors ? ?viter un autre ?cueil non moins grave que le premier. Cet ?cueil est pr?cis?ment celui dont le fonctionnalisme montre qu'il fait ?chouer le physicalisme. Par ? physicalisme ?, j'entends la doctrine This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions 340 Ned Block selon laquelle la douleur, par exemple, est identique ? un ?tat physique (ou physiologique)1. Comme Tont soutenu de nombreux philosophes (en particulier Fodor, 1965 ; Putnam, 1966 ; voir aussi Block et Fodor, 1972), si le fonctionnalisme est vrai, alors le physicalisme est probable ment faux. C'est particuli?rement clair en ce qui concerne les versions du fonctionnalisme formul?es en termes de machine de Turing. N'importe quelle machine de Turing abstraite peut ?tre r?alis?e par une grande vari?t? de m?canismes physiques ; il est plausible en effet que, pour quel que correspondance ?tablie entre un ?tat de machine de Turing et un ?tat physique configurationnel (ou physiologique) que ce soit, il existe une r?alisation possible de cette machine de Turing qui constitue un contre exemple de cette correspondance (voir Kalke, 1969 ; Gendron, 1971 ; et Mucciolo, 1974 pour une d?monstration peu convaincante du contraire ; voir aussi Kim, 1972). Par cons?quent, si la douleur est un ?tat fonction nel, elle ne peut, par exemple, ?tre un ?tat du cerveau, parce que les cr?a tures sans cerveau peuvent r?aliser la m?me machine de Turing que des cr?atures avec cerveau. Je dois insister sur le fait que l'argument fonctionnaliste contre le physicalisme ne fait pas simplement appel au fait qu'une machine de Turing abstraite peut ?tre r?alis?e par des syst?mes dont la composition mat?rielle est diff?rente (vois, m?tal, verre...). Car ce serait alors comme arguer que la temp?rature ne peut pas ?tre une grandeur microphysique de ce que plusieurs objets avec des microstructures diff?rentes peuvent avoir la m?me temp?rature (1972). Les objets avec des microstructures diff?rentes, tels que les objets faits en bois, m?tal, verre, etc., peuvent avoir plusieurs propri?t?s microphysiques en commun, telle qu'une ?ner gie cin?tique mol?culaire de m?me valeur moyenne. L'argument fonc tionnaliste contre le physicalisme est bien plut?t qu'il est difficile de voir comment // pourrait y avoir une propri?t? physique de premier ordre (cf. n. 4) qui ne soit pas triviale et qui soit commune ? toutes les r?alisa tions physiques d'un ?tat de machine de Turing et ? elles seules. Essayez de trouver un candidat m?me vaguement plausible ! A tout le moins, il 1. Je parle ici de types d'?tats mentaux et non d'?tats particuliers. Dans tout cet ar ticle, j'entends par ? physicalisme ? la doctrine qui affirme que chaque type d'?tat mental est identique ? un certain type d'?tat physique ; par exemple que la douleur (l'universel) est un ?tat physique. Le physicalisme des particuliers, d'autre part, est la doctrine (plus faible) selon laquelle chaque douleur particuli?re donn?e est un ?tat de tel ou tel type physique. Le fonctionnalisme montre que le physicalisme des types est faux, mais non pas que le physicalisme des particuliers l'est. Quand je parle de ? physicalisme ?, j'entends un physicalisme du premier ordre, c'est ?-dire la doctrine que la propri?t? d'?prouver de la douleur est par exemple une propri?t? physique de premier ordre (au sens de Russel et Whitehead). (Une propri?t? de premier ordre est une propri?t? dont la d?finition n'exige pas de quantifier sur les propri?t?s ; une propri?t? de second ordre est une propri?t? dont la d?finition exige de quantifier sur les propri?t?s de premier ordre.) La th?se que le fait d'?prouver de la douleur est une propri?t? physique de second ordre est en fait une forme (physicaliste) de fonctionna lisme, voir Putnam, 1970. This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions Le fonctionnalisme face au probl?me des qualia 341 incombe ? ceux qui pensent que de telles propri?t?s physiques sont concevables de montrer comment nous pouvons en concevoir une. En d'autres termes, selon le fonctionnalisme, le physicalisme est une doctrine chauviniste : il refuse d'accorder des propri?t?s mentales ? des syst?mes qui en sont en fait dot?es. En disant que les ?tats mentaux sont des ?tats du cerveau, par exemple, les physicalistes excluent injustement les pauvres cr?atures d?pourvues de cerveau et n?anmoins dot?es d'esprit. La seconde th?se principale de cet article est que l'argument au nom duquel le fonctionnalisme condamne le physicalisme peut ?tre aussi bien employ? pour r?futer le fonctionnalisme lui-m?me ; en fait, n'importe quelle version du fonctionnalisme qui ?vite le lib?ralisme succombe, comme le physicalisme, au chauvinisme. Cet article a trois parties. La premi?re soutient que le fonctionnalisme est coupable de lib?ralisme, la seconde que l'une des mani?res d'?viter le lib?ralisme est de resserrer ses liens avec la psychologie empirique, et la troisi?me qu'aucune version du fonctionnalisme ne parvient ? ?viter ? la fois le lib?ralisme et le chauvinisme. 1.1. Pr?cisions sur la nature du fonctionnalisme Pour tenter d'introduire un peu d'ordre au sein de la vari?t? d?concertante de th?ories fonctionnalistes, on peut distinguer entre celles qui sont formul?es en termes de machine de Turing et celles qui ne le sont pas. Une table de machine de Turing ?num?re un ensemble fini d'?tats, Si ... S, ; d'inputs, It ... I* ; et d'outputs, Ot ... Ô. Elle sp?cifie en outre un ensemble de conditionnels de la forme : si la machine est dans l'?tat Si et re?oit l'input IJf elle ?met l'output Ok et passe dans l'?tat Si. En d'autres termes, ?tant donn? un ?tat quelconque et un input, la table sp?cifie un output et l'?tat suivant. Tout syst?me dot? d'un ensemble d'inputs, d'outputs et d'?tats reli?s de la mani?re sp?cifi?e par la table est d?crit par cette table et est une r?alisation de l'automate abstrait qu'elle sp?cifie. Pour ?tre capable de calculer n'importe quelle fonction r?cursive, une machine de Turing doit pouvoir contr?ler son input de certaines fa?ons. Dans les formulations habituelles, l'output de la machine de Turing est consid?r? comme ayant deux composants. La machine imprime un sym bole sur un ruban, puis fait bouger le ruban, mettant ainsi un nouveau symbole sous l' il du d?tecteur d'inputs. Pour qu'une machine de Turing soit aussi puissante que possible, le ruban doit ?tre sans fin dans l'une au moins de ses deux directions et doit pouvoir ?tre boug? dans les deux sens. Si la machine n'exerce aucun contr?le sur le ruban, c'est un ? transducteur fini ?, une esp?ce assez limit?e de machine de Turing. Il This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions 342 Ned Block n'est pas m?me besoin de consid?rer que les transducteurs finis sont dot?s d'un ruban. Ceux qui croient ? la v?rit? du fonctionnalisme machi nique doivent consid?rer la question de la nature exacte du type d'auto mates que nous sommes comme une question empirique importante. Si nous sommes des machines de Turing dot?es du maximum d? puissance possible, l'environnement doit faire partie du ruban. Dans l'une de ses plus simples versions, le fonctionnalisme machini que (cf. Block et Fodor, 1972) affirme que chaque syst?me dot? d'?tats mentaux peut ?tre d?crit par au moins une table de machine de Turing d'une esp?ce qui peut ?tre sp?cifi?e, et que chaque type d'?tat mental du syst?me est identique ? l'un des ?tats de la table de machine. Soit par exemple la machine de Turing d?crite dans la table 1 (cf. Nelson, 1975). Input : pi?ce de 5 cents Table 1 Si N'?mettre aucun output Passer en S2 Emettre un coca-cola Passer en oi Input : pi?ce de 10 cents Emettre un coca-cola Rester en Si Emettre un coca-cola et une pi?ce de 5 cents Passer en Si Il est possible de se faire une id?e approximative de cette version sim ple du fonctionnalisme machinique en consid?rant que Si = le d?sir de pi?ce de 5 cents, et S2 = le d?sir de pi?ce de 10 cents. Bien entendu, aucun fonctionnaliste ne pr?tend qu'un distributeur de coca-cola ne d?sire quoi que ce soit. La version simple du fonctionnalisme machini que que je viens de mentionner fait appel ? une table de machine hypo th?tique bien plus complexe. Il faut souligner que le fonctionnalisme machinique caract?rise les inputs et les outputs de mani?re explicite, et les ?tats internes de mani?re implicite. (Putnam, 1967, cf. supra, p. 329) ?crit : ? Les S? une fois encore, ne sont sp?cifi?s qu'implicitement par la description, c'est-?-dire qu'ils ne sont caract?ris?s que par l'ensemble des probabilit?s de transition donn? par la table de machine. ? Pour ?tre d?crit par cette table de machine, un m?canisme doit accepter les pi?ces de 5 et 10 cents comme inputs et fournir des pi?ces de 5 cents et des coca-cola comme outputs. Mais les ?tats Si et S2 peuvent avoir ? peu pr?s n'importe quelle nature (m?me non physique), aussi longtemps que cette nature permet de connecter les ?tats les uns avec les autres ainsi qu'avec les inputs et les outputs sp?cifi?s dans la table de machine. Nous ne This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions Le fonctionnalisme face au probl?me des qualia 343 connaissons de Si et de S2 que ces relations ; on peut donc dire que le fonctionnalisme r?duit le mental ? des structures input-output. Cet exemple permet de faire sentir la force de l'argument fonctionnaliste contre le physicalisme. Essayez de trouver une propri?t? physique de premier ordre (voir n. 4) qui puisse ?tre partag?e par toutes les r?alisa tions (et elles seules) de cette table de machine ! On peut aussi prendre pour crit?re de classification des fonctionnalistes le fait qu'ils consid?rent ou non les identit?s fonctionnelles comme rele vant de la psychologie a priori ou de la psychologie empirique... Les fonc tionnalistes a priori (tels que Smart, Armstrong, Lewis, Shoemaker) sont les h?ritiers des b?havioristes logiques. Il tendent ? consid?rer les analyses fonctionnelles comme des analyses du sens des termes mentaux, tandis que les fonctionnalistes empiriques (tels que Fodor, Putnam, H?rman) consi d?rent les analyses fonctionnelles comme des hypoth?ses scientifiques. Dans ce qui suit, je d?signerai la premi?re de ces deux positions au moyen du terme de Fonctionnalisme, et la seconde au moyen du terme de Psycho fonctionnalisme). (J'emploierai celui de fonctionnalisme ? avec un/mi nuscule ? pour d?signer une position neutre sur la question qui divise le Fonctionnalisme et le Psychofonctionnalisme. Quand il sera indispensable de distinguer entre les deux, j'emploierai toujours les majuscules.) Les notions de Fonctionnalisme et de Psychofonctionnalisme, ainsi que la diff?rence qui les s?pare, peuvent ?tre clarifi?es au moyen de la notion d'?nonc? de Ramsey d'une th?orie psychologique. Les termes d?signant des ?tats mentaux qui apparaissent dans une th?orie psychologi que peuvent ?tre d?finis de diff?rentes fa?ons au moyen de l'?nonc? de Ramsey d'une th?orie. Toute th?orie fonctionnaliste de l'identit? peut ?tre consid?r?e comme d?finissant un ensemble d'?tats fonctionnels au moyen de l'?nonc? de Ramsey de cette th?orie psychologique ? chaque ?tat men tal correspondant ? un ?tat fonctionnel. L'?tat fonctionnel correspondant ? la douleur sera appel? le ? corr?lat fonctionnel de Ramsey ? de la douleur par rapport ? la th?orie psychologique en question. La distinction entre le Fonctionnalisme et le Psychofonctionnalisme peut alors ?tre red?finie en termes de corr?lat fonctionnel de Ramsey par rapport ? une th?orie : le Fonctionnalisme identifie un ?tat mental S avec le corr?lat fonctionnel de Ramsey par rapport ? une th?orie psychologique du sens commun ; le Psy chofonctionnalisme identifie S avec le corr?lat fonctionnel de Ramsey par rapport ? une th?orie psychologique scientifique. Cette diff?rence entre le Fonctionnalisme et le Psychofonctionnalisme entra?ne une diff?rence au niveau de la sp?cification des inputs et des out puts. Les Fonctionnalistes adoptent une sp?cification qui puisse raisonna blement faire partie de la connaissance du sens commun ; les Psychofonc tionnalistes ne sont soumis ? aucune restriction de ce genre. Quoique les deux groupes insistent sur le caract?re physique ? ou du moins non men tal ? de la sp?cification des inputs et des outputs, les Fonctionnalistes doi vent faire appel ? des classifications qui soient observables d'un point de This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions 344 Ned Block vue externe (par exemple, caract?risation des inputs en termes d'objets pr? sents dans l'entourage de l'organisme, caract?risation des outputs en termes de mouvements des parties du corps). Les Psychofonctionnalistes, d'autre part, ont la possibilit? de sp?cifier les inputs et les outputs en termes de param?tres internes, tels que des signaux dans les neurones d'inputs et dans les neurones d'outputs. Soit une th?orie psychologique soit du sens commun, soit de la psychologie scientifique. contient des g?n?ralisations de la forme : toute personne qui est dans l'?tat w et re?oit un input ?met un out put j, et passe dans l'?tat ^. Ecrivons sous la forme suivante : T(Si ... S? Ii ... I?, Oi ... O,) o? les S sont les ?tats mentaux, les I les inputs, et les O les outputs. Les ? S ? doivent ?tre compris comme des constantes d'?tats mentaux, et non comme des variables, et il en va de m?me pour les ? I ? et les ? O ?. Aussi pourrait-on encore ?crire sous la forme suivante : (douleur..., lumi?re de 400 nm entrant dans l' il gauche..., le gros orteil gauche se d?place de 1 cm vers la gauche...). Pour obtenir l'?nonc? de Ramsey de T, il faut remplacer les termes d?signant les ?tats mentaux ? mais non pas les termes d?signant les inputs et les outputs ? par des variables, et pr?fixer un quantificateur existentiel pour chaque variable : 3 F, ... 3F?T(Fi ... F,Ii ... I,, Ox ... Om). Si ? Fi7 ? est la variable qui a remplac? le mot ? douleur ? lors de la for mation de l'?nonc? de Ramsey, la douleur peut alors ?tre d?finie comme suit : ?prouve de la douleur o 3 Fi ... 3 F, T[Fi ... F?,Ii ... ,, ... O.) et xaF17]. Le corr?lat fonctionnel de Ramsey de la douleur est la propri?t? expri m?e par le pr?dicat ? la droite de ce biconditionnel. Il faut souligner que ce pr?dicat contient des constantes d'inputs et d'outputs, mais aucune constante mentale, puisque les constantes mentales ont ?t? remplac?es par des variables. Le corr?lat fonctionnel de Ramsey de la douleur est d?fini en termes d'inputs et d'outputs, mais non pas en termes mentaux. Admettons par exemple que soit la th?orie que la douleur est cau s?e par une blessure de la peau et cause une irritation ainsi que l'?mission de ? A?e ?, et que l'irritation, ? son tour, cause un froncement des sour cils. La d?finition ? la Ramsey serait alors la suivante : ?prouve de la douleur o II y a deux ?tats (propri?t?s), dont le premier est caus? par une blessure de la peau et cause ? la fois l'?mission de ? a?e ? et le second ?tat, et le second ?tat cause un froncement des sourcils, et est dans le premier ?tat. This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions Le fonctionnalisme face au probl?me des qualia 345 Le corr?lat fonctionnel de Ramsey de la douleur par rapport ? cette ? th?orie ? est la propri?t? d'?tre dans un ?tat caus? par la blessure de la peau et qui cause l'?mission de ? a?e ? et un autre ?tat, qui ? son tour cause le froncement des sourcils. (Notez que les mots ? douleur ? et ? souci ? ont ?t? remplac?s par des variables, mais que les termes d?si gnant les inputs et les outputs demeurent.) Le corr?lat fonctionnel de Ramsey d'un ?tat S est un ?tat qui a beau coup en commun avec S. A savoir les propri?t?s structurales sp?cifi?es par la th?orie T. Mais, il y a deux raisons pour lesquelles il est naturel de supposer que S et son corr?lat fonctionnel sont distincts. En premier lieu, le corr?lat fonctionnel de Ramsey par rapport ? la th?orie ne peut ? inclure ? au mieux que les aspects de S dont rend compte ; tout as pect qui n'est pas saisi par est laiss? de c?t?. En second lieu, le corr?lat fonctionnel de Ramsey peut m?me laisser de c?t? certains aspects dont rend compte, car la d?finition de Ramsey ne contient pas le vocabulaire ? th?orique ? de T. La th?orie mentionn?e en guise d'exemple au para graphe pr?c?dent n'est vraie que des organismes qui sentent la douleur ? mais simplement en raison de la mani?re dont elle fait usage du mot ? douleur ?. Cependant, le pr?dicat qui exprime le corr?lat fonctionnel de Ramsey ne contient pas ce mot (puisqu'il a ?t? remplac? par une variable), et peut donc ?tre vrai de choses qui ne ressentent pas la dou leur. Il serait facile de fabriquer une machine simple qui ait une peau artificielle, un sourcil, un ? a?e ? enregistr?, et deux ?tats qui satisfont les relations causales mentionn?es, mais qui ne ressente pas la douleur. L'hypoth?se audacieuse du fonctionnalisme est que pour une certaine th?orie psychologique, cette supposition naturelle qu'un ?tat et son cor r?lat fonctionnel de Ramsey sont distincts est fausse. Le fonctionnalisme dit qu'il existe une th?orie telle que la douleur, par exemple, est le corr? lat fonctionnel de Ramsey par rapport ? cette th?orie. Un dernier point pr?liminaire : j'ai donn? l'impression trompeuse que le fonctionnalisme identifie tous les ?tats mentaux avec des ?tats fonc tionnels. C'est ? l'?vidence une exag?ration. Admettons que X soit une r?plique cellule par cellule de vous qui vient d'?tre cr??e (qui, bien entendu, est fonctionnellement ?quivalente ? vous). Peut-?tre vous sou venez-vous de votre Bar-Mitzvah. Mais X ne s'en souvient pas. En effet, quelque chose peut ?tre fonctionnellement ?quivalent ? vous et cepen dant ne pas r?ussir ? savoir ce que vous savez, ou ? (verbe) ce que vous (verbe), pour un grand nombre de verbes signifiant la ? r?ussite ?. Pire encore, si Putnam (1975?) a raison de dire que ? les sens ne sont pas dans la t?te ?, des syst?mes fonctionnellement ?quivalents ? vous peuvent, pour des raisons similaires, ne pas r?ussir ? avoir plusieurs de vos atti tudes propositionnelles. Supposons que vous croyiez que l'eau est mouil l?e. Selon les arguments plausibles avanc?s par Putnam et Kripke, une des conditions qui vous permettent de croire que l'eau est mouill?e est une certaine connexion causale entre vous et l'eau. Votre ? jumeau ? sur This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions 346 Ned Block la terre jumelle, qui est li? de mani?re similaire ? du XYZ plut?t qu'? du H20, ne croirait pas que l'eau est mouill?e. Le fonctionnalisme ne peut ?tre d?fendu que si on consid?re qu'il ne s'applique qu'? une sous-classe des ?tats mentaux, ces ?tats ? ?troits ? qui sont tels que les conditions de v?rit? de leur application sont en quelque sorte ? dans la personne ?. Mais m?me si l'on suppose qu'il est possible de formuler une notion satisfaisante d'?troitesse d'un ?tat psychologique, l'int?r?t du fonctionnalisme s'en trouve peut-?tre diminu?. Je ne men tionne ce probl?me que pour le laisser de c?t?. Dans ce qui suit, je consid?rerai le fonctionnalisme comme une doc trine concernant tous les ?tats mentaux ? ?troits ?. 1.2. Les robots dirig?s par des homoncules Dans cette section, je vais d?crire une classe de m?canismes qui constituent apparemment une source d'embarras pour toutes les versions du fonctionnalisme que j'ai mentionn?es, en ce qu'ils r?v?lent que le fonctionnalisme est coupable de lib?ralisme ? c'est-?-dire qu'il consi d?re comme dot?s de propri?t?s mentales des syst?mes qui en sont d?pourvus. Soit la version simple du fonctionnalisme machinique qui a ?t? d?crite plus haut. Elle affirme que chaque syst?me dot? d'?tats mentaux est d?crit par au moins une table de machine de Turing, et que chaque ?tat mental de ce syst?me est identique ? l'un des ?tats sp?cifi?s par la table de machine. Je supposerai que les inputs et les outputs sont caract?ris?s en termes d'im pulsions neuronales dans les organes sensoriels et dans les neurones moteurs. Ce qui n'implique nullement que les consid?rations qui vont sui vre valent plus pour le Psychofonctionnalisme que pour le Fonctionna lisme. Ainsi que je l'ai d?j? indiqu?, toute version du fonctionnalisme sup pose une certaine caract?risation des inputs et des outputs. On pourrait donc aussi bien retenir ici une caract?risation fonctionnaliste. Imaginez un corps qui soit ext?rieurement comme un corps humain, par exemple le v?tre, mais int?rieurement tout ? fait diff?rent. Les neu rones des organes sensoriels sont reli?s ? une rang?e de voyants lumi neux dans une cavit? de la t?te. Une s?rie de boutons sont par ailleurs connect?s avec les neurones moteurs. A l'int?rieur de la cavit? si?ge un groupe de petits hommes. Chacun a une t?che tr?s simple : impl?menter un des ? carr?s ? d'une table de machine qui est une description ad?quate de vous. Sur un mur est accroch? un tableau sur lequel on a affich? une carte des ?tats ; c'est-?-dire une carte sur laquelle figure un symbole d?si gnant l'un des ?tats sp?cifi?s par la table de machine. Voici ce que font les petits hommes : quand sur la carte figure un ? G ?, les petits hommes qui impl?mentent les carr?s G ? ils s'appellent eux-m?mes ? les hommes G ? ? sont alert?s. Supposons que la lumi?re repr?sentant l'in This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions Le fonctionnalisme face au probl?me des qualia 347 put In s'allume. L'un des hommes G a pour seule t?che la mission sui vante : quand sur la carte figure ? G ? et que la lumi?re In s'allume, appuyer sur le bouton d'output Oi9t et inscrire un ? M ? sur la carte des ?tats. Cet homme G n'est que rarement appel? ? exercer sa mission. En d?pit du faible niveau d'intelligence exig? de chacun de ces petits hommes, l'ensemble du syst?me parvient ? simuler votre comportement parce que l'organisation fonctionnelle qu'ils ont ?t? entra?n?s ? r?aliser est la v?tre. Une machine de Turing peut ?tre repr?sent?e par un ensem ble fini de quadruples (ou de quintuples si l'input est divis? en deux par ties) : l'?tat actuel, l'input actuel ; l'?tat suivant, l'output suivant. Chaque petit homme a une t?che correspondant ? un seul quadruple. Gr?ce ? leurs efforts, le syst?me r?alise la m?me table de machine (raisonnable ment ad?quate) que vous et est donc fonctionnellement ?quivalent ? vous1. Je vais maintenant d?crire un cas de stimulation par homoncules qui ait plus de chance d'?tre nomologiquement possible. Combien d'homon cules sont n?cessaires pour cela ? Un milliard sera peut-?tre suffisant. Supposons que nous parvenions ? convertir le gouvernement de Chine au fonctionnalisme et ? le convaincre de r?aliser un esprit humain pendant une heure. Nous fournissons ? chacun des mille millions de Chi nois (c'est pour cela que j'ai choisi la Chine) un ?metteur-r?cepteur sp? cialement con?u ? cet effet qui les relie les uns avec les autres et avec le corps artificiel mentionn? dans l'exemple pr?c?dent. Nous rempla?ons donc chacun des petits hommes par un citoyen chinois ?quip? d'une radio. Les lettres sont dispos?es non plus sur un tableau mais sur une s?rie de satellites visibles de n'importe quel endroit de la Chine. Le syst?me form? par ce milliard d'individus communiquant les uns avec les autres et par les satellites joue le r?le d'un ? cerveau ? externe connect? au corps artificiel par radio. Il n'y a rien d'absurde ? imaginer quelqu'un reli? ? son cerveau par radio. Le jour viendra peut-?tre o? nos cerveaux pourront ?tre p?riodiquement d?tach?s pour ?tre r?par?s et net toy?s. Cela pourrait ?tre r?alis? par exemple en traitant les neurones qui relient le corps au cerveau au moyen d'une substance chimique qui leur permette d'?tre aussi flexibles que des ?lastiques, de fa?on ? ce qu'aucune connexion entre le corps et le cerveau ne soit rompue. Mais des hommes d'affaires intelligents s'apercevraient vite qu'il est plus facile d'attirer le client en rempla?ant les neurones ?lastifi?s par des liaisons radio, de sorte que les cerveaux puissent ?tre nettoy?s sans que le corps de son posses seur ait l'inconv?nient d'?tre immobilis?. Il n'est pas du tout ?vident que le syst?me chinois soit physiquement 1. L'id?e qui est ? la base de cet exemple a sa source dans l'article de Putnam, 1967. Elle a fait l'objet de maintes conversations avec Harty Field que je tiens ici ? remercier. J'examine dans la section 1.3 la fa?on dont Putnam tente de d?fendre le fonctionnalisme contre les probl?mes pos?s par ce genre de cas. This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions 348 Ned Block impossible. Et il pourrait ?tre fonctionnellement ?quivalent ? vous pen dant une courte p?riode de temps, disons une heure. ? Mais, pourriez-vous objecter, comment quelque chose pourrait-il ?tre fonctionnellement ?quivalent ? moi pendant une heure ? Mon organi sation fonctionnelle ne d?termine-t-elle pas la fa?on dont, par exemple, je r?agirai au fait de ne rien faire d'autre pendant une semaine que de lire le Reader's Digest? ? Souvenez-vous qu'une table de machine sp?cifie un ensemble de conditionnels de la forme : si la machine est dans l'?tat S, et re?oit un input Iy, elle ?met un output Ok et passe dans l'?tat Si. Ces conditionnels doivent ?tre entendus en un sens subjonctif. Ce qui conf?re une organisation fonctionnelle ? un syst?me ? un moment donn? n'est pas seulement ce que le syst?me fait ? ce moment-l?, mais ce sont aussi les contrefactuels qui sont vrais de lui ? ce moment-l? : c'est-?-dire ce qu'il aurait fait (et ce que ses transitions d'?tat ? ?tat auraient ?t?) s'il avait re?u un input diff?rent ou s'il avait ?t? dans un ?tat diff?rent. S'il est vrai de ce syst?me au moment / qu'il ob?irait ? une certaine table de machine quel que soit son ?tat et quels que soient ses inputs, alors le sys t?me est d?crit ? / par cette table de machine (et r?alise un automate abs trait sp?cifi? par la table), m?me s'il n'existe que pendant un instant. Au cours de l'heure pendant laquelle le syst?me chinois est ? branch? ?, il a effectivement un ensemble d'inputs, d'outputs et d'?tats dont de tels conditionnels subjonctifs sont vrais. C'est par l? que n'importe quel ordinateur r?alise l'automate abstrait qu'il r?alise. Il y a bien entendu des signaux auxquels le syst?me r?pondrait et aux quels vous ne r?pondriez pas ? par exemple une interf?rence radio mas sive ou un d?bordement de la rivi?re Yangtze. De tels ?v?nements pour raient entra?ner un dysfonctionnement, et par l? faire ?chouer la simulation, tout comme une bombe plac?e dans un ordinateur peut l'em p?cher de r?aliser la table de machine que sa construction doit le faire r?aliser. Mais de m?me que l'ordinateur sans la bombe peut r?aliser la table de machine, de m?me le syst?me compos? du peuple chinois et du corps artificiel peut r?aliser la table de machine aussi longtemps que ne survient aucune interf?rence catastrophique, telle qu'une inondation, etc. ? Mais, pourriez-vous encore objecter, il y a une diff?rence entre met tre une bombe dans un ordinateur et mettre une bombe dans un syst?me chinois ; dans le second cas (mais pas dans le premier), les inputs tels qu'ils sont sp?cifi?s dans la table de machine peuvent ?tre la cause du dysfonctionnement. Une activit? neuronale inhabituelle dans les organes sensoriels des r?sidents de la province de Chungking caus?e par une bombe ou un d?bordement du Yangtze peut par exemple d?traquer le syst?me. ? R?ponse : En sp?cifiant de quel syst?me on parle, il faut aussi d?limi ter une certaine cat?gorie d'inputs et d'outputs. Je ne consid?re comme inputs et outputs que l'activit? neuronale normale du corps artificiel connect? par radio au peuple de Chine. Les signaux neuronaux dans le This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions Le fonctionnalisme face au probl?me des qualia 349 peuple du Chungking ne sont pas plus des inputs de ce syst?me qu'un ruban coinc? entre les relais ? l'int?rieur d'un ordinateur ne compte comme un input de cet ordinateur. Bien entendu, l'objet form? par le peuple de Chine et le corps artifi ciel peut recevoir d'autres descriptions en termes de machine de Turing en vertu desquelles les signaux neuronaux des habitants de Chungking compteraient comme des inputs. Un tel syst?me (c'est-?-dire le syst?me caract?ris? par cette nouvelle description en termes de machine de Turing) ne serait pas fonctionnellement ?quivalent avec vous. Pareille ment, n'importe quel ordinateur commercial peut ?tre red?crit de telle fa?on qu'un ruban coinc? ? l'int?rieur compte comme l'un de ses inputs. En d?crivant un objet comme une machine de Turing, on trace une fron ti?re entre l'int?rieur et l'ext?rieur. (Si nous ne comptons que des impul sions neuronales comme inputs et outputs, cette ligne passe ? l'int?rieur du corps ; si au contraire nous ne comptons que des stimulations p?ri ph?riques comme inputs, cette ligne co?ncide avec la peau.) En d?crivant le syst?me chinois comme une machine de Turing, j'ai trac? cette ligne de mani?re ? satisfaire un certain type de description fonctionnelle ? un type que vous satisfaites ?galement et qui, selon le fonctionnalisme, justifie l'attribution de propri?t?s mentales. Le fonctionnalisme ne pr?tend pas qu'il existe pour chaque syst?me mental une table de machine d'une esp?ce qui justifie des attributions de propri?t?s mentales par rapport ? n'importe quelle sp?cification d'inputs et d'outputs, mais seulement par rapport ? une certaine sp?cification. Objection : Le syst?me chinois fonctionnerait trop lentement. Le type d'?v?nements et de processus avec lesquels nous sommes normalement en contact sont bien trop rapides pour que le syst?me puisse les d?tecter. Par cons?quent nous serions incapables de converser avec lui, de jouer au bridge avec lui, etc. R?ponse : On ne voit pas pourquoi la rapidit? du syst?me importerait d'une mani?re ou d'une autre. Est-il v?ritablement contradictoire ou insens? de supposer que nous pourrions rencontrer une race d'?tres intel ligents avec lesquels nous ne pourrions communiquer qu'au moyen d'un proc?d? comme l'acc?l?r? ? Quand nous observerions de telles cr?atures, elles sembleraient presque inanim?es. Mais quand nous les verrions sur des films projet?s en acc?l?r?, nous les verrions converser les unes avec les autres. En fait, nous nous apercevrions qu'elles disent pr?cis?ment que nous n'avons du sens pour elles que sur des films regard?s au ralenti. Accorder une telle importance ? la rapidit? du syst?me, c'est semble-t-il faire preuve d'un b?haviorisme ?l?mentaire. Ce qui fait du syst?me dirig? par des homoncules (les deux exemples ne devant ?tre consid?r?s que comme des variantes d'un seul syst?me) qui vient d'?tre d?crit en apparence un contre-exemple du fonctionnalisme (machinique), c'est qu'il y a semble-t-il une raison de douter qu'il ait le moindre ?tat mental ? en particulier qu'il ait ce que les philosophes ont This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions 350 Ned Block appel? tant?t des ? ?tats qualitatifs ?, tant?t des ? sensations brutes ?, tant?t des ? qualit?s ph?nom?nologiques imm?diates ?. (Si vous me demandez : qu'est-ce que ce que les philosophes ont appel? des ?tats qua litatifs ? Je vous r?pondrai ? ? demi s?rieusement : Comme disait Louis Armstrong quand on lui demandait ce qu'?tait le jazz, ? si vous le demandez, c'est que vous ne le saurez jamais ?.) Dans la terminologie de Nagel (1974), on est en droit de douter qu'?tre comme un syst?me dirig? par des homoncules ressemble ? quoi que ce soit1. 1.3. ha proposition de Putnam L'un des moyens qui s'offrent aux fonctionnalistes pour r?soudre le probl?me pos? par le contre-exemple du syst?me dirig? par des homon cules consiste tout simplement ? l'?carter par une stipulation ad hoc. Un fonctionnaliste pourrait par exemple stipuler que deux syst?mes ne peuvent ?tre fonctionnellement ?quivalents si l'un contient des parties qui ont une organisation fonctionnelle caract?ristique de celle des ?tres sensibles et l'autre non. Dans l'article o? il fait l'hypoth?se que la douleur est un ?tat fonctionnel, Putnam stipule ? qu'aucun organisme capable de sentir de la douleur ne peut ?tre d?compos? en parties qui poss?dent chacune s?par?ment des Descriptions ? (en termes de machine de Turing pouvant ?tre dans l'?tat fonctionnel que Putnam identifie avec la douleur). Le but de cette restriction est ? d'?carter la possibilit? pour des "organismes" (pour autant qu'on puisse les consi d?rer comme tels) tels que les essaims d'abeilles de la classe des indivi dus capables de ressentir de la douleur ? (Putnam, 1967, cf. supra, p. 330). La condition pos?e par Putnam pourrait par exemple ?tre satisfaite de la mani?re suivante : un organisme qui sent la douleur ne peut ?tre d?compos? en parties qui ont toutes une organisation fonctionnelle caract?ristique de celle des ?tres sensibles. Mais cela ne permettrait pas d'?carter l'exemple du syst?me dirig? par des homoncules, vu qu'il poss?de des parties sensibles et non sensibles, tels que le corps m?cani que et les organes sensoriels. Il ne servirait ? rien d'adopter la th?se compl?tement oppos?e et d'exiger qu'aucune partie r?elle ne soit sensi ble. Sans quoi les femmes enceintes et ceux qui ont des parasites sensi bles ne pourraient ?tre consid?r?s comme des organismes capables de ressentir la douleur. Ce qui semble important dans la simulation par 1. Shoemaker (1975) soutient (en r?ponse ? Block et Fodor, 1972) que l'absence de qualia est logiquement impossible ; c'est-?-dire qu'il est logiquement impossible que deux syst?mes soient dans le m?me ?tat fonctionnel et que cependant seul l'un des deux ait un contenu qualitatif. N.d.T. : L'article auquel il est ici fait allusion ? What is it like to be a bat ? ? figure dans le recueil Mortal Questions, traduction fran?aise de C. Engel-Tierce lin et P. Engel sous le titre Questions mortelles, puf, ? Philosophie d'aujourd'hui ?, 1983. This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions he fonctionnalisme face au probl?me des qualia 351 homoncules que j'ai d?crite, c'est le fait que des ?tres sensibles jouent un r?le crucial dans l'organisation fonctionnelle que poss?de la machine. Ce qui sugg?re une version de la proposition de Putnam qui exige qu'un organisme capable de ressentir la douleur ait une certaine organisation fonctionnelle, mais n'ait par contre aucune partie 1 / qui poss?de elle-m?me la m?me sorte d'organisation fonctionnelle, et 2 / qui joue un r?le crucial dans le fait que l'ensemble du syst?me ait l'organisation fonctionnelle qu'il a. Quoique cette proposition fasse appel ? la notion vague de ? r?le crucial ?, elle est assez pr?cise pour permettre de voir qu'elle ne marche pas. Supposons qu'il y ait une partie de l'univers qui contienne une mati?re tout ? fait diff?rente de la n?tre, une mati?re qui est ind? finiment divisible. Dans cette partie de l'univers se trouvent des cr?a tures intelligentes de toutes sortes de tailles, y compris des cr?atures anthropomorphes beaucoup plus petites que nos particules ?l?men taires. Au cours d'une exp?dition intergallaxique, ces ?tres d?couvrent l'existence de notre type de mati?re. Pour des raisons qui ne sont connues que d'elles seules, elles d?cident de consacrer une centaine d'ann?es ? cr?er, ? partir de leur mati?re, des substances qui aient les caract?ristiques chimiques et physiques de nos ?l?ments (sauf au niveau des particules infra-?l?mentaires). Elles construisent des hordes de vais seaux spatiaux de diff?rentes esp?ces, ? peu pr?s de la taille de nos ?lec trons, protons, et autres particules ?l?mentaires, et les conduisent de fa?on ? imiter le comportement de ces derniers. Les vaisseaux contien nent ?galement des g?n?rateurs qui produisent le m?me type de radia tion que nos particules ?l?mentaires. Sur chaque vaisseau se trouve une ?quipe d'experts sur la nature de ces particules. Tout ceci a pour fin de produire une quantit? ?norme (selon nos standards) de substances dot?es des caract?ristiques chimiques et physiques de l'oxyg?ne, du car bone, etc. Peu apr?s l'ach?vement de ce projet, vous menez une exp? dition jusqu'? la partie en question de l'univers et vous d?couvrez ? l'oxyg?ne ?, ? le carbone ?, etc. Ignorants de la v?ritable nature de ces ?l?ments, vous d?cider d'y ?tablir une colonie et vous les utilisez pour faire pousser des plantes capables de nourrir cette colonie, de produire ? l'air ? qui lui permet de respirer, etc. Puisque les mol?cules qui composent un individu sont constamment ?chang?es avec celles de son environnement, les membres de la colonie et vous-m?mes finissez par ?tre compos?s (en quelques ann?es) essentiellement de la ? mati?re ? compos?e des individus minuscules qui sont dans les vais seaux spatiaux. Seriez-vous moins capables de ressentir de la douleur, de penser, etc., simplement parce que la mati?re dont vous ?tes compos? contient (et d?pend pour ce qui est de ses caract?ristiques) des ?tres qui eux-m?mes ont une organisation fonctionnelle caract?risti que de celle des ?tres sensibles ? Je ne crois pas. Les m?canismes ?lec trochimiques de base au moyen desquels la synapse op?re sont mainte This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions 352 Ned Block nant assez bien compris. Autant que je sache, les changements qui n'af fectent pas ces m?canismes n'affectent pas les op?rations du cerveau, et donc les propri?t?s mentales. Les m?canismes ?lectrochimiques de vos synapses ne subiraient aucune modification si la nature de votre mati?re changeait1. Il est int?ressant de comparer cet exemple avec les pr?c?dents. D'o? nous vient notre intuition que les simulations dirig?es par des homon cules n'ont pas de propri?t?s mentales ? Il est naturel de supposer que c'est parce que nous pensons qu'ils ont trop de structure mentale interne. Les petits hommes peuvent parfois s'ennuyer, parfois s'exciter. Nous pouvons m?me imaginer qu'ils d?lib?rent quant ? la meilleure fa?on de r?aliser l'organisation fonctionnelle qu'ils r?alisent, et qu'ils y apportent des modifications destin?es ? leur donner plus de loisir. Mais l'exemple des particules ?l?mentaires compos?es d'homoncules sugg?re que cette supposition naturelle est en fait fausse. Ce qui semble important c'est la question de savoir comment les propri?t?s mentales des parties contri buent au fonctionnement du tout. Il y a une diff?rence majeure entre la premi?re et la seconde esp?ce de cas. Dans la seconde, le changement qui s'op?re en vous quand votre mati?re se transforme progressivement en homoncules n'entra?ne de dif f?rence ni dans votre traitement psychologique (c'est-?-dire dans votre traitement de l'information) ni dans votre traitement neurologique, mais seulement dans votre traitement microphysique. Aucune technique pro pre ? la psychologie ou ? la neurophysiologie humaine ne r?v?lerait une quelconque diff?rence en vous. Cependant, les simulations dirig?es par des petits hommes du d?but de cet article ne sont pas des choses aux quelles les th?ories neurophysiologiques qui sont vraies de nous s'appli quent, et si elles sont interpr?t?es comme des simulations Fonctionnelles (plut?t que Psychofonctionnelles), il n'est pas n?cessaire que ce soit des choses auxquelles les th?ories psychologiques (th?ories du traitement de l'infor mation) qui sont vraies de nous s'appliquent. Cette diff?rence sugg?re que nos intuitions sont en partie contr?l?es par l'id?e, qui n'est pas d?raisonnable, que nos ?tats mentaux d?pendent du fait que nous avons la psychologie et/ou neurophysiologie que nous avons. De sorte qu'une chose qui diff?re nettement de nous de ce double point de vue (il s'agit je le rappelle d'une simulation Fonctionnelle, plut?t que Psychofonction nelle) ne doit pas ?tre suppos?e pourvue de propri?t?s mentales pour la simple raison qu'elle est d?sign?e comme fonctionnellement ?quivalente avec nous. 1. Puisqu'il y a une diff?rence entre le r?le des petits hommes dans la production de votre organisation fonctionnelle dans le cas que je viens de d?crire et dans celui qui pr? c?de, la condition pos?e par Putnam doit vraisemblablement pouvoir ?tre reformul?e de mani?re ? ?liminer le pr?c?dent sans ?liminer celui-ci. Mais ce serait l? une man uvre parfaitement ad hoc. This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions Le fonctionnalisme face au probl?me des qualia 353 1.4. Le doute n'est-il qu'apparent ? L'argument de l'absence des qualia fait appel ? l'intuition que les simulations par homoncules manquent de propri?t?s mentales ou du moins de qualia. J'ai dit que cette intuition conduisait apparemment ? mettre en doute la v?rit? du fonctionnalisme. Mais des intuitions que ne viennent ?tayer aucun argument de principe peuvent difficilement ?tre consid?r?es comme fond?es. En fait, des intuitions incompatibles avec une th?orie bien ?tay?e (telle que l'intuition pr?copernicienne que la terre ne se meut pas) ne tardent pas, Dieu merci, ? s'?vanouir. M?me dans des domaines comme la linguistique, o? les donn?es sont essentiellement faites d'intuitions, on rejette souvent (pour des raisons th?oriques) les intuitions que des phrases comme les suivantes ne sont pas grammaticales : ? The horse raced past the barn fell. ? The boy the girl the cat bit scratched died1. Ces phrases sont en r?alit? grammaticales, quoique difficiles ? traiter2. Les appels ? l'intuition, quand il s'agit de juger de la possession de propri?t?s mentales, sont cependant particuli?rement sujets ? caution. Intuitivement, aucun m?canisme physique ne semble ?tre un candidat tr?s plausible au titre de si?ge des qualia, le cerveau moins que tout autre. Est ce qu'un bout de mati?re grise tremblotante est intuitivement plus ad?quat qu'une compagnie de petits hommes ? Mais alors, le doute que les syst?mes dirig?s par des cerveaux aient des qualia n'est-il pas qu'apparent ? Il existe toutefois une diff?rence importante entre les syst?mes dirig?s par des cerveaux et les syst?mes dirig?s par des homoncules. Puisque nous savons que nous sommes des syst?mes dirig?s par des cerveaux et que nous avons des qualia, nous savons que les syst?mes gouvern?s par des cer veaux ont des qualia. De sorte que, m?me si nous n'avons aucune th?orie des qualia qui permette d'expliquer comment cela est possible, nous avons pourtant de tr?s fortes raisons de rejeter tous les doutes qui pourraient surgir ? l'?gard des syst?mes dirig?s par des cerveaux. Bien entendu, mon argument est par l? rendu en partie empirique ? il d?pend de la connaissance de ce qui nous fait fonctionner. Mais puisque c'est l? une 1. Litt?ralement : 1) Le cheval entra?n? devant la grange est tomb? ; 2) Le gar?on, que la fille, que le chat a mordue, a griff?, est mort. (N.d.T.) 2. Comparez la premi?re phrase avec ? Le poisson mang? ? Boston puait ?. La rai son pour laquelle raced fait difficult? est qu'on l'interpr?te naturellement plut?t comme une forme active que passive. Voir Fodor et al., 1974, p. 360. Pour une justification de la grammaticalit? de la seconde phrase, voir Fodor et Garrett, 1967 ; Bever, 1970 ; et Fodor et al, 1974. This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions 354 Ned Block connaissance que nous poss?dons, cette d?pendance ne saurait ?tre consi d?r?e comme un d?faut1. Il existe une autre diff?rence entre ceux qui, comme nous, ont la t?te pleine de mati?re grise et ceux qui ont la t?te pleine d'homoncules : ces derniers sont des syst?mes con?us pour nous imiter, mais nous ne sommes aucunement con?us pour imiter quelque chose (je m'appuie ici une fois encore sur un fait empirique). Ce qui pr?vient toute tentative de faire appel ? la notion d'inf?rence ? la meilleure explication pour le cas des qualia des t?tes remplies d'homoncules. La meilleure fa?on d'expli quer les cris et les grimaces des t?tes pleines d'homoncules n'est pas qu'elles ressentent de la douleur, mais qu'elles ont ?t? con?ues pour imi ter nos cris et nos grimaces. Certains semblent penser que le comportement subtil et complexe des t?tes pleines d'homoncules (comportement tout aussi complexe et subtil ? m?me aussi ? sensible ? aux caract?ristiques de l'environne ment, humain et non humain ? que votre comportement) constitue une raison suffisante pour ?carter le doute que celles-ci ont des qualia. Mais ce n'est l? rien d'autre que du b?haviorisme ?l?mentaire. Mon argument contre le Fonctionnalisme d?pend du principe sui vant : si une doctrine conduit ? une conclusion absurde qu'il n'y a aucune raison ind?pendante d'adopter, et s'il n'y a aucun moyen de dis siper cette absurdit? ou de montrer qu'elle est erron?e ou sans perti nence, ni aucune bonne raison de croire ? la doctrine qui conduit ? une telle absurdit? en premier lieu, alors cette doctrine doit ?tre r?cus?e. Je pr?tends qu'il n'y a aucune raison ind?pendante de croire qu'une t?te pleine d'homoncules ait des propri?t?s mentales, et je ne vois aucun moyen d'?carter la conclusion absurde qu'elle en soit dot?e (quoique bien entendu, mon argument perde de sa validit? si quelqu'un en trouve le moyen). Le probl?me est donc de savoir s'il y a une bonne raison de croire au Fonctionnalisme. L'un des arguments en faveur du Fonction nalisme est qu'il constitue actuellement la meilleure solution du pro bl?me des rapports entre l'esprit et le corps. Je pense que c'est un mau vais argument, mais puisque je pense par ailleurs que le Psychofonctionnalisme est pr?f?rable au Fonctionnalisme (pour les rai sons mentionn?es pr?c?demment), je renvoie l'examen de sa validit? ? la discussion du Psychofonctionnalisme. Le seul autre argument que je connaisse en faveur du Fonctionna lisme est celui que la v?rit? des identit?s Fonctionnelles peut ?tre ?tablie 1. Nous ?chouons souvent ? concevoir comment quelque chose est possible parce que les concepts th?oriques ad?quats font d?faut. Avant la d?couverte du m?canisme de la duplication g?n?tique, Haldane soutenait par exemple de fa?on tr?s persuasive qu'au cun m?canisme physique concevable ne pouvait effectuer un telle t?che. Mais au lieu d'encourager les savants ? d?velopper les id?es qui nous auraient permis d'imaginer un tel m?canisme physique, il en concluait qu'il devait s'agir d'un m?canisme non-physique. (L'exemple m'a ?t? fourni pas R. Boyd.) This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions Le fonctionnalisme face au probl?me des qualia 355 par le biais de l'analyse du sens des termes mentaux. De sorte que les identit?s Fonctionnelles doivent ?tre justifi?es de la m?me fa?on que peut l'?tre l'affirmation que l'?tat de c?libataire est identique ? l'?tat d'homme non mari?. Il existe aussi un argument similaire qui fait lui appel aux pla titudes du sens commun relatives aux ?tats mentaux et non ? l'id?e de v?rit? en vertu du sens des termes. Lewis dit que les caract?risations fonctionnelles des ?tats mentaux rel?vent de ? la psychologie du sens commun ? de la science populaire, plut?t que de la science profession nelle ? (Lewis, 1972, p. 250) (voir aussi Shoemaker, 1975 et Armstrong, 1968. Armstrong fait une erreur sur la question de l'analyticit?, voir ibid., p. 84-85 et 90). Plus encore, il insiste aussi sur le fait que les caract?risa tions Fonctionnelles ne devraient ? inclure que des platitudes qui sont connues de tous ? chacun les conna?t, chacun sait que chacun les conna?t, etc. ? (Lewis, 1972, p. 256). Je m'attacherai ici essentiellement ? la version ? plate ? de l'argument. Celle qui fait appel ? la notion d'ana lyticit? est sujette aux m?mes objections, outre le fait qu'elle pr?te le flanc aux doutes qu'a soulev?s Quine ? propos de l'analyticit?. Je suis pr?t ? conc?der, pour les seuls besoins de l'argumentation, qu'il est possible de d?finir n'importe quel ?tat mental en termes de pla titudes concernant d'autres termes mentaux; des termes d'input et des termes d'output. Mais cela ne m'engage nullement ? cette sorte de d?fi nition des ?tats mentaux qui ?limine toute esp?ce de terminologie men tale au moyen de la Ramsification ou d'un autre proc?d?. Il est tout sim plement erron? de supposer que si chaque ?tat mental peut ?tre d?fini en termes d'autres ?tats mentaux (ainsi que d'inputs ou d'outputs), alors chaque ?tat mental peut ?tre d?fini de fa?on non mentale. L'exemple pr? c?dent permettra d'?clairer ce point. Pour simplifier les choses, je laisse rai de c?t? ici les inputs et les outputs. D?finissons la douleur comme la cause du souci, et le souci comme l'effet de la douleur. M?me une per sonne assez na?ve pour admettre cela n'a pas besoin d'accepter une d?fi nition de la douleur qui fasse de celle-ci la cause de quelque chose, ou une d?finition du souci qui fasse de celui-ci l'effet de quelque chose, Lewis pr? tend que c'est une v?rit? analytique que la douleur est ce qui d?tient un certain r?le causal. M?me s'il a raison s'agissant d'un r?le causal caract? ris? de fa?on partiellement mentaliste, on ne peut en conclure que c'est une v?rit? analytique que la douleur est ce qui d?tient un certain r?le causal quand ce r?le est caract?ris? de fa?on non mentaliste. Je ne vois aucun argument d?cent en faveur du Fonctionnalisme qui puisse ?tre fond? sur des platitudes ou sur l'analyticit?. De plus, le Fonc tionnalisme fond? sur des platitudes conduit ? des difficult?s dans les cas o? de telles platitudes font d?faut. Souvenez-vous de l'exemple des cer veaux que l'on extrait pour les nettoyer et les remettre ? neuf et o? les connexions entre le cerveau et le corps sont maintenues par radio. Ce processus prend quelques jours et une fois qu'il est achev?, le cerveau est r?ins?r? dans le corps. Il peut arriver que le corps d'une personne soit This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions 356 Ned Block d?truit accidentellement au cours de l'op?ration. Si on le rattachait ? des organes sensoriels d'inputs (mais non pas d'outputs), le cerveau ne mani festerait alors aucune des connexions habituelles mentionn?es par le sens commun entre le comportement et les groupements d'?tats mentaux et d'inputs. Si, comme cela est ? premi?re vue plausible, ce cerveau pouvait avoir presque tous les m?mes ?tats mentaux (?troits) que nous avons (et puisque cet ?tat de choses pourrait devenir typique), le Fonctionnalisme est faux. Il est instructif de comparer ce cas avec la mani?re dont le Psycho fonctionnalisme tente de rendre compte des cerveaux enferm?s dans des bocaux. Selon le Psychofonctionnalisme, ce qui compte au nombre des inputs et des outputs d'un syst?me est une question empirique. Consid? rer les impulsions neuronales comme des inputs et des outputs permet trait d'?viter les probl?mes auxquels il vient d'?tre fait allusion, puisque les cerveaux dans les bocaux et les paralytiques peuvent avoir les bonnes impulsions neuronales sans avoir de mouvements corporels. Objection : Une paralysie pourrait affecter le syst?me nerveux, et donc les impulsions neuronales, de sorte que le probl?me qui fait obstacle au Fonctionnalisme affecte tout autant le Psychofonctionnalisme. R?ponse : Les maladies du syst?me nerveux peuvent effectivement modifier les pro pri?t?s mentales : elles peuvent par exemple rendre leurs victimes incapa bles de ressentir de la douleur. De sorte qu'il pourrait en fait ?tre vrai qu'une maladie g?n?ralis?e ayant entra?n? une paralysie intermittente du syst?me nerveux rende les gens incapables d'avoir certains ?tats mentaux. Selon les versions plausibles du Psychofonctionnalisme, le probl?me de savoir quels processus neuronaux peuvent ?tre consid?r?s comme des inputs ou des outputs revient en partie ? se demander quels dysfonctionne ments doivent ?tre consid?r?s comme des modifications de propri?t?s mentales et quels dysfonctionnements doivent ?tre consid?r?s comme de simples modifications des inputs p?riph?riques et des connexions avec les outputs. Le Psychofonctionna lisme a plus de ressources que le Fonctionnalisme, puisqu'il nous permet de faire co?ncider la ligne de partage entre les deux avec la limite entre l'int?rieur et l'ext?rieur de l'organisme, et d'?viter par l? les probl?mes qui viennent d'?tre examin?s. L'erreur de toutes les versions du Fonc tionnalisme est de tenter de tracer cette ligne sur la seule base de la connaissance du sens commun ; et celle des versions ? analytiques ? du Fonctionnalisme est plus pr?cis?ment de tenter de le faire a priori. 2.0. Le Psychofonctionnalisme Ma critique du Fonctionnalisme repose sur le principe suivant : si une doctrine conduit ? une conclusion absurde qu'il n'y a aucune raison ind?pendante de croire, et si il n'y a aucun moyen de dissiper cette absur dit? ou de montrer qu'elle est erron?e ou sans pertinence, ni aucune This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions Le fonctionnalisme face au probl?me des qualia 357 bonne raison en premier lieu de croire en la doctrine qui conduit ? une telle absurdit?, alors cette doctrine doit ?tre r?cus?e. J'ai affirm? qu'il n'y avait aucune raison ind?pendante de croire que les simulations Fonction nelles par homoncules ont des propri?t?s mentales. Cependant, ily a une raison ind?pendante de croire qu'une simulation Pj^?fonctionnelle a des ?tats mentaux y ? savoir le fait qu'une simulation Psychofonctionnelle de vous serait Psychofonctionnellement ?quivalente avec vous, de telle sorte que n'importe quelle th?orie psychologique qui serait vraie de cette simulation serait ?galement vraie de cette simulation. Quelle meilleure raison peut-il y avoir de lui attribuer n'importe lequel des ?tats mentaux qui appartiennent au domaine de la psychologie ? Cet argument montre qu'une simulation psychofonctionnelle quel conque de vous partage vos ?tats mentaux non qualitatifs. Je vais toute fois tenter de montrer dans la section suivante qu'il est permis de douter qu'elle partage vos ?tats mentaux qualitatifs. 2.1. Les qualia sont-ils des ?tats Psychofonctionnels ? J'ai commenc? cet article en d?crivant un syst?me dirig? par des homoncules et en affirmant qu'il y a apparemment des raisons de dou ter qu'un tel syst?me poss?de des ?tats mentaux, et en particulier des ?tats mentaux tels que la douleur, la d?mangeaison ou la sensation de rouge. Peut-?tre est-il possible d'expliquer le doute particulier que sou l?vent les qualia en consid?rant non pas le ph?nom?ne de l'absence des qualia, mais celui de leur inversion. Il n'est pas insens?, ou du moins il ne semble pas insens? de supposer que les objets que vous et moi appelons verts m'apparaissent en fait de la mani?re dont les objets que nous appelons rouges vous apparaissent. Il semble que nous puissions ?tre fonctionnellement ?quivalents m?me si la sensation que les extinc teurs ?voquent en vous est qualitativement la m?me que la sensation que l'herbe ?voque en moi. Imaginez une lentille qui, lorsqu'on la place sur l' il d'un sujet am?ne celui-ci ? prof?rer des exclamations du genre : ? Les choses rouges m'apparaissent maintenant de la fa?on dont les choses vertes m'apparaissaient ayant, et vice versa. ? Imaginez de plus deux jumeaux identiques sur l'un desquels de telles lentilles ont ?t? pos?es ? la naissance. Les jumeaux grandissent normalement, et ? 21 ans sont fonctionnellement ?quivalents l'un avec l'autre. Il est assez probable que le spectre de l'un est l'inverse de celui de l'autre (la pos sibilit? d'une inversion de spectre au sein d'une m?me personne a ?t? d?fendue de mani?re assez convaincante par Shoemaker, 1975, . 17). Par contre, on ne voit pas quel sens il peut y avoir ? faire une supposi tion analogue dans le cas des ?tats non qualitatifs. Imaginer deux per sonnes dont l'une croit que p est vrai et que q est faux, tandis que l'au tre croit que p est faux et que q est vrai. Ces deux personnes This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions 358 Ned Block pourraient-elles ?tre fonctionnellement ?quivalentes ? On ne voit pas comment cela serait possible1. Il est en effet extr?mement difficile de concevoir comment deux personnes pourraient n'avoir d'autre diff? rence entre leurs croyances que celle-ci et ne manifester cependant aucune diff?rence de comportement. Les qualia semblent ?tre dans une relation de d?pendance asym?trique2 vis-?-vis de l'organisation fonc tionnelle dont les croyances sont exemptes. Il y a une autre raison pour distinguer fermement entre les ?tats qua litatifs et les ?tats non qualitatifs ? propos des th?ories fonctionnalistes : ? savoir le fait que le Psychofonctionnalisme ?vite les probl?mes aux quels se heurte le Fonctionnalisme ? propos des seconds ? par exemple, les attitudes propositionnelles comme les croyances et les d?sirs. Mais le Psychofonctionnalisme n'est peut-?tre gu?re plus capable que le Fonc tionnalisme de rendre compte des ?tats qualitatifs. La raison en est qu'il se peut que les qualia ne tombent pas dans le domaine de la psychologie. Essayons en effet d'imaginer ? quoi ressemblerait une r?alisation de la psychologie humaine par un syst?me dirig? par des homoncules. La recherche psychologique actuelle semble essentiellement tourn?e vers la 1. Supposez qu'une personne qui a une bonne vision des couleurs utilise par erreur ? rouge ? pour d?noter le vert et ? vert ? pour d?noter le rouge. Elle confond simple ment les deux mots. Puisque cette confusion est purement verbale, quoiqu'elle dise d'une chose verte qu'elle est rouge, elle ne croit pas plus qu'elle l'est qu'un ?tranger qui confond ? aschan ? et ? sandwich ? ne croit que les gens d?jeunent d'aschans plut?t que de sandwichs. Disons que la personne qui a ainsi confondu ? rouge ? et ? vert ? est vic time d'une commutation de mots. Consid?rons maintenant un mal diff?rent : des lentilles qui inversent le rouge et le vert ont ?t? plac?es sur les yeux de quelqu'un ? son insu. On dira que cette personne est victime d'une commutation de stimuli. Comme la victime pr?c?dente, celle-ci applique le mot ? rouge ? aux choses vertes et vice versa. Mais ? la diff?rence de la premi?re, elle a des croyances fausses ? propos des couleurs. Si vous lui montrez une tache verte, elle dit et croit qu'elle est rouge. Supposons maintenant qu'une personne victime de commutation de stimuli soit aussi victime d'une commutation de mots. (Supposons ?galement qu'il s'agit d'un habitant d'un village perdu de l'Arctique qui n'a pas de croyances du genre ? l'herbe est verte ?, ? les extincteurs sont rouges ?, etc.). Elle parle normalement, appliquant ? vert ? aux taches vertes, et ? rouge ? aux taches rouges. En fait, elle est fonctionnellement normale. Mais ses croyances sont tout aussi anormales qu'elles l'?taient avant qu'elle ne soit victime d'une commutation de mots. Avant elle confondait les mots ? vert ? et ? rouge ?, elle ap pliquait ? rouge ? ? une tache verte, et croyait de fa?on erron?e que la tache ?tait rouge. Maintenant, elle emploie (correctement) ? rouge ?, mais sa croyance reste erron?e. Donc deux individus peuvent ?tre fonctionnellement identiques et avoir cependant des croyances incompatibles. Le probl?me de l'inversion des qualia affecte donc tout au tant les croyances que les qualia (encore qu'il ne s'agisse, semble-t-il, que des croyances qualitatives). Ce qui devrait inqui?ter non seulement ceux qui d?fendent une th?orie qui identifie la croyance ? un ?tat fonctionnel, mais aussi ceux qui sont attir?s par les expli cations du sens en termes de r?le fonctionnel ? la Harman. Notre double victime consti tue un contre-exemple de telles th?ories. Car son mot ? vert ? joue normalement son r?le dans ses raisonnements et ses inferences, et pourtant, quand elle dit d'une chose qu'elle est ? verte ?, elle exprime la croyance qu'elle est rouge, mais elle emploie ? vert ? dans un sens anormal. Je suis reconnaissant envers Sylvain Bromberger pour l'aide qu'il m'a ap port?e sur ce point. 2. Supervenience. This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions Le fonctionnalisme face au probl?me des qualia 359 description de relations de flux d'information entre des m?canismes psy chologiques. Son but est de d?composer ces m?canismes en d'autres m?canismes psychologiquement plus simples, des ? bo?tes noires ? dont la structure interne rel?ve de la physiologie plut?t que de la psychologie (voir Fodor, 1968 ; Dennett, 1975, et Cummins, 1975 , Nagel, 1969, sou l?ve d'int?ressantes objections contre une telle perspective). Un m?ca nisme qui compare deux ? deux les ?l?ments d'un syst?me repr?senta tionnel et d?termine s'ils sont des occurrences du m?me type est un exemple de m?canisme quasi primitif. Les m?canismes primitifs peuvent ?galement ?tre similaires ? ceux que l'on trouve dans un ordinateur digi tal ? ils peuvent par exemple : a) ajouter 1 ? un registre donn?, et b) soustraire 1 d'un registre donn?, ou, si ce registre contient 0, passer ? la #-i?me instruction indiqu?e. (N'importe quelle op?ration r?alisable par un ordinateur digital peut ?tre r?alis?e par une combinaison de telles op?rations, cf. Minsky, 1967, p. 206). Consid?rez un ordinateur dont le langage machine ne contient que deux instructions correspondant ? a) et b). Si vous vous demandez comment il peut faire des multiplications ou r?soudre des ?quations diff?rentielles ou ?tablir des listes de salaires, on peut vous r?pondre en vous montrant un programme ?crit au seul moyen de ces deux instructions. Mais si vous vous demandez comment il additionne 1 ? un registre donn?, la r?ponse ad?quate n'est pas un pro gramme mais un diagramme de c?blage. La machine additionne les 1 en vertu de ses circuits. Quand l'instruction correspondant ? a) appara?t dans un certain registre, les contenus d'un autre registre changent ? automatiquement ? d'une certaine fa?on. La structure computation nelle d'un ordinateur est d?termin?e par un ensemble d'op?rations primi tives et par la mani?re dont les op?rations non primitives sont construites ? partir de ces derni?res. Il est donc sans importance pour la structure computationnelle d'un ordinateur que ses m?canismes de base soient r?alis?s par des circuits de tubes, de transistors ou des relais. Pareillement, il est sans importance pour la psychologie d'un syst?me mental que ses circuits primitifs soient r?alis?s par tel ou tel m?canisme neurologique. Consid?rez qu'un syst?me est ? une r?alisation de la psy chologie humaine ? si chaque th?orie psychologique qui est vraie de ce syst?me est vraie de nous. Soit une telle r?alisation o? les op?rations psy chologiques primitives sont accomplies par de petits hommes, comme dans le cas des simulations dirig?es par des homoncules. Admettons par exemple qu'un petit homme extraie des ?l?ments d'une liste, un par un, qu'un autre petit homme les compare avec d'autres repr?sentations pour voir s'ils concordent, etc. Il y a de bonnes raisons de supposer que ce syst?me a des ?tats men taux. Les attitudes propositionnelles en sont un exemple. Une th?orie psychologique identifiera peut-?tre le fait de se souvenir que avec le fait d'avoir ? stock? ? un objet de type phrastique qui exprime la propo sition que (cf. Fodor, 1975). Si donc l'un des petits hommes a stock? This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions 360 Ned Block un certain objet de type phrastique, nous avons de bonnes raisons de consid?rer que le syst?me se souvient que P, mais ? moins qu'avoir des qualia ne soit rien d'autre qu'une certaine fa?on de traiter de l'informa tion (ce qui est au mieux une hypoth?se contestable), il n'y a aucune raison th?orique de consid?rer qu'un tel syst?me a des qualia. En bref, les qualia de ce syst?me dirig? par des homoncules sont peut-?tre aussi douteux que ceux de la simulation Fonctionnelle dirig?e par des homoncules. Mais le syst?me dont il est ici question est ex hypothesi quelque chose dont n'importe quelle th?orie psychologique qui est vraie est vraie. Donc toute raison de douter qu'il poss?de des qualia est aussi une raison de douter que les qualia appartiennent au domaine de la psychologie. A quoi on pourrait faire l'objection suivante : ? Le type de psycholo gie que vous avez en t?te est la psychologie cognitive, c'est-?-dire la psy chologie des processus de pens?e ; et il n'y a pas ? s'?tonner que les qualia ne rel?vent pas de la psychologie cognitive ! ? Mais ce n'est pas l? la psycho logie ? laquelle je pense, et si je donne cette impression, c'est simplement parce que rien dans ce que nous savons des processus psychologiques sous-jacents ? notre vie mentale consciente n'a quoi que ce soit ? voir avec les qualia. Ce qui passe pour de la ? psychologie ? de la sensation ou de la douleur, par exemple, est soit a) de la physiologie, soit h) de la psy chophysique (c'est-?-dire l'?tude de fonctions math?matiques reliant des variables de stimulus ? des variables de sensation ; par exemple la d?pen dance fonctionnelle entre l'intensit? du son et l'intensit? des ondes sonores) ou c) un pot-pourri d'?tudes descriptives (voir Melzack, 1973, chap. 2). Et bien entendu seule la psychophysique pourrait en fait ?tre con?ue comme traitant des qualia per se. Il est non moins clair que la psy chophysique ne s'int?resse qu'au seul aspect fonctionnel de la sensation, et non ? son aspect qualitatif. Les exp?riences psychophysiques que l'on peut faire sur vous auraient les m?mes r?sultats que celles faites sur un quelconque syst?me psychofonctionnellement ?quivalent avec vous, m?me en cas d'absence ou d'inversion de qualia. Or si les r?sultats exp? rimentaux demeurent inchang?s, qu'il y ait ou non absence ou inversion de qualia, on peut difficilement s'attendre ? ce qu'ils jettent un peu de lumi?re sur les qualia. En fait, ?tant donn? le type d'appareil conceptuel sur lequel s'appuie la psychologie contemporaine, je ne vois pas comment elle pourrait expliquer les qualia. Nous ne pouvons aujourd'hui concevoir comment la psychologie aurait la capacit? d'expliquer les qualia, quoique nous puis sions concevoir comment elle peut expliquer la croyance, le d?sir, l'es poir, etc. (cf. Fodor, 1975). Qu'une chose soit pour le moment inconce vable ne constitue en aucune fa?on une bonne raison de penser que cette chose soit impossible. Des concepts pourraient demain voir le jour qui permettront de rendre concevable ce qui est aujourd'hui inconcevable. Mais il faut pour le moment faire avec ce que l'on a, et ?tant donn? ce This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions Le fonctionnalisme face au probl?me des qualia 361 dont nous disposons, il semble que les qualia n'appartiennent pas au domaine de la psychologie. Que les qualia soient en fait le paradigme de ce qui appartient au domaine de la psychologie ne constitue aucunement une objection vala ble contre l'hypoth?se que ce ne sont pas des entit?s psychologiques. Ainsi qu'on l'a fait ? maintes reprises remarquer, la question de savoir si quelque chose appartient ou non au domaine de telle ou telle branche de la science est une question en partie empirique. Il s'av?re que la liquidit? de l'eau n'est pas explicable par la chimie, mais plut?t par la physique subatomique. Les branches de la science font ? tout moment face ? un ensemble de ph?nom?nes qu'elles cherchent ? expliquer. Mais il peut fort bien se faire qu'un ph?nom?ne qui semblait central ? telle branche de la science rel?ve aujourd'hui de telle autre. L'argument de l'absence des qualia exploite la possibilit? que l'?tat Fonctionnel ou Psychofonctionnel que les Psychofonctionnalistes ou les Psychofonctionnalistes voudraient identifier avec la douleur puisse se produire sans qu'aucun quale ne se produise. Il semble ?galement conce vable que le second puisse survenir sans le premier. Certains faits vont dans ce sens. Apr?s des lobotomies frontales, les patients rapportent en g?n?ral qu'ils ?prouvent toujours de la douleur, quoique cette douleur ne les g?ne plus (Melzack, p. 95). De tels patients manifestent tous les signes ? sensoriels ? de la douleur (tels que reconna?tre les piq?res d'?pingles comme d?sagr?ables), mais ils ne manifestent le plus souvent que peu ou aucun d?sir d'?viter les stimuli ? douloureux ?. Ces observations sugg?rent notamment que chaque douleur est en fait un ?tat composite dont les ?l?ments sont un quale et un ?tat fonctionnel ou Psychofonctionnel1. Ou, ce qui revient ? peu pr?s au m?me, que chaque douleur est un quale qui joue un certain r?le Fonctionnel ou Psychofonc tionnel. Si cela est vrai, on peut alors comprendre comment on a pu croire ? tant de th?ories diff?rentes sur la nature de la douleur et d'autres sensa tions ; c'est qu'on a en fait mis l'accent sur tel ou tel composant aux d?pens de l'autre. Les tenants du b?haviorisme et du fonctionnalisme avaient un composant en t?te ; les tenants de la d?finition priv?e ostensive, l'autre. Les deux approches ont commis l'erreur de donner une explication unilat?rale de quelque chose qui a deux composants de nature tout ? fait diff?rente. 3.0. Chauvinisme versus lib?ralisme Il est naturel de comprendre les th?ories psychologiques vers les quelles se tourne le Psychofonctionnalisme comme des th?ories de la 1. Ce quale peut ?tre identifi? ? un ?tat physico-chimique. Un tel point de vue rejoint une suggestion faite par Putnam ? la fin des ann?es 60 dans un s?minaire de philosophie de l'esprit ; voir ?galement Gunderson, 1971, chap. 5. This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions 362 Ned Block psychologie humaine. Selon le Psychofonctionnalisme ainsi entendu, il est impossible pour un syst?me d'avoir des croyances, des d?sirs, etc., sauf si les th?ories psychologiques qui sont vraies de nous sont vraies de ce sys t?me. Le Psychofonctionnalisme (ainsi entendu) stipule qu'il doit y avoir une ?quivalence Psychofonctionnelle avec nous pour qu'on puisse parler de propri?t?s mentales. Mais m?me si l'?quivalence Psychofonctionnelle avec nous est une condition de la reconnaissance des propri?t?s mentalesy quelle raison y a-t-il de penser que c'est une condition de la possession de telles propri?t?s ? Ne pourrait-il se faire qu'il existe une grande vari?t? de processus psycholo giques qui soient compatibles avec la possession de propri?t?s mentales, et que nous n'exemplifions qu'un type particulier de ces processus ? Sup posons que nous rencontrions des Martiens et que nous nous aperce vions qu'ils sont ? peu pr?s ?quivalents ? nous Fonctionnellement (mais non pas Psychofonctionnellement). Apr?s avoir appris ? mieux les conna?tre, nous d?couvrons qu'ils ne sont pas plus diff?rents de nous que les ?tres humains que nous connaissons. Nous d?veloppons toutes sortes de relations culturelles et commerciales avec eux. Nous ?tudions leurs journaux scientifiques et philosophiques et eux les n?tres, nous al lons voir leurs films et eux les n?tres, nous lisons leurs romans et eux les n?tres, etc. Les psychologues martiens et terrestres comparent ensuite leurs notes et finissent par s'apercevoir que, psychologiquement, mar tiens et terriens sont en r?alit? plus diff?rents qu'il n'y para?t. Ils finissent ?galement par convenir que cette diff?rence peut ?tre d?crite de la mani?re suivante. Imaginez que les martiens et les terriens soient le pro duit d'un projet conscient. En ?laborant un tel projet il faut n?cessaire ment op?rer un certain nombre de choix. Certaines capacit?s peuvent ?tre int?gr?es dans leur nature (?tre inn?es), d'autres apprises. Le cerveau peut ?tre con?u de mani?re ? accomplir des t?ches en utilisant le maxi mum de m?moire afin de minimiser l'utilisation de sa capacit? de calcul ; ou au contraire de mani?re ? ?conomiser sa m?moire et ? faire essentiel lement appel ? sa capacit? de calcul. Les inferences peuvent ?tre accomplies par des syst?mes qui utilisent peu d'axiomes et beaucoup de r?gles d'inf?rence, ou, au contraire, peu de r?gles et beaucoup d'axiomes. Imaginez maintenant que les psychologues martiens et terriens, quand ils comparent leurs notes, s'aper?oivent que martiens et terriens diff?rent autant que s'ils ?taient le point d'aboutissement de projets aussi diff? rents que possible (mais compatibles avec une ?quivalence Fonctionnelle approximative au niveau des adultes). Devons-nous pour autant rejeter notre hypoth?se que les martiens peuvent appr?cier nos films, croire ? nos r?sultats scientifiques, etc. ? Devraient-ils ? rejeter ? leur ? hypo th?se ? que nous ? appr?cions ? leurs romans, ? apprenions ? dans leurs livres, etc. ? Peut-?tre n'ai-je pas fourni suffisamment d'informations pour que nous puissions r?pondre ? ces questions. Apr?s tout, il y a peut-?tre plusieurs mani?res de d?crire les diff?rences entre martiens et This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions Le fonctionnalisme face au probl?me des qualia 363 terriens qui rendraient raisonnable de supposer qu'il n'y a tout simple ment pas de diff?rence objective, ou m?me de supposer que les martiens ne m?ritent pas que leur soient attribu?s des ?tats mentaux. Mais il y a certainement aussi plusieurs mani?res de d?crire la diff?rence indiqu?e ci-dessus qui rendraient parfaitement clair que m?me si les martiens se comportaient de fa?on diff?rente de nous dans certaines exp?riences psy chologiques subtiles, ils n'en penseraient, d?sireraient, appr?cieraient pas moins, etc. Supposer qu'il en aille autrement ne serait rien d'autre que du chauvinisme ?l?mentaire. (Je rappelle qu'une th?orie est chauviniste quand elle nie ? tort que des syst?mes ont des propri?t?s mentales, et lib?rale quand elle attribue ? tort des propri?t?s mentales.) Pour ?chapper ? cette difficult?, il vient naturellement ? l'esprit de tenter d'identifier les ?tats mentaux avec des ?tats Psychofonctionnels, en supposant que toutes les cr?atures dot?es de propri?t?s mentales} y compris les martiens, sont incluses dans le domaine de la psychologie. Ce qui revient ? d?finir ? le Psychofonctionnalisme ? en termes de psychologie ? uni verselle ? ou ? intersyst?mes ? plut?t qu'en termes de psychologie hu maine comme pr?c?demment. La psychologie universelle est toutefois une entreprise suspecte. Car comment d?cider si un syst?me doit ?tre inclus dans le domaine de la psychologie universelle ? L'une des fa?ons possibles de d?cider si un syst?me a des propri?t?s mentales, et par cons?quent s'il rel?ve de la psychologie universelle, consisterait ? recou rir ? une autre th?orie des propri?t?s mentales telle que le b?haviorisme ou le Fonctionnalisme. Mais ce genre de proc?dure a aussi peu de l?giti mit? que la th?orie utilis?e. De plus, si le Psychofonctionnalisme doit pr?supposer une autre th?orie de l'esprit, autant se contenter de cette autre th?orie. Peut-?tre la psychologie universelle parviendra-t-elle ? ?viter ce pro bl?me de ? domaine ? de la m?me fa?on que les autres branches de la science. Celles-ci commencent par d?limiter approximativement leur domaine en s'appuyant sur des versions intuitives et pr?scientifiques des concepts qu'elles sont cens?es expliquer. Elles s'efforcent ensuite de d?velopper des esp?ces naturelles permettant la formulation de g?n?rali sations nomiques qui s'appliquent ? toutes ou presque toutes les entit?s qui figurent dans les domaines pr?scientifiques. Dans bien des cas ? y compris pour les sciences biologiques et sociales telles que la g?n?tique et la linguistique ? il s'av?re que le domaine scientifique permet l'arti culation de g?n?ralisations nomiques. Il se pourrait toutefois que nous soyons un jour capables de d?velop per une psychologie universelle comme nous avons ?t? capables de d?ve lopper une psychologie terrienne. Nous d?ciderons sur une base intuitive et pr?scientifique des cr?atures qui devront ?tre incluses dans un premier temps dans son domaine, et nous travaillerons ? d?velopper les esp?ces naturelles de la th?orie psychologique qui s'appliqueront ? elles ou du moins ? la plupart d'entre elles. Peut-?tre l'?tude d'une grande vari?t? This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions 364 Ned Block d'organismes d?couverts dans des mondes diff?rents conduira-t-elle un jour ? des th?ories capables de d?terminer des conditions de v?rit? pour l'attribution d'?tats comme la croyance, le d?sir, etc., applicables ? des syst?mes qui, ? un niveau pr?th?orique, sont tout ? fait diff?rents de nous. En fait, il est certain qu'une telle psychologie intermondaine exi gera toute une cat?gorie de concepts mentalistes. Peut-?tre y aura-t-il des familles de concepts correspondant au d?sir, ? la croyance, etc. : c'est-? dire une famille de concepts du type croyance, une famille de concepts du type d?sir, etc. La nature de cette psychologie universelle d?pendra alors des nouveaux organismes que nous d?couvrirons en premier lieu. M?me si une psychologie universelle est effectivement possible, il y aura certainement de nombreux organismes possibles dont le statut mental restera ind?termin?. D'un autre c?t?, il se peut que cette psychologie universelle ne soit pas possible. Peut-?tre la vie dans l'univers est-elle ainsi faite que fera d?faut toute base raisonnable pour d?cider quels syst?mes appartiennent au domaine de la psychologie et quels syst?mes n'y appartiennent pas. Si une psychologie universelle est possible, le probl?me que j'ai sou lev? dispara?t. Le Psychofonctionnalisme universel ?vite le lib?ralisme du Fonctionnalisme et le chauvinisme du Psychofonctionnalisme humain. Mais la question de savoir si une psychologie universelle est possible est certainement une question que nous n'avons pas les moyens de r?soudre pour le moment. En r?sum? mon argument a donc ?t? jusqu'ici le suivant : 1 / Le Fonctionnalisme a cette cons?quence bizarre qu'une simulation de vous dirig?e par des homoncules a des qualia. Il ?choit donc au Fonc tionnaliste de fournir une raison de croire ? la th?orie qu'il propose. Mais l'argument fourni ? cet effet par la litt?rature Fonctionnaliste n'est pas valable, et le Fonctionnaliste semble donc incapable de se justifier. 2 / Les simulations Psychofonctionnalistes de nous partagent avec nous tous les ?tats mentaux qui figurent dans le domaine de la psycholo gie ; de sorte que la t?te Psychofonctionnelle remplie d'homoncules ne jette pas le doute sur les th?ories Psychofonctionnelles des ?tats cognitifs, mais seulement sur les th?ories Psychofonctionnelles des qualia, c'est-?-dire sur le fait que les qualia puissent appartenir au domaine de la psychologie. 3 / Les th?ories Psychofonctionnalistes des ?tats mentaux qui figurent dans le domaine de la psychologie sont cependant d?sesp?r?ment chauvinistes. Une des versions du fonctionnalisme se heurte donc au lib?ralisme, et l'autre au chauvinisme. Quant aux qualia, s'ils appartiennent au domaine de la psychologie, alors le Psychofonctionnalisme est aussi chauviniste ? leur ?gard qu'il l'est ? l'?gard de la croyance. D'autre part, si les qualia ne rel? This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions Le fonctionnalisme face au probl?me des qualia 365 vent pas de la psychologie, la t?te Psychofonctionnelle remplie d'homon cules peut ?tre utilis?e contre le Psychofonctionnalisme. Car la seule chose qui prot?ge le Psychofonctionnalisme de l'argument de la t?te remplie d'homoncules par rapport ? un ?tat mental S est que si vous avez S, alors n'importe quelle simulation Psychofonctionnelle de vous a S, parce que la th?orie de S vraie de S s'applique aussi bien ? elle qu'? vous. 3.1. Le probl?me des inputs et des outputs J'ai suppos? tout au long (ainsi que le font souvent les Psychofonc tionnalistes ? voir Putnam, 1967) que les inputs et les outputs peuvent ?tre caract?ris?s au moyen de descriptions d'impulsions neuronales. Mais c'est l? une supposition chauviniste, puisqu'elle exclut que les orga nismes d?pourvus de neurones (telles que les machines) puissent avoir des descriptions fonctionnelles. Comment ?viter le chauvinisme dans la description des inputs et des outputs ? Une premi?re fa?on d'y parvenir serait de caract?riser les inputs et les outputs uniquement en tant 'inputs et outputs. De sorte que la description fonctionnelle d'une personne pourrait simplement distinguer les outputs les uns des autres par des nombres : output 1, output 2... Un syst?me pourrait alors ?tre fonction nellement ?quivalent ? vous ? condition qu'il ait un ensemble d'?tats, d'inputs et d'outputs causalement reli?s les uns aux autres de la m?me mani?re que les v?tres le sont, quelle que soit leur nature. Et de fait, quoiqu'une telle solution ne respecte pas l'exigence pos?e par certains fonctionnalistes de caract?riser les inputs et les outputs en termes physi ques, d'autres fonctionnalistes ? ceux qui exigent seulement que les inputs et les outputs soient caract?ris?s de mani?re non mentale ? ont peut-?tre quelque chose de ce genre en t?te. Cette version du fonctionna lisme n' ? ?pingle ? pas les descriptions fonctionnelles ? la p?riph?rie au moyen de descriptions relativement pr?cises des inputs et des outputs : en fait, elle traite plut?t les inputs et les outputs exactement de la m?me fa?on dont toutes les versions du fonctionnalisme traitent les ?tats internes. C'est-?-dire qu'elle se contente, dans sa sp?cification des ?tats, des inputs et des outputs, d'exiger que ce soit des ?tats, des inputs et des outputs. Le probl?me de cette version du fonctionnalisme est qu'elle est sau vagement lib?rale. Les syst?mes ?conomiques ont des inputs et des out puts, tels que les flux de cr?dits et de d?bits. Et les syst?mes ?conomi ques ont aussi une grande vari?t? d'?tats internes, tels par exemple que celui d'avoir un taux d'augmentation du PNB ?gal au double du taux de base. Il n'est pas impossible qu'un cheikh richissisme puisse gagner le contr?le de l'?conomie d'un petit pays, par exemple la Bolivie, et mani puler son syst?me financier de fa?on ? le rendre ?quivalent ? une per sonne, par exemple ? lui-m?me. Si cela ne vous semble gu?re plausible, This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions 366 Ned Block souvenez-vous que les ?tats ?conomiques, les inputs et les outputs qui, selon le cheikh, correspondent ? ses ?tats mentaux, ses inputs et ses out puts, n'ont pas besoin d'?tre des grandeurs ?conomiques ? naturelles ?. Notre cheikh pourrait choisir n'importe quelle grandeur ?conomique ? par exemple, la d?riv?e cinqui?me de la balance des paiements. La seule contrainte qu'il doive respecter est que les grandeurs qu'il choisit soient des grandeurs ?conomiques, que les valeurs qu'elles ont consti tuent autant d'inputs, d'outputs et d'?tats, et qu'elles lui permettent de mettre sur pied une structure financi?re adaptable au moule formel en question. La mise en relation des grandeurs psychologiques et des gran deurs ?conomiques peut ?tre aussi bizarre qu'il le souhaite. Cette version du fonctionnalisme est bien trop lib?rale et doit par cons?quent ?tre rejet?e. S'il y a quelques ?l?ments de certitude dans le d?bat autour des rapports entre l'?me et le corps, l'un d'eux est certaine ment que l'?conomie de la Bolivie ne peut avoir d'?tats mentaux, quelle que soit la fa?on dont de riches amateurs puissent s'y essayer. Il nous faut ? l'?vidence ?tre beaucoup plus pr?cis dans nos descriptions des inputs et des outputs. Le probl?me est alors le suivant : y a-t-il une des cription des inputs et des outputs qui soit assez pr?cise pour ?viter le lib?ralisme et cependant assez g?n?rale pour ?viter le chauvinisme ? Pour ma part, j'en doute. Toutes les propositions de descriptions des inputs et des outputs que j'ai pu rencontrer ou auxquelles j'ai pu penser se rendent coupables soit de chauvinisme, soit de lib?ralisme. Quoique j'ai surtout insist? ici sur le lib?ralisme, le chauvinisme est le probl?me le plus difficile ? circonscrire. Les Psychofonctionnalistes tendent ? caract?riser les inputs et les outputs ? la mani?re des b?havioristes : c'est-?-dire ? caract?riser les seconds en termes de mouvements des bras et des jambes, de sons ?mis et de choses du m?me genre, et les premiers en termes de lumi?re et de sons qui affectent les oreilles et les yeux. De telles descriptions sont ? l'?vidence sp?cifiques ? une esp?ce. Les humains ont des bras et des jambes, mais pas les serpents ? et que les serpents aient ou non des propri?t?s men tales, on peut ais?ment imaginer des cr?atures du type serpent qui en ont. En fait, il est possible d'imaginer des cr?atures avec des m?canismes d'input et d'output de toutes sortes, par exemple, des cr?atures qui communiquent et man uvrent par ?mission de champs magn?tiques puissants. Bien entendu, il serait possible de formuler des descriptions Fonctionnelles pour chacune de ces esp?ces, et quelque part au paradis des disjonctions, il existe une description disjonctive permettant de trai ter de toutes les esp?ces qui ont jamais exist? dans l'univers (cette des cription pouvant ?tre infinie). Mais m?me un appel ? des entit?s aussi douteuses que des disjonctions infinies ne permettra pas de sauver le Fonctionnalisme, puisque nous n'apprenons rien par l? sur ce qui est commun ? tous les organismes qui ressentent de la douleur et en vertu de quoi ils ressentent tous de la douleur. Et cela ne permet pas non plus This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions Le fonctionnalisme face au probl?me des qualia 367 d'attribuer de la douleur ? d'hypoth?tiques (mais non existantes) cr?a tures qui ressentent de la douleur. De plus, c'est pr?cis?ment ce au nom de quoi les fonctionnalistes rejettent en g?n?ral d'un ton acerbe les th?o ries disjonctives avanc?es parfois par des physicalistes d?sesp?r?s. Si les fonctionnalistes accueillaient soudain ? bras ouverts des ?tats sauvage ment disjonctifs pour ?chapper au chauvinisme, ils deviendraient incapa bles de se d?fendre contre l'accusation de physicalisme. Les descriptions Psychofonctionnalistes habituelles (par exemple en termes d'activit? neuronale) des inputs et des outputs sont tout aussi sp? cifiques ? notre esp?ce et par cons?quent tout aussi chauvinistes. Le chauvinisme des descriptions habituelles d'inputs et d'outputs n'est pas difficile ? expliquer. Le nombre d'?tres intelligents possibles est ?norme. Etant donn? une quelconque description d'inputs et d'outputs suffisamment pr?cise, n'importe quel ?colier ?pris de science-fiction sera en mesure de d?crire un ?tre capable de conna?tre et de sentir dont les inputs et les outputs ne pourront satisfaire cette description. A mon avis, toute description physique des inputs et des outputs (souve nez-vous que beaucoup de fonctionnalistes ont insist? sur la n?cessit? d'avoir des descriptions physiques) conduit ? une version du fonctionna lisme qui est in?vitablement chauviniste ou lib?rale. Imaginez que vous soyez si gravement br?l? dans un incendie que votre meilleur moyen de communiquer avec le monde ext?rieur soit de recourir ? une transcrip tion en morse de votre ?lectroenc?phalogramme. Avoir une pens?e amu sante produit en vous une certaine configuration ?lectrique que ceux avec qui vous communiquez d?cident d'interpr?ter par un point, et une pens?e d?primante est interpr?t?e par un ? trait ?. Cette fiction n'est pas si loin de la r?alit?. Selon un article r?cent {Boston Globe, 21 mars 1976), ? des scientifiques de UCLA travaillent sur l'utilisation de l'?lectroenc? phalogramme pour contr?ler les machines... Un sujet place des ?lec trodes sur son cuir chevelu et pense ? un objet ? travers un labyrinthe ?. Le processus ? inverse ? est en apparence ?galement possible : des per sonnes peuvent communiquer avec vous en code Morse en envoyant des d?charges ?lectriques qui affectent votre cerveau (par exemple, en cau sant une image r?manente de courte ou de longue dur?e). Pareillement, si les c?r?broscopes dont les philosophes ont souvent r?v? devenaient une r?alit?, il serait possible de lire vos pens?es directement ? partir de votre cerveau. L? encore, le processus inverse semble ?tre possible. Dans tous ces cas, le cerveau lui-m?me devient une partie essentielle des m?canismes d'input et d'output. Cette possibilit? a d'embarrassantes cons?quences pour les fonctionnalistes. Vous vous rappelez que les fonctionnalistes font valoir que le physicalisme est faux parce qu'un seul acte mental peut ?tre r?alis? par un nombre infiniment grand d'?tats physiques qui n'ont pas de caract?risation physique n?cessaire et suffisante. Mais si cet argument fonctionnaliste contre le physicalisme est juste, il s'applique ?galement aux inputs et aux outputs, puisque la r?alisation physique des ?tats mentaux This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions 368 Ned Block peut constituer une partie essentielle des m?canismes d'inputs et d'out puts. En d'autres termes, quelle que soit l'interpr?tation du terme ? phy sique ? qui rend la critique du physicalisme correcte, // ne peut y avoir de caract?risationphysique des inputs et des outputs qui s'applique ? eux tous et ? eux seuls. Par cons?quent, toute tentative de formuler une description fonc tionnelle au moyen d'une caract?risation physique des inputs et des out puts soit exclut in?vitablement des syst?mes dot?s de propri?t?s men tales, soit inclut des syst?mes qui en sont d?pourvus. En d'autres termes, les fonctionnalistes ne peuvent ?viter et le chauvinisme et le lib?ralisme. Les descriptions physiques des inputs et des outputs ne font donc pas l'affaire. De plus, on ne peut non plus recourir ? des termes mentaux ou ? des termes d'action (tels que ? frapper du poing la personne qui vous a offens? ?), puisque cela reviendrait ? abandonner le programme fonction naliste de caract?risation du mental en termes non mentaux. D'autre part, comme vous vous en souvenez peut-?tre, caract?riser des inputs et des out puts simplement comme des inputs et des outputs est in?vitablement lib?ral. Pour ma part, je ne vois aucun vocabulaire pour d?crire les inputs et les outputs qui ?vite ? la fois le lib?ralisme et le chauvinisme. Je ne pr?tends pas que ce soit l? un argument d?cisif contre le fonctionnalisme. J'y vois plut?t, comme dans le cas de la critique fonctionnaliste du physicalisme, un argument qui met la balle dans le camp du fonctionnaliste. Le fonctionna liste dit au physicaliste : ? Il est tr?s difficile de voir comment il pourrait y avoir une seule caract?risation physique des ?tats internes de toutes les cr?atures dot?es de propri?t?s mentales et d'elles seules. ? Je dis au fonc tionnaliste : ? Il est tr?s difficile de voir comment il pourrait y avoir une seule caract?risation physique des inputs et des outputs de toutes les cr?a tures dot?es de propri?t?s mentales et d'elles seules. ? Dans les deux cas, il est clair qu'il incombe maintenant ? ceux qui croient que de telles caract?ri sations sont possibles d'indiquer comment elles pourraient l'?tre ?\ Ned Block. BIBLIOGRAPHIE Armstrong D. (1968), materialist theory of Mind, London, Routledge & Kegan Paul. Bever T. (1970), The cognitive basis for linguistic structures, in J. R. Hayes (ed.), Cognition and the Development of language, New York, Wiley. Block N. et Fodor J. (1972), What psychological states are not, Philosophical Review, 81, 159-181. Chisolm R. (1957), Perceiving, Ithaca, Cornell University, Press. 1. Je remercie Sylvain Bromberger, Hartry Field, Davil Hills, Paul Horwitch, Bill Lycan, Georges Rey et David Rosenthal pour leurs commentaires d?taill?s de Tune ou l'autre des versions pr?liminaires de cet article. Certaines de ces versions pr?liminaires ont fait, ? partir de l'automne 1975, l'objet d'expos?s ? Tufts University, Princeton Uni versity, The University of North Carolina de Greensboro et The State University of New York de Binghamton. This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions Le fonctionnalisme face au probl?me des qualia 369 Cummins R. (1975), Functional Analysis, journal of Philosophy, 72, 741-764. Dennett D. (1969), Content and Consciousness, London, Routledge & Kegan Paul, 1969. Dennett D. (1975), Why the law of effect won't go away, Journal of the Theory of Social Behavior, 5, 169-187. Dennett D. (1978?), Brainstorms, Montgomery, Vt, Bradford. Fodor J. (1965), Explanations in Psychology, in M. Black (ed.), Philosophy in America, London, Routledge & Kegan Paul. Fodor J. (1968), The appeal to tacit knowledge in psychological explanation, journal of Philosophy, 65, 627-640. Fodor J. (1974), Special Sciences, Synthese, 28, 97-115. Fodor J., Bever T., Garrett M. (1974), The psychology of Language, New York, McGraw Hill. Geach P. (1957), Mental Acts, London, Routledge. Gendron H. (1971), On the relation of neurological and psychological theories : A criti que of the hardware thesis, in R. C. Buck and R. S. Cohen (eds), Boston, Studies in the Philosophy of Science VIII, Dordrecht, Reidel. Grice H. P. (1975), Method in philosophical psychology (from the banal to the bizarre), Proceedings and Adresses of the American Philosophical Association. Gunderson K. (1971), Mentality and machines, Garden City, Doubleday Anchor. H?rman G. (1973), Thought, Princeton, Princeton University Press. Kalke W. (1969), What is wrong with Fodor and Putnam's functionalism ?, Nous, 3, 83 93. Kim J. (1977), Phenomenal properties, psychophysical laws, and the identity theory, The Monist, 56 (2), 177-192. Lewis D. (1972), Psychophysical and theoretical identifications, Australasian Journal of Philosophy, 50 (3), 249-258. Locke D. (1968), Myself and others, Oxford, Oxford University Press. Melzack R. (1973), The pu%%le of pain, New York, Basic Books. Minsky M. (1967), Computation, Englewood Cliffs nj, Prentice-Hall. Mucciolo L. F. (1974), The identity thesis and neuropsychology, Nous, 8, 327-342. Nagel T. (1969), The boundaries of inner space, Journal of Philosophy, 66, 452-458. Nagel T. (1974), What is it to be like a bat ?, Philosophical Review, 83, 435-450. Nelson R. J. (1969), Behaviorism is false, Journal of Philosophy, 66, 417-452. Nelson R. J. (1975), Behaviorism, finite automata and stimulus response theory, Theory and Decision, 6, 249-267. Pitcher G. (1971), A theory of perception, Princeton, Princeton University Press. Putnam H. (1963), Brains and Behavior, Philosophical Papers, vol. II, London, Cambridge University Press. Putnam H. (1966), The mental life of some machines, ibid. Putnam H. (1967), The nature of mental states, ibid. Putnam H. (1970), On properties, ibid., vol. I. Putnam H. (1975*z), Philosophy and out mental life, ibid., vol. II. Putnam H. (1975?), The meaning of meaning, ibid. Sellars W. (1968), Science and Metaphysics, London, Routledge. Shoemaker S. (1975), Functionalism and qualia, Philosophical Studies, 27, 271-315. Smart J. J. C. (1971), Reports of immediate experience, Synthese, 22, 346-359. Wiggins D. (1975), Identity, designation, essentialism and physicalism, Philosophia, 5, 1-30. This content downloaded from 128.122.149.154 on Sun, 8 Jun 2014 16:55:03 PM All use subject to JSTOR Terms and Conditions

Denial of Responsibility and Normative Negation Federico L.G. Faroldi University of Pisa, Pisa, Italy University of Florence, Florence, Italy [email protected] Abstract. In this paper I provide some linguistic evidence to the thesis that responsibility judgments are normative. I present an argument from negation, since the negation of descriptive judgments is structurally different from the negation of normative judgments. In particular, the negation of responsibility judgments seem to conform to the pattern of the negation of normative judgments, thus being a prima facie evidence for the normativity of responsibility judgments. I assume - for the argument's sake - Austin's distinction between justification and excuse, and I sketch how to accommodate the distinction between internal (justification) and external (excuse) negation of responsibility within a language with a second-order analogous of existential generalization and λ operator. In the end I confront with and refute some objections against this argument. Keywords: Responsibility, Negation, Excuses, Justifications. 1 Introduction In this paper I suggest that negations of responsibility judgments are isomorphic to the negation of normative sentences. It is only external negation that inverts the value of responsibility judgments, thus providing a a prima facie evidence to consider responsibility judgments non-descriptive and normative-like. First, I contrast two sorts of negation: negation of normative sentences and negation of descriptive sentences, pointing out where they differ. My provisional hypothesis is that internal negation and external negation work in opposite ways for descriptive sentences and normative sentences. (i) In descriptive sentences internal negation inverts their (truth) value;1 whereas (ii) in normative sentences it is external negation that changes their (normative) value. Second, I consider denials of responsibility. I show that negation of responsibility judgments falls under case (ii). It is only external negation that inverts the value of responsibility judgments, thus suggesting at least an analogy between responsibility judgments and normative judgments. 1 Of course the value inversion occurs only in classical two-valued logic. In multivalued logics, it assigns its complement. This observation applies every time I mention truthvalues. F. Cariani et al. (Eds.): DEON 2014, LNAI 8554, pp. 81–94, 2014. c⃝ Springer International Publishing Switzerland 2014 82 F.L.G. Faroldi In the end of this paper I confront with three apparently possible objections to my argument: first, I have begged the question in the definition of responsibility judgments; second, all I have shown is that external negation inverts the value of sentences if they are not descriptive, but this tells nothing about the exact nature of those sentences; third, that these features of negation hold for other modalities, so there is nothing special about normativity. This paper aims at clarifying the various kinds of negation in logic and natural language (in §2). It then advances an interpretation of normative negation (§3) and considers how this model might shed light on responsibility judgments and in particular on negative responsibility judgments (§4). 2 Negation, Negations I shall now briefly introduce some concepts I use in this paper, namely: (i) the difference between negation, denial and rejection; (ii) the difference between logical negation, and natural language negations, including internal vs external negation and metalinguistic negation.2 2.1 Negation, Denial, Rejection For the purposes of this chapter, I shall adopt the now common distinction among negation, denial and rejection.3 While these definitions are apodictically stated, nothing significant for my arguments relies on them. Very roughly, negation acts on contents. For instance, 'unhappy' is the negation of 'happy'.4 Denial is, instead, an act. It can be either a linguistic act, or a non-linguistic act (for instance: shaking one's head). Rejection is, instead, a mental attitude.5 2.2 Internal vs. External Negation Due to a felicitous intuition in [26],6 the well-known sentence: (1) The King of France is not bald can be given two readings, usually paraphrased as follows:7 2 For an engaging yet theory-driven introduction to negation, see [17]. 3 For a survey on the matter, see [24]. The paper discusses even some theories about the respective relationships among negation, denial and rejection. 4 We got 'un', in English, from a reconstructed *en-, from Proto-Indoeuropean *n- (probably zero grade of *ne-), prefix usually found in most Indo-European tongues, cf. at least [6,23,38]. 5 On rejection, see [12,19,31,34]. 6 As far as I am aware, [10] (for instance in [10]) did not notice this phenomenon or shunned it. 7 For instance by [17, §6]. Denial of Responsibility and Normative Negation 83 (1a) INTERNAL: The King of France is not -bald (is un-bald);8 (1b) EXTERNAL: It is not the case (true)9 that the King of France is bald.10 The former (1a) is usually read as an example of internal negation; whereas the latter (1b) is usually read as an example of external negation.11 In propositional logic internal negation and external negation are equivalent, that is, they both equally invert the logical value of a given sentence.12 So, for instance: (2) Maria is brunette changes its truth-value both in (2a) and (2b), examples of internal negation and external negation, respectively: (2a) INTERNAL: Maria is not brunette; (2b) EXTERNAL: It is not the case (true) that Maria is brunette. Please keep this point in mind because it will become handy infra at §3, when we shall see that internal negation and external negation are not equivalent in normative sentences.13 2.3 Metalinguistic Negation Metalinguistic negation is defined14 as a formally negative utterance used to object to a previous utterance on any grounds (even of intonations, assertability, and so on). 8 ∃x(∀y(Kxf ↔ y = x) ∧ ¬Bx) 9 'True' was proposed by [20]. 10 ¬∃x(∀y(Kxf ↔ y = x) ∧ Bx) 11 [17, §6] questions the use of 'true' and underlines how no known natural language employ two distinct negative operators corresponding directly to internal and external negation, even if a given language employs two (or more) negative operators, for instance (former: declarative negation; latter: emphatic negation): Ancient Greek: 'ou' vs. 'mē'; Modern Greek: 'den' vs. 'me'; Hungarian: 'nem' vs. 'ne'; Latin: 'non' vs. 'nē'; Irish: 'nach' vs. 'gan'; Sanskrit: 'na' vs. 'mā'. There is another 'un-' in English which is not a negative operator, but it is analogous to German 'ent-' as in 'un-fold', 'ent-falten'. See Horn's interesting list of languages with distinct negative operators at p. 366. 12 But please keep in mind that duplex negatio affirmat only in propositional logic and some natural languages, for instance contemporary standard English. Both in Old and Middle English, along with contemporary languages such as Italian, Portuguese and many others, duplex negatio n e g a t. 13 This point was noticed also by St. [2]: "dicimus etiam nos "non debere peccare" pro "debere non peccare". Non enim omnis, qui facit, quod non debet, peccat, si proprie consideretur." Cf. [28, p. 36]. For an interesting survey of modal logics in Anselm, see [15] and [36,37]. 14 For instance by [18,17]. 84 F.L.G. Faroldi Here is an example of metalinguistic negation: (3) John didn'tmanage to pass his viva - it was quite easy for him. (Emphasis signals stressed intonation here.) (4) Ben is meeting a man this evening. No, he's not- he's meeting his brother. So one does not object to the truth of a sentence, but to its (felicitous, appropriate) assertability. Another interesting feature of metalinguistic negation is its inability to be incorporated prefixally: (5) The King of France is not happy (*unhappy) - in fact there isn't any king of France.15 2.4 Illocutionary or Neustic Negation Introduced as "neustic" negation by Hare ([14, p. 21] [13, p. 35]) and later called "illocutionary" negation (originally by Searle, cf. [22,29]), it should apply to what expresses illocutive force in a sentence or the neustic. Here it is an example. (8) I promise to come. (9a) I promise not to come. (9b) I don't promise to come. According to Searle, (9a) is simply a propositional (or internal) negation, whereas (9b) is an example of illocutionary negation: one denies the very linguistic act, not its content. (9a) and (9b) are not equivalent. Illocutionary negation, if it exists, seems non-truth conditional. Is it assimilable to metalinguistic negation? As [21] maintains, not always: in fact metalinguistic negation need not to be expressed linguistically, whereas illocutionary negation is necessarily linguistic. Some doubts about the very existence of illocutionary (or neustic) negation are expressed by [7,11,16] and [21]. [16] has proposed a very interesting reading of illocutionary negation not as external or metalinguistic negation (ie, a negation of the whole speech-act), but simply as an internal negation. According to him, (10) It is not the case (that) I promise to come it is not equivalent to (9b). But (9b) must be read not as the internal negation of the coming, but as the negation of promise (as in not-promise): (9b) I don't promise to come. (9b) would be - at most - the negation of a preceding speech-act, rather the negation of that very speech-act produced by uttering (9b). 15 [17, p. 392]. Denial of Responsibility and Normative Negation 85 To Sum Up. First, there is logical negation. Logical negation is a logical operator (for instance: '¬') which is unambiguous: it always inverts the truth-value of a given sentence p.16 Moreover, internal (logical) negation and external (logical) negation are functionally equivalent.17 Second, there is natural negation, ie negation in natural languages. As we have seen supra, negation in natural languages is much more complex a phenomenon than logical negation. Firstly, it may be pragmatically ambiguous (as [17, §6] and [32] masterly argued); secondly, other than descriptive negation, natural negation can be realized externally or metalinguistically, and it is not the case that it be always used to act on the truth of a given sentence; thirdly, nondescriptive negation cannot always be semantically analyzed in terms of external or metalinguistic negation, because there are pragmatic phenomena (intonation, phonetics, etc.) involved: external or metalinguistic negation can be realized implicitly, without fixed semantic features ('it is not the case that', 'it is not true that', etc.). Fourthly, not all (negated) sentences in natural language are truth-functional, but they may be commands, prayers, wishes or insults. Third, natural negation, for instance via metalinguistic negation, can be used not only to invert the truth-value of a sentence, but also to reject or question its assertability. 3 Normative Negation Last section was devoted to analyze different kinds of negation in logic and natural languages. In this section I try to give an account of normative negation. I maintain that it can be differentiated from non-normative negation because normative negation cancels (at least) one of its presuppositions, whereas non-normative negation preserves the presuppositions of the negated sentence. I have argued elsewhere that is not possible to have distinct species of negation for descriptive and normative language, but only different realizations of a single attitude.18 I therefore propose to extend the model we have sketched in the preceding sections to normative language. We have seen that logical negation, although unambiguous, is quite limited. Natural negation is instead a complex phenomenon, it does not always act on truth-values and it can be pragmatically ambiguous, divided among at least internal and external or metalinguistic negation. Moreover, following [17, §6], we have noticed that at least metalinguistic negation is a formally negative utterance used to object to a previous utterance on any grounds, especially its assertability. 16 In many-valued logics, it assigns p's truth-value complement. In logics with more than one negation, they are nonetheless unambigous. 17 Of course I am referring here to classical propositional logic. Intuitionistic logics do not accept the equivalence of internal and external negation, nor the law of double negation: ¬¬B = B. 18 See [9, §5]. 86 F.L.G. Faroldi I propose to extend this model also to normative language. To stick to a logical level, even [25, §§31-2] noticed that while external and internal negation are functionally equivalent in propositional logic, internal negation and external negation differ quite radically in deontic logic: the fact you are under an obligation not to teach deontic logic (O¬δ), for instance, it is quite different from the fact you are not under an obligation to teach deontic logic (¬Oδ). In an analogous fashion, I maintain that internal normative negation keeps the sentence binding or, so to speak, normative, only to invert its deonticity: from obligatory to forbidden, and so on.19 (Please note that I am not forced to assign normative sentences truth-aptness, because truth does not tell us the whole story even when (non-normative) natural language is concerned.) External or metalinguistic negation is a rejection of the assertability (lato sensu) of a prima facie, allegedly normative sentence. Specifically, though, rejection of the assertability of a normative sentence is (implicity, I maintain) not a normative judgment, but a judgment on its normativity (or bindingness, or you name it). If a speaker feels20 a given (non-normative) proposition unassertable, he rejects it metalinguistically; if he feels a given (prima facie normative) proposition not binding or not normative, he rejects it metalinguistically or externally, canceling its presupposition of normativity.21 Consider the following normative sentence: (1) Abortion is wrong and its prima facie negation: (2) Abortion is not wrong. Both are moral (normative) judgments, and share - among others - the following presupposition: (0) Abortion can be an object of a genuine moral judgment. Now consider external negation of (1):22 (3) It is not the case that abortion is wrong. Now, while (2) is still a normative judgment, (3) seems intuitively a judgment on the normativity of (1). (3) cancels (1)'s and (2)'s presupposition (0), because it simply rejects that abortion can be object of (that) moral judgment. Let's now make a comparison with internal and external negation of nonnormative sentences. Let's consider (4) He stopped beating his wife 19 I am well aware that not all normative sentences (or propositions) are in deontic terms. This was only an example to illustrate the general principle I want to bring forth. 20 Please note that 'to feel' here is used generally has no intended reference to emotivism or expressivism. 21 I am using this as a sort of a term of art, in order to make a general point without supporting a substantive theory of normativity either in terms of reasons (cf. for instance [27,30]), good (cf. [35]) or oughts. 22 Of course it can be realized also metalinguistically. Denial of Responsibility and Normative Negation 87 its internal negation: (5) He didn't stop beating his wife and its external negation: (6) It's not true that he stopped beating his wife. Neither (5) nor (6) modify the ("factive") presuppositions of (4) such as that he has a wife, and he used to beat her. Since - as we have seen - not every instance of external or metalinguistic negation is analyzable with distinct semantic (or, for that matter, syntactic) features, I assume a paraphrase in terms of external negation will account for the phenomenon, at least for our present purposes. Considering the problem with normative negation only from the point of view of truth is quite limited, because truth does not tell the whole story even in nonnormative negation, as I pointed out in the case of metalinguistic negation. This turns out to be a plus, because normative sentences are usually not considered truth-apt.23 In this section I contrasted descriptive and normative sentence by considering negation. I showed that normative negation, usually realized externally or metalinguistically, cancels its presupposition(s) of normativity. Next section applies this conclusion to judgments of responsibility, showing that their structure with respect to negation is akin to normative sentences. 4 Denial of Responsibility In last section I contrasted descriptive and normative sentence by considering negation. I suggested that normative negation, usually realized externally or metalinguistically, cancels its presupposition(s) of normativity. In this section, I apply these results to judgments of responsibility, in order to provide an argument to the thesis that responsibility judgments are normative, 23 And consequently one may maintain that (a) what you negate is not their truth; or that (b) norms cannot be negated. (a) was the position of the very first philosopher known to have written on this topic: Jerzy Sztykgold. In [33], he argued that you cannot negate the truth of norms, but only their righteousness [s"luszność] in terms of non-righteousness [nies"luszność]. (Righteousness and unrighteousness are, for Sztykgold, the strict análogon of truth and falseness.) (b) was instead the position of Karel Englǐs ([8]), according to which: (i) logical operations are possible only for "descriptive judgments" [soudy ]); (ii) negation [popřeńı ] is a logical operation; (iii) norms [normy ] and postulates [postuláty ], although sentential, are not "descriptive judgments"; and therefore (iv) logical operations don't apply to norms and postulates. In particular: (v) norms cannot be negated. Of course Englǐs' argument shows - at most, if premise (i) holds - that negation as a logical operator doesn't apply to norms. But negation is not exclusively a logical operator. Negation exists outside logic, in natural language, with different characteristics. 88 F.L.G. Faroldi and namely an argument "from negation". I shall show that when one denies responsibility, what happens is (a) what happens when one denies normative statements; (b) what happens is the case only when normative entities are concerned. This might show that judgments of responsibility are normative.24 Here is a more schematic version of my fourth argument: 1. when you deny a responsibility judgment, what happens (what obtains) is a cancelation of its presuppositions; 2. canceling of presuppositions obtains only when normative judgments are negated; 3. Therefore, responsibility judgements are normative judgments. Let's begin. I shall use negation a test to isolate a normative entity. We have seen back in §3 that negation of descriptive and normative entities differs in at least one substantial point: internal and external negation work in opposite ways. Here is an example for descriptive statements: Internal negation (1) "John isn't tall' vs. External negation (2) "It's not the case that John is tall" Now, let's take a normative statement (for simplicity's sake, I shall consider an imperative): Internal negation O(¬W ) (3): "Don't shut the window!" (that is: "Shut not the window"). Note that (3) and its "positive" (3a) Shut the window share a presupposition of normativity. Now, (3a)'s external negation: External negation ¬O(W ) (4a) "I do not accept that is the case of shutting the window"/ (4b) "I do not accept the command 'Shut the window"'/ (4c) "I don't care".25 instead, rejects (cancels) the presupposition of normativity that both (3) and (3a) shared. 24 This is by no means the standard theory. When judgments of responsibility are kept separate from responsibility or concepts of responsibility, they are usually considered non-normative; for example, judgments of responsibility are considered explanatory by [5,4]. Anderson ([1, §3.1] and p.c.) considers responsibility judgments to be normative, even though he does not provide any arguments for this thesis. 25 Of course I am aware these are only some possible paraphrases - there might be many more. The most important fact is that internal and external negation can be consistently kept separable. Denial of Responsibility and Normative Negation 89 As I explained in §2, for descriptive sentences it is internal negation that might change their truth-value (from truth to false and viceversa); vice versa, for normative (imperative, in this case) sentences, it is external negation that changes their normativity-value, by rejecting the presupposition of normativity. Now, let's apply this test to responsibility. Internal negation (5) "He is not responsible for killing A, because. . . " vs. External negation (6) "It is not the case that he is responsible for killing A, because. . . "/ Now, if (5) stands to (1) as (6) stands to (2), we can confidently conclude that (5) and (6) are statements analogous to (1) and (2), that is, non-normative. Quite on the contrary, if (5) stands to (3) as (6) stands to (4), we can confidently conclude that (5) and (6) are statements analogous to (3) and (4), that is, broadly normative. It turns out, unfortunately, that you cannot really tell if (5) - internal negation of responsibility - tells us something of significance, for the very simple reason that its interpretation requires an understanding of responsibility. If you think responsibility is an objective state-of-affairs, that can be somehow empirically ascertained, then you would interpret (5) as a descriptive statement, whose truth-value is to be checked against the world; and vice versa. Therefore, let's turn to (6) to seek some clarification of the matter. My hypothesis is that a statement such as (5) stands for a justification; while (6) stands for an excuse. I take advantage of the paradigm excuse vs. justification developed in [3]. With a justification, I maintain, we remain in the domain of the normative: we accept A, and even add some reasons for it. The presupposition of normativity is kept. Quite on the contrary, an excuse, in a way, suspends what was going on, it makes "normativity freeze" because it refers to conditions other than the very act A, conditions that (by definition) rule out responsibility (duress, infancy, mental incapacity, maybe psychopathy for moral responsibility). The presupposition of normativity is canceled. In the words of Austin: [i]n the one defence [= justification], briefly, we accept responsibility but deny that it was bad: in the other [= excuse], we admit that it was bad but don't accept full, or even any, responsibility ([3]). it is not quite fair or correct to say baldly "X did A". We may say it isn't fair just to say X did it; perhaps he was under somebody's influence, or was nudged. Or, it isn't fair to say baldly he did A; it may have been partly accidental, or an unintentional slip. Or, it isn't fair to say he did simply A – he was really doing something quite different and A was only incidental, or he was looking at the whole thing quite differently ([3, p.2]). 90 F.L.G. Faroldi First, excuses are denial of responsibility because, in giving excuses, a person contests or opposes a previously ascribed responsibility, by rejecting constitutive elements of the accusation: for instance, by denying having committed anything. He simply denies that the previous ascription of responsibility is sound. Second, excuses are rhetic (and not thetic) negations (denials) of responsibility because they do not seek to cancel or nullify responsibility, since they assume that there is no responsibility whatsoever. Absence of responsibility is constitutive of excuses: if there were responsibility, they would not be excuses but - at most - justifications. Excuses do not presuppose responsibility, but only ascription of responsibility.26 Justifications, instead, are not at all negations of responsibility because justifications presuppose responsibility: justifications affirm responsibility, but deny it is responsibility for something bad. (A paradigmatic example seems to me "self defense": a admits to having killed b, but b was assaulting him with a knife, for instance.) Negative Properties and Existential Generalization. A possible way to account for the difference between internal and external negation, and the existence of a given property is to consider a plausible analogous of Existential Generalization at the second order (I am not arguing for it at this point; I shall only make my point with a somewhat sloppy notation). (EG1) Fa |= ∃xFx (1) ¬Fa |= ∃xFx But with a λ operator we can gain negative properties: (2) λx(¬Fx)a |= ∃x(¬Fx ∧ x = a) Likewise, it is plausible to hold the following: (EG2) Fa |= ∃P∃x(Px ∧ x = a ∧ P = F )27 (3) ¬Fa |= ∃P∃x(Px ∧ x = a ∧ P = F ) but (4) λPλx(¬Fx)a |= ∃P∃x(Px ∧ x = a ∧ P = λx(¬Fx)) While both (1) and (3) are plain external negations and don't license any inference to the existence of either something or some property; (2) and (4) can, with the use of λ-abstraction, represent internal negation. Internal negation seems to license an inference to the existence of some property of sort. The connection with internal and external negation of responsibility, while stretched, is significant: in fact, we suggested that with external negation of responsibility (excuse) there is no more responsibility (and normativity) involved, whereas with internal negation of responsibility (justification) the normativity is kept. 26 As I noted with accusations, not all excuses are pled using a verb like 'to excuse' or 'scusare'; in an analogous fashion, it is not only the use of 'to excuse' or 'scusare' that can make an excuse. 27 Of course one needs to explain what '=' among P and F means. I thank Tim Williamson for discussion on this point. Denial of Responsibility and Normative Negation 91 Two Examples: Excuses vs. Justifications. I am going to illustrate the difference between justification and excuses. I ask the reader to imagine two fictional criminal cases (both involve a death), and to abstract from particular legal systems in order to focus on the general point. In the first, let's call it WIFE, a man comes back home and sees an intruder trying to rape or kill his wife. By chance, there is the intruder's loaded gun at hand. The man takes it up, aims and finally shoots the intruder down - killing him. In court, he admits the murder and puts forward his reasons. His lawyer says: "Look, he is not responsible for the killing, because that was self-defence: he was trying to defend and save his own wife." This is a justification: you admit your deed (there are all the relevant required elements: actus reus, mens rea, volition, intention, knowledge and so on to make that killing a murder) but you have a (good) reason for you action. In the second, let's call it MAD, a mentally-ill man escapes from a psychiatric hospital, manages to get a gun, and shoots down a random passer-by. His lawyer says: "Look, he is not responsible for the killing, because it is not the case he is (= can be) responsible at all : he is mad (under duress, in infancy. . . )." This is an excuse: you may admit the deed, but it was done without the relevant required conditions: without mens rea, for instance, or without those capacities required for a death or a killing to be a murder. To sum up, with a justification you deny your responsibility for that deed qua a particular action (but you admit, nonetheless, that you are under the domain of responsibility, that you can be responsible); with an excuse you deny your responsibility tout court, you deny that you are under the very domain of responsibility. The lawyer's sentence in WIFE: "he is not responsible for the killing" is comparable to (3): "Don't shut the window" and (5): "He was not responsible", inasmuch as they are internal negations. On the contrary, the lawyer's sentence in MAD: "it is not the case he is (= can be) responsible at all" seems to me analogous to (4): "It is not the case you order me to shut the window" and (6): "It is not the case that he is responsible for A, because. . . " As (3) conserved the imperative nature of the sentence, so WIFE conserved the domain of responsibility. As (4) instead went out the domain of the imperative, to make a non-imperative claim, in the same way MAD appealed to a condition - in a way a non-normative, even factual condition - to be excluded from the domain of responsibility. This linguistic evidence is consistent with the conceptual arguments I put forward earlier in this section: while justifications aren't at all denial of responsibility because they presupposes responsibility, excuses are in fact denial of responsibility, because they reject it. With justifications and excuses, negation of responsibility coincides both with a linguistic act (denial) and a mental state (rejection). 92 F.L.G. Faroldi We suggested that – (i) when we deny responsibility, we have (at least) two cases: internal negation (which stands for a justification) and external negation (standing for an excuse). Then, we have seen that – (ii) internal negations of responsibility do not exit the domain of responsibility (they presuppose responsibility); whereas external negations do (they reject the presupposition of responsibility). But this was exactly what happened with normative sentences (as I showed in §3): internal negation keeps the sentence normative (it keeps the presupposition of normativity), whereas external negation rejects it (it cancels the presupposition of normativity). If we suppose that this kind of negation is at work only with non-descriptive (and namely, normative statements), we can therefore conclude that – (iii) since judgments denying responsibility are structurally akin to normative sentences, responsibility judgments are akin to normative sentences. Caveats and Assumptions. Now, some caveats. I have limited my discussion to the word (and the concept) of responsibility in the proper, fuller sense. I am very well aware that there may be pragmatical ways to express a responsibility judgment without mentioning the word 'responsibility' or any related. I am also aware that we may get indicative (or descriptive) sentences (to express/ascribe responsibility). For this (and for other) reasons linguistic arguments are interesting but not conclusive. I offer more (non linguistic) arguments for the thesis that responsibility is normative in [9]. Last but not least, my argument makes the following assumption: there are only two kinds of language relevant to our investigation here: descriptive and normative language. This may not be the case: there are several other language domains I am not considering: prayers, exclamations, insults, whose "status" with regard to negation is unclear. Therefore, it might be the case that the different ways negation works (in descriptive and normative domains) is not exclusive: negation might work in prayers as in normativity, and the second premise of my argument would be factually undermined. Assuming the prima facie evidence I discussed as conclusive might be too strong, and other interpretations are certainly possible depending on substantive theories of normativity, modality, and responsibility. But even if in general this argument does not prove to be conceptually unassailable, I think it is still very telling. Objections. I consider here three possible objections to my argument: first, I have begged the question in the definition of responsibility judgments; second, all I have shown is that external negation inverts the value of sentences if they are not descriptive, but this tells nothing about the exact nature of those sentences; third, these features of negation may be shared by other kinds of modality, so there is not special about normativity. To the first objection, I put forward a twofold reply: first, there is no shared consensus either on what responsibility is or on what responsibility judgments Denial of Responsibility and Normative Negation 93 are: a degree of arbitrariness is needed anyway; second, there is no conceptual reason precluding my analysis to be extended further, given the right premisses. To the second objection, I reply that I have at least shown that responsibility judgments are not descriptive; nonetheless I believe a linguistic test such as mine cannot exhaust the richness of human practices - in other words, normativity is not a sheer linguistic notion. To the third objection, I reply that, examples with "oughts" notwithstanding, it is not clear whether normativity is a modality or not (it may be a property, for one). Moreover, other modalities may be normative as well (recently [30] so argued for necessity, the a priori, and other modalities), and thus these features of negation shouldn't come as a surprise. Acknowledgements. For discussion on various points, suggestions, and critiques I thank Amedeo Giovanni Conte, Mattia Bazzoni, Guglielmo Feis, Sergio Filippo Magni, Timothy Williamson and two anonymous referees. References 1. Anderson, S.: Coercion. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (Winter 2011) 2. Anselm of Canterbury. "Lambeth's Fragments". In: Schmitt, F.S. (ed.) Ein neues unvollendetes Werk des hl. Anselm von Canterbury. Aschendorff, Münster i W (1936) 3. Austin, J.L.: A Plea for Excuses: The Presidential Address. In: Proceedings of the Aristotelian Society, vol. 57, pp. 1–30 (1956) 4. Björnsson, G., Persson, K.: A Unified Empirical Account of Responsibility Judgments. In: Philosophy and Phenomenological Research (forthcoming) 5. Björnsson, G., Persson, K.: The Explanatory Component of Moral Responsibility. Noûs 46(2), 326–354 (2012) 6. Brückner, A.: S"lownik etymologiczny jezyka polskiego. Wiedza Powszechna, Warszawa (1957) 7. Cohen, L.J.: Do Illocutionary Forces Exist? The Philosophical Quarterly 14, 118–137 (1964) 8. Englǐs, K.: Postulát a norma nejsou soudy. In: Casopis pro právńı a státńı vědu XXVIII, pp. 95–113 (1947) 9. Faroldi, F.L.G.: The Normative Structure of Responsibility. Law, Ethics, Neuroscience (forthcoming) 10. Frege, F.L.G.: Der Gedanke: Eine logische Untersuchung. In: Beiträge zur Philosophie des Deutschen Idealismus I, pp. 58–77 (September 1918) 11. Garner, R.T.: Some Doubts about Illocutionary Negation. Analysis 31, 106–112 (1971) 12. Gomolińska, A.: On the Logic of Acceptance and Rejection. Studia Logica 60(2), 233–251 (1998) 13. Hare, R.M.: Some Sub-Atomic Particles of Logic. Mind 98, 23–37 (1989) 14. Hare, R.M.: The Language of Morals. Clarendon Press, Oxford (1952) 15. Henry, D.P.: St. Anselm on the Varieties of 'Doing'. Theoria 19, 178–183 (1953) 16. Hoche, H.-U.: Do Illocutionary, or Neustic, Negations Exist? Erkenntnis 43, 127–136 (1995) 94 F.L.G. Faroldi 17. Horn, L.R.: A Natural History of Negation. University of Chicago Press, Chicago, Illinois (1989) 18. Horn, L.R.: Metalinguistic Negation and Pragmatic Ambiguity. Language 61(1), 121–174 (1985) 19. Incurvati, L., Smith, P.: Rejection and Valuations. Analysis 70(1), 3–10 (2010) 20. Karttunen, L., Peters, S.: Conventional Implicature. In: Oh, C.-K., Dinneen, D. (eds.) Syntax and Semantics 11: Presupposition, pp. 1–56. Academic Press, New York (1979) 21. Moeschler, J.: Negation, Scope and the Descriptive/Metalinguistic Distinction. Generative Grammar in Geneva 6, 29–48 (2010) 22. Peetz, V.: Illocutionary Negation. Philosophia: Philosophical Quarterly of Israel 8, 639–644 (1979) 23. Pokorny, J.: Indogermanisches etymologisches Wörterbuch. Francke (1994) 24. Ripley, D.: Negation, Denial, and Rejection. Philosophy Compass 6(9), 622–629 (2011) 25. Ross, A.C.N.: Directives and Norms. Humanities Press, New York (1968) 26. Russell, B.: On Denoting. Mind 14(56), 479–493 (1905) 27. Scanlon, T.M.: Being Realistic about Reasons. Oxford University Press, Oxford (2014) 28. Schmitt, F.S.: Ein neues unvollendetes Werk des hl. Anselm von Canterbury. Aschendorff, Münster i W (1936) 29. Searle, J.R., Vanderveken, D.: Foundations of Illocutionary Logic. Cambridge University Press, New York (1985) 30. Skorupski, J.: The Domain of Reasons. Oxford University Press, Oxford (2010) 31. Smiley, T.: Rejection. Analysis 56(1), 1–9 (1996) 32. Speranza, J., Horn, L.R.: A Brief History of Negation. Journal of Applied Logic 8(3), 277–301 (2010) 33. Sztykgold, J.: Negacja normy. Przeglad filozoficzny 39, 492–494 (1936) 34. Tamminga, A.: Logics of Rejection: Two Systems of Natural Deduction. Logique et Analyse 146, 169–208 (1994) 35. Thomson, J.J.: Normativity. Open Court, Chicago, Illinois (2008) 36. Uckelman, S.L.: Anselm's Logic of Agency, Amsterdam: Institute for Logic, Language and Computation (ILLC). University of Amsterdam (2007) 37. Uckelman, S.L.: Modalities in Medieval Logic. Institute for Logic, Language and Computation, Amsterdam (2009) 38. Vasmer, M.: Russisches etymologisches Wörterbuch. C. Winter, Heidelberg (1958)

booK reVIeWS AND NotICeS470 TONY BOLOS University of Edinburgh Rolfe King. Obstacles to Divine Revelation. Continuum, 2009. If there is a God, then there seems to be significant hindrances or obstacles in the way in which God reveals himself to his creation. What are these obstacles? Can these obstacles be overcome? Are there necessary limits to the way in which God must operate when it comes to divine revelation? If there are necessary limits, is this a feature of the created order or is this a feature of God himself? Given these limitations, what sort of divine acts must God use in order to reveal himself (pp. 1-3)? In Obstacles to Divine Revelation, rolfe King provides some stimulating answers to the above questions and, interestingly, brings the discussion to the centre of epistemology. It is the latter contribution, I think, that is unique to King's project. His project begins, however, by answering an essential question: What is revelation? King defines revelation as: "God's self-disclosure, in any form, leading to some kind of awareness, or knowledge, of him" (p. 5). From this, one might assume that God's options for revealing himself are limitless. However, King rightly notes that God's options are limited given that the people that he wishes to communicate with are also limited (p. 54). If God, for example, chose to reveal himself with distinct clarity and undeniable evidence in the uDFy galaxy (13 billion light years away) it would do little in convincing his creation of himself. So if God does indeed wish to reveal himself (something traditional theism affirms), then he must do so in such a way that reflects the capacities of his limited creation (e.g., by not revealing himself in galaxies to which we will never have access). So if God's options for revelation are indeed limited, this might be considered the first obstacle to his revelatory plan. This obstacle, as King notes, says nothing about God and everything about his creation. This is why King understands obstacles to divine revelation as: "Any feature of the created order that may either block or hinder a form of divine disclosure, or has in some way to be overcome in order for God to disclose himself." (p. 5). booK reVIeWS AND NotICeS 471 These features, whatever they might be, must be located within the created order. but what, exactly, are these features? And how exactly does one go about identifying these features? The two principles that King suggests to identify the features that hinder revelation are the epistemic-revelatory principle (erP) and the revelatory-context principle (rcP) (pp. 39-40). The former principle claims that epistemic problems concerning evidence (interpreted quite broadly) are an obstacle to divine revelation (p. 9), while rcP claims that a specific context is required in order for God to reveal himself in a way that is comprehensible by his creatures. All obstacles to revelation, according to King, fall into the above categories – they are either epistemic problems or contextual problems (pp. 40-41). I find erP more interesting, and more controversial I might add. erP, for example, makes the claim that God might have difficulty revealing himself insofar as our epistemic position lacks complete discernment of the available evidence. The "eden parable" illustrates this nicely (pp. 44-47). In the parable the angels are talking amongst themselves about all the different ways in which God might reveal himself to his new creation. Despite the available options, they all seem plagued by the fact that there is no assurance the creatures will trust the revelation (some experience for example) to be evidence of what God is trying to reveal (perhaps that he exists and that he loves them). Scepticism looms, despite the potentially good revelatory intentions. but, King suggests, there is another option available to God – one that isn't dependent on the difficult task of matching belief with the available evidence. King notes that this option, which he calls direct cognition, might potentially undermine his central claim that there are indeed obstacles to revelation (p. 60). This worry, however, seems unnecessary. even if there is direct cognition, one might assume that this feature has been corrupted (by sin for example) and thus doesn't always function the way it was intended. This aside, it is King's understanding of Plantinga (as it relates to direct cognition) that deserves more attention. King's discussion of Plantinga is very interesting, but ultimately, I think, mistaken. King's key claim is that Plantinga's model is (1) a form of direct cognition and that (2) direct cognition should be understood as divine self-testimony. And that in order for testimony to be trusted, (3) there needs to be decisive evidence that the testifier can in fact booK reVIeWS AND NotICeS472 be trusted (pp. 76-77, 100). If (1)–(3) is true, then the problem with Plantinga's model seems obvious. After all, if belief in God is directly acquired by some properly functioning faculty, the belief still wouldn't be warranted, according to King, since there is no evidence that the faculty is in fact functioning properly (p. 78). Without getting into all the nuances of the debate here, I think the point to stress is that (3) is false. even if Plantinga's model is a form of testimony (which is questionable), not all models in the current epistemology of testimony would concede (3) given that it's not at all clear that there needs to be decisive evidence that the testifier can in fact be trusted. You might think that testimony is noninferential, which is consistent with Plantinga's claim that belief in God is properly basic. Thus, the claim would simply be that there couldn't be any defeating evidence against the testimony of the testifier. This is known as the defeater clause, which Plantinga's model rightly incorporates. At any rate, however one feels about the above understanding, the point that King is trying to make is that any account of direct cognition is plagued by the issue of trust. And the requirements concerning evidence and the necessity of trust bring us closer to King's position. King's position, then, is that evidence and trust are both necessary given that any (special) revelation necessarily involves testimony (p. 194). This being the case, King provides a solution to the trust problem and claims, as was seen in the critique of Plantinga, that sufficient evidence is needed for trust. King calls this trust-evidentialism (p. 176). The obvious question is whether God can provide such evidence. but in asking this question, we are immediately faced with a dilemma. The dilemma, as described by King, is that "God cannot give us any evidence for special revelation independently of self-testifying in some way that the evidence is due to him. but we need independent evidence to rationally trust that this purported revelation is from God" (his emphasis, p. 251). King's response to this is that "although God cannot give us evidence independent of his self-testimony we may be able to find such evidence" (p. 197). This point isn't as confusing or controversial as it sounds. For example, it was argued by King that there are certain limitations in the ways that God can reveal himself given our limitations. This being the case, there must be some necessary structure in which God will reveal himself. And the necessary structure of this revelation must take into account our limitations. So if we can discern what exactly the necessary booK reVIeWS AND NotICeS 473 structure of revelation is, we can then know what kind of evidence to look for (see pp. 176-178). King gives several suggestions that I won't recount here (see pp. 177-181, 197, 201-205), but the point that should be taken is that this evidence would provide the basis for what King claims to be central to divine disclosure – trust. So, then, there is the initial evidence that King finds necessary for trust. but, it seems, we are still plagued with the problem of properly evaluating the evidence that would lead to knowledge of God. After all, it's not objectively clear that the evidence does anything beyond giving the initial trust or confidence to think some divine testimony might be compelling. This is where King's journey-epistemology becomes becomes important. As King notes, "all I can do is to try to find the best grounds on which to base my trust" (p. 200). And this initial trust will be subject to reasons of the heart (suspicions, fears, personal goals, etc. p. 214). So it's a journey in that the evidence (the evidence that initiates trust) is not sufficient to know (in this case perhaps God's existence), but it is both necessary and sufficient to get you on your journey. While the significance of the arguments presented above depend, I think, on the truth of (3), I find King's Obstacles to Divine Revelation both interesting and compelling. Students and scholars who work in religious epistemology and philosophy of religion will find King's work to be of value as it examines and advances many contemporary issues in those fields.

Criminal Law News Issue 105 July 2017 Analysis of R v H [2014] EWCA Crim 1555 -pp 2-26 Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 2 Analysis of R v H [2014] EWCA Crim 1555 Picture: United Kingdom's Supreme Court. Source: Google Hasty conviction 'up north' 1. This is an analysis proposing laissez faire approach to investigating child sex abuse in the North of England, United Kingdom, without reference to any governance policies; statutes; protocol; disclosure and much more. This is occurring within families, and this paper and this specific case law of a young, well-liked, hard-working, senior General Medical Practitioner, destroyed through hearsay, ignorance about mental illness and officials walking through the law 'as if they owned it', with no care for human rights or dignity, on the sayso of a very mentally ill youth who imagined her good father doing bad things to her and on the hearsay of an alleged confession made by one mentally ill youth allegedly to another mentally ill youth whilst both were incarcerated in a longstay mental institution. Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 3 2. This paper seeks to warn of the consequences of lack of careful and diligent respectful following of the criminal laws in haste to arrive at a conviction. This paper well illustrated government agencies back-footing and using their might to change regulations so as not to be found wanting at some future time when this case returns to haunt them. There is a saying in England, "You can run, but you cannot hide". 3. Expert2 evidence is admissible only if it provides the court with scientific information likely to be outside the experience and knowledge of a judge or jury. In other words, expert evidence will be restricted to that which in the opinion of the court is necessary to assist the court to resolve the proceedings. 3 The idea that a young girl's father had sexual intercourse with her and therefore cause her to become psychotic and a diagnosed schizophrenic must not be tolerated and is the reason for Dr Stephen Hamilton's very lengthy imprisonment, a sentence more befitting a convicted murderer. 4. There is abounding evidence of real sex offences caused to youth who do not become psychotic and schizophrenic. As Thomas Szasz stated, the controversy about mental illness still rages, and the nature of controversy is still stubbornly misunderstood because mental health professionals and lay persons alike seem to fail too understand that mental illness is not a disease brought on by someone, something or genetics, but that if mental illness is misbehaviour, then it is behaviour, not disease. 1 1 See Stasz, T.S., (1961) The myth of mental illness: foundations of a theory of personal conduct, New York: Hieber-Harper. See also, Szasz, T. (1989) Law Liberty and psychiatry, United States: Syracuse University Press. Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 4 This case must be urgently referred to the Criminal Appeals Review Commission 5. It has become apparent that this appeal decision is flawed. This appeal (Case No 2013 03235-B2) against the decision by HH Judge Mansell, QC, at Manchester Crown Court (Case T20127444) goes to criminal court procedure. This case has the background of a bitter divorce case and in the course of time, the child of the family, now no longer under 16 years of age, revealed to a schoolchild who herself allegedly suffered sexual abuse, that her father, Dr Stephen Hamilton, a senior medical General Practitioner at GP Surgery at 2 Lucy Street, Manchester, aged 43 at the time of her revelation, had committed sex offences against her. The friend allegedly told the police. Dr Stephen Hamilton was arrested and charged with 3 counts of cruelty to a person under 16, 6 counts of rape of a child under 13 and 3 counts of sexual assault of a child under 13. He was convicted and sentenced to a total of 18 years imprisonment. Expert witness evidence in the case of Stephen Hamilton v The Queen [2014] 1 Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 5 6. Expert2 evidence is admissible only if it provides the court with scientific information likely to be outside the experience and knowledge of a judge or jury. In other words, expert evidence will be restricted to that which in the opinion of the court is necessary to assist the court to resolve the proceedings. 3 This case must be urgently considered by the Criminal Appeals Review Commission as it becomes apparent that this court of appeal decision is flawed, not surprising when one considers the mountain of miscarriages of justice in the United Kingdom criminal justice system. 7. This appeal (Case No 2013 03235-B2) against the decision by HH Judge Mansell, QC, at Manchester Crown Court (Case T20127444) goes to criminal court procedure. This case has the background of a bitter divorce case and a mother, also a medical doctor, who was abusing alcohol use and using anti-depressants whilst herself also practicing as a general medical practitioner. In the course of time, the older of the two children in this dysfunctional family who sought to selfmedicate to solve their older child's mental illness, now no longer under 16 years of age, but diagnosed a severely ill schizophrenic, revealed to another mentally ill youth incarcerated at a long-stay mental institution, that her father had allegedly raped her after hearing from the other girl that her father had raped her. The 'friend' allegedly told the police. 8. Dr Stephen Leonard Hamilton, who was a senior partner at Heaton Medical Centre in Lucy Road, appeared in court to face 14 charges of raping a child under the age of 13; administering a poison or noxious substance; three counts of cruelty to a person under 16, six counts of rape of a child under 13 and 3 counts of sexual assault of a child under 13. He was tried in Manchester Crown Court, was convicted and sentenced to a total of 18 years imprisonment, judge Mansell presiding. 2 2 Editor, 'GP faces 14 child rape charges', The Bolton News, 19 June 2012. See also Editor, 'Doctor jailed for sex offences launches bid to clear his name', Manchester Evening News, 12 June 2014. Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 6 Judge Mansell QC. Source: Google. The daughter of the Appellant, anonymised as X 9. Although the 'victim' was anonymised as X, all the local newspapers revealed Dr Hamilton's medical practice address, his photograph and his full name. After Dr Stephen Hamilton was convicted and imprisoned for 18 years and his unsuccessful appeal the General Medical Council struck off Mrs Hamilton, Stephen Hamilton's ex-wife, herself also a medical doctor, for bringing shame on the profession, having been charged with the repeated road traffic offence of driving whilst drunk. It is even easier to identify this youth and her family, all lives now in total ruins. Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 7 PictureHeaton Medical Centre where Dr S Hamilton was a senior partner. Source: Google Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 8 Timeline to this serious criminal case against Dr Hamilton 1. Dr H is the father of X. Dr H's wife was also a medical doctor. Dr & Mrs H have two daughters X and a younger daughter. 2. 2011; November: X first tells her mother, Mrs H, of sexual abuse by Dr H, her father. X is 15 years old. 3. 2007: X said sexual abuse started. 4. 2007: Mother of X, Dr H's wife, also a doctor, said X became ill. 5. 2007: Mother of X , Mrs H, said she took X to a General Practitioner and then to hospital consultants. 6. 2007: Mother confirmed that X's hospital consultants advised treatment from psychiatrist and appointment followed. 7. 2007: X missed 1st appt. with Psychiatrist. New appt. made. X missed 2nd appt. New appt. made. X missed 3rd appt. 8. 2007: Dr H tries to convince mother of X , Mrs H, also a medical doctor, to take X to child psychiatrist. Appointment made but X did not attend because neither Mrs H nor daughter X wished to attend a psychiatrist. 9. 2007:Dr H wrote out doctor's prescription for anti-depressants including Citalopram. 10. 2008: February & March: Dr H asked X to take the prescription medicine Escitlopam instead of the prescription Citalopram. 11. 2008, January: Mother of X, a doctor herself, was also using prescription antidepressant medicine Citalopram prescribed by her husband, Dr H. 12. 2008, July: Dr H, X, X's mother, Mrs F and X's sister, Miss H holidayed in Egypt. 13. 2009, January: X said that sexual abuse stopped. X was 13. 14. 2009, February: H leaves the marital home. 15. 2009, March: X's mother and Dr H were divorced. 16. 2010, October: X hospitalized in a psychiatric hospital for nearly one year to October 2011. 17. 2010, October: X was asked by one of the hospital's medics to draw her timeline but included nothing about sexual even though this, is true, would have been a major trauma for four years. Note that X is a daughter of two medical doctors, so is not malnourished, neglected, or lacking any comfort and privileges as would a half- starved child from a poor council estate with inadequate clothes, food, heat or hobbies. Note that X frequently "TRASHED" HER BEDROOM and all the furniture in her bedroom, which Dr H had to have replaced at much financial cost to him. Note that X enjoyed holidays abroad. Note that Dr H's wife left Dr H to cope with X's tantrums. Note that Dr H's wife at the time was abusing alcohol. Note that the prosecution could find no other witnesses to call except for the interested party, Dr H's Note that X was the first patient of a trainee psychiatrist, Dr Lauren McKeown. 18. 2011, January: It is now 2 years since Dr H left the marital home. 19. 2013: At trial, no psychiatrist would tell the court their diagnosis of X (patient confidentiality, judge said). This is a farce. Therefore no real medical evidence was heard at trial at Manchester Crown Court. Trial Judge Mansell summed up at Manchester Crown Court. 20. 2013: Dr H was convicted and received an18-year prison sentence. 21. 2014: Dr H appealed on the point of law that judge wrongly disallowed the Defence Expert Witness' evidence of Dr Janet Boakes (three reports) and by stating that X had recovered her memory during counselling. 22. 2014: Appeal case heard in London. Louise Blackwell, QC for the Crown, stated that Dr Janet Boakes' eports were mere commentary because she had not interviewed X. She was not allowed to interview X. No other family member ever saw or heard this sexual abuse taking place The prosecutor called only two witnessesX's mother and X's sister. Mother of X, said she noted an odd relationship between X and her father, in conflict. Mother of X also had violent relationship with X. X had been violent to her mother and mother, a medical doctor, had in turn been violent to X. X was "anorexic". X often screamed at both parents, both medical doctors. X did not keep her bedroom clean and tidy. X was aware that her mother, a medical doctor, was abusing alcohol. It is not known if X also was abusing alcohol in this affluent family, where money was not scarce. X spent much time in her bedroom of her own volition. When police arrested Dr H , they did not seize the daughter's computer to see if she was viewing Pornography or buying illegal drugs or if she had a sexual boyfriend. No investigation was made as to whether X was having a sexual relation with another juvenile during this time. No school reports on X's behaviour were submitted to the court to show that X was not doing homework and failing assignments and perhaps consuming alcohol at school. X barred mother and Dr H and sister from her bedroom. No inspection was done to find out if X was abusing drugs or alcohol or what she was doing on the her computer. Mother of X heard much banging as X trashed her bedroom on occasion and sister of X confirmed X's violent outbursts. Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 9 Legal defence team for Dr H called several witnesses to the trial One defence witness was medically qualified. A second defence witness who was a long-standing friend; A third defence witness was a medical patient of the doctor, Dr H. A fourth witness is the spouse of another of Dr H's medical patients. The daughter of the family, daughter of the Appellant, anonymised as X Although the 'victim' was anonymised as X, all the local newspapers revealed his photograph and his medical practice and its address. When the General Medical Council struck off Mrs H, a doctor also, for repeated drunk driving, it was even easier to identify this youth and her family, now in total ruins. Case illustrates loopholes in English criminal law This paper uses this caselaw H v R as a vehicle to illustrate the loopholes that might have been used by law enforcement and the prosecutor and the court to valiantly strive for a conviction, instead of valiantly striving for the absolute truth, however complex or unpleasant it might have turned out to be. Youth conduct disorder Putting aside emotional reactions to this case, as reported officially, we are reminded of youth conduct disorder which this crown court trial and appeal at the Royal Courts of Justice in London was completely and significantly silent on, and which the writer contends is the crux of this case, and not the technical matters of 'expert witnesses'. The official view on 'youth conduct disorder' is as follows:- 'Conduct disorders are characterised by a repetitive and persistent pattern of anti-social, aggressive or defiant behaviour. Young people with conduct disorder may exhibit excessive levels of fighting or bullying, cruelty to other people and to animals, severe destructiveness to property, repeated lying, unusually frequent and violent temper tantrums, and defiant provocative Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 10 behaviour. The behaviours that are associated with conduct disorder major violations of age-appropriate social expectations and are more severe than ordinary childish mischief or adolescent rebelliousness. The diagnostic criteria for conduct disorder are similar but not identical to anti-social personality disorder'. Conduct disorder during childhood and adolescence According to the International Classification of Diseases (ICD 10) (WHO 1994) and DSM-IV (APA 1994) diagnostic criteria), conduct disorder usually occurs during childhood or adolescence, whereas anti-social personality disorder is not diagnosed in people under the age of 18. This is early onset conduct disorder Furthermore, according to ICD-10 and DSM-IV criteria, any diagnosis should distinguish between early-onset (symptoms present at age 10) and late-onset conduct disorder (absence of symptoms before age 10). The diagnostic criteria are also similar to oppositional defiant disorder ('ODD'), which according to ICS-10 usually occurs in younger children and 'does not include delinquent acts or the more extreme forms of aggressive or dissocial behaviour (WHO 1994). ODD is generally seen as milder than, and a risk factor to developing conducts disorder. H v R: Further reading recommended A serious and diligent study of the case report of H v R immediately brought the 20 texts listed as recommended further reading to mind. This case immediately reminded the writer of a very sad and serious set of events, the Cleveland child abuse scandal of 1987, which occurred in Cleveland, England, UK, almost three decades ago, in which a young female doctor incorrectly diagnosed a baby with child sex abuse which led to all the children in the particular village being examined and diagnosed with having been sexually molested and all the children, like the story of the Pied Piper, were wickedly Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 11 removed from their parents' homes, fostered and adopted. Those young children who had been put up for adoption, the law has been interpreted, cannot return to their innocent parents because a piece of paper states that they now belong to another couple. 2014-immediate rule changes; Cleveland miscarriages had no such legal changes Just as the way that the UK brought in immediate change of criminal procedure rules concerning expert evidence after Dr Hamilton's case in 2014, the Cleveland miscarriages of justice cases of wrongful charges of sex abuse could have brought immediate legislation passed right then to enable those Cleveland children to all be reunited with their their genetic families. How stupid is the law that stops a parent from having a child returned when the fault lay with incompetent social services. Why was no-one sacked or charged for gross negligence, distress, and upheaval and destruction of these families? It was said that those children who had already been adopted were not reconciled with their families because it was too later as they were now legally children of other peopleadopted – and therefore too late to mend the broken vessels that constituted that community when years later, with much zeal, heart-ache and cost, these diagnoses were proved wrong. Breach of Dr Hamilton's article 6 European Convention on Human Rights This case can be seen as an abuse of a man's Human Rights as per Article 6 of the European Convention on Human Rights 1948, which, like the abuses of the Cleveland parents' human rights, were it to have happened in the United States of America ('US'), the parents concerned would have litigated in a class action lawsuit in tort for their distress and children's disrupted lives, and very probably received one billion pounds sterling in compensation for hasty, neurotic, criminal and cruel acts caused by one female doctor who herself should have been at least examined for mental illness and possibly brought before the UK General Medical Council ('GMC') and possibly struck off. The Appellant in H v R may, if his case is prepared for the CCRC by some courageous and brilliant defence litigator, can also bring a lawsuit against the police of Yorkshire and the relevant CPS. The UK has a statute of limitation of six years for tort as per the Statute of Limitation Act 1980 does not apply as per recent precedent case law when it Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 12 was decided that the limitation does not apply if there is crime involved, including abuse of process. Britain's noted historical child abuse no reason to destroy men today In a knee-jerk reaction to historical abuse, authorities today are over-eager to imprison people for such abuse, notwithstanding the fact that modern children have access to high technology toys, the Internet where they can view pornography freely available and where they are savvy about their rights, without being taught their responsibilities also. For centuries child sex abuse has taken place in Britain. Centuries ago the English law was changed to allow children from age 13 to be sexual partners. It was therefore a crime to have sex with a girl under the age of thirteen. In those days, circumstances were of abject poverty for the masses; poor sanitation, and cramped living conditions of a family of ten sleeping together in just one rented room. Child sex abuse was not a term that was documented and was not a crime. It is only in recent years that domestic violence against women by boyfriends, husbands and partners had been given priority in the UK criminal justice system. Prior to this time, it was seen as a private matter and was rarely prosecuted. Similarly domestic murder by women of their partners or husbands was allowed a defence of provocation until recently and now such murders can only have provocation as a partial defence if it occurred immediately prior (in the heat of the moment) to the act or response by the woman of murder of the man. Similarly, the violence of rape of a wife within a marriage in English law was as per precedent case law of R v R [1992] and subsequently by the 1994 Criminal Justice and Public Order Act, s.142, which applied to the case of R v C [2005]. Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 13 Harm to another A continuation of the examination of a crime and of what constitutes harm to another leads to the consideration of white-collar crimes of insider trading, computer misuse and city frauds which were civil offences until recent years. Stop and Search causes harm to harassed ethnic minorities It must be admitted that the police in the UK cause much harm in their stop and search tactics, especially targeted to ethnic minorities by using the Vagrancy Act 1924, a very vague offence, and since 2001, anti-terrorism statutes, which allows carte blanche behaviour and huge infringements of people's liberties. Using the media to spread publicity of alleged crimes In many cases, such harm constitutes the complete destruction of people's lives through irreparable damage to their businesses because of widespread media publicity. Lip service to the Rule of Law It has been illustrated that lip service is paid to the 'rule of law', which dictates equality before the law. These examples show that some groups continue today to still receive less protection under the criminal law than other groups. Invasion of human privacy English law invades human privacy in areas of:  abortion,  age of consent,  drug use,  prostitution,  homosexuality,  incest,  masochistic homosexual encounters, Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 14  pornography and  Euthanasia, (to mention but a few areas). As regards consenting masochistic encounters as listed above, if A wounds or assaults B occasioning him actual bodily harm in the course of a sado-masochistic encounter, the prosecution must prove lack of consent on the part of B before they can establish A's guilt under s. 20 and s. 47 of the 1861 Offences Against the Person Act ('OAPA'). Abuse of process In his treatise, John Stuart Mill in 1859 gave what, in his opinion, was the traditional rationale, in that era, for criminalising conduct. In English law today, the use of lethal force by the police against civilian citizens is largely unpunished and not prosecuted as murder or corporate manslaughter. Nor are police officers charged with the crime of manslaughter when they cause deaths in police custody. Even in the globally infamous case, when police shot an unarmed stationary man, John Charles De Menezes in the year 2003 on an underground train at Stockwell Tube Station in London, the Commissioner of the Metropolitan Police was charged with a low-level strict liability offence of 'endangering the public' under the 1974 Health and Safety at Work Act ('HSA') section 3. The case was heard in 2007, giving four years for the public's memory to be dulled. Note that by virtue of the European Convention on Human Rights ('ECHR') article 2, the State has a duty to protect the life of the citizen. The State, as per article 8, ECHR, also has a duty to protect the private life of the citizen. The duty is known as the Osman duty. Sex abuse due to indulgence of Internet Pornography Sex abuse does occur in the United Kingdom ('UK') especially with acknowledged wide usage of Internet pornography which, over time, seems to have dulled many British men's sense of morals and ethics. Prosecutions over the past 16 years display evidence Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 15 that law enforcement is very enthusiastic in efforts to bring alleged sex offenders to justice, since the 1998 report Speaking up for Justice, which made 78 proposals to encourage and support vulnerable or intimidated witnesses and to help them give their best evidence in criminal cases. Extra police funding might motivate enthusiasm to log sex offences It appears that law enforcement is so enthusiastic that they suppress evidence, fail to comply with criminal disclosure rules, hone the case to omit any controversial evidence, are silent on real evidence; 'tip off' the media to whip up hatred for the defendant, just to bring about a successful conviction. This unjust manipulation of the criminal justice system But look at the knitting and weaving together of this case, H v R, and discover the unjust manipulation of the criminal justice system. For the Yorkshire police to charge a senior professional medical family doctor with a crime is to attract scrutiny to what is understood as crime. To say that a crime is causing harm to others is a vague statement, for such laws are to protect us from violent aggressors, including the police. Swamping the defence with impressive low-grade paperwork It causes huge worries that the UK criminal justice system in the year 2013 and 2014 still resembles the justice system of the eighteenth century, with lots of paperwork provided by the prosecution to swamp the poorly-paid legal defence team and lodging such mass of documents in court is often unwarranted, hearsay, and poor quality evidence, illustrating omissions, manipulations, fraud and forgery, machinations and much more. 'Tip of the iceberg' in UK criminal justice system The extremely worrying thing is that this is but the 'tip of the iceberg' in the UK criminal justice system, where today, as centuries ago, those with power twist the law to fit their cases. Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 16 A miscarriage of justice One need only examine the myriad of miscarriages of justice in the UK, which only come to light if the media favours one, and decides to run the story. The media has too much power. Maligned father and former family doctor However, this father and former family doctor, now cut off from his caring profession in his prime, has had no evidence in court of any pornography on his computers or in his house. It appears that the local police were enthusiastic to see a middle class professional medical doctor brought down. This was evidenced in the new statistics in Police Professional, February 2015 Issue. The UK now has one of the highest statistics in convictions for sex abuse. Whether this is all sex abuse or partly motivated by extra funding for constabularies in certain parts of the country has yet to be proved. Money trumps everything The picture overall seems somewhat confused. We live in a schizophrenic country where money trumps this matter because the UK Internet servers have license to provide pornography to adult viewers who pay for the service; free pornography to others; hundreds of pole dancing strip clubs around the UK where businesses, provided they pay the very large licence fee, are free to trade as such; pornography publications abound in the United Kingdom; sex shops openly advertise sex 'toys' on national television in the United Kingdom; there are national television channels which, if a customer pays, can join a sex channel which reveals naked women talking sexily and in pornographic poses, etc. Child violence in the UKassault on parents To return to the matter of child violence, a BBC programme on Thursday evening, 12 February 2015, revealed the horrendously frequent incidents of children who continuously assault their parents in their violent rages against parental control of bedtimes etc. One child took a cleaver knife to his poor mother. When one considers the positions of the parents of child C in R v H, one can understand their reluctance and Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 17 feelings of failure and shame as being the cause of not calling in professional help because they themselves were doctors. One feasible factorindulgent parents Some of the children with conduct disorder are simply ones who have grown progressively out-of-control by indulgent parents. The consequence of indulgence and no discipline results in a complex mix of:  power over the parents by such a child;  the thrill of wielding such power over adults;  and the progression to psychopathy in such children.  Bad children become bad adults and this phenomenon stretches across all strata of society.  Furthermore it has been established that there is a link between age and crime over the life span.  Bearing in mind that the legal age of criminal liability in England is age 10, we find that since the year 2013, the statistics of the children under age 14 who had been prosecuted and those between ages 14 to 17 were as follows: Child domestic violence offences prosecuted Age range Time period Prosecutions Under 14 2010-2011 216 Under 14 2011-2012 148 Under 14 2012-2013 118 14-17 years 2010-2011 3,144 14-17 years 2011-2012 2,643 14-17 years 2012-2013 3,144 Source: Google. However, no-one has studied the types of children prosecuted or discover how many, if any, are from rich or professional, middle-class families. Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 18 Parentline Plus Between the years 2008 to 2010, the charity Parentline Plus reported that it had received 22,537 telephone calls from mothers and fathers who were struggling to cope with their children's extremely violent behaviour. According to research, the violence in children who abuse their parents peak from age 13 to 15. According to Parentline Plus, their own research of such child violence occurs every single day; 50 percent of such violent children destroyed property and 20 percent were drinking alcohol. R v Hpolice did not check for child alcohol abuse Yet, not one police officer checked to see if this child was abusing alcohol and if this is why she locked herself in her bedroom so that she would not be found out. We see teenagers and younger children turn very violent whilst drunk, breaking mirrors, attacking furniture, etc, for no good reason than that they are drunk. British children today: well fed and heavier According to an Independent newspaper article one of the factors why children beat up their parents is their size. Today, children in the UK are usually very well fed and some of the assaulted parents complained that even as young as 11 years old, their daughters were almost impossible to handle physically. Mental illness, alcohol and substance abuse Other factors for child violence include early signs of mental illness, alcohol and other substance abuses. Some experts say that there is a collapse in social authoritative boundaries today, as children are pampered and given access to the Internet at a young age. Mimic of alcoholic parent If a girl sees her mother abusing alcohol, she loses respect for that parent and may also begin practicing alcohol abuse herself, especially in a middle class family where the Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 19 problem is not one of lack of finances. The recorded phenomenon of children beating up their parents is most likely the 'tip of the iceberg' in the UK. Hand in Hand and ParentlinePlus Because sociologists are aware that this explosion of parent abuse is shameful and largely kept within the family, several Internet Help Centres have emerged to help parents to cope, Parentline Plus and Hand in Hand parenting being two such websites. Child disorders that manifest themselves in property damage, parental assault and selfassault are often treated medically as follows: Drug brand name usual dosage comment Fluoxetine Prosac 1mg/kg of body weight Can be dissolved in water Paroxetine Paxil 20-60 mg per day Dissolved in water. May cause weight gain. Citalopram Celoxa 20-40 mg per day Not soluble Sertraline Zoloft 3mg per 1 kg of body weight Soluble Fluvoxamine Luvox 3 mg per 1 kg of body weight Not soluble Escitalopram Cipralex 10-20 mg per day No studies in children Duloxetine Cymbalta 30-60 mg per day Few Case Studies Source: Google Child sexuality The trial R v H [2014] never questioned whether this youth could possibly have been having sex with another youth. The dearth of literature about this subject creates gap in public knowledge about the development of such sexually assaultive behaviour and the professional and legal issues accompanying this little spoken-of violent yputh behaviour. No UK research interest in sexually assaultive behaviour of juveniles For decades there has been much interest in the juvenile sex offender in the United States but not at all in the United Kingdom. Interest in the sexually assaultive behaviour of juveniles has a long history (Atcheson and Williams, 1954; Cook, 1934; Doshay, 1943; Waggoner and Boyd, 1941). In 1964, a study by Mohr, Turner and Jerry (1964) showed that child sex offenders pose a longterm risk. Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 20 Ignorance about child sexual development Initially child sex was seen not as violence but as innocent behaviour and this misconception was due to a profound lack of knowledge concerning social and psychological aspects of sexual development in adolescence. Available estimates show that juveniles commit 20% of rapes with penetration in 59% of juvenile sex offences. Glossed-over educational non-progress of X The caselaw report of R v H [2014] glossed over the lack of educational progression of child X. The caselaw report incorrectly painted a picture of a good and virtuous child X for whom 'butter would not melt' which is inconceivable of a strong young girl who could destroy all the furnishings in her bedroom in one angry outburst. For this serious prosecution, there was NO mention of any tests for alcohol abuse by X. Cunning use of rape charges instead of incest charge in order to avoid all evidence The prosecution decided to bring rape charges against the accused because of recent changes of anti-cross-examination of the accuser 16 year old daughter at trial. The accuser x displayed prolonged anti-social behaviour over many years. X's alarming and distressing conduct to her father, her mother and her sister could have been brought to a stop by an anti-social behaviour order, had the ex-wife called in social services years before. Assault and battery of Dr Stephen Hamilton by X Clearly X had assaulted and battered her father Dr Hamilton on very many occasions in front of Dr Hamilton's ex-wife, No evidence offered by schoolteachers or friends or other family members This trial of Dr Hamilton called no teachers to give evidence of X's behaviour at school. Did X's behaviour drive her mother to drink? Did X terrorise her parents to the extent that her father, a very senior medical doctor, returned home in fear of what he would find? All of this and more is admissible evidence deliberately withheld by the prosecution. Admissibility in the law of evidence is the concept that determines whether or not Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 21 evidence can be received by the court. The evidence must be relevant, but even relevant evidence will be tested for its admissibility. Forced to leave the marital home-after which time X became psychotic Most tellingly, the caselaw report mentioned that after Dr Hamilton was driven to leaving the marital home and agreeing to a quick divorce, breaking up the family, (an occurrence that is caused by most of such violent child behaviour), that X behaved even worse after her father left hometo such an extent that she was a full-time patient in a mental hospital for two years. There was no alleged sexual abuse of X during this timeher behaviour was just a progression of the behaviour she wreaked on her family from the beginning. Order for assessment of X A court could have issued orders for the assessment and treatment of X, as also a general medical practitioner ('GP') in an emergency situation could. This is a very serious criminal case and inequality of arms is not allowable These extremely serious criminal charges by police to this very senior and likeable family doctor beggars belief as to the way the defendant was treated. It is for the prosecution to prove their case beyond reasonable doubt and yet Dr Stephen Hamilton was treated by the court as though he had no right to his human rights as per the European Convention of Human Rights 1948 and the UK Human Rights Act 1998. Never forget that beyond a reasonable doubt is the standard of [roof in criminal cases in the UK. This standard is higher than the civil standard of the 'balance of probabilities'. It is not a matter of weighing up both sides and deciding who has won. Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 22 Ex-wife could be an accomplice If X's mother has aided and abetted her to hold a steadfast memory that her father abused her, then her mother could eventually be proved to have been an accomplice, participating in X's lies about her father. All that is necessary is for the ex-wife to have some degree of guilty knowledge be charged as an accomplice. An accomplice is a person who participates in a crime, either by accession or as a perpetrator, before or after the fact, by committing, procuring, or aiding and abetting. A trainee female psychologist destroyed the medical career of this senior GP So why did it take a non-experienced female newly qualified psychologist and an allegedly sexually abused female's hearsay to bring this Senior General Practitioner to his demise and subsequently financially ruining the whole family's future permanently? Was the alleged sexually abused X other girl at the special school where child X was sent to after two years? Or was she a 'patsy' put there by local police to say those things? Pre-divorce situation may be reason for this criminal case Dr Hamilton's wife could be said to have been consuming alcohol to such a regularity and length of years that Dr Stephen Hamilton could himself have filed for divorce as he could not be expected to live with such behaviour. Instead Mrs Hamilton struck first and went 'full hog; with incest accusations also, perhaps hoping to acquire all of Dr Hamilton's then considerable assets. Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 23 X –vivid tales The Appellant, Dr Hamilton, sought to introduce false memory as to his daughter's tales of explicit sexual abuse. Nobody, not even Dr Hamilton's now ex-wife found him in flagrante delecto with X. X was suffering from serious psychosis and imagined it. Eventually, X had been removed, after her father was made to leave the marital home, compulsorily from the same family home, into the care of the local authority, into a mental institution, by way of a place of safety order. The trial court disallowed the false memory syndrome However, it is established that the evidence of an expert is admissible if it provides the Court with scientific information likely to be outside the experience and knowledge of a judge or jury: R v Turner [1975] QB 834; [1974] 60 Cr.App.R. (S) 80. In addition, the witness must possess the relevant expertise, and the subject matter and nature of the expertise must be of sufficient standing to be reliable as evidence. The Defendant medical doctor, Dr Hamilton should have sought an extension of time in order to bring such an expert into the United Kingdom to give evidence. Other proofsX's computer; porn; etc Proof should have been vigorously sought from the girl's computers to see what, if any pornography she was viewing or reading about because it sounds so incredulous. It is noted that the case report spoke of the girl as being very right, even though no evidence or proof was forwarded for such a statement. The seemingly very biased and non-neutral caselaw report noted that X said that she wanted to kill herself if she felt that no-one would believe her in court. Yet it is noted that for two years se told no-one of sexual abuse apart from a mentally disordered girl she met whilst at a special school. Why was the other girl not a witness for the prosecution? Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 24 The case report had a tone of utter sympathy from the court to X, yet, looking coldly at X's statement, any reasonable person would conclude that X is mentally unstable. Promises of compensation and own council flat to live in? By this time, all X's father's finances and pension pot had been spent on legal fees and X's mother was also struck off by the General Medical Council for repeated drunk driving road traffic offencesso no income was forthcoming from X's mother or father, meaning that X had caused the total annihilation and destitution of her own family. It is noted that no psychologist interviewed X regarding her plans for the future unless an unqualified social worker had promised her criminal compensation award of £19,000 pounds of government finances plus her own council flat to live in as she pleased. It is a fact that social workers often tell young people about the many state resources they will be entitled to, not realising themselves that even if a lump sum compensation was given to X, she would very probably spend it very frivolously and it would have dissipated in a short while, as is usual in these cases. Meanwhile, X's mother and sister would have no funds available, unless they too, claimed criminal compensation, because the GP's funds have all been used up in legal fees. Conclusion of analysis As the English criminal justice system trundles along incrementally improving by hits and by misses, it has been announced that expert evidence is subject to a new procedure, as per three primary sources: Stephen Hamilton v R [2014] EWCA Crim 1555 (22nd July 2014); the Criminal Procedure Rules 2014 (laid before Parliament on the 25th June 2014 -in force since 6th October 2014); and the Criminal Practice Directions Amendment No. 2 (published 23rd July 2014 –in force since 7th October 2014). Expert reports will have to contain more detail if admissions are not to be made. Courts are enjoined to achieve as much agreement as possible in advance, this time with Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 25 the force of statutory instrument. The Practice Direction gives a checklist of factors for reliability and is the first port of call in any argument about admissibility. At paragraph 26 of the caselaw report, the Court stated that the fact of mental ill health does not mean that the witness cannot accurately be describing what has happened to her or that it would prevent her from (or make her incapable of) being reliable in her account. These issues of fact are not for resolution by doctors but are to be determined by the jury, the court stated: as Kay LJ put it in R v Bernard that evidence is admissible when it is necessary 'to inform the jury of experience of a scientific and medical kind of which they might be unaware, which they ought to take into account when they assess the evidence in the case in order to decide whether they can be sure about the reliability of a particular witness.' The Court took the opportunity, however, to herald the forthcoming changes that will be brought about by the Practice Direction and the 2014 Rules. There is noted real concern about the use of unreliable or inappropriate expert evidence, necessitating a new and more rigorous approach on the part of advocates and the courts to the handling of expert evidence and a simple list was constructed that an expert must: *make clear which of the facts stated in the report are within the expert's own knowledge; *say who carried out any examination, measurement, test or experiment which the expert has used for the report and * give the qualifications, relevant experience and accreditation of that person, and say whether or not the examination, measurement, test or experiment was carried out under the expert's supervision, and summarise the findings on which the expert relies. However, this is a farse because the Defence expert witness was not allowed to examine or interview X and X's medical notes were not made available to the defence. The farce and travesty in this case is the creation of correctness when in fact the case was machinated and manipulated to whatever financial funding reasons. It is hoped that this doctor can bring his case to the attention of the CCRC. The accused Dr Hamilton did not get a fair trial. A fair trial is a human right to a trial that provides Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 26 certain practical protections for the citizen. A fair trail is made up of an impartial judge; effective legal representation; a lack of undue delay and freedom from selfincrimination. Bibliography Arrigo, B.A. (1999) Social Justice/ criminal justice: the maturation of critical theory in law, crime and deviance, London: International Thomson Publishing Europe. Bartol, C.R. (2002) Criminal behavioura psychosocial approach, 6 th Edn., New Jersey: Prentice Hall Inc. Bogan, P. and Roberts, A. (2011) Identificationinvestigation, trial and scientific evidence, Bristol, UK: Jordans. Herring, J. (2012) Criminal lawgreat debates, 2 nd Edn, Hampshire, UK: Palgrave Macmillan. Kapardis, A. (2003) Law and psychologya critical introduction, 2 nd Edn. Cambridge: Cambridge University Press. Muncie, J. (2015) Youth crime, London: SAGE. Stasz, T. (1989) Law, liberty and psychiatryan inquiry into the social uses of Mental Health Practices, United States: Syracuse University Press. Tepperman, L. (2006) Deviance, crime and controlbeyond the straight and narrow, Oxford: Oxford University Press. ENDS+ Criminal Law News Issue 105July 2017 (Print version) ISSN 1758-8421 27 Printed and published by SALLY RAMAGE ®, Copehale, Coppenhall, Stafford, ST18 9BW, UK. Registered as a Newspaper at the Post Office. © Sally Ramage® 2017.All Rights Reserved. No part of this publication may be reproduced in any material form (including photocopying or storing it in any medium by electronic means and whether or not transiently or incidentally to some others use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Design and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency, Saffron House, 6-10 Kirby Street, London, England EC1N 8TS. Application for the copyright owner's written permission to reproduce any part of this publication should be addressed to the publisher. Warning: the doing of an unauthorised act in relation to a copyright work may result in both a civil claim for damages and criminal prosecution. ISSN 1758-8421.

This article was downloaded by: [J. S. Swindell Blumenthal-Barby] On: 03 May 2013, At: 09:12 Publisher: Routledge Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK The American Journal of Bioethics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/uajb20 On Nudging and Informed Consent-Four Key Undefended Premises J. S. Swindell Blumenthal-Barby a a Baylor College of Medicine To cite this article: J. S. Swindell Blumenthal-Barby (2013): On Nudging and Informed Consent-Four Key Undefended Premises, The American Journal of Bioethics, 13:6, 31-33 To link to this article: http://dx.doi.org/10.1080/15265161.2013.781717 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Nudging and Informed Consent manipulate them, trust might well break down, and the desire for a second, third, or fourth opinion would become commonplace.  REFERENCES Cohen, S. 2013. Nudging and informed consent. American Journal of Bioethics 13(6): 3–11. Mitchell, G. 2005. Libertarian paternalism is an oxymoron. Northwestern University Law Review 99(3): 1245–1277. Ploug, T., S. Holm, and J. Brodersen. 2012. To nudge or not to nudge: Cancer screening programmes and the limits of libertarian paternalism. Journal of Epidemiology and Community Health 66: 1193–1196. Sunstein, C. R., and R. H. Thaler. 2003. Libertarian paternalism is not an oxymoron. University of Chicago Law Review 70(4): 1159–1202. On Nudging and Informed Consent-Four Key Undefended Premises J. S. Swindell Blumenthal-Barby, Baylor College of Medicine In his article "Nudging and Informed Consent," Shlomo Cohen (2013) argues, among other things, that 1) "to the extent that the nudge-influenced decision making is rational-in whatever sense," there is no opposition on grounds of respect for autonomy; 2) even nudges that take advantage of or induce epistemically or rationally flawed processes are ethically permissible so long as patients are not deceived; 3) preferences that result from taking advantage of or inducing such flawed processes are still normatively valuable because they are the patient's actual preferences; and 4) nudges that take advantage of these processes are more ethical when they exert their impact on the environment rather than the patient directly. In this commentary, I argue that 1) premise 1 begs the key question of what exactly rational decision making is, 2) premise 2 is certainly false, 3) premise 3 fails to recognize that if the preferences are caused by a nudger who induces processes to get the patient to adopt that preference then in fact there is a real sense in which it is not the patient's (think meddling, preference-implanting neurosurgeon), and 4) premise 4 draws a false dichotomy between impact on patient's choice architecture and patient directly in addition to failing to defend the normative importance of indirect rather than direct impact. PREMISE 1: NUDGES, RATIONAL DECISION MAKING, AND AUTONOMY Cohen notes the challenge that autonomy is preserved either by noninterference or by interference through "rational persuasion alone," and that exploitation of imperfections in judgment and decision making is prima facie threatening to liberty or autonomy. In response to this challenge, he responds that nudges often do not elicit irrational decision making and thus are not a threat to autonomy. He then goes Address correspondence to J. S. Swindell Blumenthal-Barby, PhD, Assistant Professor, Center for Medical Ethics and Health Policy, Baylor College of Medicine, One Baylor Plaza, Room 310D, Houston, TX 77030, USA. E-mail: [email protected] on to give several examples where nudges do not (so he claims) elicit irrational decision making. The problem with this line of argument is that 1) insofar as it succeeds, it only succeeds in showing that this very limited set of examples pose no threat to autonomy; 2) it does not succeed in that some of the examples are examples of eliciting irrational decision making (insofar as I can intuit his view of irrational decision making); and 3) the core problem is that he fails to offer an account of rational versus irrational decision making that would tell us when nudges pose a threat to autonomy. So, for example, he tells of a doctor who frames a woman's cancer risk comparatively (compared to other women) instead of simply absolutely (her risk alone). He characterizes this nudge as "providing relevant information," "adding a comment," simply "adding information" (5), and inducing rational deliberation. But certainly it induces emotions as well. The whole reason that giving comparative information is an effective nudge is because it induces fear, alarm, and sensitivity to the normal. Would decision making driven by alarm and fear count as rational decision making for Cohen? He gives us no account of what rational decision making is, only saying that whatever it is, as long as nudging elicits it, or rather avoids eliciting "irrational decision making," it poses no threat to autonomy (5). PREMISE 2: NUDGES, DECEPTION, AND ETHICAL PERMISSIBILITY Cohen goes on to argue that even nudges in informed consent that take advantage of or induce epistemically or rationally flawed processes (and subvert autonomy) could still be ethically permissible as long as patients are not deceived. He argues that according to Onora O'Neill, the function of informed consent is to prevent patients from being coerced June, Volume 13, Number 6, 2013 ajob 31 D ow nl oa de d by [ J. S . S w in de ll B lu m en th al -B ar by ] at 0 9: 12 0 3 M ay 2 01 3 The American Journal of Bioethics or deceived. I'm not sure whether Cohen has O'Neill quite right here, and I'm sure that many other scholars have different accounts of the purpose of informed consent (and he does not indicate why he favors O'Neill's theory over those), but regardless, the premise that a nudge is morally permissible as long as it is not deceptive seems to me to be obviously false. Simply imagine that a physician offers a patient $100,000 to choose A over B, or yells at a patient until she chooses A over B. No deception, yet still morally problematic. Cohen might claim that these are problematic because they are coercive, but that won't work here, as he is trying to make the argument that even if nudges subvert autonomy they can still be ethically permissible. PREMISE 3: NUDGES AND LOCATION OF PREFERENCES Cohen further argues that preferences that result from nudging that takes advantage of or induces flawed processes are still normatively valuable simply because they are the patient's actual preferences. But this is an odd way to look at it. I would say that preferences that are caused by a nudger who induces processes to get the patient to adopt that preference are in a very real sense not the patient's. Imagine a meddling neurosurgeon who implants preferences into an agent (à la Harry Frankfurt: Frankurt 1969), or an evil demon who plays a piano, the keys of which are attached to an agent's brain cells, with the chords inducing certain neurological processes, resulting in certain behaviors or decisions (à la Peter van Inwagen: Van Inwagen 1983). Externalists about autonomy would deem such cases to be cases where the agent fails to act autonomously because they are not acting on the agent's preferences. But even internalists like Frankfurt who would allow for the possibility of autonomous action in such cases would require a further story to be told, namely, that the agent identifies with the preference that she finds herself having.1 PREMISE 4: NUDGES AND CAUSE AND EFFECT Another major premise in Cohen's argument is that nudges that take advantage of these processes are more ethical when they exert their impact on the environment rather than the patient directly. Ones that do the reverse (impact the person directly) are less ethically permissible. On the nudge's causation being direct versus indirect, Cohen writes of the importance of "the extent to which a nudge exerts its influence on the environmental circumstances of choice as distinct from the chooser itself" (8, emphasis in original). This line of argument draws a false dichotomy between impact on the patient's choice architecture and the patient directly. What impacts the patient's choice architecture will necessarily impact the patient-this is not an either/or. Moreover, it is not clear what the normative importance is of an indirect impact versus a direct impact. As James Rachels (1975) persuasively argued years ago, if Smith directly kills 1. For an extremely helpful discussion of all of this see Mele (2001), particularly chapters 8–11. his cousin by drowning him and Jones indirectly kills his cousin by happily watching him drown when he could have easily saved him (and let us say arranging the environment to make his accident likely), Jones is just as morally responsible as Smith. The direct versus indirect mechanism is irrelevant; what matters is that both men intended for their cousin to die and the outcome was the same in both cases. INTER ALIA Cohen mentions several times that the literature on nudging in health care deals almost exclusively with health policy, but this is not accurate, as my colleagues and I have written extensively on nudging and shaping decisions at the level of individual to individual (even in clinical contexts) in previous and forthcoming work (Blumenthal-Barby 2012; 2013; in press; Blumenthal-Barby and Burroughs 2012; BlumenthalBarby, McGuire, and Halpern 2010; 2011; Blumenthal-Barby et al. 2013a; 2013b). Some of these articles may be helpful to those interested in thinking about this debate. Second, Cohen characterizes the use of nudges during informed consent as a "new model" between autonomy and paternalism, but to claim it is a new model is quite a stretch. Physicians are acutely aware of the fact that they shape decisions during the shared decision-making encounter. Third, Cohen's characterization of shared decision making (SDM; which is, in fact, an "in between model," a third option) is highly problematic. Cohen characterizes SDM as the patient bringing knowledge of his or her values, and the physician having "the trivial [role] of offering technical knowledge" (4; Cohen cites a paper from 1990 for this characterization). The concept has evolved a great amount since 1990. Alex Kon (2010), for example, has recently argued that SDM is a continuum concept with five key points, and that one of those points (the middle one) is where the physician and patient are "equal partners," working together to reach a decision. Cohen also claims that SDM promotes a meager ideal of professionalism, divesting the doctor–patient relationship of depth and meaning. These claims seem to me to be completely unsupported and misguided, though no doubt due to a certain understanding (which I hope to have shown to be incorrect) of SDM. Recent studies have illustrated the promise of SDM for increasing patients' satisfaction with decisions, increasing patients' understanding, improving health outcomes, and better aligning care with patients' values (Lee and Emanuel 2013).2  REFERENCES Blumenthal-Barby, J. S. 2012. Between reason and coercion: Ethically permissible influence in health care and health policy contexts. Kennedy Institute of Ethics Journal 22(4): 345–366. Blumenthal-Barby, J. S. 2013. Choice architecture: A mechanism for improving decisions while preserving liberty? In Paternalism: 2. See also the February 2013 issue of Health Affairs on "patient engagement," an issue entirely devoted to the data supporting shared decision making. 32 ajob June, Volume 13, Number 6, 2013 D ow nl oa de d by [ J. S . S w in de ll B lu m en th al -B ar by ] at 0 9: 12 0 3 M ay 2 01 3 Nudging and Informed Consent Theory and practice, ed. C. Coons and M. Weber, 178–196. Cambridge, UK: Cambridge University Press. Blumenthal-Barby, J. S. In press. A framework for assessing the moral status of manipulation. In Manipulation, ed. C. Coons and M. Weber. New York, NY: Oxford University Press. Blumenthal-Barby, J. S., and H. Burroughs. 2012. Seeking better healthcare outcomes: The ethics of using the "nudge." American Journal of Bioethics 12(2): 1–10. Blumenthal-Barby, J. S., S. Cantor, A. Naik, H. Russell, and R. Volk. 2013a. Decision aids: When nudging patients to make a particular choice is more ethical than balanced, nondirective content. Health Affairs 32(2): 303–310. Blumenthal-Barby, J. S., L. McCullough, H. Krieger, and J. Coverdale. 2013b. A typology and ethical analysis of methods of influence in psychiatric decision making. Harvard Review of Psychiatry, in press. Blumenthal-Barby (Swindell), J. S., A. L. McGuire, and S. D. Halpern. 2010. Beneficent persuasion: techniques and ethical guidelines to improve patients' decisions. Annals of Family Medicine 8(3), 2010: 260–264. Blumenthal-Barby (Swindell), J. S., A. L. McGuire, and S. D. Halpern. 2011. Shaping patients' decisions. Chest 139(2): 424–429. Cohen, S. 2013. Nudging and informed consent. American Journal of Bioethics 13(6): 3–11. Frankfurt, H. 1969. Alternate possibilities and moral responsibility. Journal of Philosophy 66(23): 829–839. Kon, A. 2010. The shared decision making continuum. Journal of the American Medical Association 304(8): 903–904. Lee, E., and E. Emanuel. 2013. Shared decision making to improve care. New England Journal of Medicine 368(1): 6–8. Mele, A. R. 2001. Autonomous agents: From self-control to autonomy. New York, NY: Oxford University Press. Rachels, J. 1975. Active and passive euthanasia. New England Journal of Medicine 292(2): 78–80. Van Inwagen, P. 1983. An essay on free will. Oxford, UK: Clarendon Press. Nudging in Interpersonal Contexts Yashar Saghai, Johns Hopkins University In "Nudging and Informed Consent," Shlomo Cohen (2013) attempts to address the common objection against nudges that they are autonomy-thwarting because they foster irrationality. He explicitly focuses on informed consent, which he contrasts with the policy context in which health nudges are usually discussed. I think Cohen's rich article is a significant contribution to the nudge literature. However, I have some concerns with the way he frames and motivates his inquiry. Cohen states that his ambition is to examine the ethics of nudging in the context of informed consent rather than health policy and public health. He maintains that "one may arguably detect an anomaly, in that the [nudge] theory has been developed in the context of policy when it focuses essentially on individual choice" (4). I disagree. It is precisely because health policy and public health interventions do not need to be coercive that nudge is a useful concept in those contexts. Population-wide nudges have the potential to generate significant group-level effects without coercively influencing individuals' choices. There is, therefore, no reason to complain about the focus of the nudge literature on public health and health policy. A related issue with Cohen's piece is that he replaces the relatively clear distinction between a policy and a one-time action with an odd distinction between policy Address correspondence to Yashar Saghai, Johns Hopkins University, Berman Institute of Bioethics, 1809 Ashland Avenue, Baltimore, MD 21205, USA. E-mail: [email protected] and informed consent. Although it is true that the implementation of many health policies and public health interventions bypasses individual express and informed consent, this is not necessarily the case (Berg 2012). For instance, all members of a population can be offered routine screening (opt-in). They are informed of the procedure and have the right to give or withhold consent (Gostin 2008, 402). This is clearly an intervention quite typical of public health and it involves individual express and informed consent. Of course, opt-in policies are not nudges, but my point is that the right distinction cannot be between policy and informed consent. Perhaps Cohen's focus should be on the ethics of using nudges in interpersonal contexts. Every interpersonal context raises problems of its own due to the particular type of relationship in which different parties stand. I read Cohen's piece as a helpful attempt to shed light on the special obligations and responsibilities involved in the patient–physician relationship. Should we, however, conclude with Cohen that nudges constitute a whole new "model" of the patient–physician relationship overcoming the gap between paternalism and autonomy? The answer is clearly, no. Nudges, I suggest, are a type of influence, not a model of the patient–physician relationship. I have argued elsewhere that a nudge is best characterized as an influence June, Volume 13, Number 6, 2013 ajob 33 D ow nl oa de d by [ J. S . S w in de ll B lu m en th al -B ar by ] at 0 9: 12 0 3 M ay 2 01

Earlier Visual N1 Latencies in Expert Video-Game Players: A Temporal Basis of Enhanced Visuospatial Performance? Andrew J. Latham1,2*, Lucy L. M. Patston1,2, Christine Westermann3, Ian J. Kirk1,2, Lynette J. Tippett1,2 1 School of Psychology, the University of Auckland, Auckland, New Zealand, 2 Centre for Brain Research, the University of Auckland, Auckland, New Zealand, 3 Department of Biopsychology, Ruhr-University, Bochum, Germany Abstract Increasing behavioural evidence suggests that expert video game players (VGPs) show enhanced visual attention and visuospatial abilities, but what underlies these enhancements remains unclear. We administered the Poffenberger paradigm with concurrent electroencephalogram (EEG) recording to assess occipital N1 latencies and interhemispheric transfer time (IHTT) in expert VGPs. Participants comprised 15 right-handed male expert VGPs and 16 non-VGP controls matched for age, handedness, IQ and years of education. Expert VGPs began playing before age 10, had a minimum 8 years experience, and maintained playtime of at least 20 hours per week over the last 6 months. Non-VGPs had little-to-no game play experience (maximum 1.5 years). Participants responded to checkerboard stimuli presented to the left and right visual fields while 128-channel EEG was recorded. Expert VGPs responded significantly more quickly than non-VGPs. Expert VGPs also had significantly earlier occipital N1s in direct visual pathways (the hemisphere contralateral to the visual field in which the stimulus was presented). IHTT was calculated by comparing the latencies of occipital N1 components between hemispheres. No significant betweengroup differences in electrophysiological estimates of IHTT were found. Shorter N1 latencies may enable expert VGPs to discriminate attended visual stimuli significantly earlier than non-VGPs and contribute to faster responding in visual tasks. As successful video-game play requires precise, time pressured, bimanual motor movements in response to complex visual stimuli, which in this sample began during early childhood, these differences may reflect the experience and training involved during the development of video-game expertise, but training studies are needed to test this prediction. Citation: Latham AJ, Patston LLM, Westermann C, Kirk IJ, Tippett LJ (2013) Earlier Visual N1 Latencies in Expert Video-Game Players: A Temporal Basis of Enhanced Visuospatial Performance? PLoS ONE 8(9): e75231. doi:10.1371/journal.pone.0075231 Editor: Elkan Akyürek, University of Groningen, Netherlands Received March 22, 2013; Accepted August 14, 2013; Published September 18, 2013 Copyright: © 2013 Latham et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Funding: This research was supported by a University of Auckland FRDF Large Project Grant (3624428) awarded to Lynette J. Tippett. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist. * E-mail: [email protected] Introduction Playing video-games has become a past-time of choice for current generations, allowing individuals to engage both socially and competitively with other players across the globe. Today's modern action video games (e.g., CounterStrike: Global Offensive, StarCraft II, Defense of the Ancients 2, Guildwars 2) present players with complex visual environments that require flexible attentional resources. Multiple items must be simultaneously processed and identified as either relevant or irrelevant to in-game goals. Furthermore, objects in modern video games are not passive, but rather, through the integration of artificial intelligence and multiplayer capabilities, learn and adjust to the player as the game progresses. Success is contingent on the player's ability to execute precise bimanual motor movements in response to these complex visual cues. Cumulating evidence suggests that extensive video-game play may lead to enhanced visual attention and executive control, generalising beyond the context of the video game (e.g., [1,2,3]). To date, however, few studies have assessed the underlying neural basis of the enhanced cognitive abilities of video game-players. Beginning with the seminal findings of Green and Bavelier [1], expert video-game players (VGPs) have been shown to have superior performance on a wide-range of visuospatial and attentional tasks. These include: Flanker, Enumeration, Useful Field of View and Attentional Blink tasks [1], superior stimulusresponse mapping [4], superior visual sensitivity [5], superior cross-modal sensory precision [6], reduced backwards masking [7], reduced task-switching costs [8], superior endogenous attention [9], and superior resolution for stored visual information [10]. Feng and colleagues have shown these PLOS ONE | www.plosone.org 1 September 2013 | Volume 8 | Issue 9 | e75231 enhancements last for at least four months after video-game play has completely ceased [11,12]. The paradigms typically utilized in these studies are computerized and require rapid target detection with manipulation of distractor difficulty and target eccentricity. Arguably such paradigms may simply measure the specific skills trained by video-games, and as a result, do not provide direct evidence that the cognitive proficiency of VGPs generalizes beyond the general training environment. A small number of studies, however, appear to demonstrate that videogame play may shape some fundamental aspects of the visual system. Green and Bavelier [13] found that expert VGPs were able to discriminate the correct orientation of significantly smaller Ts than non-VGPs during a visual crowding paradigm, suggesting they may possess superior visual acuity. Li, Polat, Makous and Bavelier [14] found that expert VGPs were significantly more accurate than non-VGPs on a standard contrast sensitivity paradigm. Finally, Buckley, Codina, Bhardwaj and Pascalis [15] found that central and peripheral visual fields measured using Goldmann kinetic perimetry were around 1000deg2 larger than non-VGPs. Very few neuroimaging studies have examined the potential neural correlates of superior visuospatial and attentional performance of VGPs. In the first reported neuroimaging study, 8 non-VGPs underwent PET scanning while playing the videogame Tetris both before, and after, daily practice sessions of 30-45 minutes for 4 to 8 weeks and were compared with 16 control participants passively viewing visual stimuli [16]. Individuals who played video-games showed a significant decrease in whole brain glucose metabolism in the second scan, with improved Tetris performance inversely correlated with levels of glucose metabolism, suggesting more efficient utilization of neural circuitry. Mishra, Zinni, Bavelier and Hillyard [17] compared expert VGPs and matched non-VGPs on a selective attention task while recording a 62-channel EEG. Participants were required to respond to numerical presentations in a cued letter stream, while ignoring distractor steams. Mishra et al. examined the steady state visual evoked potential (SSVEP), an EEG component thought to reflect the attentional demands of an attended stimuli, and found no significant group differences for attended stimuli, but significantly reduced SSVEP toward unattended stimuli in the expert VGPs. This suggests that expert VGPs may show a superior ability to suppress, or disregard, irrelevant stimuli. Expert VGPs also showed a significantly larger P300 component in response to numerical targets under a high perceptual load. Mishra and colleagues suggest that this result may show expert VGPs possess a greater sensitivity to task-relevant stimuli under high load. Bavelier, Achtman, Mani and Föcker [18] compared the activation between expert VGPs and non-VGPs in the frontoparietal attentional network and motion sensitive visual area, MT, on a visual search task. Participants indicated whether a diamond or square was in a ring of shapes surrounding a central fixation cross. This occurred under both low and high perceptual loads. Only non-VGPs showed a significant increase in the activation of the fronto-parietal attentional network in response to greater perceptual load. Bavelier and colleagues suggest that expert VGPs may be able to rely on automatized attentional allocation even under high perceptual load. Conversely, non-VGPs may need to switch to a more 'online' approach to attentional allocation in order to successfully perform the same task. Expertise in humans can be accompanied by structural changes in the specific brain regions and pathways associated with that expertise (e.g., taxi drivers [19]; bilingual individuals [20]; musicians [21,22]). Similarly, inter-individual variation in task performance can be reflected in variation in specific related white matter pathways (e.g., mental rotation [23]; bimanual co-ordination [24]). Expert musicians, who show enhanced performance on visuospatial tasks (e.g., [25,26,27]) also have reduced asymmetry in the inter-hemispheric transfer time (IHTT) of visual information measured by visual evoked potentials, in contrast to the IHTTs of matched non-musicians [28]. Although there are clear differences in the skills involved in expert musicianship and expert video-game play, both proficiencies require the translation of complex visual cues into precise bi-manual motor movements and the adjustment of performance based on sensory feedback. In both forms of expertise, training often begins during early childhood and continues through much of adolescence, a period in which the brain is most malleable and continues to develop The objective of the following study was to use EEG to examine components of the underlying neurophysiological basis of visuospatial and attentional performance of expert VGPs. Specifically we examined callosal functioning of expert VGPs on a task that required transfer of simple visual information as well as absolute occipital N1 latencies. We predicted that expert VGPs would have a more equal speed of transfer of visual information between the two hemispheres than non-VGPs, perhaps facilitating a greater ability to recruit the left hemisphere during tasks that require visuospatial attention. We used a standard task in which participants respond when detecting simple stimuli that are presented to each visual field individually, while a concurrent 128-channel EEG was recorded continuously. The latencies of occipital event-related potentials (ERPs) in the hemisphere contralateral (direct pathway) to the stimuli are subtracted from that in the hemisphere ipsilateral (callosal pathway), providing a measurement of IHTT. This methodology makes it possible to compare IHTT in two directions (i.e., left-to-right hemisphere transfer; right-to-left hemisphere transfer). Numerous studies of IHTT have shown that neural information travels more quickly from the right hemisphere of the brain to the left hemisphere, than from left hemisphere of the brain to right hemisphere in neurologically healthy adults (e.g., [29,30,31,32]). Miller [33] has proposed that the right hemisphere of the brain contains a greater number of heavily myelinated axons than the left hemisphere, enhancing its performance in fast parallel processes. Studies combining electrophysiology and anatomical imaging have shown that the speed of hemispheric transfer is inversely correlated with fractional anisotropic values in the posterior corpus callosum [34], suggesting that greater callosal integrity may result in quicker hemispheric transfer. In this study we predicted that the asymmetry between left-to-right IHTT and right-to-left IHTT Visual N1 Latencies and Expert Video-Game Players PLOS ONE | www.plosone.org 2 September 2013 | Volume 8 | Issue 9 | e75231 would be reduced, or non-existent in expert VGPs, reflecting a more balanced connectivity between the left and right hemisphere. Consistent with previous literature, non-VGPs were expected to show a quicker right-to-left IHTT than left-toright IHTT. Electrophysiological studies also allow for the study of absolute N1 latencies. The absolute latency is the latency of the evoked potential appearing in the hemispheric contralateral to the visual field in which the stimulus was presented. N1 latencies along this direct pathway have been suggested to reflect the time taken to discriminate visual stimuli [35] and lengthen as the attentional burden of an experimental task increases [36]. Previous research on expert VGPs consistently shows that VGPs possess quicker responses to visuospatial stimuli. Due to the link between the absolute N1 latency of the direct visual pathway and the discriminative processing of attended stimuli [34], it is possible that commonly observed quicker behavioural responses shown by expert VGPs may be partially underpinned by more rapid low-level visual processing. We also predicted that any changes to electrophysiological measures shown by expert VGPs would be the result of extensive video-game play. Consequently we also calculated correlations between electrophysiological measures (IHTT asymmetry; absolute N1 latency) and video-gaming characteristics (age begun; years of experience; hours per week). Finally we compared the reaction times of expert VGPs and non-VGPs to the visual stimuli. This was primarily to test whether this sample of VGPs showed the usual advantage in speed of responding and reduced effects of hand dominance on response speed. Methods 2.1) Ethics Statement Ethics approval for this study was obtained from the University of Auckland Human Participants Ethics Committee. Written informed consent was obtained from all participants prior to testing. All participants were naïve to the study's hypotheses. The experiment was performed in the Research Center for Cognitive Neurosciences high density EEG facility within the School of Psychology of The University of Auckland. 2.2) Participants Fifteen male expert video-game players and 16 matched non-VGPs participated in the study. Participants were recruited through public advertisement and via advertisement of opportunities to participant in research in the School of Psychology at the University of Auckland. Video-gamers were defined as expert if they began playing before the age of 10, had a minimum 8 years of experience and a minimum play time of 20 hours per week over the last 6 months. Game types currently played by VGPs were restricted to first-person shooters (e.g., CounterStrike: Global Offensive), real-time strategy (e.g., StarCraft II), action real-time strategy (e.g., Defense of the Ancients 2), massively multiplayer online roleplaying (e.g., Guild Wars 2), and others which utilize the mechanics of the aforementioned game genres but are not aggressive (e.g., the Portal series). These game genres require swift bimanual movements in response to complex in-game visual cues and commonly employ advanced artificial intelligence and multiplayer capabilities. The expert VGP group on average began playing at the age of 5.80 (SE = .42), with 17.47 (SE = .97) years of experience and a mean 34.67 hours (SE = 5.01) of play-time per week over the last 6 months. NonVGPs were required to have little-to-no video-game experience (maximum of 1.5 years). Individuals were excluded if they were left handed. Although females were eligible to participate, no females were recruited who met the criterion to be included as an expert VGP. There were no statistical differences between the expert VGPs and non-VGPs for age (expert VGPs: M = 23.27, SE = .88; non-VGPs: M = 25.69, SE = 1.19; t(29) = 1.62, p = .12), years of education (expert VGPs: M = 16.33, SE = 1.19; non-VGPs: M = 16.53, SE = .71; t(29) = .15, p = .89) or handedness as established by the Edinburgh Handedness Inventory [37] (expert VGPs: M = 93.87, SE = 1.68; non-VGPs: M = 91.35; SE = 2.62, t(25.27) = .81, p = .43). There were also no significant differences between the two groups on measures of estimated verbal IQ (expert VGPs: M = 111.00, SE = 2.12; non-VGPs: M = 113.50, SE = 3.10; t(26) = .67, p = .51), performance IQ (expert VGPs: M = 120.86, SE = 1.52; nonVGPs: M = 118.53, SE = 2.39; t(27) = -.81, p = .43) or estimated full scale IQ (expert VGPs: M = 117.50, SE = 1.79; non-VGPs: M = 118.07, SE = 2.71; t(26) = .18, p = .86), as assessed by the Wechsler Abbreviated Scale of Intelligence (WASI) [38]. 2.3) Materials and general procedure Stimuli were black/white checkerboard circles with a diameter of 3° visual angle and presented for 92 ms against a gray background using E-Prime. At the widest diameter of the circle there were 17 checkerboard squares. Stimuli appeared in either the left visual field or right visual field with their midpoint 6° from the central fixation cross. EEG was recorded continuously at a 1 kHz sampling rate (0.1-400 Hz analogous band pass) using a high density 128channel Ag/AgCl electrode net (Electrical Geodesics Inc., Eugene, OR, USA). Impedances for all electrode channels were kept below 40 kΩ. Data were acquired using a common reference electrode (Cz), positioned anatomically, and were later re-referenced to the average. Participants were tested within a quiet, electrically shielded Faraday chamber and were seated 57 cm away from a 22 inch Samsung computer monitor (1920x1080 pixel resolution; 60 Hz refresh rate; 13 ms lag in display) on which stimuli were presented. Throughout the experiment participants were instructed to maintain their gaze on the centrally located fixation cross. An initial block of 12 practice trials was followed by four experimental blocks, with hand order set to: right hand, left hand, left hand, right hand. Stimuli were preceded by a variable interstimulus interval of 542 ms, 742 ms or 942 ms. Following presentation of a stimulus participants responded by pressing the space bar. Each block contained a total of 70 trials which were randomized between 30 presentations to the left visual field, 30 presentations to the right visual field and 10 catch trials (no stimulus). Catch trials were included to ensure Visual N1 Latencies and Expert Video-Game Players PLOS ONE | www.plosone.org 3 September 2013 | Volume 8 | Issue 9 | e75231 participants maintained focus on the task. Participants were provided with an opportunity to rest at the beginning of each block, where they were also instructed on which hand to use next. 2.4) Analyses 2.4.1) Behavioural data. Reaction data was collected at a 1 ms resolution. Correct responses were defined as those key presses that occurred after a stimulus presentation. Response errors occurred when participants did not respond to a stimulus presentation or responded to a catch trial. Participant accuracy was calculated as the number of correct responses divided by the total number of stimulus, expressed as a percentage. Analysis of reaction time and accuracy data were evaluated with separate repeated measures ANOVA with group (expert VGPs; non-VGPs) as the between-subjects factor and hand (left hand; right hand), and visual field (left visual field; right visual field) as within-subject factors. 2.4.2) EEG data. EEG was segmented into epochs 100 ms pre-stimulus onset to 400 ms post-stimulus onset. Electrodes located at the outer canthi and above and below the left and right eyes were used to calculate horizontal and vertical electrooculogram (EOG). Recordings contaminated by vertical eye-movements and eye blinks (vertical EOG amplitudes exceeding ±70 μV), and horizontal eye-movements (horizontal EOG amplitudes exceeding ±70 μV) were discarded from the analysis. The remaining trials were corrected for residual eye movement artifacts using procedures from Jervis et al. (1985) [39]. The mean number of epochs remaining for expert VGPs was 92.47 (SE = 3.54) for the left visual field and 94.60 (SE = 3.65) for the right visual field, and for non-VGPs was 72.38 (SE = 5.83) for the left visual field and 73.69 (SE = 6.05) for the right visual field. Independent samples t-tests revealed the expert VGPs had significantly more epochs for analyses than the non-VGPs for both the left visual field, t(24.51) = -2.95, p = .007, and the right visual field, t(24.45) = -2.96, p = .007. Data were re-filtered to 30 Hz lowpass offline and average evoked potentials were constructed for left visual field and right visual field conditions. The N1 component of the evoked potential was defined as the greatest peak of the first negative wave that occurred at least 140 ms following stimulus presentation. N1 latencies were recorded from each participant using a cluster of seven lateral occipital electrodes (chosen a priori) centered between P3, T5 and O1 in the left hemisphere and between P4, T6 and O2 in the right hemisphere (standard 10-20 system) and averaged (see Figure 1). Interhemispheric transfer time estimates were calculated for each individual participant by subtracting the latency of the contralateral N1 (direct) from the latency of the ipsilateral N1 (following callosal transfer) for both left visual field and right visual field conditions. Results 3.1) Behavioural Analyses Analysis of reaction time data revealed a significant main effect of group, F(1, 29) = 4.24, p = .049. As expected, expert VGPs responded (M = 274.50, SE = 8.16) significantly more quickly than non-VGPs (M = 297.88, SE = 7.90) to the presentation of lateralized visual stimuli. There was a significant interaction between hand and visual field, F(1,29) = 15.28, p = .001, with responses to stimuli in the right visual field made significantly more quickly with the right hand than the left hand (p = .005), while responses made to stimuli in the left visual field were made significantly faster with the left hand than the right hand (p = .03). Responses made with the right hand were significantly faster to stimuli in the right visual field than the left visual field (p = .01). Contrary to predictions, however, there was no significant difference between reaction times to stimuli presented in the left visual field between the left hand and the right hand (p = .92). There were no other significant main effects or significant interactions although the main effect of hand approached significance, F(1,29) = 3.21, p = .08, as did the hand by group interaction, F(1,29) = 3.41, p = .08. Figure 2 shows expert VGPs responded as quickly with their left as their right hand, whereas Non-VGPs tended to respond more quickly with their right hand. Accuracy data for expert VGPs and non-VGPs were analysed to make sure that expert VGPs and non-VGPs performed the task accurately and comparably. This was confirmed. Accuracy for both expert VGPs (M = 98.22, SE = . 28) and non-VGPs (M = 98.98, SE = .27) was high and no significant main effects for group, hand or visual field, or significant interactions were found (all p-values > .05). 3.2) EEG 3.2.1) IHTT. Effects for IHTT were analyzed using a repeated measures ANOVA with group (expert VGPs; nonVGPs) as the between-subjects factor and direction (left-toright transfer; right-to-left transfer) as a within-subjects factor. Grand mean wave forms for contralateral (direct) and ipsilateral (indirect pathway) N1s elicited by left and right visual field stimuli in expert VGPs and non-VGPs are shown in Figure 3. The ANOVA for IHTT did not reveal a significant main effect of group, F(1,29) = 1.56, p = .22, nor, surprisingly, a significant main effect of transfer direction, F(1,29) = 1.33, p = .26. Furthermore, although non-VGPs appeared to show the expected faster transfer from right-to-left than left-to-right in Figure 4, there was no significant interaction between group and transfer direction, F(1,29) = .57, p = .46. Together this suggests that neither expert VGPs, nor non-VGPs, had faster IHTTs when transferring visual information from the right-to-left hemisphere. 3.2.2) Absolute latency of the N1. N1 latencies for direct pathways only (contralateral visual fields and hemispheres) were analysed with a repeated measures ANOVA with group (expert VGPs; non-VGPs) as the between-subjects factor and hemisphere (left hemisphere; right hemisphere) as a withinsubjects factor. The main effect of group was significant, F(1,29) = 4.87, p = .04 (see Figure 5), with expert VGPs displaying significantly earlier N1 latencies (M = 182.92, SE = 3.60) than non-VGPs (M = 193.99, SE = 3.49). No other effects were significant (all p-values > 0.5). If earlier N1 latencies influence speed of behavioral responding, then these two variables should be positively Visual N1 Latencies and Expert Video-Game Players PLOS ONE | www.plosone.org 4 September 2013 | Volume 8 | Issue 9 | e75231 correlated. In other words as absolute N1 latency increases, so too should behavioural response times. This was confirmed by a significant Pearson's correlation between the grand average of absolute N1 latency and response time, r = .39, p = .03. Finally, Pearson's correlation coefficients were calculated within the expert VGP group to assess the relationship between video gaming characteristics (age began regular gaming; years of experience; hours per week) and mean absolute N1 latency. It was predicted that for expert VGPs, as years of experience and number of hours gaming increased, absolute N1 latency should show a relative decrease. Contrary to predictions no Pearson's correlation was found to be significant. Figure 1. Diagram of Electrical Geodesic 128-electrode net (standard 10-20 system). Black circles and black line connectors show electrode clusters used for left and right hemisphere. doi: 10.1371/journal.pone.0075231.g001 Visual N1 Latencies and Expert Video-Game Players PLOS ONE | www.plosone.org 5 September 2013 | Volume 8 | Issue 9 | e75231 Discussion This study is the first electrophysiological investigation of IHTT, a measure of callosal function, and absolute occipital N1 latencies of expert VGPs. Using the latencies of N1 responses we measured IHTT speeds in both directions for both expert VGPs and non-VGPs. As predicted, expert VGPs showed no directional advantage for IHTT, indicating relatively equilateral transfer of visual information across the corpus callosum in the two directions (right-to-left hemisphere and left-to-right). Contrary to predictions, however, non-VGPs showed a similar degree of symmetry in their IHTTs for transfer of visual information. Notably expert VGPs showed significantly earlier absolute N1 latencies in both hemispheres than non-VGPs. Figure 2. Mean reaction time by hand used for expert video-game players and non-video-game players. Error bars represent standard error. doi: 10.1371/journal.pone.0075231.g002 Figure 3. Grand mean waveforms in left and right hemisphere occipital electrode clusters for non-video-game players and expert video-game players during stimulus presentation in the left and right visual field. doi: 10.1371/journal.pone.0075231.g003 Visual N1 Latencies and Expert Video-Game Players PLOS ONE | www.plosone.org 6 September 2013 | Volume 8 | Issue 9 | e75231 As expected the behavioural responses of expert VGPs were quicker than those of non-VGPs. Expert VGPs responded equally quickly with both hands, while non-VGPs tended to respond slightly more quickly with their right than left hands. These findings are consistent with previous studies investigating VGPs (e.g., [1,2,3]), and in this sense confirm that this sample of VGPs has behavioural characteristics in-line with those of VGPs in previous research. Modern video-game play requires players to translate complex visual cues into precise and rapid bimanual movements. Performance is continually adjusted based on incoming sensory feedback and occurs under a constant time pressure. Extended game-play often begins during early childhood and continues through much of adolescence, a period in which the brain is developing and at its most malleable. Thus the ability of expert VGPs to make quicker behavioural responses may result from prolonged training. An alternative account, however, is that the quicker responses of expert VGPs are not the result of video-game play, but instead reflect a pre-existing characteristic that allows these individuals to become expert VPGs. Another behavioural difference between the groups was that expert VGPs had significantly more good EEG epochs available for analyses, although the numbers available for both groups were high. This difference reflects fewer trials were contaminated (and rejected) by eye movements, blinks or facial musical contractions in expert VGPs, suggesting gamers are better at staring impassively at the computer screen, which again may reflect prolonged game-play. The key finding in the current study is that expert VGPs have earlier N1 latencies than non-VGPs in the direct pathways of both hemispheres. The latency of the N1 component of a visual event related potential has been suggested to reflect the time taken for an individual to discriminate visually-attended stimuli [35] and typically lengthens as the attentional demands of an experimental task increase [36]. Thus shorter N1 latencies may enable expert VGPs to discriminate attended visual stimuli significantly earlier than non-VGPs and contribute to faster responding in visual tasks. Additionally as expert VGPs can successfully perform the same visual task as non-VGPs with a shorter N1 latency, they may be able to make more efficient use of limited attentional resources in the visual domain. Certainly there is existing literature showing enhanced performance by VGPs on tasks involving attentional abilities in the visual domain (e.g., [1,2,3]). The earlier N1 time-course in expert VPGs may underpin some of the temporal enhancements seen in tasks such as attentional blink, temporal order judgment and backwards masking (e.g., [5,6,7]). The significant correlation between N1 latency and response time also suggests shorter N1 latencies may contribute to the faster responses of expert VGPs during visual tasks. One explanation for this electrophysiological enhancement in expert VGPs is that it results from training in the form of sustained video-game play. Successful video-game play requires players to continuously attend to and classify visual stimuli as relevant or irrelevant to in-game goals while under constant time pressure. Constant play while under these conditions, overtime, may facilitate an ability to identify attended visual stimuli earlier and make more efficient use of attentional resources in the visual domain. An alternative account, however, is that individuals who become expert VGPs do so because they can discriminate Figure 4. Mean interhemispheric transfer time for each direction for expert video-game players and non-video-game players. Error bars represent standard error. doi: 10.1371/journal.pone.0075231.g004 Visual N1 Latencies and Expert Video-Game Players PLOS ONE | www.plosone.org 7 September 2013 | Volume 8 | Issue 9 | e75231 visual stimuli earlier and use attentional resources in the visual domain more efficiently than their peers, resulting in more successful game-play. Estimates of IHTT are constructed by subtracting the latency of evoked potentials in the hemisphere contralateral to the visual field in which the stimulus is presented (direct pathway) from the latency of evoked potentials ipsilateral to the presented stimulus (indirect or callosal pathway). This methodology also makes it possible to compare IHTT in two directions (i.e., left-to-right transfer; right-to-left transfer). Previous IHTT literature has frequently shown that neural information transfers faster right-to-left than from left-to-right in healthy adults (e.g., [29,30,31,32]). In the current study we predicted that this typical asymmetry in expert VGPs relative to non-VGPs would be reduced, or non-existent, reflecting more balanced neural connectivity between the left and right hemisphere. Contrary to predictions there was no significant difference in the speed of callosal transfer between expert VGPs and non-VGPs, nor in the relative speed of transfer in the two directions. Although the pattern of transfer speeds in the non-VGP group was in the expected direction of faster transfer from right-to-left, this was not significant; both expert VGPs and non-VGPs showed relatively equilateral speed of transfer of visual information across the corpus callosum. No evidence in the current study exists to suggest that there are any significant group differences between expert VGPs and non-VGPs in callosal function. The absence of the predicted asymmetry in IHTT times in our non-VGP group was somewhat surprising. However, while a number of studies have shown the expected IHTT asymmetry (quicker right-to-left IHTT than left-to-right IHTT) in adult populations, other patterns of IHTT have also been reported. A number of sub-populations appear to show a reduced or absent asymmetry in IHTT, for example, expert musicians [28], individuals with attention deficit hyperactivity disorder both inattentive and combined subtypes [40], females [41] and lefthanders [42]. Additionally, Whitford and colleagues [43] failed to find an IHTT asymmetry in their healthy controls using N1 latencies derived from current source density headmaps. Surprisingly, a reversed IHTT asymmetry (i.e., quicker left-toright IHTT than right-to-left IHTT) was found when calculated using P1 latencies. The non-VGPs in this study provide another example of absent asymmetry in IHTT in a healthy adult sample. We failed to find any significant correlations between videogaming characteristics and absolute N1 latencies within the expert VGP group. This may reflect the low variability within our expert VGP sample in both the age at which they began gaming, years of gaming and hours per week spent gaming. If the criteria for inclusion in the VGP sample was expanded to include VGPs who started to play after the age of 10 and played casually for less than 20 hours per week there may have been a higher likelihood of significant correlations. Many video-game training studies have shown that even a small Figure 5. Mean absolute N1 latency for direct pathways for each hemisphere for expert video-game players and non-videogame players. Error bars represent standard error. doi: 10.1371/journal.pone.0075231.g005 Visual N1 Latencies and Expert Video-Game Players PLOS ONE | www.plosone.org 8 September 2013 | Volume 8 | Issue 9 | e75231 period of video-game play can result in behavioural enhancements in the same direction as those seen in expert VGPs (e.g., [1,44,45,46,47]) and these enhancements last for at least 5 months [11,12]. In conclusion, this study provides evidence, for the first time, that expert gamers have faster neural processing of visual stimuli than non-VGPs. 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Whitford TJ, Kubicki M, Ghorashi S, Schneiderman JS, Hawley KJ et al. (2011) Predicting inter-hemispheric transfer time from the diffusion properties of the corpus callosum in healthy individuals and schizophrenia patients: A combined ERP and DTI study. NeuroImage 54: 2318-2329. doi:10.1016/j.neuroimage.2010.10.048. PubMed: 20977941. 44. Basak C, Boot WR, Voss MW, Kramer AF (2008) Can training in a realtime strategy video game attenuate cognitive decline in older adults? Psychol Aging 23: 765-777. doi:10.1037/a0013494. PubMed: 19140648. 45. Clark JE, Lanphear AK, Riddick CC (1987) The effects of videogame playing on the response selection processing of elderly adults. J Gerontol 42: 82-85. doi:10.1093/geronj/42.1.82. PubMed: 3794204. 46. Goldstein JH, Cajko L, Oosterbroek M, Michielsen M, van Houten O et al. (1997) Video games and the elderly. Soc Behav Personal 25: 345-352. doi:10.2224/sbp.1997.25.4.345. 47. Orosy-Fildes C, Allan RW (1989) Psychology of computer use: XII. Video game play: Human reaction time to visual stimuli. Percept Mot Skills 69: 243-247. doi:10.2466/pms.1989.69.1.243. Visual N1 Latencies and Expert Video-Game Players PLOS ONE | www.plosone.org 10 September 2013 | Volume 8 | Issue 9 | e

The
Price
of
Truth 1 Draft: No citation without permission The
Price
of
Truth M.P.
Lynch [email protected] 1. Introduction William
James
once
said
that
truth
is
"the
good
in
the
way
of
belief".1 For James,
this
not
only
signaled
that
truth
was
a
value,
it
told
us
what
truth
was. Truth's
being
a
value,
in
his
view,
was
a
fact
about
the
nature
of
truth
itself. Now
days,
those
of
us
working
on
truth
for
a
living
mostly
ignore
James' insight.
The
currently
dominant
philosophical
view
among
the
specialists holds
that
the
concept
of
truth
is
nothing
but
an
expressive
device,
one
that allows
us
to
overcome
our
merely
medical
limitations
and
generalize
over infinitely
large
chains
of
propositions.
On
this
view,
truth
may
be
a
value,
but that
fact
is
superfluous
to
our
understanding
of
what
truth
is. I
aim
to
make
three
points.
First,
I
will
argue
that
talk
of
the
value
of
truth often
slides
back
and
forth
between
talk
of
two
very
different
values.
Second, I'll
urge
that
reflection
on
those
values
helps
us
to
understand
what
truth
is. In
this
respect,
I
agree
not
only
with
James,
but
with
the
more
recent
work
of Huw
Price,
who
has
argued
that
a
"norm
of
truth
plays
an
essential
and
little‐ recognized
role
in
our
assertoric
dialogue".2
For
both
Price
and
myself,
to understand
truth
one
must
understand
what
it
does,
its
role
in
our
cognitive economy,
and
truth's
normative
dimension
is
an
integral
part
of
that
role.
But Price
also
thinks
that
we
should
regard
truth-conceived
of
as
property
of our
beliefs-as
something
like
a
metaphysical
myth.
Here
I
disagree.
My
third 1
James,
Pragmatism
and
The
Meaning
of Truth
(Cambridge:
Harvard
University
Press,
1942) p.
42. 2 Price,
Naturalism
without
Mirrors
(Oxford:
Oxford
University
Press,
2011)
p.
165.
The quote
is
originally
form
"Truth
as
Convenient
Friction"
Journal
of
Philosophy
100
(2003): 167‐190.
All
citations
from
Price
in
this
paper
will
referee
to
the
aformentioned
book. The
Price
of
Truth 2 point
is
that
reflection
on
truth's
values
pushes
us
in
a
slightly
different direction,
one
that
opens
the
door
to
certain
metaphysical
possibilities
that even
a
Pricean
pragmatist
can
love. 2. Two
Values To
say
that
truth
is
a
value
can
mean
at
least
two
things.
First,
it
can
mean that
truth
is
the
answer
to
this
question:
in
virtue
of
what
are
beliefs
in
good standing,
or
correct?
Thus,
we
might
say
for
example:
that
a
belief
qua
belief is
correct
just
when
its
content
is
true. To
endorse
this
thought
is
to
endorse
that,
in
one
sense
of
the
word
"norm", that
truth
is
the
basic
norm
of
belief.
It
needn't
be
the
only
norm,
of
course,
or the
only
way
in
which
a
state
of
belief
can
be
correct.
Beliefs
can
also
be correct
when
they
are
justified,
for
example,
and
because
of
that
we presumably
sometimes
correctly
believe
what
is
false.
Thus
the
correctness of
believing
what
is
true
must
be
pro
tanto:
it
is
always
correct
but
not always
correct
all
things
considered.
In
this
way,
cognitive
norms
like justification
and
truth
are
no
different
than
most
norms
or
values.
Keeping one's
promises
is
always
pro
tanto
right,
but
it
is
not
always
right
all
things considered,
as
everyone
knows.
Sometimes
a
pro
tanto
value
is
outweighed by
other
values,
and
so
it
is
with
truth. The
analogy
with
promising
illustrates
a
further
point.
When
we
talk
about truth
being
the
norm
of
belief,
what
we
are
saying
is
something
more
like: truth-that
is,
the
truth
of
the
content-is
the
right,
not
the
good,
of
belief. The
proposition's
being
true
makes
it
correct
to
have
the
attitude
of
belief toward
that
proposition;
it
provides,
as
it
were,
a
definitive
reason
to
believe it.
As
Shah
has
argued,
this
is
because
the
belief
that
p
is
indirectly
responsive to
<p>'s
truth.
In
the
typical
conscious,
deliberative
case,
it
is
so
by
via
being directly
responsive
to
evidence
for
<p>
(Shah,
2003).
And
this
suggests,
in turn,
that
truth
is
not
just
a
norm
of
belief,
it
is
a
more
basic
norm
than justification
or
warrant.
For
we
take
it
to
be
correct
to
believe
what
is
based on
evidence
because
beliefs
based
on
evidence
are
likely
to
be
true,
and
thus the
value
of
truth
in
this
sense
is
more
basic
than
the
value
of
believing
what is
based
on
evidence. The
idea
that
truth
is
the
norm
of
belief
is
one
thing
that
is
sometimes
meant when
we
say
that
truth
is
a
value.
But
there
is
another
thing
we
might
have
in mind
as
well.
To
see
what
it
is,
consider
this:
you
might
think
that
being
a The
Price
of
Truth 3 winning
move
is
what
makes
a
particular
move
in
a
game
correct,
but wonder
why
we
should
care
about
winning.
In
the
same
way,
you
might
think that
being
true
is
what
makes
a
belief
correct
but
wonder
why
we
should care
about
having
correct
beliefs.
You
might
wonder
why
having
true
beliefs is
valuable. The
fact
that
we
can
even
raise
this
question
points
at
the
second
thing
we typically
mean
when
we
say
that
truth
is
a
value.
The
state
of
affairs
of
having true
beliefs
is
a
good. If
so,
then
we
should
strive
to
reach
it
while
engaging in
any
practice
aimed
at
producing
beliefs.
Inquiry
is
one
such
practice.
So having
true
beliefs
is,
as
it
were,
a
worthy
goal
of
inquiry,
the
practice
of figuring
out
what
to
believe.
This
is
a
thought
that
again
is
often
identified with
the
pragmatists,
who
in
some
moods
go
so
far
as
to
identify
truth
with that
towards
which
we
strive
during
inquiry. So
here
are
two
different
thoughts
that
folks
have
in
mind
when
they
talk about
the
value
of
truth: NORM:
Beliefs
are
(pro
tanto)
correct
just
when
their
contents
are true. GOAL: Having
true
beliefs
is
of
value,
and
therefore
should
be
the
goal of
inquiry. There
is
of
course,
much
debate
–
and
there
should
be
–about
how
to
exactly characterize
either
of
these
values.
One
might
wonder,
for
example,
whether NORM
underwrites
an
equivalence,
like: NORM*
The
belief
that
p
is
correct
if
and
only
if
the
proposition
that
p is
true.
3 The
idea
here
would
be
to
see
NORM
as
expressing
the
Socratic
reading
of NORM*:
as
telling
us
that
the
right
hand
side
has
explanatory
priority
over the
left.
On
the
other
hand,
you
might
think
–
notwithstanding
my
remarks about
pro
tanto
correctness
above-that
NORM*
is
too
strong.
If
so,
then
you might
say
that
NORM
only
implies
that
truth
is
a
sufficient
condition
for correctness
of
belief. Moreover,
there
is
the
question
of
how
NORM
or 3
See,
e.g.
P.
Horwich
"The
Value
of
Truth"
Nous,
40:
347‐40;
N.
Shah,
"How
Truth
Governs Belief"
Philosophical
Review
112
(4):
447‐82. The
Price
of
Truth 4 NORM*
are
connected
to
the
idea
–
explored
by
Price
and
others-that
truth is
also
a
norm
of
assertion.
4 Parallel
questions
of
detail
arise
with
GOAL.
For
example,
you
might
wonder whether
the
value
of
having
true
beliefs
entails
that
we
should
strive
to
have all
and
only
true
beliefs
(be
like
God
in
effect)
or
whether
a
more
humdrum end-like
having
true
beliefs
about
matters
we
care
about-would
do.5
And again,
you
might
wonder
how
GOAL
is
connected
to
other
values
in
the vicinity:
most
notably
the
value
of
knowledge.
(Some
believe
that
knowledge is
more
valuable
than
mere
true
belief,
for
example,
although
I
am
not
among them). In
any
event,
in
these
remarks,
I
won't
be
fussing
too
much
over
those particular
details. I
want
to
turn
instead
to
the
question
of
whether
GOAL and
NORM
tell
us
anything
about
truth. 3. But
what
does
this
tell
us
about
the
nature
of
truth? It
seems
to
me
that
NORM
and
GOAL
do
tell
us
something
about
truth.
In
the first
instance,
they
tell
us
about
true
beliefs,
or,
to
speak
more
precisely, about
beliefs
in
true
propositions.
If
having
true
beliefs
is
valuable,
then GOAL
tells
us
that
true
propositions
are
those
we
should
aim
at
believing during
inquiry.
NORM
tells
us
that
true
propositions
are
those
that
are correct
to
believe.
Now
while
these
truisms
are
explicitly
about
true propositions,
they
implicitly
describe
the
property
in
virtue
of
which propositions
are
true.
That
is,
we
can
take
them
to
describe
the
property
that plays
the
truth-role.
That
is: N1:
If
a
property
plays
the
truth‐role,
then
it
is
had
by
propositions that
are
correctly
believed. 4
One
thought
is
that
they
are
connected
because
one
ought
to
assert
what
one
believes
and one
should
believe
what
is
true. 5
For
discussion
of
this
sort
see
Kvanvig,
J.
The
Value
of
Knowledge
and
the
Pursuit
of Understanding.
(Cambridge:
Cambridge
University
Press,
2003)l
M.
P.
Lynch,
True
to
Life (Cambridge:
MIT
Press,
2004),
"Replies
to
Critics"
in
Philosophical
Books
46:
331‐42. M. David.
"Truth
as
an
Epistemic
Goal"
in
M.
Steup
(ed).
Knowledge,
Truth
and
Duty
(Oxford: Oxford
University
Press,
2001). The
Price
of
Truth 5 G1:
If
a
property
plays
the
truth‐role,
then
it
is
had
by
propositions
we aim,
during
inquiry,
to
believe. To
the
extent
to
which
N1
and
G1
tell
us
about
the
property
that
plays
the truth‐role,
they
also
tell
us
about
that
role
itself:
its
shape
and
structure,
as
it were.
Of
course,
it
is
an
open
question
how
"deep"
the
information
they
give us
about
the
truth‐role
really
is.
I'll
have
more
to
say
about
that
in
a
moment. Right
now,
I
am
content
point
out
that
however
you
conceive
of
the information
GOAL
and
NORM
give
us
about
truth,
that
information
has
a normative
character.
And
this
normative
character
is
what
suggests
that
the information
contained
in
NORM
and
GOAL
is
distinct
from
the
information contained
in
purely
"descriptive"
platitudes,
such
as ES:
The
proposition
that
p
is
true
iff
p. Arguably,
we
can't
derive
GOAL
or
NORM
from
the
purely
descriptive
ES.
And if
this
is
right,
then
at
least
one
version
of
deflationism-minimalism-is mistaken. The
label
"deflationism",
much
like
the
word
"realism"
or
"naturalism"
has come
to
describe
a
family,
or
at
best,
a
spectrum
of
views. There
are
at
least three
dimensions
along
which
a
theory
can
be
labeled
deflationary.6 EQUIVALENCE:
Ascribing
the
truth
concept
or
predicate
to
a contentful
item
is
equivalent
to
just
expressing
that
content. CONCEPTUAL: Appealing
to
ES
is
sufficient
to
account
for
the
concept of
truth. EXHAUSTION:
An
account
of
the
concept
of
truth
is
sufficient
to account
for
its
nature,
or
the
property
of
truth:
all
the
essential
facts about
the
property
are
exhausted
by
(an
account
of)
the
concept and/or
meaning
of
the
truth
predicate. Not
all
views
that
might
be
called
deflationary
will
endorse
all
of
these
views. And
specific
deflationary
views
will
differ
in
how
they
explain
each
of
them. Views
may
differ,
for
example,
over
how
to
understand
the
equivalence relationship
between
a
truth
ascription
and
a
contentful
item
given
in 6
For
a
similar
account
of
deflationism,
see
Burgess
and
Burgess,
Truth
(Princeton:
Princeton University
Press,
2010). The
Price
of
Truth 6 EQUIVALENCE;
or
they
may
differ
on
what
it
is
to
give
an
account
of
a concept.
But
what
we
might
call
fully
deflationary
views
endorse
all
three. The
combination
of
CONCEPTUAL
and
EQUIVALENCE
tell
us
that,
for
the
full deflationist,
the
truth‐role
is
very
simple.
Truth
functions
as
an
expressive device
and
only
as
such
a
device.
EXHAUSTION
tells
us
that
there
is
nothing more
to
say
metaphysically
about
truth. Perhaps
the
most
prominent
example
of
a
fully
deflationary
view,
and
the one
I'll
take
as
my
stalking
horse
here,
is
Paul
Horwich's
minimalism. According
to
Horwich,
the
concept
of
truth
is
a
logical
device
for generalization:
it
allows
us
to
overcome
our
merely
medical
limitations
and make
generalizations
(such
as
"every
proposition
is
either
true
or
false").
We grasp
this
concept
by
grasping
instances
of
the
equivalence
schema.
An
when we
do
so
we
know
all
the
essential
facts
about
truth
–
any
other
fact
about the
truth‐role
can
be
deduced
from
ES
together
with
some
non‐truth‐ theoretic
fact.
Consequently
,
even
if
we
grant
that
truth
is
a
property
–
if
a metaphysically
transparent
one,
like
the
concept
of
being
a
logical conjunction-it
is
a
property
does
no
significant
explanatory
work.7 If
NORM
tells
us
something
distinctive
about
truth,
then
a
fully
deflationary view
like
minimalism
is
in
trouble
for
two
reasons.
First,
it
won't
be
the
case that
we
can
know
all
the
facts
about
the
truth‐role
just
by
grasping
the instances
of
ES.
Hence
what
we
labeled
CONCEPTUAL
above
will
be mistaken.
And
second,
truth
will
apparently
do
some
explanatory
work.
In particular,
it
explains
when
tokens
of
a
certain
kind
of
mental
state
are
in good
standing. Here
we
face
an
obvious
objection
from
the
minimalist.
For
ease
of exposition,
I'll
put
these
objections
in
terms
of
NORM,
but
I
think
a
similar debate
can
occur
over
GOAL.
The
objection
is
that
we
can
derive
NORM*,
for example,
from
ES,
as
long
as
we
add
a
further
non
truth‐theoretic
fact.
What we
can
do
is
to
endorse
instances
of (B):
It
is
correct
to
believe
<p>
iff
p. And
then
if
we
add
instances
of ES:
<p>
is
true
iff
p. 7
The
locus
classicus
of
minimalism
is
Paul
Horwich's
Truth
(Oxford:
Oxford
University
Press, 1998). The
Price
of
Truth 7 Presto,
we
derive: NORM*:
It
is
correct
to
believe
<p>
iff
<p>
is
true. In
short,
the
thought
goes,
NORM
doesn't
tell
us
anything
new
about
truth.
It merely
demonstrates
the
expressive
use
of
the
truth
predicate. Let's
look,
however,
at
this
supposed
derivation.
I
don't
find
it
persuasive, because
it
is
difficult
to
see
why
we
would
accept
instances
of
(B)
in
the absence
of
already
being
committed
to
the
implicit
generalization
NORM*.8 For
the
list
of
(B)'s
instances
is
an
infinite
list
of
normative
prescriptions:
a list
of
little
belief
norms
as
it
were:
it
is
correct
to
believe
snow
is
white
iff snow
is
white,
correct
to
believe
roses
are
red
iff
roses
are
red
and
so
on.
But now
we
are
faced
with
an
obvious
question:
Why
should
we
accept
each
of these
individual
norms?
Individual
normative
prescriptions
are
justified
by general
normative
principles. Consider
promising:
it
is
correct
to
keep
your promise
to
Tom
for
the
same
reason
that
it
is
correct
to
keep
your
promise
to Bridget:
because
it
is
correct,
other
things
being
equal,
to
keep
your promises.
So
too
with
truth:
it
is
prima
facie
correct
to
believe
that
grass
is green
for
the
same
reason
it
is
correct
to
believe
that
snow
is
white:
because it
is
correct
to
believe
what
is
true.
The
general
normative
principle-NORM* in
this
case-is
in
the
epistemic
driver's
seat. Consequently
we
are
justified in
accepting
instances
of
(B)
only
in
virtue
of
accepting
instances
of
NORM*. An
aspect
of
my
point
here
was
made
familiar
by
Gupta:
namely,
you
can't
get generalizations
from
schemas.9
We
might
put
it
this
way:
the
generalization problem,
even
if
it
could
be
overcome
elsewhere,
is
damning
when
it
comes
to generalizations
over
norms. Why? The
reason
concerns
the
epistemology
of
norms.
In
ethics,
there
is
a position
known
as
particularism
according
to
which
there
are
no
true
general moral
principles,
or
if
there
are,
no
such
principle
does
any
epistemic
work
in our
moral
reasoning.
Perhaps
particularism
about
morality
is
true
–although I
doubt
it.
(It
has
always
been
hard
for
me
to
take
seriously
the
idea
that 8
The
following
argument
originally
appeared
in
"Minimalism
and
the
Value
of
Truth" Philosophical
Quarterly
54
(2004):
497‐517. 9
See
Anil
Gupta,
"A
Critique
of
Deflationism."
Philosophical
Topics
21
(1993)
2:
57‐81. The
Price
of
Truth 8 there
are
no
true
generalizations
about
some
matter).
My
present
point
is that
even
if
it
is
true
for
morality,
it
seems
very
implausible
when
applied
to cognitive
norms
like
truth.
It
seems,
rather,
very
plausible
that
the
reason
it is
correct
to
believe
that
snow
is
white
when
snow
is
white
is
the
very
SAME reason
it
is
correct
to
believe
that
oysters
are
tasty
when
they
are
tasty. Namely:
it
is
correct
to
believe
what
is
true.
You
ought
to
endorse
the individual
prescriptions
because
of
a
general
normative
truth. One
way
to
avoid
the
argument
would
be
to
go
find
yourself
a
logical apparatus
that
allows
you
to
generalize
over
(B). Something
like substitutional
quantification
might
come
to
mind,
for
example.
10 But
one might
wonder
how
much
ground
one
would
gain
from
such
a
maneuver. First,
note
that
by
appealing
to
a
device
like
substitutional
quantification,
we deprive
the
minimalist
of
one
of
their
key
points:
that
the
reason
we
have
a truth
concept
at
all
is
for
generalizing-that
is
truth's
job
as
it
were. Appealing
to
substitutional
quantification
effectively
outsources
that
job. More
importantly,
appealing
to
substitutional
quanitification
in
particular leaves
the
problem
that
caused
Horwich-rightly
in
my
view-to
appeal
to schemas
in
the
first
place:
namely,
that
it
is
hard
to
see
how
to
define substitutional
quantifiers
without
appealing
to
truth.
Indeed,
if
we
appeal
to substitutional
quantification
to
generalize
over
(B)
then
we
would
be,
in effect,
trying
to
explain
away
one
putative
fact
about
truth
by
appeal
to
a device
that
is
itself
explained
in
terms
of
truth,
thereby
demonstrating
that truth
has
more
than
an
expressive
role
to
play
after
all. But
even
if
some
such
maneuver
were
to
work,
it
doesn't
settle
the
issue. That
issue
is
whether
CONCEPTUAL
is
true:
whether
or
not,
in
coming
to grasp
NORM
or
NORM*,
we
learn
a
new
fact
about
truth,
or,
as
I've
put
it,
the truth‐role.
The
argument
we've
been
considering
is
that
CONCEPTUAL
is
true because
we
can
deduce
NORM*
from
ES
and
some
non‐truth
theoretic
fact. And
there's
the
rub.
Whether
this
strategy
works
will
depend
on
whether
(B) (in
either
the
generalized
or
schema
form)
really
is
"non‐truth‐theoretic". That
is,
it
will
depend
on
what
(B)
is
"about"-or
what
explains
why
(B)
is true. 10
See
Chris
Hill's
Thought
and
World
(Cambridge:
Cambridge
University
Press,
2002)
for
one strategy
for
example. And
of
course
one
might
pursue
completely
different
versions
of deflationism,
such
as
a
prosentential
theory
like
Brandom's
or
the
"pure
disquotational" theory
of
Field. The
Price
of
Truth 9 Some
might
think
that
just
because
(B)
doesn't
use
the
word
"true",
it
is obvious
(B)
isn't
about
truth
or
the
truth‐role.
But
words-or
the
lack
of them-are
cheap.
If
a
generalized
(B)
were
said
to
be
a
good
paraphrase
of NORM*,
then
we
would
have
just
as
much
reason
to
think
generalized
(B) conveys
information
about
the
truth‐role
as
we
do
for
thinking
NORM*
does. Good
paraphrases
carry
their
ontological
commitments
with
them. (Otherwise
they
wouldn't
be
good
paraphrases).
And
it
seems
that
the advocate
of
generalized
(B)
would
have
to
say
that
is
such
a
paraphrase, precisely
because
it
is
part
of
the
standard
deflationary
position
that
the
right hand
sides
of
(B)
and
NORM*
are-by
dint
of
(ES)-intersubstitutable (ignoring
as
always
opaque
and
pathological
contexts).
11 So
again:
what
makes
(B)
true?
One
thought
is
that
it
is
true
by
virtue
of
the concept
of
belief.
It
is
a
conceptual
truth
about
belief,
that
tells
us
something about
what
beliefs
are.
This
seems
very
plausible. Indeed,
it
seems
plausible that
both
(B)
and
NORM
are
tell
us
something
about
belief.
We
might
even
go so
far
to
say
that
the
fact
that
beliefs
are
correct
in
virtue
of
their
contents being
true
is
partly
constitutive
of
what
is
to
be
a
belief.
12
Believing,
unlike hoping
desiring
or
assuming,
is
a
propositional
attitude
that
can
go
right
or wrong
by
way
of
the
proposition
believed
being
true
or
false.13 11
Indeed,
this
fact
is
what
the
appeal
to
(B)
trades
on:
the
idea
is
that
we
can
paraphrase away
the
seeming
commitment
to
the
"truth‐property"
by
showing
that
we
can
express whatever
is
said
by
NORM*
with
a
generalized
version
of
(B),
which
lacks
even
the appearance
of
referring
to
the
alleged
property.
And
there
ihe
fact
that
"the
average
family has
1.5
children"
is
semantically
equivalent
to
"the
number
of
children
divided
by
the number
of
families
is
1.5"
doesn't
all
by
itself
tell
you
which
is
the
more
perspicuous statement.
One
must
supply
some
additional
theory-such
as
reasons
to
think
that
the material
world
contains
no
such
things
as
"average
families"-in
order
to
make
that
call. Likewise
with
truth:
mere
paraphrase
alone
shows
nothing
about
what
exists.
It
only
shows that
we
can
talk
either
way. 12
For
similar
pronouncements
see:
D.
Vellemann,
The
Possibility
of
Practical
Reason
(Oxford: Oxford
University
Press,
2000),
16.
P.
Boghossian,
''The
Normativity
of
Content'', Philosophical
Issues,
13
(2003),
31–45;
M.
P.
Lynch,
''The
Values
of
Truth
and
the
Truth
of Values'',
in
D.
Pritchard
(ed.),
Epistemic
Value
(Oxford:
Oxford
University
Press);
N.
Shah, ''How
Truth
Governs
Belief" Philosophical
Review,
112
(2003),
447–83;
R.
Wedgwood,
''The Aim
of Belief
''
Philosophical
Perspectives,
36
(2002),
267–297. 13
Of
course,
to
say
this
much
isn't
to
say
very
much
about
belief.
One
would
like
an explanation.
But
it
is
worth
pointing
out
that
philosophers
of
mind
have
tried
to
given
such explanations:
one
explanation,
for
example,
is
that
beliefs
have
a
distinctive
direction
of
fit
– beliefs
try
to
fit
how
things
are.
Another
explanation
is
that
having
a
true
content
is
a
belief‐ state's
proper
function,
what
it
was
designed
to
do.
Whatever
the
explanation,
the
point
is The
Price
of
Truth 10 But
acknowledging
this
fact
leaves
open
the
obvious
possibility
that
what explains
why
NORM,
NORM*
and
(B)
are
true
are
conceptual
facts
about belief
and
truth.
That
is,
it
leaves
open
the
possibility
that
in
the
relevant sense
of
"about"
NORM
and
(B)
are
about
the
concepts
of
belief
and
truth. Given
the
connection
between
belief
and
truth
noted
above,
this
seems
very plausible
–
even
unavoidable.
And
it
is
hardly
surprising
given
the
standard functionalist
analysis
of
belief.
Consider,
for
example,
the
thought
that If
one
believes
that
X'ing
will
get
you
Y,
and
one
desires
Y,
then
one
is likely
to
X. Is
this
a
truth
about
belief
or
desire?
The
standard
answer
is:
both.
It
is
a truism
that
shows
the
connection
between
these
two
concepts
and
is
true
in virtue
of
them.
I
take
NORM*
and
(B)
to
be
similar:
they
tell
us
something about
truth
and
belief.
This
was
something
Dummett
pointed
out
a
long
time ago
by
way
of
a
famous
analogy
with
winning
and
truth.
One
doesn't understand
what
a
competitive
game
is
if
one
doesn't
understand
that
the point
of
the
game
is
to
win.
And
one
wouldn't
understand
winning
if
one didn't
understand
that
it
was
the
point
of
a
competitive
game. Nonetheless,
if
ALL
that
NORM
or
NORM*
were
to
tell
us
about
truth
is
that
it is
what
makes
beliefs
correct,
one
might
feel
rightly
cheated.
That,
one
might think,
is
hardly
substantive
information
about
truth's
nature
or
essence.
But in
fact,
NORM
and
NORM*
tell
us
more
than
that,
as
I'll
now
go
on
to
argue. What
NORM
tells
us
is
that
(a)
truth,
and
therefore
any
property
that
plays that
role,
is
a
fundamental
norm;
(b);
that
said
property
is
regulative
of
any practice
aimed
at
producing
beliefs;
and
(c)
that
it
is
an
unabstractable
norm. 4. Truth
as
a
fundamental
and
regulative
norm
of
belief. To
appreciate
the
specific
information
NORM
tells
us
about
the
truth‐role, let's
turn
to
a
thought
experiment
of
Huw
Price's-who,
along
with
Crispin Wright,
has
repeatedly
and
influentially
stressed
the
importance
of understanding
the
value
of
truth. not
epistemological.
It
is
not
about
how
we
tell
whether
a
propositional
attitude
is
a
belief. We
do
that,
presumably,
by
looking
at
the
person's
behavior.
The
idea,
rather,
is
that
part
of what
it
is
to
be
a
belief
is
that
a
belief
is
correct
just
when
true. The
Price
of
Truth 11 Price
asks
us
to
imagine
a
community
of
speakers
who
don't
recognize
a truth
norm
of
assertion,
one
that
is
parallel
to
the
truth
norm
of
belief
we've been
considering.
The
imagined
community-he
calls
them
the
"Mo'ans"
‐‐ instead
takes
itself
to
operate
by
two
weaker
norms.
The
first
is
a
norm
of sincerity:
one
ought
to
assert
what
one
believes.
The
second
is
a
norm
of warranted
assertibility,
according
to
which
what
makes
an
assertion
correct or
incorrect
is
whether
it
is
warranted.14
Presumably,
this
norm
itself
is
not understood
by
the
members
of
the
community
as
being
defined
in
terms
of truth.
For
them,
"being
based
on
adequate
grounds"
is
just,
say,
"being coherent
with
one's
previous
assertions".
So
for
the
Mo'ans,
when
one
person asserts
that,
e.g.
"Sarah
Palin
is
a
genius"
what
they
are
asserting
is
correct (according
to
the
norms
of
the
community)
just
when
they
believe
it,
and
it
is coherent
with
their
previous
assertions. And
someone
who
asserts
the negation
of
the
above
is
likewise
correct
just
when
her
assertion
is sincere and
coherent
with
her
past
assertions. As
Price
has
noted,
the
Mo'ans
needn't
lack
a
disquotational
truth
predicate or
the
minimalist
concept
of
truth.
We
can
still
imagine
them
using
the concept
of
truth
to
generalize
over
their
"assertions".
Someone
who
asserts "Everything
Sarah
Palin
says
is
true"
(shudder)
can
be
thought
to
be asserting
something
like
"if
SP
says
that
x,
then
x,
and
if
SP
says
that
y,
then y..."
and
so
on.
Of
course,
the
conjuncts
here,
if
asserted,
are
correct, according
the
community,
only
in
virtue
of
whether
they
are
warranted
in the
sense
in
which
they
understand
that
term.
But
that
doesn't
mean
they can't
use
a
disquotational
truth
predicate
or
a
minimalist
concept
of
truth. So
what
is
lacking
in
this
society?
Well
one
thing
that
seems
to
be
lacking, Price
notes,
is
conversational
engagement
in
fundamental
disagreement
or criticism.
The
Mo'ans
seems
to
be
blind
to
the
fact,
so
obvious
to
us,
that assertions
can
be
criticized
as
being
right
or
wrong
independently
of whether
those
assertions
are
warranted
by
the
community's
standards. And
surely
this
is
right:
there
is
no
disagreement
in
assertion
amongst
the Mo'ans.
But
that
doesn't
mean
there
is
no
disagreement.
It
just
means
that they
are
like
a
bunch
of
freshman
who
say
"whatever,
dude"
when confronted
with
anyone
else's
claims.
They
can't
be
bothered
to
really
dig
in, to
confront
one
another
just
so
long
as
consistency
is
gained.
But
they
still might
believe
each
other
to
be
mistaken
even
if
they
tolerate
one
another's 14
See
Price,
2011,
chapter
8.
See
also
"Three
Norms
of
Assertibility"
in
J.
Tomberlin
(ed). Philosophical
Perspectives.
Vol.
XII,
Language,
Mind
and
Ontology,
Blackwell,
241‐54. The
Price
of
Truth 12 assertions. So
let's
modify
Price's
thought
experiment. Let's
imagine
that
in
addition
to lacking
a
truth
norm
for
assertion,
our
imaginary
community-call
them
the Ultra‐Mo'ans-also
lacks
one
for
belief.
Let's
imagine,
in
short,
that
they
are not
governed
by
NORM
(whether
or
not
they
recognize
this
fact).15
We
can imagine
that
the
members
of
our
community
still
"accept"
certain propositions
and
"reject"
others. These
mental
acceptances,
are
correct,
let's say,
just
when
they
are
coherent
with
what
they've
accepted
in
the
past.
So they
have
mental
states.
They
have
norms
operating
on
those
mental
states. And
again,
it
seems
possible
that
they
have
a
truth
predicate
allows
them
to generalize
over
infinite
strings
of
the
contents
of
such
states. Price's
community
lacked
the
ability
to
disagree
in
assertion.
But
it
seems clear
our
new
community
lacks
quite
a
bit
more. The
Ultra‐Mo'ans
lack
the ability
to
disagree
in
belief.
And
there
is
an
obvious
reason
why:
they
don't have
beliefs.
In
accepting
what
they
accept
and
rejecting
what
they
reject,
the Ultra‐Mo'ans
are
governed
only
by
whether
or
not
those
acceptances
are coherent
or
incoherent
with
what
a
given
member
has
accepted
in
the
past.
If so,
and
if
NORM
is
constitutive
of
belief,
then
they
lack
beliefs. Where
beliefs
vanish,
so
does
inquiry.
Inquiry
is
just
the
practice
of
trying
to figure
out
what
to
believe.
Sure,
they
may
LOOK
like
they
inquire.
They'll
ask questions,
and
give
answers.
They
might
go
through
the
motions
of
what looks
like
belief‐formation-at
least
sometimes.
But
they
wouldn't
be
trying to
answer
the
questions
correctly.
They'd
just
be
looking
to
accept
answers that
are
consistent
with
their
previous
acceptances.
This
is
hard
to
imagine.
It is
hard
to
imagine
because
it
sounds
to
us
like
such
people
would
really
only be
engaged
in
an
elaborate
game
of
wishful
thinking. What
the
case
of
the
Ultra‐Mo'ans
tells
us
is
that
if
we
disagree,
and
are believers,
we
must
be
governed
by
NORM.
But as
Price
himself
notes
(2011, 180)
the
thought
experiments
also
tell
us
something
else:
namely,
that
the values
in
question
can't
be
reduced
to
"purely"
epistemic,
non‐truth conducive
values.
For
the
members
of
our
second
community
have
norms that
express
such
values,
but
they
lack
the
goal
or
norm
of
truth. Consequently,
the
truth
norm
is
more
basic
than
a
warrant
norm.
Note
that this
is
even
more
obvious
when–unlike
our
modified
Mo'ans
–we
take 15
Here
I
draw
on
a
previous
paper
of
mine,
"Truth,
Value
and
Epistemic
Expressivism" Philosophy
and
Phenomenological
Research,
2006. The
Price
of
Truth 13 warrant
to
be
defined
as
that
which
makes
our
beliefs
likely
to
be
true. Here we
take
warrant
to
be
subservient
to
truth
by
definition:
it
is
correct
to believe
what
is
warranted
because
doing
so
is
likely
to
result
in
having
true beliefs. Let's
bring
this
back
around
to
the
point.
Above,
we
imagined
that
someone could
respond
to
our
earlier
arguments
by
claiming
that,
while
NORM
is
true, and
distinct
from
ES,
it
really
tells
us
nothing
about
the
truth‐role.
It
only
tells us
about
belief.
But
we
now
have
a
reason
to
doubt
this.
The
reason
is
that,
in grasping
NORM,
we
already
grasp
something
about
the
truth‐role:
namely that
the
property
in
virtue
of
which
it
is
correct
to
believe
a
proposition
is
a more
basic
norm
than
justification
or
warrant.
Consequently,
NORM,
or
even a
generalized
version
of
NORM*,
isn't
just
about
belief.
To
grasp
such
facts
is to
grasp
something
more
specific
about
truth.
It
is
to
learn
not
only
that
truth is
a
norm
of
belief,
but
that
truth
is
the
most
fundamental
norm
of
belief. A
consequence
of
truth
being
the
fundamental
normative
standard
of
belief
is that
the
having
of
that
property
plays
a
regulative
role
for
any
practice
that aims
at
producing
belief.
A
property
P
plays
a
regulative
role
in
a
practice when,
just
by
virtue
of
participating
in
that
practice,
one
is
normatively committed
to
regulating
one's
moves
in
the
practice
by
one's
judgments about
what
has
or
lacks
that
property
(Wedgewood,
2002,
268).
Thus
the property
of
being
a
winning
chess
move
is
regulative
of
chess:
in
playing chess
I
am
committed
to
regulating
my
moves
by
my
judgments
of
what
is
or isn't
a
winning
move. Likewise,
in
figuring
out
what
to
believe
–
that
is,
when engaging
in
inquiry
–
I
am
committed
to
regulating
my
doxastic
practices
by my
judgments
about
what
is
or
isn't
true.
Indeed,
I
am
regulated
by
the
truth in
inquiry
in
the
most
direct
possible
way:
the
recognition
that
p
is
true
is
a decisive
reason
to
believe
it
(Shah,
2003). In
sum,
a
further
fact
about
the truth‐role
that
we
learn
from
NORM
is
that
truth
is
a
regulative
property
of inquiry. In
sum,
I
think
we
should
acknowledge
that
the
normative
truisms
NORM and
GOAL
tell
us
something
important
about
the
truth‐role,
something distinct
from
what
we
learn
about
truth
from
platitudes
like
ES. CONCEPTUAL
therefore
is
unwarranted,
and
so
is
any
fully
deflationary
view like
minimalism
that
is
committed
to
it. The
Price
of
Truth 14 5. Truth
as
Unabstractable So
if
our
Pricean
reflections
are
right,
a
fully
deflationary
view
is
implausible. But
as
Price
himself
would
no
doubt
point
out,
that
still
leaves
a
more moderate
deflationary
position
on
the
table. Earlier
I
noted
three
respects
in
which
a
view
can
be
assessed
as deflationary:
EQUIVALENCE;
CONCEPTUAL
and
EXHAUSTION.
To
embrace EXHAUSTION
is
to
embrace
the
idea
that
we
can
know
all
the
facts
about truth
just
by
virtue
of
grasping
the
ordinary
concept
of
truth.
Being
true,
on such
an
account,
is
a
lot
like
being
a
conjunction.
You
understand
everything about
being
a
conjunction
just
by
grasping
the
truth‐tables.
Nothing
is
left out. Price
can
say
his
view
is
still
deflationary
in
this
respect.
We
might
put
it
like this.
To
understand
what
truth
is
to
understand
that
it
is
a
property
that plays
a
particular
functional
role
in
our
cognitive
life.
But
that
role
is exhausted
by
some
simple
truisms.
One
sort
of
deflationist-the
full deflationist-thinks
it
is
exhausted
by
ES.
Another
sort-Price's
sort-thinks we
need
to
add
more,
NORM
and
GOAL,
for
example.
But
both
think
there
is nothing
else
to
say
about
truth
once
we've
grasped
these
truisms.
As
Price provocatively
puts
it,
his
thought
is
that
we
can
acknowledge
that
truth
is
a norm
but
still
regard
talk
of
a
"property"
of
truth
as
something
akin
to
a metaphysical
fiction.
By
this,
he
doesn't
mean
that
it
is
true
that
there
is
not truth
or
some
such.
Rather,
he
says
he
is
like
the
person
who,
rather
than saying
there
is
no
God,
rejects
the
theological
language‐game
entirely
(2011, 181). In
short,
his
thought
is
that
we
can
acknowledge
that
truth
is
a
norm without
having
to
say
anything
"metaphysical"
about
what
that
norm consists
in. This
is,
on
the
face
of
it,
a
curious
position.
If
I
tell
you
that
x
is correct/right/of
value
just
when
it
is
y,
then
it
is
surely
reasonable
to
ask about
what
y
IS,
or
failing
that,
about
the
conditions
under
which
things
are or
are
not
y. And
if
no
answer
is
forthcoming,
one
might
well
suspect
one's right
to
make
the
normative
claim.
Indeed,
one
might
think
that
this
sort
of reasoning
is
part
of
what
has
led
deflationists
like
Horwich
to
resist
the thought
that
truth
is
a
norm
in
the
first
place-they
take
it
that
once
you
let that
camel's
nose
under
the
tent,
it
will
walk
away
with
the
whole
thing. The
Price
of
Truth 15 So
why
does
Price
think
that
he
can
acknowledge
that
truth
is
a
norm without
saying
anything
more
about
what
it
consists
in?
He
sees
the conclusion
as
dropping
out
of
the
Mo'ans
thought
experiment.
That's because,
in
his
view,
what
that
experiment
shows
is
that
"what
matters
is that
speakers
take
there
to
be
a
norm
of
truth,
not
that
there
actually
be
such a
norm,
in
some
speaker‐independent
sense
(2011,
180;
emphasis
added). Yet
whether
this
is
convincing
hinges
on
what
is
meant
by
"taking
there
to
be a
norm". Price
suggests
this
is
a
practical
or
pragmatic
matter:
"I
think
in practice
we
find
it
impossible
to
stop
caring
about
truth"
(Ibid.).
This
makes it
sound
as
if
the
modality
here
is
very
weak:
what
matters
is
for
us
to
act
as if
we
are
governed
by
NORM,
not
that
we
are
so
governed.
But
the
case
of
the Ultra‐Mo'ans
suggests
that
the
modality
is
conceptual,
not
practical.
Indeed, this
falls
out
of
the
very
set‐up
of
the
ultra‐Mo'ans
thought
experiment. By stipulation,
they
aren't
governed
by
NORM
–
whether
or
not
they
recognize that
fact. And
in
not
being
governed
by
NORM,
they
lack
beliefs.
Hence,
it seems
deeply
plausible
that,
in
order
for
the
ultra‐Mo'ans
to
start
to
have beliefs,
it
will
not
be
sufficient
for
them
to
just
start
acting
as
if
they
are governed
by
NORM. To
see
this,
imagine
a
variant:
a
group
of
ultra‐Mo'ans that
are
not
governed
by
NORM
but
act
as
if
they
do.
Perhaps
they
even
have convinced
themselves
they
do
take
beliefs
to
be
correct
when
true
–
even when,
really,
all
they
do
is
take
beliefs
to
be
correct
when
they
are
consistent with
their
past
asssertions
or
flatter
their
"side"
of
a
"debate.
In
effect,
they still
pay
lip
service
to
the
idea
that
it
is
correct
to
believe
when
true.
But
that won't
make
them
have
beliefs,
any
more
than
saying
your
are
honest
means that
you
are
honest. These
considerations
suggest
a
further
fact
about
the
truth‐role
that
we
learn from
NORM:
namely,
that
truth
is
an
unabstractable
norm
of
belief.16 Here's what
I
mean.
Skeptical
views
about
value
generally
presuppose
that
we
can take
what
Mark
Timmons
has
called
a
disengaged
standpoint
towards
those values.
That
is,
that
we
can
distinguish
the
standpoint
from
which
we
make our,
e.g.
moral
commitments
and
the
standpoint
from
which
we
explain
the point,
nature
and
truth
of
those
commitments. In
the
case
of
moral
values,
it is
not
difficult
to
take
this
standpoint.
We
do
so
by
noticing
that
human beings
can
engage
in
actions
but
take
moral
values
different
from
our
own
to govern
those
actions.
We
note
that
engaging
in
action
doesn't
necessitate 16
The
following
paragraph
summarizes
arguments
made
in
much
more
detail
in
"Truth Value
and
Epistemic
Expressivism"
and
"The
Values
of
Truth
and
the
Truth
of
Values"
in Epistemic
Value.
Ed.
By
a.
Haddock,
A.
Millar,
D.
Pritchard
(Oxford:
Oxford
University
Press, 2010). The
Price
of
Truth 16 having
particular
moral
values
to
evaluate
those
actions. But
note
that
something
very
different
is
going
on
in
the
case
of
the
cognitive values
picked
out
by
NORM
And
GOAL.
For
our
Pricean
thought
experiment suggests
that
we
can't
abstract
in
the
requisite
way
–
and
this unabstractablity
is
due
to
the
very
concepts
involved.
That's
why
we
can't even
imagine
a
community
of
believers
and
inquirers
not
governed
by
NORM And
GOAL. We
can
imagine
people
doing
things
and
having
mental
states that
are
somewhat
like
believing
and
inquiring,
but
that
is
it.
We
can't
seem to
sufficiently
disengage
from
those
values
to
be
skeptical
about
them. The
conceptual
unabstractability
of
the
truth
norm
should
make
us suspicious
of
Price's
implicit
view
that
we
can
retain
EXHUASTION
whilst still
holding
to
the
idea
that
truth
is
a
norm.
If
our
concept
of
truth
requires us
to
regard
it
as
a
norm
of
belief
(and
not
just
to
act
as
if
it
is)
then
our question
above
remains.
In
what
does
this
norm
consist? 6. Pragmatism
and
Pluralism Price's
over‐all
picture
of
truth
is,
as
he
says,
"hard
to
find
on
contemporary maps"
because
its
elements
are
not
normally
thought
to
be
compatible: In
one
sense
it
is
impeccably
pragmatist,
for
example,
for
it
appeals
to nothing
more
than
the
role
of
truth
in
linguistic
practice.
Yet
it
rejects the
pragmatist's
ur‐urge,
to
try
to
identify
truth
with
justification. Again,
it
defends
a
kind
of
truth
commonly
seen
as
realist,
but
does
so from
a
pragmatist
starting
point,
without
the
metaphysics
that typically
accompanies
a
realist
view
of
truth
(2011,
182). I
am
sympathetic
with
this
picture.
According
to
Price,
if
we
adopt
his
view
of truth,
we
can
avoid
what
he
calls
the
placement
problem-the
problem
of placing
"various
kinds
of
truth
in
a
natural
world"
(2011,
6).
Price
diagnoses this
problem
as
resting
on
the
assumption
that
truth
is
always
and everywhere
a
matter
of
representation.
Think
that,
and
you'll
find
yourself puzzled
about
how
our
beliefs
about
numbers
and
norms
can
be
true:
For what
items
do
they
represent?
Dealing
with
this
problem
is
the
bread
and butter
of
much
modern
analytic
philosophy.17
But
Price
suggests
a
different solution,
or
dissolution:
give
up
on
the
idea
that
truth
always
consists
in representation.
Instead,
endorse
a
form
of
what
he
has
famously
called 17
See
Lynch
(2009). The
Price
of
Truth 17 "discourse
pluralism"
"the
view
that
philosophy
should
recognize
an irreducible
plurality
of
kinds
of
discourse-the
moral
as
well
as
the
scientific, for
example."
(2011,
36). Not
all
truths
need
to
represent
the
world. I'm
down
with
pluralism.
But
I
worry
about
Price's
way
into
it.
In
his
view, representationalism
is
wrong
not
because
truth's
nature
is
sometimes
to
be understood
in
terms
other
than
correspondence,
but
because
truth
has
no nature
at
all. That
is,
Price
marries
pluralism
to
a
kind
of
delfationism.
But
as I've
argued,
his
deflationism-his
endorsement
of
EXHAUSTION
in particular-is
in
tension
with
the
view
that
truth
is
a
norm.
The
problem, again,
is
that
embracing
NORM's
lessons
begs
the
question
of
what
this
norm consists
in.
But
accepting
EXHAUSTION
means
embracing
a
dispirited quietism. In
short,
here's
the
scoreboard
as
I
see
it:
Price
is
right
that
truth
has
a normative
character.
He
is
right
that
not
all
truths
represent
the
world.
But he
is
wrong
to
think
that
a
deflationary
view
which
accepts
EXHAUSTION
is the
right
way
to
combine
the
first
view
with
the
second. What
we
need
is
a
way
of
securing
pluralism
while
being
able
to
say something
explanatory
about
the
nature
of
the
truth
norm.
The
key,
as
I've argued
elsewhere,
lies
in
embracing
the
thought
that
truth
is
a
functional property.
Like
Price,
I
think
that
to
understand
the
concept
of
truth
is
to
grasp a
particular
functional
role-this
indeed,
is
the
lesson
I
took
from
his
brilliant Facts
and
the
Function
of
Truth.
But
unlike
Price,
I
think
that
this
tells
us something
about
what
truth
is. To
define
truth
functionally
in
the
sense
I
am
interested
in
is
to
define
it
by way
of
its
connections
to
other
related
concepts.
These
connections
are embodied
in
certain
common
truisms
that
have
played
a
central
role
in
the historical
discussions
over
truth.
One
of
these
truisms
in
NORM;
another
is GOAL.
A
third
is: OBJECTIVITY:
P
is
true
if
and
only
if
were
P
to
be
believed,
things
would be
as
they
are
believed
to
be. The
idea
is
that
these
truisms,
or
ones
very
much
like
them,
together
with certain
obvious
platitudes
connecting
truth
with
validity,
knowledge
and
the like,
jointly
pick
out
the
truth‐role.
They
do
so
by
describing
the
conceptually essential
features
of
the
truth
property.
We
might
put
it
like
this:
truth
just
is The
Price
of
Truth 18 the
property
that
has
these
features
essentially.
Having
that
property
is
what constitutes
a
proposition's
being
true.
It
is
in
this
sense
that
the
functionalist thinks
that
truth
is
one,
and
in
it
is
in
this
sense
that
the
functionalist
view
is most
akin
to
Price's.
For
like
Price,
the
functionalist
takes
the
truth‐role
to
be more
complex
than
the
full
deflationist
(she
rejects
CONCEPTUAL),
and
like Price,
she
takes
the
truth
property
itself
to
be
a
holistically
defined
affair. But
unlike
Price,
my
kind
of
alethic
functionalist
can
allow
for
pluralism without
having
to
accept
EXHAUSTION.
For
truth's
being
a
functional property
is
consistent
with
the
idea
that
there
is
more
than
one
property
in virtue
of
which
propositions
are
true.
18
That
is,
it
is
consistent
with
the
idea that
whether
a
given
proposition
is
true
is
determined
by
its
having
some other,
ontologically
distinct
property
–
perhaps
a
representational
property, like
correspondence,
or
perhaps
an
epistemically
constrained
property
such as
Wright's
superassertibility.
19According
to
the
functionalist,
we
can
pick out
a
property
that
determines
whether
a
proposition
is
true
by
seeing whether
it
plays
the
truth‐role.
Such
properties
should
not
be
confused
with truth
itself,
but
they
could
be
described
as
properties
that,
under
certain conditions,
and
for
certain
kinds
of
content,
realize
truth.
And
of
course,
the fact
that
truth
is
multiply
realizable
in
this
way
is
not
something
that
we
can read
off
from
the
truth
concept.
To
see
whether
it
is
true,
we
must
do
some "metaphysical"
work. So
it
seems
to
me
that
this
is
just
the
sort
of
view
the
Pricean
pragmatist needs.
It
allows
us
to
say,
against
traditional
theorists,
that
truth
as
such
may be
a
very
thin
property.
Beliefs
are
true
just
when
they
are
correct,
when they
are
the
sort
of
beliefs
we
aim
to
have
during
inquiry,
when
the
world
is as
they
portray
it
as
being,
That
is
the
job‐description
of
truth,
as
it
were.
But seeing
truth
as
a
functional
property
in
this
sense
is
entirely
consistent
with there
being
OTHER
properties,
distinct
from
truth,
which
satisfy
this description
and
so
realize
that
thin
property
–
which,
in
short,
play
the
truth‐ role. And
that
is
a
helpful
thought,
because
it
opens
to
door
to
appealing
to these
other
properties
to
explain
what
grounds
the
norm
in
different
domains or
discourses.
It
allows
us
new
tools
for
addressing
what
plays
the
truth‐role 18
This
is
the
view
first
defended
by
Crispin
Wright,
Truth
and
Objectivity
(Cambridge: Harvard
University
Press,
1992);
see
also
Lynch,
Truth
in
Context
(Cambridge:
MIT
Press, 1998)
and
Lynch,
Truth
as
One
and
Many
(Oxford:
Oxford
University
Press,
2009). 19
See
Wright,
1992
for
discussion. The
Price
of
Truth 19 for
different
sorts
of
belief,
and
therefore,
what
makes
particular
beliefs
of those
kinds
correct.
It
opens
new
theoretical
doors
rather
than
closing
them. Now
it
might
be
said
that
pragmatists
aren't
supposed
to
like
Metaphysics. But
(to
modify
a
famous
bit
of
Rorty's)
there
is
Metaphysics
and
then
there
is metaphysics.
There
is
nothing
spooky
about
investigating
the
properties
that play
the
truth‐role.
In
conjunction
with
cognitive
science,
for
example,
we
can ask
how
some
mental
states
might
represent
certain
objects
and
properties. That
is
a
perfectly
naturalistic
enterprise
to
all
save
those
who
think
they
can build
an
a
priori
wall
against
the
onslaught
of
empirical
science
that
says
that such
representation
is
possible
(Good
luck,
I
say:
a
priori
walls
only
look impressive
from
the
inside).
Nor
is
there
anything
particularly
recherché
in the
task
of
figuring
out
what
it
means
to
say
a
proposition
would
be warranted
relative
to
all
future
stages
of
information. My
present
goal
is
not
to
conduct
these
investigations
of
course
(although I've
made
some
headway
elsewhere)
but
to
simply
recommend
them
as reasonable
lines
of
inquiry,
and
ones
that
can
be
approached
from
a naturalistic,
pragmatist
perspective.20
Indeed,
to
pursue
such
investigations is
to
follow
an
important
pragmatic
maxim,
famously
invoked
by
James,
who urged
us
to
know
the
"particular
go
of
it"
when
it
came
to
truth.
It
is
not enough,
James
warned,
to
endorse
platitudes
and
truisms
to
understand what
truth
is;
not
enough
to
simply
say
that
"true
beliefs
correspond
to reality"
or "it
is
correct
to
believe
what
is
true".
What
the
pragmatist demands,
or
should
demand,
is
some
understanding
of
what "correspondence"
and
"reality"
are,
what
makes
for
"correctness"
and
so
on. That's
what
is
involved
in
figuring
out
what
plays
the
truth‐role: understanding
the
particular
go
of
it. Another
thing
I've
always
thought
the
pragmatists
had
right,
and
which
Price in
particular
has
stressed,
is
that
if
you
want
to
know
what
something
is
then you
need
to
understand
what
it
does,
why
it
matters
in
our
cognitive economy,
why
we
care
about
it. The
pragmatists'
big
idea
was
that
this
held not
just
for
ideas
like
belief
and
assertion,
but
for
truth
itself.
And
their
idea still
seems
right,
at
least
in
spirit,
if
not
in
detail.
If
you
want
to
understand truth,
you
must
understand
truth's
values.
Yet
the
cost
of
doing
so
is
not 20
For
discussion
of
both
representation
and
superwarrant/superassertibility,
see
Lynch, 2009. The
Price
of
Truth 20 quietism;
it
is
further,
metaphysical,
work.
Happily,
this
is
not
too
high
a
price to
pay.
21 21
Thanks
to
audiences
at
Princeton
University,
the
University
of
Aberdeen
and
the conference
on
Price's
Naturalism
Without
Mirrors
at
the
University
of
Zurich.
I've
benefited greatly
from
discussions
and
comments
from
Douglas
Edwards,
Casey
Johnson,
Paul Horwich,
Huw
Price,
Steven
Gross,
Lionel
Shapiro
and
Crispin
Wright.

Axel HonnetH Riconoscimento e conflitto di classe scritti 1979-1989 cura e traduzione di eleonora Piromalli mimesis © 2011 – MiMesis edizioni (milano – Udine) collana: ??? www. mimesisedizioni. it / www. mimesisbookshop. com Via Risorgimento, 33 – 20099 sesto san Giovanni (mi) Telefono: +39 02 24861657 / +39 02 24416383 Fax: +39 02 89403935 Via chiamparis, 94 – 33013 Gemona del friuli (Ud) E-mail: [email protected] indice Fonti p. 7 introduzione di Eleonora Piromalli p. 9 Riconoscimento e conflitto di classe lA «biogrAFiA lAtente» dei giovAni dellA clAsse lAvorAtrice p. 33 lAvoro e Azione struMentAle. ProbleMi cAtegoriAli Per unA teoriA criticA dellA società p. 43 coscienzA MorAle e doMinio di clAsse p. 91 consenso MorAle e senso di ingiustiziA. sullo studio di bArrington Moore Le basi sociaLi deLL'obbedienza e deLLa rivoLta p. 111 l'onore Ferito – ForMe quotidiAne dell'esPerienzA MorAle p. 121 eticA del discorso e concetto iMPlicito di giustiziA p. 129 lA logicA dell'eMAnciPAzione – sull'eredità FilosoFicA del MArxisMo p. 139 bibliogrAFiA p. Fonti: – La «biografia latente» dei giovani della classe lavoratrice, di axel Hon neth, Birgit mahnkopf e Rainer Paris, è la traduzione di Zur «latenten Biographie» von Arbeiterjugendlichen, in Soziologische Analysen. Referate aus den Veranstaltungen der Sektionen der Deutschen Gesellschaft für Soziologie und der ad-hoc-Gruppen beim 19. Deutschen Soziologentag, a cura di Rainer mackensen e felizitas sagebiel, tUB-dokumentation, Berlin 1979, pp. 930-939. – Lavoro e azione strumentale. Problemi categoriali per una teoria critica della società è la traduzione di Arbeit und instrumentales Handeln. Kategoriale Probleme einer kritischen Gesellschaftstheorie, in Arbeit, Handlung, Normativität, a cura di axel Honneth e Urs Jaeggi, suhrkamp, frankfurt a. m. 1980, pp. 185-233 1. – Coscienza morale e dominio di classe è la traduzione di Moralbewusstsein und soziale Klassenherrschaft. Einige Schwierigkeiten in der Analyse normativer Handlungspotentiale, in «leviathan», iX (1981), n. 3-4, pp. 556-570. – Consenso morale e senso di ingiustizia. Sullo studio di Barrington Moore «Le basi sociali dell'obbedienza e della rivolta» è la traduzione di Moralischer Konsens und Unrechtsempfindung. Zu Barrington Moores Untersuchung «Ungerechtigkeit», in Suhrkamp Wissenschaft. Weisses Programm, almanach, frankfurt a. m. 1984, pp. 108-114. – L'onore ferito. Forme quotidiane dell'esperienza morale è la traduzione di Die verletzte Ehre. Zur Alltagsform moralischer Erfahrungen, in «li teraturmagazin», XVi (1985), pp. 84-90. 1 Una precedente, parziale traduzione in italiano di questo saggio, fatta da marina calloni (con revisione di mauro Protti), è presente in Dopo la Scuola di Francoforte. Studi su J. Habermas, a cura di m. Protti, Unicopli, milano 1984, pp. 143-169. 8 Riconoscimento e conflitto di classe – Etica del discorso e concetto implicito di giustizia è la traduzione di Diskursethik und implizites Gerechtigkeitskonzept, in Moralität und Sittlichkeit. Das Problem Hegels und die Diskursethik, a cura di Wolfgang Kuhlmann, suhrkamp, frankfurt a. m. 1986, pp. 183-193. – La logica dell'emancipazione – sull'eredità filosofica del marxismo è la traduzione di Logik der Emanzipation. Zum philosophischen Erbe des Marxismus, in Wege ins Reich der Freiheit. André Gorz zum 65. Geburtstag, a cura di Hans leo Krämer e claus leggewie, Rotbuchverlag, Berlin 1989, pp. 86-106.

Michael Baur* In this Essay, I examine some apparent difficulties with what I call the "actualization criterion" connected to Rawls's notion of public reason, that is, the criterion for determining when Rawlsian public reason is concretely actualized by citizens in their deliberating and deciding about constitutional essentials and matters of basic justice. While these apparent difficulties have led some commentators to reject Rawlsian public reason altogether, I offer an interpretation that might allow Rawlsian public reason to escape the difficulties. My reading involves the claim that Rawlsian public reason is to be understood essentially as an imperative or an ideal, and as not necessarily grounded in any stock of existing beliefs or opinions. I make this claim on the basis of the seemingly counterintuitive observation that it is possible for citizen-interlocutors to know that public reason has been violated without necessarily knowing who the violator is (and thus without being able to foreclose the possibility that the violator may even be oneself). This observation is based in turn on my analysis of the necessary reciprocity and self-referentiality built in to the very concept of public reason as such. I. THE APPARENT PARADOXES OF PUBLIC REASON In Lecture VI of Political Liberalism, Rawls tells us that his notion of "public reason" is suggested by Immanuel Kant's distinction between public and private reason in the 1784 essay, What Is Enlightenment?, and is related to Kant's discussion in the Critique of Pure Reason regarding "The Discipline of Reason with Regard to Its Polemical Use."' But Rawls's overt reference to Kant in this regard immediately raises some questions, since the Rawlsian notion of public reason seems to be at odds with what Kant actually says. This is because the Rawlsian notion of public reason focuses on the limits that are to be placed on the kinds of reasons to which citizens may legitimately appeal when deliberating and making decisions publicly about constitutional essentials and matters of basic justice. By contrast, Kant seems to argue that reason itself is essentially public 2153 * Associate Professor of Philosophy and Director of the Natural Law Colloquium, Fordham University; Ph.D., University of Toronto (Philosophy); J.D., Harvard Law School. 1. John Rawls, Political Liberalism 213 n.2 (1996). ON ACTUALIZING PUBLIC REASON FORDHAM LAW REVIEW and unrestricted and that no limits whatsoever should be placed on what might be debated and decided by free citizens. In An Answer to the Question: What is Enlightenment?, Kant clearly states that citizens should be free to discuss and deliberate about anything, provided only that they remain obedient to existing law.2 For Kant, the proper imperative for citizens should be: "Argue as much as you will, and about whatever you will, but obey!"3 Furthermore, in the section of Kant's Critique of Pure Reason to which Rawls refers, Kant explains that reason consists essentially in the freedom to subject all things whatsoever to unrestricted critique, and that such critique allows no room for any prohibited topics or sacred cows, even if personal sensitivities or social utility might seem to be threatened by such far-reaching critique.4 Kant writes: Reason must subject itself to critique in all its undertakings, and cannot restrict the freedom of critique through any prohibition without damaging itself and drawing upon itself a disadvantageous suspicion. Now there is nothing so important because of its utility, nothing so holy, that it may be exempted from this searching review and inspection, which knows no respect for persons. The very existence of reason depends upon this freedom, which has no dictatorial authority, but whose claim is never anything more than the agreement of free citizens, each of whom must be able to express his reservations, indeed even his veto, without holding back.5 In an earlier and parallel passage found in the Preface to the 1781 edition of The Critique of Putre Reason, Kant makes a similar point about the right-and even the duty-of reason to subject all things (including those pertaining to religion and public legislation) to unrestricted critical scrutiny: Our age is the genuine age of criticism, to which everything must submit. Religion through its holiness and legislation through its majesty commonly seek to exempt themselves from it. But in this way they excite a just suspicion against themselves, and cannot lay claim to that unfeigned respect that reason grants only to that which has been able to withstand its free and public examination. 6 Thus we have here what might be called a textual or bibliographical paradox, since the Kantian passages to which Rawls refers seem to contradict the spirit of Rawls's own notion of "public reason." But the Rawlsian account of public reason seems to involve more than just 2. Immanuel Kant, An Answer to the Question: What is Enlightenment?, Berlinische Monatsschrift, Dec. 1784, reprinted in Immanuel Kant: Practical Philosophy 15, 18 (Mary J. Gregor trans. & ed., 1996). 3. Id. 4. Imimanuel Kant, Critique of Pure Reason 643 (Paul Guyer & Allen W. Wood trans. & eds., Cambridge University Press, 1997) (1781 & 1787). 5. Id. 6. Id. at 100 n.*. 2154 [Vol. 72 ONACTUALIZING PUBLIC REASON this textual paradox. As Rawls himself recognizes, there also seems to be a theoretical paradox surrounding his notion of public reason. How can it be reasonable or rational, Rawls asks, to say that citizens should appeal "only to a public conception of justice and not to the whole truth as they see it"7 when they discuss and decide on matters as important as constitutional essentials and basic questions of justice? "Surely," Rawls writes rhetorically, "the most fundamental questions should be settled by appealing to the most important truths"s-and yet it is these most important truths that apparently must be declared off limits or taken off the table because of the requirements of public reason. In addressing the paradox, Rawls reminds us that the exercise of public reason has to do with the reason of free and equal citizens who, as a collective body, deliberate about and decide on constitutional essentials and matters of basic justice. And it is through such deliberation and decision making (e.g., in enacting laws and in amending their constitution) that such free and equal citizens "exercise final political and coercive power over one another."9 The idea here is that, because the power being exercised is final and coercive, it should not be imposed on those subject to it in an external or dogmatic fashion, but instead should be exercised on the basis of principles and ideals that can be made justifiable to the persons who are subject to it. As Kent Greenawalt has written, the idea behind the requirement that the coercive power of the law should be based on public reason alone is "that people should not be compelled on the basis of reasons that are not persuasive for them."'° Along these lines, Rawls holds that political power should be exercised in accordance with what he calls "the liberal principle of legitimacy," which states: "[O]ur exercise of political power is proper and hence justifiable only when it is exercised in accordance with a constitution the essentials of which all citizens may reasonably be expected to endorse in the light of principles and ideals acceptable to them as reasonable and rational."'" And to the extent that citizens are reasonable and rational, they "should be ready to explain the basis of their actions to one another [e.g., their decisions about constitutional essentials and 7. Rawls, Political Liberalism, supra note 1, at 216. 8. Id. 9. Id. at 214 (emphasis added). 10. Kent Greenawalt, Natural Law and Public Reasons, 47 Vill. L. Rev. 531, 535 (2002). 11. Rawls, Political Liberalism, supra note 1, at 217. But see id. at 137 (stating a slightly different formulation of the liberal principle of legitimacy: "[O]ur exercise of political power is fully proper only when it is exercised in accordance with a constitution the essentials of which all citizens as free and equal may reasonably be expected to endorse in light of principles and ideals acceptable to their common human reason"). 2004] 2155 FORDHAM LAW REVIEW basic issues of justice] in terms each could reasonably expect that others might endorse as consistent with their freedom and equality."'12 In addition to the liberal principle of legitimacy, the ideal of democratic citizenship implied by the notion of public reason entails a (moral) duty of civility among citizens. This duty of civility includes not only the duty to "to be able to explain to one another on those fundamental questions how the principles and policies they advocate and vote for can be supported by the political values of public reason"; it also includes the duty to be willing to listen to others and to be fair-minded "in deciding when accommodations to their views should reasonably be made."' 3 Rawls goes on to explain how the apparent theoretical paradox concerning public reason can be resolved, once one understands what is meant by an overlapping consensus of reasonable comprehensive doctrines. An overlapping consensus is not a mere modus vivendi; that is to say, it is not a mere compromise that is struck on the basis of selfor group-interest.14 Rather, an overlapping consensus remains stable-in spite of changing compromises and shifts in the distribution of political power-because the political conception of justice that belongs to an overlapping consensus is genuinely supported by the individual citizens themselves, and the citizens endorse such a conception "on moral grounds" in spite of their differing and even conflicting comprehensive views.15 For Rawls, then, an overlapping consensus exists when "citizens who affirm reasonable but opposing comprehensive doctrines" endorse a properly political "conception of justice as giving the content of their political judgments on basic institutions" and when "unreasonable comprehensive doctrines... do not gain enough currency to undermine society's essential justice."' 6 If there exists an overlapping consensus of reasonable comprehensive doctrines and if each of these reasonable comprehensive doctrines can support and include within itself (as a kind of "module") a properly political conception of justice, then there is nothing odd or paradoxical about public reason and the limits prescribed by it. Indeed, the idea of an overlapping consensus-when properly understood-entails that public reason must place certain limits on the kind of principles to which citizens may appeal in deliberating and deciding on constitutional essentials and matters of basic justice. Thus Rawls writes: [W]hen the political conception [of justice] is supported by an overlapping consensus of reasonable comprehensive doctrines, the 12. Id. at 218. 13. Id. at 217. 14. Id. at 147. 15. Id. 16. Id. at 39. 2156 [Vol. 72 ONACTUALIZING PUBLIC REASON paradox of public reason disappears. The union of the duty of civility with the great values of the political yields the ideal of citizens governing themselves in ways that each thinks the others might reasonably be expected to accept; and this ideal in turn is supported by the comprehensive doctrines reasonable persons affirm.17 Given the normative significance of the idea of an overlapping consensus, there is nothing strange about holding that the most fundamental matters affecting citizens in a liberal democracy (e.g., matters pertaining to constitutional essentials and basic justice) should not be decided on the basis of "the most important truths" or "the whole truth" as citizens might see it from the differing perspectives of their comprehensive doctrines. Rawls seeks to illustrate why the "'paradox of public reason" disappears by referring to familiar political and legal situations in which "we recognize a duty not to decide in view of the whole truth" and which show "how it is often perfectly reasonable to forswear the whole truth" in some instances.'8 For example, the rules of evidence place limits on the kinds of testimony that may be introduced at a criminal trial-and thus place limits on our otherwise acceptable urge to know and to act on the basis of the whole truth."9 As Rawls puts the point in his 1997 essay, The Idea of Public Reason Revisited: "I propose that in public reason comprehensive doctrines of truth or right be replaced by an idea of the politically reasonable addressed to citizens as citizens.... The zeal to embody the whole truth in politics is incompatible with an idea of public reason that belongs with democratic citizenship."20 II. IS THE RAWLSIAN "ACTUALIZATION CRITERION" INTERNAL/SUBJECTIVE OR EXTERNAL/OBJEcTIvE? The preceding part displayed how a cluster of inter-related concepts is integral to a proper understanding of Rawls's account of public reason. This cluster of concepts included: a political conception of justice, an overlapping consensus of reasonable comprehensive doctrines, the liberal principle of legitimacy, the duty of civility, and the ideal of democratic citizenship. Even without delving further into 17. Id. at 218. 18. Id. at 219. 19. Id. at 218. There are some commentators who would reject this analogy between public reason and the rules of evidence. Leif Wenar, for example, argues that Rawlsian public reason does more than simply require that citizens appeal in public to only part of what they believe. For Wenar, Rawlsian public reason also requires (even if implicitly) that citizens appeal in public to reasons that in some cases actually contradict their comprehensive doctrines. Leif Wenar, Political Liberalism: An Internal Critique, 106 Ethics 32, 62 (1995). 20. John Rawls, The Idea of Public Reason Revisited, 64 U. Chi. L. Rev. 765 (1997), reprinted in John Rawls: Collected Papers 573, 574 (Samuel Freeman ed., 1999). 21572004] FORDHAM LAW REVIEW the meaning and inter-relations of these concepts, it is possible to raise some illuminating questions about the notion of Rawlsian public reason. I wish to raise such questions by asking about the Rawlsian criterion for determining whether or not in any particular concrete situation public reason is being properly actualized in the deliberations and decision making of the citizens (I shall call this the "actualization criterion"). In The Idea of Public Reason Revisited, Rawls seems to provide a relatively clear and straightforward statement of the actualization criterion. He writes: [a] A citizen engages in public reason, then, when he or she deliberates within a framework of what he or she sincerely regards as the most reasonable political conception of justice, [b] a conception that expresses political values that others, as free and equal citizens might also reasonably be expected reasonably to endorse. Now we can ask a number of important questions about this criterion. For example, under the criterion, is it sufficient that a citizen deliberate within a framework that he or she sincerely regards as the most reasonable political conception of justice? Or is sincere belief insufficient, and must it also be the case that the citizen deliberate within a framework that as a matter of fact is the most reasonable, or that at least is a reasonable conception of justice? The first part of the criterion given above (starting at the letter [a]) seems to support what we might call the "internal" or "subjective" reading of the actualization criterion: According to this reading, the idea of public reason requires only that the citizen deliberate within a framework of what he or she internally (or subjectively) and sincerely regards as reasonable. But the second, appositional part of the criterion given above (starting at the letter [b]) seems to support what we might call an "external" or "objective" reading of the criterion: According to this reading, it is not sufficient that the citizen deliberate within a framework of what he or she sincerely regards as reasonable; the proper exercise of public reason also requires that this framework actually be a reasonable and properly political conception of justice, one that expresses political values that others might reasonably (and not just sincerely) be expected reasonably to endorse. The ambiguity indicated above, it seems, cannot be resolved on strictly textual grounds alone, since Rawls's various statements on the matter are not univocal. Some passages in Rawls seem to support an internal/subjective reading of the actualization criterion, while others seem to support an external/objective reading. For example, in support of the internal/subjective reading, one can turn to the following passage in Political Liberalism: 21. Id. at 581. 2158 [Vol. 72 ONACTUALIZING PUBLIC REASON [P]ublic reason does not ask us to accept the very same principles of justice, but rather to conduct our fundamental discussions in terms of what we regard as a political conception. We should sincerely think that our view of the matter is based on political values everyone can reasonably be expected to endorse. For an electorate thus to conduct itself is a high ideal the following of which realizes fundamental democratic values not to be abandoned simply because full agreement does not obtain. A vote can be held on a fundamental question as on any other; and if the question is debated by appeal to political values and citizens vote their sincere opinion, the ideal is sustained.22 For further support in favor of an internal/subjective reading, one can refer also to The Idea of Public Reason Revisited, in which Rawls states that the ideal of public reason is realized, or satisfied, whenever judges, legislators, chief executives, and other government officials, as well as candidates for public office [and also citizens acting and thinking of themselves as legislators], act from and follow the idea of public reason and explain to other citizens their reasons for supporting fundamental political positions in terms of the political conception of justice they regard as the most reasonable. 23 But several other passages in Rawls's work seem to support an external/objective reading of the actualization criterion. For example, in a subsequent passage from The Idea of Public Reason Revisited, Rawls states that the proper exercise of political power supported by public reason requires not just subjectively sincere belief, but also objectively reasonable belief insofar as the reasons offered by one citizen can also be reasonably endorsed by other citizens: "Our exercise of political power is proper only when we sincerely believe that the reasons we would offer for our political actions-were we to state them as government officials-are sufficient, and we also reasonably think that other citizens might also reasonably accept those reasons."' In spite of the textual ambiguity on the matter, there are good systemic and theoretical reasons to opt for an external/objective reading of the actualization criterion. After all, a merely internal/subjective criterion would seem to entail the utter privatization or subjectivization of public reason. For under a merely internal/subjective reading of the actualization criterion, citizens could be said to meet the requirements of public reason just so long as they sincerely believed that they were operating with a conception of justice that expressed political values that others might also reasonably be 22. Rawls, Political Liberalism, supra note 1, at 241 (emphasis added). 23. Rawls, The Idea of Public Reason Revisited, supra note 20, at 576 (emphasis added). 24. Id. at 578 (emphasis added). 2004] 2159 FORDHAM LAW REVIEW expected reasonably to endorse (even if such sincere belief were not, as a matter of fact, "objectively" reasonable). But even if one were to opt for an external/objective reading of the actualization criterion, further problems remain. As we have seen, under the external/objective reading of the actualization criterion, it is not sufficient that the citizen deliberate within a framework of what he or she sincerely regards as the most reasonable political conception of justice. The proper exercise of public reason also requires that this framework actually be a properly political conception of justice, one which expresses political values that others might reasonably (and not just sincerely) be expected reasonably to endorse. When is a citizen being not only sincere, but also "objectively reasonable" in his or her expectation that others will endorse the values expressed by his or her own political conception of justice? In The Idea of Public Reason Revisited, Rawls provides what we-at least for our present purposes-might regard as a "reasonableness criterion." Rawls writes: Citizens are reasonable when, viewing one another as free and equal in a system of social cooperation over generations, they are prepared to offer one another fair terms of cooperation according to what they consider the most reasonable conception of political justice; and when they agree to act on those terms, even at the cost of their own interests in particular situations, provided that other citizens also accept those terms.2` Notice that the central condition specified in this criterion (namely, that citizens be "prepared to offer one another fair terms of cooperation according to what they consider the most reasonable conception of political justice")26 is a merely subjective or internal condition. But furthermore, the seemingly "external" or "objective" condition specified in this criterion ("provided that other citizens also accept those terms") qualifies only the requirement that citizens ''agree to act" on the terms of cooperation that they themselves have offered.27 In other words, Rawls adds this qualifying requirement in order to make clear that citizens are obligated to act on the terms that they themselves have offered as "fair terms of cooperation," 2 8 but that they are thus obligated only so long as other citizens also accept those terms. Accordingly, this seemingly "external" or "objective" condition (that other citizens also accept the terms offered) does not yield the position that a citizen is being "reasonable" only if other citizens actually accept the terms of cooperation offered by him or her. (Even if it did yield this position, this position would be problematic for yet other reasons: Why should one citizen's 25. Id. (emphasis added). 26. Id. 27. Id. 28. Id. 2160 [Vol. 72 ONACTUALIZING PUBLIC REASON reasonableness depend on the fact that other citizens happen to agree with the terms of cooperation offered by him or her?) Thus, when all is said and done, this Rawlsian reasonableness criterion seems to require little more than sincere, subjective belief. What might have initially appeared to be an external/objective requirement within the reasonableness criterion given above is shown upon further analysis to be reducible to a merely internal/subjective requirement. And the same sort of difficulty seems to threaten even Rawls's criterion of reciprocity. Explaining the criterion of reciprocity, Rawls writes: The criterion of reciprocity requires that when those terms are proposed [by citizens] as the most reasonable terms of fair cooperation, those proposing them must also think it at least reasonable for others to accept them, as free and equal citizens, and not as dominated and manipulated, or under the pressure of an inferior political or social position.29 As Rawls later observes, "the criterion of reciprocity is an essential ingredient specifying public reason and its content." 30 But while the criterion of reciprocity is crucial to public reason, it nevertheless seems to depend ultimately on a nierely "internal" or "subjective" requirement, namely that a citizen "must also think it at least reasonable" for others to accept the terms of cooperation which he or she has offered them." In light of the preceding observations, it is not clear how the Rawlsian actualization criterion, reasonableness criterion, or criterion of reciprocity can really provide more than essentially "internal" or "subjective" requirements. Samuel Freeman, in his contribution to this issue of the Fordham Law Review, correctly notes that, in Rawlsian political liberalism, the notion of reasonableness "stands in for" or takes the place of the notion of truth.32 But as we have just seen, Rawls's account of public reason seems to come dangerously close to allowing the notion of "subjective sincerity" to stand in for the notion of "publicness" or "reasonableness." Freeman goes on to assert that for Rawls, ''reasonable principles" are those principles that "are generally acceptable to conscientious, informed, and morally motivated moral agents."33 But this seemingly external/objective set of requirements for reasonableness does not address the problem at hand. First of all, the generic requirement that "reasonable principles" be "generally acceptable" cannot-on its own-provide any grounds for distinguishing principles that a broad majority happens to find 29. Id. (emphasis added). 30. Id. at 609. 31. Id. at 578. 32. Samuel Freeman, Public Reason and Political Justifications, 72 Fordham L. Rev. 2021, 2023 (2004). 33. Id. 2004] 2161 FORDHAM LAW REVIEW acceptable, and those that are genuinely reasonable. And as we know from the history of slavery and racial discrimination in our own country, what may be "generally acceptable" is not co-extensive-or at least should not be co-extensive-with what we take to be "reasonable." Indeed, if the "generally acceptable" were supposed to be co-extensive with "the reasonable," then a political liberalism based on such a view of "reasonable principles" would be virtually indistinguishable from sheer majoritarianism: The majority's will would always determine "the reasonable." Freeman's further requirement that reasonable principles be generally acceptable to persons who are "morally motivated"34 does not offer any helpful new content beyond a mere "subjective sincerity" requirement. Finally, the additional requirement that reasonable principles be generally acceptable to "informed" persons remains normatively vacuous as long as it is not tied to some normatively significant "truths" or "facts" by means of which persons could be said to be "informed" in the relevant respects. But as Freeman has correctly noted, Rawls aims to provide a political conception of justice and an account of public reason that does not rely on any comprehensive doctrines or normatively significant "truths."3 5 In summary, the attempt to identify a plausible "actualization criterion" for Rawlsian public reason seems doomed to founder between two unpalatable options: the requirement of mere "subjective sincerity" (on the one hand) and the requirement of some sort of normative "truth" (on the other hand).36 III. FURTHER QUESTIONS ABOUT DETERMINING WHEN PUBLIC REASON HAS BEEN VIOLATED The preceding part raised some critical questions about identifying a criterion for determining whether, in any particular concrete situation, Rawlsian public reason is being properly actualized by the deliberations and decision-making of the citizens (the "actualization criterion"). In this part, I intend to raise some further and related questions. My ultimate aim is not to reject Rawlsian public reason as such, but rather to suggest a different way of understanding it, a way that may render Rawlsian public reason more plausible and defensible, even in light of the critical questions being raised here. 34. Id. 35. See id. at 2038-43. 36. This critical conclusion about the "actualization criterion" roughly echoes Onora O'Neill's observation that Rawlsian constructivism in general seems uncomfortably trapped 4[b]etween realism and relativism." See Onora O'Neill, Constructions of Reason: Explorations of Kant's Practical Philosophy 218 (1989). O'Neill writes: "Rawlsian constructivism has ended up on an uncomfortable knife edge, and teeters between idealizing and relativized conceptions of ethics. The idealized readings demand proofs of a moral reality Rawls does not discern; the relativized readings can only offer an internal critique of the justice of [existing] modern liberal societies." Id. 2162 [Vol. 72 ONACTUALIZING PUBLIC REASON The two questions with which I would like to begin in this part are related to the questions already raised about the actualization criterion. The two inter-related questions are as follows: 1) If the Rawlsian actualization criterion is indeed problematic for the reasons discussed above, then on what grounds-if ever-may one legitimately complain that another citizen's attempts to persuade or influence others within a public forum (and on matters pertaining to constitutional essentials or basic justice) are unreasonable or exceed the limits of public reason, if that citizen (let us call him "George" henceforth) subjectively and sincerely believes that he is being reasonable and is operating within the limits prescribed by Rawlsian public reason? 2) Conversely, if the Rawlsian actualization criterion is problematic for the reasons discussed above, then on what grounds-if ever-may one legitimately complain that another citizen's refusal to be open to persuasion within a public forum (and on matters pertaining to constitutional essentials or basic justice) is unreasonable or violates the duty of civility, if that citizen (let us call her "Judith" henceforth) subjectively and sincerely believes that she is being open and honoring the duty of civility, as prescribed by Rawlsian public reason? The questions sketched above help to highlight one of the central difficulties of Rawlsian public reason: Just as two or more citizeninterlocutors can debate and disagree on constitutional essentials and matters of basic justice, so too can they debate and disagree on the question of whether any particular citizen (either oneself or another) is genuinely living up to the requirements of public reason. In other words, citizen-interlocutors can debate and disagree not only about fundamental political matters, but also about the very terms of their debating and disagreeing (qua citizens). More specifically, they can debate and disagree on whether the terms of discourse presupposed by one or another citizen-interlocutor do or do not satisfy the requirements of public reason. Now, it is obvious that the question of whether a particular citizen's discursive practices do or do not satisfy the requirements of public reason cannot be settled simply on the basis of whether a certain critical mass of other citizens happens to agree or disagree with that citizen (either on basic political matters or on the meaning of "public reason" itself). For the notion of "public reason"' is a normative notion: Instead of simply mirroring actual agreements or disagreements among citizens, the notion of public reason is supposed to specify normatively significant guidelines for determining how agreement and/or disagreement among citizens (qua citizens) ought to take place to begin with. The issue is complicated by the fact that an individual's own understanding of public reason (and in turn, his or her sincere belief that he or she is satisfying the requirements of public reason) may depend indirectly on some portion of his or her comprehensive view. 2004] 2163 FORDHAM LAW REVIEW Consider the following analysis involving George and Judith. George's sincere, subjective beliefs about whether he is being reasonable and is operating within the limits prescribed by public reason seem to depend in part on his views regarding whether he really can reasonably expect others reasonably to endorse the positions that he himself holds on constitutional essentials and matters of basic justice (let us refer to these views as his views on the "reasonable endorsability" of his basic political positions). In turn, George's sincere, subjective beliefs about the reasonable endorsability of his basic political positions seem to depend in part on his views regarding the nature and general accessibility of the evidence that might lead another person to accept George's basic political positions (let us refer to these views as his views on the "evidentiary grounds" of his basic political positions). Finally, George's sincere, subjective beliefs about the evidentiary grounds of his basic political positions seem to depend in part on his beliefs about matters such as the nature and capacity of human reason, the possible objectivity or "knowability" of moral truth, the normative significance and value of certain facts of nature, and so forth. But beliefs about these matters are philosophical, epistemological, and/or axiological in nature, and would be placed by Rawls "outside" the scope of public reason, insofar as they belong to a (fully or partially) comprehensive doctrine.37 To summarize this analysis: George's sincere, subjective beliefs about whether he is being reasonable and operating within the limits prescribed by public reason seem to depend on his views regarding "reasonable endorsability," which seem to depend in turn on his views regarding "evidentiary grounds," which seem to depend in turn on his comprehensive view.38 Thus George's comprehensive view seems (at least in part) to determine his sincere, subjective beliefs about whether he is or is not being reasonable and operating within the limits prescribed by public reason when he attempts to persuade 37. See Rawls, Political Liberalism, supra note 1. at 175. 38. Against Rawls, it is tempting to argue along the following lines: "If a person happens to believe some proposition (let's call it X), then it is obvious that he also believes in the reasonable endorsability of X (since he clearly considers himself to be reasonable in endorsing X if he did not consider himself reasonable in doing so, then he would cease to endorse X!). And if the person believes in the reasonable endorsability of X, then he must also believe that it is reasonable for him to try to convince others of X." My claims here have not been based on this sort of argument, and my notion of "reasonable endorsability" here pertains to the question of what can be reasonably endorsed, not just by myself or by some generic, imaginary substitute for myself, but by persons who may actually be quite different from myself. It seems to me that the sort of argument given above does not make an adequate case against Rawls. After all, a person may very well consider himself to be reasonable in endorsing X, even though he might also hold (for any number of other reasons, e.g., for reasons connected to his views on the "evidentiary grounds" of his belief in X) that he would be unreasonable if he tried to persuade others of X. [Vol. 722164 ONACTUALIZING PUBLIC REASON others (on matters pertaining to constitutional essentials or basic justice) within a public forum.39 This sort of analysis has led some thinkers to conclude that Rawls's notion of public reason is ultimately untenable and must be rejected in favor of a position that more overtly allows-or perhaps even encourages-comprehensive doctrines to play a determining role in public deliberation and decision making on constitutional essentials and matters of basic justice.40 But for now at least, I would like to steer this analysis in a slightly different direction. And in order to do this, I would like to begin by considering matters from the perspective of Judith as well. According to a corollary requirement of public reason, Judith has a duty of civility, a duty to listen and be open to George's attempts at persuasion, provided that George himself is being reasonable and observing the limits prescribed by public reason. Now Judith's sincere, subjective beliefs about whether or not she is being genuinely open and honoring the duty of civility-even if it turns out that she is not persuaded by George-seem to depend in part on her own views regarding the reasonable endorsability of George's basic political positions (i.e., his positions on constitutional essentials and matters of basic justice). In turn, Judith's sincere, subjective beliefs about the reasonable endorsability of George's basic political positions seem to depend in part on her views regarding the "evidentiary grounds" of George's basic political positions. And finally, Judith's sincere, subjective beliefs about the evidentiary grounds of George's basic political positions seem to depend in part on her own (fully or partially) comprehensive (i.e., philosophical, epistemological, and/or axiological) views. Thus we see that Judith's position on the question of whether she is being reasonable and honoring the duty of civility is the mirror image of George's position on the question of whether he is being reasonable and operating within the limits prescribed by public reason. Each person's answer to the question at hand (regarding whether he or she is himself or herself living up to the requirements of public reason) seemingly depends (at least in part) on his or her own comprehensive view. But there is an additional symmetry here: Judith's beliefs about whether George is being reasonable and honoring the requirements of public reason seem to depend (at least in part) on her own comprehensive view; and George's beliefs about whether Judith is being reasonable and honoring the requirements of public reason seem to depend at least in part on his own 39. In a somewhat similar vein, Kent Greenawalt has argued that "comprehensive views can influence someone's sense of the application of fundamental values"; thus comprehensive views can influence a person's ideas regarding which sorts of issues actually do (or don't) belong to what Rawls calls "constitutional essentials." Kent Greenawalt, Private Consciences and Public Reasons 117 (1995). 40. See, e.g., Robert George, In Defense of Natural Law 196-227 (1999); see also John Finnis, Abortion, Natural Law, and Public Reason, in Natural Law and Public Reason 75 (Robert P. George & Christopher Wolfe eds., 2000). 2004] 2165 FORDHAM LAW REVIEW comprehensive view. In short, George's judgments about whether public reason is properly being actualized by himself and/or by Judith apparently depend (at least in part) on his own comprehensive views; and Judith's judgments about whether public reason is being properly actualized by herself and/or by George apparently depend (at least in part) on her own comprehensive view. Now let us suppose that George and Judith not only disagree about some fundamental political issue (e.g., abortion), but also disagree about the nature, scope, and limits of public reason. What ought to happen if George and Judith start to suspect that their disagreement about the very terms of their disagreement (their disagreement about public reason itself) depends on the different and even opposing comprehensive views that they each hold? It is clear that the two cannot simply "agree to disagree," since that would reduce their relationship to a mere modus vivendi, which-as we have already seen-is excluded by the Rawlsian account of public reason as sustained by an overlapping consensus of reasonable comprehensive views. But it also seems that the two cannot simply agree (e.g., through a common moral commitment to ongoing civil discourse) that certain topics defy reasonable agreement and thus should be excluded from what counts as "public reason" for them. For the disagreement in which the two are already engaged is itself precisely about the nature, scope, and limits of "public reason"! Let us further suppose that in his further attempts to persuade Judith, George begins to "dig deeper" into his repertoire of comprehensive views, hoping to give a more complete and compelling account of his positions, and hoping to demonstrate the sincerity of his commitment to these positions. In the midst of his "deeper digging," George remains convinced that he is successfully translating the lessons of his comprehensive views into properly political reasons; and so George remains convinced that he has satisfied "the proviso" requirement of Rawlsian public reason.41 Now, at what point in George's ongoing attempts can Judith legitimately complain that George is not being reasonable and is overstepping the limits prescribed by public reason? And conversely, at what point in this interchange can George legitimately complain that Judith-in spite of George's sincere attempts to satisfy the proviso-is not being openminded and thus is not honoring the duty of civility required by public reason? Before going further, it will be helpful to reinforce the point that George and Judith are to be understood as ideal types. They represent the situation of any citizen insofar as he or she-qua 41. Rawls. The Idea of Public Reason Revisited, supra note 20, at 591-94. For more on Rawls's "inclusive view" of public reason (according to which citizens may sometimes introduce their own comprehensive views into public reason), see Rawls, Political Liberalism, slcpra note 1, at 247-54. 2166 [Vol. 72 ONACTUALIZING PUBLIC REASON citizen-seeks to live up to the requirements of public reason while attempting (like George) to persuade others, or while being (like Judith) at the receiving end of such attempts at persuasion. George and Judith should not be understood individualistically, but may refer to entire groups of citizens who are bound together insofar as they are participants (speakers or listeners) in a broad public conversation about constitutional essentials and matters of basic justice, and perhaps even about public reason itself. Finally, this broad public conversation should not even be understood as if it were unidirectional; for as we all know from our own experience, it is quite possible to be a George and a Judith at once, even in the midst of a single conversation. In applying these ideal types, we have seen that a major difficulty arises in connection with George's ongoing attempts to persuade Judith. George may continue trying to persuade Judith about his basic political positions; he may also continue trying to persuade her about the limits and scope of public reason itself (about the very terms of their discursive practice). Indeed, George's very commitment to public reason may lead him quite reasonably to hold that he may not legitimately give up on his attempts to persuade Judith regarding the proper meaning of public reason; for-he may reasonably believe-to renounce the imperative to reach agreement on the meaning of public reason itself is to allow his relationship with Judith to devolve into a mere modus vivendi whereby the two simply "agree to disagree" for external, utilitarian reasons. But if George and Judith already disagree about the limits and scope of public reason, then is it not the case that George's ongoing attempts at persuasion (especially if they involve any "deeper digging" into his own comprehensive views) will appear to Judith as themselves a violation of public reason (the very opposite of what George takes them to be)? At what point can Judith legitimately complain that George is overstepping the limits of public reason; at what point can George legitimately complain that Judith is not genuinely open and is thus violating her duty of civility? It is obvious that there must be at least some point at which either George or Judith may legitimately complain that the other is not living up to what is required by public reason. For if Judith could not legitimately complain that George is exceeding the proper limits of public reason, or if George could not legitimately complain that Judith is not honoring the duty of civility (e.g., if it were Judith's unreasonable close-mindedness that led her to accuse George of exceeding the limits of public reason), then public reason would be devoid of all meaningful normative content for George and Judith. In order to have meaningful normative content for them, the notion of public reason (and its corollary principles) must disallow both George and Judith from acting in certain ways. Furthermore, if the notion of public reason is to have any meaningful normative content for them, 2004] 2167 FORDHAM LAW REVIEW then it must be possible for George and Judith to disagree on which of the two of them is failing to live up to what is required by the ideal of public reason and yet nevertheless-in the midst of such disagreement-for one of them to be correct. For if it were not possible for them to disagree on which of the two of them is failing to live up the requirements of public reason, then the need to appeal to the notion of public reason (in order to settle such a disagreement) would never arise for them in the first place; in that case, the notion of public reason would be normatively meaningless, superfluous, and useless for them. Furthermore, if it were not possible for one of them to be correct in this disagreement, then-once again-the notion of public reason would be devoid of meaningful normative content for them; for if it were not possible for one of them to be correct in saying that the other has violated the requirements of public reason (i.e., if it were not possible for one of them actually to have violated the requirements of public reason), then once again the notion of public reason would not disallow them from acting in certain ways, in which case it would be normatively vacuous for them. IV. SELF-REFERENTIALITY, RECIPROCITY, AND How ONE CAN KNOW ON IMMANENT GROUNDS THAT PUBLIC REASON HAS BEEN VIOLATED In the preceding part we began to see that the notion of public reason can involve a certain kind of self-referentiality: citizeninterlocutors may debate and disagree not only about constitutional essentials and matters of basic justice, but also about the very terms of their debate and disagreement. That is, in the midst of their attempts to discuss fundamental political issues within the limits prescribed by public reason, they can also debate and disagree about the very nature, scope, and limits of public reason itself. But public reason can also involve a further, related kind of self-referentiality: If citizens (represented here by George and Judith) happen to disagree about the nature, scope, and limits of public reason, then one citizen's sincere commitment to living up to the requirements of public reason may lead him or her to engage in discursive behavior that, from the perspective of another citizen, will count as evidence that he or she is precisely nlot committed to public reason but indeed acting in violation of it. For example, if George and Judith disagree fundamentally about the nature and requirements of public reason, then George's subjectively sincere commitment to public reason and his ongoing attempts to prevent his relationship with Judith from devolving into a mere modus vivendi (i.e., his ongoing attempts to persuade Judith, either about political issues or about public reason itself) may appear to Judith as evidence that George is being unreasonable and precisely not living up to the requirements of public reason. Conversely, Judith's subjectively sincere commitment to public reason and her [Vol. 722168 ON ACTUALIZING PUBLIC REASON insistence that George should be disallowed from making certain kinds of arguments in a public forum may appear to George as evidence that Judith is being uncivil and precisely not living up to the requirements of public reason. But there is a further issue here, connected to the traditional problem of ideology or the apparently self-validating character of the views or stances that a citizen adopts: A citizen's existing stance will incline him or her to regard new evidence (including evidence presented by sincere others that could potentially disconfirm one's existing stance) as evidence that confirms the rightness of the stance that one has taken. So in the context of their ongoing disagreement, George's existing stance regarding public reason will incline him to regard Judith's attempts to limit his public arguments as evidence of her incivility, and thus as confirming his view that he is justified in dismissing her protestations as unreasonable. Conversely, Judith's existing stance regarding public reason will incline her to regard George's ongoing attempts at persuading her as evidence that he is unreasonable, and thus confirming her view that she is justified in dismissing his ongoing attempts at persuasion as unreasonable. Or to state the matter the other way around: Judith's discursive behavior will tend to confirm for George the rightness of his own position on public reason, but only because George himself has already taken the particular stance that he has; and George's discursive behavior will tend to confirm for Judith the rightness of her own view on public reason, but only because Judith herself has already taken the particular stance that she has.42 Is there a way to give a fair and nonprejudicial answer to the question of which of the two is failing to live up to the requirements of public reason, given the apparently selfvalidating character of the stances adopted by George and Judith (and the citizens represented by them)? 42. With these observations about the seemingly self-validating character of the particular stances that one takes, we are-interestingly enough-echoing Immanuel Kant. For Kant showed in his Critique of Pure Reason that the difficulties to which human reason falls prey will continue to seem irresolvable only if one persists in adhering to the pre-critical stance or view that human reason is passive in relation to the things it knows. For Kant, the fact that human reason continually falls into such difficulties leads reason to think that it is confirmed in its view that it is, indeed, the passive "victim" of mysterious forces and causes beyond itself; but reason regards such difficulties as confirmatory evidence only because of the particular stance or view that it has already taken with regard to itself. As Kant argues, if one were to take an altogether different stance and initiate a shift in one's thinking (i.e., if one were to perform a "Copernican revolution"), then the seemingly irresolvable difficulties and thus the seemingly confirmatory evidence about reason's passivity would altogether disappear. For Kant, things appear as they do to the pre-critical mnind, precisely because of the stance that the pre-critical mind takes in relation to things; if one were to adopt a different (critical) stance regarding things, then things would also appear differently to the mind. Kant, supra note 4, at 106-24. 21692004] FORDHAM LAW REVIEW We saw in the preceding part that if public reason is to have any normatively significant content for George and Judith there must be some point at which Judith could legitimately complain that George is exceeding the proper limits of public reason, and some point at which George could legitimately complain that Judith is not honoring the duty of civility, as required by public reason. Now in the light of the issues just addressed, it is tempting to think that the disagreement between George and Judith (on the question of who is in violation of public reason) would have to be settled by reference to some standard or criterion external to their own interchange on the issue. For example, it is tempting to think that the disagreement has to be settled by reference to some stock of existing beliefs or opinions that can "externally" or "objectively" determine whose view and which person (i.e., George or Judith) should be regarded as "reasonable" (and "unreasonable") in the interchange. But there are problems with any such appeal to a stock of existing beliefs or opinions. First of all, if such appeal were made, then the person eventually judged to be unreasonable and in violation of public reason would be quite likely to reject that judgment (and reject the alleged normativity of the existing beliefs appealed to), just as he or she has already rejected the judgment of his or her opposing citizeninterlocutor. In other words, there is no reason to believe that George (or Judith) would willingly accept the judgment that he or she is being unreasonable, if that judgment is made on the basis of a stock of existing beliefs which themselves happen to contradict the beliefs held by George (or Judith). However, there is a further problem: On what basis is some set of existing beliefs to be taken as normative for determining who or what counts as "reasonable" in the context of a disagreement about public reason? It seems that such existing beliefs could be taken as normative for two possible reasons: either because these existing beliefs happen to be held by an identifiable majority of other persons (beyond George and Judith themselves), or else because these beliefs are deemed to be uncontentiously "true" in some sense. But as we have already seen, the nature, scope, and limits of Rawlsian public reason cannot be determined simply on the basis of "majority opinion" or on the basis of "truth." Of course, one could respond here that the nature, scope, and limits of public reason may indeed be determined by appealing to the majority opinion, if this majority opinion is the opinion of reasonable people (and not just any people). Unfortunately, this response does not ultimately resolve the issue at hand, but only pushes it back one step further. For the very thing at issue is the question of who counts as a "reasonable person'" (within the context of public reason), and-in the absence of any further qualifying conditions-it is normatively vacuous to say that the opinions of a majority of "reasonable" people can determine who (in the disagreement between George and Judith) is being "reasonable." 2170 [Vol. 72 ONACTUALIZING PUBLIC REASON In the remainder of this Essay, I would like to suggest a way in which it may be possible to think meaningfully about the requirements imposed by public reason, but without appealing (problematically or illicitly) to a stock of existing beliefs which are supposed to serve (because they are either "true" or held by a majority of people) as an external standard for determining who is actually correct when there is disagreement about who is or is not acting in violation of public reason. In other words, I would now like to suggest that a meaningful sense of public reason can be derived from what is immanent to the very interchange and disagreement between George and Judith themselves (and by implication, between any citizens represented by them). If one reflects further on the implications of the self-referentiality that (as we have begun to see) characterizes public reason, then an important inference can be drawn: If there is actual disagreement between George and Judith on what is demanded of them by public reason, and if each claims that the other has violated the requirements of public reason, then-we can infer-the requirements of public reason really have been violated by at least one of them. In other words, if each of them claims that the other has acted in violation of public reason, then at least one of them is correct and the other really has violated the requirements of public reason. (Of course, both can be correct about the other's having violated the requirements of public reason, but they cannot both be correct about some particular alleged violation about which they disagree.) The important point here is that it is possible to know on immanent grounds alone (i.e., simply through the fact that George and Judith disagree about which of them has violated public reason, and with no necessary reliance on an external standard such as "majority opinion" or "truth") that there has indeed been a violation of public reason. The demonstration is as follows. If Judith accuses George of having violated public reason with respect to a particular discursive practice of his (e.g., she objects to his ongoing attempts to persuade her, since she believes that his arguments appeal to reasons that should be disallowed from the public sphere), then she is either right or wrong in that accusation. If she is right, then there has indeed been a violation of public reason, since (as she correctly claims) George has violated public reason. If, on the other hand, she is wrong in making that accusation, then her own discursive practice (i.e., her very own accusation) itself constitutes a violation of public reason, in which case-once again-there has indeed been a violation of public reason. This is because her accusation entails the claim that George should be disallowed from making certain arguments in the public sphere. But if George insists that his arguments do not violate public reason, and if he is correct in so insisting, then Judith's claim that George should be disallowed from making such arguments in the public sphere is itself ----- ------ - ---21712004] FORDHAM LAWREVIEW unfair and a violation of the duty of civility, and thus it is contrary to the requirements of public reason (and reciprocally, George is correct to claim that it is Judith who has violated public reason). In short, if George and Judith disagree on which of the two of them is in violation of public reason, then Judith's accusation about George entails that there has actually been a violation of public reason (if not by George, then by Judith herself). Since the stances taken by George and Judith are the mirror images of one another (each claims that the other is in violation of public reason), a similar analysis emerges if one focuses on George's claims about Judith. If George and Judith disagree on which of the two of them is in violation of public reason, then the very fact that George accuses Judith of violating public reason (and Judith disagrees and regards the accusation as unfair) entails that there actually has been a violation of public reason (if not by Judith, then by George himself).43 Notice that-whether applied to George or to Judith-this analysis works because the concept of public reason is an intrinsically "intersubjective" or "reciprocal" concept. The concept simply has no normative significance if there are not two or more citizens who are able to disagree about the requirements of public reason and thus disagree about who is or is not living up to the requirements of public reason. Furthermore, as soon as there is any actual disagreement between citizens regarding who is or is not in violation of public reason, the very concept of public reason acquires a kind of selfreferentiality. When such disagreement exists, the concept takes on a self-referential character because any actual claim that some other citizen has acted in violation of public reason is by its very nature a discursive practice that either (a) accurately describes some actual violation of public reason by another citizen, or (b) itself constitutes a violation of public reason. Because of the intersubjectivity and selfreferentiality built into the very idea of public reason, it follows that if there is actual disagreement between citizens about what is required of them under public reason and if one citizen accuses another of violating public reason (in which case the accused citizen would claim that he or she is being unfairly accused), then-as a matter of factpublic reason has actually been violated. Stated differently, there is 43. In this analysis, I do not consider the possibility that George and Judith-in spite of their reciprocal charges against one another-are both completely misinformed about the nature and requirements of public reason. I do not consider this possibility here, because there is no need to do so. As I argued earlier, if the notion of public reason is to have any meaningful normative content for George and Judith, then it must be possible for them to disagree on which of the two of them is in violation of public reason and yet nevertheless-in the midst of such disagreementfor one of them to be correct. See supra Part III. But if one starts with the hypothesis that George and Judith are both completely misinformed about public reason, thencontrary to a necessary condition of our analysis here-it is no longer possible for one of them to be correct in the midst of their disagreement. 2172 [Vol. 72 ONACTUALIZING PUBLIC REASON something self-confirming or self-validating about any accusation within the context of actual disagreement about public reason itself that some other citizen has violated the requirements of public reason.4 4 The foregoing observations show that it is possible to claim meaningfully and correctly that there has been a violation of public reason, without having to justify this claim by showing that the violator's discursive practices contradict norms that have been gleaned from some stock of existing beliefs. In other words, the meaningfulness of public reason does not have to be determined by reference to some set of existing beliefs which somehow establish the terms of what counts as reasonable discourse (either because such beliefs are true or because they are held by a majority of people). Rather, the meaningfulness of public reason-and the possibility of its being violated-can be grounded in the simple fact of disagreement between citizens about the nature and requirements of public reason itself. Thus, the foregoing observations about public reason may help open the way towards an understanding of Rawlsian public reason that is not tied to any set of existing beliefs at all and that as a result is more genuinely sensitive to the fact of radical pluralism and disagreement. But there is a further, perhaps more extraordinary implication to the preceding analysis. This analysis shows that in the midst of actual disagreement about public reason, it is possible for a citizen (e.g., George, Judith, or anyone represented by them) to be sure that there has indeed been a violation of public reason, yet without being sure that one's own discursive practice was not itself the cause of the violation. In other words, it is possible for a citizen to know that a violation of public reason has actually occurred, but without being able to foreclose the possibility that he (or she) himself (or herself) was not the violator. In turn, this result shows just how the ideal of public reason can remain a truly demanding imperative for every citizen, even in the midst of that citizen's being absolutely certain that a violation of public reason has occurred, and being firmly convinced that the violator was someone else. For if in the midst of actual disagreement a citizen could be certain not only that public reason had been violated but also that he or she was not the violator, then public reason-in this particular instance at least-would no longer specify a genuine imperative for that citizen. After all, this citizen would have already satisfied the imperative of public reason in this 44. In a similar vein, Michael Dummett observes that there is something selfconfirming or self-validating about the belief that evil exists: if one believes in the existence of evil when in fact there is no evil, then this very belief is an illusion, in which case there is evil. Michael Dummett, A Defence of McTaggart's Proof of the Unreality of Time, 69 Phil. Rev. 497 (1960), reprinted in Truth and Other Enigmas 356 (1978). 21732004] FORDHAM LAW REVIEW particular instance and the only further action required of him or her would be to encourage or demand that some other person or persons stop acting in violation of public reason. But if the spirit of Rawlsian public reason teaches us anything, it teaches us that we should think of public reason as imposing a constant and continuouis imperative on us, an imperative that calls for ongoing openness, even (and perhaps especially) in the midst of our certain knowledge that a violation of public reason has taken place and our firm conviction that we ourselves were not the violator. 45 What is meant by public reason, then, may not be fixed or determinate at all, but may be perpetually "in the making."4 6 Thus it is perhaps best to think of public reason, not in terms of any alreadyexisting set of beliefs or states of affairs, but simply as a task, an ideal, an imperative. Conversely, it is probably altogether wrong-or at least highly misleading-for Rawls to say (as he sometimes does) that the normative content of public reason must be determined in advance by reference to existing beliefs or "truths now widely accepted."47 It should be clear by now that the interpretation suggested here would greatly expand the scope of public reason. But it would not do so by claiming that the stock of existing beliefs upon which public reason is grounded must be broadened so as to include beliefs that Rawls and Rawlsians have typically excluded. Rather, my interpretation would expand the scope of public reason by regarding 45. In some respects, then, the imperative imposed by the ideal of public reason is similar to Kant's notion of a categorical imperative. For as Kant makes clear, it is possible for me to know with assurance that the categorical imperative has been violated in a particular instance, but never possible for me to know with assurance that I myself have successfully lived up to what the categorical imperative requires of me. See Immanuel Kant, Grounding for the Metaphysics of Morals (1785), reprinted in Immanuel Kant: Ethical Philosophy 19 (James Ellington trans., 1983). 46. Rawls, The Idea of Public Reason Revisited, supra note 20, at 582 ("Political liberalism, then, does not try to fix public reason once and for all ..... 47. Rawls, Political Liberalism, supra note 1, at 225. As we have said, on matters of constitutional essentials and basic justice, the basic structure and its public policies are to be justifiable to all citizens, as the principle of political legitimacy requires. We add to this that in making these justifications we are to appeal only to presently accepted general beliefs and forms of reasoning found in common sense, and the methods and conclusions from science when these are not controversial. The liberal principle of legitimacy makes this the most appropriate, if not the only, way to specify the guidelines of public inquiry.... This means that in discussing constitutional essentials and matters of basic justice we are not to appeal to comprehensive religious and philosophical doctrines... nor to elaborate economic theories of general equilibrium, say, if these are in dispute. As far as possible, the knowledge and ways of reasoning that ground our affirming the principles of justice and their application to constitutional essentials and basic justice are to rest on the plain truths now widely accepted, or available, to citizens generally. Id. at 224-25 (emphases added). 2174 [Vol. 72 ONACTUALIZING PUBLIC REASON public reason as an essentially practical idea or imperative, and thus by asserting the radical primacy of the practical.48 In so doing, my interpretation would altogether deny that the scope of public reason is to be established by reference to any antecedently determinable, theoretically ascertainable, or empirically observable state of affairs or set of existing beliefs. Following Aristotle-and following one of Rawls's own observations4 9 -my suggestion is that public reason is simply the imperative or ideal that properly governs the practical activity known as "civic friendship." And as Aristotle indicates, civic friendship consists not in any already-established state of affairs, but in an activity; and ideally it is the activity of ruling and being ruled in turn50 whereby no individual knows with assurance and in advance of actual enquiry and debate whether his or her own view should serve as the "measure or rule" for others or rather should be modified to conform to the "rule or measure" offered by others. In conclusion, I would like to return to the discussion with which I began, namely Kant's discussion in the Critique of Pure Reason regarding "The Discipline of Reason With Regard to Its Polemical Use" (and it is this discussion to which Rawls refers when he first introduces his account of public reason in Political Liberalism)."1 In this discussion, Kant argues that reason is essentially dialogical, and as a result "it is quite absurd to expect enlightenment from reason and yet to prescribe to it in advance on which side it must come out."52 If there is a lesson to be extracted from this Kantian passage, then perhaps it is that Rawlsian public reason-like Kantian reason-is not tied to any stock of existing beliefs, but is essentially an ideal, an imperative, a task that cannot be pre-determined in advance of its own self-unfolding. And in one of his more powerful statements on the matter, Rawls seems to agree: "As an ideal conception of citizenship for a constitutional democratic regime, [public reason] presents how things might be, taking people as a just and well-ordered society would encourage them to be. It describes what is possible and can be, yet may never be, though no less fundamental for that."53 48. Elsewhere I have offered a reading of Rawlsian contractualism that similarly asserts the radical primacy of the practical. Michael Baur, Reversing Rawls: Criteriology, Contractualism, and the Primacy of the Practical, in Phil. & Soc. Criticism, May 2002, at 251-96. 49. Rawls, Collected Papers, supra note 20, at 579. 50. Aristotle, The Politics 92 (Carnes Lord trans., 1984) (n.p., n.d.). 51. See supra notes 1-6 and accompanying text. 52. Kant, supra note 4, at 647. 53. Rawls, Political Liberalism, supra note 1, at 213. 21752004] COPYRIGHT INFORMATION TITLE: On actualizing public reason SOURCE: Fordham Law Rev 72 no5 Ap 2004 WN: 0409403192034 The magazine publisher is the copyright holder of this article and it is reproduced with permission. Further reproduction of this article in violation of the copyright is prohibited. To contact the publisher: http://law.fordham.edu/publications/index.ihtml?pubid=500 Copyright 1982-2004 The H.W. Wilson Company. All rights reserved.

Akademie Verlag Wirklich. Wirklichkeit. Wirklichkeiten Nietzsche über „wahre" und „scheinbare" Welten Herausgegeben von Renate Reschke ISBN: 978-3-05-005742-2 Nietzscheforschung Jahrbuch Band 20 der Nietzsche-Gesellschaft Sonderdruck aus: Inhaltsverzeichnis Siglenverzeichnis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 I. Der Nietzsche-Preis Ludger Lütkehaus Laudatio auf den Preisträger . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Andreas Urs Sommer Philosophie als Wagnis. Festrede aus Anlass der Verleihung des Friedrich-Nietzsche-Preises des Landes Sachsen-Anhalt am 13. Oktober 2012 . . . 19 II. Wirklich. Wirklichkeit. Wirklichkeiten. Friedrich Nietzsches „wahre" und„scheinbare" Welten Peter Peinzger Interpretation und Machtwillen. Nietzsches Denkwirklichkeiten als fiktive Welten . 31 Damir Barbarić Glück des Kreises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Pietro Gori Nietzsche on Truth: a Pragmatic View? . . . . . . . . . . . . . . . . . . . . . . 71 Jutta Georg Existenzielle Bewegung: Scheinwelten, Rauschwelten, Tanzwelten . . . . . . . . 91 Carlo Gentili Kant, Nietzsche und die ‚Philosophie des Als-Ob' . . . . . . . . . . . . . . . . . 103 Pirmin Stekeler-Weithofer Wirklichkeit als bewertete Möglichkeit Zum Problem allgemeiner Wahrheiten und möglicher Welten . . . . . . . . . . . 117 Alexander Aichele An sich kein Ding: Nietzsches Wirklichkeiten . . . . . . . . . . . . . . . . . . . 139 6 Inhaltsverzeichnis Christian Niemeyer Wie wirklich ist Der Wille zur Macht? Über die Frage, warum sich Nietzsche fünf Jahre an einem Text versuchte, der primär rhetorisch gemeint war . . . . . . . . . . . . . . . . . . . . . . . . . 163 Helmut Heit „... was man ist"? Zur Wirklichkeit des Subjekts bei Nietzsche . . . . . . . . . . 173 Axel Pichler ‚Den Irrtum erzählen' Eine Lektüre von „Wie die ‚wahre Welt' endlich zur Fabel wurde . . . . . . . . . 193 Friedrich Kittler (†) Wie man abschafft, wovon man spricht: Der Autor von Ecce homo . . . . . . . . . 211 III. Nietzsches Philosophie des Geistes. 20. Nietzsche Werkstatt Schulpforta, 11.–14. September 2012 Wiss. Leitung: Marco Brusotti und Helmut Heit Marco Brusotti, Helmut Heit Nietzsches Philosophie des Geistes. Einleitung . . . . . . . . . . . . . . . . . . 231 Manuel Dries The Feeling of Doing – Nietzsche on Agent Causation . . . . . . . . . . . . . . . 235 Luca Lupo „Ein phantastischer Commentar über einen ungewussten Text". Zu einem Bild des Bewusstseins in Morgenröthe . . . . . . . 249 Claudia Rosciglione Conscious and Uncoscious Mental States in Nietzsche's Philosophy of Mind . . . 259 Selena Pastorino Die Rolle der Interpretation bei psycho-physiologischen Prozessen im Nachlass (1885–1886) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Christoph Schuringa Nietzsche and the Unfolding of Mind . . . . . . . . . . . . . . . . . . . . . . . 279 Thomas Land Zur Autonomie des Geistes in Nietzsches Philosophie . . . . . . . . . . . . . . . 289 Sarah Bianchi „Wahrhaft gerecht" urteilen. Zu den Dimensionen einer ‚sinnsetzenden Anerkennung' in Nietzsches zweiter Unzeitgemässen Betrachtung . . . . . . . . . 301 7Inhaltsverzeichnis IV. Beiträge Anatoly Livry Mandelstam, un dionysiaque nietzschéen . . . . . . . . . . . . . . . . . . . . . 313 Detlef Thiel Die Magie des Extrems und die Magie der Mitte. Nietzsche im Urteil Salomo Friedlaender/Mynonas . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Walter Martin Rehahn „Der bewunderungswürdige Logiker des Christenthums" Friedrich Nietzsche und Blaise Pascal . . . . . . . . . . . . . . . . . . . . . . . 343 Michael Skowron Dionysischer Pantheismus. Nietzsches Lenzer Heide-Text über den europäischen Nihilismus und die ewige Wiederkehr/-kunft . . . . . . . . . . . . . 355 Jonas Holst Ethik der Freundschaft. Über eine nachgelassene Idee im Werk Friedrich Nietzsches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 V. Rezensionen Im Wirbel der Metaphysik – und kein Ende in Sicht. Rezension zu: Damir Barbarić, Im Angesicht des Unendlichen (Sören Reuter) . . . . . . . . . . 393 Figuren im Spiel der Philosophie. Rezension zu: Peter Peinzger, Masken, Joker, Parasiten (Knut Ebeling) . . . . . . . . . . . . . . 397 Ein bildmächtiger Gang durch Nietzsches Leben. Rezension zu: Michel Onfray, Maximilien Le Roy, Nietzsche (Ralf Eichberg) . . . . . . . . . . . 400 Synthese aus Kant und Clown. Rezension zu: Salomo Friedlaender (Mynona), Friedrich Nietzsche – eine intellektuale Biographie (Steffen Dietzsch) . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 Wie erzeugt man Nietzsche-Kitsch? Rezension zu: Ludger Lütkehaus, Die Heimholung, Nietzsches Jahre im Wahn (Klaus Goch) . . . 407 Mehr Tod geht immer. Rezension zu: Katarina Botsky, In den Finsternissen (Renate Reschke) . . . . . . . . . . . . . . 412 Personenverzeichnis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 Autorenverzeichnis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 Pietro Gori Nietzsche on Truth: a Pragmatic View? I. Nietzsche's utilitarian view of truth The critique to the notion of truth is one of the most important (and most discussed) topics of Nietzsche's thought. This critique sustains his philosophical views, and with them it changes, from the unpublished writing on Truth and Lie in an Extra-moral Sense to the late observations concerning the superfluousness of the ‚true world' in Twilight of the Idols. During the years, Nietzsche's concerning with truth left the pure theoretical plane, and started involving the moral one, that of human agency. In so doing, Nietzsche shows the philosophical relevance of his critique, i. e. its dealing with the core of the Metaphysical worldview, of the Western (both Platonic and Christian) thought.1 Since the notion of truth is strictly related to those of good and ‚full of value' (NL 14[103], KSA 13, 280), the awareness of its inner lack of content, and consequently the rejection of its absolute value as reference point for the human existence, leads to the well known disorientation of the human being about which the madman warns people in FW 125. Nietzsche's dealing with truth has been stressed by many scholars during the last decades2, and I believe that there is no need of treating once more the basic questions concerning that topic. My aim in this paper is rather to stress both one fundamental character of Nietzsche's theory of knowledge, in order to show its compliance with some 19th and 20th century philosophical views, and the role that his epistemology played in Nietzsche's late thought. The topic with which I basically deal in this paper is Nietzsche's stating a mere relative character of truth. This idea follows from Nietzsche's fundamental view according to which human knowledge is interpretation. This is a well known feature of Nietzsche's thought, a topic that many scholar stressed for its philosophical relevance.3 There is no 1 See Werner Stegmaier, Nietzsches Neubestimmug der Wahrheit, in: Nietzsche-Studien, 14 (1985), 69–95. 2 Among the others, two fundamental studies on this topic are Ruediger H. Grimm, Nietzsche's Theory of Knowledge, Berlin, New York 1977 and Maudemarie Clark, Nietzsche on Truth and Philosophy, Cambridge 1990. 3 See e. g. Günter Abel, Nietzsche. Die Dynamik der Willen zur Macht und die ewige Wiederkehr, Berlin, New York 1998, Chapter VI, and Johan Figl, Interpretation als philosophisches Prinzip. Friedrich Nietzsches universale Theorie der Auslegung im späten Nachlass, Berlin, New York 1982. 72 Pietro Gori „knowledge in itself" – states Nietzsche in GM III 24;4 moreover, there is no knowledge of a „thing in itself": what can be known by the human beings is only something conditioned – something that presupposes a relationship between the object and the knowing subject (NL 2 [154], KSA 12, 141 f.). If we consider the word ‚interpretation' in a wide sense – so to say, both in a physiological and in a hermeneutic sense –, it is easy to see that Nietzsche's principle characterizes his whole theory of knowledge. His first statements concerning the metaphorical value of language (WL); his early observations concerning the physiological ‚falsification'5 of reality; his later perspectivism6 – all these topics can be related to the idea that to know is to interpret, that the knowing subject plays an active role in her relationship with things, and she thus ‚creates' something, instead of merely replicate a state of affairs.7 What follows from that view is first of all a rejection of the traditional character of truth, i. e. its being absolute and undisputable. On the contrary, as we all know, according to Nietzsche there is no ‚Truth', but only an indefinite amount of world-interpretations, of world-descriptions, of viewpoints that cannot be a priori rejected as ‚absolutely false'.8 Most important, anything that one can say about reality falls within these interpretations, i. e. „the ‚apparent' world is the only world", while the „true world" reveals its inner lack of content (GD, KSA 6, 75). According to this view, if we want to keep on describing the world in terms of ‚true' and ‚false' (a very useful attitude, as I will later show), that must be done within the pure human realm of the ‚appearances'. In JGB 34 Nietzsche stresses this point, by arguing that „if, with the virtuous enthusiasm and inanity of many philosophers, someone wanted to completely abolish the ‚world of appearances', – well, assuming you could do that, – at least there would not be any of your ‚truth' left either! Actually, why do we even assume that ‚true' and ‚false' are in4 See also NL 15 [9]: „Our knowledge is not knowledge in itself, moreover it is not even knowledge, but rather a chain of deductions and spider's webs: it is the result of thousands years of necessary optical errors – necessary, since we basically want to live –, errors, since any perspectival law is basically an error" (KSA, 9, 637). 5 The interpretation of Nietzsche's ‚falsification thesis' is probably the most discussed topic of Clark's book from 1990. See Mattia Riccardi, Il tardo Nietzsche e la falsificazione, in: Pietro Gori, Paolo Stellino (eds.), Teorie e pratiche della verità in Nietzsche, Pisa 2011, 57–73. 6 Although Nietzsche's perspectivism is strictly related with his theory of knowledge, and chiefly with his ‚falsification thesis', its meaning is not merely theoretic. I think that the meanings of the word ‚interpretation' in Nietzsche can be understood by focusing on the differences between the purely physiological processes of both the sense organs and the human intellect (which ‚falsify' the world), and the perspectival knowledge of the world peculiar to the human being. I recently dealt with the practical value of Nietzsche's perspectivism in Pietro Gori, Il „prospettivismo". Epistemologia ed etica, in: Pietro Gori-Paolo Stellino (eds.), Teorie e pratiche della verità in Nietzsche, 101–123. 7 In 1885 Nietzsche defined the human being „a shapes and rhythms moulding creature" (NL 38 [10], KSA, 11, 608). 8 Although – as Volker Gerhardt argues in his Die Perspektive des Perspektivismus (Nietzsche-Studien, 18 (1989), 260–281, 279) – Nietzsche mostly focuses on the human being, it is not clear who really is the subject of his perspectivism. In his writings Nietzsche refers to the species, to the human being, and also to the indefinite number of „centres of force", each of whom „adopts a perspective toward the entire reminder, i. e., its own particular valuation, mode of action, and mode of resistance" (NL 14 [184], KSA, 13, 371). See Pietro Gori, Il „prospettivismo", 111 ff., and Christoph Cox, The „Subject" of Nietzsche's Perspectivism, in: Journal of the History of Philosophy, 35 (1997), 269–291. 73Nietzsche on Truth: a Pragmatic View? trinsically opposed? Isn't it enough to assume that there are levels of appearance and, as it were, lighter and darker shades and tones of appearance [...]?" (JGB, KSA 5, 53 f.). The last line of this quote is of the greatest importance, since it characterizes Nietzsche's attitude towards a potential nihilism. The elimination of the ‚true world' leaves an empty space, that cannot be filled with anything, since every absolute reference point of human knowledge (and agency) is rejected as ‚unattainable', ‚unprovable', and ‚superfluous' (GD, KSA 6, 80). By completely eliminating that metaphysical plane, and therefore by referring only to the ‚apparent world', Nietzsche makes a step beyond the traditional dualistic view, but exposes his own philosophy to the danger of becoming a mere relativistic worldview. If the world is painted only with human colours (MA I, KSA 2, 16), how can we find the reference points to orient ourselves to the world, both in a theoretical and in a practical sense? Nietzsche seems to be aware of this danger, and his aim is clearly to help the human being to find a way out of the maze of nihilism. Mostly in his later writings Nietzsche clearly shows how to manage the disorientation that follows from the ‚dead of God', and furthermore how to turn it in a strongly positive attitude towards life. In his view, the emptiness of the ‚true world' becomes the open see for a new navigation, and the prevailing feeling of the philosopher is the ‚cheerfulness' (Heiterkeit) to which Nietzsche refers in FW 343 and in the Preface of GD. Nietzsche's relativistic view is therefore not nihilistic at all, since he leaves the space for a determination of both ‚true' and ‚false' within the limits of the ‚apparent world'. The lack of content is indeed attributed only to that absolute and immutable Truth that is traditionally seen as an eternal idol,9 but not to the many truths which represent the necessary reference points of human life (since they make it manageable). That is a very important – but often neglected – point. Some scholars indeed focus on Nietzsche's critique of metaphysics, and stress only the pars destruens of his theory of knowledge (his falsification thesis). By so doing, they pretend that Nietzsche's use of the word ‚true' is always the same, and don't consider that he is rather moving on two different planes, involving two different meanings of the same word.10 More specifically, as we read in JGB 34, Nietzsche's rejection of truth as correspondence to reality does not entails that we must live without referring points. Nietzsche is rather aimed at re-defining the relationship between ‚true' and ‚false', and tries to find the principles of an evaluation that would not consider them as in opposition anymore. Moreover, if we accept that knowledge is interpretation, and therefore that the world we know is basically ‚erroneous'11, then it is only possible to temporarily adopt some of these errors as the ground of our world-description (and, consequently, of our agency). 9 See EH, Twilight of the Idols 1: „What the word ‚idol' on the title page means is quite simply what had been called truth so far. Twilight of the Idols – in plain language: the end of the old truth..." (KSA, 6, 354). 10 According to Maudemarie Clark, both Wolfgang Müller-Lauter and Ruediger Grimm „think that Nietzsche discards our ordinary concept of truth and replaces it with a new use of ‚true' and ‚false'" (Maudemarie Clark, Nietzsche on Truth and Philosophy, 33). Clark, on the contrary, focuses on Nietzsche's falsification thesis, and criticizes his making reference to truth in the later period of work, after his rejection of the thing in itself. 11 This erroneousness must be understood in terms of the old notion of truth, i. e. of truth as agreement with reality. Nietzsche's thesis is that our knowledge is erroneous, since we only know things after having modified them, and never as they are in themselves. That is something that has been dis74 Pietro Gori What we now need is the criterion of this new evaluation of the referring points of our world-description. Since we cannot make reference to a ‚true world' anymore, our evaluation must remain on the plane of our knowledge itself. Moreover, since our knowledge is a fundamental tool for us in order to manage things, one possibility is to evaluate it in utilitarian terms – i. e. in terms of its usefulness for our practical purposes. That is what Nietzsche argues in a very important note from 1888, where he deals with the origin of human belief in the explanatory power of his knowledge and its adaptive role: „The aberration of philosophy is that, instead of seeing in logic and the categories of reason means toward the adjustment of the world for utilitarian ends (basically, toward an expedient falsification), one believed one possessed in them the criterion of truth and reality. The ‚criterion of truth' was in fact merely the biological utility of such a system of systematic falsification; and since a species of animal knows of nothing more important than its own preservation, one might indeed be permitted to speak here of ‚truth'. The naiveté was to take an anthropocentric idiosyncrasy as the measure of things, as the rule for determining ‚real' and ‚unreal': in short, to make absolute something conditioned" (NL 14 [153], KSA 13, 336). This note represents the last stage of a reflection that started in the 1870s, and that Nietzsche carried on until his mental collapse. The basic idea, published with particular clarity in FW 110, is that human knowledge has been fundamental for the preservation of our species. According to Nietzsche, the schematization and simplification of reality provided by our intellect, its modifying that reality, is an unavoidable tool for the human being, in order to win the struggle for life. From the first stages of the human life, the usefulness of the categories of reason has been taken as sign of its truthfulness, and Nietzsche seems to agree with that evaluation – provided that this truthfulness does not mean ‚agreement with reality'. Since „we cannot cut off [our] head" (MA I, KSA 2, 29), and therefore there is no way to compare the reality processed by our intellect with any ‚thing in itself', the only plane we can refer to, in order to define ‚true' and ‚false', is that of our own world-representation:12 „Truth does not signify the antithesis of error but the status of certain errors vis-à-vis others, such as being older, more deeply assimilated, cussed, especially from Clark, since Nietzsche cannot know if our sense organs or our intellect give us the reality as it is (even thought it is highly improbable that they do so). Moreover, what Clark chiefly criticizes is Nietzsche's talking about ‚erroneousness' and ‚falsification' in his later writings, after his rejection of the thing in itself. According to Clark, if there is no thing in itself, than it is not possible to say that a knowledge is ‚erroneous', since there is nothing to which it can be compared. That is true, but Clark fails in claming that Nietzsche rejects the existence of a thing in itself. Nietzsche in fact never does that; he never denies that there is something beyond our perception and our intellectual knowledge of things. Nietzsche simply rejects the usefulness of referring to reality in itself, since he believes that we cannot know anything out of both the testimony of the senses and their intellectual modification. Thus, our knowledge is with all likelihood something different from how reality is in itself, but at the same time that reality is something ‚unattainable', ‚unprovable', and ‚superfluous'. Moreover, as Nietzsche writes, „to err is the condition of living. So strongly err, in fact. To know the error does not delete it! That is nothing bitter! We must love and improve our errors, since they are the basis of our knowledge" (NL 11 [162], KSA, 9, 504). 12 The ‚aberration of philosophy' which Nietzsche talks about comes from the idea that the usefulness of the categories of reason are a sign of their reality, i. e. that their truthfulness is metaphysical (that they describe the things as they are in themselves), and not merely fictitious. 75Nietzsche on Truth: a Pragmatic View? our not knowing how to live without them, and so on. [...] The valuations must stand in some kind of relation to the conditions of existence, but by no means that of being true, or exact. The essential thing is precisely their inexactitude, indeterminacy, which gives rise to a kind of simplification of the external world – and precisely this sort of intelligence favours survival" (NL 34 [247], KSA 11, 503 f.).13 This utilitarian view of truth is biologically grounded, but under its surface we find a more general idea. Nietzsche's aim, in dealing with human knowledge, is primarily to understand how the metaphysical worldview has been generated, and he therefore focuses on the adaptive role of the intellect. According to him, our world-picture evolved during the development of our species (MA I, KSA, 2, 16), and what we now believe to be real is only a very useful interpretation of things. That utilitarian view can be applied not only to a long-lasting event such as the development of the human brain, but also to any worlddescription and world-interpretation, in both a theoretical and a practical sense. Thus, Nietzsche's statements on the possibility of assuming ‚true' and ‚false' as ‚levels of appearance' does not only pertain to the biological level of human knowledge, but it rather can be adopted as a more general evaluation principle. ‚True' is therefore what is relatively more useful, what ‚pays', whose effects are ‚better', depending on what we need. Of course, what is important at the most is human life, and that is what Nietzsche stresses in his writings (mostly notebooks. See e. g. NL 6 [421], KSA 9, 306). But the utilitarian principle can be applied for example to science, and a scientific world-description can be ‚true' for it is the most explanatory and economic – but not since it reveals the inner structure of reality. This outcome of Nietzsche's theory of knowledge – the idea that the truth-value is merely relative, and that it is the result of an utilitarian evaluation – persuaded some scholars to compare Nietzsche's view with the pragmatism of William James.14 As I will show in what follows, there are many elements supporting this comparison, which can be properly understood in the light of the late 19th century philosophy of science. During the last decades, many scholars stressed the tight connection between Nietzsche's thought and some of the most significant outcomes of the science of his time.15 With regard to his theory of knowledge, and the view of truth which follows from that, it is possible to 13 Nietzsche also states the biological value of truth in NL 6 [421], KSA 9; NL 25 [372], KSA 10; NL 14 [105], KSA 13. 14 See e. g. Ludwig Marcuse, Nietzsche in Amerika, in: Harold von Hofe (hg.), Essays, Porträts, Polemiken aus vier Jahrzenten, Zürich 1950, 91–103; id., Amerikanisches Philosophieren. Pragmatisten, Polytheisten, Tragiker, Hamburg 1959; Kai-Michael Hingst, Perspektivismus und Pragmatismus. Ein Vergleich auf der Grundlage der Wahrheitsbegriffe und der Religionsphilosophien von Nietzsche und James, Würzburg 1998; Id., Nietzsche Pragmaticus. Die Verwandtschaft von Nietzsches Denken mit dem Pragmatismus von William James, in: Nietzscheforschung, 7 (2000), 287–306; Rossella Fabbrichesi, Nietzsche and James. A Pragmatist Hermeneutic, in: European Journal of Pragmatism and American Philosophy, 1 (2009), 25–40. 15 The first important studies concerning the scientific sources of Nietzsche's thought has been carried out by Alwin Mittasch, in his Friedrich Nietzsches Naturbeflissenheit, Heidelberg 1950, and Nietzsche als Naturphilosoph, Stuttgart, 1952. More recently, studies on that topic have been published in Thomas Brobjer, Gregory Moore (eds.), Nietzsche and Science, Aldershot 2004; Helmut Heit, Günter Abel, Marco Brusotti (Hg.), Nietzsches Wissenschaftsphilosophie, Berlin 2012; Helmut Heit, Lisa Heller (Hg.), Handbuch Nietzsche und die Wissenschaften, Berlin 2013. 76 Pietro Gori stress that Nietzsche shares with the 19th century science the idea that ‚truth' is something that we establish from a theoretical point of view, and not discover into things.16 During the second half of the 19th century, the scientists discussed about the explanatory power of their own discipline, and some of them rejected the old mechanistic worldview, since they found it marked by a bad metaphysics. More specifically, the Newtonian physics was too deeply grounded on the belief in unchanging and absolute principles, while the new (non-Euclidean) mathematics showed the possibility of calculating the world with different ones – all of them equally ‚true' on the theoretical plane. These discovers contributed to change the view of the power of scientific investigations, and in particular led to a rejection of the belief in an absolutely certain truth attainable through mathematics. A pioneer of this rejection of the explanatory power of mechanism has been Ernst Mach, a scientist whose epistemological studies strongly influenced 20th century science and philosophy, and whose name during the last decades has been quoted in several studies on Nietzsche.17 James explicitly refers to him many times, and it is possible to argue that his pragmatism is grounded on a pure Machian view. In the following sections I will first consider the fundamental ideas of Mach's epistemology, and compare them with some basic outcomes of Nietzsche's theory of knowledge. I will then turn to James, in order to deal with his pragmatism in the light of the new philosophy of science, and show the similarity between his view of truth and that of Nietzsche.18 In the final section I will 16 See Werner Stegmaier, Nietzsches Neubestimmung der Wahrheit, 89. A comparison between Nietzsche's view and that of Heisenberg has been carried out by Jochen Kirchoff, Zum Problem der Erkenntnis bei Nietzsche, in: Nietzsche-Studien, 5 (1977), 17–44. 17 On Mach's influence on 20th century scientific and philosophical thought see Phillip Franck, Modern Science and its Philosophy, Cambridge 1949; John Blackmore, Ernst Mach. His Work, Life, and Influence, Berkeley, Los Angeles, London 1972; Friedrich Stadler, Von Positivismus zur „Wissenschaftlichen Weltauffassung", Wien, München 1982. As regards the comparison between Nietzsche and Mach, the most recent studies on that topic showed that we cannot consider Mach as a direct source of Nietzsche. Nietzsche in fact bought and read Mach's Beiträge zur Analyse der Empfindungen in 1886, but most of the ideas that he shared with Mach can be found in earlier writings (see Pietro Gori, The Usefulness of Substances. Knowledge, Science and Metaphysics in Nietzsche and Mach, in: Nietzsche-Studien 38 (2009), 111 ff.). Most likely, they both shared a common debate (thinkers such as Fechner, Spencer, and Lichtenberg), and from that source they developed their comparable theories of knowledge. I dealt more exhaustively with that topic in Pietro Gori, Il meccanicismo metafisico. Scienza, filosofia e storia in Nietzsche e Mach, Bologna 2009. Nadeem Hussain also dealt with a comparison between Nietzsche and Mach in his Reading Nietzsche through Ernst Mach, in: Gregory Moore, Thomas Brobjer (eds.), Nietzsche and Science, 111–129. 18 At the beginning of the 20th century Hans Kleinpeter, a pupil and friend of Mach, first argued that Nietzsche shared some of the basic features of Mach's epistemology. He particularly highlighted the connection between Nietzsche, Mach and Pragmatism, claiming that they all played a leading role in overcoming Kantian philosophy by taking on a biological theory of truth (Kleinpeter pays particularly attention to Kant's view of the absolute value of concepts. See e. g. Hans Kleinpeter, Nietzsche als Schulreformer, in: Blätter für deutsche Erziehung 14 (1912), 100, and id., Der Phänomenalismus, eine naturwissenschaftliche Weltauffassung, Leipzig 1913). Kleinpeter also stressed the similarity between Nietzsche's philosophy and Pragmatism in a letter to Mach sent on 22. 12. 1911 (See Pietro Gori, Drei Briefe von Hans Kleinpeter an Ernst Mach über Nietzsche, in: Nietzsche-Studien, 40 (2011), 290–298). I dealt with Kleinpeter's interest in Nietzsche in Pietro Gori, Nietzsche as Phenomenalist?, in: Helmut Heit, Günter Abel, Marco Brusotti (Hg.), Nietzsches Wissenschaftsphilosophie, 345–355. 77Nietzsche on Truth: a Pragmatic View? consider Nietzsche's ‚pragmatic' view, and focus on the role that it played in his late philosophy. II. Elements of a new epistemology The guiding lines of Mach's epistemology are well expressed in a lecture held at the Royal Bohemian Society of Sciences in 1871, titled The History and the Root of the Principle of Conservation of Work (or energy, as one reads in some translations).19 Here we find the fundamental ideas of a quite new perspective, whose basic statement is that both physical concepts and laws have a mere relative and historical value, and in no way can be seen as an absolutely ‚true' description of the world. This view undermines the very ground of the science of mechanics by revealing its metaphysical character. According to Mach, ‚metaphysical' are indeed the scientific notions assumed with no reference to their genealogical development. In the opening pages of that work, Mach indeed states that „we accustomed to call concepts metaphysical, if we have forgotten how we reached them" (GEA, 17).20 With these words he does not reject the usefulness of these concepts; he only stresses their relative value, and warns the scientists not to mistake the pure logical function of the notions they use with their ontological content.21 In Mach's view, the way to get the scientific knowledge rid of any dogmatic heritage is therefore an inquiry concerning the genesis of the notions daily used in physics, psychology etc., since it reveals their development during the history of thought and culture.22 Mach first presented the idea of the usefulness of a retrospective look some years before, in his Über die Definition der Masse (1868), by suggesting what he later wrote in this terms: „One can never lose one's footing, or come into collision with facts, if one 19 Mach published that lecture the following year (Prague, 1872). Nietzsche wrote the title of this essays in a note from 1882, among other books that he read some years before. Alwin Mittasch first discovered it, as we read in his Friedrich Nietzsches Naturbeflissenheit, 186. Nietzsche's note has been then published in the critical apparatus of KGW (VII/4/2, 67). 20 I will use the following abbreviations for Mach's works: GEA = History and Root of the Principle of Conservation of Energy, Chicago 1911; ME = The Science of Mechanics. A Critical and Historical Account of its Development, Chicago, London 1919; AE = The Analysis of Sensations, and the Relation of the Physical to the Psychical, Chicago, London 1914. To avoid any misunderstanding, it is important to say that Mach's view of this topic is far from the idea of a logical analysis of the scientific notions; his aim is rather to show the importance of working with concepts whose origin is known, or can at least be found through a genealogical reconstruction. Mach's epistemology should thus not be interpreted in an analytical way, so to say, à la Wittgenstein or – better – à la Carnap. Even though Carnap directly referred to Mach in carrying on his new philosophical perspective, the latter was a pure ‚continental' philosopher with a peculiar interest in the history of his own discipline, in its genealogical development. 21 Mach's way of reasoning here is notably close to Nietzsche's late critical remarks to the philosophers who ‚dehistoricize' the concepts of reason, and „turn them into a mummy" (GD, KSA 6, 74). I will come back to this at the end of this section. 22 In GEA, Mach writes: „Quite analogous difficulties lie in wait for us when we go to school and take up more advanced studies, when propositions which have often cost several thousand years' labour of thought are represented to us as self-evident. Here too there is only one way to enlightenment: historical studies" (GEA, 16). 78 Pietro Gori always keeps in view the path by which one has come" (GEA, 17). He soon developed his idea in a wider project concerning the historical explanation of both scientific concepts and laws, a project briefly outlined in the conference from 1871, and with which Mach dealt in his later works, e. g. in the books concerning The Science of Mechanics (1883) and the Analysis of Sensations (1886). Within these texts, the first one is probably the most important, since it clearly shows that in dealing with his own discipline Mach aimed to carry on „a critical and historical account of its development" (as the subtitle of his Science of Mechanics indicates). Moreover, according to him, a ‚critical' analysis of the principles of the Newtonian physics can be provided only through a historical reconstruction of their genesis. In the opening pages of this work, Mach writes that the aim of his volume is „to clear up ideas, expose the real significance of the matter, and get rid of metaphysical obscurities" (ME, x). Thus, his critical aim has a deep „anti-metaphysical" value, as he himself sometimes states.23 Moreover, Mach explains that „the gist and kernel of mechanical ideas has in almost every case grown up in the investigation of very simple and special cases of mechanical processes; and the analysis of the history of the discussions concerning these cases must ever remain the method at once the most effective and the most natural for laying this gist and kernel's bare. Indeed, it is not too much to say that it is the only way in which a real comprehension of the general upshot of mechanics is to be attained" (ME, x–xi). According to Mach the historical analysis allows us to get „the positive and physical essence of mechanics" rid of the „mass of technical considerations" beneath which it's buried, and which conceals how the principles of mechanics „have been ascertained, from what sources they take their origin, and how far they can be regarded as permanent acquisitions" (ME, x). This analysis therefore shows the inner side of the scientific notions, and reveal their being mere ideas, thought symbols (Gedankensymbol)24 that human beings created during their development, and that change together with the ‚paradigm shifts' (to use Kuhn's words). Mach dealt for the first time with the use of history for science in his lecture on the Principle of Conservation, by claiming that this is the only tool we have to see the frequently changing of views, concepts, and theories, and thus to let us „get used to the fact that science is unfinished and variable" (GEA, 17). In stressing the historical nature of science, Mach also argues the inner impermanence of its notions, since they are only the result of an ever changing and improving description of the natural world. Thus, as regards this topic, Mach clearly assumes the concepts to be only resting points of our mind, thought symbols that a scientist temporarily adopt as the best result that until now has been reached in his field of study. These ‚labels' are first of all useful to save experience and let the scientists communicate the results of their studies to other researchers who will carry on the formers' work. That is what Mach thinks in talking about an ‚economical' office of science: „Science is communicated by instruction, in order that one man may profit by the experience of another and be spared the trouble of accumulating it from himself" (ME, 481). In the 1871 conference he pre23 Mach explicitly declares that in the Preface to ME. Furthermore, the opening paragraph of the first chapter of AE is titled „first anti-metaphysical principles". 24 Mach uses the word Gedankensymbol for example in AE, 254 and 296. This is a fundamental concept of Mach's analysis of the scientific world-description, as James himself highlights in his lectures on Pragmatism. I will turn to this in the following section. 79Nietzsche on Truth: a Pragmatic View? sented the same idea by stating that a formula, a scientific law has „no more real value than the aggregate of the individual facts" explained by it. „Its value for us lies merely in the convenience of its use: it has an economical value" (GEA, 55).25 It is easy to see that this perspective directly follows from Mach's view on the development of his own discipline, since he thinks that the physicists (but the same can be said for what concerns the researchers working in other fields) keep on creating new concepts that would adapt in a better way to the objects or to the processes they want to explain.26 On the philosophical plane, that leads to a new evaluation of the results of scientific investigation: even though the practical usefulness of the concepts daily used cannot be denied, one must say that they have a mere relative value on the ontological plane, and thus reject (or at least limit) the ‚truthfulness' of scientific knowledge. According to Mach, unlike both 17th and 18th century scientists, any concept has to be defined only as a methodological reference point to describe and manage the natural world. Again, the scientific notions are mere thought symbols which do not lead to something stable and permanent under the becoming surface of our sensations. In a way very close to Nietzsche's perspective, Mach rejects the reference to any kind of thing in itself: even though he never claims that it does not exist, he states that it is not important to make reference to it, in order to investigate our own reality (since that is a pure phenomenal world. See AE, 29–31). The best starting point to stress the similarity between Mach's view of truth and that of Nietzsche is the definition of ‚metaphysical concepts' published in GEA. The idea that we „call concepts metaphysical, if we have forgotten how we reached them" can indeed be compared to Nietzsche's well-known statement in On Truth and Lie in an Extra-moral Sense, according to which „truths are illusions of which one has forgotten that they are illusions" (WL, KSA 1, 881). In this unpublished work Nietzsche calls ‚truth' a schematization of the external data which value is related (or even mistaken) with both its practical usefulness and its having been helpful for the preservation of the species. Therefore, ‚truth' is a concept that has never been brought into question and, after a long time, has been adopted with no reference to its origin. In particular, Nietzsche talks about „metaphors that have become worn-out and deprived of their sensuous force, coins that have lost their imprint and are now no longer seen as coins but as metal" (ibid.). As well as the metaphysical notions with which Mach deals in GEA, the truths that Nietzsche describes in WL are the result of a wrong judgment, since they're isolated from the process of becoming which they are part of. On the contrary, both the scientific concepts and these truths can be properly described only through a historical analysis. Nietzsche's early criticism towards the notion of truth follows from his idea that a genealogical reconstruction tracing the development of human thought is the only tool we have to enlighten the character of the notions that we usually adopt, the „mobile army 25 In the endnote to this claim Mach writes that „in science we are chiefly concerned with the convenience and saving of thought", and that „the moment of inertia, the central ellipsoid, and so on, are simply examples of substitutes by means of which we conveniently save ourselves the consideration of the single mass-points" (GEA, 88). 26 In 1910 Mach summed up this ‚evolutionary' interpretation of the investigating process talking about the „adaptation of the ideas to the facts and the adaptation of the ideas to themselves" (Ernst Mach, Die Leitgedanken meiner naturwissenschaftlichen Erkenntnislehre und ihre Aufnahme durch die Zeitgenossen, in: Scientia, 7 (1910), 225–240). 80 Pietro Gori of metaphors, metonyms, and anthropomorphisms" which are nothing but illusions of knowledge. Nietzsche clearly states that in the first section of Human, all too Human, where he deals with many questions first treated in WL (but left unpublished). As I pointed out in the first section of this paper, in MA 16 Nietzsche argues that the concepts commonly used to describe the external world are a gradually evolved and still evolving product of our intellect. According to him, „it is the human intellect that has made appearance appear and transported its erroneous basic conceptions into things. Late, very late – it has reflected on all this: and now the world of experience and the thing in itself seem to it so extraordinarily different from one another and divided apart that it rejects the idea that the nature of one can be inferred from the nature of the other" (MA I, KSA 2, 37). Moreover, the world of phenomena is an „inherited idea, spun out of intellectual errors" (ibid.). This way of treating this problematic relationship between the appearances and the thing in itself directly leads to a possible solution, since if one admits that the world we know is a mere product of our intellect generated during the development of the species, then a genealogical analysis can easily show its inner lack of content. Nietzsche indeed goes on by stating that „with all these conceptions the steady and laborious process of science, which will one day celebrate its greatest triumph in a history of the genesis of thought, will in the end decisively have done; for the outcome of this history may well be the conclusion: that which we now call the world is the outcome of a host of errors and fantasies which have gradually arisen and grown entwined with one another in the course of the overall evolution of the organic being, and are now inherited by us as the accumulated treasure of the entire past – as a treasure: for the value of our humanity depends upon it" (ibid.).27 The view of human knowledge that Nietzsche presents in Human, all too Human is the ground of his later criticism towards the notion of truth, and more widely towards the Western metaphysics. His statements indeed concern the metaphysical realm of absolute and unchanging concepts, that realm that he will later call ‚true world'. In this realm we find all the ‚eternal idols', the hypostatized world-schemes that our intellect created, and that are commonly seen as a „criterion of truth and reality", instead of mere „means toward the adjustment of the world for utilitarian ends" (NL 14 [153], KSA 13, 336). In MA Nietzsche detects the reason of this mistake in our language, in its being an essential tool for us, in order to orient ourself to the world and make it manageable. In that book Nietzsche indeed writes that „the shaper of language was not so modest as to think that he was only giving things labels; rather, he imagined that he was expressing the highest knowledge of things with words; and in fact, language is the first stage of scientific effort" (MA I, KSA, 2, 30 f.). This statement is coherent with the note from 1888 quoted above, and confirms Nietzsche's idea that the rejection of the metaphysical worldview only involves our belief in the absolute value of our knowledge, and not our use of it for practical purposes. As I briefly pointed out earlier, Nietzsche is well aware that our intellect's fallibility is physiological, and therefore that the human beings cannot live without referring to the intellectual ‚errors' (see e. g. MA I, KSA 2, 9). Thus, according to him, the plane of fixed, unchanging shapes (thoughts, symbols, bodies, subjects, and things) 27 In order to stress even more the similarity between Nietzsche's and Mach's view of truth, it should be noted that they both carried on a biological theory of knowledge. See on this topic Milič Čapek, Ernst Mach's Biological Theory of Knowledge, in: Synthese,18/2–3 (1968), 171–191, and Pietro Gori, Il meccanicismo metafisico, Chapter 1. 81Nietzsche on Truth: a Pragmatic View? must not be completely rejected, and we can look at it as a temporarily reference for our world-description and world-interpretation. That can be done only by changing the traditional philosophical perspective, and becoming historians, as Nietzsche points out in a note from 1885: „What distinguishes us in the deepest way from all the Platonic and Leibnitzean way of thinking, is this: we do not believe in eternal concepts, eternal values, eternal shapes, eternal souls; and philosophy, as far as it is science and not legislation, is for us just the broadest extension of the concept of ‚history'" (NL 38 [14], KSA 11, 613).28 Nietzsche's dealing with the categories of reason is therefore very similar to Mach's view of the scientific notions. The characters of these notions are basically the same as that of human ‚truths': they both have indeed a mere relative and historical value, but they are so useful for us, that it is not possible to live (or work) without them. This practical usefulness is what avoids the nihilistic drift of this relativistic view. Since both the human truths and the scientific notions make the world manageable, their being relative does not lead to an indifferentism according to which it is not possible to choose any option, since their truth-value is the same. Our need of reference points for our agency (on the scientific side: the researcher's need of reference points for his world-description) forces us to find a criterion of truth. Since there is no reference out of the plane of the human knowledge (there is no dualism between the ‚apparent' and the ‚true' world anymore), this criterion must be found on that plane itself. This criterion, as we saw above, is a utilitarian one, and that leads to a pragmatic view of truth. As I tried to show by stressing the parallelism between the philosophical and the scientific views, this is true both for Nietzsche and for the scientists who accept Mach's principles. Moreover, this is true for James, whose pragmatism is explicitly grounded on the historical description of the scientific investigation. III. ‚Denkmittel' and common sense: William James on truth James' pragmatism is strictly related with Machian empiricism and his epistemological views29. James' theory of truth, in particular, follows from the scientific worldview of the late 19th century, and can be evaluated as an attempt to answer to the crisis of contemporary science. In what follows I shall argue that the several similarities between James' view of truth and that of Nietzsche can be understood in the light of that context. Even though 28 This exhortation to develop a ‚historical philosophy' recalls the opening of MA, where Nietzsche complains the „lack of historical sense" of the philosophers (MA, KSA 2, 24). The same complaining is later repeated in GD, where Nietzsche stresses the importance of looking at the concept of reason as mere tools to manage the world, and deplores the inability of the philosophers to see human knowledge as part of a still becoming process (GD, KSA, 6, 74). I dealt with Nietzsche's reference to history as tool to enlighten the hollowness of the idols in Pietro Gori, „Sounding Out Idols". Knowledge, History and Metaphysics in Human, All Too Human and Twilight of the Idols, in: Nietzscheforschung, 16 (2009), 239–247. 29 See e. g. Gerald Holton, From the Vienna Circle to Harvard Square: The Americanization of a European World Conception, in: Friedrich Stadler (ed.), Scientific Philosophy: Origins and Development, Dodrecht, Boston, London 1993, 47–73. James explicitly mentions Mach and his school both in Pragmatism (32 and 89) and in The Meaning of Truth (MT 178), but several of his statements are clearly references to Mach's ideas. I will later deal with some of them. 82 Pietro Gori Nietzsche never refers to the same authors quoted by James (e. g. Henri Poincaré, Pierre Duhem, Wilhelm Ostwald), beyond his theory of knowledge we find people to whose outcomes these authors themselves made reference.30 Moreover, as I argued in the previous section, Nietzsche's theory of knowledge is comparable with Mach's epistemology. Even though we cannot take Mach as a common source between James and Nietzsche, that similarity can anyway be the sign of a shared view of epistemological questions. The starting point of James' dealing with truth is the rejection of the ‚correspondence theory', i. e. the idea that truth expresses what reality is in itself.31 In the opening of his lecture on Pragmatism's Conception of Truth, James contrasts „the popular notion that a true idea must copy its reality", and reject this claim as a bad interpretation provided by the intellectualists of the definition of truth as ‚agreement' with ‚reality' (P, 92 f.).32 James' thesis, on the contrary, is that truth cannot be considered as a ‚static' predicate of things, but rather as a becoming property of them. According to him „the truth of an idea is not a stagnant property inherent in it. Truth happens to an idea. It becomes true, is made true by events. Its verity is in fact an event, a process: the process namely of its verifying itself, its veri-fication" (P, 93). This idea that „truth is simply a collective name for verification-processes" (P, 101) can be compared with some Nietzsche's statements on truth, and actually follows from a quite similar view of the truth-value of the ‚facts'. That is particularly clear if we just consider this excerpt from Nietzsche's Nachlass: „Truth is not something that's there and must be found out, discovered, but something that must be made and that provides the name for a process – or rather for a will to overcome, a will that left to itself has no end: inserting truth as a processus in infinitum, an active determining, not a becoming conscious of something that is ‚in itself' fixed and determinate" (NL 9 [91], KSA 12, 385). The ground idea of this view, that both Nietzsche and James state, is the inexistence of a thing in itself to which we can refer, in order to provide a description of the world. More precisely, if a thing in itself exists (they both never reject its existence, but only its theoretical value!), it is neither ‚true' nor ‚false'. Reality have indeed no truth value in itself, and the facts only acquire truthfulness from us, from our knowledge of them. In a way very similar to Nietzsche's well known statement according to which there are no facts, but only interpretations (NL 7 [60], ibid., 315), James argues that „the ‚facts' themselves are not true. They simply are", and furthermore „truth is the function of the beliefs that start and terminate among them" (P, 104).33 It is notably that this idea arises from a very Machian remark, a sensualist account that Nietzsche (apparently) shares, too. James indeed argues that „the first part of reality [...] is the flux of our sensations. Sensations are forced upon us, coming we know not whence. 30 That is very similar to what happens in the case of Nietzsche's affinity with Mach's view, since that can be only understood by referring to the scientific debate they both referred to. See Pietro Gori, The Usefulness of Substances, 112 ff. 31 See Kai-Michael Hingst, Nietzsche Pragmaticus, 293 ff. 32 In this section I will use the following abbreviations for the two works of James that I will chiefly quote: P = Pragmatism. A New Name for some Old Ways of Thinking; MT = The Meaning of Truth. I quote from Pragmatism & The Meaning of Truth, Seaside 2011. 33 In MT the same claim is stated with reference to reality: „Realities are not true, they are; and beliefs are true of them" (MT 233). Rossella Fabbrichesi stressed the similarity between James' theory of truth and Nietzsche's perspectivism in her paper on Nietzsche and James, 26 ff. 83Nietzsche on Truth: a Pragmatic View? Over their nature, order, and quantity we have as good as no control. They are neither true nor false; they simply are. It is only what we say about them, only the names we give them, our theories of their source and nature and remote relations, that may be true or not" (P, 112). That is exactly one of the basic topics with which Mach deals in his Analysis of Sensations, and that can be compared with Nietzsche's late view.34 In GD Nietzsche indeed states that senses „do not lie at all", and goes on by claming that „what we do with the testimony of the senses, is where the lies begin. [...] ‚Reason' makes us falsify the testimony of the senses" (GD, KSA 6, 75).35 This statement concerns the interpreting role of human knowledge, its adding something to a theoretically ‚neutral' element. That is what also James argues, when he quotes Ferdinand Schiller's Humanism and his idea that „our truths are a man-made product" (P, 111),36 or in stating that „in our cognitive as well as in our active life we are creative. We add, both to the subject and to the predicate part of reality" (P, 118). In so doing, James stresses the necessity of focusing to the human side of knowledge in order to talk about truth, to that ‚apparent world' whose role of exclusive reference point of our world-description Nietzsche emphasizes in Twilight of the Idols. The reference to GD can be further stressed, since in the opening of the lecture on Pragmatism and Humanism where James deals with the perspectival character of truth, we find some claims concerning the rationalistic view on truth (which James is aimed at contrasting) that are comparable with Nietzsche's late critique to the ‚prejudices of reason'. „The notion of the truth, conceived as the one answer, determinate and complete, to the one fixed enigma which the world is believed to propound", is defined by James as 34 On Nietzsche's sensualism, and its relationship with Mach's view, see Pietro Gori, The usefulness of substances, 114 ff. and 123 ff.; Nadeem Hussain, Reading Nietzsche through Ernst Mach; id., Nietzsche's Positivism, in: European Journal of Philosophy, 12/3 (2004), 326–368. 35 Nietzsche's positive attitude towards sensualism is also expressed in JGB 15 and FW 272. See on this topic Robin Small, Nietzsche in Context, Aldershot 2001, Chapter 9. 36 The name of Schiller deserves a short digression. In 1982 George Stack dealt with Nietzsche's influence on Pragmatic Humanism, and suggested that some fundamental statements of Schiller could not have been completely original. Stack noticed the influence that Nietzsche had on Schiller, and stressed the almost totally absence of explicit references to Nietzsche in the latter's work. The main outcome of Stack's investigation is that Nietzsche is probably a direct (but hidden) source of Schiller, and therefore his Humanism is grounded on a pure Nietzschean ground (see George Stack, Nietzsche's Influence on Pragmatic Humanism, in: Journal of the History of Philosophy, 20/4 (1982), 339–358). If Stack is right, that shed a new light on our research. We should indeed evaluate Nietzsche's role in the development of James' theory of truth, and not just consider the similarity between their views. If we follow Kleinpeter's view, according to which „in defining the notion of truth, Nietzsche completely agrees with Pragmatism" of both James and Schiller (Hans Kleinpeter, Die Erkenntnislehre Friedrich Nietzsches, in: Wissenschaftliche Rundschau, 3 (1912), 9), our basic assumption is that the two pragmatists developed their own views independently from Nietzsche's theory of knowledge. On the contrary, if Schiller assimilated some of Nietzsche's ideas, and James makes reference to Schiller in developing his pragmatism, then Nietzsche could have played a role in James' philosophy, even if an indirect and quite limited one. Moreover, what is notably here is that Schiller is most probably the person who suggested to Kleinpeter that Nietzsche's theory of knowledge was comparable to the modern epistemology, and to pragmatism itself. It is a fact that Kleinpeter started dealing with Nietzsche only in 1911, after the International Congress of Philosophy held in Bologna, when he first met Schiller (see Pietro Gori, Drei Briefe von Hans Kleinpeter an Ernst Mach, in: Nietzsche-Studien, 40 (2011), 290). 84 Pietro Gori a „typical idol of the tribe" (P, 110, my emphasis in the last part of the quotation). Moreover, he argues that „by amateurs in philosophy and professional alike, the universe is represented as a queer sort of petrified sphinx whose appeal to man consists in a monotonous challenge to his divining powers. The truth: what a perfect idol of the rationalistic mind!" (ibid.). With all likelihood, in talking about ‚idol of the tribe' James simply quotes Bacon's idola tribus, but his view on truth is nevertheless comparable with Nietzsche's late statements. The eternal idols Nietzsche deals with in GD are exactly those old truths that constituted the metaphysical realm, the reference points of the Western worldview. Moreover, Nietzsche's idols are peculiar to the ‚philosopher's idiosyncrasy', and have been generated by those philosophers' having trusted in the „prejudices of reason" (GD, KSA 6, 74 ff.). Finally, as I argued in the previous section, Nietzsche's idols are the human beliefs that have lost their historical character; concepts developed during the long (both biological and cultural) history of the human being, and that are now seen as fixed, immutable, non-becoming attributes of the world. From what I showed until now, it is arguable that James' view is in compliance with the theoretical disorientation peculiar to the late 19th century both scientific and philosophical worldview, which followed from the discover of the inadequacy of the traditionally adopted reference points. Let's now see how deeply his pragmatism is connected with contemporary epistemology. According to James, truth is something that does not belong to things. There is nothing to discover, and what we can define in terms of true and false is only a human view of reality, his interpretation of it. The role played by the concepts in this picture follows explicitly from the main outcome of Mach and his school (e. g. Duhem and Ostwald). „All our conceptions are what the Germans call Denkmittel, means by which we handle facts by thinking them. Experience as such doesn't come ticketed and labelled, we have first to discover what it is" (P, 81). These Denkmittel can easily be Mach's Gedankensymbol, as much as Nietzsche's labels, whose usefulness is merely practical, since they make the world manageable. James calls indeed the concepts „artificial short-cuts for tacking us from one part to another of experience's flux", and – with clear reference to Mach – „sovereign triumph of economy in thought" (P, 89). The similarity with Nietzsche's view is not limited to that, and concerns the relationship between these Denkmittel and our ‚common sense'. James indeed states that the Denkmittel have become the ground concepts of our common worldview, since they were useful, and played a fundamental role in the development of the human race. Moreover, James defines the ‚common sense' as „a perfect definite stage in our understanding of things, a stage that satisfies in an extraordinarily successful way the purposes for which we think" (P, 85). What forms this „great stage of equilibrium in the human mind's development" are „our fundamental ways of thinking about things", which are „discoveries of exceedingly remote ancestors, which have been able to preserve themselves through the experience of all subsequent time" (P, 80). James argues this, and then adds: „We are now so familiar with the order that these notions have woven for us out of the everlasting weather of our perceptions that we find it hard to realize how little of a fixed routine the perceptions follow when taken by themselves".37 Here, again, James' view is very close 37 See also MT 177: „Experience is a process that continually gives us new material to digest. We handle this intellectually by the mass of beliefs of which we find ourselves already possessed, as85Nietzsche on Truth: a Pragmatic View? to that of Nietzsche. Let me just recall FW 110, where Nietzsche deals with the adaptive role of human knowledge, and states that some errors produced by our intellect „through immense periods of time [...] turned out to be useful and species-preserving [...]. Such erroneous articles of faith, which were passed on by in inheritance further and further, and finally almost became part of the basic endowment of the species, are for example: that there are enduring things; that there are identical things; that there are things, kinds of material, bodies; that a thing is what it appears to be; that our will is free; that what is good for me is good in and for itself" (FW, KSA 3, 469).38 In the light of what I stated in the previous sections, I believe that the similarity between James and Nietzsche on this topic is self-evident.39 James explicitly reveals his epistemological ground at the end of his lecture on Pragmatism and Common Sense, where he shows as possible outcome of 20th century philosophy the enlightenment of the pure practical value of the common-sense concepts. First, he deals with the „naïf conception of things", and argues that it can get „superseded, and a thing's name [can be] interpreted as denoting only the law or Regel der Verbindung by which certain of our sensations habitually succeed or coexist" (P, 87).40 Moreover, he goes on in claming that „science and critical philosophy burst the boundaries of common sense" (ibid.). Then, James states, with a pure Nietzschean language: „Scientific logicians are saying on every hand that these entities and their determinations, however definitely conceived, should not be held for literally real. It is as if they existed; but in reality they are like co-ordinates or logarithms, only artificial short-cuts for taking us from one part to another of experience's flux. [...] Just now, if I understand the matter rightly, we are witnessing a curious reversion of the common-sense way of looking at physical nature, in the philosophy of science favoured by such men as Mach, Ostwald and Duhem. According to these teachers no hypothesis is truer than any other in the sense of being a more literary copy of reality. They are all but ways of talking on our part, to be compared solely from the point of view of their use" (P, 89). According to James, the late 19th century epistemology contributed in changing the basic elements of our worldview. In so doing, it undermined that view, in fact. The outcomes of Mach's, Duhem's, and Ostwald's investigations force us to find a new perspective, in order to give value to our practical need to handle the world. Our evaluation of it, in both a theoretical and a moral sense, similating, rejecting, or rearranging in different degrees. Some of the apperceiving ideas are recent acquisitions of our own, but most of them are common-sense traditions of the race. [...] All these were once definite conquests made at historic dates by our ancestors in their attempt to get the chaos of their crude individual experiences into a more shareable and manageable shape. They proved of such sovereign use as Denkmittel that they are now a part of the very structure of our mind". 38 According to James, the most important concepts inherited, and which now form the common-sense belief, are: „thing; the same or different; kinds; minds; bodies; one time; one space; subjects and attributes; causal influence; the fancied; the real" (P, 81). 39 I'd like to say the same with regards to Mach, who also shares a biological and evolutionary view of human knowledge. Unfortunately I had no space do develop this topic in the previous section, and now I can only refer to the same studies I quoted above (see footnote 27). 40 Here, again, we find an implicit reference to Mach. In James' view that should most likely be selfevident – at least, to anyone who knows Mach's basic writings, as he did (see Gerald Holton, From the Vienna Circle to Harvard Square, 50 f., and Massimo Ferrari, Well, and Pragmatism?, in: Friedrich Stadler (ed.), The Present Situation in the Philosophy of Science. Vienna 2010, 78). 86 Pietro Gori cannot be grounded on the possibility of discovering the character of things; we rather must consider our creative attitude to them, and thus define a new criterion of truth. Before coming to what follows from that all, i. e. James' definition of the ‚pragmatic method', let me briefly explain my previous reference to the Nietzschean language. The last excerpt from James' Pragmatism can indeed be compared with those writings in which Nietzsche refers to his contemporaries, and argues that the development of both logic and physics would lead to a new evaluation of the explanatory power of science. In the first book of Beyond Good and Evil, for example, Nietzsche deals with the prejudices of philosophers, and focuses on some outcomes of the scientific worldview. He particularly contrasts the mechanistic view grounded on the belief in material things (e. g. JGB, Aph. 12 and 17), and then argues that „now it is beginning to dawn on maybe five or six brains that physics too is only an interpretation and arrangement of the world [...] and not an explanation of the world" (JGB, KSA 5, 28). This ‚dawning' (which recalls the same dawning „on man that in their belief in language they propagated a tremendous error" stated in MA I, KSA 2, 31) is Nietzsche's word for the change that he founds out in 19th century thought. A change that he believes not to be purely theoretical, but which can also have a transformative power on human life. The interpretation of common-sense concepts as Denkmittel, and the refusal of the correspondence theory (both in compliance with the outcomes of 19th century epistemology), are the grounds of James' ‚pragmatic method'. Since there is no Truth to refer to, nothing that we can simply discover into things, James suggests to pay attention to the practical plane, and more precisely to the effects that our believing something to be true has on our life. „Pragmatism asks its usual question. ‚Grant an idea of belief to be true', it says, ‚what concrete difference will its being true making in anyone's actual life? How will the truth be realized? What experiences will be different from those which could obtain if the belief were false? What, in short, is the truth's cash-value in experiential terms?'" (P, 93). As Rossella Fabbrichesi sums up, James argues that „a belief counts as true when it satisfies us, it pays, also, in the cash-value of the word, it gratifies us, is held as true, proves itself useful if considered true, functions in orienting us along the road of research, that is, is advantageous as related to our vital power".41 James thus focuses on the human being. His view in fact shifts from the known reality to the knowing subject, and James particularly stresses the practical plane of human agency. In so doing, he shares Nietzsche's view, according to which both our theoretical and moral evaluations belong to the interpretative plane. ‚True' and ‚false' – as much as ‚good' and ‚bad' – are only „levels of appearance and, as it were, lighter and darker shades and tones of appearance" (JGB, KSA 5, 14). James' pragmatisms can be therefore compared with that of Nietzsche, and we can particularly stress the interest in human agency which characterizes them. Both the thinkers start indeed from the same epistemological principles, and find in the practical usefulness of our knowledge a criterion of truth. But this usefulness can be understood in many ways, and the reference to human life that we find in Nietzsche's writings may be different to that of James. As I shall argue in the next and final section, Nietzsche's ‚pragmatism' goes beyond a mere utilitarian principle, and involves a modification of human life itself. What is ‚true', in Nietzsche's view, is something that can have 41 Rossella Fabbrichesi, Nietzsche and James, 31. 87Nietzsche on Truth: a Pragmatic View? a transformative effect on the human being, i. e. whose ‚cash-value' must be evaluated not only in experiential, but also (and chiefly) in existential terms. IV. Nietzsche's ‚pragmatism' In the first section of this paper I quoted the note 14 [153] from 1888, which I find particularly clear in displaying the ground of Nietzsche's ‚pragmatism'. In that text, Nietzsche defines the categories of reason as „means toward the adjustment of the world for utilitarian ends (basically, toward an expedient falsification)", and complains the philosophers' belief of possessing „in them the criterion of truth and reality. The ‚criterion of truth'" – Nietzsche goes on – „was in fact merely the biological utility of such a system of systematic falsification; and since a species of animal knows of nothing more important than its own preservation, one might indeed be permitted to speak here of ‚truth'". According to this excerpt, Nietzsche accepts James' pragmatic method. He indeed states that to speak of ‚truth' it is permitted, if we make reference to a useful knowledge – a knowledge that has a significant cash-value. Nietzsche here holds a biological perspective, and talks about ‚truth' as that falsification which permitted the conservation of the species. As I argued in the first section, Nietzsche is interested in the highest cash-value for us – our own life preservation – but we can take a general criterion of truth out of his statements on the adaptive role of knowledge. In that note from 1888 Nietzsche argues that there can be something one calls ‚true'; what is there concerned is in fact only the kind of ‚truth' one can talk about. That must be stressed, since most of the time Nietzsche's relativism is interpreted in a nihilistic way, as if he stated that there can be no truth at all. On the contrary, his perspectivism – the idea that all the world-desscriptions, being only interpretations, in principle have the same truth-value, and it is not possible to evaluate them on the metaphysical plane – does not lead to an indifferentism. Nietzsche's rejection of the ‚true world' indeed leaves the space for a new definition of both ‚true' and ‚false'. He never rejects the possibility of that definition: he rather only limits the plane into which we can evaluate true and false, and shows us the criterion of that evaluation. The plane is that of our world-interpretation, and we can sum up Nietzsche's view in that way: we cannot know anything in itself, anything unconditioned (since we condition what we know); thus, it's extremely highly probable that our knowledge does not correspond to reality; thus, our evaluation of true and false must be in terms of ‚more or less false'. Moreover, that ‚more or less' must not be understood in terms of the correspondence theory (that is what Nietzsche first rejects), and the criterion of truth of the late Nietzsche is therefore the usefulness of knowledge for our practical life, i. e. a concept will be ‚truer' inasmuch as it helps human orientation. That view is open to the objection according to which in rejecting the old criterion of truth Nietzsche replaces it with another one – his own.42 Someone can thus ask: isn't the old criterion as arguable as Nietzsche's perspectivism? Even though on the theoreti42 Maudemarie Clark assumes „that Nietzsche claims superiority for his own perspective" in her Nietzsche of Truth and Philosophy, 140 f. See also Brian Leiter, Nietzsche's Metaethics: Against the Privilege Readings, in: European Journal of Philosophy, 8/3 (2000), 277–297. 88 Pietro Gori cal plane the answer to that question is ‚yes, it is!', the things change if we look at them from Nietzsche's perspective. More precisely, my suggestion is to look at Nietzsche's perspectivism in the light of his ‚pragmatism' itself, with particular reference to the aim of his later writings. During his last years of thought, Nietzsche focused on a diagnosis of his era, with particular reference to the type of man generated by the Western metaphysics: the 19th century European. As we all know, one of Nietzsche's most important contribution to philosophy has been his having traced the genealogical development of our culture, and thus shown the seeds of our attitude towards both the world and ourselves. In few words, Nietzsche sees in the belief in a ‚true world' the basis of the décadence, of the declining type of life peculiar to the 19th century Europe.43 The philosopher's „lack of historical sense", their having „turned into a mummy" our concepts (GD, KSA 6, 74); moreover, their having mistaken a mere falsification for the knowledge of reality in itself (NL 14 [153], KSA 13, 336 ff.) generated a metaphysical worldview, full of ‚eternal idols' to which both our knowledge and our agency must comply. Nietzsche's alternative is to reject the absolute value of these idols – the old truths – to hit them with the hammer of history, and therefore show their inner becoming nature. The idea that any truth is relative is therefore the basis for a new worldview, from which follows another human type. Here we find the transformative value of Nietzsche's perspectivism, since in his view a man who holds this theory of truth will act in a different way, and thus become a ‚higher' human being, compared to the declining one of the late 19th century Europe. We can evaluate this as the ‚cash-value' of Nietzsche's perspectivism, and thus consider his view of truth as involved in the same pragmatic method that rises from it. That view is not ‚truer' than the old one, at least not in terms of the correspondence theory. It is not true at all, in fact, and Nietzsche never claims it to be. Nietzsche's perspectivism is as relative as any other theory of truth. It simply is a worldview alternative to the old one, whose effects on the human being are therefore different. What can let us choose it (as more useful, as having a higher ‚cash-value' – i. e. as being ‚truer' in a pragmatic sense) are exactly these effects, and nothing more. That is what really interested Nietzsche. According to him, one of the fundamental questions of philosophy (maybe the basic one) is not ‚What is truth?', but rather ‚What do we do with our truths?', ‚Which are the effects of our truths on us?'. Nietzsche thus modifies the core of the old worldview, which was grounded on the first of these questions, on the belief that there was one absolute Truth, and that it could be discovered. Nietzsche rejects that metaphysical principle, but in so doing he only criticizes the character that we traditionally attribute to truth. If we closely consider his criticism towards truth, we never find the rejection of it as tool for the human being's orientation. Nietzsche indeed never thought that we could live and act without referring points; during his whole life he just stressed that these referring points are not absolute and unchanging. His heavy attack against the Western metaphysics is thus aimed at finding an alternative way to the nihilistic drift of 19th century Europe. That way starts from the detection of the relative value of the old truths, without involving a rejection of their usefulness. Nietzsche's pragmatism is therefore necessarily related to his perspectivism, and that must be stressed in order to contrast the interpretations of his thought which make him a relativist in the negative 43 GD is basically devoted to that topic. 89Nietzsche on Truth: a Pragmatic View? sense of this word. In a way similar to James, Nietzsche finds in the evaluation of the practical effect of a concept the tool to give to the human being the referring points that he needs in order to live. Both Nietzsche and James react to the 19th century crisis that involved the whole European culture, and they both find a way to avoid the danger of a complete disorientation. In so doing, Nietzsche and James follow the example of the new epistemology, which was aimed at helping science in carrying on its task of providing a highly explanatory world-description. Even though thinkers such as Mach revealed the relative value of the scientific notions, they still needed a criterion to evaluate the result of their researches, in order to avoid the whole building's collapse. That is exactly what we can find beneath Nietzsche's theory of knowledge, and that is why we can talk about pragmatism in referring to it. As a conclusion, let me just stress that the difference between Nietzsche's pragmatism and that of James can be evaluated in their interest in human life. The aims of these thinkers are different: while James is primarily a scientist, and thus shows interest for the purely theoretical side of the theory of truth (which, then, has important consequences on the practical plane), Nietzsche is chiefly interested in the effect that a worldview can have on the human being. We must therefore refer to the existential plane, in order to assess the truth-value of a worldview: the ‚truer' one – that which we should assume as ground of our agency – is the worldview which helps us becoming ‚who we are', and thus makes possible the development of a higher type of man.

Book Review: The Species Problem by Richard Richards∗ (Forthcoming in Mind) Makmiller Pedroso [email protected] Philosophy & Religious Studies, Towson University Richards' book provides an account of how we should divide the organic world into species, such as Homo sapiens and Escherichia coli. This is a timely topic with substantial ramifications inside and outside of philosophy. Our understanding of the history of life is often expressed in terms of the origin and the extinction of species. In policymaking, the Endangered Species Act understands species as the units of biodiversity and conservation. A conception of species even affects the issue of whether there is such a thing as human nature. Yet, despite the significance of the concept of species, there is no consensus on what species are. Over 20 species definitions are in circulation, and they often disagree with each other over which groups are species. This is the species problem: "there are multiple, inconsistent ways to divide biodiversity into species on the basis of multiple, conflicting species concepts, without any obvious way of resolving the conflict" (p. 5). And Richards' book offers no less than a solution to the species problem. Richards' book contains two parts. In the first part Richards introduces the species problem through its history, from Aristotle to the modern species concepts (chapters 2 to 4). In the second half of his book, Richards develops his own solution to the species problem (chapters 5 to 7). The historical chapters criticize the "Essentialist Story," a view about the history of the species problem widely endorsed by both philosophers and biologists. The Essentialist Story has two tenets. The first one is that the pre-Darwinian species concepts ∗I would like to thank Marc Ereshefsky, Kerry McKenzie, Frank Jankunis and Justin Caouette and for their helpful suggestions. 1 were mostly essentialist: species were defined by a set of properties that all and only members of a species must have. The second tenet is that one of Darwin's main contributions was to show that species essentialism is inadequate. While essentialists viewed species as discrete and unchanging units, Darwin regarded species as composed of variable populations that gradually evolve over time. Because of the essentialist consensus before Darwin, the Essentialist Story implies that the current species problem is partly due to the rejection of essentialism after Darwin. Richards offers an alternative picture. He contends that the often cited examples of species essentialists before Darwin, including Aristotle and Linnaeus, were not essentialists after all (chs. 2, 3). Rather than being confronted with an essentialist consensus, Darwin's challenge was adjudicating conflicting non-essentialist conceptions of species. Darwin and the naturalists before him were facing a similar species problem that we are today. Richards' solution to the species problem builds on Richard Mayden (1997) and Kevin de Queiroz' (1999) distinction between operational and theoretical species concepts (ch. 5). Theoretical concepts describe what species are; operational concepts tell us how to identify them. Richards contends that the resilience of the species problem is partly because we use the same standards to evaluate operational and theoretical species concepts (pp. 143– 144). The theoretical concept should be universal, "applying across biodiversity as much as possible, to sexual and asexual species organisms, vertebrates, invertebrates, bacteria and fungi" (p. 142). In contrast, there should be as many operational concepts as we can come up with. "The more, the merrier," says Richards (p. 139). Briefly, Richards is a monist about theoretical concepts but a pluralist regarding operational concepts. Following de Queiroz and Mayden, Richards endorses the theoretical concept according to which species are segments of population lineages-i.e., species are formed by a line of ancestry and descent of populations. Richards is not committed to a particular account of how 'population lineages' evolve and how we identify them. For him this is the role of operational concepts: operational concepts distinguish population lineages through organismal 2 features such as morphology, genotype, and mating preferences (p. 135ff.). By not specifying the evolutionary processes that produce lineages, Richards ensures the universality of his theoretical species concept: "[t]he idea is that in order to accommodate all kinds of organisms, the primary theoretical concept must not specify which processes are responsible for the populations that form the lineages and segment them into species" (p. 134). Richards claims that the adequacy of his position should be a result of biological practice rather than a priori philosophizing (p. 132, 135). Richards' maneuver is compelling, but it raises further questions concerning how biological practice can warrant his view. As Richards admits, it is not clear that microbial species satisfy his species concept (p. 142). An ongoing debate in microbiology is whether or not species should be defined in terms of lineage segments (Ereshefsky, 2010). Members of different microbial species can exchange genes by a mechanism called 'lateral gene transfer.' Depending on the proportion of genes laterally acquired from other species, species lineages may only track the history of a small portion of a species' genome, casting doubt on whether species are best viewed as lineage segments (Doolittle and Zhaxybayeva, 2009). However, even if you do not accept Richards' position, his book provides a valuable framework for examining how species concepts and evolutionary theory should relate to each other-including the relevance of lateral gene transfer to species definitions. Richards' book makes a significant contribution to the species problem debate, by integrating both the historical and contemporary developments of this debate. Richards has kept his book accessible to newcomers by carefully introducing the species problem. While developing his own view on the subject, he also discusses major topics in philosophy, such as theories of reference and meaning, unification in science, and naturalistic metaphysics. His book illustrates how philosophical theories can be fruitfully applied to conceptual problems outside of philosophy. Given the wide range of issues discussed in the book (from theories of reference to human nature), Richards' book will be of interest to not only philosophers of biology but philosophers more generally. 3 References de Queiroz, K. (1999). "The general lineage concept of species and the defining properties of the species category." In: Species: New Interdisciplinary Essays. Ed. by R. Wilson. Cambridge: MIT Press, pp. 49–90. Doolittle, W. and O. Zhaxybayeva (2009). "On the origin of prokaryotic species." Genome Research 19, pp. 744–56. Ereshefsky, M. (2010). "Microbiology and the species problem." Biology and Philosophy 25, pp. 553–568. Mayden, R. (1997). "A hierarchy of species concepts: the denouement in the saga of the species problem." In: Species: the Units of Biodiversity. Ed. by M. Claridge, A. Dawah, and M. Wilson. London: Chapman & Hall, pp. 381–424. Richards, R. (2010). The species problem: a philosophical analysis. Cambridge: Cambridge University Press.

Bayesianism for non-ideal agents Mattias Skipper ⋅ Jens Christian Bjerring Penultimate draft, forthcoming in Erkenntnis Abstract: Orthodox Bayesianism is a highly idealized theory of how we ought to live our epistemic lives. One of the most widely discussed idealizations is that of logical omniscience: the assumption that an agent's degrees of belief must be probabilistically coherent to be rational. It is widely agreed that this assumption is problematic if we want to reason about bounded rationality, logical learning, or other aspects of non-ideal epistemic agency. Yet, we still lack a satisfying way to avoid logical omniscience within a Bayesian framework. Some proposals merely replace logical omniscience with a different logical idealization; others sacrifice all traits of logical competence on the altar of logical non-omniscience. We think a better strategy is available: by enriching the Bayesian framework with tools that allow us to capture what agents can and cannot infer given their limited cognitive resources, we can avoid logical omniscience while retaining the idea that rational degrees of belief are in an important way constrained by the laws of probability. In this paper, we offer a formal implementation of this strategy, show how the resulting framework solves the problem of logical omniscience, and compare it to orthodox Bayesianism as we know it. Keywords: Bayesianism ⋅ Logical omniscience ⋅ Bounded rationality ⋅ Logical learning 1 Introduction Keep your degrees of belief probabilistically coherent at all times, and update them by conditionalization as new information comes in. So the orthodox 1 Bayesian story goes. The story is picture-perfect. It draws the contours of an ideal epistemic life. A life where all tautologies are believed with certainty, and where an agent's confidence never drops across entailment; one where logical perfection is just a matter of good epistemic housekeeping. But for ordinary humans like you and me, life isn't perfect. For us, and other imperfect beings like us, probabilistic coherence remains an unattainable ideal. As hard as we may try, we will never unveil all tautologies or recognize all entailment relations. Inevitably, we will fall short of logical perfection. Not by choice, to be sure. Most of us would take great pride in being able to prove the Riemann hypothesis or decide whether P = NP. We simply can't. After all, we are only human; logical imperfection is part of our condition as cognitively limited beings. Does this elementary fact about our epistemic predicament spell doom for orthodox Bayesianism? Not straightforwardly. Epistemic ideals are interesting in their own right, and deserve our attention no less than than do moral or political ideals.1 But if we want to reason about bounded rationality, logical learning, or other aspects of non-ideal epistemic agency, the message seems clear: 'probabilistic coherence must go!' The message has not gone unheard. Many formally inclined epistemologists have viewed the commitment to probabilistic coherence as one of the most serious problems for orthodox Bayesianism-one that has come to be known as the problem of logical omniscience.2 Don't get us wrong: Ordinary humans need not be careless or epistemically irresponsible. We often engage successfully in logical reasoning, not only when we sweat over a logic exam, but also when we deliberate about which decisions to make in day-to-day life. Suppose you ponder whether to ask your boss for a pay raise today. You know that your boss is in a generous mood only if she took the bike to work. Yet, you don't see her bike in the bike rack. Thus, you decide to defer your request for another day. This sort 1Although it is a matter of contention whether probabilistic coherence is indeed a rational ideal. For more on this point, see Christensen (2007), Smithies (2015), and Titelbaum (2015). 2Cf. Easwaran (2011) and Talbott (2016). See also Fagin et al. (1995) for a discussion of logical omniscience as it arises in epistemic and doxastic logic. 2 of basic ability to engage in logical reasoning should, we submit, feature in a solution to the problem of logical omniscience. It is not enough to model agents who fall short of logical omniscience; we also need to capture the sense in which such agents nevertheless display some level of logical competence. Over the past half century, several attempts have been made at solving the problem of logical omniscience. But none has gained widespread acceptance. Some proposals merely replace logical omniscience with a different logical idealization; others sacrifice all traits of logical competence on the altar of logical non-omniscience (§2). We think a better strategy is available: by enriching the Bayesian framework with tools that allow us to capture what agents can and cannot infer given their limited cognitive resources, we can avoid logical omniscience while retaining the idea that rational degrees of belief are in an important way constrained by the laws of probability (§3). As we will see, this 'dynamic' approach to the problem is not without substantive commitments; it forces us to reconsider some fundamental principles of orthodox Bayesianism (§4). Yet it offers what we take to be a principled and intuitive solution to a long-standing problem in Bayesian epistemology. 2 Existing approaches to logical omniscience Let us begin by taking a closer look at the two main sources of logical omniscience within orthodox Bayesianism:3 Classical Preservation: For any two propositions, A and B, if A logically entails B, then Cr(A) ≤ Cr(B). Classical Normality: For any tautology T , Cr(T ) = 1. According to Classical Preservation, an agent's credences never drop across entailment. For example, someone who is 80% confident that "it's raining" will be at least 80% confident that "it's raining or snowing." More generally, whenever a proposition, A, entails another proposition, B, an agent will be at least as confident of B as of A, however difficult it may be to see that 3Both principles follow from the Kolmogorov axioms; see Earman (1992) and Titelbaum (forthcoming) for relevant background. 3 B follows from A. According to Classical Normality, agents are certain of all tautologies. For example, they will be certain that "it's either raining or not." More generally, for any tautology, T , an agent will be certain of T , however difficult it may be to see that T is tautological. Anyone who seeks to avoid logical omniscience must find a way to drop (or relax) Classical Preservation and Classical Normality. But how? In the rest of this section, we survey and criticize three prominent answers. 2.1 Garber on 'Local Bayesianism' In his "Old Evidence and Logical Omniscience in Bayesian Confirmation Theory" (1983), Daniel Garber attempts to solve the problem of logical omniscience by making room for logical learning in the Bayesian framework.4 Logical learning is not possible within orthodox Bayesianism, since Classical Preservation and Classical Normality tell us that agents already know everything there is to know about logic; and you cannot learn what you already know. Thus, while orthodox Bayesianism can accommodate empirical learning (in terms of conditionalization on incoming evidence), it rules out the possibility of logical learning. Garber seeks to rectify this situation by developing what he calls a local version of the Bayesian framework. The idea is to think of an agent's credence function as being defined over a problem-relative language, which consists of (the truth-functional closure of) those sentences that the agent is concerned with when being engaged in a given "inferential problem" (Garber 1983, p. 111). For example, the local language of a 17th-century physicist might contain vocabulary from Newtonian mechanics, but will presumably not contain vocabulary from quantum mechanics. In addition to these problem-relative sentences, Garber also includes sentences of the form 'A ⊢ B' in the agent's local language. Informally, these sentences are to be interpreted as 'A entails B'. But they are treated as atomic 4In a broader perspective, Garber's approach to logical omniscience forms part of an attempt to solve the so-called 'problem of old evidence' introduced by Glymour (1980). Related approaches to this problem can be found in Gaifman (2004), Good (1968), and Jeffrey (1983). See also Fitelson and Hartmann (2015) and Sprenger (2015) for more recent developments in this direction. 4 sentences in the formalism, and may therefore be assigned non-extreme credences. The idea, then, is to make room for logical learning by allowing agents to update on sentences of the form 'A ⊢ B' that they are not already certain of. For example, a physicist might at one point have a non-extreme credence that quantum mechanics entails such-and-such evidence, but might later discover that this entailment relation in fact obtains. For present purposes, let us say that A locally entails B iff B can be derived from A in the agent's local language, and let us say that T is a local tautology iff T is a tautology in that language. We can then think of Garber's proposal as replacing Classical Preservation and Classical Normality with the following two principles: Local Preservation: For any two sentences, A and B, if A locally entails B, then Cr(A) ≤ Cr(B). Local Normality: For any local tautology T , Cr(T ) = 1. Assuming that all local entailments are logical entailments, but not vice versa, Local Preservation is strictly weaker than Classical Preservation. Likewise, assuming that all local tautologies are logical tautologies, but not vice versa, Local Normality is strictly weaker than Classical Normality. Thus, Garber's proposal offers a way around the classical assumption of logical omniscience. But although Local Bayesianism does not give rise to full-blown logical omniscience as we know it from orthodox Bayesianism, it still carries a commitment to a problematic kind of logical idealization. Ellery Eells puts the point as follows: "[T]here will always be extremely complex logically true sentences of the local language [for which] it will be inappropriate to insist on probability 1" (Eells 1985, p. 241). The observation here is that the local language is still closed under truth-functional operations, which means that even a very sparse characterization of the local language will give rise to very complicated logical relations that go far beyond the cognitive reach of ordinary agents. As such, Local Bayesianism merely replaces the classical assumption of logical omniscience with a different logical idealization. It is also worth pointing out that even if we could define a local language 5 without any overly complex logical relations, there are general grounds for doubting that the resulting model would adequately capture the sense in which ordinary humans fall short of logical omniscience. The central reason why such agents fail to be logically omniscient, we take it, is not that they operate with a restricted language, but rather that they have limited cognitive resources available to reason logically in that language. Thus, it seems to us that the strategy of defining an agent's credence function over a restricted language does not ultimately get at the heart of the matter. 2.2 Hacking on 'Personal Probability' In his "Slightly More Realistic Personal Probability" (1967), Ian Hacking attempts to solve the problem of logical omniscience by developing a 'personalized' version of the Bayesian framework. The idea is to replace the classical entailment relation with a 'personal' entailment relation, which is strictly weaker (that is, all personal entailments are logical entailments, but not vice versa). Personal entailment is defined in terms of a notion of personal possibility: just as A logically entails B iff A∧¬B is not logically possible, A personally entails B (for a given agent) iff A ∧ ¬B is not personally possible (for that agent). When is a proposition personally possible for a given agent? Intuitively, says Hacking, when the agent does not know that the proposition is false (Hacking 1967, p. 318). In other words, A is personally possible for an agent iff A cannot be ruled out by the agent given his or her empirical information and deductive abilities. For example, suppose that B is a highly complex logical consequence of A, which lies far beyond the cognitive reach of any human. Given this, the notion of personal possibility will come apart from logical possibility: although A ∧ ¬B is not logically possible, it will be personally possible for agents who are unable to recognize that A and ¬B are in fact logically incompatible. Accordingly, personal entailment comes apart from logical entailment in this case: while A logically entails B, it does not personally entail B. For present purposes, let us supply the notion of personal entailment with 6 a notion of a personal tautology: T is a personal tautology (for an agent) iff ¬T is not personally possible (for that agent). We can then think of Hacking's proposal as replacing Classical Preservation and Classical Normality with the following two principles: Personal Preservation: For any two sentences, A and B, if A personally entails B, then Cr(A) ≤ Cr(B). Personal Normality: For any personal tautology T , Cr(T ) = 1. Assuming that all personal entailments are logical entailments, but not vice versa, Personal Preservation is strictly weaker than Classical Preservation. Likewise, given that all personal tautologies are logical tautologies, Personal Normality is strictly weaker than Classical Normality. Thus, like Garber's proposal, Hacking's proposal offers a way around the classical assumption of logical omniscience. Just how weak are Hacking's principles? The answer is: extremely weak! The reason for this is that Hacking treats personal entailment as a "degenerate concept with no closure conditions" since such closure conditions (say, closure under modus ponens) tend to "lead disastrously near the divine sense of knowledge" (Hacking 1967, p. 319). That is, to eliminate all traits of logical omniscience, Hacking assumes that no logical entailment, however trivial, need count as a personal entailment. The result is a model that shows no sign of logical competence. In response to this sort of worry, Hacking submits that the notion of personal probability is not, in fact, as 'laissez faire' as one might think. He writes: [D]oes not the slightly more realistic theory excuse a man from any cogent reasoning whatsoever? No. In the classical [Bayesian] theory, the Dutch Book argument is used to club a man into reasoning. There may be a better club to hand. (Hacking 1967, p. 322) The thought here is that, even if ordinary humans cannot be faulted for being susceptible to a dutch book made by a logically omniscient Bookie, they can 7 be faulted for being susceptible to a dutch book made by a Bookie who is epistemically on a par with them. If so, we can still rely on Dutch book considerations to justify certain non-ideal rationality constraints. Unfortunately, we don't think that the appeal to non-ideal Bookies provides us with the necessary resources to model agents who are logically competent in the relevant sense. As Ellery Eells rightly points out: [Hacking's proposal] has the consequence that if the agent is unaware of an incoherence in his subjective probabilities, then so must be an appropriate betting opponent. But this means that a person will turn out to be rational in the Bayesian sense as long as the person is not aware of an incoherence. (Eells 1985, p. 217) In other words, as long as there are no general constraints on what can be personally possible for an agent, there is no incoherence so obvious that an agent must be aware of it to be rational. As such, nothing in Hacking's framework reflects the sense in which ordinary humans are logically competent. Nevertheless, we think that Hacking is halfway right. He is wise not to impose any substantive closure constraints on epistemic states, since these give rise to a problematic kind of logical omniscience. What is missing from his picture is a way of capturing what ordinary humans can and cannot infer given their limited cognitive resources. 2.3 Hintikka on 'Impossible Possible Worlds' In his "Impossible Possible Worlds Vindicated" (1975), Jaakko Hintikka tries to solve the problem of logical omniscience by developing an impossible-worlds model of belief, which extends his earlier work on doxastic and epistemic logic.5 While Hintikka's own exposition centers around "all-or-nothing" belief rather than credences, we can straightforwardly translate it into Bayesian terms. To put Hintikka's proposal in its proper context, let us begin by considering his original possible-worlds model for belief: 5Hintikka (1962). See also von Wright (1951) for an early precursor to Hintikka. 8 Belief: An agent believes a proposition, A, iff A is true at all possible worlds that are doxastically possible for the agent. As Hintikka himself recognized, this model carries a commitment to logical omniscience: it implies that any agent believes all logical consequences of what they believe, including all tautologies. To see this, suppose that you believe A, and let B be any logical consequence of A. Since you believe A, A is true at all possible worlds that are doxastically possible for you. And since A entails B, all possible worlds that verify A also verify B. Thus, B must be true at all doxastically possible worlds for you, which means that you believe B. To avoid this result, Hintikka suggests that we extend the set of possible worlds with a set of impossible worlds: worlds that "look possible and hence must be admissible as epistemic alternatives but which none the less are not logically possible" (Hintikka 1975, p. 477).6 The basic idea is that, for limited agents, the space of doxastic possibilities is larger than the space of logical possibilities. For example, even if you believe each of the Peano Axioms, you might well fail to believe Goldbach's Conjecture, even if the former in fact entail the latter. In other words, even if the Axioms are true at all doxastically possible worlds for you, the Conjecture need not be. Motivated by this thought, Hintikka suggests the following modification of the original possible-worlds model of belief: Belief-impossible: An agent believes a proposition, A, iff A is true at all worlds (whether possible or impossible) that are doxastically possible for the agent. By quantifying over impossible as well as possible worlds, we can make room for agents who believe the Peano Axioms, but fail to believe Goldbach's Conjecture. For even if the Axioms are true in all doxastically possible worlds, the Conjecture need not be, as long as impossible worlds are allowed to verify the Axioms without verifying the Conjecture. When translated into a credence framework, Hintikka's proposal becomes: 6For other early discussions of impossible worlds, see Cresswell (1973) and Rantala (1982). See also Berto and Jago (2018; 2019) for relevant background. 9 Credence-impossible: An agent's credence in a proposition, A, is x iff the probabilities of those A-worlds (whether possible or impossible) that are doxastically possible for the agent sum up to x. Like Belief-impossible, Credence-impossible offers a way around the assumption of logical omniscience. Even if you are certain that the Peano Axioms are true, you need not, according to Credence-impossible, be certain that Goldbach's Conjecture is true. After all, there might be doxastically possible (but logically impossible) worlds that verify the Axioms, but not the Conjecture. As stated, Credence-impossible says nothing about the nature of impossible worlds. Hence, we can get different versions of the impossible-worlds model by adopting different underlying conceptions of impossible worlds. Following Berto and Jago (2018), we can distinguish between two general such conceptions-what they call the 'Australasian stance' and the 'American stance' on impossible worlds. According to the Australasian stance, impossible worlds are allowed to violate the laws of classical logic, but must still respect the laws of some non-classical logic (say, intuitionistic or paraconsistent logic).7 According to the American stance, impossible worlds are not subject to any closure constraints whatsoever, but may be arbitrarily logically ill-behaved.8 Both these stances give rise to problems that are structurally very similar to those that led us to reject the proposals by Garber and Hacking. Consider first the Australasian stance. Suppose that all impossible worlds must respect the laws of some non-classical logic, L. Given this, we can replace Classical Preservation and Classical Normality by the following two principles: Non-classical Preservation. For any two propositions A and B, if A entails B in L, then Cr(A) ≤ Cr(B). Non-classical Normality. For any tautology T in L, Cr(T ) = 1. Since L is assumed to be strictly weaker than classical logic, these principles do not imply that agents are omniscient within classical logic. But they do 7See, e.g., Fagin et al. (1995), Levesque (1984), and Lakemeyer (1987). 8See, e.g., Nolan (1997). 10 imply that agent are omniscient within the chosen non-classical logic, L. For example, if we understand L as an intuitionistic logic, agents are assumed to be certain of all intuitionistic tautologies and entailment relations. Yet, just as ordinary humans fall short of classical omniscience, they fall short of intuitionistic omniscience as well. Indeed, even if we grant that a particular non-ideal agent reasons intuitionistically rather than classically, the agent obviously cannot reason unlimitedly in that logic. Thus, the Australasian stance faces much the same problem as Garber's Local Bayesianism: it merely replaces logical omniscience with a different logical idealization. To add fuel to the fire, we do not think the Australasian stance provides us with the resources to capture the sense in which ordinary humans are logically competent. The central reason why we fall short of logical omniscience, it seems, is not that we operate with a non-classical notion of entailment, but rather that we have limited cognitive resources available for logical reasoning. Thus, it seems that the Australasian stance, like Garber's approach, does not ultimately get at the heart of the problem. What about the American stance? Suppose that impossible worlds are not subject to any closure constraints whatsoever. That is, impossible worlds are allowed to verify A without verifying B, for any A and B. Given this, we can eliminate all traits of logical omniscience: an agent's confidence need not be preserved across any entailments, however trivial; and the agent need not be certain (or even moderately confident) of any tautologies, however obvious. The problem, of course, is that logical anarchy not only eliminates logical omniscience, but also breeds logical incompetence. More specifically, if our model allows an agent's credences to be arbitrarily logically ill-behaved, we are left with no way of capturing the logical reasoning abilities of ordinary humans. Thus, the American stance faces much the same problem as Hacking's approach: it sacrifices all traits of logical competence on the altar of logical omniscience. In light of these problems, it is very natural to search for a 'middle way' between the Australasian stance and the American stance on impossible worlds. In particular, it is natural to think that we should try to close 11 an agent's doxastic state under a notion of logical entailment that-unlike classical entailment and, say, intuitionistic entailment-reflects the agent's limited cognitive resources. At the level of worlds, this amounts to saying (roughly) that impossible worlds should be closed under a notion of partial logical consequence: they should verify everything that lies within the agent's cognitive reach, and nothing more. The hope is to thereby be able to replace Classical Preservation and Classical Normality with the following two principles: Partial Preservation. For any propositions A and B, if A partially entails B, then Cr(A) ≤ Cr(B). Partial Normality. For any partial tautology T , Cr(T ) = 1. As stated, these principles are obviously not fully precise, since we have not given a precise definition of partial entailment. But the idea is clear enough: the principles are supposed to avoid the assumption of logical omniscience while retaining an appropriate level of logical competence. Unfortunately, the natural idea cannot be made to work. The problem, in a nutshell, is that any notion of partial entailment 'collapses' into full entailment. Here is a way of illustrating the mechanism behind this sort of collapse: Consider a very minimal partial closure constraint, which says that impossible worlds should at least obey those entailment relations that are trivial or obvious for ordinary humans to recognize. That is, if A trivially entails B, every impossible world that verifies A must verify B as well. What counts as 'trivial' or 'obvious' is clearly a vague matter. But nothing turns on the vagueness: regardless of how we make the notion of a 'trivial logical entailment' precise, it turns out that we cannot close impossible worlds under trivial logical consequence without closing them under full logical consequence. Here is why: let w be any world that is not closed under full logical consequence. That is to say, there exists two propositions, A and B, such that the following three conditions are met: (i) A entails B; (ii) w verifies A; and (iii) w does not verify B. By (i), we can consider a sequence of propositions A, S1, S2, . . . , B corresponding to a step-by-step inference from A to B in 12 terms of simple logical rules such as conjunction elimination, modus ponens, and the like. By (ii) and (iii), we know that w must violate at least one step in this inference. Yet, each step in the inference is exceedingly simple, and so must count as trivial for ordinary humans, if anything does. It follows, then, that w cannot be closed under trivial logical consequence. Upshot: if w is closed under trivial logical consequence, w must be closed under full logical consequence. Intuitively, it collapses under its own deductive weight.9 In sum, there is no stable 'middle way' between the Australasian stance and the American stance. Even the most minimal closure constraints on impossible worlds are too strong. What to do about this? 3 A Dynamic Bayesian Framework The foregoing considerations urge us to choose between two evils: logical omniscience or logical incompetence-which shall it be? Neither! Or so we submit. The dilemma arises when we try to model logical competence in terms of closure constraints on doxastic states. But we need not restrict ourselves to this 'static' way of modeling logical competence. A better strategy is available: if we enrich the Bayesian framework with tools that allow us to model how an agent's doxastic state can change as a result of engaging in logical reasoning, we can steer clear of logical omniscience and logical incompetence at the same time. In previous work, we have used this 'dynamic' strategy to solve the problem of logical omniscience as it arises within doxastic and epistemic logic (Bjerring & Skipper forthcoming). Here we want to show how the same basic strategy can be extended to a Bayesian context. Before we get into the details, let us clarify how the positive proposal outlined below should be seen in relation to the collapse result discussed in the previous section. Our claim is not that we can avoid the collapse result by moving into a dynamic framework. Indeed, if the foregoing considerations are right, this cannot be done. Rather, our dynamic framework will offer a 9For related discussions of similar collapse results that arise in the contexts of epistemic logic, decision theory, and formal semantics, see Bjerring (2013), Bjerring and Skipper (forthcoming), Bjerring and Schwarz (2017), Elga and Rayo (ms.), Jago (2013), and Rasmussen (2015). 13 way out of the more generic dilemma between logical omniscience and logical incompetence, which is compatible with the collapse result. That is the claim, anyway. The rest of the section makes the case. Our first task is to define a quantitative measure of an agent's cognitive resources, which can be implemented in a Bayesian framework. The exact choice of measure is of little importance for present purposes. Our aim is not to give an empirically accurate representation of how human beings reason logically. Rather, we are looking to provide a general, empirically noncommittal way of capturing the elementary fact that some logical inferences are more complex or difficult to perform than others. That is, we need a way of distinguishing the Kurt Gödels of this world from his less resourceful fellow earthlings. In principle, many different measures could do this job, whether they appeal to time consumption, neural activity, or some third quantity. However, in line with previous work, we will use a simple step-based model of bounded logical reasoning.10 The basic idea is to represent an agent's cognitive resources by a number, n, which corresponds to the number of inference steps that the agent is able to perform. By varying the value of n, we can then generate a whole spectrum of agents with different levels of cognitive resources. When n = 0, no chain of logical reasoning, however simple, lies within the agent's cognitive reach. Intuitively, the agent has no cognitive resources whatsoever. When n approaches infinity, no chain of logical reasoning, however complex, lies beyond the agent's cognitive reach. Intuitively, the agent has unlimited cognitive resources. For intermediate values of n, some but not all chains of logical reasoning lie within the agent's cognitive reach. Intuitively, the agent is neither logically omniscient nor logically incompetent. This is the part of the spectrum that we will mainly be interested in. The step-based model goes along with a broadly rule-based picture of logical reasoning. On such a picture, agents reason logically by applying rules from a designated set, R, of available inference rules. For illustrative 10Bjerring & Skipper (forthcoming). The step-based model was initially inspired by work in active logic; see Elgot-Drapkin and Perlis (1990) for background. 14 purposes, we will think of R as containing familiar inference rules such as conjunction elimination, modus ponens, disjunction introduction, and the like. But on the official story, we do not presuppose any particular specification of R. There are two reasons for this. First, by leaving the specification of R open, our framework will be applicable to different contexts, which call out for different specifications of R. In particular, our framework will not be limited to contexts that call out for a sound and complete proof system of classical logic. Second, our framework is not going to stick its neck out with respect to substantive, empirical questions about human cognition. This is not to say that such questions are entirely orthogonal to the present project. Indeed, we suspect that certain applications of our framework may require an empirically informed reasoning mechanism. But for the purposes of laying out the basic framework, there is no reason to sacrifice generality for empirical accuracy. Henceforth, then, we will think of logical competence as the ability to perform up to n steps of reasoning using the rules in R. In the formalism, we will implement this idea by defining a 'non-ideal' notion of logical entailment, which we call n-entailment: n-entailment: A set of sentences, Γ, n-entails a sentence, A, (written 'Γ ⊢nR A') iff A can be inferred from Γ within n applications of the inference rules in R. The role of the ⊢nR-relation is to capture what logical entailments lie within an agent's cognitive reach; and which do not. To get a feel for the definition, suppose that R contains just a single rule: conjunction elimination. Given this, A is 1-entailed by A ∧B, and A ∧B is 1-entailed by (A ∧B) ∧ (A ∧B), but A is not 1-entailed by (A ∧B) ∧ (A ∧B), since it takes two applications of conjunction elimination to infer A from (A ∧B) ∧ (A ∧B). Three properties of the ⊢nR-relation are worth noting. First, as illustrated by the example above, n-entailment is not transitive (in contrast to classical entailment): even if A n-entails B, and B n-entails C, A need not n-entail C. Such transitivity failures are going to play an important role in our solution to the problem of logical omniscience. 15 Second, the ⊢nR-relation is monotonic in n: if A i-entails B, A also jentails B, for i ≤ j. The reason is trivial: any inference that can be carried out in i steps or less can also be carried out in j steps or less, for i ≤ j. The opposite is obviously not the case. So, even if A j-entails B, A need not i-entail B. Third, n-entailment is equivalent to classical entailment in the special case where n goes to infinity and R is a sound and complete proof system of classical logic. Although not our main target, this effectively means that our framework is general enough to model agents who are logically omniscient in the classical sense. Our next task is to define the formal language over which we will define our models. So, let L be a probabilistic modal language with atomic sentences p1, p2, . . . , negation ¬, conjunction ∧, weak inequality ≤, a credence function Cr, and a countably infinite set of 'dynamic' operators ⟨n⟩ and [n] (for n = 0, 1, 2, . . . ).11 In addition to the atomic sentences, L contains the following sentence types: ¬A ∣ A ∧B ∣ Cr(A) ≤ x ∣ ⟨n⟩A ∣ [n]A, where x is a real number in the unity interval [0, 1], and A and B are arbitrary sentences of L. We will help ourselves to other familiar connectives (∨,→, . . . ) and (in)equalities (=,<, . . . ), which can be defined in the usual way from the primitive language. The dynamic operators have the following intended readings: ⟨n⟩A: after some n-step reasoning process, A is the case. [n]A: after any n-step reasoning process, A is the case. By combining the dynamic operators with the credence function, we can write things like '⟨n⟩(Cr(A) < .7)' to say that the agent is less than 70 % confident of A after some n-step reasoning process, or '[n](Cr(A) = 1)' to say that the agent is certain of A after any n-step reasoning process. 11Slightly abusing notation, we will use 'L' both as the name of our object language and as a variable that ranges over all sentences of that language. 16 To develop a semantics for L, we will combine some familiar tools from probabilistic modal logic with some more recent developments in dynamic epistemic logic.12 We begin with the notion of a subjective probability space: Subjective probability space: Let W P and W I be finite, non-empty sets of possible and impossible worlds, and let W = W P∪W I . A subjective probability space is a pair, (S, Pr), where S ⊆ W is a non-empty set of worlds, and Pr ∶ S ↦ [0, 1] is a distribution over S such that ∑S Pr(S) = 1. The set of all subjective probability spaces is denoted by S. As usual, we think of a subjective probability space as a representation of an agent's doxastic situation at a given time. The worlds in S are those that are doxastically possible for the agent. That is, for each w ∈ S, w might be the actual world for all the agent can tell given his or her cognitive resources and empirical information. The distribution, Pr, encodes information about how probable the agent takes it to be that a given world is actual. For example, if Pr(w) < Pr(w′), the agent takes it to be more probable that w′ is actual than that w is. Since the agent is certain that some world in S is actual, we require that ∑S Pr(S) = 1. We extend the notion of a subjective probability space to a full probabilistic model as follows: Probabilistic model: A probabilistic model is a tuple, M = (W P , W I , f, V ), where f ∶ W ↦ S assigns a subjective probability space to each world in W , and V ∶ W ↦ 2L assigns a set of sentences in L to each world in W . The function f tells us what the agent's subjective probability space looks like at different worlds. In general, f will assign different subjective probability spaces to different worlds, since an agent's doxastic situation differs 12For background on dynamic epistemic logic, see Ditmarsch et al. (2008) and Baltag and Renne (2016). 17 from world to world. The function V serves as a 'labeling device' that associates each world in W with a set of sentences in L. As we will see, V is going to behave as a standard valuation function at possible worlds, but will behave non-standardly at impossible worlds. For ease of exposition, we will henceforth say that w n-entails A iff V (w) n-entails A. In light of the collapse result, we do not want to impose any closure constraints on impossible worlds. Doing so would result in a problematic kind of logical omniscience. To avoid any such problems, we will instead adopt a highly liberal comprehension principle, as Nolan (1997) calls it, according to which no set of sentences is too logically ill-behaved to count as an impossible world. More precisely: Comprehension principle: For any incomplete and/or inconsistent set of sentences Γ ⊆ L, there is a world w ∈ W I such that V (w) = Γ. This effectively means that we will take an 'American stance' on impossible worlds. However, while the American-style approach discussed in the previous section fails to retain a proper measure of logical competence, our approach is going to avoid this pitfall. We now turn to the dynamic part of our semantic framework. To make it easier to parse the definitions below, it will be helpful to have the intuitive picture in mind. So, consider a simple sentence of the form '⟨n⟩(Cr(A) = 1)'. The semantics below will tell us that this sentence is true iff A is n-step inferable from every doxastically possible world. This is meant to capture the idea that the agent is in a position to become certain of A after having performed some n-step reasoning process. But note that even if the agent is capable of performing n steps of reasoning, she need not be in a position to become certain of A after some n-step reasoning process. After all, she might be uncertain of one or more of the premises involved in the relevant reasoning process. At the level of worlds, this amounts to saying that one or more of the premises might fail to be true at one or more doxastically possible worlds. More generally, then, our semantics will say that '⟨n⟩(Cr(A) = x)' is true iff the probabilities of those doxastically possible worlds that n-entail A sum up to x. 18 This informal characterization of our semantics already shows that the truth-conditions for '⟨n⟩(Cr(A) = x)' will be weaker than those for 'Cr(A) = x': while the semantics for 'Cr(A) = x' requires that the probabilities of those doxastically possible worlds that verify A sum up to x, the semantics for '⟨n⟩(Cr(A) = x)' merely requires that the probabilities of those doxastically possible worlds that n-entail A sum up to x. So, for example, the truthconditions for '⟨n⟩(Cr(A) = 1)' will be weaker than those for 'Cr(A) = 1': while the semantics for 'Cr(A) = 1' requires that A is true at every doxastically possible world, the semantics for '⟨n⟩(Cr(A) = 1)' merely requires that every doxastically possible world n-entails A. That's the intuitive picture; now for the formal details. We begin by defining a formal device that allows us to capture what is n-step inferable from a given world: n-radius: The n-radius of a world, w, is written 'wn' and is defined as follows: wn = ⎧⎪⎪⎪ ⎨ ⎪⎪⎪⎩ {w} for w ∈ W P . {w′ ∈ W I ∶ V (w) ⊆ V (w′) and V (w) ⊢nR V (w′)} for w ∈ W I . Each member of wn is called an n-expansion of w. Let us unpack this definition a bit. The idea is that any given world is associated with an 'n-radius', which is the set of 'n-expansions' of that world. Each n-expansion is itself a world; so the n-radius of a world is a set of worlds. Now, which worlds count as an n-expansion of w depends on whether w is possible or impossible. If w is possible, w is its own unique n-expansion, and so the n-radius of w is a singleton set: wn = {w}, for any n. This reflects the fact that possible worlds are deductively closed entities that already verify everything that follows from them in any number of steps. More interestingly, if w is impossible, w′ is an n-expansion of w iff the following three conditions are satisfied: (i) w′ is impossible; (ii) w′ verifies everything that w verifies; and (iii) everything that w′ verifies is n-step inferable from w. It follows that every impossible world is an n-expansion of itself, just as every 19 possible world is an n-expansion of itself. However, in contrast to possible worlds, impossible worlds generally have more than one n-expansion. For example, suppose that V (w) = {A→ B,¬B}, V (w1) = {A→ B,¬B,¬A} and V (w2) = {A → B,¬B,¬B ∨C}. Here w1 and w2 both count as 1-expansions of w (assuming that R contains modus tollens and disjunction introduction). Since the ⊢nR-relation is monotonic in n, any i-expansion of w is also a j-expansion of w, for i ≤ j. The opposite is obviously not the case. Thus, we can think of w0, w1, w2, . . . as a sequence of concentric circles that stand in the following subset relations: w0 ⊆ w1 ⊆ w2, . . . Assuming, as we will henceforth do, that no inference can be carried out in zero steps (except for the trivial inference 'A, therefore A'), the 0-expansion of a world contains just the world itself. That is: w0 = {w}, for any w ∈ W . In addition to the n-radius of a world, we will also need a way to pick out exactly one n-expansion from the n-radius of each doxastically possible world. The following choice function allows us to do just that: Choice function: Let C ∶ 22W ↦ 22W be a function that takes a set W = {W1, . . . , Wm} of sets of worlds as input and returns the set C(W ) of sets of worlds that results from all the ways in which exactly one element can be picked from each Wi ∈ W . Each member of C(W ) is called a choice of W . An example will help illustrate this somewhat cumbersome definition. Let W = {{w1},{w2, w3}}. A choice of W is a set of worlds formed by picking exactly one world from each member of W . In the case at hand, there are precisely two ways of doing so: we can either pick w1 and w2 or w1 and w3. Accordingly, the choice function maps W to a set containing those two choices: C(W ) = {{w1, w2},{w1, w3}}. While the choice function is defined in a highly general way, it will serve a much more specific purpose in what follows: it will allow us to capture all the different ways in which one can pick exactly one n-expansion of each 20 doxastically possible world from a given world. For present purposes, then, we can think of a choice as a set of worlds formed by picking exactly one n-expansion of each world in S. The reason why choices are important is that they will allow us to distinguish semantically between the dynamic operators, ⟨n⟩ and [n]. Here is the rough idea: consider again a simple sentence of the form '⟨n⟩(Cr(A) = 1)'. For such a sentence to be true, we do not want to require that every nexpansion of each doxastically possible world verifies A. We only want to require that at least one n-expansion of each doxastically possible world verifies A. By contrast, for '[n](Cr(A) = 1)' to be true, we do want to require that every n-expansion of each doxastically possible world verifies A. The same goes for all sentences of the form '⟨n⟩(Cr(A) = x)' and '[n](Cr(A) = x)'. Next up is the most central notion of our semantic framework, which will govern the truth-conditions of the dynamic operators. Let 'n∼' be a binary relation that holds between pairs of pointed models (where, as usual, a pointed model consists of a model and a world). If the relation holds between two pointed models, (M, w) and (M ′, w′), we write '(M, w) n∼ (M ′, w′)' and say that (M ′, w′) is n-accessible from (M, w). Since the formal definition of naccessibility will get a bit ugly, it will be helpful to begin with a sketch of the intuitive idea: Suppose that the pointed model (M, w) characterizes an agent's doxastic state at a given time. We then want to say that (M ′, w′) is n-accessible from (M, w) iff (M ′, w′) characterizes a doxastic state that the agent can enter by performing some n-step reasoning process. At the level of worlds, this amounts to saying that (M ′, w′) is n-accessible from (M, w) iff the set of doxastically possible worlds at w′ in M ′ corresponds to a choice of n-expansions of the doxastically possible worlds at w in M . To ensure that the n∼-relation behaves in the desired way, we need a way to replace a set of doxastically possible worlds with a choice of n-expansions of those worlds. The notion of an n-variation will allow us to do just that: n-variation: Let M = (W P , W I , f, V ) be a model, and let f(w) = (S, Pr) be the subjective probability space associated with (M, w). The function V arn (for n = 0, 1, 2, . . . ) associates (M, w) with a set of subjec21 tive probability spaces: V arn(M, w) = ⎧⎪⎪⎪ ⎨ ⎪⎪⎪⎩ f ′ ∈ S ∶ ⎧⎪⎪⎪ ⎨ ⎪⎪⎪⎩ f ′(w′) = fc(w′) for w′ = w f ′(w′) = f(w′) for w′ ≠ w ⎫⎪⎪⎪ ⎬ ⎪⎪⎪⎭ , where fc(w) = (c, Prc) is a subjective probability space such that c ∈ C({w′n∣w′ ∈ S}), and Prc is the unique probability distribution over c such that, for each w′ ∈ S, Prc(w′c) = Pr(w′), where w′c is the n-expansion of w′ in c. Each member of V arn(M, w) is called an n-variation of (M, w). The way to understand this definition is as follows: We start with a pointed model, (M, w), which is associated with a subjective probability space, f(w) = (S, Pr). This subjective probability space consists of a set of doxastically possible worlds, S, and a distribution over those worlds, Pr. Now we modify S in a particular way: we replace each doxastically possible world with an n-expansion of that world. That is, we replace S with a choice, c, of n-expansions of the worlds in S. We keep the distribution fixed: each of the chosen n-expansions is assigned the same probability that was assigned to its corresponding world in S. More precisely, for any v ∈ S, if v′ is the chosen n-expansion of v, we let Prc be a distribution such that Prc(v′) = Pr(v). We then define a function, f ′, such that f ′ is identical to f except at w, where f ′(w) = (c, Prc). What we have ended up with is an n-variation of (M, w). Three properties of the n-variation of a pointed model are worth keeping in mind. First, there are in general many different n-variations of a given pointed model; indeed, as many as there are ways of forming choices of nexpansions of the doxastically possible worlds. Second, if f ′ is an i-variation of (M, w), f ′ is also a j-variation of (M, w), for i ≤ j. Again, this follows from the fact that the ⊢nR-relation is monotonic in n. Third, since every world is an n-expansion of itself, f will itself be an n-variation of (M, w). We can now define the n∼-relation: n-accessibility: Let M = (W P , W I , f, V ) and M ′ = (W ′P , W ′I , f ′, V ′) be two models. Then (M, w) n∼ (M ′, w′) iff w′ = w, W ′P = W P , W ′I = W I , V ′ = V , and f ′ ∈ V arn(M, w). 22 According to this definition, (M ′, w′) is n-accessible from (M, w) iff the following two conditions are met: (i) f ′ is an n-variation of (M, w); and (ii) (M ′, w′) is otherwise identical to (M, w). By construction, the definition reflects the intuitive role that we wanted the n-accessibility relation to play: if (M, w) characterizes an agent's doxastic state at a given time, the n-accessible pointed models from (M, w) represent all the different doxastic states that the agent can enter by performing some n-step reasoning process. As such, the n∼-relation gives us a formally precise way of capturing what the agent can and cannot infer given her limited cognitive resources. The three properties that we highlighted about the notion of an n-variation carry over to the notion of n-accessibility as well. First, there are in general many different n-accessible pointed models from a given pointed model; as many as there are n-variations of that pointed model. Second, if (M ′, w′) is i-accessible from (M, w), (M ′, w′) is also j-accessible from (M, w), for i ≤ j. Third, (M, w) is always n-accessible from itself, since the 'empty' line of reasoning-not performing any inference at all-is always within the agent's cognitive reach, for any value of n. This puts us in a position to complete our semantics for L. As usual, sentences are evaluated for truth and falsity at pointed models. We write '⊧' and 'â' for verification and falsification, respectively. For any possible world, w ∈ W P : (P1) M, w ⊧ p iff p ∈ V (w), where p is an atomic sentence. (P2) M, w ⊧ ¬A iff M, w /⊧ A. (P3) M, w ⊧ A ∧B iff M, w ⊧ A and M, w ⊧ B. (P4) M, w ⊧ Cr(A) ≤ x iff ∑Q Pr(Q) ≤ x, where Q = {v ∈ S ∶ M, v ⊧ A}. (P5) M, w ⊧ ⟨n⟩A iff M ′, w′ ⊧ A for some (M ′, w′) ∶ (M, w) n∼ (M ′, w′). (P6) M, w ⊧ [n]A iff M ′, w′ ⊧ A for all (M ′, w′) ∶ (M, w) n∼ (M ′, w′). (P7) M, w â A iff M, w ⊭ A. For any impossible world, w ∈ W I : (I1) M, w ⊧ A iff A ∈ V (w). (I2) M, w â A iff ¬A ∈ V (w). 23 For the central results below, logical validity is defined in terms of truthpreservation across all possible worlds: Γ ⊧ A iff every possible world in every model is such that it verifies every sentence in Γ only if it verifies A. A few remarks about the various satisfaction clauses are in order. First, note that falsehood behaves classically at both possible and impossible worlds: a sentence is false iff its negation is true. But, in contrast to possible worlds, impossible worlds can contain truth-value gaps (sentences that are neither true nor false) and truth-value gluts (sentences that are both true and false). Second, note that (P1)-(P4) are identical to a standard possible-worlds semantics for probabilistic modal logic, except the semantics for the credence operator, (P4), which quantifies over both possible and impossible worlds. This is what allows us to steer clear of logical omniscience. Third, (P5) says that sentences of the form '⟨n⟩A' are true at a pointed model, (M, w), iff A is true at some n-accessible pointed model from (M, w). In particular, ⟨n⟩(Cr(A) = x) is true at (M, w) iff Cr(A) = x is true at some n-accessible pointed model from (M, w). This reflects the idea that the agent can come to have a credence of x in A by performing some n-step reasoning process provided that there is an n-accessible doxastic state from her current doxastic state at which she has a credence of x in A. Since the ⊢nR-relation is monotonic in n, any pointed model that verifies ⟨i⟩A will also verify ⟨j⟩A, for i ≤ j. Finally, (P6) says that sentences of the form '[n]A' are true at (M, w) iff A is true at every n-accessible pointed model from (M, w). In particular, [n](Cr(A) = x) is true at (M, w) iff Cr(A) = x is true at every n-accessible pointed model from (M, w). This reflects the idea that the agent will come to have a credence of x in A regardless of which n-step reasoning process she performs. Since a pointed model is always n-accessible from itself, the semantics for [n](Cr(A) = x) is equivalent to that of Cr(A) = x. That is, [n](Cr(A) = x) and Cr(A) = x are true under exactly the same circumstances. This might seem to deprive the [n]-operator of much of its interest. Indeed, our main focus in what follows will be on the ⟨n⟩-operator. But the semantics for the [n]-operator captures the aforementioned idea that the 'empty' line of reasoning is always within an agent's cognitive reach, for any 24 value of n. With our semantics in place, we can establish the first of our main results (all proofs can be found in the Appendix): Theorem 1 (n-preservation) If A ⊢nR B, then Cr(A) = x ⊧ ⟨n⟩(Cr(B) ≥ x). This result says that, if A n-entails B, and the agent's credence in A is x, there is an n-step inference such that, after having performed that inference, the agent's credence in B is at least x. For example, if the agent is 70% confident that "it rains," she will be at least 70% confident that "it rains or snows" after having performed some 1-step inference (assuming that R contains disjunction introduction). We can think of n-preservation as a non-ideal analogue of Classical Preservation. In contrast to Classical Preservation, n-preservation does not carry any commitment to logical omniscience: it does not describe an agent's credences as being preserved across logical entailment. In fact, for all npreservation says, an agent's credences need not be preserved across any logical entailments. Yet, n-preservation allows us to retain a central trait of logical competence: it describes agents as being in a position to preserve their credences across those entailments that lie within their cognitive reach. The second of our main results is a non-ideal analogue to Classical Certainty: Theorem 2 (n-certainty) If ⊢nR A, then ⊧ ⟨n⟩(Cr(A) = 1). According to n-certainty, if A is an 'n-step tautology'-that is, if A is nstep inferable from the empty set-then an agent can come to be certain of A after having performed some n-step reasoning process. For example, the agent can come to be certain that "it's either raining or not" after some 1-step reasoning process (assuming that A ∨ ¬A is 1-step provable in R). In contrast to Classical Certainty, n-certainty does not carry any commitment to logical omniscience: it does not describe agents as being certain of all tautologies. In fact, for all n-certainty says, agents need not be certain of 25 any tautologies. Yet, n-certainty allows us to retain a central trait of logical competence: it describes agents as being in a position to become certain of any tautology that lies within their cognitive reach. Together, n-preservation and n-certainty show how our dynamic framework avoids the problems that faced the static approaches discussed in the previous section: it allows us to model agents who are logically competent despite falling short of logical omniscience. But we are not home free yet. The unorthodox nature of our approach gives rise to a number of questions that need to be addressed. That is the task of the next section. 4 Damage Control: Beyond Orthodox Bayesianism It is not cost-free to give up the assumption of logical omniscience. Without it, many fundamental results of orthodox Bayesianism do not go through. But all is not lost. Just as our dynamic framework provides us with non-ideal analogues of Classical Preservation and Classical Certainty, so it provides us with non-ideal analogues of various other centerpieces of orthodox Bayesianism. Here we focus on two in particular. First up is the notion of a conditional credence. Bayesians typically define conditional credences in terms of ratios of unconditional credences:13 Ratio Formula: Cr(A ∣B) = Cr(A∧B)Cr(B) This definition is sensible as long as conjunctions relate to their conjuncts in the usual, truth-functional way. But conjunctions do not behave in the usual, truth-functional way in our framework: impossible worlds show no respect for classical truth-functional dependencies between conjunctions and their conjuncts. In particular, A∧B need not be true just because A and B are. Hence, the Ratio Formula makes little sense in our framework. The stakes are high: without the Ratio Formula, the standard derivation of Bayes' theorem is blocked. Suddenly it looks like we are throwing out the baby with the bathwater. 13A notable exception is Hájek (2003). 26 But a fix is available. Instead of the Ratio Formula, we can define conditional credences as follows: (P8) M, w ⊧ Cr(A ∣B) ≤ x iff ∑Q P r(Q)∑Q′ P r(Q′) ≤ x, where w ∈ W P , Q = {v ∈ S ∶ M, v ⊧ A and M, v ⊧ B}, and Q′ = {v ∈ S ∶ M, v ⊧ B}. This definition captures much the same idea as the Ratio Formula: to determine an agent's credence of A conditional on B, we look at those Bworlds that are doxastically possible for the agent and check which of those worlds verify A. Furthermore, it is easily verified that (P8) and (P4) jointly entail Bayes' theorem. The danger is averted. Our definition of conditional credence also allows us to establish a third main result: Theorem 3 (n-conditionality) If A ⊢nR B, then ⊧ ⟨n⟩(Cr(B ∣A) = 1). According to n-conditionality, if A n-entails B, then an agent can become certain of B conditional on A after having performed some n-step reasoning process. For example, the agent can become certain that "it rains or snows" conditional on "it rains" after some 1-step reasoning process (assuming that R contains disjunction introduction). We can think of n-conditionality as a non-ideal analogue of the following Bayesian principle: Classical Conditionality: If A entails B, then Cr(B ∣A) = 1. This principle captures yet another way in which orthodox Bayesianism gives rise to logical omniscience: intuitively, it describes agents as being certain of all entailment relations. By contrast, n-conditionality carries no such commitment. Indeed, for all n-conditionality says, agents need not be certain of any entailment relations. Yet, n-conditionality allows us to retain a central trait of logical competence: it describes agents as being in a position to become certain of those entailment relations that lie within their cognitive reach. The second aspect of orthodox Bayesianism that we want to focus on is 27 its algebraic structure. The story is familiar: by defining a Boolean algebra on the set of possible worlds, we can understand logical operations in terms of set-theoretic ones. For example, we can understand conjunction and disjunction in terms of intersection and union (where '∣A∣' denotes the set of possible worlds that verify A): Classical Conjunction: ∣A ∧B∣ = ∣A∣ ∩ ∣B∣ Classical Disjunction: ∣A ∨B∣ = ∣A∣ ∪ ∣B∣ These principles do not generally hold in our framework. More specifically, they hold at the level of possible worlds, but fail at the level of impossible worlds. The reason, once again, is that impossible worlds show no respect for classical, truth-functional dependencies between complex sentences and their parts. For example, A ∨B need not be true just because A is. However, we can still formulate non-ideal analogues to Classical Conjunction and Classical Disjunction (where '∣A∣n' denotes the set of worlds in W that have at least one n-expansion that verifies A):14 n-conjunction: ∣A ∧B∣n ⊆ ∣A∣n+1 ∩ ∣B∣n+1 ∣A∣n ∩ ∣B∣n ⊆ ∣A ∧B∣n+1 n-disjunction: ∣A ∨B∣n ∩ ∣A∣n ⊆ ∣B∣n+1 ∣A∣n ⊆ ∣A ∨B∣n+1 These principles show that our dynamic framework, while not truth-functional in the classical sense, still allows us to associate set-theoretic properties with various logical connectives; properties that nicely capture the roles that such connectives play in our cognitive lives. This strikes us as an interesting result in its own right. 14Here is a sketch of a proof of the first part of n-conjunction: suppose w ∈ ∣A ∧ B∣n, for any w ∈W . Assuming that R contains standard introduction and elimination rules for conjunction, ∣A∧B∣n ⊆ ∣A∣n+1 and ∣A∧B∣n ⊆ ∣B∣n+1. Hence, w ∈ ∣A∣n+1 ∩ ∣B∣n+1. The other subset relations can be established in similar ways. 28 5 Concluding remarks We began this paper with a critical discussion of three existing approaches to the problem of logical omniscience in the Bayesian literature. Some proposals merely replaced logical omniscience with a different logical idealization; others sacrificed all traits of logical competence on the altar of logical omniscience. The collapse result made the waters hard to navigate. But in diagnosing why, a new 'dynamic' approach emerged: by enriching the Bayesian framework with tools that allowed us to model what agents can and cannot infer given their limited cognitive resources, hope remained to circumvent collapse. We went on to develop this dynamic approach in formal detail, and showed how the resulting Bayesian framework allows us to model agents who are logically competent despite falling short of logical omniscience. Let us close by addressing a residual worry about our dynamic approach, due to Berto & Jago (2019, §5.5). The worry goes as follows: while our framework allows us to model what agents can infer given his cognitive resources, it does not allow us to model what they should infer given those resources (since the semantics for '[n](Cr(A) = x)' is equivalent to that of 'Cr(A) = x'). Yet it is the job of a theory of non-ideal rationality to tell us how non-ideal agents should live their epistemic lives. After all, rationality is a normative notion; not a descriptive one. We want to offer two remarks in reply. First, it is worth noting that there is at least a weak sense in which our framework is normative. If we accept that 'ought' implies 'can' in the domain of epistemic rationality (which is obviously a big 'if'), then agents will not be required to live their epistemic lives in ways that are incompatible with their cognitive abilities. Thus, insofar as our dynamic framework allows us to represent an agent's cognitive abilities, it will at least arguably place negative requirements on how agents ought to live their epistemic lives. Second, and perhaps more importantly, we are doubtful that a formal theory of non-ideal rationality should indeed place any positive demands on which inferences ordinary agents should perform. After all, if an agent performed every inference within her cognitive reach, she would end up 'clut29 tering her mind with trivialities,' to use a rubric from Harman (1986, p. 12). The situation seems analogous to that of evidence-gathering: if an agent gathered every piece of evidence within her practical reach, she would most likely end up with a massive pile of useless junk. Yet, it is presumably not the task of formal epistemology to say which pieces of evidence, among the practically feasible ones, the agent should gather. Likewise, we do not consider it the task of our dynamic Bayesian framework to say which inferences, among the epistemically feasible ones, agents should perform. Acknowledments. An earlier version of this paper was presented at the "Normative Notions Formalized" Workshop in Munich. Many thanks to the audience on that occasion. Thanks also to two anonymous referees from Erkenntnis for very detailed and helpful comments. Appendix This appendix contains proofs of three main results of the paper. All definitions can be found in §3. The results are repeated here for convenience. Theorem 1 (n-preservation) If A ⊢nR B, then Cr(A) = x ⊧ ⟨n⟩(Cr(B) ≥ x). Proof. Suppose that A ⊢nR B and consider any pointed model, (M, w), such that M, w ⊧ Cr(A) = x, where M = (W P , W I , f, V ) and w ∈ W P . We must show that M, w ⊧ ⟨n⟩(Cr(B) ≥ x). We proceed by defining a suitable naccessible pointed model from (M, w). Let M ′ = (W ′P , W ′I , f ′, V ′) be a model such that W ′P = W P , W ′I = W I , and V ′ = V . Since A ⊢nR B, we can let f ′ be an n-variation of (M, w) for which it holds that f ′(w) = (c, Prc), where M ′, v ⊧ B, for all v ∈ {v′ ∈ c ∶ M ′, v′ ⊧ A}. By the definition of naccessibility, then, (M, w) n∼ (M ′, w). Since M, w ⊧ Cr(A) = x, (P4) tells us that ∑Q Pr(Q) = x, where Q = {v ∈ S ∶ M, v ⊧ A}. Hence, ∑Q′ Prc(Q′) ≥ x, where Q′ = {v ∈ c ∶ M ′, v ⊧ B}. By another application of (P4), M ′, w ⊧ Cr(B) ≥ x. So, by (P5), it follows that M, w ⊧ ⟨n⟩(Cr(B) ≥ x). 30 Theorem 2 (n-certainty) If ⊢nR A, then ⊧ ⟨n⟩(Cr(A) = 1). Proof. Suppose that ⊢nR A and let (M, w) be any pointed model such that M = (W P , W I , f, V ) and w ∈ W P . We must show that M, w ⊧ ⟨n⟩(Cr(A) = 1). We proceed by defining a suitable n-accessible pointed model from (M, w). Let M ′ = (W ′P , W ′I , f ′, V ′) be a model such that W ′P = W P , W ′I = W I , and V ′ = V . Since ⊢nR A, we can let f ′ be an n-variation of (M, w) such that f ′(w) = (c, Prc), where M ′, v ⊧ A, for all v ∈ c. Hence, ∑Q′ Prc(Q′) = 1, where Q′ = {v ∈ c ∶ M ′, v ⊧ A}. By (P4), M ′, w′ ⊧ Cr(A) = 1. By the definition of n-accessibility, (M, w) n∼ (M ′, w). So, by (P5), it follows that M, w ⊧ ⟨n⟩(Cr(A) = 1). Theorem 3 (n-conditionality) If A ⊢nR B, then ⊧ ⟨n⟩(Cr(B ∣A) = 1). Proof. Suppose that A ⊢nR B and let (M, w) be any pointed model such that M = (W P , W I , f, V ) and w ∈ W P . We must show that M, w ⊧ ⟨n⟩(Cr(B ∣A) = 1). Let M ′ = (W ′P , W ′I , f ′, V ′) be a model such that W ′P = W P , W ′I = W I , and V ′ = V . Since A ⊢nR B, we can let f ′ be an n-variation of (M, w) such that f ′(w) = (c, Prc), where M ′, v ⊧ B, for all v ∈ {v′ ∈ c ∶ M ′, v′ ⊧ A}. Hence, ∑Q Prc(Q) = ∑Q′ Prc(Q ′), where Q = {v ∈ c ∶ M ′, v ⊧ A and M ′, v ⊧ B} and Q′ = {v′ ∈ c ∶ M ′, v′ ⊧ q}. By (P8), M ′, w ⊧ Cr(B ∣A) = 1. By the definition of n-accessibility, (M, w) n∼ (M ′, w). 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A supressão transitória dos piores demônios da nossa natureza - uma revisão de Steven Pinker ' os melhores anjos da nossa natureza: por que a violência declinou ' ('The Better Angels of Our Nature: Why Violence Has Declined') (2012)(revisão revisada 2019) Michael Starks Abstrata Este não é um livro perfeito, mas é único, e se você desnatado o primeiro 400 ou assim páginas, o último 300 (de alguns 700) são uma tentativa muito boa para aplicar o que é conhecido sobre o comportamento de mudanças sociais na violência e maneiras ao longo do tempo. O tema básico é: como o nosso controle genético e limitar a mudança social? Surpreendentemente, ele não descreve a natureza da seleção de parentes (aptidão inclusiva), o que explica grande parte da vida social animal e humana. Ele também (como quase todos) carece de um quadro claro para descrever a estrutura lógica da racionalidade (LSR - termo preferencial de John Searle) que eu prefiro chamar a psicologia descritiva do pensamento de ordem superior (DPHOT). Ele deveria ter dito algo sobre as muitas outras maneiras de abusar e explorar as pessoas e o planeta, uma vez que estes são agora muito mais graves para tornar outras formas de violência quase irrelevante. Estender o conceito de violência para incluir as conseqüências globais a longo prazo da replicação dos genes de alguém, e ter um entendimento da natureza de como a evolução funciona (ou seja, a seleção de parentes) fornecerá uma perspectiva muito diferente sobre a história , eventos atuais, e como as coisas são susceptíveis de ir nas próximas centenas de anos. Pode-se começar por notar que a diminuição da violência física sobre a história tem sido combinada (e tornou possível) pelo estupro constantemente crescente impiedoso do planeta (ou seja, pela destruição das pessoas do seu próprio futuro descendente's). Pinker (como a maioria das pessoas a maior parte do tempo) é muitas vezes distraído pelas superficialidades da cultura, quando é a biologia que importa. Veja meus comentários recentes de Wilson ' a conquista social da terra ' e Nowak e Highfield ' SuperCooperators ' aqui e na net para um breve resumo da vacuidade de "verdadeiro altruísmo" (seleção de grupo), e a operação de seleção de parentes e a inutilidade e superficialidade de descrever o comportamento em termos culturais. Esta é a natureza clássica/nutrir a questão ea natureza supera nutrir-infinitamente. O que realmente importa é a violência feita para a terra pelo aumento implacável da população e destruição de recursos (devido à medicina e tecnologia e supressão de conflitos por policiais e militares). Cerca de 200.000 mais pessoas por dia (outra Las Vegas a cada 10 dias, outro Los Angeles a cada mês), as 6 toneladas ou assim de solo superior indo para o mar/pessoa/ano -cerca de 1% do total do mundo desaparecendo anualmente, etc. significa que, a menos que algum milagre acontece que a biosfera e 2 a civilização vão desmoronar em grande parte durante os próximos dois séculos, e haverá fome, miséria e violência de todo tipo em uma escala escalonamento. As maneiras, as opiniões e as tendências das pessoas para cometer atos violentos não são relevantes a menos que possam fazer algo para evitar essa catástrofe, e eu não vejo como isso vai acontecer. Não há espaço para argumentos, e nenhum ponto ou (Sim, eu sou um fatalista), então eu vou fazer apenas alguns comentários como se fossem fatos. Não imagine que eu tenha uma participação pessoal na promoção de um grupo à custa de outros. Eu sou 78, não têm descendentes e não parentes próximos e não se identificam com qualquer grupo político, nacional ou religioso e consideram os que eu pertenço por padrão como tão repulsivo como todo o resto. Os pais são os piores inimigos da vida na terra e, tomando a visão ampla das coisas, as mulheres são tão violentas quanto os homens, quando se considera o fato de que a violência das mulheres (como a maior parte do que feito pelos homens) é largamente feito em câmera lenta, a uma distância no tempo e no espaço e principalmente realizado por procuração por seus descendentes e por homens. Cada vez mais, as mulheres carregam crianças, independentemente de terem um companheiro e o efeito de parar uma mulher de reprodução é, em média, muito maior do que parar um homem, uma vez que são o gargalo reprodutivo. Pode-se ter a visão de que as pessoas e seus descendentes merecem ricamente qualquer miséria que vem seu caminho e (com raras exceções) os ricos e famosos são os piores infratores. Meryl Streep ou Bill Gates ou J. K Rowling e cada um de seus filhos podem destruir 50 toneladas de solo superior cada por ano para gerações no futuro, enquanto um agricultor indiano e seu pode destruir 1 tonelada. Se alguém nega que é bom, e aos seus descendentes eu digo "bem-vindo ao inferno na terra" (Welcome To Hell On Earth) (WTHOE). A ênfase hoje em dia é sempre sobre direitos humanos, mas é claro que se a civilização é ter uma chance, as responsabilidades humanas devem substituir os direitos humanos. Ninguém recebe direitos sem ser um cidadão responsável e a primeira coisa que isto significa é a destruição ambiental do minimo. A responsabilidade mais básica não é nenhuma criança, a menos que sua sociedade lhe peça para produzi-las. Uma sociedade ou um mundo que permite que as pessoas se reproduzem aleatoriamente serão sempre exploradas por genes egoístas até que ela desmorona (ou atinge um ponto onde a vida é tão horrível que não vale a pena viver). Se a sociedade continua a manter os direitos humanos como primários, a seus descendentes pode-se dizer com confiança "WTHOE". Aqueles que desejam um quadro até à data detalhado para o comportamento humano da opinião moderna dos dois sistemas consultar meu livros Falando Macacos 3a Ed (2019), A Estrutura Lógica da Filosofia, Psicologia, Mente e Linguagem em Ludwig Wittgenstein e John Searle 2a Ed (2019), Suicídio Pela Democracia,4aEd(2019), Entendendo as Conexões entre Ciência, Filosofia, Psicologia, Religião, Política e Economia Artigos e Análises 2006-2019 (2019), Ilusões Utópicas Suicidas no 21St século 5a Ed (2019), A Estrutura Lógica do Comportamento Humano (2019), e A Estrutura Lógica da Consciência (2019) e outras. 3 Este não é um livro perfeito, mas é único, e se você desnatado o primeiro 400 ou assim páginas, o último 300 (de alguns 700) são uma tentativa muito boa para aplicar o que é conhecido sobre o comportamento de mudanças sociais na violência e maneiras ao longo do tempo. O tema básico é: como o nosso controle genético e limitar a mudança social? Surpreendentemente, ele não descreve a natureza da seleção de parentes (aptidão inclusiva), o que explica grande parte da vida social animal e humana. Ele também (como quase todos) carece de um quadro claro para descrever a estrutura lógica da racionalidade (LSR - termo preferencial de John Searle) que eu prefiro chamar a psicologia descritiva do pensamento de ordem superior (DPHOT). Principalmente as críticas dadas por outros são mesquinharia e irrelevante e, como disse Pinker, ele não poderia escrever um livro coerente sobre "coisas ruins", nem poderia dar todas as referências possíveis e ponto de vista, mas ele deveria ter dito pelo menos algo sobre os muitos outras formas de abusar e explorar as pessoas e o planeta, uma vez que estas são agora muito mais severas para tornar irrelevante outras formas de violência. Estender o conceito de violência para incluir as conseqüências globaisa longo prazo da replicação dos genes de alguém, e ter uma noção da natureza de como a evolução funciona (ou seja, a seleção de parentes) fornecerá uma perspectiva muito diferente sobre a história, eventos atuais, e como as coisas são susceptíveis de ir nos próximos cem anos. Pode-se começar por notar que a diminuição da violência física sobre a história tem sido combinada (e tornou possível) pelo estupro constantemente crescente impiedoso do planeta (ou seja, pela destruição das pessoas do seu próprio futuro descendente's). Pinker (como a maioria das pessoas a maior parte do tempo) é muitas vezes distraído pelas superficialidades da cultura, quando é a biologia que importa. Veja meus comentários recentes de Wilson ' a conquista social da terra ' e Nowak e Highfield ' SuperCooperators ' para um breve resumo da vacuidade do altruísmo e da operação de seleção de parentes e da inutilidade e superficialidade de descrever o comportamento em cultural de acordo com os termos. Esta é a natureza clássica/nutrir a questão e a natureza supera nutrir-infinitamente. O que realmente importa é a violência feita para a terra pelo aumento implacável da população e destruição de recursos (devido à medicina e tecnologia e supressão de conflitos por policiais e militares). Cerca de 200.000 mais pessoas por dia (outro Las Vegas a cada 10 dias, outro los Angeles a cada mês), as 6 toneladas ou assim de solo superficial in do para o mar/pessoa/ano etc. significa que, a menos que algum milagre acontece a biosfera e a civilização será em grande parte colapso nos próximos dois séculos e haverá fome, miséria e violência de todos os tipos em uma escala escalonamento. As maneiras, as opiniões e as tendências das pessoas para cometer atos violentos não são relevantes a menos que possam fazer algo para evitar essa catástrofe, e eu não vejo como isso vai acontecer. Não há espaço para argumentos, e nenhum ponto ou (Sim, eu sou um fatalista), então eu vou fazer apenas alguns comentários como se 4 fossem fatos. Não imagine que eu tenha uma participação pessoal na promoção de um grupo à custa de outros. Eu sou 75, não têm descendentes e não parentes próximos e não se identificam com qualquer grupo político, nacional ou religioso e consideram os que eu pertenço por padrão como tão repulsivo como todo o resto. Os pais são os piores inimigos da vida na terra e, tomando a visão ampla das coisas, as mulheres são tão violentas quanto os homens, quando se considera o fato de que a violência das mulheres (como a maior parte do que feito pelos homens) é largamente feito em câmera lenta, a uma distância no tempo e no espaço e principalmente Carri por procuração por seus descendentes e por homens. Cada vez mais, as mulheres carregam crianças, independentemente de terem um companheiro e o efeito de parar uma mulher de reprodução é, em média, muito maior do que parar um homem, uma vez que são o gargalo reprodutivo. Pode-se ter a visão de que as pessoas e seus descendentes merecem ricamente qualquer miséria que vem seu caminho e (com raras exceções) os ricos e famosos são os piores infratores. Meryl Streep ou Bill Gates ou J. K. Rowling e cada um de seus filhos podem destruir 50 toneladas de solo superficial cada por ano para as gerações no futuro, enquanto um agricultor indiano e seu pode destruir 1 tonelada. Se alguém nega que é bom, e aos seus descendentes eu digo "bem-vindo ao inferno na terra" (WTHOE). A ênfase hoje em dia é sempre sobre direitos humanos, mas é claro que se a civilização é ter uma chance, as responsabilidades humanas devem substituir os direitos humanos. Ninguém recebe direitos (ou seja, privilégios) sem ser um cidadão responsável e a primeira coisa que isso significa é a destruição ambiental mínima. A responsabilidade mais básica não é nenhuma criança, a menos que sua sociedade lhe peça para produzi-las. Uma sociedade ou um mundo que permite que as pessoas se reproduzem aleatoriamente serão sempre exploradas por genes egoístas até que ela desmorona (ou atinge um ponto onde a vida é tão horrível que não vale a pena viver). Se a sociedade continua a manter os direitos humanos como primários, isso é bom e para seus descendentes pode-se dizer com confiança "WTHOE". "Ajudar" tem de ser visto a partir de uma perspectiva globala longo prazo. Quase toda a "ajuda" que é dada por indivíduos, organizações ou países prejudica os outros e o mundo a longo prazo e só deve ser dada após uma consideração muito cuidadosa. Se você quer entregar o dinheiro, a comida, a medicina, etc., você precisa de perguntar o que as conseqüências ambientais a longo prazo são. Se você quiser agradar a todos o tempo todo, novamente para seus descendentes eu digo "WTHOE". Disgenics: trilhões intermináveis de criaturas que começam com formas bactérias-like mais de 3.000.000.000 anos atrás morreram para nos criar e toda a vida atual e isso é chamado de eugenia, evolução por seleção natural ou seleção de parentes (aptidão inclusiva). Todos nós temos "genes maus", mas alguns são piores do que outros. Estima-se que até 50% de todas as concepções humanas terminam em aborto espontâneo devido a "genes ruins". A civilização é disgênica. Este problema é atualmente trivial em comparação com superpopulação, mas piorar a cada dia. A medicina, o bem-estar, a democracia, a igualdade, a justiça, os direitos humanos e a 5 "ajuda" de todos os tipos têm conseqüências ambientais e desenfreadas a longo prazo globais que desmoronarão a sociedade mesmo se o crescimento da população parar. Mais uma vez, se o mundo se recusa a acreditar ou não quer lidar com isso que é bom e para os seus (e todos) descendentes podemos dizer "WTHOE". Cuidado com os cenários utópicos que sugerem que o juízo final pode ser evitado pela aplicação criteriosa das tecnologias. Como eles dizem que você pode enganar algumas das pessoas o tempo todo e todas as pessoas um pouco do tempo, mas você não pode enganar a mãe natureza a qualquer momento. Deixo-vos com apenas um exemplo. O famoso cientista Raymond Kurzweil (veja minha resenha de ' como criar uma mente ') propôs nanorobôs como salvadores da humanidade. Eles fariam tudo o que precisávamos e limpavam cada bagunça. Eles iriam mesmo fazer versões cada vez melhores de si mesmos. Eles nos manteriam como animais de estimação. Mas pense em quantas pessoas tratam seus animais de estimação, e animais de estimação estão sobrepovoando e destruindo e tornando-se disgênico quase tão rápido quanto os seres humanos (por exemplo, gatos domésticos e selvagens sozinhos matam talvez 100.000.000.000 animais silvestres por ano). Animais de estimação só existem porque nós destruímos a terra para alimentá-los e temos clínicas de neutro e eutanizar os doentes e indesejados. Nós praticamos rigoroso controle populacional e Eugenia sobre eles deliberadamente e por omissão, e nenhuma forma de vida pode evoluir ou existir sem esses dois controles - nem mesmo robos. E o que é para impedir que os nanorrobôs evoluam? Qualquer mudança que facilitou a reprodução seria automaticamente selecionado para e qualquer comportamento que desperdiçou tempo ou energia (ou seja, cuidar dos seres humanos) seria fortemente selecionado contra. O que impediria o programa de robôs controlados por ai de se transformar em uma forma homicida e explorar todos os recursos da Terra causando colapso global? Não há almoço livre para bots ou para eles também podemos dizer com confiança "WTHOE". Este é o lugar onde qualquer pensamento sobre o mundo eo comportamento humano deve levar uma pessoa educada, mas Pinker não diz nada sobre isso. Assim, as primeiras 400 páginas deste livro podem ser ignoradas e o último 300 lido como um sumário agradável do EP (Pssicologia Evolucionário) até à data de 2011. No entanto, como em seus outros livros e quase universalmente nas ciências comportamentais, não há um amplo quadro claro para a intencionalidade como pioneira por Wittgenstein, Searle e muitos outros. Eu apresentei uma tal estrutura em meus muitos comentários de obras de e sobre esses dois gênios psicológicos naturais e não vai repeti-lo aqui.

KRITIKE VOLUME FIVE NUMBER TWO (DECEMBER 2011) 112-122 © 2011 Koshy Tharakan http://www.kritike.org/journal/issue_10/tharakan_december2011.pdf ISSN 1908-7330 Article Questioning the Body: From Technology towards a Sense of Body Koshy Tharakan Abstract: Many attempts of contemporary philosophers to reduce 'mind' to 'body' notwithstanding, where the 'body' is understood in the Cartesian framework, the continental philosophers in general repeatedly remind us that body has a significance that goes beyond its materiality as a bio-chemical physical substance. In "questioning body," we wish to take up the philosophical underpinnings of the significance of body as a framework or tool to understand 'technology'. By doing so, we are able to see the link between technology and body as more than a fortuitous relation. Relying on the writings of Heidegger, Merleau-Ponty and Ihde, the paper attempts to show how a "sense of body," particularly the notion of "agentive body" as distinguished from the "symbolic body," hermeneutically evolves from the way in which it is entangled in the technological matrix. Key words: Heidegger, Merleau-Ponty, body, technology Introduction: Technology and Body any attempts of contemporary philosophers to reduce 'mind' to 'body' notwithstanding, where the 'body' is understood in the Cartesian framework, the continental philosophers in general repeatedly remind us that body has a significance that goes beyond its materiality as a bio-chemical physical substance. In "questioning body," we wish to take up the philosophical underpinnings of the significance of body as a framework or tool to understand 'technology'. In the process, the paper attempts to show how a "sense of body" evolves hermeneutically. It is otiose to state that technology is the most visible thread by which the modern societies are connected with each other in their economic and other socio-cultural spheres. However, a philosophical take on technology inevitably leads us to probe our own 'body'. The link between 'technology' and 'body' is more obvious than any philosophical arguments that purport to establish the 'truth' of 'individualism' against 'holism' as an ontological statement about society-'society' as a conglomeration of individual 'bodies.' Rather, as Fortunati et al. point out, ." . . [T]he body is increasingly running the risk of becoming an appendix to the machine, despite there being, as Nietzsche M K. THARAKAN 113 wrote . . . more reason in the body than in our best wisdom."1 Those who apprehend this possibility argue that, technology has become an end in itself rather than a mere means for human ends. As an end in itself, technology has succeeded in "invading the body." Thus, Fortunanti et al. remarks: The human body is undergoing the same processes today that nature once underwent. In fact, whereas initially technology turned to nature, today it has become very interested in the human body . . . Communication technologies have extended the boundaries of the body, increasing the capacity to transmit information. Technology has progressively grown closer to our bodies, approaching through first clothing, then synthetic clothing fibers, and finally "smart fabrics," wearable computers, and communicative machines embedded . . . into the body.2 With the advent of the Cyborg, the technological invasion of body is complete! Philosophy of Technology: Heidegger's Questioning Philosophical reflection on technology seems to be as old as philosophy itself. However, as Ihde remarks, though ." . . there is a vast literature concerning technology, rarely has it been the primary theme of philosophers . . . even within the plethora of books concerning the human impact of technology, few are concerned with the nature of technology per se. Either the literature tends to focus upon effects of technology or it may itself be technical literature."3 Ihde himself provides an illuminating reason for this inexplicable silence on the part of philosophers: Part of the silence concerning technology comes from within philosophy itself. Philosophy usually conceives of itself more as a type of "conceptual" engineering than as a "material" engineering. Here there is a deeper set of relationships between science and technology as they emerge both in ancient and contemporary thought in philosophy. This symptomatology points to the dominance of a long "Platonistic" tradition with respect to science and technology, a tradition which, with respect 1 Leopoldino Fortunanti, James E. Katz and Raimonda Riccini, introduction to Mediating the Human Body: Technology, Communication and Fashion (Mahwah, New Jersey: Lawrence Erlbaum Associates, 2003), 2. 2 Ibid., 3-5. 3 Don Ihde, Instrumental Realism: The Interface Between Philosophy of Science and Philosophy of Technology (Bloomington and Indianapolis: Indiana University Press, 1991), 3. 114 QUESTIONING THE BODY to science and technology, turns out to be "idealistic." . . . a "Platonistic" tradition is one which negatively judges, or at least evaluates, perception and embodiment as lower on the scale of human activity than what is presumed to be a "pure" conceptuality.4 As against the "Platonistic" tradition, the "Praxis" tradition acknowledges the primacy of a theory of action that attaches a positive value to "perception" and "embodiment." Ihde reads Heidegger as a pioneer who belonged to the praxis tradition. Heidegger is foremost amongst the twentieth Century philosophers who reflected critically on technology. As some recent philosophers of technology note, for Heidegger the pivotal role of technology in modern society is symptomatic of a "wrongheaded attitude towards Being."5 For Heidegger, the purpose of philosophy is to concern itself with 'Being'. However, in thinking of 'Being qua Being' or the 'being of beings', its relatedness to man's nature has been already implicated. So an existential analytic of Dasein must proceed from its average everydayness. The world of everyday Dasein, which is closest to it, is the environment. In the environing world, we do not encounter 'mere things' per se. Such things come across to us with regard to our interests. Our dealings in the world and with things in the world are by way of 'concern'. Heidegger calls those entities, which we encounter in concern "equipment" or "tool" (de Zeug). Equipment always belongs to the horizon of a totality of equipments, their interrelatedness. There is not just a thing as 'equipment'; rather there always exists a totality of equipments. What an item of equipment depends on the way it is related to or assimilated into the total equipmental context. In other words, an object derives its essence from its functional role as well as its actual existence. Heidegger terms the entities encountered as equipments as 'ready-to-hand' (Zuhanden) as against those encountered in theoretical cognition, 'the presentat-hand' (Vorhanden). For Heidegger, the level of 'Zuhandenheit' is basic and 'Vorhandenheit' is a second level abstraction. These are not two distinct types of ontic entities; rather it refers to a distinction in Dasein's way of viewing the entities. When Dasein 'views' with circumspection the entities show themselves as ready-to-hand equipments. When Dasein adopts an attitude of merely observing, the very same entities appear as merely present at hand.6 In advocating a "praxis-perception model" for both philosophy of science and technology, Ihde emphasizes the significance of Heidegger's "tool analysis" for philosophy of technology. According to Ihde, the decisive shift that Heidegger brought to our view may be termed as "materialist." The "materialist" shift that Heidegger inaugurated in his Being and Time, has 4 Ibid., 5. 5 Maarten Franssen et al., "Philosophy of Technology," in Stanford Encylopedia of Philosophy <http://plato.stanford.edu/entries/technology/#MetIssStaChaArt>, 5 June 2010. 6 Martin Heidegger, Being and Time, trans. by John Macquarrie and E. Robinson (New York: Harper and Row, 1962), 95-102. K. THARAKAN 115 "inverted the standard view of the science-technology relation to that of technology-science."7 It was a long held view that technology is mere "applied science." The standard view thus accords primacy to science construed in the "Platonistic" tradition of "pure conceptuality." Heidegger's famous example of hammer serves our purpose here. As the "tool" analysis shows, a hammer is not first known as an "object" having certain weight such that it qualifies as a heavy object with such and such a shape or extension. Rather, we encounter hammer as an "embodiment which extends some human activity into its pragmatic context within an immediate environment."8 While we use the hammer, its "cognitive properties" are secondary and as an "object," the hammer "withdraws" itself. As Heidegger says: The peculiarity of what is proximally ready-to-hand is that, in its readiness-to-hand, it must, as it were, withdraw . . . in order to be ready-to-hand quite authentically. That with which our everyday dealings proximally dwell is not the tools themselves . . . On the contrary, that with which we concern ourselves primarily is the work.9 While using the hammer, say to drive a nail, if we cognitively attend to the hammer, then we make wrong nailing. In use, the tools possess a "dynamic" being of their own and they cease to be objects "known." Again, the "dynamic being" of the tool is contextual. The tool belongs to a "toolcontext." In the case of hammer, it would be the nails, the wood, etc. In fact, the tool context leads to an entire environment and implicitly with it a "world." Heidegger writes: Any work with which one concerns oneself is ready-tohand not only in the domestic world of the workshop but also in the public world. Along with the public world, the environing Nature . . . is discovered and is accessible to everyone. In roads, streets, bridges, buildings, our concern discovers Nature as having some definite direction. A covered railway platform takes account of bad weather; an installation for public lighting takes account of the darkness . . . When we make use of the clock-equipment, which is proximally and inconspicuously ready-to-hand, the environing Nature is ready-to-hand along with it. Our concernful absorption in whatever work-world lies closest to us, has a function of discovering; and it is essential to this function that, depending upon the way in which we are absorbed, those 7 Ihde, op cit., 48. 8 Ibid., 52. 9 Heidegger, op cit., 99. 116 QUESTIONING THE BODY entities within-the-world which are brought along . . . in the work and with it . . . remain discoverable in varying degrees of explicitness and with a varying circumspective penetration.10 Thus, the ready-to-handiness of the tool opens up a wider environment and finally leads to Dasein as "for the sake of whom" the "inorder-to" structure of the tool points. As Ihde says, this "praxical perceptual" dimension of human experience is available in some way or other to all human communities. Such experience occurs both "without" science as well as "within" science. This implies that "technology" is not to be taken as "applied science," rather "technology is broader than an explicit science."11 "The Heideggerian inversion," effected by the "tool" analysis makes a fundamental move in ontology as it makes "readiness-to-hand" as the basic ontological category by which entities are defined as they are "in themselves." It is only when the "tool-hood" is broken that the "readiness-to-hand" turns into a "present-at-hand" entity amenable to a sort of "theoretical" knowledge. Thus, as Ihde observes: This derivation of the occasion of "knowledge" makes the totality of the objects of knowledge not only derivative but special cases of human concern and activity. "Observer" consciousness is a particular development of actional, prior concerns. Thus, underneath the presumed disinterestedness of observation lies the engagement of praxis.12 In other words, technological praxis is not an application of science; rather science is now seen as "the tool of technology."13 This inversion is explicitly stated in Heidegger's later reflection on technology: It is said that modern technology is something incomparably different from all earlier technologies because it is based on modern physics as an exact science. Meanwhile we have come to understand more clearly that the reverse holds true as well: modern physics, as experimental, is dependent upon technical apparatus and upon progress in the building of apparatus.14 10 Ibid., 100. 11 Ihde, op cit., 53. 12 Ibid., 54-55. 13 Ibid., 55. 14 Martin Heidegger, "The Question Concerning Technology," in Martin Heidegger: Basic Writings, ed. by David Farrell Krell (London: Routledge and Kegan Paul, 1978), 295-96. K. THARAKAN 117 However, as Heidegger himself points out, the above statement only makes certain "facts" about technology. What is more significant is to grasp the "essence" of technology. Thus, Heidegger asks, "What is the essence of technology?" On first count, technology shows up as a "means to an end" and as a "human activity." This definition of technology, Heidegger calls the instrumental and anthropological definition. The instrumental-anthropological definition, though is correct does not reveal the essence of technology as Heidegger says, "the merely correct is not yet the true." Thus, he further asks, "What is the instrumental itself? Within what do such things as means and end belong?" Heidegger's questioning traces instrumentality back to "fourfold causality." Further interrogation reveals that the four causes-the formal, the material, the efficient and the final- "are the ways, all belonging at once to each other, of being responsible for something else."15 The "four ways of being responsible" together bring forth something into appearance. According to Heidegger, even the coming of something from out of itself, physis, is also a bringing forth: Not only handcraft manufacture, not only artistic and poetical bringing into appearance and concrete imagery, is a bringing-forth, poiēsis. Physis also, the arising of something from out of itself, is a bringing-forth, poiēsis.16 The growing things of nature as well as the products of arts and crafts thus make their presence through bringing-forth. But, bringing-forth happens only if "something concealed comes into unconcealment." The "revealing" of the concealed is "aletheia," the Greek term that got translated as "truth." Thus, Heidegger's search for the essence of technology thus shows that technology is no mere means; rather technology is a way of revealing. Technology as a mode of revealing characterizes not only earlier technologies of handcrafts. Even modern technology is also a mode of revealing, albeit a different one. As Heidegger says: . . . the revealing that holds sway throughout modern technology does not unfold into a bringing-forth in the sense of poiēsis. The revealing that rules in modern technology is a challenging . . . in that the energy concealed in nature is unlocked, what is unlocked is transformed, what is transformed is stored up, what is stored up is, in turn, distributed and what is distributed is switched about ever anew. Unlocking, transforming, storing, distributing, and switching about are ways of revealing.17 15 Ibid., 290. 16 Ibid., 293. 17 Ibid., 296-98. 118 QUESTIONING THE BODY The revealing, however, does not come to a standstill. The manifold ways of revealing and setting upon of the challenging-forth is gathered together under the rubric of "Ge-stell" or "Enframing." However, Heidegger's revealing of the essence of technology as "enframing" renders technology aporetic. As Belu and Feenberg point out: . . . enframing is not simply a widespread "problem" we could solve with appropriate remedies, but the underlying structure of being in our time. It is ontological rather than ontic . . . However, the universality of enframing would seem to block knowledge of it. The enframed subject should not be able to understand or to have a sense of her own enframing.18 As seen above, either Heidegger's characterization of the essence of technology as enframing is only a partial enframing and to that extent, the essence of technology is compromised or it is total enframing in which case human beings too are enframed and to that extent no theory of enframing is conceivable.19 The aporetic understanding of technology emanating from Heidegger's questioning may be rendered less aporetic if we begin our questioning with body rather than technology per se. Merleau-Ponty, through his ontology of flesh, offers us precisely such a possibility. Philosophy of Body: Merleau-Ponty's Ontology of Flesh According to Merleau-Ponty, our relation to the world is not one between the thinker and his object of thought. The world we actually perceive is not the 'objective' world, rather it is the world of our everyday life, the one in which we 'live-through'. In the 'lived world' one deals with objects that are 'situated' in relation to specific human actions. In other words, it is the human body as subject of action which determinates the objects as situated in its field of action. Merleau-Ponty writes: Our bodily experience of movement is not a particular case of knowledge; it provides us with a way of access to the world and the object, with a 'praktognosia', which has to be recognized as original and perhaps as primary. My body has its world, or understands its world, without 18 Dana S. Belu and Andrew Feenberg: "Heidegger's Aporetic Ontology of Technology," in Inquiry, 53 (No.1, 2010), 2. 19 Ibid., 8. K. THARAKAN 119 having to make use of my 'symbolic' or 'objectifying function.'20 Thus, the subject, by its very nature as embodied consciousness, right from its beginning is oriented towards the world. One's body and the world are not to be understood as objects coordinated together by a functional relationship that objective thought establishes. The relation between my body and the world rather should be understood in terms of a real implication. Merleau-Ponty understands that subject's interactions with the world are not primarily through the intellectual powers but through habits. When the body inhabits the space, it is done through habits and body has habits through inhabiting space. Habits and inhabiting are mutually implicatory.21According to him, the habit's contributions to inhabiting is sedimented generalized possibilities for inhabiting and inhabiting's contribution is pre-reflectively knowing one's way around enabling habitual actions. As Merleau-Ponty says, the expressive spatiality is the projection of body-consciousness to another object in space. He gives the example of a typewriter's keyboard as an instance of specific spatiality. While one types, one's body spatiality gets merged with that of the keyboard's spatiality. The "subject who learns to type incorporates the key-bank space into his bodily space."22 Another example that Merleau-Ponty gives is a blind man's cane: The blind man's stick has ceased to be an object for him, and is no longer perceived for itself; its point has become an area of sensitivity, extending the scope and active radius of touch, and providing a parallel to sight. In the exploration of things, the length of the stick does not enter expressly as a middle term: the blind man is rather aware of it through the position of objects than the position of objects through it. The position of things is immediately given through the extent of the reach which carries him to it, which comprises besides the arm's own reach the stick's range of action . . . The points in space do not stand out as objective positions in relation to the objective position occupied by our body; they mark, in our vicinity, the varying range of our aims and our gestures. To get used to a hat, a car or a stick is to be transplanted into them, or conversely, to incorporate them into the bulk of our own body.23 20 Maurice Merleau-Ponty, Phenomenology of Perception, trans. by Colin Smith (London: Routledge and Kegan Paul, 1962), 140-141. 21 Ibid,. 304-305. 22 Ibid., 145. 23 Ibid., 143. 120 QUESTIONING THE BODY The possibility of thus extending one's body space to other objects through expressive spatiality, suggests a new way to understand technology. As Ihde points out, these are examples of "embodiment relations." Such relation, Ihde notes, "are existential (bodily-sensory), but they implicate how we utilize technologies and how such use transforms what it is we experience through such technologies." In the Visible and the Invisible, Merleau-Ponty provides a new ontology by introducing the notion of 'flesh'. The flesh is a primal 'element' and both the subject and the world are born out of it. It is neither a mind nor a material substance. The distinguishing characteristic of flesh is its 'intertwining' relations. Using the example of one hand touching the other hand and being touched in turn, Merleau-Ponty says that the body can play the role of both the perceiver and the perceived. As he points out, there is an identity-in-difference when the two hands touch. The two hands are never, with regard to one another, "touched and touching at the same time." The notion of "identity-in-difference" is fundamental to Merleau-Ponty's ontological description of the visible. It is not only applicable to the experience of touch, but is relevant in the way in which the body is related to the world. Both my body as well as the world is flesh. The flesh of my body perceives the world as flesh. This new ontology negates the dualistic ontology that institutes a separation between my mind and body and between my body and the world. Conclusion: Body/Technology Dreyfus attempts to free the aporetic nature of Heideggerian ontology of technology by invoking marginal practices as the source of resistance to the enframing. Thus, Dreyfus writes: . . . although a technological understanding of being is our destiny it is not our fate. That is, although our understanding of things and ourselves as resources to be ordered, enhanced and used efficiently has been building up since Plato, we are not stuck with that understanding. Although the technological understanding of being governs the way things have to show up for us, we can be open to a transformation of our current cultural learning.24 As Belu and Feenberg show, the textual inspiration for suggesting marginal practices as offering resistance to enframing comes from Heidegger's remark in "Question Concerning Technology" that the saving power is to be 24 Hubert Dreyfus, "Heidegger on the Connection Between Nihilism, Art, Technology and Politics," in The Cambridge Companion to Heidegger, ed. by Guignon C.B. (London: Cambridge University Press, 1998), 307. Quoted in Dana S. Belu and Andrew Feenberg, op cit., 10. K. THARAKAN 121 found "[h]ere and now and in little things."25 However, this option seems to be doubtful as such a practice as Belu and Feenberg say remains an ontic solution whereas enframing is ontological. This perhaps necessitates a new ontology of technology and body. The ontology of flesh as well as Merleau-Ponty's characterization of the "mutual implication" of 'habit and 'inhabiting', I suggest mitigate the aporia of Heidegger's notion of "enframing" as the essence of technology. Perhaps, what is aporetic is not the ontology of technology, but the body as flesh. To flush out the aporia of body as flesh, we may focus on the two conceptions of body available in anthropological discourse: body as "symbol" and body as the "agent." Mary Douglas was one of the pioneers who articulated the notion of body as symbol. According to Douglas, the social situation is "replicated" symbolically by our body. The notion of "symbolic body" renders the body to be "viewed metaphorically as a text that can be "read" as a symbol or signifier of the social world that it inhabits."26 MerleauPonty provides a succinct account of the "agentic body" as well as how body extends itself through technology, when he says that: The body is our general medium for having a world. Sometimes it is restricted to the actions necessary for the conservation of life, and accordingly it posits around us a biological world; at other times, elaborating upon these primary actions and moving from their literal to a figurative meaning, it manifests through them a core of new significance: this is true of motor habits such as dancing. Sometimes, finally, the meaning aimed at cannot be achieved by the body's natural means; it must then build itself an instrument, and it projects thereby around itself a cultural world. At all levels it performs the same function which is to endow the instantaneous expressions of spontaneity . . . 27 (italics mine) Following Merleau-Ponty, we may thus say that technology is an "expression" of the body, an extension of the body and body is the "medium" of a technological world. This reciprocal "envelopment" of body and technology would make sense if we construe the body that extends itself not as a Cartesian res extensa but as an "embodied mind" manifested by ontology of "flesh." In other words, technology may be now understood as extension of the "agentic body." Such an understanding of technology as an extension of agentic body not only alleviates the Heideggerian aporia to a great extent but also projects hermeneutically the sense of the body as agentic in contrast to the 25 Heidegger, "Question Concerning Technology," 315. 26 Erica Reischer and Kathryn S. Koo: "The Body Beautiful: Symbolism and Agency in the Social World," in Annual Review of Anthropology, 33 (2004), 300. 27 Merleau-Ponty, op cit., 146. 122 QUESTIONING THE BODY symbolic sense of the body. The apprehension expressed at the outset, of the body being invaded by technology would be true of the "symbolic body," whereas for the "agentic body" technology would be a tool to "extend" agency. Thus, even while technology invades the symbolic body, there remains an element of agentic body beyond technology through the power of its agency, an agency that is capable of the "saving power" that Heidegger had alluded to. Department of Philosophy, Goa University, India References Belu, Dana S. and Feenberg, Andrew, "Heidegger's Aporetic Ontology of Technology," in Inquiry, 53:1 (2010). Fortunanti, Leopoldino, Katz , James E and Riccini, Raimonda ed., Mediating the Human Body: Technology, Communication and Fashion (Mahwah, New Jersey: Lawrence Erlbaum Associates, 2003). Franssen, Maarten, Lokhorst, Gert-Jan and Poel, Ibo van de, "Philosophy of Technology," in The Stanford Encyclopedia of Philosophy (Spring 2010 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/spr2010/entries/technology/>. Guignon, C.B. ed., The Cambridge Companion to Heidegger (London: Cambridge University Press, 1998). Heidegger, Martin, Being and Time, trans. by John Macquarrie and E. Robinson (New York: Harper and Row, 1962). Ihde, Don, Instrumental Realism: The Interface Between Philosophy of Science and Philosophy of Technology (Bloomington and Indianapolis: Indiana University Press, 1991). ___________, Technics and Praxis: A Philosophy of Technology (Dordrecht: D. Reidel Publishing Co., 1979). Krell, David Farrell (ed.), Martin Heidegger: Basic Writings (London: Routledge and Kegan Paul, 1978). Merleau-Ponty, Maurice, Phenomenology of Perception, trans. by Colin Smith (London: Routledge and Kegan Paul, 1962). Reischer, Erica and Koo, Kathryn S., "The Body Beautiful: Symbolism and Agency in the Social World," in Annual Review of Anthropology, 33 (2004).

THE MATHEMATICAL FACTS OF GAMES OF CHANCE BETWEEN EXPOSURE, TEACHING, AND CONTRIBUTION TO COGNITIVE THERAPIES: PRINCIPLES OF AN OPTIMAL MATHEMATICAL INTERVENTION FOR RESPONSIBLE GAMBLING CĂTĂLIN BĂRBOIANU  Infarom, Applied Mathematics Division Abstract On the question of whether gambling behavior can be changed as result of teaching gamblers the mathematics of gambling, past studies have yielded contradictory results, and a clear conclusion has not yet been drawn. In this paper, I bring some criticisms to the empirical studies that tended to answer no to this hypothesis, regarding the sampling and laboratory testing, and I argue that an optimal mathematical scholastic intervention with the objective of preventing problem gambling is possible, by providing the principles that would optimize the structure and content of the teaching module. Given the ethical aspects of the exposure of mathematical facts behind games of chance, and starting from the slots case – where the parametric design is missing, we have to draw a line between ethical and optional information with respect to the mathematical content provided by a scholastic intervention. Arguing for the role of mathematics in problem-gambling prevention and treatment, interdisciplinary research directions are drawn toward implementing an optimal mathematical module in cognitive therapies. Cuvinte cheie: matematica jocurilor de noroc; aparate de sloturi; teoria probabilitatilor; interpretarile probabilitatii; psihologia probabilitatii; etica jocurilor de noroc. Keywords: gambling mathematics; slot machines; probability theory; interpretations of probability; psychology of probability; gambling ethics. Autor corespondent: Cătălin Bărboianu. Email: [email protected] 26 1. INTRODUCTION Problem gambling is one of the fields patronized exclusively by psychology, as it came about naturally as one of the social effects of the gambling phenomenon. Mathematics is strongly connected to gambling through the mathematical models underlying any game of chance. Games of chance are developed structurally and physically around abstract mathematical models, which are their mere essence, and the applications within these mathematical models represent the premises of their functionality (for instance, the house edge is ensured through precise calculations regarding expected value; if such calculations were not possible, the game would never run). Since in problem-gambling research, treatment, and prevention we cannot separate the gambler from the game he plays, it follows that an optimal psychological intervention cannot disregard mathematics. Call this the gamblingmath indispensability principle. Thus far, the mathematics of gambling has been a subject of interest more for gamblers than psychologists, and the plethora of literature on gambling mathematics for the popular audience in the last decade confirms that observation. As regards psychology, the role of mathematics has been limited to providing odds (of winning/losing) and statistical indicators, and adjusting erroneous beliefs and fallacies related to probability and randomness. Empirical studies have been conducted testing hypotheses related to how gambling behavior changes with this mathematical knowledge, but those studies did not yield conclusive results. The relationship mathematics has developed with psychology in the course of such research is a indirect one – the mathematical intervention is addressed exclusively to gamblers via a third-party resource, and psychology only conducted the empirical studies and interpreted the results in terms of predicted behavior. In conclusion, the direct contribution of mathematics to psychological intervention in problem gambling was reduced to facing the odds and correcting misconceptions. However, these interventions are not enough, and some of the past empirical studies have confirmed that statement. Following the gambling-math indispensability principle, mathematics can go deeper into the gambler's mind with the help of psychology (or conversely) and its contribution can extend further to cognitive therapies, going beyond Probability & Statistics and incorporating knowledge from adjacent domains such as mathematical modeling, decision theory, theory of representation, and even epistemology (Bărboianu, 2013c). The current paper is focused on the indirect contribution of mathematics to responsible gambling through an optimal scholastic intervention and draws on the further research needed for establishing and implementing a direct contribution of it into the psychological interventions. 27 2. EXPOSURE OF THE MATHEMATICAL FACTS OF A GAME AS AN ETHICAL OBLIGATION 2.1. The slots case – an unjustified secrecy of their parametric design The slot games have gained and maintained top popularity despite their nontransparency with respect to parametric configuration, as this information is not exposed. Slots remains the only game in which players are not aware of the essential parameters of the game, such as number of stops of the reels, number of symbols, and their distribution on the reels. Obviously, the lack of data regarding the configuration of a machine prevents people from computing the associated odds of winning as well as other mathematical indicators. The so-called PAR sheets, exposing few of the parameters of the machines and probabilities associated with the winning combinations, are kept secret by game producers and can be retrieved only upon request via legal action in some jurisdictions–for example, through the Freedom of Information and Protection of Privacy Act, in Canada (Harrigan & Dixon, 2009). Fortunately, mathematics provides us with statistical methods of retrieving the missing parametric data based on long-run observation, as approximations (refined through methods based on numerical analysis and pattern recognition); however, such methods require considerable effort to put into practice (for instance, recording the outcome of thousands of spins for each reel, done by volunteers) (Bărboianu, 2013a). In fact, the existence and theoretical applicability of these methods of retrieving the missing data are in and of themselves arguments for the insubstantiality of the secrecy of slot producers on their PAR sheets. Nor do slot producers have a valid justification with respect to the company's interests. In the appeal decisions of the Information and Privacy Commissioner (IPC) in Ontario, Canada, with respect to declined PAR sheets requests, producers who declined the requests invoked the exemption set forth for scientific and technical information, considering that PAR sheets are trade secrets in the gaming industry and their exposure can significantly prejudice the competitive position of the company. (Information and Privacy Commissioner [IPC], 2009, 2010). The slot producers' reasons, shown in the IPC's appeal decisions, seem to be judicially formal rather than factual because: a) the trade secret and intellectual ownership reasons fail against the generality of the math formulas and equations since although the parametric details vary from game to game, the mathematical results are just applications of general formulas that are publicly available in mathematics and common across all slot machines; b) the competitive prejudice reasons fail against the open possibility for all slot producers to configure, test, and use any parametric design for their slot machines, which can be manipulated in unlimited ways, so as to obtain the desired statistical indicators for the house. 28 Finally, slot producers have no valid justification with respect to their players. The hypothetical reason of being afraid of losing players who face the real odds and expected values of their games fails against the a priori expectation of the players for low and very low odds of winning induced by the secrecy of PAR sheets that they have encountered, and against the lottery example, in which lottery players keep playing against the (well-known) lowest odds of winning due to other addictive elements that slots also hold (Bărboianu, 2013b, forthcoming). 2.2. The mathematical facts between ethically required and optional information The slots case raises the problem of the obligation to expose the parametric configuration of any existent or forthcoming game of chance, even though it currently applies only to slots. The goal of exposing the parametric configuration of the slot games is not necessarily to acquire for slots the same status as other games of chances in this respect, but rather, in the respect of ethics. Exposing the parametric configuration of a game to the player prior to playing is an ethical obligation in two aspects – one commercial and the other humanitarian. The commercial aspect treats the game as any commercial service, for which full technical specifications are required from the producer to the customer; a bet is still a purchased service once the player inserts a non-returnable coin in the machine. The humanitarian aspect is related first to the free will of thought and second, to the limitation of the risk factors through further improved knowledge. Being informed on all parameters of a game one plays is a condition for unconstrained personal thinking leading to personal actions. It is as if someone asks you to bet you can jump from a high place and land on your feet; of course, if you know in advance the height from which you will jump, or measure it before you bet, you might decline the bet or propose another one for a certain measurement, and this means free decision. Regarding the limitation of risk factors through further improved knowledge, acquired either as pre-calculated numerical results such as winning odds and other statistical indicators, or by learning theoretical and applied probability theory basics, that is the subject of the next section. The information required to be exposed as parametric configurations would be in the form of a technical/mathematical sheet specific to each game, consisting of those parameters of the mathematical design of that game that define the sets of possible outcomes and are essential toward probability and statistical computations. For example, in slots the parametric-configuration sheet must show the number of distinct symbols, number of stops of each reel, and the symbol distribution (weighting) of each reel. In a card game, the number of decks used, the number of 29 cards in each deck, and the composition of each deck (numbers of card values and symbols) are known. With a drawing machine (for example, lottery or bingo), the total number of numbers/balls, their value interval, the number of numbers/balls to be drawn, and so on are likewise known. The ethical obligation being established, the question arises as to whether this obligation should remain simply the parametric configuration of the game or be extended to include basic or advanced mathematical results coming from applications worked out on the mathematical model of that game. The extension could consist of basic pre-calculated numerical results, such as probabilities of the basic winning events and expected value, or stretch further to more complex gaming events and other statistical indicators, and the interpretation of these results. The latter variant already assumes a new level of mathematical knowledge, attainable only through scholastic intervention. For the parametric-configurationonly variant, which is merely informative and either provided by the game producer or retrieved by third parties, it would remain for the player to inquire further for the mathematical results as an optional action. The question, then, clearly becomes where to draw the line between ethically required and optional information on the mathematical facts of games of chance. Once the line drawn, the obligation would be imposable only by law, since game operators, like game producers, might consider that it is not to their advantage to provide such technical/mathematical sheets to their customers. On the entire range of mathematical information possible to be exposed, as the amount of information increases, there are three specific levels as seen in the next figure: parametric configuration, basic numerical results (odds of winning and EV), and knowledge of the mathematics of gambling presented in a specific teaching module. Interval I from the first to the second level does not have intermediary values, while interval II – marked with a continuous line in the figure – could have very many intermediary values, depending on the amount and structure of the exposed mathematical information. Figure. Dividing the range of the mathematical information for a game of chance between ethically required and optional information. 30 If assigning the two aspects – commercial and humanitarian – of the ethical obligation over the range of the mathematical information, the commercial one covers only interval I, while the humanitarian aspect could stretch theoretically to the endpoint of interval II, if enhanced mathematical knowledge can have an impact on gambling behavior. Indeed, we cannot extend the coverage of the commercial component beyond the exposure of the basic numerical results, since imposing this extra effort on game producers would be unethical/unfair, given that they already provided the parametric configuration and the basic results that can be further developed optionally by players with the help of other qualified entities. Such a requirement would be comparable to requiring drug producers to expose on their informational leaflets not only the substances in the composition of the drug and the numerical results of the statistical studies on drug's side effects, but also medical information on the possible side effects, while this information is available optionally through medical consultation. Examples aside, the practice of exposing a product/service's information based on commercial ethics confirms this limitation in any economic/commercial field. As for the humanitarian component, even though its coverage could be established only through subjective criteria coming from the involved entities, there is also an objective limitation imposed by the extra time and effort that attendance for a course requires from gamblers. In addition, imposing on game producers a requirement to maintain, sustain, or support such courses would also be unethical/unfair, for the same reason presented for the commercial component. In conclusion, if reducing the ethical obligation to its commercial component, there are two options for drawing the line between ethical and optional information: at parametric configuration only for ethics (position 1 in the above figure) or parametric configuration plus basic numerical results for ethics (position 2 in the above figure), and a final choice can be made only by legislators. Choosing position 2 would be somehow in the vein of the ethical information exposed on cigarette packs, where not only the substances contained in tobacco smoke are mentioned (equivalent of position 1 in our account), but also the warning on health injuries caused by cigarettes and sometimes statistical data on cancers caused by smoking (equivalent of position 2). Of course, in many respects the two situations are not equivalent. If keeping the humanitarian component as necessary for defining the ethics in this particular domain, the line would lie in interval II (including position 2), at a position yet to be established through the consultation of the communities involved (gamblers, game producers, problem-gambling scientific communities, and other specialists) before a choice is made by legislators. Compromise options would still place the line in interval II, in the proximity of position 2. For instance, such an option would be the exposure of the parametric configuration, basic winning odds, expected value, warnings toward gambling 31 fallacies, misconceptions and misinterpretations of the exposed results, and optional recommendations to attend gambling mathematics courses for a better understanding and interpretation of the mathematical facts that govern the game. At first glance, the best option seems to lean toward position 2, which also has the highest number of corresponding examples from other domains; however, further interdisciplinary research is necessary for a rigorous standard, including how "best" should be defined in this particular ethical context. 3. THE OPTIMAL MATHEMATICAL SCHOLASTIC INTERVENTION FOR RESPONSIBLE GAMBLING It is necessary before proceeding toward an optimal mathematical scholastic intervention in gambling to decide whether such an intervention would accomplish the goal of limiting the risk factors and result in a significant desirable change in gamblers' behavior. Regarding the setting in which such an intervention could take place, there are three non-exclusive options: in secondary to post-secondary public schools as optional course or module attached to the probability/statistics courses, within private companies or institutions dealing with gambling and problem gambling, and within cognitive therapy sessions for pathological gambling, strongly reduced to conclusive knowledge and guidelines implemented by the therapist and applied through psychological counseling. The principles stated in the later section Principles of an optimal mathematical scholastic intervention to gamblers apply to the first two of these options. 3.1. Theoretical versus empirical studies on the impact of scholastic intervention In the literature on this matter, contradictory results have been published and a clear conclusion has not yet been drawn. Most of the results were based on statistical studies of college-student gamblers who received a scholastic intervention; some of these results were declared by their authors as "paradoxical" or "unexpected," as they did not confirm the expectation of a significant change in the gambling behavior of the subjects. Thus, Hertwig et al. (2004) found that students who received education on probability gambled on low-odds events more than the students who did not know the actual odds; Steenbergh et al. (2004) found that students who were taught about and given warning about gambling fallacies and mathematical expectation gained superior knowledge on these matters, but were just as likely to play roulette as students who did not receive this intervention; Williams & Connolly (2006) found that students who received instruction on probability theory applied in gambling demonstrated superior ability to calculate gambling odds, as well as resistance to gambling fallacies, but this enhanced knowledge was not associated with any decreases in actual gambling behavior. On 32 the other side, additional theoretical studies proved that post-secondary statistics education developed critical thinking, which also applied to gambling, and gamblers who get such education tend to have significantly lower rates of problem gambling (Gray & Mill, 1991; Gerstein et al., 1999; Abbot & Volberg, 2000). I think that the approach to the problem of changing gambling behavior as result of the mathematical scholastic intervention must be more theoretical than empirical, even though it assumes the use of psychological tools of evaluation. The main reason is that a proper testing of the hypotheses or expectations of the empirical studies is only marginally attainable – if not impossible – since gamblers much be monitored over a long time with no constraints on their actions; the monitoring period should take into account each gambler's own frequency of playing and other personal parameters; therefore, a unique overall monitoring period for the entire sample group cannot be determined with respect to the relevance of the results – a very long monitoring period is needed; as for the constraints, participation itself in the study apprises gamblers of the expectations of the study, which from the outset becomes a constraint that might influence his/her actions. For example, given the newly acquired mathematical knowledge, a gambler could be keen to see whether this knowledge can be applied strategically in the games of chance, resulting in profits, and this attitude could result in an initial increase of his/her gambling activity after the intervention – which could also decrease later in the absence of the anticipated results. (Even though the mathematical facts were taught with the goal of limiting gambling activity, such a trial period on gambler's side could result in moving from one type of game to another more suitable for probability-based strategies.) This potential behavior toward the strategic use of mathematics could explain the "unexpected" results of the studies mentioned above, among which the reported increase in the gambling activity was consistent. Besides the monitoring period, there are also several issues with the conditions of testing the hypotheses of the empirical studies (for instance, the gambling activity being measured in money or time spent, the trustworthiness of the gamblers' reports given the pathological aspect of problem gambling, the quantification in case of playing more than one type of game, etc.). All of the studies mentioned had a laboratory-based evaluation of gambling behavior for testing the hypotheses, which cannot reproduce real-world gambling activity – filling a questionnaire on future intentions can neither substitute for nor predict real actions. Another criticism of the performed empirical studies concerns the sampling. All mentioned studies were undertaken on groups of college students, which is not a representative sample for the gambler population with respect to age. The argument for choosing that category of gamblers was that official reports have found the rate of problem and pathological gambling to peak in ages 18 to 24. My argument for extending the age criterion beyond 24 is twofold: first, the final goal 33 of a scholastic intervention is to limit the risk factors for all gamblers, already manifested or potential, given that in time, a non-problem gambler can become a problem one; second, the age interval 18 24 assumes a particular psychological profile whose features could affect the outcome of the intervention. It is well known that young persons – although more open to learning than older persons– are interested in filling their spare time with entertaining activities more so than their elders, and gambling seems to be one such activity. This status of their gambling activity could prevail over the other main reasons for gambling that older persons may have, of which winning money is the most important. In addition, older gamblers have experienced the gambling failure (money and time spent versus profits gained) more than the younger ones and scholastic intervention could find a more favorable ground in the age range over 24. For these reasons, I expect to see different results on the impact of the mathematical scholastic intervention from empirical studies using a representative sample of the gambling population with respect to the age criterion. One can object that a mathematical background is essential for application of the intervention, and college students are the most likely to have such a background. I answer that in case of a non-math gambler, the intervention can be reduced to the simple delivery of numerical odds and statistical indicators, along with a basic interpretation of them (this is actually the practical side and main goal of any advanced learning) and the studies can test the same hypotheses under this condition. I also claim that the sample should be representative for the locale, as different economic environments can affect the intensity of gambling activity where money is involved, of course if such studies have an international focus and sampling. The structure and content of the teaching module is also important toward the effects of the intervention, and we cannot draw a complete conclusion on the similarity of the reported empirical results if the teaching module for each intervention has a different structure. The structure of the teaching modules is also the subject of the next section on the optimal mathematical scholastic intervention. In conclusion, further theoretical interdisciplinary research is needed on the impact of the mathematical intervention on gambling behavior and also on the optimal conducting of the statistical studies on representative samples from the gambler population; these enhanced empirical studies could confirm the theoretical results. I am inclined to think that a decrease in the gambling behavior can be the result of an optimal intervention (despite the reported results of past statistical studies). Overall, to the question of whether a mathematical scholastic intervention is worth studying, developing, and putting into practice, I answer positively. Even considering as nonconclusive the studies to date on the impact of such an intervention, the intervention falls partially within the ethical obligation to expose 34 the mathematical facts behind games of chance, and an optimal exposure assumes not only numbers, but also interpretations and warnings, which have a scholastic component. 3.2. Principles of an optimal mathematical scholastic intervention to gamblers Although over the last two decades, probability and statistics were present in the curricula of most of the secondary schools as well as some 7 th and 8 th grades worldwide, studies have indicated a decrease in the role of probability and a greater focus on data processing at these educational levels (Borovcnik, 2006). Among the reasons given for this decrease (one of which is that probability is oriented too much toward advanced mathematics, which makes it a difficult topic to teach at the secondary school level), there is the puritanistic view that probability is too closely connected to games of chance, which are seen as plagues of contemporaneous society even in the jurisdictions where they are legal (Borovcnik, 2006). Of course, this principle directly affects the structure of the respective teaching modules, unfortunately lacking sufficient examples and applications from the games of chance, which are essential for a good understanding of probability theory. With this trend, probability theory came to be taught in schools only because it is necessary to justify the methods of inferential statistics. Besides the contradiction with the genesis of probability theory and the concept of probability itself (which were born in the 17th century from games of chance), and with the optimality of the learning process that according to my view, knowledge of the mathematics of gambling can have an impact on gambling behavior if properly taught, that puritanistic principle becomes paradoxical: teaching is modified to avoid mentioning games of chance as much as possible, while on the contrary, understanding the mathematical facts of these games can have a decisive role in limiting the risk factors. However, this principle is not applied in all countries. The best example is Australia, where not only do syllabi outline the role of probability in everyday life and decision making, but teaching modules on the mathematics of gambling have been implemented with success in the secondary schools. In 2008, in the state of Queensland, mathematician Robert Peard developed and helped to implement through governmental intervention a teaching unit called The Mathematics of Responsible Gambling (Peard, 2008). The current research is not focused on probability and statistics courses from the school curricula with respect to problem gambling (which still can remain a good background for further learning), but on developing an optimal teaching module on the mathematics of gambling which will be applicable for both potential and experienced gamblers, with the goal of limiting gambling risk factors and controlling gambling behavior against its pathological side. Such a teaching module would remain optional for the gamblers, as it cannot be imposed through regulations, and offered in both governmental and private venues. However, the 35 possible imposition by law of the exposure of the parametric configuration plus the basic numerical results (such as probabilities of the main winning events and expected values) as an ethical obligation for all games of chance might encourage gamblers to attend such a teaching module on their own for a better understanding and use of the exposed results. When defining the optimality of the mathematical scholastic intervention to gamblers through principles, we should relate them to the main goals of the intervention, which are: 1. All gamblers should be able to attend the teaching module and understand the basic knowledge taught, regardless of their level of mathematical education. 1 2. The gambler will understand the nature and interpretations of the probability concept, its relativity toward the practical aspects of its use in making decisions, and the relation between the probabilistic models and the real world; and he/she will have a clear image of the concepts of randomness and independence. 3. The gambler will be able to perform basic probability computations, evaluations, and approximations for the various gaming events encountered, expected values, and to search for pre-calculated results from available resources. 4. The gambler will show resistance to all gambling fallacies specific to any game. 5. The gambler will evaluate mathematically his gambling activity for short and long term; he/she will finally have in mind an abstract representation of the games he/she plays by reducing them to their mathematical models, and thereby ignoring their addictive elements added in the physical state. These goals can be accomplished mainly through (but not limited to) the structure and content of the teaching module. The following principles are important for an optimal structure and content, and the purpose of this paper is just to claim them as decisive toward the proposed aim without generating the entire detailed structure of the module. This structure and the inference on why and how these principles can induce the sought-after effect on gambler's behavior will be the matter of a forthcoming interdisciplinary research and scholastic project. These principles are stated below: a) The teaching module must be adapted to all levels of background mathematical education, which will be established through preliminary tests. For the lower levels, the module will be extended with additional preliminary lessons as the level requires, having topics such as real numbers, numeric calculus, functions, 1 A degree of a secondary school is required to attend this teaching module, so the use of the terms "all gamblers" and "gamblers" assumes this prerequisite. 36 algebraic calculus, and set theory basics. Also for the lower levels, the lessons within the probability and statistics parts will be enhanced with more examples and a more extensive interactive component. Some lessons will be split and completed with extra examples. This approach requires teaching a large part of the module in separate groups of different levels and reduces the risk of rejection and abandonment of the intervention by the gambler for reasons of incomprehension and inadaptability. This principle is related to goal 1. b) The theoretical parts should be limited in generality strictly to cover through application the games of chance, which means teaching only in discrete probability and only those results facilitating understanding of the basic concepts and the computational purpose. Exceptions are applicable if serving the purpose of clarifying a concept (for instance, probability as a measure requires a good deal of generality). Any added advanced mathematics not serving the purposes of the intervention could result in a break in the student's connection with the teaching flow. This principle is related to goals 1, 3, 4, and 5. c) The module should have a strong applicative character, showing the student how to frame each game and gambling problem within the suitable probabilistic model to which theory is applied, and conversely, each theoretical asset should be followed by solved applications from gambling. For the student to acquire computational skills, an algorithmic approach of the applications is required. This principle is related to goals 1, 3, and 5. d) The module should have compact sub-modules, each dedicated to one major game of chance, where the most important applications specific to that game are presented. Such sub-modules should also have collections of pre-calculated numerical results for the student to study and assign to imaginary gaming situations, thereby facing as many probabilities as possible. Students who practice only one game may attend only the sub-module dedicated to that game. This principle is related to goals 1, 2, 3, 4, and 5. e) The module should have a compact sub-module dedicated to gambling fallacies, misconceptions and erroneous interpretations of theoretical and numerical results from probability and statistics, even though these subjects are touched upon in other theoretical lessons. This principle is related to goals 2 and 4. f) Applicative lessons and seminars should contain recommendations and instructions on the choosing and use of external resources on gambling 37 mathematics, given the wide exposure on the internet and the large number of book titles on this topic. This principle is related to goal 3. g) The applications toward strategy and optimal play, presented in a sub-module dedicated to a specific game, will be limited and focused on winning odds and the long-run play of that game. The student will be taught that an optimal play would give him/her advantage in a game against opponents, never in a game against the house, but the winnings are still governed by the odds, under the luck factor. This principle is related to goals 2, 4, and 5. h) The module should contain a sub-module dedicated to the interpretations, relativities, psychology, and philosophy of the probability concept, placed at the end of the module. The lessons of this sub-module should be developed by a mathematician assisted by a psychologist and will be a popular presentation of the probability concept in all major views surrounding mathematical probability – classical, inductive, subjective, frequentialist, propensitistic – adapted for the non-mathematician. Students will be walked through the philosophy of probability with no reserve, touching the ontological status of probability and passing through the entire range of interpretations; he or she will be shown the differences between the common-language term and the scientific concept in its various interpretations, the view of probability as both objective and subjective, the difference between possible and probable, the relationship of probability with the individual psychological degree of belief in the occurrence of an uncertain event. This principle is related to goals 2, 4, and 5. Besides lectures, the teaching module will have interactive sessions consisting of seminars in which to clarify issues with understanding, solve problems and applications, and perform knowledge tests. As a principle for the interactive portion: The interactive sessions should also contain discussions on the ongoing gambling experience of the students. These discussions should be focused on the mathematical analysis of the gambling stories, which should be framed within a general model where they are treated as simply one experiment from a series of independent experiments. This principle is related to goals 2, 3, 4, and 5. The scholastic intervention could have an impact on development of the pathway of gambling in only one type of its processes, namely, the influence of classical and operant conditioning, as this process is common to all models of gambling pathways (Blaszczynski & Nower, 2002); however, the impact could be decisive, since this type of processes corresponds to an early state of the pathway. While such an intervention will remain optional for gamblers (and future studies can determine the extent to which gamblers will attend an optional 38 intervention), there is still a way of using it as non-optional, namely, as implemented in a proper reduced form in cognitive therapy sessions. Such a direct contribution of mathematics to the psychological intervention can extend beyond providing mathematical information, which is the subject of the next section. 3.3. Further research on the direct contribution of mathematics to psychological interventions Principle (h), as stated in the previous section is what is missing in the current curricula on either probability theory or mathematics of gambling and I claim it as essential toward an optimal intervention. Such a conceptual component is given the lowest priority, when it is included at all, in the probability courses of secondary and even post-secondary schools, but – paradoxically – could have a role in preventing problem gambling. The conceptual components across all principles stated in the previous section make the transition from an indirect to a direct contribution of mathematics to psychological interventions. The main subject of further investigation is based on the following premise: The surplus (physical) structure added in reality to the abstract mathematical model of a game (the game in its consumable, commercial, casino form, the environments in which games are running, gambling industry) is that which contains the addiction elements and not the game itself as mathematical model. For instance, the near-miss effect on slots, obtained through a progressive visualization of a beginning part of a winning combination does not exist in the probabilistic model of that slot game. In that model, only the combinations of stops (holding the symbols) of the reels as elementary events of the probability field do exist, and the expected winning combination has a certain probability. Therefore, there is no near miss in the mathematical model, but only the probability of that "near-missed" combination. If the player, through psychological counseling, would reach a state where to have a representation of the game only as mathematical model, the nearmiss effect would vanish and, with it, an important addictive element. A similar observation applies for the illusion of control. It is such feature through which gambling addiction is different from other types of addiction (for instance, from smoking, where the addictive elements are in the cigarette itself, which cannot be reduced any more to an essential model). Then further research is needed for proving theoretically and empirically that an optimal structure and content of a mathematical module following to be implemented into cognitive therapies should contain the reduction-to-models conceptual approach, together with the facing-the-odds component. Given the interdisciplinary aspects of the processes involved in acquiring the aims of the mathematical intervention, an elaborated research project is needed, which will be the topic of a forthcoming paper. Projects of developing the entire 39 structure and content of both the teaching module and the module for cognitive therapies should also follow the results of this further research. 4. CONCLUSIONS Past studies on the role of mathematics in problem gambling prevention and treatment have been limited to isolated empirical researches focused on the indirect contribution of mathematics as scholastic intervention. Such empirical researches did not yield conclusive results. Optimizing such mathematical scholastic interventions is possible, by following certain principles that relates to the goals of the intervention. Moreover, given the strong relation of mathematics with the gambling activity, as underlying the games, the potential of mathematics extends beyond the indirect contribution, to a direct one, namely its inclusion into cognitive therapies as a proper mathematical module. Further research is needed for proving theoretically and testing empirically that such a module, basing on the two main principles facing the odds and reduction-to-models, can improve decisively the cognitive therapies for problem gambling. REFERENCES Abbott, M.W. & Volberg, R.A. (2000). Taking the Pulse on Gambling and Problem Gambling in New Zealand: A Report on Phase One of the 1999 National Prevalence Survey. Department of Internal Affairs, Government of New Zealand. Bărboianu, C. (2013a). How to estimate the number of stops and the symbol distribution on a reel. In Infarom (Ed.), The Mathematics of Slots: Configurations, Combinations, Probabilities (pp. 46-63). Craiova: Infarom. Bărboianu, C. (2013b). Is the secrecy of the parametric configuration of slot machines rationally justified? The exposure of the mathematical facts of games of chance as an ethical obligation. Manuscript submitted for publication. Bărboianu, C. (2013c). Mathematician's call for interdisciplinary research effort. Forthcoming in International Gambling Studies. Blaszczynski, A. & Nower, L. (2002). A pathways model of problem and pathological gambling. Addiction, 97 (5), 487-499. DOI: 10.1046/j.1360-0443.2002.00015.x Borovcnik, M. (2006). Probabilistic and statistical thinking. In M. Perpinan, & M. A. Portabella (Eds.), Proc. of the Fourth Congress of the European Society for Research in Mathematics Education (pp. 484-506). Sant Feliu de Guixols, Spain : ERME Gerstein, D., Volberg, R.A., Murphy, S., Toce, M., et al. (1999). Gambling impact and behavior study. Report to the National Gambling Impact Study Commission. Chicago: National Opinion Research Center at the University of Chicago. Gray, T. & Mill, D. (1991). Critical abilities, graduate education, and belief in unsubstantiated phenomena. Canadian Journal of Behavioral Science, 22, 162-172. 40 Harrigan, K.A., & Dixon, M. (2009). PAR Sheets, probabilities, and slot machine play: Implications for problem and non-problem gambling. Journal of Gambling Issues, 23, 81110. DOI: 10.4309/jgi.2009.23.5 Hertwig, R., Barron, G., Weber, E.U., Erev, I. (2004). Decisions from experience and the effect of rare events in risky choice. Psychological Science, 15 (8), 534-539. Information and Privacy Commissioner, Ontario, Canada (2009). Decisions and resolutions: PO-2744. Retrieved from http://www.ipc.on.ca/images/Findings/PO-2774.pdf Information and Privacy Commissioner, Ontario, Canada (2010). Decisions and resolutions: PO-2744. Retrieved from http://www.ipc.on.ca/images/Findings/PO-2903.pdf Peard, R. (2008). Teaching the Mathematics of Gambling to Reinforce Responsible Attitudes towards Gambling. Retrieved from http://www.stat.auckland.ac.nz/~iase/publications/icme11/ICME11_TSG13_15P_peard.pdf Steenbergh, T.A., Whelan, J.P, Meyers, A.W., May, R.K., & Floyd, K. (2004). Impact of warning and brief intervention messages on knowledge of gambling risk, irrational beliefs and behavior. International Gambling Studies, 4 (1), 3-16. Williams, R.J., Connolly, D. (2006). Does learning about the mathematics of gambling change gambling behavior? Psychology of Addictive Behaviors, 20 (1), 62-68. REZUMAT La întrebarea dacă se poate schimba comportamentul practicanţilor jocurilor de noroc ca rezultat al învăţării matematicii jocurilor de noroc, studiile efectuate până în prezent au oferit răspunsuri contradictorii, nefiind trasă o concluzie clară. În acest articol aduc unele critici cercetărilor empirice care au înclinat să răspundă nu acestei ipoteze, în ceea ce priveşte eşantionarea şi testarea de laborator, şi susţin că o intervenţie scolastică matematică având ca scop prevenţia jocului problematic este posibilă, enunţând principiile care optimizează structura şi conţinutul modulului didactic. Plecând de la aspectele etice ale expunerii faptelor matematice din spatele jocurilor de noroc şi de la cazul jocurilor de sloturi – unde configuraţia parametrică nu este vizibilă – trebuie să tragem linia de demarcaţie între informaţia etică şi cea opţională provenind dintr-o intervenţie scolastică matematică. Susţinând rolul matematicii în prevenţia şi tratamentul jocului problematic, sunt trasate direcţii de cercetare interdisciplinară privind implementarea unui modul matematic adecvat in terapiile cognitive.

Canadian Journal of Philosophy Kripke's Objections to Description Theories of Names Author(s): Michael McKinsey Reviewed work(s): Source: Canadian Journal of Philosophy, Vol. 8, No. 3 (Sep., 1978), pp. 485-497 Published by: Canadian Journal of Philosophy Stable URL: http://www.jstor.org/stable/40231052 . Accessed: 02/11/2012 11:35 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Canadian Journal of Philosophy is collaborating with JSTOR to digitize, preserve and extend access to Canadian Journal of Philosophy. http://www.jstor.org CANADIAN JOURNAL OF PHILOSOPHY Volume VIII, Number 3, SeptemberJl978 Kripke's Objections to Description Theories of Names MICHAEL McKINSEY, Wayne State University In "Naming and Necessity" Saul Kripke describes some cases which, he claims, provide counterexamples both to cluster theories and, more generally, to description theories of proper names. My view of these cases is that while they do not provide counterexamples to cluster theories, they can be used to provide evidence against single-description theories. (I count as singledescription theories both "short-for-descriptions" theories of the Frege-Russell sort and what I shall call below "fixed-by-attributes" theories.) In this paper I shall defend both of the claims involved in my view. 1. Kripke' s cases. Although it is somewhat of an oversimplification to do so, I will take all of Kripke's cases as directed against a single principle endorsed by every cluster theorist. The principle is that for every speaker s, token a of a proper name, individual x, and time t , 485 Michael McKinsey (CD) ifs utters a at t and s is not immediately experiencingx at t, then a refers to (denotes) x only if there is a property F such that: (i)x is the one and only individual that isF; (ii)at t, s believes that there is just one individual that is F; and (iii) the property of being F is not question-begging with respect to s's use of a at t . Let us understand that on cluster theories of names, the referent of a given name-use (or token) is supposed to be usually determined by a non-empty cluster of properties which the user associates with the name. Cluster theorists like P. F. Strawson and John Searle have not provided any clear answer to the question of how a given cluster determines the referent of its associated name-use. They have instead been content to say that a name-use refers to an object if the object satisfies a "sufficient" number of the properties in the associated cluster (see [8], p. 490, for instance). But they have clearly demanded that for an object to be the referent of a name-use, it is necessary that it uniquely satisfy at least one of the properties in the cluster. Why have they demanded this? Each property F in a reference determining cluster is supposed to provide a non-question-begging answer of the form 'the one and only F which the name-user would give to the question 'Who or what are you referring to?' A question-begging answer to this question would be given by a speaker if he said in response, for instance, 'The one I am now referring to with a ' or 'The one I now have in mind' or 'The one I mean (by a )'.1 The cluster theorist claims that the referent of a name-use must uniquely satisfy a member of the associated cluster since (a) for an object to be the referent of a name-use, the speaker must mean or intend to refer to that object with the use, and (b) for a speaker to mean, or intend to refer to, a given object to the exclusion of all others, there must be at least one non-questionbegging property F uniquely satisfied by this object which the speaker would use to pick out or identify the object to which he intends to refer as 'the one and only F'. (See Searle [9], p. 87, and Strawson [10], p. 185). This is basically why cluster theorists have endorsed (CD). But Kripke argues that a speaker may refer to an object with a name, even if no object satisfies any of the properties in the associated cluster. For example, where the <?'s are the properties in a given cluster, he says: 1 In calling such descriptions "question-begging" I am following terminology which Donnellan uses in [2], p. 365. 486 Description Theories of Names suppose.. .that nothing satisfies most, or even any substantial number of the ^'s. Does this mean the name doesn't refer? No: ...you may have false beliefs that are true of absolutely no one. And these may constitute the totality of your beliefs. ([4], p. 295) If this is so then of course the cluster theory principle (CD) is false. One case which Kripke gives in arguing for the just-quoted claim is that of a speaker s who uses the name 'Godel' and whose sole answer to the question 'Who are you referring to?' would be 'the man who discovered the incompleteness of arithmetic'. Kripke asks us to suppose that no one discovered the incompleteness of arithmetic. Perhaps the proof miraculously appeared on a sheet of paper. Perhaps a subtle error in Gbdel's argument has not yet been noticed. Nevertheless, Kripke claims, our speaker s would still be referring to Gbdel with his name. If this claim is correct, then (CD) is false. Another case which Kripke gives is one in which again a speaker uses 'Godel' and associates with this name only the description 'the man who discovered the incompleteness of arithmetic'. But in this case, we are to suppose that it was not Godel but someone else who first proved incompleteness. As Kripke puts it, we are to suppose that "a man named 'Schmidt', whose body was found in Vienna under mysterious circumstances many years ago, actually did the work in question. His friend Godel somehow got hold of the manuscript and it was thereafter attributed to Godel." Kripke then claims: "So, since the man who discovered the incompleteness of arithmetic is in fact Schmidt, we, when we talk about 'Godel', are in fact always referring to Schmidt. But it seems to me that we are not. We simply are not" ([4], p. 294). Apparently, Kripke means that in this case, we are referring to Godel. If so, and if he is right, then again (CD) is false. However, his main point in giving this example is to show that cluster theories do not provide the correct sufficient conditions for name reference. (All we know if (CD) is false is that they do not provide the correct necessary conditions.) We obtain this result since in this case we are dealing with a one-membered cluster. Here, if Kripke is right, an individual's uniquely satisfying all the members of a name-use's associated cluster is not sufficient for that individual to be the use's referent. Kripke considers a reply that might be made to his Godel-Schmidt case, namely, that the speaker might have had some other description in mind which Godel does satisfy. Suppose he had in mind the description 'the man to whom the discovery of incompleteness is commonly attributed'. Kripke answers this reply by saying that the same sort of counterexample as he has already 487 Michael McKinsey given applies here as well. The speaker might still be referring to Codel even if, unbeknownst to him, the discovery is now commonly attributed to Schmidt ([4], p. 296). There is something puzzling about this objection which Kripke imagines might be made to his case. For how exactly is it supposed to be relevant? After all, the initial case was one in which there were no descriptions which the speaker associated with 'Coder other than 'the man who discovered the incompleteness of arithmetic'. So what is the point of suggesting that the speaker might have associated some other description with 'Codel'? Surely, the point is not that some case other than the one Kripke gives would not provide a counterexample to cluster theories. For this point, though true, has no bearing on the issue of whether the case Kripke does give provides a counterexample. Perhaps the point of the reply is this. In the sort of case Kripke imagines, one in which the user of 'Codel' believes he is referring to the man who discovered arithmetic's incompleteness, it is natural to assume that the user would also have various other beliefs about the referent, beliefs yielding further properties in the cluster associated with the use. Consider for instance the properties mentioned in the descriptions: (a) the man to whom the discovery of arithmetic's incompleteness is commonly attributed; (b) the man of whom I have heard (read) that he discoverd the incompleteness of arithmetic; (c) the only man named 'Codel' of whom I have heard; (d) the man named 'Codel' of whom I have heard (read) that he discovered the incompleteness of arithmetic. Since the descriptions (b), (c), and (d) are descriptions which the speaker would give, the pronoun 'I' represents a first-person reference on his part. Notice how extremely likely it is that a typical user of 'Godel' would associate descriptions of this sort with his use of 'Codel' when he also intends to refer with this name to the discoverer of incompleteness. But then the sinister possibility arises that when Kripke claims that in his case the speaker is referring with 'CSdel' to CSdel and not Schmidt, his claim seems intuitively correct only because we tacitly assume that the speaker has at least four other ways (represented by (a)-(d) of picking out the referent of his use. Then, since Codel, and not Schmidt, in fact uniquely satisfies four out of five of the properties in the use's associated cluster, Kripke's claim 488 Description Theories of Names might seem intuitively correct only because it is the correct claim to make on the cluster theory. This point is well taken. For suppose that Kripke had described his case so as to explicitly rule out certain of (a)-(d) as being in the cluster which s associates with 'Godel'. Imagine, for instance, thats uses 'Godel' with the intention of referring to the discoverer of incompleteness, but s believes both that he has never in his life heard of anyone named 'Godel' and that he has never heard the proof of incompleteness attributed to anyone named 'Godel'. What would s be doing using 'Godel' in such circumstances? We can only assume that by some wild coincidence s just happened to pick the name 'Godel' and decided to use it to refer to the discoverer of incompleteness (perhaps he just happened to like the sound of 'Godel ). If we assume that Schmidt rather than Godel proved incompleteness, who is s referring to with 'Godel'? The intuitively correct answer now is Schmidt, not Godel. Or, if we assume that no one proved incompleteness, it is now intuitively correct that s is referring to no one with 'Godel'. It is clear, then, that Kripke has given no counterexample to cluster theories at all. For suppose, on the one hand, that the only assumption of Kripke's Godel-cases is that the cluster associated with the speaker's use of 'Godel' is one-membered and contains just the property of having discovered the incompleteness of arithmetic. Then, as we've just seen, it is natural to suppose that the speaker is referring with 'Godel' to whomever made this discovery, and we have no counterexample to (CD). Suppose, on the other hand, that Kripke makes other unmentioned assumptions about his cases, assumptions which lead him to reach different conclusions than the ones we reached in the previous paragraph. For all we know these assumptions are such that if they were made explicit, they would yield cases in which the claim that Godel is the referent of 'Godel' is consistent with (CD). Again we have no counterexample to (CD). (No doubt Kripke's main assumption is that the uses of 'Godel' in his cases are typical, similar to ones that he would make or that members of his audience would make. But if the uses are typical, properties like (a)-(d) are in the associated clusters, and again we have no counterexample.) Kripke does raise some objections to claims that in his Godel-cases the speakers would probably have had other properties in mind which Godel does uniquely satisfy. But these objections are unconvincing. One such objection is that we can take a description like (a) and construct a case in which Godel is the referent of a use of 'Godel' even though he does not satisfy (a) ([4], p. 296). I fail to see the point of this objection. Even if we can take each description proposed as one a speaker probably had in mind and show that the 489 Michael McKinsey referent might not have satisfied it, we have in doing this still not constructed a counterexample to (CD). What is necessary to refute (CD) is a case in which it is intuitively obvious that a given object is the referent of a name-use even though it uniquely satisfies none of the properties in the use's associated cluster. It is a sufficient reply to Kripke's cases to point out that they do not have this feature; Kripke's point about (a) has no bearing on this reply. Another of Kripke's objections is to the sort of property to which a description theorist would apparently have to appeal in reply to his cases. For an individual to satisfy descriptions (a)-(d), there must have been other references to this individual which the speaker in question has witnessed. When a speaker's reference with a name is determined by such properties, we might say, as Strawson does ([10], p. 185), that the speaker's reference "borrows its credentials" from other persons' references (though note that in the case of (a)-(d), not all of these other references need have been made with the name in question, or with any name at all). Of course if one person's use of a singular term borrows its reference-credentials from a second person's use, which borrows from a third person's, and so on, the chain of reference-borrowings cannot be infinite or circular, if the first person's use is to have a referent. For instance, if I borrow my reference from another's, who borrows his from another's, who borrows his from mine, none of these references will be successful. Kripke comments: Is one sure that this won't happen?. ..[although in general such chains do exist for a living man, you won't know what the chain is. You won't be sure what descriptions the other man is using, so the thing won't go in a circle, or whether by appealing to [the other speaker's reference] you won't get back to the right man [i.e., Godel] at all. So you cannot use this as your identifying description with any confidence. ([4], p. 298). In the "Addenda" to his paper Kripke makes it clear that he meant this remark as an objection to the assumption by cluster theorists like Strawson that "buck-passing" properties could play a role in determining reference ([4], p. 766). But I find it hard to see the force of the objection. What, for instance, is the difficulty if a buck-passing property sometimes fails to lead back to the "right" man? Here, Kripke seems to have in mind the point he made earlier concerning the use of (a) to determine the reference of 'Godel': the speaker might refer to Godel even though at the time, and unbeknownst to the speaker, most people attribute the discovery of incompleteness to Schmidt. But surely, a cluster theorist could allow such a possibility and at the same time allow buck-passing properties to play a role in determining name-reference. For instance, the cluster (a)-(d) might 490 Description Theories of Names yield Godel as the referent of a use of 'Godel', even if (a) happens to be satisfied by Schmidt. Here, (a) leads back to the "wrong" man, though the cluster of which (a) is a part yields the "right" man, and the other properties in the cluster are buck-passing. There is no difficulty for cluster theories here. Perhaps Kripke has in mind a case in which (a) is the sole member of a cluster determining reference for 'G6del\ But in such a case (as I argued earlier), if Schmidt satisfies (a), it is far from clear that Schmidt would in fact be the "wrong" man. Kripke objects that one who uses a name on the basis of buck-passing properties would not really know whether he has borrowed his reference from a reliable source. Perhaps the chain of reference-borrowings goes in a circle, or perhaps at the far end of the chain no original reference was made to any individual at all. This might be a problem for individual name-users; a given person might be totally unjustified in believing that any individual at all satisfies the buck-passing description(s) with which he would try to identify the referent of his name-use. But this fact does not pose any obvious theoretical difficulty for cluster theories. Even if it were true, as Kripke seems to think it is, that such beliefs are almost always unjustified (and this suggestion is extremely dubious) there would still be no particular difficulty for cluster theories, since it is certainly not necessary for a proponent of such a theory to endorse the implausible thesis that a speaker's use of a name has a referent only if the speaker knows it does. Finally, Kripke claims that "Strawson apparently must require that the speaker know from whom he got his reference, so that he can say : 'By "Godel" I mean the manyones calls "Godel" ([4], p. 299). Then Kripke objects that "If the speaker has forgotten his source, the device is unavailable to Strawson; if he misremembers it, Strawson's paradigm in his footnote can give the wrong results" ([4], p. 300). But Kripke doesn't explain why Strawson must require that the speaker remember from whom he got the reference. Why should he require this? After all, there are other ways in which one can borrow reference without depending on such memories (see descriptions (b)-(c)). The claim that Strawson's device might yield the "wrong" results when the speaker misremembers from whom he got his reference was considered before in connection with Kripke's objection to the use of (a). We may conclude that Kripke has offered no relevant objection to the reply to his 'Godel'-cases that the user of 'Godel' would probably have had other properties in mind that Godel does uniquely satisfy. Thus his cases do not pose conclusive counterexamples to the cluster theory principle (CD). Nor does his 491 Michael McKinsey Cbdel-Schmidt case show that no cluster theory will provide the sufficient conditions for name-reference. Searle has said that the clusters of properties which determine the reference of a name are composed of those properties most "commonly attributed"to the referent ([7], p. 160). Similarly, Strawson suggests that the cluster will provide a "composite description incorporating the most frequently mentioned facts" ([10], p. 196). But Kripke's 'Godel'-cases show that one who wishes to maintain a cluster theory of names will have to allow that properties which are not frequently attributed by use of the name, properties such as that of being a man named 'Godel' of whom a particular speaker has heard, may nevertheless play a decisive role in determining reference. Kripke has shown that the StrawsonSearle variety of cluster theory is strictly false, since his cases show that a person may be the referent of a name even if that person does not possess the characteristics most commonly attributed to him. But this is a minor difficulty, one which is easily repaired by allowing properties such as those mentioned in (a)-(d) to be members of reference-determining clusters. Certainly, Kripke's examples do not, as he claims show that "the whole picture given by this theory of how reference is determined seems to be wrong from the fundamentals" ([4], p. 300). 2 2. The effect of Kripke's cases on single-description theories. By a single-description theory of names, I mean a theory which endorses (at a minimum) the following thesis: 2 Thus Kripke's /C6del'-cases are valuable, not because they show that no cluster theory is correct, but because they suggest ways in which previous cluster theories must be revised in order to be made correct. I have proposed such a revised cluster theory in my paper [5]. Kripke makes several other valuable points in his discussion of names in [4], the most important of which is that proper names are rigid designators, that is, are terms which denote the same individual in every possible world. The sort of view I am defending here, on which the referent of a name in the actual world is determined by a cluster of definite descriptions, is of course consistent with Kripke's idea that names are rigid designators. For a concise and comprehensive description of Kripke's views on reference, see R. B. De Sousa's [1]. This paper also contains criticisms of Kripke's view that theoretical identities in science are necessary if true, since they are composed of rigid designators. For a reply to these criticisms, see R.M. Yoshida's[11]. 492 Description Theories of Names (SD) If s is not immediately experiencing x at t and s utters a token a of a proper name /3 at t, then a refers to (denotes) x only if there are a proposition p, a non-question-begging property F, and a sentence A such that: (i) x is the one and only individual that is F; (ii) s utters a in the course of uttering A; (iii) s's uttering A att expresses s's thinking that p; and (iv) the proposition that p is expressible by a (possible) sentence /\*such that /\*may be obtained (at least in part) from A by replacing each occurrence of 0 in A by a definite description whose matrix expresses the property of being F. (SD) is meant to capture a principle once endorsed by Bertrand Russell, a principle which he expressed as follows : Common words, even proper names, are usually really descriptions. That is to say, the thought in the mind of a person using a proper name correctly can generally only be expressed explicitly if we replace the proper name by a description. Moreover, the description required to express the thought will vary for different people, or for the same person at different times. ([6], p. 54). In addition to holding (SD), Russell also seems to have believed that a name, as used on a given occasion, has the same meaning as the definite description which could be used to express the proposition of which the speaker is thinking at the time of use. That is, Russell seems to have held a "short-for-descriptions" theory of names. But one can hold (SD) without believing that names are used as short for definite descriptions. Theories which hold (SD) while denying that names are short for descriptions I call "fixed-by-attributes" theories. As we have seen, the reason why Kripke's cases are ineffective against the cluster theory principle (CD) is that these cases show at most that an object may be the referent of a name-use without uniquely satisfying a particular property in the use's associated cluster, while to show (CD) is false, it is necessary to show that an object may be the referent of a name-use while uniquely satisfying none of the properties in the cluster. However, to show that (SD) is false, it is only necessary to show that the referent of a name-use may fail to uniquely satisfy one particular member of the cluster, namely, the property mentioned in the description by use of which the speaker's thought at the time would be expressed. Thus, for all we know so far, Kripke's cases might prove effective against single-description theories. Let us consider this possibility with respect to the Godel-Schmidt case. 493 Michael McKinsey Given Kripke's official description of this case, it has no effect on single-description theories either, since on this description, we are to make the special assumption that the only way in which the speaker would attempt to pick out the referent of his use would be as "the discoverer of the incompleteness of arithmetic/' This assumption makes Kripke's case an atypical use of 'G6del', so that, as we have seen, it is far from clear that Godel would in fact be the referent of such a use had Schmidt proved incompleteness. However, it is apparent that Kripke makes the unofficial assumption that in his case, the user of 'Godel' is a typical user of this name, someone who has heard and read of a certain famous logician named 'Godel', who has heard the incompleteness proof attributed to this man by use of this name, and so on. Suppose Jones is such a typical user of 'Godel' who on a given occasion says (1) Godel resides in Princeton. Jones, let us assume, utters (1) in atypical communication situation; that is, Jones intends to make by use of (1) an assertion about a given individual and to express a given belief of his concerning this individual. If asked who he means by 'Godel', Jones would respond by using descriptions (a)-(d) as well as (e) the discoverer of the incompleteness of arithmetic. Descriptions (a)-(d) as used by Jones are, assume, satisfied uniquely by a certain logician (whom we shall call 'Godel') who does in fact reside in Princeton; suppose, however, unbeknownst to Jones, we have incontrovertible evidence that it was not Godel who first proved the incompleteness of arithmetic, but an unknown Viennese high school teacher named Schmidt who died in 1930 under mysterious circumstances. Knowing this fact, it is nevertheless clear that we would correctly take Jones to have referred with 'Godel' to Godel, and not to Schmidt. Now the fact that Godel would be the referent of Jones's use of 'Godel' does not by itself controvert or support any particular view of names. However, the fact that we know that Jones would be referring to Godel in this case indicates something about our concept of reference which is difficult to reconcile with single-description views. For on such views, we of course cannot know which individual a speaker is referring to with a name until we know which individual it is that the speaker is thinking of when he uses the name. But the assumptions of the Godel-Schmidt case do not tell us which individual it is that the thought expressed by the 494 Description Theories of Names speaker's use is about. For these assumptions concern only what the speaker meant, or the intentions with which the speaker used the name, not what the speaker was thinking at the time of use. For instance, it is consistent with our description of this case that the thought in Jones's mind when he says (1) is his thought that (2) The discoverer of incompleteness resides in Princeton. It is also consistent with this description that at the time he says (1), Jones is thinking of nothing at all that is expressible by replacing 'Godel' in (1) by a definite description. For instance, suppose Jones reads off (1) from a list of answers he has written down to questions in a parlor game called "Residences of the Renowned", and while uttering (1), Jones is thinking only of where he should eat lunch. Surely, this is consistent with Jones's having used 'Godel' with the intentions I ascribed to him, but it is not consistent with his thinking of a proposition having the form of (2). Thus for all our assumptions tell us Jones is thinking of Schmidt not Godel, or thinking of neither Schmidt nor Godel, when he says (1). The assumptions of our case, then , do not provide evidence that Jones is thinking of a proposition about Godel having the form of (2) when he says (1). But given these assumptions, it is intuitively correct that Jones is referring to Godel in his utterance of (1). Consequently, referring to an object with a name does not entail thinking of a proposition expressible by use of a definite description which refers to that object. Otherwise, evidence that a person is referring to an object with a name would always be evidence that the person is thinking of a proposition expressible by a description which refers to that object, and as we have seen, this is not always the case. Hence, (SD) is false, and no single-description view is true. I believe that the argument just given from the Godel-Schmidt case against single-description theories captures one of the dominant motives lying behind the unwillingness of many, myself included, to embrace a single-description view, In fact, I think, cases of the Godel-Schmidt sort have historically provided one of the main reasons why many philosophers have rejected single-description views in favor of a cluster theory. It is therefore not surprising that this type of case should prove effective only against single description views and not against cluster theories. I have tried in this paper to defend cluster theories of names by defending the principle (CD) against Kripke's objections. I have not tried to argue here that (CD) is true and so I have not given reasons for believing that the correct theory of names is a description theory 495 Michael McKinsey as opposed to a causal theory of the sort Kripke and others have endorsed as an alternative to description theories. (See [4], pp. 298-303). 3 However, I think that a primary consideration which has led many, including Kripke, to endorse a causal theory of names, has been the belief that Kripke's examples show that no description theory of names can be true, plus the fact that causal theories are consistent with these examples. If I am right, this is not a good reason for believing that the correct theory of names is a causal theory. February 1978 3 According to the sort of causal theory Kripke proposes, a name-use's referent is typically determined by a causal chain of communication which links the use to an initial baptism of the referent with the name. In his excellent paper [3], Gareth Evans argues that the sort of causal theory suggested by Kripke is inadequate because it does not take into account the role played by speakers' intentions in determining the reference of the names they use. However, Evans also believes that (CD) is false; so he proposes an anti-descriptionist causal theory on which (roughly) a name-use's referent is in part determined by what the speaker intends to refer to, and what a speaker intends to refer to is in turn determined by a cluster of causal connections between the speaker and the sources of the (perhaps totally incorrect) information which the speaker associates with his name-use. In my paper [5], I have argued that causal theories of the sort which Evans proposes are false. The argument of [5] also supports (CD) and thus supports the view that the correct theory of names is a description theory and not a causal theory of any sort. 496 Description Theories of Names REFERENCES [1] De Sousa, R. B. "Kripke on Naming and Necessity." Canadian Journal of Philosophy 3 (1973-4), 447-464. [2] Donnellan, Keith S. "Proper Names and Identifying Descriptions." Synthese 21 (1970), 335-58. [3] Evans, Gareth. "The Causal Theory of Names." Part I of a symposium with J. E. J. Altham. Proceedings of the Aristotelian Society, Supplementary Volume 47 (1973), 187-208. [4] Kripke, Saul A. "Naming and Necessity," and "Addenda." In Semantics of Natural Language, D. Davidson and C. Harman, eds. (Dordrecht: D. Reidel, 1972): 253-355; 763-69. [5] McKinsey, Michael. "Names and Intentionality.'The Philosophical Review 87 (1978). [6] Russell, Bertrand. The Problems of Philosophy (New York: Galaxy Books, 1959). [7] Searle, John. "Proper Names." Reprinted in Philosophy and Ordinary Language, Charles Caton, ed. (Urbana: University of Illinois Press, 1963), 154-61. [8] _____ . "Proper Names and Descriptions." In The Encyclopedia of Philosophy, Paul Edwards, ed. Volume 6 (New York: Collier-MacMillan, 1967), 487-92. [9] . Speech Acts (Cambridge, England : Cambridge University Press, 1969). [10] Strawson, P. F. Individuals (Garden City, N.Y.: Anchor Books, 1963). [11] Yoshida, R. M. "De Sousa on Kripke and Theoretical Identities." Canadian Journal of Philosophy 5 (1975), 137-41.

International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 2 Issue 12, December – 2018, Pages: 59-66 www.ijeais.org 59 Use of Blockchain in Strengthening Cybersecurity And Protecting Privacy Arif Sari MIS Department, Girne American University Canterbury, United Kingdom [email protected] Abstract-The purpose of this study is to highlight and prove the positive impact in which blockchain could have on today's IoT environment in terms of providing Cybersecurity for not just organizations, but other individuals who share data via the internet. The current IoT environs operates on a centralized cloud based server, meanwhile block chain operates on a decentralized server. The differentiation between the both plays a major role in the level of security they both provide; whereby, decentralized systems are less vulnerable to cyber-attacks and centralized ones are not. In this report, real life illustrations are used to justify the knowledge that the block chain security could serve as a saving grace for internet users. Also, this form of security seems to be the only way to eradicate all forms of insecurity in supply chain and IoT breaches for businesses. An in depth understanding on how the block chain system operates would show why it's held with such high hopes and how it can be beneficial for a wide range of industries. However, this research also highlights some difficulties that might arise in implementing the block chain system. It also gives ways in which the system can be gradually implemented. Firstly, the government needs to make it mandatory for businesses that deal with national based confidential data to implement block chain in their supply chain system to boost security. Secondly, top company officials like C.E. O's need to take genuine interest in the block chain technology by investing in its development and making it a part of employee training, which in the long run would benefit the economy and business in return. Thirdly, a merge of more public and private businesses would help push the use of block chain further. Lastly, the government should make the block chain technology easily accessible, making the official permission to implement -easy and cost effective. Keywords-Blockchain, Cybersecurity, privacy, Internet of things, cyber attacks 1. INTRODUCTION Without a shadow of doubt, eye brow raising tragedies related to poor Cybersecurity has influenced businesses and customers, making them more aware of the high possibility of security breach when personal data or company data is uploaded. With the mentality most customers have now, they hesitate before they put out their information. However, most applications and cloud based networks require this personal information in order to gain users access. Therefore, it becomes a helpless situation for internet users, who have to just comply and hope their information is not being misused. Reason being that Cybersecurity is an alarming issue which constantly poses a threat to the cyber-world. Ironically, this issue happens to be one of the promoting factors for block chain security. It is deemed as a possible rescue against cyber-attacks [1-2]. The block chain technology is a not a new introduction to the cyber world, it was first conceptualized in 1991 but it is just recently gained its popularity amongst people worldwide, especially after the lunch of the cyber currency-bitcoin on its platform. The block chain technology is simply a melting pot whereby transactions are put in form of a puzzle. Users need to agree on a transaction; by making sure that each cryptographic harsh aligns. A cryptographic hash is more or less like a confirmation key. Those confirming this transaction cannot see the details, such as how much wealth the sender or receiver has but they see enough cryptocurrency proof to confirm the validation of the transaction. The application of the block chain technology is endless. It is a secure route in this packed cyber threaten era of the internet. The entire system works with a distributed ledger technology that operates on a decentralized pattern which makes it safe and cost effective because companies don‟t take the risk alone. Meaning that, other partners are involved in ensuring security through the multiple signature protection. The behind the scenes logistics seems complicated but executing transactions on this platform is not. The only tasking thing is left to the hackers trying to breach the system. A January 2017 World Economic Forum report predicts that most likely, "by 2025 about Ten percent of global GDP will be stored on block chains or block-related technology." [1-2]. Many Organizations are beginning to gradually implement this system to safeguard data. For example, Guard-time is a software security company that developed a digital signature system based on block chain technology like the company safe guards‟ data by spreading it all to nodes throughout the system. Therefore, if any hack attempt is made to manipulate data, the whole mass of chains then makes a compassion to metadata packet and then excludes any that do not match up. This makes it almost impossible to hack. Guard-Time has been doing great with this technique as they have worked with reputable companies such as: Sony Ericsson on a cloud computing project [3]. International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 2 Issue 12, December – 2018, Pages: 59-66 www.ijeais.org 60 However, many organizations still believe their data can only be secured when the design or implementation of that system is kept a secret- "security through obscurity." Unfortunately, this is not always the case, as we know the tech world is fast developing daily, so is the knowledge of its usersincluding hackers. Cybersecurity report shows 320% increase hacking attacks in 2016, which is a clear illustration of how dangerous the cyber world is. The bad news for companies relying on security through obscurity; is that once access to the technique of the system is gained by a hacker the whole system is at risk with low chances of rejuvenation [4]. Using Microsoft as an example, when its system was hacked through the use of QAZ Trojan horse which is done when the hacker sends a random email attached with an unseen program. This program enables the hacker control the system of the receiver when the mail is opened, which gives them access to surf through for log in passwords and information. Microsoft security discovered this laps in their security from suspected emails containing several secret passwords used to transfer source codes (languages used to build word and office) from US company‟s computer to Russia. However, if the implementation of a block chain system was done, they would never have had such problems. A typical example of its excellence is the bitcoin. Although, they have been attempts to hack the system no proven record shows success, no exposure of users‟ anonymity to hackers of any such [5]. Privacy protection is a pressing issue for a majority of people. As mentioned previously in this research, most people do not want their personal space to be infiltrated and they feel sharing too much personal information would lead to that. Also, most organizations use customer‟s personal information for secondary purposes which aids to the discomfort of customers to share information on them. For example, a company that‟s about to close down, in order to present its self as being expensive for sale it offers its customers information as part of the package that comes with buying the company. (Nigerian ODB Block industriesa construction company. sold 1200 customer information to CEKA insurance LTD when turning over the company. Over 6 customers complained about a new insurance companyceka, calling their direct lines and showing up at their door steps proposing insurance packages) [6]. Block-chain security is a type of technology that only gives company‟s access to information the user chooses to share. If customers of ODB companies used this they wouldn‟t be any infiltration of space. For example, Secure-Key already has experience with this system. The company uses triple blind authentication for its network for over 7 years now. Block chains role for secure-key is to maintain that security but for the sake of identity only whereby no data is being put in the block chain, only evidence of the data. "There‟s no personally identifiable information in the block-chain at all. The block-chain is being used for evidence and integrity, not for PII." [7]. Before we dive into other aspects of this research it is important you note that although the popularity of the block chain system is fast growing, there is still no concrete evidence to prove that it would be efficient than the current working system. Reason being that, everything has its pros and cons and most of the block chain cons cannot be pinpointed down at the moment until its fully implemented like the current system – new problems arise every day in the IoT world. Therefore, block-chain eradicating cybersecurity is just an optimistic thought judging by its current works. The relationship between block-chain and cloud computing is one that cannot be easily ignored, the two-work hand in hand with each other. "Cloud computing is simply an information technology paradigm that enables ubiquitous access to shared pools of configurable system resources and higher-level services that can rapidly be provisioned with minimal management effort"- (Wikipedia). Both block chain and cloud were both designed to maximize security, but one just does it better. 2. THE SET UP OF CLOUD AND BLOCKCHAIN AND ITS USES The setup of the both is different. The cloud adoption is one thing another is to choose which type of cloud to use. It is categorized into public and private clouds. The main difference between these two is that the user is not responsible for any form of management of a public cloud hosting solution. Because the data is stored in the provider‟s data center and the provider is the one in charge of management of the data center. Organizations use these cloud types depending on the different level of security and management required for that organization, some even use a hybrid cloud. A public cloud is simply the internet (e.g Google, amazon), a private cloud is dedicated to one organization with strict security controls, it is commonly used amongst medical offices, banking institutions and organizations who are required to meet federal and state guidelines for data controls use, meanwhile, hybrid is a combination of both public and private. (e.g apple "We use private infrastructure for compute and energy, but we also use public options from AWS, Microsoft Azure and google."Weinman [2]. Meanwhile, Blockchain can be categorized into the permission and permission-less chain. As the name suggests the permissionless chain requires no acceptance to join, anyone can join e.g. bitcoin. However, the permission chain requires the user to authorize your access, which logically is more secure. International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 2 Issue 12, December – 2018, Pages: 59-66 www.ijeais.org 61 3. SECURITY FOR CLOUD AND BLOCKCHAIN NETWORKS It shares some similar properties. Blockchain just like cloud, the security of the block model is strong. Cloud constantly monitors suspicious activities in real time firewall. Which most organizations love to deploy but is not the best option in terms of security because hackers have been successful in the past penetrating these firewalls. On the other hand, block-chain uses cryptographic hash functions which make it almost impossible to hack the entire system. Because once one of the systems is hacked, the hacker still needs to hack all systems connected to that network in order to get full access (and each system needs a key to align). The public and private key cryptography makes sure data is not breached and security in this aspect is quite high [2]. 4. CHALLANGES Cloud networks are not the most trust worthy regardless of the model used, public or private. Its cost of maintenance on the other hand is much, and a common alarming factor of this network is its security and confidentiality. August 2013, demonstrated that no company is safe from hack when the tech giants like Oracle, Sony, T-Mobile and Dropbox dealt with massive hacks and breaches of customer data, and these companies make use of cloud networks [3]. However, block chain has no record of a successful hack. Although the slow deployment of this network is because not too many mechanisms have been invented to support it. As for cloud networks, it is important to know cyber-attacks are a pressing issue in this tech world and it becomes riskier knowing that, "organizations in most networks run the same code." [3-4]. What the above statement implies is that if hackers get a loop hole that gives them little access to a cloud based network the entire system faces intense risk, therefore increasing the organizations cyber-vulnerability. In summary, cloud adoption is a strategic move of reducing cost, mitigating risk and achieving scalability of data base capabilities – HCL technologies. This method has some downsides attached to it as mentioned previously therefore it is advisable that organizations should be more conscious of the content they store rather than the medium used in storing this content. Block chain serves as a better option. 5. SECURITY AND PRIVACY IN BLOCKCHAIN HEALTH CARE INDUSTRY AND MARKET PERSPECTIVE Without a shadow of doubt the healthcare industry is one that is as important as any top leading industry in a country. Many people flood into hospitals daily. Over 141.4 million people visit the hospital in the space of 5months in the U.S and 7.9% result in hospital admission. This outrageous number would have you thinking how the health care industry keeps its patient‟s data safe and up-to-date. In the current system, security and trust are the most valuable assets for business; talk less about the health care industry – a more delicate industry dealing with life and death. Information needs to be shared and documented during the line of hospital communication and this leads to trust issues. Many hospitals hold different records of one patient that are not validated, which leads to several errors and incompetency in the health care industry. Between 2005-2001, 140 million patient records were breached according to Protenus Breach Barometer report and it was reported in 2014 that more than 750,000 consumer devices were compromised to distribute phishing and spam emails [5-6]. Having said this, we look into the possible contributions block chain could make in the health care industry in terms of protecting patients‟ identity and information with its decentralized and encrypted way of sharing, distributing and sharing information which gives maximum security and protects identity. Speaking of identity, we need to look into the ways in which block chain handles identity exchange compared to the traditional way of identity document exchange in the world [7-9]: • The typical procedure of this is identifying the person or asset, • an official government agency approves or notarizes the document • and lastly an investigation is held on the individual to confirm legitimacy of the money. However, when it comes to block chain the process is not that tasking. The transfer just involves just two distinct ledgers with a key, one encrypted and the other not. The encrypted key is where you get access to information and it has to be approved for your view by the individual. In block chain this ledger contains all information needed about the individual – it is known as key rings. "But it‟s also a problem that looks tailor made for a block chain to solve"John Halamka chief information officer at Beth Israel Deaconess Medical Centre in Boston. „The healthcare industry is packed with a lot of cyber related issues. These issues range from malware that compromises the integrity of systems and privacy of patients to distributed denial service (DDos) attacks that disrupt facilities ability to provide health care"Centre of internet security [10-13]. Critical Healthcare information is often scattered across multiple facilities as mentioned previously. Some of this information need to be retrieved constantly. This process cost money and sometimes even lives. Lack of protection for patient files has been misused. Therefore, better security is needed that‟s where block chain comes in with its interoperability, integrity, security, and portable user owned data [13-15]. International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 2 Issue 12, December – 2018, Pages: 59-66 www.ijeais.org 62 6. ADVANTAGES OF BLOCKCHAIN IN HEALTH CARE AND WHO USES THE SYSTEM CURRENTLY •Medical records would be accessed securely by authorized persons with permission from providersaves time, money, duplicated records and most especially lives. • Those patients who want to give medical records for research purposes can now freely participate without the fear of being known. • Could help reduce counterfeit drug implications that currently cost pharmaceutical companies and estimated 200 billion dollars yearly. For example, "Connecting Care is our current revenue-generating, block chain-backed platform," SimplyVital Health CTO Lucas Hendren said in an interview with the author. "It uses care coordination and financial forecasting to help providers in bundled payments get insight into what happens to patients when they leave the hospital. It is a strategic early use case for block chain in healthcare because it uses block chain as an immutable audit trail." [16-19] . 200 healthcare executives were surveyed and 16 percent say they would adopt the commercial block by the year end. The platforms for this tech are gradually getting available, For example: Block chain Health (San Francisco). Block chain Health is a software company that provides healthcare organizations with HIPAA-compliant block chain solutions. In conclusion, the advantages to block chain are evident but however, it is a question of if hospitals are ready to adapt to this technology and go against the traditional methods. Are they willing to try workers on how this works?[20-22]. 7. BLOCKCHAIN AND IOT SECURITY Block chain and IoT are two top networks that perfectly complement each other. However, they both have challenges. IoT networks have several loop holes and one major one is its vulnerability to cyber-attacks. IoT hacks are one of the major cyberattacks that keep companies on their toes. For example, back in October 2016, one of the largest DDoS attack ever was lunched on service provider Dyn using an IoT botnet. This was made possible by a malware called Mirai- (surfs the internet for weak IoT devices and try default usernames and password to login.) Hackers sense vulnerability in an IoT device and they use the front end, back end or middle (Man-in the middle attack) to execute their task. In 2011, tech giants Sony were hit by a security breach by a lone hacker. The hacker infiltrated the network and gained access to 77 million customers info such as usernames, security questions and passwords. Sony‟s chief information officer (CIO) shinji Hasejima, went to the media to explain the situation. Hasejima believed that the weakest link in Sony‟s network was the application server and the hacker took advantage of the vulnerability of the company [23-24]. "An application server is a software framework that resides in the middle-tier of a server-centric architecture and provides an environment where an application can run." Having shown how vulnerable the IoT system is, it is reasonable for companies to seek a new means of protecting data. Which leads us to putting it side by side with the block chain network which could fill up some of its loop holes? However, block chain itself has its own challenges to deal with. This system is not all too new in the industry but its adaptation is a slow process, therefore, they are not too many supporting applications for it. Besides that, block chain is the most appropriate network used in tackling privacy and security problems associated with IoT. In the subsequent explanation, we touch on the mechanisms and key processes that would help achieve a stronger IoT security with block chain. 8. INTEGRATED BLOCKCHAIN IN IOT SECURITY According to the company's website, "the modum sensors record environmental conditions during shipments. When shipped goods change ownership, the collected data is checked against a specific smart contract in the block-chain. The contract validates that the transaction meets all of the standards set out by the customer, their clients or the regulator and triggers various actions: notifications to sender and receiver, payment, or release of goods, etc." The above is an example of a company who has deployed the use of block chain in its IoT network in order to transfer of IoT data without central control and management. This makes the entire procedure more secure. Not with holding the fact that to run a block chain is cheaper if integrated into the IoT, it saves the company from cyber-attacks, also time to retrieve files and maintain them. "Block-chain and IoT together create a 'sweet spot' that form an Internet of Value which allows secure value flow across a range of industry segments," said Cisco Senior Vice President Hilton Romanski. " [25]. The integrated block chain in IoT security makes a stronger network even tech giants like IBM also uses large cloud infrastructure to provide block chain services used to track high value items along the supply chain line. In addition, block chain has International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 2 Issue 12, December – 2018, Pages: 59-66 www.ijeais.org 63 an option of smart contracts for an IoT system. Smart contracts help you exchange money, property, shares or valuable things without the middle man, which means cheaper cost. For example the company Gartner has estimated that by 2022, so-called ratified unbundled (i.e. defined impact) smart contracts will be in use by more than 25% of global organizations. Companies like IBM and Microsoft word are already ahead in this aspect [26]. In summary, IoT and block chain work better hand in hand with each other. The launch of public block chains like Ethereal and Hyper ledger has enabled more block chain adopters as developers can freely build their own applications on top of them. 9. CENTRALIZED CLOUD MODEL VS DECENTRALISED BLOKCHAIN MODEL A centralized cloud is one which has a history of security flaws and breaches. As we touched on earlier in this research a decentralized block chain always serves as a better option to avoid these security breaches; however, we must understand how bad the situation with a centralized cloud model is for IoT. • In 2018, approximately 3.6 billion internet users are projected to access cloud computing services, up from 2.4 billion users in 2013. This serves as concrete evidence that the internet users increase daily. Therefore, companies running on a centralized model transferring confidential information and keeping up with customer demand face great risk of cyber-attacks and increase working cost. Reason being that, the centralized system just needs one access and the whole system can be infuriated. The block chain model would be a better solution in achieving IoT effectiveness. "Traditional, centralized databases are like castles with moats," said Chronicled CEO Ryan Orr. "You can fortify them as much as you want, but a hacker will always find a clever way to sneak inside the castle. Block-chain introduces a whole new paradigm. It's a distributed network, data is cryptographically secured, a breach in one node has no effect on the whole, and the consensus mechanism prevents malicious actors from tampering the system. That's one of the things that's really revolutionary about this technology [26]. " • IoT operate through cloud servers and since users of these network increases daily, some concerns come to mind on how to keep up with the growth. Like any constant growing program or platform – the more users involve, the more capacity needs to be created for them. The IoT is a concept built on networking nodes all linked together via a global network used for (media surfing, streaming, data transfer and so on). As the device grows in reputation amongst people and is being used more than often it becomes harder and expensive to manage communication especially with current centralized model. "There will be 8.4 billion connected things in 2017, setting the stage for 20.4 billion Internet of Things (IoT) devices to be deployed by 2020, according to analyst firm Gartner. The installed base of hard-to-secure smart things, such as TVs, fridges, and security cameras, is expected to grow 31 percent this year to reach 8.4 billion devices, or around a billion more than the world's total population." Zdnet [27]. • The centralized cloud model of IoT can be manipulated, which possess a threat to accuracy of information. Whereby, data can be accessed easily. For example, in Nigeria EFCC are responsible for tracking political member‟s wealth and tracing its legitimacy, however, the network in which these confidential findings and information are documented are most times manipulated in favor of the politician. This is because third-party-go-betweens are possible. If the system ran on a block chain smart contract where by any found records of looted money automatically triggers an action of the police force. In conclusion, to get the best security and authorize all information passed through and fro the IoT, block chain technology is the one for the job. 10. BLOCK CHAINS ROLE IN ENSURING SECURITY OF SUPPLY CHAIN The supply chain is becoming increasingly dependent on secure digital technology. Without a shadow of doubt, supply chain has started tilting towards operating on a more technological level such as the Internet of things (IoT). This is a game changer on how goods are produced and distributed. For example, global retailer Walmart uses block chain to track scales of pork meat in china. Its processing and storage, and sellby date. In the event of product recall, the company can also see which batches are concerned and who bought them. Also, with the use of smart contracts deliveries can be made and when not made the system can take action to give penalties or alert to sender. From conducting payment and audits, to tracing inventory and assets, block chain technology would definitely enable greater supply chain efficiency than ever. With the help of block chain each time a product changes hands it can be documented creating history of the seller to manufacturer of the product. This would save time and cost. Below are the advantages of block chain to supply chain security. International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 2 Issue 12, December – 2018, Pages: 59-66 www.ijeais.org 64 Regardless of the application, block chain offers shippers the following advantages [28]: • Enhanced Transparency. Documenting a product‟s journey across the supply chain reveals its true origin and touch points, which increases trust and removes all forms of bias behaviors. • Better security: due to the use of decentralized and key codes of the block chain better security is being made possible. From a company‟s stand point, no mix up in data or cyber theft of customer payments etc. "Many larger producers do not want to reveal provenance of their goods for fear of losing a competitive advantage. Blockchain allows information to be transferred in a trustworthy and anonymous way, essentially providing a trust network that allows information to cascade down the chain from raw material onwards, without revealing who people are." – web magazine innovation enterprise. • Transparency: take for example the food industry, Blockchain supply chain management programs aim to change the way food is tracked by creating an auditable chain of custody from raw materials to the consumer. Whereby, each step of this requires data to be entered and updated and real time tracing of delivery. 11. BLOCK-CHAIN AND THE FAIR INFORMATION PRACTICES (FIPS) "Fair Information Practices are a set of principles and practices that describe how an information-based society may approach information handling, storage, management, and flows with a view toward maintaining fairness, privacy, and security in a rapidly evolving global technology environment.". In the global village in which we live in, information is being uploaded on the internet daily involving confidential messages or public messages. Big data is a pool of these information. Many concerns have arisen over the years regarding the handling of data in big data environment. Most organizations are found guilty to contributing to the misuse of these data. Most organizations take customers information and use them for secondary unauthorized purposes. It almost feels like customers have shown their "nakedness" giving a lot of information up on the internet. Unlike back in those days whereby computers had very little ability to recognize an individual besides their log in used. Meanwhile, now the advancement of technology in IoT enables voice recognition, websites know your biometrics through a wearable device, conversations, best visited sites, etc. Most times this data is stored to help you work faster and feel a sense of ease [29-30]. For example, if an individual visits Facebook daily on his or her system, the system automatically presents an option to save log in details if you agreed on, it then suggests on ways to give the person notifications and keep them involved (reminding about your own birthday even) Information is power that‟s why organizations want it. But when it stars becoming an issue is when its misused. Especially when it goes against the fair information conducts. For example, Acxiom is one of the largest data-brokering firms in the world, is known for collecting data on costumers and selling them to companies. It‟s an ongoing argument on if its sales of customer information are legal and authorized by the customers [32]. Meanwhile, in China 361 criminal cases came up involving violation of personal data, up from 176 in 2015, said Xie Yongjiang, associate director for the Institute of Internet Governance and Law at the Beijing University of Posts and Telecommunications. Apple on the other hand, were caught up in this scandal in china whereby The Chinese police said Apple contractors had been arrested, 22 of them suspected of selling the personal data of an unspecified number of Apple customers. The police, in Cangnan County in the eastern province of Zhejiang, said the thieves had reaped 50 million renminbi, or about $7.3 million, over an unspecified period [33]. Most customers have their information documented and they have no idea. Block chain technology promises to help solves these issues. With block chain technology data is controlled with private and public keys there is no custodian of user data. The user chooses to whom to release the information to. Also, identification does not need to be an issue; animosity is at its highest [34]. Taking for example the bitcoin, coins containing info, money etc., are transferred from one individual to another without their identity exposed. Same thing would apply to this identification whereby users can log in a system and the owners of the site would have no idea about their identity. Also an audit trail is another block chain feature that would help accountability. Therefore, block chain would essentially help promote fair information practices [35]. International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 2 Issue 12, December – 2018, Pages: 59-66 www.ijeais.org 65 12. CONCLUSION Block chain has proven to be a-go-to tech for better security. The technology would be a challenge for cybercriminals where individuals have control of their own data. Some cloud challenges can be solved through the use of block chain to ensure privacy and security. Its cryptographic verification feature would help stop Mitm. It would help supply chain in terms of cost, speed, safety and transparency. Its function is endless and it can fit in all industries. The future with this technology might solve a lot of problems and also bring about other problems unknown yet. REFERENCES [1] Sharma, P.K., Moon, S.Y., Park, J.H., 2017. Block-VN: a distributed blockchain based vehicular network architecture in Smart City. Journal of Information Processing Systems 13 (1), 184–195. [2] Lee, B., Lee, J.H., 2017. Blockchain-based secure firmware update for embedded devices in an internet of things environment. J. Supercomput. 73 (3), 1152–1167. [3] Huckle, S., Bhattacharya, R., White, M., et al., 2016. Internet of things, blockchain and shared economy applications. Procedia Computer Science 98, 461–466. [4] Sari, A.; Rahnama, B., (2013) "Simulation of 802.11 Physical Layer Attacks in MANET," Computational Intelligence, Communication Systems and Networks (CICSyN), 2013 Fifth International Conference on , vol., no., pp.334,337, 5-7 June 2013, http://dx.doi.org/10.1109/CICSYN.2013.79 . [5] Sari, A., Rahnama, B (2013). "Addressing security challenges in WiMAX environment". In Proceedings of the 6th International Conference on Security of Information and Networks (SIN '13). ACM, New York, NY, USA, 454-456. DOI=10.1145/2523514.2523586 http://doi.acm.org/10.1145/2523514.2523586 [6] Sari, A., Kilic, S., (2017); Exploiting Cryptocurrency Miners with OSINT Techniques, Transactions on Networks and Communications. Volume 5 No. 6, December (2017); pp: 62-76. http://dx.doi.org/10.14738/tnc.56.4083 [7] Sari, A.,, Qayyum, Z.A, Onursal, O. (2017) The Dark Side of the China: The Government, Society and the Great Cannon, Transactions on Networks and Communications. Volume 5 No. 6, December (2017); pp: 48-61. http://dx.doi.org/10.14738/tnc.56.4062 [8] Sari, A. (2017); The Blockchain: Overview of "Past" and "Future", Transactions on Networks and Communications. 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(2016) Quality Assurance Issues in Higher Education Sectors of Developing Countries; Case of Northern Cyprus, Procedia Social and Behavioral Sciences, Volume 229, 19 August 2016, Pages 326-334, ISSN 1877-0428, http://dx.doi.org/10.1016/j.sbspro.2016.07.143. [13] Sopuru, J., Sari, A., (2016) When Technologies Manipulate our Emotions – Smell Detection in Smart Devices. International Journal of Scientific & Engineering Research, Vol.7, No.4, pp. 988-991, ISSN 2229-5518. [14] Kirencigil, B.Z., Yilmaz, O., Sari, A., (2016) Unified 3-tier Security Mechanism to Enhance Data Security in Mobile Wireless Networks. International Journal of Scientific & Engineering Research, Vol.7, No.4, pp. 1001-1011, ISSN 2229-5518. 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(2014); "Security Approaches in IEEE 802.11 MANET – Performance Evaluation of USM and RAS", International Journal of Communications, Network, and System Sciences, Vol.7, No.9, pp. 365-372, ISSN: 1913-3723; ISSN-P: 1913-3715, DOI: http://dx.doi.org/10.4236/ijcns.2014.79038. [29] Megdadi, K., Akkaya, M., & Sari, A. (2018). Internet of Things and Smart City Initiatives in Middle Eastern Countries. In P. Raj, & A. Raman (Eds.), Handbook of Research on Cloud and Fog Computing Infrastructures for Data Science (pp. 289-311). Hershey, PA: IGI Global. ISBN: ISBN13: 9781522559726, doi: https://doi.org/10.4018/978-1-5225-5972-6.ch014. [30] Rahnama, B., Sari, A., & Ghafour, M. Y. (2016). Countering RSA Vulnerabilities and Its Replacement by ECC: Elliptic Curve Cryptographic Scheme for Key Generation. In D. G., M. Singh, & M. Jayanthi (Eds.) Network Security Attacks and Countermeasures (pp. 270-312). Hershey, PA: Information Science Reference. Doi: https://doi.org/10.4018/978-1-4666-87615.ch012 [31] Sari, A., (2015), "Security Issues in Mobile Wireless Ad Hoc Networks: A Comparative Survey of Methods and Techniques to Provide Security in Wireless Ad Hoc Networks", New Threats and Countermeasures in Digital Crime and Cyber Terrorism, (pp. 66-94). Hershey, PA: IGI Global. doi: https://doi.org/10.4018/978-1-4666-8345-7. ISBN: 978146668345, April 2015. [32] Sari, A., Akkaya, M., Abdalla B., (2017) "Assessing e-Government systems success in Jordan (e-JC): A validation of TAM and IS Success model". International Journal of Computer Science and Information Security, Vol.15, No.2, pp.277-304, ISSN:1947-5500. [33] Sari, A. (2016); "E-Government Attempts in Small Island Developing States: The Rate of Corruption with Virtualization", Science and Engineering Ethcis, Springer , Vol 23., No. 6, pp.1673-1688, ISSN-O: 1353-3452,DOI: http://dx.doi.org/10.1007/s11948-016-9848-0 [34] Sari, A. (2018) "Context-Aware Intelligent Systems of Fog Computing For Cyber-Threat Intelligence" Springer International Publishing, Springer Book on "Fog Computing: Concepts, Frameworks and Technologies", In: Mahmood Z. (eds). Online ISBN: 978-3-319-94890-4, Print ISBN: 978-3-319-94889-8., doi: https://doi.org/10.1007/978-3-319-94890-4_10. [35] Sari, A., Alzubi, A., (2017) Path Loss Algorithms for Data Resilience in Wireless Body Area Networks for Healthcare Framework, In Intelligent Data-Centric Systems, edited by Massimo Ficco and Francesco Palmieri, Academic Press, 2018, Pages 285-313, Security and Resilience in Intelligent Data-Centric Systems and Communication Networks, ISBN 9780128113738, doi: https://doi.org/10.1016/B978-0-12-811373-8.00013-6 .

MAY 2009 DRAFT-NOT TO BE QUOTED WITHOUT PERMISSION SAVAGES, WILD MEN, MONSTROUS RACES: THE SOCIAL CONSTRUCTION OF RACE IN THE EARLY MODERN ERA forthcoming (2009) in BEAUTY REVISITED, Peggy Zeglin Brand, ed. (Indiana University Press) by Gregory Velazco y Trianosky, Professor of Philosophy, California State University, Northridge Abstract: This paper is a case study in the social construction of race. It focuses in particular on the development in the Renaissance and early modern era of the new images of "the savage" as they appeared during the first forty or fifty years of the European conquest and colonization of the Americas, and their racialization during this same period. I propose a genealogy of these racialized images that traces them to transformations in traditional medieval and Renaissance images of the Wild Man and the Monstrous Races that occur both before and during the early period of conquest and colonization, in response to very specific cultural, social and political pressures, including in particular changing conceptions of space that are created before and during the colonial enterprise. The modern conception of race is often thought by philosophers to have developed during the 18 th and 19 th centuries in response to a unique confluence of scientific, philosophical, and imperial forces; and in recent decades some impressive work has been done to excavate the details of its construction during this period. 1 Of course philosophers generally acknowledge, if only in passing, that this late modern construction has its roots in various medieval traditions, among which are typically mentioned the medieval versions of the Noachic legend, according to which the peoples of the earth can be divided into three broad groups (African, Asian, and European), each descended from one of the sons of Noah. 2 2 I will argue, however, that an analysis of the visual images created by Europeans during the first half-century after 1492 reveals that the essential elements of the late modern conception of race are put into place during that period. In brief, the tremendous social, economic and political pressures that culminate in this comparatively brief moment yield the modern notion of the savage. I will suggest that from its inception this notion is an inherently racialized one; and that it is the nodal point from which, in broad outline and in much of its detail, the template is drawn for the more familiar 18 th and 19 th century understandings of non-European races. Moreover, I will show that the modern notion of the savage is synthesized, not directly from the Noachic legends, but from images drawn, sometimes literally, on the margins of medieval understandings of humanity: powerful and deeply-entrenched images of the wild man and the monstrous races. This synthesis is made possible by the ways in which the discovery and colonization of the New World simultaneously expand the boundaries of the world defined as European to include far-way places known previously only through rumors, legends, and the testimonies of medieval authorities; and at the same time realign these newly-familiar and redefined boundaries with the old margins of medieval mythology. It is this transfer of old, mythologized concepts to newly-discovered, living peoples, virtually completed in the very moment of discovery, that is the fountainhead of late modern conceptions of race. My investigation is preliminary, and much work remains to be done; but I hope it will be enough to convince aestheticians and historians of philosophy to venture beyond the well-worn paths of the late modern era to uncover the origins of modern ideas and images of race. It is important to place the artwork discussed here in its art-historical context. The high art of the sixteenth century, often referred to as "mannerism," has been characterized by one art historian as showing "an insistently cultured grace and accomplishment..accompanied by the 3 kindred qualities of abstraction from natural behavior and appearances, bizarre fantasy, complexity and invention." 3 The most well-known images of the early colonization and conquest of the Americas, produced by the De Bry family at the end of the 16 th century, fit this characterization well. Even a cursory review of a representative selection of de Bry's engravings, as they are published in Michael Alexander's Discovering The New World: Based on The Works of Theodor de Bry, for example, will confirm that author's claim that de Bry is "a mannerist in the heyday of mannerism." 4 Our attention here, however, will be on the earlier images of America, produced under rather different circumstances. The history of the production of these earlier, pre-De Bry images points to artists and artisans whose work is, if not necessarily more immediate, then at least more free of the dominant mannerist conventions. Most of the images of America and American Indians discussed here are woodcuts, produced by artisans trained as cabinet-makers, rather than engravings, produced by artists or goldsmiths trained or supervised by artists. 5 With some notable exceptions, the work of these artisans was not regarded as highly among patrons of the arts, nor expected to conform closely to the high-art standards of the day. In many early cases it was intended only to increase the attractiveness of a book or pamphlet to a mass audience. These early pictures are often drawn hastily, in many cases by artisans whose only guide was the text, working under the commercial pressures of producing mass-market publications like the various editions of Columbus's 1493 letter. One need not embrace a naïve realism to point out that it is characteristic of these works that they pay much less attention to fidelity to the text being illustrated (whatever one makes of the issue of its factual legitimacy), than they do to the exigencies of the printing process and, perhaps above all, to the marketability of what is produced. 4 The point of focusing on this earlier, artisanal work is not that, in so doing, we can catch a glimpse of what American Indian people were "really like;" but rather, one might say, that in so doing, we can catch a glimpse of what European people were "really like." We glimpse, that is, famliar and less polished, workaday European preconceptions about strange peoples, not so heavily influenced, perhaps, by the dominant high-art conventions of the time. To be sure, many of the earliest drawings already display the same political and economic motivations that are often drawn to our attention in commentaries on the later, more polished work of de Bry and others. 6 In particular, the explicit interest in colonization and subjugation is already present from the very beginning. This is clear, for example, in an image representing the reactions of American Indians upon first encountering Columbus and his ships that was repeated frequently with some variation or other in various editions of Columbus's letter, which letter was itself widely and quickly reprinted in a number of European languages: 5 Here we see the islands described in the letter presented literally from the point of view of Europe, with Ferdinand himself in the foreground, seated upon his throne, holding his badge of authority and reaching out in an imperial gesture directly aligned with the gaze and movement of Columbus, who can be seen almost in the center of the frame disembarking from the largest of his three ships. The accompanying text remarks upon the richness of the land, the fearfulness of the natives, their rustic simplicity and childlike credulity, their willingness to give away gold, silver, and items that the Spaniards regard as valuable or have need of in return for trifles, and their potential as loyal subjects of the Spanish monarch and converts to Catholicism. The contrast between clothed Europeans and unclothed, long-haired and bearded natives, together with the related contrast between their naturally timid response and the authoritative and 6 unhesitating stance of Ferdinand and his representative, Columbus, coalesce in the his repeated comments in the letter and in his log about the Indian belief that the Spaniards had descended from heaven, underlining the Spanish commercial and political interests in a fantasy of blatantly self-serving religious terms. 7 Thus it would not be accurate to describe these early drawings as more "innocent", or less interested. What is true, however, is that, in addition to these sorts of often crudely explicit interests that echo the accompanying texts, many of the early drawings also reveal European attempts to assimilate the new and unfamiliar to old and familiar European ideas, expressed in tropes, symbols and metaphors that were pervasive throughout the Middle Ages and early Renaissance. This attempt, understood as itself an interested one, is all the more powerful because the viewer, as a member of a broadly-shared European culture, is encouraged to respond to what is represented as though it were just an instance of the old myths, familiar but now rendered incarnate. Consider for example, the long, unkempt hair of the Indians in the 1493 woodcut just presented-a feature that reappears in several of the variant woodcuts representing this same scene. Olive Dickason remarks on the persistent European tendency to describe, and to portray, American Indians as hairy despite the near-unanimity of eyewitness reports to the contrary. 8 While some have claimed, I think rightly, to find more subtle and attenuated references to hairiness in many of the later engravings (in de Bry, for example), in the earlier representations that concern us here the hairiness of the Indians is a commonplace. 9 It is perhaps as a result of the circumstances of production that these early woodcuts present American Indian figures explicitly as elaborations on familiar and easily-recreated icons such as the Wild Man. In the current case, as many authors have pointed out, depicting American Indians 7 as hairy and unkempt is a way of representing wildness that draws upon a centuries-old convention about those Others who live in the woods and mountains of Europe, where civilized people venture only when they have lost, or are in danger of losing, what makes them civilized. The same is often done by calling on the tradition of the monstrous races. For example, the use of the cynocephali, the monstrous race of dog-headed people (described by Pliny and dozens of medieval texts that follow him) to depict the alleged cannibalism of American Indians in this illustration from a 1530 edition of Vespucci's Carta Marítima, published in Strasbourg, instantly made it a familiar instance of the medieval trope of the horrific practices of far-away and alien peoples. 10 A great deal is communicated in these two pictures by a few simple strokes of the artisan's pencil, with no annotation or explication required, and no artistic sophistication or classical knowledge on the part of the observer presupposed. The stage is thus quickly set for the ordinary 8 Europeans (working-class and small-business owners) who make up many of the early colonizers to see American Indians as characteristically lacking in one or more of the qualities necessary for interactions governed by the jus gentium, or perhaps even the lex naturalis itself-in short, as "savages", as they quickly came to be called in a variety of European languages. These two pictures thus serve as an introduction en breve to the transfer of mythic concepts to living peoples that is encapsulated in the idea of the savage, and that lays the groundwork for the racialized treatment of American Indians as essentially and irredeemably Other. It is a more detailed examination of this transfer that will occupy us here. 1. Old images We may begin to understand these details by examining briefly two strands deeply woven into the European mythology of the Middle Ages: the stories of the monstrous races and the images of the wild man. 1. a. The monstrous races 1.a.i Character of the monstrous races The monstrous races first make their appearance in ancient Greek writings; but the most influential source of information about them during the Middle Ages is unquestionably the Historia Naturalis of Pliny the Elder (23-79 C.E.) 11 Identical descriptions or obvious variations on Pliny's appear repeatedly and with nearly-identical illustrations, from the important and widely-read work of Isidorus of Seville (560-636 C.E.) to the margins of the Hereford Mappa mundi (1290), to the Travels of Sir John Mandeville (circa 1357) , which is sometimes claimed to have been an important reference work for Columbus. 12 They continue to appear with frequency in translations of Pliny and other works contemporary with Columbus' voyages, as for example in this frontispiece from the German work, Das Buch Der Croniken Und Geschichten, published in 1500: 9 10 There are many monstrous races; but even a modest effort at cataloguing the variations show the consistency of their descriptions across a great range of medieval sources. 13 To understand how the tradition of the monstrous races provided material for the emergent notion of the savage, I propose that we divide the monstrous races into two groups: those whose phenotype is identical to that of known humans, but whose behavior or culture is represented as strange and alien to European audiences; and those whose phenotype itself is so strange and bizarre as to be beyond the range of what is regarded as familiarly human, typically signaling bizarre or outré behavior as well. In the former category one group is of particular interest for the European response to the New World, namely, the anthropophagi. The anthropophagi are cannibals who are variously described as eating their enemies, their friends, their family members, and certainly any available strangers, as in this image of a phenotypically-ordinary anthropophagus from the Sion College Bestiary of 1277, produced in France: 11 Depictions of cannibalism are of course standard fare in representations of New World peoples; but the earliest pictures published after 1492 are quite striking in their fairly direct reliance on the earlier traditions. The earliest extended account of cannibalism in the New World, and the earliest of the captivity narratives that later formed a staple of Puritan literature, is the story offered by Hans von Staden, who claimed to have been kidnapped by the Tupinambá of Brazil, and held captive by them for six months. A series of woodcuts depicting his experiences were executed in 1557, apparently under his direct supervision. 14 The following is representative: 12 Though published a little later than the period under consideration here, this woodcut and its companions repeatedly re-enact the familiar device found in Mandeville's 14 th -century narrative (widely reprinted and available during the period under discussion), though not in its accompanying illustrations, of the single, shocked European observer who encounters customs and cultures far removed from his own. In the von Staden illustrations, as often in Mandeville's tales, this device is used to frame the depiction of cannibalism. In many early depictions New World cannibals are represented as a group, to convey that their behavior is characteristic of a certain culture; and their behavior is interwoven with scenes of family and friendship, emphasizing the "normality" of these horrific activities for this monstrous race, as is suggested 13 by the presence of the child carrying a severed head in the lower right-hand corner of the Von Staden engraving, and also in this 1505 German woodcut by Johann Froschauer, accompanying a German edition of Vespucci's Carta: Note that even when the Indians are portrayed as doing horrific things they are nonetheless represented at the same time as "normal" or even beautiful in appearance, as one would expect from a monstrous race whose distinguishing feature is behavior and culture, not phenotype. 15 Thus Columbus says, referring to information he claims to have garnered from his Taino informants: In these islands I have found no human monstrosities, as many expected, but on the contrary the whole population is very well formed....Thus I have found no monsters, nor had any report of any, except in an island [called] 'Carib'...which is inhabited by people who are regarded in all the islands as very fierce and who 14 eat flesh....They are no more malformed than are the others, except that they have the custom of wearing their hair long like women... 16 This dissonance between phenotype and behavior may appear startling at first, until one realizes that it invokes the frequently-depicted contrast between two types of monstrous races adumbrated here. A Dutch woodcut accompanying the 1520 English version of Jan van Doesborch's Of The Newe Landes (originally published 1508-1510), reputed to be the first English book about the Americas, is typical: 17 In the latter category of monstrous races-those who are phenotypically dissimilar as well as culturally or behaviorally alien-may be found the donestre, lion-headed people who pretend to understand the foreign languages of travelers, and then kill them, devour their bodies, and mourn over their heads, as in this 11 th -century version of The Marvels Of The East: 18 15 The particular traits and behaviors represented do not always remain firmly on one side or the other of the distinction I have drawn. The pretense of linguistic familiarity, and the friendliness it is used to express, reappear in one of the most familiar and horrific vignettes described by early European explorers of the New World, namely Vespucci's 1501 report on the seductive American Indian women, hardly monstrous in appearance, who lured one of his sailors into conversation, killed him, and ate him. Here is a representation of the scene from a 1509 German translation of Vespucci's letter: 16 Of even greater interest among the phenotypically-alien monstrous races, because of their more frequent appearance in New World settings, are the cynocephali, the dog-headed people, illustrated above in the frontispiece to the 1500 Das Buch Der Croniken Unnd Geschichten, and pictured in their New World habitat in the earlier illustration from the 1530 edition of Vespucci's Carta Marítima. Traditionally the cynocephali displayed two contradictory features. On the one 17 hand they were often described as cannibals, as they are here represented in the Siena College Bestiary of 1277: On the other hand they were also often described either as Christians or as a people capable of being converted to Christianity. 19 One or the other or both of these features are, of course, often taken over in depictions and descriptions of American Indians, both earlier and later, a point to which I will return in section iii.c. below. Geography of the monstrous races Together with the distinction just discussed between two types of monstrous races, the geography of the monstrous races is key to understanding the transformation involved in the modern deployment of these stock figures of the European imaginary for the understanding of the living peoples of the Encounter itself. Even as late as the 15 th century, the monstrous races were typically located in "the East", a region that seems to have included Central, Southern, and Eastern Asia. 20 This location is reinforced by medieval readings of the extremely popular fourth18 century Alexander Romance, which was interpreted as describing a wall built by Alexander the Great, or sometimes a chain of mountains, that blocked off the known world from the monstrous races to the east and north of Europe, the Middle East, and Asia Minor. 21 The location of the monstrous races in the East places them on the margins, far from the center of the world as Europeans lived it, in places with which Europe has no large-scale human interaction. More precisely, they are located in places regarding which Europeans have no imperial interests. That is to say, whatever Europeans want from the East during this period, they do not desire to occupy it; and whatever resources Europeans draw from it are drawn by trade or by travel, and not by conquest. 22 Moreover, this lack of imperial interests is reciprocated. From the 5 th -century invasions of the Huns until the 13 th -century invasions by the Tatars and Mongols, the residents of central, southern, and eastern Asia generally have no substantial imperial interests of their own in Europe. It is because of this reciprocal imperial disinterest that these places can easily be imagined by most Europeans as utterly alien and distant. And, in turn, it is because these locations are so alien that locating the monstrous races there allows them to function as a locus for imaginative European exploration and definition of the boundaries of being human. This placement allows the monstrous races to be imagined, not just as moderately different with respect to custom and phenotype, as, say, the Germans might differ from the French, but as radically different kinds of beings, who live in ways that are shockingly different and horrendous, and whose phenotype is often shaped in equally alien or monstrous ways. 1.b. The wild man The second inhabitant of the medieval European imaginary who comes somehow to life in the Encounter with the New World is the Wild Man. The Wild Man is described in some medieval sources as one of the monstrous races; but there are crucial differences. Setting aside 19 the special case of the ordinary person who becomes wild for some period of time, 23 it is clear that there is a long tradition of seeing the Wild Man as a naturally solitary figure who is violent and lascivious, often preying upon travelers or innocent women. 24 Moreover, his geography is unlike that of the monstrous races, for he inhabits the woods or the mountains comparatively close to European towns and villages. 25 Indeed, it is this bestial solitude in proximity to European culture and society that is the defining feature of the Wild Man, as Mary Shelley's use of the trope in Frankenstein reminds us. 26 A heraldic image from the 1480s shows a typical member of the genre: These images of the wild man are very frequently taken over without significant modifications into early depictions of American Indians, where the standard medieval weapon of the Wild Man, namely, a club or tree limb, becomes the club or spear, or sometimes a bow and arrow, as in this 1505 Florentine edition of Vespucci's letter, De Novo Mundo: 20 In the 15 th and 16 th centuries, pictures of Wild Families become more common; and the figure of the Wild Woman, which had always had a separate trajectory with a rather different significance, now becomes part of the family ensemble, as in this 1500 edition of the French The Four Conditions of Society, "Ballade of a Wild Man": 27 21 Unlike the monstrous races, therefore, the Wild Man, whether accompanied or not, lives in relatively close proximity to Europeans. Like the monstrous races, however, the places he occupies are not the objects of imperial European interests. He lives in the mountains, or in the old growth forests that still covered much of Europe at least until 1250 C.E. or so. These are from the point of view of most Europeans terra incognita, vast and largely unknown islands, 22 right in the midst of Europe. They are therefore, like "the East", suitable sites for the imaginative explorations of human nature. 2. The transformation of europe Even a modest survey of the history of the three hundred years preceding the New World encounter will show that Europe underwent several radical changes during this period that directly impacted European thinking about the Other, as embodied in the mythologies of the monstrous races and the Wild Man. Taken together, these changes pave the way for an extension of the lives of the monstrous races and of the Wild Man from the realm of fantasy onto the realities of New World peoples. 2.a. Invasions from "the east" First, during the 13 th -15 th centuries Europe is repeatedly under attack by waves of invaders from the east and the southeast. In the years 1220-1225 the Mongols encircled the Caspian, sacking the Genoese outpost in Crimea. They invaded Armenia and Azerbaijan, and they completed the conquest of Bulgaria and Ukraine by 1240. No sooner were the Mongol invasions over than the Ottoman Turks begin their rise to military and political prominence, circa 1299. In 1389 the Ottomans defeated the Serbs at the Battle of Kosovo, thus opening routes of military expansion from the East into Europe. In 1453 Constantinople fell to the Ottomans; and by 1529 they were at the gates of Vienna. From 1423 to 1571 (the Battle of Lepanto) they were a constant threat to the merchant states of Italy, particularly to Venice. I have suggested that, prior to the 13 th century, at least, the Eastern location of the monstrous races ensured that encounters with them were largely imaginary rather than Encounters with living beings. As a result of the Tatar, Mongol, and Turkish incursions, however, the Eastern lands come alive for Europeans. This is certainly not to say that Europeans suddenly began to believe that these lands were really there and really inhabited. Because of the trade in silk and 23 spices, as well as the much older westward movement of religious ideas from India, Europeans were long aware of the reality of China, India, and the lands between there and Europe. But for a thousand years and more most of the peoples of what we now think of as Europe were familiar with peoples east of Asia Minor primarily through traveler's tales, or perhaps through the signs and symbols of the goods and ideas that came with those tales. 28 It is this dependence on travel reports, real and fictional, together with deeply-entrenched assumptions about how distance and travel unmoor the familiar, that make the location of the monstrous races in "the East" feasible. 29 What is new in the 13 th through 15 th centuries, as I have already hinted, is that peoples from these places now make an undeniable and widespread appearance in Europe itself. In general it is difficult to overstate the importance of the centuries-long experience of the Ottoman presence in redefining European images and European geography of "the Orient;" but the present point is that, from the perspective of the three hundred years immediately preceding the New World encounter, the Ottomans constitute simply one more great wave of peoples from the East forcibly presenting themselves in Europe. 30 Together the Mongols, the Tatars, and the Ottomans irrevocably alter European understandings of the character of those regions. At first European observers try to place the invaders into the familiar categories. Consider for example this depiction of the Tatars from Matthew of Paris' Chronica Majora for the year 1240 C.E., which reflects common belief in much of the Europe of his time, and which appears in a section significantly entitled, "An Irruption of the Tatars": In this year, that human joys might not long continue, and that the delights of this world might not last long unmixed with lamentation, an immense horde of that detestable race of Satan, the Tatars, burst forth from their mountain-bound regions, and making 24 their way through rocks apparently impenetrable, rushed forth, like demons loosed from Tatarus (so that they are well called Tatars, as it were inhabitants of Tatarus); and overrunning the country, covering the face of the earth like locusts, they ravaged the eastern countries with lamentable destruction, spreading fire and slaughter wherever they went. Roving through the Saracen territories, they razed cities to the ground, burnt woods, pulled down castles, tore up the vine-trees, destroyed gardens, and massacred the citizens and husbandmen....The men are inhuman and of the nature of beasts, rather to be called monsters than men, thirsting after and drinking blood, tearing and devouring the flesh of dogs and human beings....they drink the blood which flows from their flocks, and consider it a delicacy....They have no human laws, know no mercy, and are more cruel than lions or bears....and when they have no blood, they greedily drink disturbed and even muddy water....[T]hey know no other country's language except that of their own, and of this all other nations are ignorant. For never till this time has there been any mode of access to them, nor have they themselves come forth, so as to allow any knowledge of their customs or persons to be gained through common intercourse with other men....The Saracens, therefore, desired and begged to be allowed to enter into alliance with the Christians, in order that they 25 might, by multiplying their forces, be enabled to resist these human monsters. 31 Similar descriptions and depictions of the Turks as a monstrous race may be found in Montaigne and Rabelais several centuries later. 32 Moreover, despite early attempts to demonize their outward appearance, the Tatars, the Mongols, and the Turks were initially understood as monstrous races of our first type, whose distinguishing characteristics are extremes of behavior-cannibalism and extreme cruelty, for example-rather than bizarre or alien phenotype. 33 For these very reasons, after the invasions of the Tatars, Mongols, and Turks the peoples of Asia can no longer be seen as mythological, but must be regarded instead as some type of really existent beings. When the monstrous races from the east are incarnated in phenotypicallyfamiliar forms, and relocated from Asia to Europe itself, they are after the first shock no longer monstrous races but real and horrible enemies. These invasions from the East thus force a new place in the European imaginary: a place for living beings from the East who leave their distant abodes and intrude into Europe itself. The humanity of these invaders may or may not continue to be in question as they are re-constructed as hordes, heathens, idolaters, barbarians. That is a question for another occasion; but they clearly are monstrous races no longer. It is precisely the distance from direct observation and constant contact which permits imagination to be the definer of the monstrous races. The monstrous races, whether through behavior or through phenotype or through their juxtapositions, serve as vehicles for imaginative probings of the limits of humanity. This is what Matthew of Paris' first-generation description of the Mongols does. But over time, those who are phenotypically familiar and who irrupt into the 26 central places of Europe, can no longer be constituted entirely by the imagination. The places they come from therefore can no longer be properly regarded as the source of monstrousness. In short, the comparatively familiar phenotypes of the invaders from the East, together with their presence in Europe itself, disrupt their identity as monstrous; and so, after the 13 th century, it becomes more and more difficult to define the East as the home of monstrous races. If the function of the monstrous races as a locus for the European imagination's meditation on the limits of being human is to be maintained, therefore, the monstrous races have to be relocated away from the center of European experience to some new and distantly-imagined periphery from which, as before the invasions of the 13 th and 14 th centuries, they do not threaten the European center. The monstrous races from the East thus gradually become mythologies in need of a new location-a location which the Americas must have seemed, at least at first blush, to offer. 2.b. Deforestation of europe With respect to medieval ideas of the Wild Man, there is another significant set of pressures at work. Beginning in the 11 th century, but peaking in the period 1250-1500 C.E. , much of Europe experiences deforestation on a massive scale as a result of population growth and the widespread use of wood as fuel. Deforestation involves the destruction of many of the places in which the wild man abides. 34 At the same time, the urbanization of Europe is underway by 1300. Indeed, "the 12 th and 13 th centuries ...saw the founding of more new towns than any time between the fall of Rome and the Industrial Revolution." 35 Deforestation and urbanization, taken together, remove the experience of living near and around wild places from the quotidian lives of more and more Europeans. In sum, during the 13 th -15 th centuries the old imagined places become more and more populated. In the case of the local forests they are now populated with familiar peoples and ways 27 of life, as the forests shrink and the cities expand; and in the case of the lands to the East, they are populated with strange peoples who are actively and aggressively involved (as opposed merely to being imaginatively represented) in the lives of Europeans. The result is that both the wild men and the monstrous races are in different ways displaced. They began the 13 th century as mythologies that were each in their own way essentially tied to certain kinds of locations; but during the period in question they become dislocated mythologies. The wild man's "habitat" vanishes as the nearby forest shrinks, and the monstrous races ride the roads of Europe instead of remaining far away, where they belong. Both have become mythologies in need of relocation. 3. The construction of "the savage" 3.a. The relocation of the monstrous races We are now in a position to appreciate how it is precisely such a relocation that is provided by the New World Encounter. Consider first the monstrous races. If they are relocated to the New World, they revert at first glance, at least, to the status of strange, far-away beings who have no imperial interests in Europe. Perhaps this is why it is so natural for Columbus and other early explorers to speculate, as they do constantly, about which of the monstrous races they will find in various parts of the Americas. 36 In the New World the monstrous races are properly restored to their natural place, so to speak: far away, setting boundaries to the human world, its geography, and the diversity of its inhabitants-boundaries that are, at least in the first decades after 1492, explored largely through the work of the European imagination. The newly-emerging logic of colonization quickly complicates this relocation, however. For the Americas are not simply a substitute for the central Asian steppes, or the river valleys of India. Instead Europeans come to see various parts of the New World as objects of imperial interest. The "newe landes" become, not simply places to be visited, travelled through, or traded with, but instead colonies, places to be acquired. The relevant use of the term "colony" in 28 English begins in the mid-16 th century. In this usage, the Oxford English Dictionary says, a colony is "a settlement in a new country; a body of people who settle in a new locality, forming a community subject to or connected with their parent state; the community so formed, consisting of the original settlers and their descendants and successors, as long as the connexion with the parent state is kept up." 37 In the 16 th century, then, at least in English, a colony becomes literally a place that is now claimed and settled by Europeans. Moreover, the "connexion" with the parent state is not a merely formal one. One recalls that the irresolvable conflict generated by the pied noir's dual loyalties to France and to Algeria were grounded in one fundamental fact, namely, that Algeria was a department of France, and so, literally, French soil, a part of France. There is a way in which all colonies are like this, however; whether they are formally named as lands "subject to or connected with" the "parent state", or whether they are formally understood to be part of it, like unfortunate Algeria. This is the social and cultural significance of the elaborate disembarkation ceremonies, in which land is claimed "in the name of the King and Queen." 38 It comes out with particular clarity in that strange and ephemeral Spanish institution, the requerimiento, by which Indians were informed (in Spanish or Latin) that they were subjects and their lands were possessions of the Spanish crown, and that violent resistance would result in destruction and enslavement, as constituting rebellion upon the part of "vassals" from whom loyalty was to be expected. 39 The logic was the same as the logic the Castilian monarchs might have imposed on a rebellious town in Extremadura or León. A colony is in this way a reflection of the metropolis (literally, the "mother country") itself. The imperial interests that create the colony draw the metropolis out of its original location; and its peoples, its revenues, and its culture are adapted, if not replicated, in a new place. The new place, distant though it may be, is therefore no longer the proper possession of the Other. It is no 29 longer "theirs", but, in the eyes of the colonizers, "ours." Seen in this way, the logic expressed in the requerimiento is much easier to understand. It is therefore all but inevitable that, in the first New World Encounters, American Indians are frequently portrayed as "human monsters", to use Matthew of Paris' vivid phrase quoted above, who invade the newly-established European spaces. 40 Like the Tatars and the Turks they are cannibals, they easily become violent, and they lack an understanding of the morés of "civilized" peoples. 41 The paradox of seeing the original inhabitants of "the newe lands" as intruders or invaders of European space is no doubt striking to us; but the logic of colonial discourse makes it self-evident to the European writers and illustrators of the period. 42 Indeed, it is precisely this idea for which a familiar European argument of the sixteenth and seventeenth century serves as propaedeutic, namely, the argument that, as Robert Gray put it in 1609, prefiguring Locke's notorious arguments in the Second Treatise of Government, "these Sauages have no particular proprietie in any part of parcel of that Countrey, but only a generall residencie there, as wild beasts haue in the forrest, for they range and wander up and downe the Countrey..." 43 3.b. The relocation of the wild man The identification of the Indians as monstrous races is very quickly placed in tension with their very proximity to newly-Europeanized space, however. Because of the logic of "the colony", they are now located in spaces that have been redefined as immediately adjacent to, if not overlapping with, "our" places. They are now as close to "our" spaces as the forest at the edge of the settlement. In this respect they are better understood as Wild Men; but only because "our" places have been extended to include the colony. On the other hand, unlike the Wild Man himself, the Indians are social beings. They do not live as solitary creatures of the woods, but have their own communities, traditions and values, as 30 the Europeans understood from the very beginning. Nor are the Indians a fantasy like the Wild Man. They do not magically retreat as the land is deforested, or as it comes under "civilized" European control, leaving behind nothing but legends. Instead, they increasingly come out of the forest, into the settlement itself, bringing their raw and wild habits with them, just as Europeans always feared that the Wild Man might. In the European imaginary, the fantasy in Lucas Cranach's horrific depiction of the "cannibal or werewolf", published in 1510, has become real. 44 31 At the same time, because it is the Europeans who have the imperial interests this time, the colonizers must see the Indians in a different light, namely, in relation to the imperial resources they are seeking. At first the Indians are seen as the key to the exploitation of resources, as the 32 early entries of Columbus's log illustrate so powerfully. Then, over time, they come to be seen themselves as a natural resource that must be tamed to be utilized. In either case their wild behavior is a real threat to imperial interests; and they must be either Christianized and civilized, or, ultimately, exterminated. 45 In short, the identity of the Indians cannot easily be fitted into the traditional categories of the European imaginary that we have discussed. They are too close-by to be constituted as the far-away monstrous races, and at the same time too highly-social, and too active in the lives of European colonies, to be classified simply as the Wild Men of old. The circumstances and the logic of colonization create a tension in the application of the old categories that is resolved by the invention of new ones. 3.c. Race and the concept of the savage It is during this time that the concept of the savage emerges as a new idea, a resolution of these tensions, built out of old materials, recent pressures, and new experiences. It should come as no surprise that the term "savage", in its uses as a label for a kind of person or group of people, emerges for the first time during the 16 th century, in English, Spanish, and probably other Romance languages as well. 46 The "savage" is constituted not as a solitary individual, but as a member of a culture or "race" who is set apart from Europeans, typically by extremes of behavior and character, including cannibalism, sexual perversion, and the propensity to violence. The savage naturally occupies lands far from the center of civilization, Europe itself; but through the mystification of colonization, appears from the forests and the hills to directly threaten European places, peoples, and interests. The tem "race" (in the usages relevant to racial ideology) emerges for the first time during the 17 th century; 47 but here the concept named, that of a group originating from a particular geographical area, united by descent, and characteristically displaying certain phenotypic or 33 characterological traits, is certainly far older. 48 Nonetheless, the power of the concept, once "named", is greatly augmented. Beginning in the 17 th century, the term "race" can be used to draw attention to the cluster of features just mentioned to make them salient in explanation and understanding, and to tie them to other explanatory concepts and hypotheses. The key seventeenth-century transformation engendered by this early modern naming of old race concepts is what I would call the interiorization of race, and the image of the savage plays a crucial role in this transformation. It is to a brief discussion of this role that I now turn. The imperial confrontation with the now-savage races of "the newe lands" raises a question about their humanity that is much more immediate and pressing than questions about the humanity of the monstrous races. It is almost unanimously agreed from the time of the very earliest reports that American Indians are not significantly different from Europeans with respect to phenotype. 49 On the other hand, that there are vast differences in culture and behavior that distinguish them from Europeans is generally taken to be indisputable. But are these behavioral and cultural differences superficial, or do they indicate some deep essential contrast between known peoples and the inhabitants of "the newe landes"? The older version of this question is raised frequently during the Middle Ages in debates over the human status of monstrous races. On the one hand, the behavior of monstrous races like the cynocephali is marked by habitual, even uncontrolled, violence, cannibalism, and sexual perversion. On the other hand, even the cynocephali can be converted to Christianity, and so redeemed. Indeed, one medieval tradition has it that St. Christopher himself was a cynocephalus prior to his conversion. 50 Indeed, Augustine says, "whoever is anywhere born a man, that is, a rational, mortal animal, no matter what unusual appearance he presents in color, movement, sound, nor how peculiar he is in some power, part, or quality of his nature, no Christian can 34 doubt that he springs from that one protoplast [Adam's seed]. We can distinguish the common human nature from that which is peculiar, and therefore wonderful." 51 Their status as imagined groups whose function it is to allow the free play of questions about the nature of humanity ensures that in the case of the monstrous races this question remains fundamentally contentious. In the case of the savage, however, the imperial pressures of dealing with living beings who have ideas of their own about how the Encounter should proceed mean that the question is quickly closed. The general supposition of the imperial enterprise is that Augustine is mistaken, and American Indians are, though phenotypically within the range of the familiar, radically different, and inferior, in their deeper nature. To be savage is to have a certain essential nature, which defines American Indians as sub-human, or, at best, a lesser breed of humans. To be sure, commingled with imperial projects of enslavement, exploitation, and extermination, is an imperial project of what we may call mass Europeanization that attempts radical religious and cultural rehabilitation of savage life. Although it has its precedents (e.g. in the Spanish colonization of the Canary Islands in the 15 th century), the scale of this project is probably unequaled in European history. 52 But even supposing the rehabilitation of "savage" behavior, language, and morés is successful, 53 the question remains of how the relation between this change in outward behavior and the inward nature of the savage thus rehabilitated is to be understood. The common view that generally grounds the imperial projects of this and indeed much of the succeeding imperial and post-imperial periods is that it is impossible even by Europeanization to transform (or to transform fully) savage nature. 54 On this view, savage nature is irredeemably savage; and even changes in language use and outward behavior should not be 35 taken to indicate that what lies beneath the rehabilitated surface has truly changed at all. 55 Here another element has been added to the Frankensteinian creation that is the idea of the savage: the medieval incarnation of the very old idea of an inherited curse that cannot be altered, perhaps not even through the magical transformative power of conversion and baptism. 56 Through the mediation of this notion, the idea of the savage is completed. It is not merely that their appearance belies their monstrous behavior, as in the case of monstrous races like the anthropophagi. Instead their appearance, as well as their Europeanized behavior and morés all give the lie to the monstrousness they carry within. 57 The liminal status of the monstrous races, underwritten by their wholly-imaginary existence, gives way to the notion, shaped by imperial interests, of the fixed status of the savage. Their status as humans is no longer uncertain and debatable, but fixed as inferior by something they carry within, something that cannot be changed. In short, the newly-minted idea of the savage, completed by the familiar idea of an ineradicably depraved inward nature, enables Europeans to relocate or reconceive the traits that mark the savage as inward when, thanks to the project of Europeanization, they are not observable in outward comportment. This relocation is marked by the deployment of an old idea, race, newly-articulated in the 17 th century and reinscribed in succeeding eras as interiorized savagery. "Race" draws attention precisely to this understanding of the people of the Americas: it picks out a savage nature that is heritable and unalterable, no matter how outward behavior is changed, that is revealed to the eye, at least, perhaps only by an unchanged and unchanging skin color. The savage of the New World thus becomes a Caliban indeed, the doomed offspring of the 36 wild man and the monstrous races, midwifed by imperial interest, cursed, marooned in isolation from the metropolis and enslaved by his putative master through the magic invocation of "race." Endnotes 1 I have in mind particularly the important work of Robert Bernasconi. See for example his "Who Invented The Concept of Race? Kant's Role in the Enlightenment Construction of Race", in Robert Bernasconi, ed., Race (Blackwell Readings in Continental Philosophy) (Wiley Blackwell, 2001). See also his magisterial Concepts of Race in the Eighteenth Century (New York: Thoemmes Continuum, 2001), and American Theories of Polygenesis (Concepts of Race in the Nineteenth Century) (New York: Thoemmes Continuum, 2002). 2 A notable exception to this cursory philosophical treatment of the pre-18 th century development of the idea of race is the work of Alison Bailey. See her "Thinking About Race and White Supremacy as if Gender Mattered," presented to the California Roundtable on Philosophy and Race, 2007. 3 John Shearman, Mannerism (London: Penguin Books, 1967), p. 23. The term "mannerist" is a vexed one among art historians, however, as Linda Murray notes. She herself restricts its use to "works produced in certain parts of Italy." The High Renaissance and Mannerism (London: Thames & Hudson, 1977), pp. 124-125. 4 Michael Alexander, ed., Discovering the New World: based on the works of Theodor de Bry (New York: Harper and Row, 1976), ref. on p. 10. 5 See Alen Shestack, Fifteenth Century Engravings of Northern Europe From The National Gallery of Art: December 3, 1967-January 7, 1968 (Washington, D.C.: National Gallery of Art), preface by Lessing Rosenwald, n.p., and introduction by Alen Shestack, n.p. Compare Arthur M. Hind, A History of Engraving & Etching From the 15 th Century to The Year 1914 (New York: Houghton Mifflin, 1923, reprinted by Dover Publications, 1963), p. 20. 6 See for example Kim Sloan's discussion of the propaganda agenda behind de Bry's renderings of the John White watercolors. Kim Sloan and others, A New World: England's first view of America (Chapel Hill, NC: University of North Carolina Press, 2007), pp. 86-92. This is the catalogue accompanying a recent American showing of the British Museum's collection of John White's drawings, which the catalogue often juxtaposes with the corresponding de Bry engravings. 7 See S. Lyman Tyler's edition of the log, Two Worlds: The Indian Encounter with the Europeans 1492-1509 (Salt Lake City, UT: University of Utah Press, 1988), ref. on p. 41. It is shortly after making this claim in the log that Columbus comments several times that he and his crew do not understand the language of their Indian hosts, and that many mistakes and misunderstandings occur as a result. See id. p. 65; cf. pp. 48, 53. 8 Olive Dickason, The Myth of the Savage (Edmonton: University of Alberta Press, 1984), pp. 77-80. 9 Peter Mason, Deconstructing America: Representations of The Other (New York: Routledge, 1990), pp. 49-50, provides a series of examples in which printers of the day simply borrow images out of context when it suits their purposes. 10 A very useful survey of the presence of the monstrous races in the New World, may be found in Mason, ibid., ch. 4. 37 11 The canonical work on the history of the monstrous races remains John Block Friedman's The Monstrous Races in Medieval Art and Thought (Syracuse, NY: Syracuse University Press, 2000). Pliny and his Greek and Roman antecedents are discussed in ch. 1. Pliny's catalogue of the monstrous races appears in Book VII of the Historia Naturalis. A translation of Book VII roughly contemporary to the period under discussion is The Naturall Historie of C. Plinius Secundus, Translated into English by Philemon Holland, 1601 [book on-line] (accessed April 22, 2008); available from http://penelope.uchicago.edu/holland/pliny7.html. 12 The Travels of Sir John Mandeville (Mineola, NY: Dover Publications, 2006). This edition includes a series of woodcuts from the second Augsburg edition, 1481. Mason, op. cit., p. 92 n.1 discusses the evidence and concludes that it is very likely that Columbus was familiar with Mandeville. Flint confirms that Columbus read (and annotated) a 1489 Italian translation of Pliny, even if he had not read Mandeville. Flint, op. cit., pp. 46, 53. 13 For one catalog, along with very interesting speculations about the underlying principles of classification, see Mason, op. cit., chs. 5 and 6. 14 See Alexander, op. cit. n. 3, p. 8. 15 Compare the woodcuts of some of the monstrous races in the 1481 second Augsburg edition of Mandeville, op. cit., who are very ordinary in appearance. E.g. p. 120 (the Lamarians, who "go all naked and they scorn when they see any strange folk going clothed....they eat more gladly man's flesh than any other flesh....Thither go merchants and bring with them children to sell to [the Lamarians]....and if they be fat they eat [these children] anon"); p. 130 (the Tracodans, who "eat flesh of serpents, and they eat but little. And they speak nought but they hiss as serpents do"). 16 Quoted by Flint, op. cit., ref. on p. 144. 17 For a description of the contents and provenance of Of the Newe Lands, see Donald F. Lach, Asia in the Making of Europe, Volume I: The Century of Discovery (Chicago: University of Chicago Press, 1994), pp. 163-164. 18 See Jerome Cohen's discussion of this picture, "Monsters, Cannibalism, and the Fragile Body in Early England" (accessed May 28, 2008); available from http://www.gwu.edu/~humsci/facpages/cannibal.html. For an analysis of a well-known 19 th century version of this traveler's tale of pretended friendship, betrayal, and ritualized killing by a strange race, see Martine van Woerken's discussion of the cult of Thuggee. Martine van Woerken, and Catherine Tihanyi, transl., The Strangled Traveler: Colonial Imaginings and the Thugs of India (Chicago: University of Chicago Press, 2002). 19 See Friedman, op. cit., ch. 4. 20 See Friedman, id., pp. 8, 39. Cf. Benjamin Braude, "The Sons of Noah and the Construction of Ethnic and Geographical Identities in the Medieval and Early Modern Periods", The William and Mary Quarterly 54 (January 1997): 103-142, p. 109. Both these sources point out that the monstrous races were also sometimes located in "Ethiopia", used as a general term for those parts of Africa known or imagined to be inhabited by black Africans. In this paper I concentrate on their location in the East. 21 David Gordon White says that "the Alexander Romance, or the Pseudo-Callisthenes, first composed in Alexandria before the fourth century, was the blockbuster bestseller of the entire Middle Ages," and "the prime vehicle for the medieval lore of the monstrous races." He claims that the descriptions of "barbarian tribes" in the Romance were conflated with the ancient "commonplace" of barbarian races who are walled off from the civilized world by a mountain 38 range "running from the Caucasus in the West across all of Asia to an Eastern Sea, identified with the coast of India." David Gordon White, Myths of the Dog-man (Chicago: University of Chicago Press, 1991), pp. 52-54. 22 The Middle East and portions of Asia Minor are probably the first areas in which Europeans had imperial interests. For this reason, among others, the contrast between European relations with the East and their relations with the Middle East, Asia Minor, and northern Africa during the same period is dramatic. Because the latter regions are and always have been part of the Mediterranean world, they are to my knowledge rarely if ever described as possible locations for the monstrous races. They are too close and too intertwined with European experience to serve that function, either before or after the rise of Islam. 23 I have in mind here both the saints in the very early "Hairy Anchorite" tradition, as well as the individuals in somewhat later medieval tales, for example Orlando in Ariosto's Orlando Furioso who go mad from being spurned in love, or for other reasons. For a useful introduction to the early thematic of the "wild hairy anchorite", see Roger Bartra, Wild Men In the Looking Glass: The Mythic Origins of European Otherness (Ann Arbor, MI: University of Michigan Press, 1994), pp. 53f. See also Richard Bernheimer, Wild Men In The Middle Ages: A Study in art, Sentiment, and Demonology (New York: Octagon Books, 1970), p. 8. (Bernheimer actually thinks that, in the Middle Ages, the wild man is understood in general to be nothing more than a human being who has suffered some kind of (reversible) degeneration or derangement.) 24 See Bartra, ibid., p. 100f; Timothy Husband, The Wild Man: Medieval Myth and Symbolism (New York: Metropolitan Museum of Art, 1980), p. 3. Bartra summarizes the character of the wild man neatly as, "gloomy isolation, unrestrained aggression, and perverse lasciviousness." Op. cit., p. 117. 25 See Bernheimer, op. cit., p. 23; and Husband, id., pp. 2, 119. 26 In Shelly's novel, "The Monster" lurks alone in the woods for several years, and first speaks with his creator in the Alps. Compare Sir Walter Scott's use of the trope of the Wild Man in his story, "The Black Dwarf", [online short story] (accessed February 23, 2009); available from http://www.gutenberg.org/etext/1460. The figure of the solitary Wild Man is powerfully persistent, as the contemporary myths of the solitary Bigfoot or Sasquatch should suggest. In fact, these creatures are often labeled "Wild Man" or "Wild Man of The Woods" in contemporary popular accounts, as even a casual search on the Web will verify. 27 See Bartra, op. cit., p. 104, and the woodcuts and engravings following; and Husband's comments on late fifteenth and early sixteenth-century engravings of the wild family, op. cit., pp. 131-133. On the medieval history of the image of the wild woman, see Bernheimer, op. cit., pp. 33-40. 28 Consider for example the medieval association of myrrh, one of the traditional gifts of the Magi to the infant Jesus, with the Far East. See Valerie I.J. Flint, The Imaginative Landscape of Christopher Columbus (Princeton: Princeton University Press, 1992), pp. 123-125. 29 Nothing in my discussion should be taken to imply that "the East" was univocally or solely understood as the abode of the monstrous races. Thus the narrative, geographical, and social-political distance I have described also sustains a variety of other images of "the Orient." In particular, the "Marvels of the East" tradition represented "the Orient" as containing fabulous wealth, exotic (but not monstrous) phenotypes, cultures and practices recognizable as variants on familiar European ones. The Prester John tradition and the notion that the earthly paradise lay in "the East" should also be mentioned. These images of a paradisiacal Orient form the main 39 subject of discussion in Moffitt and Sebastián, op. cit. These images are sometimes mixed and blended into the imaginary of the monstrous races in the descriptions of groups like the brahmani or bragmanni, "a race of naked wise men who spend their days in caves," discussed by Friedman, op. cit., pp. 12, 164-166. 30 The alienness of the Turks-in contrast, say, with the familiarity of their Byzantine predecessors-has of course to do in large measure with their status as Muslims invading what is seen as a Christian world. See Robert I. Moore, The Formation Of A Persecuting Society: Power And Deviance In Western Europe, 950-1250 (London: Wiley-Blackwell, 2001) on the integration of Christianity into European identity. In this respect, the rise of the Ottomans clearly redefines for Europeans of the 15 th and 16 th century where "the East" begins, as Asia Minor is transformed from a place of trade and quotidian interaction and into a site of military conflict. But an examination of the place of Islam and the Ottoman Empire on European conceptions of the Other must await another occasion. 31 J.A. Giles, transl. & ed., Matthew Paris's English History from the year 1235 to 1273 (London: George Bell and Sons, 1889), vol. I, pp. 312-313. This vivid description is accompanied by a lurid woodcut, frequently reprinted, of Tatar cannibalism and rapine. See Suzanne Lewis, The Art of Matthew Paris in the Chronica Majora (Berkeley: University of California Press, 1987), pp. 282-287. Cf. Debra Higgs Strickland, Saracens, Demons, and Jews: Making Monsters in Medieval Art (Princeton, NJ: Princeton University Press, 2003), pp. 192194. 32 I am told that there are still seaside villages in Italy where small children are induced to come in from the shore with the admonition that, "it's getting dark, and if you don't come in the Turks will eat you." ("Ti mangiano i turchi.") It is claimed by some sources that Allied soldiers were told by their commanders during the Gallipoli campaign in World War I: "If the Turks catch you they will eat you." "The Gallipoli Campaign 1915: All the King's Men and 1/5 Norfolk Regiment", (accessed May 28, 2008), available from: http://www.canakkale.gen.tr/eng/closer/closer.html. 33 The very absence of references to phenotype in Matthew of Paris's description confirms this. 34 See Bartra, op. cit., pp. 81, 134. At the same time, the inevitable accompaniment of deforestation during the period in question is the near-eradication of wolves, the most powerful living representative of the wildness of the forest. This is particularly dramatic in Britain, where, notoriously, Edward I ordered in 1281 the extermination of all wolves in England. It appears that they did indeed become more and more scarce after this time. On the other hand, it must be said that the eradication or radical reduction in wolf populations occurred much later-indeed, into the 19 th century-in many other parts of Europe, including Germany, which is the source of many of the early woodcuts and drawings under discussion. See New York Times, May 25, 2001 (accessed May 29, 2008); available from http://query.nytimes.com/gst/fullpage.html?res=9E07E6D6133AF936A15752C1A9679C8B63. 35 "City", Encyclopedia Britannica Ultimate Reference Suite [CD-ROM] (Chicago: Encyclopedia Britannica, 2008). 36 See for example materials cited above, n. 10. Flint confirms that Columbus read and annotated a copy of Pierre d'Ailly's widely-distributed compilation, Imago Mundi, published between 1480 and 1483. Op. cit., p. 54. She comments that Columbus' annotations evince his particular interest in cannibalism among the monstrous races. 40 37 The Oxford English Dictionary cites the earliest relevant uses as follows: "1548-9 Compl. Scot. x. (1872) 82 To preue that scotland vas ane colone of ingland quhen it vas fyrst inhabit. 1555 Eden Decades II. I. 56 (fr. Latin of Peter Martyr 1516), Vppon the bankes...they [Pizarro, etc.] entended to playnte their newe colonie or habitacion. Ibid. 252 (fr. Italian) Which thynge they [Christian Princes] myght easely brynge to passe by assignynge colonies to inhabite dyuers places of that hemispherie, in lyke maner as dyd the Romanes in provinces newely subdued." Even the English use of the term in descriptions of Greek or Roman settlements is documented by only one reference prior to this time. Edmund C. Weiner and John Simpson, eds., Compact Oxford English Dictionary, Second Edition (Oxford: Oxford University Press, 1991), hereinafter OED. 38 See e.g. Columbus' description of his very first landfall in Tyler, op. cit. n. 7, p. 37-38. 39 The absurdity of reading such a document to Indians who spoke neither Spanish nor Latin was too much even for the Spaniards, and the practice was abandoned before the end of the period under discussion. A Spanish-language version of the requerimiento may be found at: Luis Lopez Nieves, "Requerimiento [Ficción jurídica: Texto completo] Monarquía Española, Redactado por Juan López de Palacios" (accessed May 29, 2008); available from http://www.ciudadseva.com/textos/otros/requeri.htm. An English translation is given by Wikipedia: "Requerimiento" (accessed May 29, 2008); available from http://en.wikipedia.org/wiki/Requerimiento. 40 The visual images represent the humanity of the Indians very clearly and unambiguously, as the included representations should indicate, even though contemporary commentators of the early the encounter remain divided on the question of their humanity. Needless to say there are other, conflicting images of the Indians, for example in Columbus' and Las Casas' well-known portrayals of the Caribbean Indians as innocent and childlike. Fairly early versions of the image of the "noble savage" may also be found. I do not discuss the construction of those images of the "childlike savage" or "noble savage" here, nor their complex interweaving with the images of the "bestial savage", which strictly speaking is our topic here. 41 Peter Hulme points out that the very act of violent resistance to the establishment of trading outposts and colonies was often regarded by early Europeans as a violation of the jus gentium, which required all nations to allow free entry and access to peaceful trading expeditions. Thus by resisting European settlement, the Indians placed themselves outside the protection of the law of the peoples. Colonial Encounters: Europe and the native Caribbean, 1492-1797 (London: Methuen & Co., 1986), pp. 161f. As Hulme intimates, this argument was made by Francisco de Vitoria, the great natural lawyer and a well-known defender of the rights of the Indians, among others. See Dickason, op. cit., pp. 130-131 for a succinct summary of de Vitoria's views on these matters. 42 Compare Hulme's analysis of how the 1622 massacre of Virginia colonists is understood by contemporaries as a slaughter of the natural inhabitants of the land by the Indians, who are cast as "unnatural Naturalls." ), id. p. 160. Hulme seems to me to miss the point, however, since he thinks that this reversal of the roles of native people and intruding colonists is only made possible by the massacre. It seems to me on the contrary that it is impossible to understand the response of writers like Samuel Purchas to the massacre (quoted by Hulme) unless one posits that, prior to this event, the English already saw themselves as the true "Naturalls" or native inhabitants of the land. 41 43 Quoted by Hulme, id., pp. 158. It is noteworthy that the wild man is usually conceived as having no property or possessions. 44 See Husband's brief discussion of this image and the relation between the wild man and the werewolf, op. cit., p. 110. 45 Compare Patrick Brantlinger's brilliant analysis of a much later European understanding of the logic of extermination in his Dark Vanishings: Discourse on the Extinction of Primitive Races, 1800-1930 (Ithaca, NY: Cornell University Press, 2003). 46 The OED, op. cit. n. 34, records 1588 as the first time the term is used in this way, contemporaneous with its first use as an adjective to describe "uncivilized" persons. Other related uses in reference to persons, character, or behavior are either contemporaneous as well, or later. 47 See the OED entry on "Race n 2 ", op. cit. Ivan Hannaford provides detailed examples of the use of the term as it evolves from older meanings in the 16 th and 17 th centuries. Ivan Hannaford, Race: The History of An Idea in The West (Baltimore: Johns Hopkins University Press, 1996), pp. 175ff. 48 See for example the evidence cited in Hannaford, id., chs. 1-5. Hannaford himself claims that the idea arises in the early modern period, but he appears to conflate the question of when the concept originated with the question of when the relevant usage of the term began. 49 For example, Columbus says of the people of Hispaniola: "Both sexes were handsomer than any they had hitherto seen, their color light, and, if clothed and guarded from the sun and air, would be nearly as fair as the inhabitants of Spain." quoted in S. Lyman Tyler's edition of Columbus' log, op. cit. n. 7, p. 75. 50 Friedman, op. cit., p. 72. 51 City of God, [book on-line] (accessed February 23, 2009); http://www.newadvent.org/fathers/120116.htm; Book XVI, ch. 8. See Friedman's discussion of this passage, id., pp. 90-91. 52 The project of Europeanization has its counterparts in Europe itself, most obviously in fifteenthand sixteenth-century Spain, where the reconquista involves the re-socialization and, of course, religious "re-education" of Jewish and Muslim populations. See below n. 51. 53 For a history of one of the early explicit attempts at Europeanization and the general 16 th century response, see Lewis Hanke's account of Las Casas' unsuccessful efforts to establish utopian Indian communities in Hispaniola and Cuba. Lewis Hanke, First Social Experiments in America: A Study in The Development of Spanish Indian Policy in The Sixteenth Century (Cambridge: Harvard University Press, 1935). 54 A parallel understanding obviously defines much of the trajectory of the Europeanization project in Spain, as the contemporaneous development of the idea of limpieza de sangre (literally, "purity of blood") shows. The descendants of converted Jews are excluded from public office and most universities in Spain at least through the 18 th century, merely on the grounds that they carry something "in the blood" that cannot be altered even by centuries of "civilized" behavior. 55 Interestingly, in many parts of Latin America today someone who has lost their temper will say, "Me salió el Indio"-"The Indian in me came out." 56 On the medieval and modern history of the idea of the inherited curse, see Hannaford, op. cit. n. 41, pp. 91f. 42 57 In the case of the Indians at least, the trope of skin color is sometimes employed to mark this inward continuity in monstrousness. Though Indians are almost universally not seen as black by Europeans of the period under discussion, they are nonetheless almost always marked as different in appearance from the Europeans themselves, who in this period are paradigmatically seen as white, whether they be English, Spanish, or Portuguese. Notice that here I am speaking of the deployment of this trope with respect to those non-European peoples who are not seen as "black." Columbus and other early European explorers often contrast the bronze or "red" color of American Indians (and Canary Islanders) with the black color of Africans. During his first encounter with the Taino in October of 1492, Columbus says, "...they are of the color of the Canary Islanders, neither black nor white....None of them are black...." Regarding the Taino women of Hispaniola, he remarks in a December, 1492 entry that, "As to beauty, [Columbus'] men stated that they exceeded the [women of the other islands] beyond comparison, both men and women being of a much lighter color, and that two young females were seen as white as could be found in Spain." Tyler, op. cit. n. 7, pp. 38-39, 73. In contrast, at least since the late medieval era-and perhaps well before-those who are seen as having black skin are often placed thereby beyond the pale of humanity. The significance of black skin in the history of race is of course a complex matter, deserving of its own study. See my "Images of Africans and Blackness in Renaissance and Early Modern Europe", in Peggy Zeglin Brand, ed., Beauty Revisited (Bloomington: Indiana University Press, forthcoming 2009).

Human Rights, Claimability, and the Uses of Abstraction ADAM ETINSON The Graduate Center, City University of New York This article addresses the so-called 'claimability objection' to human rights. Focusing specifically on the work of Onora O'Neill, the article challenges two important aspects of her version of this objection. First: its narrowness. O'Neill understands the claimability of a right to depend on the identification of its dutybearers. But there is good reason to think that the claimability of a right depends on more than just that, which makes abstract (and not welfare) rights the most natural target of her objection (Section II). After examining whether we might address this reformulated version of O'Neill's objection by appealing to the specificity afforded to human rights in international, regional, and domestic law (in Section III), the article challenges a second important feature of that objection by raising doubts about whether claimability is a necessary feature of rights at all (Section IV). Finally, the article reflects more generally on the role of abstraction in the theory and practice of human rights (Section V). In sum, by allaying claimability-based concerns about abstract rights, and by illustrating some of the positive functions of abstraction in rights discourse, the article hopes to show that abstract rights are not only theoretically coherent but also useful and important. Human rights culture has often been accused of a certain imbalance. For instance, it is often said that the practitioners of human rights (i.e. lawyers, politicians, judges, legislators, intellectual advocates, activists, etc.) are too quick to proclaim the existence of rights and too slow to define or allocate attendant duties.1 It is not difficult to locate the grounds of such complaints. In a manner typical of most prominent declarations, the 1948 Universal Declaration of Human Rights (UDHR) states that members of the United Nations should take 'progressive measures, national and international, to secure [the] universal and effective recognition and observance'2 of the human rights it proclaims, e.g. rights to freedom from torture, oppression, arbitrary arrest, poverty, unemployment, starvation, and exile. But the UDHR rarely goes into any serious detail about what member states are concretely required to do in order to secure such rights. And while legally binding human rights treaties and covenants are generally less reticent in this respect,3 they similarly leave a wide range of deontic questions unanswered. For one, like prominent 1 Throughout this article, I will use the terms 'duty' and 'obligation' interchangeably. 2 UDHR, Preamble. 3 For instance, the International Covenant on Economic, Social and Cultural Rights (ICESCR) is more specific about the human right to work, claiming that it can only be fully realized by steps that include, among other measures, 'technical and vocational guidance and training programs' for citizens (Article 6.2). declarations, they proclaim an abundance of rights without clearly specifying the conditions of their satisfaction.4 And second, such treaties and covenants are generally silent about what is to be done – and in particular what third-party or backup responsibilities come into effect – when signatory state parties are either unwilling or unable to abide by their treaty obligations. What might the costs of such abstraction be? Well, one such cost is sure to come at the level of the practical relevance or meaningfulness of a proclaimed right. It is normal to expect a human right (indeed, any right) to have some discernable practical bearing on our individual and institutional decision-making, even if its bearing is not always decisive against competing practical considerations, including other rights. Some such expectation follows naturally from the idea that rights represent normative standards to which our choices and actions ought to conform. And yet it is clear that the abstractness of a right can undermine its performance in this respect. This is because we need information about the content and allocation of the duties connected to a right if it is going to take up a meaningful place among the array of reasons, values, and practical considerations that bear on our personal and political conduct. Thus, as Henry Shue has argued, a claim of right is in an important sense incomplete until we have spelled out, 'at least a little bit, what it would actually mean for a certain right to be fulfilled and enjoyed.'5 But these are precisely the sorts of details that abstraction skates over. Other critics have offered more grandiose assessments of the detrimental effects of abstraction, seeing it as evidence of a commitment to questionable normative and ontological assumptions.6 Of late, however, renewed attention has been paid to a longstanding worry about abstract rights that emerges from considerations internal to the theory of rights itself. According to this concern, sometimes referred to as the 'claimability objection' to human rights,7 abstract proclamations of rights are not merely incomplete, or entangled with broader philosophical assumptions of questionable validity; they are incoherent. It is this last gloss on the importance of allocating and specifying responsibilities when we talk of rights – particularly as developed by Onora O'Neill – that shall occupy me in this article.8 4 Consider, for instance, Article 12 (b), which claims, without adding any further detail, that state parties must take the 'steps necessary for: the improvement of all aspects of environmental and industrial hygiene' (My emphasis). 5 See Henry Shue, 'Thickening Convergence', The Ethics of Assistance: Morality and the Distant Needy (Cambridge, 1996a), ed. D.K. Chatterjee, pp. 226-227; Henry Shue, Basic Rights: Subsistence, Affluence, and U.S. Foreign Policy (Princeton, 1996b), e.g. p., 157. 6 For instance, according to some, the inattention of human rights culture towards concrete matters of responsibility signals its lack of appreciation for the significance of 'community,' the 'social dimension' of human life, or the interdependence of human beings [See, e.g. Mary Ann Glendon, Rights Talk: The Impoverishment of Political Discourse (New York, 1991), Ch.4; Amitai Etzioni, The Spirit of Community (New York: Crown Publishers, 1993), pp. 1-23; Shue 1996a, pp. 218-219.]. And according to others, that inattention signals the basis of human rights in a 'Western' system of value, one that prioritizes individual well being over the greater good of the community. [See, e.g. Jack Donnelly, 'Human Rights and Asian Values: A Defense of 'Western' Universalism', The East Asian Challenge for Human Rights (Cambridge, 1999), eds. Joanne R. Bauer & Daniel A. Bell, pp. 78-79]. 7 See John Tasioulas, 'The Moral Reality of Human Rights', Freedom from Poverty as a Human Rights: Who owes What to the Very Poor? (Oxford, 2008), ed. Thomas Pogge, p. 80, 88-95. 8 The main sources I shall be referring to are: (a) Onora O'Neill, Towards Justice and Virtue: A Constructive Account of Practical Reasoning (Cambridge, 1996), pp. 128-136; (b) Onora O'Neill, 'The Dark Side of Human Rights', International Affairs, Vol. 81, No. 2 (2005), pp. 427-439. I. O'NEILL'S CRITIQUE OF WELFARE RIGHTS Those who argue that rights must be claimable tend to start from the widely accepted idea that rights are logically correlated to duties or obligations.9 This correlativity is not normally understood to be symmetric. So, for instance, the idea is not that every obligation is the logical corollary of some specifiable right, since the possibility of an imperfect obligation (e.g. of charity or benevolence) that is owed to no one in particular and irrespective of anyone's 'rights' is widely acknowledged to be a coherent one. All that the thesis of logical correlativity entails is that any genuine right be matched to some obligation, or set thereof, on the part of others. In this sense, rights are classified as a kind of claim against others. And on some such grounds, proponents of claimability argue that a right ought to be claimable, a prerequisite of which is that we be able to identify its duty-bearers. In other words, if a right is kind of claim (noun) against others, then we ought to be able to meaningfully and validly claim (verb) it, and this, it is alleged, requires knowing who these others are. This line of reasoning is especially salient in the writings of O'Neill. Consider, for example, the following quotation: Any right must be matched by some corresponding obligation, which is so assigned to others that right-holders can in principle claim or waive the right (or where not competent to do so, that others be able to at least claim it on their behalf). Unless obligation-bearers are identifiable by right-holders, claims to have rights amount only to rhetoric: nothing can be claimed, waived or enforced if it is indeterminate where the claim should be lodged, for whom it may be waived or on whom it could be enforced.10 As described in the previous section, and like other authors who argue for claimability as a necessary feature of rights,11 O'Neill suggests that talk of unclaimable rights must be understood rhetorically, i.e. as talk that is ultimately about something other than rights. But not all of human rights discourse is at risk of falling into this trap, according to O'Neill; in truth, only talk about a certain class of rights – 'welfare rights,' or rights to goods and services – is at risk of this. According to O'Neill, liberty rights – or rights that demand 'non-interference' – are different from welfare rights in that welfare rights demand a 'specific performance' on the part of obligation holders.12 And this performance-demanding feature of welfare rights creates difficulties of coordination and duty allocation that are not present (or at least not present in the same way) in the case of liberty rights. For instance, according to O'Neill, liberty rights – rights to personal security, private property, freedom of association, etc. – can be realized for all providing no one interferes with the behaviour of anyone else. Such negative duties, which in essence involve leaving one another alone, can plausibly be allocated universally, and so we can know by pure moral reasoning alone that the duties correlative to liberty rights are allocated to everyone (or at least to all agents capable of respecting them). Welfare rights, on the other hand 9 This thesis of logical correlativity is most famously attributed to Wesley Hohfeld, who advances it in his discussion of 'claim-rights', which he sees as rights 'in the strictest sense'. See Wesley Hohfeld, Fundamental Legal Conceptions (New Haven, 1923), pp. 36-38. The same idea is also implicit in Joseph Raz's influential account of rights: ''X has a right' if and only if... an aspect of X's well-being (his interest) is a sufficient reason for holding some other person(s) to be under a duty.' Raz, The Morality of Freedom (Oxford, 1986), p. 166. 10 O'Neill (1996), p. 129. 11 See Joel Feinberg, Social Philosophy (Englewood Cliffs, 1973), pp. 66-67. 12 O'Neill (1996), pp. 130, fn. 4. – rights to healthcare, education, work, an adequate standard of living, etc. – can be realized only through the expenditure of resources, the transfer of goods, and the performance of services. Such positive duties, unlike their negative counterparts, cannot reasonably be allocated to all, and so the question of who bears them is a complicated one. In fact, O'Neill argues, there is not a great deal that we can know about who bears such duties in the abstract, outside the context of a given institutional scheme. 13 Thus, when conceived of as rights held by all individuals irrespective of institutional affiliation, welfare rights are uniquely unclaimable, or so the argument goes. Now, the once fashionable notion that liberty rights are characteristically negative and welfare rights characteristically positive, in the very sense described above, has long fallen into disrepute.14 Most contemporary moral and political philosophers have come to accept the idea that both kinds of rights, in fact, generate a mixture of positive and negative duties.15 This becomes obvious once we consider that welfare rights will typically generate negative duties as well as positive duties: for instance, duties not to interfere with one's means of subsistence, or block one's access to crucial resources. Similarly, liberty rights will typically generate positive duties as well as negative ones: for instance, duties to prevent the infringement of liberty rights through the establishment of a police force and a system of justice. And so this leaves the central distinction of the preceding paragraph in doubt. O'Neill is, in fact, ready to admit that the distinction between welfare and liberty rights can be muddled in this way. But she nevertheless insists on the basic asymmetry between welfare and liberty rights by drawing on a further distinction between the primary and secondary duties associated with a right.16 To illustrate: the protection of liberty rights will indeed require governments to coordinate (e.g. through policing, legislating, judging, prosecuting, jailing, etc.) the complex allocation of secondary duties to enforce citizens' primary duties to avoid infringing upon one another's liberty rights. But this takes for granted that, in the case of such rights, we already know 'who' is bound by such primary duties and 'what' these duties involve: the 'who' includes everyone and the 'what' involves simple non-interference. Governmental agents step in to enforce (however elaborately) a straightforward normative relationship. By contrast, in the case of welfare rights, O'Neill argues, it becomes impossible to claim that institutions similarly enforce a set of known primary obligations. This is because, again, in the case of such performance-demanding rights, there is often little that we can know about their corresponding obligations before these have been organized and allocated under an institutional scheme, e.g. through welfare legislation, a healthcare program, social security provisions, council housing projects, etc. Only once these overarching institutions are in place can we know the shape of the primary obligations – i.e. who owes what to whom – that hold between agents in light of their welfare rights; hence the purported asymmetry with liberty rights.17 One uncomfortable consequence of these observations, according to O'Neill, is that we cannot claim, as we commonly do, that welfare rights are both rights and institutionally transcendent moral norms at the same time; one of these claims has to give. For, on the one hand, if we want to hold on to our understanding of welfare rights as rights, then we can only do so in 13 Ibid, pp. 130-132. 14 Maurice Cranston was the most popular early defender of this view. See Maurice Cranston, What are Human Rights? (London, 1973), pp. 66-67. 15 See Shue (1996b). 16 O'Neill (1996), p. 131. 17 O'Neill (2005), p. 432; O'Neill (1996), pp. 132-136. the case of welfare rights that have been rendered claimable by an existing institutional scheme. But this undercuts the sense in which such rights are pre-institutional, independently valid, or affirmed (rather than created) by institutions that recognize and implement them.18 On the other hand, if this is intolerable, and we opt to preserve our belief in the independent moral validity of welfare rights, then their inherent (pre-institutional) deontic indeterminacy will undermine their status as rights. Instead, we will be forced to identify them in other terms, e.g. as noble aspirations, goals, ideals, imperfect obligations, or 'manifesto rights.' 19 Our common understanding of welfare rights as moral rights that transcend and, indeed, motivate and justify their own institutional embodiment begins to look unsustainable.20 Along with this conceptual upshot of the preceding observations, O'Neill also highlights some of their moral consequences. For instance, she writes: Proclamations of universal 'rights' to goods or services without attention to the need to justify and establish institutions that identify corresponding obligation-bearers may seem bitter mockery to the poor and needy, for whom these rights matter most. When advocates of Human Rights proclaim universal rights to food or to work or to welfare, yet fail to show who has corresponding obligations, or where claims of right or redress may be lodged, they hurl a weapon that may boomerang. At best a premature rhetoric of rights may have political point and impact. An appeal to the 'manifesto rights' of the sort promulgated in Charters and Declarations invokes and highlights ideals that may guide agitation, politics and legislation in a quest for institutionalized, claimable rights... But at worst a premature rhetoric of rights can inflate expectations while masking a lack of claimable entitlements.21 There seem to be two overlapping strands of argument here. According to one strand, the proclamation of unclaimable 'rights' to goods and services is a form of 'bitter mockery to the poor and needy' because it encourages them to believe that they have a realizable or enjoyable claim to relief from (or compensation for) their predicament, when in fact this isn't so – and not because such relief (or compensation) is infeasible but because it simply isn't clear who owes it to them. Remedial action is, in effect, thwarted by ignorance. The mocking or cynical aspect of the proclamation, on such a reading, consists in its attribution of unrealizable entitlements to those who are in desperate need of real help, and certainly not of false hope.22 18 O'Neill (2005), pp. 432-433; O'Neill (1996), pp. 133-134. 19 The term originates in Feinberg (1973), p. 67. 20 It is unclear whether O'Neill's argument here is based on an epistemological claim (i.e. that we need institutions to help us identify a right's duty-bearers) or an ontological claim (i.e. that only institutions can make the identity of a right's duty-bearers determinate), or both. Ultimately, I don't think anything of serious importance hangs on the matter. If O'Neill is only making the epistemological claim then, it is true, there will be a sense in which welfare rights can exist prior to institutional specification, since such rights may still have ontologically determinate duty-bearers. But this theoretical possibility would be entirely irrelevant in practice, since our inability to actually identify the relevant duty-bearers in such a case would still leave us unable to confirm the right's ontological determinacy, and so unable to justifiably claim its existence. Thus, the basic thrust of O'Neill's objection remains the same on either claim. 21 O'Neill (1996), p. 133. 22 It is worth briefly noting here that a right or entitlement can be what I am calling 'unrealizable' for a number of distinct reasons: these include its (a) unenforceability, (b) non-justiciability, or (c) infeasibility. So, for instance, (a) a right may be unrealizable if there are no reliable and available means of enforcing compliance with its attendant obligations. Or, (b) it may be unrealizable because the courts and legislature are unwilling and/or unable to legally administer the right, in which case it is non-justiciable. In other According to a second strand in O'Neill's argument, however, such proclamations constitute bitter mockery for a different reason. If, at bottom, universal ascriptions of rights to goods and services cannot be coherently regarded as ascriptions of rights, then, according to O'Neill, at best they must be read as affirming the existence and importance of certain moral goals, considerations, ideals, aspirations, or imperfect moral obligations, i.e. obligations owed to others in general, but to no one in particular.23 While such norms and requirements can in principle be just as morally important, weighty, or urgent as rights, their importance is nevertheless distinct from that standardly attributed to rights, or to rights-based obligations. This is because rights-based obligations have a direction – i.e. they are owed to some individual or group thereof – and so when they are violated someone or some group of rights-holders is wronged, and so is understandably entitled to remedy or compensation of some form.24 By contrast, when an imperfect duty – owed to no one in particular – is violated, a wrong is committed but no particular person or group has been wronged, and so directed compensation is uncalled for. Thus, according to O'Neill, when unclaimable welfare rights are ascribed universally, the poor and needy are lured into a false sense of entitlement. Such proclamations lead the destitute to believe that the world's brutish indifference towards their misfortune constitutes an injustice done unto them: one that justifies demands for remedial action or (at least) apology and that licenses feelings of resentment and blame. On O'Neill's account, however, the deflating truth is that any injustice done is strictly impersonal, and so the poor and needy are not in fact entitled to make such demands or to legitimately harbour feelings of that sort (nor is anyone entitled to on their behalf). This is another sense – one borne in the falsity rather than the non-realizability of an entitlement – in which O'Neill urges us to see the proclamation of universal welfare rights as a reckless and even cynical act. II. NARROW AND WIDE CLAIMABILITY So far, I've portrayed O'Neill as understanding the claimability of a right to depend on whether it is possible to identify precisely whom that right can be claimed against. This portrayal requires circumstances, (c) the duties generated by a right may not be feasible in the present and for the foreseeable future (this is different from their being unenforceable), in which case we might reasonably question the existence of that right (For a useful discussion, see Pablo Gilabert, 'The Feasibility of Basic Socioeconomic Rights: A Conceptual Exploration' in The Philosophical Quarterly, Vol. 59, No. 237 (2009), pp. 651-681) In this strand of O'Neill's argument, (d) a right is deemed unrealizable not because we cannot enforce compliance with its attendant obligations, or because these are unfeasible or nonjusticiable, but because we do not know who bears these obligations. Several authors have run the issues of feasibility and claimability together, most likely because, when a right is infeasible, it is also unclaimable, i.e. it has no identifiable duty-bearers (e.g. see Feinberg (1973), pp. 66-67; Cranston (1973), pp. 68-69; Bernard Williams, In the Beginning was the Deed: Realism and Moralism in Political Argument (Princeton, 2005), ed. Geoffrey Hawthorne, p. 64). But this tendency should be resisted. This is because a right can be unclaimable but nevertheless feasible if, say, we know that it generates feasible duties but cannot yet discern who specifically bears those duties. Precisely because they fudge this distinction, it is not entirely clear whether these authors think claimability (properly understood) is an existence condition of rights. 23 See Onora O'Neill, Constructions of Reason: Explorations of Kant's Practical Philosophy (Cambridge, 1989), pp. 230-231; O'Neill (1996), pp. 136-153. 24 See, e.g. Allen Buchanan, 'What's so Special about Rights?', Social Philosophy and Policy, Vol. 2, No. 1 (1984), p. 74; O'Neill (1996), p. 139. some qualification. And that is because O'Neill sometimes suggests that the claimability of a right also depends on specifying the content (and not just allocation) of the duties that attach to it.25 For instance, in her critique of welfare rights, O'Neill at one point draws attention to the ambiguities of Article 12 of the ICESCR, which proclaims 'the right of everyone to the enjoyment of the highest attainable standard of physical and mental health.'26 As she observes, in proclaiming a right to the highest attainable standard of health, many questions are begged. For one, according to what measure is the index of attainability to be understood? Is this a local standard, in which case the right may turn out (in the context of severely impoverished states) to yield relatively little in the way of claimable healthcare? Or, is the relevant standard of attainability a global one, in which case the right may seem too demanding? Furthermore, on either interpretation, what would such a right require on the ground, e.g. on the part of those (doctors, surgeons, nurses, and physicians) who are in possession of the goods demanded by the right? It is easy to see why O'Neill might categorize these content-related indeterminacies as threats to the claimability of a right. Indeed, how can I meaningfully demand some act or omission from an agent without specifying precisely what it is that I am demanding? But once we acknowledge this much, it becomes hard to see why O'Neill does not amplify her understanding of claimability even further. This is because there is a rich plurality of information that is plausibly relevant to the claimability of a right. Along with (a) details about the bearers of a right's correlative duties and (b) about the content of those duties, this plurality includes: (c) details about the holders of the right (e.g. how can I validly claim a right that I do not hold?), and (d) details about the weight or priority of the right and its resistance to trade-offs in cases of conflict with other practical considerations, including other rights (e.g. how can I validly claim a right that has been defeated by, say, competing concerns of national security?).27 Informational gaps at any of these levels will undermine the support that a right offers claimants who seek to invoke it. Thus, at this point, it's helpful to distinguish between narrow and wide conceptions of the requirements of claimability. According to the narrow conception, which is more or less explicitly adopted by O'Neill, the claimability of a right can be fully secured providing we possess adequately specific information about (a) and (b); according to the wider one, by contrast, the claimability of a right requires specific information of all four kinds – (a), (b), (c), and (d) – and possibly more. What I am now suggesting is that the amplified notion is the more coherent one. If what one wants is to claim the object of some right from others, then indeterminacies of type (c) and (d) will prove just as debilitating as those of type (a) and (b). On the most natural understanding of O'Neill's arguments, then, what claimability actually requires is specific information about a wide range of practical matters. 25 Indeed, Elizabeth Ashford interprets O'Neill as adopting this broader conception of claimability, which requires identifying 'both the precise content of the corresponding duties and the specific agents responsible for fulfilling them.' Elizabeth Ashford, 'The Duties Imposed by the Human Right to Basic Necessities' in Freedom from Poverty as a Human Rights: Who owes What to the Very Poor? (Oxford, 2008), ed. Thomas Pogge, p. 214 (My emphasis). 26 See O'Neill (2005), pp. 427-439. 27 I recognize that, as a Kantian, O'Neill may be unwilling to acknowledge the claim that rights can conflict, and so would also be unwilling to admit that rights have weight. Nevertheless, even the most ardent absolutist about rights – e.g. see Russ Shafer-Landau, 'Specifying Absolute Rights,' in The Arizona Law Review, Vol. 35 (1995), pp. 209-225 – must admit that, until the full content of a right is worked out, there are questions about the lexical priority of rights over one another (and over non-rightsbased considerations) that remain open and important. This revision has some important ramifications. Chief among these is the fact that the set of 'unclaimable' rights is likely to expand quite drastically. So long as the claimability of a right was understood to depend exclusively (or at least predominantly) on the identifiability of its duty-bearers, O'Neill was able to point to the allegedly special difficulties involved in allocating the positive duties that attach to welfare rights in order to single out those rights as the sole objects of her critique.28 But now that it has become clear that (on pain of arbitrariness) it is wide claimability that is demanded by O'Neill's critique, those objects can no longer be so well isolated. In other words, now that the demands of claimability have been augmented, welfare rights cannot plausibly be the only rights that fall short of those demands. Even if we were to charitably concede to O'Neill her claim that the allocation and (let us also concede, for the sake of argument) content of the duties that correspond to liberty rights are determinate in a way that the duties attached to welfare rights are not, this would still not be enough to contain her critique. Liberty rights are, after all, in no way obviously immune to trade-offs or indeterminacies about their weight or priority. The interminable character of debates about the permissibility of abortion and torture – where conflicts between different rights (e.g. to life or liberty, on the one hand, and to privacy or security, on the other) are a central issue – is a testament to that. And liberty rights are equally beset by indeterminacies regarding who holds them, e.g. do infant, comatose, or psychotic individuals have a right to liberty? The answer is not so clear. What all this is pointing to is that O'Neill's critique, as expounded above, can be turned not just on welfare rights but on any right for which we lack specific information about (one or more of) details (a), (b), (c), and (d). Understood in this way, O'Neill's critique becomes what is in effect a critique of abstract rights. Abstract rights, as I shall define them, bracket or omit specific details of some kind.29 Thus, a right to healthcare is more abstract than a right to, say, a single-payer healthcare system, because it omits reference to any specific form of healthcare. And a right to liberty is more abstract than the related right to freedom from arbitrary arrest, since the latter is a right to freedom from a specific form of coercive interference. Furthermore, a right can be abstract in a variety of respects – i.e. with respect to its holders, duty-bearers, content, and/or weight – and it can be abstract in some respects but specific in others.30 Since abstract rights, so defined, are characterized by the very sorts of omissions and indeterminacies that threaten the wide claimability of a right, they constitute the chief object of O'Neill's critique – although this of course does not mean that welfare rights are no longer touched by that critique as well.31 28 I leave aside here some important doubts about the cogency of O'Neill's attempt to single welfare rights out for criticism by way of the distinction between primary and secondary duties. See Tasioulas (2008), pp. 91-92. 29 This is also how O'Neill herself understands the notion of abstraction: 'Abstraction, taken straightforwardly, is a matter of bracketing, but not of denying, predicates that are true of the matter under discussion.' O'Neill (1996), p. 40. 30 For a similar definition of abstract rights, see James W. Nickel, Making Sense of Human Rights (Oxford, 2007), 2nd Edition, pp. 23-24. Also see Raz 1986, p. 184; James Griffin, On Human Rights (Oxford, 2008), p. 50; and Waldron (1993), pp. 77-78. 31 It's worth noting here that the abstractness of a right is something different from its (i) vagueness or (ii) ambiguity, although these are also forms of indeterminacy. A right is vague if it appeals to concepts or predicates that are prone to borderline cases, i.e. cases in which they neither clearly apply nor clearly fail to apply. But not all abstract rights are vague in this sense. Moreover, not all abstract rights are ambiguous in the sense of committing homonymy, i.e. picking out categorically different objects in the way that a right to 'fence' can be both a right to engage in an Olympic sport and to engage in the activity All of this brings us back to the initial concern with abstraction. As noted above, references to abstract rights are ubiquitous in human rights discourse. Even specific human rights (e.g. the Burmese people's right to vote) are commonly thought to be derived or applied versions of more abstract counterparts (e.g. the human right to political participation). On the best understanding of O'Neill's critique, however, this widespread reliance on abstraction is a grave error. Since it is part of the very nature of abstract rights to omit fine-grained details about their content, holders, duty-bearers, and priority – details crucial to the claimability of any right – such rights cannot be seen as higher-level rights that exist prior to, or that explain and justify, their more specific counterparts.32 At best, references to markedly abstract human rights consist in rhetorical elaborations of what are, at bottom, imperfect obligations or moral goals that can be violated without anyone being wronged in the process.33 If we want to avoid this discomforting thought, there are roughly two available strategies. One is to deny that human rights discourse is anything more than superficially abstract and to claim that, underneath this façade, there is a wealth of readily accessible information that can satisfy the stipulated requirements of claimability. This is a strategy that I will briefly explore in the context of human rights law below, in the following section. Another strategy is more aggressive, and denies that the strictures of claimability – wide or narrow – are ones that need to be satisfied at all. This second strategy is, I think, more promising, and one that I shall pursue in Section IV.34 III. SPECIFICATION THROUGH LAW One question that I have postponed, up to now, is that of whether the preceding concerns about claimability apply to moral rights or legal rights, or both. Standardly understood, moral rights are rights that exist in light of moral demands and considerations; legal rights, by contrast, are rights that exist under the rules of a given legal system. Defined as such, these two categories of rights of marking boundaries between properties. See Jeremy Waldron's discussion of vagueness, ambiguity, indeterminacy, and contestability in 'Vagueness in Law and Language: Some Philosophical Issues' in California Law Review, Vol. 82, No. 3 (1994), pp. 509-540. 32 This is simply a repetition of the criticisms that O'Neill raises against welfare rights, as discussed in Section II, but now applied to abstract rights more generally. Since O'Neill lauds the use of abstraction in moral reasoning elsewhere in her writings [e.g. O'Neill (1996), Ch. 2], one way to interpret my own analysis is as pointing out an inherent tension in her views on abstraction. 33 I should mention that O'Neill might be willing to accept the existence of abstract rights if they were thought of as merely generalized assemblages of specific and claimable rights (in the way that my right to freedom of movement might be understood as nothing more than a shorthand term for an infinite number of specific and claimable rights, e.g. to stand on my head, to move my arm, to walk outside, etc.). But this is not how abstract rights are normally understood: they are often thought to have independent justificatory force as the grounds of more specific rights, in the way just described. [See also Jeremy Waldron, Liberal Rights: Collected Papers 1981-1991 (Cambridge, 1993 pp. 77-78; Raz (1986), p. 169]. That prominent and important conception of abstract rights – without which we could not affirm a human right to, say, adequate nourishment, since we cannot cite specific and claimable rights to such nourishment that hold in each individual case in advance – would still be unsustainable on O'Neill's account. 34 In this respect, the current article builds on Tasioulas' response to O'Neill's claimability objection in Tasioulas (2008). That said, my own response (in Section IV) is less dependent on the acceptance of the interest-based theory of rights than his is. are clearly distinct and potentially non-overlapping; just think of all the obviously immoral rights that have been upheld by law at some time and place (e.g. rights to hold slaves, to deport Jews, to deny women the vote, etc.). Despite this, there are two good reasons for thinking that O'Neill's critique touches both the domains of morality and law. First, her complaints about claimability and indeterminacy are directed towards human rights culture as a whole, of which human rights law forms an undoubtedly integral part. Indeed, O'Neill herself sometimes draws on legal examples in order to prove her point.35 Second, despite their differences, there are important and deep connections between moral and legal rights. It is, in particular, not at all unusual to think of human rights law as giving expression to human rights understood as moral norms.36 Since the 1960s and 1970s human rights have, after all, been expressed and articulated largely through the medium of law. And the practical requirements of such rights have been given their most elaborate and detailed formulations in domestic, regional, and international legal systems. This gives scope to the possibility that the specificity and detail with which human rights are handled in the arms of the law can be used to exonerate the whole of human rights culture (in both its moral and legal dimensions) of any blanket charge of abstract indeterminacy. At the very least, that is the possibility that I want to briefly explore here. As noted in the introduction, international covenants are chock-full of abstract rights (e.g. to food, health, political participation, etc.) that have dramatically unclear practical implications. However, if we look beyond the strict letter of major legal covenants such as the ICESCR and ICCPR (The International Covenant on Civil and Political Rights), at least two relevant modes of legal specification come into view. Perhaps the single most important of these consists in judicial efforts to apply abstract rights to individual cases or judgments, a process which inevitably involves interpretation and which produces a body of case history or precedent that can be referred to in the future. Taken on its own, an individual article in a bill of rights may be normatively opaque, but the explicit wording of that article is only the tip of a much larger legal iceberg, so to speak. With time, the background history of judicial attempts to apply, interpret, and specify the requirements of the article constitutes a vast body of legal and practical information (which includes moral reasoning about individual cases) that lend the article implicit specificity. Of course, the case histories of a given right may not add up to anything very coherent. Previous interpretations of a right may conflict, be mistaken, or suffer from internal contradictions, and (whenever necessary) it is open to judges and legislators to break away from past interpretations of a right.37 Nevertheless, case histories do give some practical shape and content to human rights as legally embodied norms. One of the difficulties of looking to case history as a source of content for human rights is that there is no single legislative and judicial body in charge of drafting, interpreting, and applying such rights. At the international level, one finds the International Criminal Court (ICC) as well as various United Nations Committees tasked with the jobs of interpreting and hearing 35 See the reference in fn. 26 above. There is, furthermore, no principled reason that I can think of for denying that claimability, if it is a necessary feature of rights at all, is a necessary feature of both moral and legal rights (at least so long as we are talking about 'claim-rights' and not some other Hohfeldian category); and so in this very direct sense claimability is as much a legal concern as it is a moral one. 36 See Gerald L. Neuman, 'Human Rights and Constitutional Rights: Harmony and Dissonance' in Stanford Law Review, Vol. 55, No. 5 (May, 2003), pp. 1868-1869. 37 For an instructive overview of the complex role of precedent in international law (including international human rights law), see Gilbert Guillaume, 'The Use of Precedent by International Judges and Arbitrators' in Journal of International Dispute Settlement, Vol. 2, No. 1 (2011), pp. 5-23. individual complaints that fall under the purview of the International Bill of Human Rights (comprised of: the ICCPR, the ICESCR, and the non-binding UDHR). But such international tribunals are designed to complement existing national and regional judicial systems as backups, so to speak. That is, aside from monitoring state behaviour and unilaterally interpreting the content of human rights provisions, such tribunals generally do not hear individual complaints unless national or regional courts are unwilling or unable to do so. Alongside these international schemes, then, one has parallel regional schemes governed by regional conventions, e.g. the European Convention on Human Rights (ECHR), which is binding for all members of the Council of Europe and is interpreted and monitored by the European Court of Human Rights. And human rights frequently have independent presence at the domestic level in the form of constitutional rights. All of this makes the case history of any particular human right a complicated affair since, in effect, a right may have parallel but disjointed legal formulations and histories. So, for instance, the right to freedom of expression has famously been given a far more demanding interpretation in the First Amendment law of the United States than it has in, say, the ICCPR, where that right has been limited by explicit prohibitions on hate speech. Or, to take another example, the Constitutional Court of South Africa has declined to adopt a UN-Committee's recommended 'minimum core' interpretation of the rights to adequate housing and healthcare proclaimed under the South African constitution, which would require that the court decide on a clear and enforceable (minimum) standard of adequacy. The court instead opted for a progressive interpretation of those rights, reviewing only the 'reasonableness' of the government's steps towards the progressive realization of the rights under the current circumstances.38 Under such fragmented circumstances, it makes more sense to talk of plural legal specifications of human rights norms – with no guarantee of convergence or even of efforts towards convergence – rather than a sustained and coordinated legal effort to interpret human rights. Still, even if legal specifications of human rights may vary according to regional and domestic jurisdiction, one can still say that, in any given jurisdiction, a claimant invoking some abstract human right can appeal to (at least) its local case history in order to lend practical determinacy to their claim. And this is of course what's important for addressing O'Neill's critique. A second way in which the legal culture of human rights contributes to their specificity is through the drafting and signing of special conventions, e.g. on the rights of the child, the rights of persons with disabilities, against racial or gender discrimination, etc. These subject-specific treaties provide crucial supplements to the main instruments of human rights law. They do this firstly by addressing special issues that core human rights have left unaddressed. This sort of supplementation is most evident in the case of the Convention on the Rights of Disabled Persons, for instance, which establishes a set of rights that address the specific needs and circumstances of disabled persons, something that the core instruments make no attempt to do. Secondly, a special convention may explore a topic that is only superficially addressed by the core conventions and declarations. So, for instance, the ICCPR proclaims a human right not to 'be subjected to torture or to cruel, inhuman or degrading treatment or punishment.' But definitions of torture, cruelty, inhumanity, and degradation are lacking here. And so are clear guidelines regarding the steps to be taken in order to enforce and promote such a right. The United Nations Convention Against Torture explicitly addresses both outstanding issues in far greater detail. At the very least, this brief legal excursus should make us hesitate before imputing the modern practice of human rights with any general disregard for detailed questions of content, 38 See Neuman (2003), p. 1878. priority, and obligation. It's clear that the formidably diverse international network of human rights law is at the vanguard of efforts to make human rights meaningful, determinate, claimable, and ultimately enforceable norms. This raises an additional point about the complementarity of philosophical and legal efforts to articulate and understand human rights. Philosophers often approach human rights law with a condescension buttressed by a logical distinction between what has been or is legally recognized as a human right and what ought to be recognized as such.39 It is important that we draw such a distinction, and it is healthy to aspire towards conceptual rigour, unity, and coherence in our proclamations of human rights. But philosophers ought to guard against the excessive condescension that these distinctions and aspirations can naturally arouse. One reason for this is that, despite occasional squabbles with the content of the law, in practice philosophers tend to work up their theories of human rights in close consultation with that content.40 Another is that there is little reason to think that the forms of practical reasoning employed by the legal authors of human rights are very different from those employed under more philosophically rigorous auspices. This is especially true given that the lawyers, scholars, judges, activists, and politicians behind the design and implementation of human rights law naturally tend to see themselves as giving legal expression to norms that have independent moral validity.41 Given all this, it is unreasonable to view the authors, practitioners, and interpreters of human rights law as engaged in a task hopelessly detached from philosophy. Rather, it seems more accurate to see them as taking on the lion's share of the moral-practical-philosophical burden: giving abstract but practicable expression to human rights in conditions of modernity, specifying them where necessary, and applying them to individual cases in ways that philosophical theory (later) plumbs for inspiration.42 Although these observations do help to absolve human rights culture of any blanket charge of cynical, naïve, or lazy abstraction, they can also play into O'Neill's hands. This is because the legal specification of human rights, at least on one understanding, is a matter of institutionalization. That is, positivization through law can be understood as a way of specifying various aspects of a right in accordance with an extant institutional scheme. Or, we might understand the legal expression of human rights as part of the creation of an institutional scheme (e.g. a treaty system, a healthcare system) that itself gives determinate and claimable shape to human rights. Human rights law may sometimes be institutional in the first sense and sometimes in the second, depending on the case. Yet all of this is entirely compatible with O'Neill's thesis 39 See, e.g. James Griffin, 'Discrepancies Between the Best Philosophical Account of Human Rights and the International Law of Human Rights' in Proceedings of the Aristotelian Society, Vol. 101, No. 1 (2001), pp. 1-28. 40 See e.g. Griffin's 'bottom-up' approach to human rights (Griffin 2008, pp. 29-30) Tasioulas' concern with 'fidelity' to the post-1948 culture of human rights (John Tasioulas, 'Are Human Rights Essentially Triggers for Intervention?' in Philosophy Compass, Vol. 4, No. 6 (2009), pp. 938-950), and Joseph Raz and Charles Beitz's explicit theoretical focus on the modern political 'practice' of human rights [e.g. Joseph Raz, 'Human Rights without Foundations' in The Philosophy of International Law (Oxford, 2010), eds. J. Tasioulas & S. Besson, esp. pp. 323-324, 334-337; Charles Beitz, The Idea of Human Rights (Oxford, 2009), pp. 7-13]. 41 See Neuman (2003), pp. 1868-1869. 42 Note I am not here suggesting that human rights are norms that are necessarily destined for embodiment in law. For a version of this kind of claim, see Jürgen Habermas, 'The Concept of Human Dignity and the Realistic Utopia of Human Rights' in Metaphilosophy, Vol. 41, No. 4 (2009), pp. 469470. that, before legal or institutional specification takes place, some (or perhaps very many) human rights will remain abstract to the point of non-claimability and thus non-existence. This only bolsters her claim that the law does not affirm such rights, but rather creates rights where none previously existed. As a form of institutionalization, the legal specification of human rights does not of itself guarantee their independent, moral specificity. In order to adequately address O'Neill's concerns, then, we need to grapple directly with the question of whether claimability is a necessary feature of rights at all. IV. MUST RIGHTS BE CLAIMABLE? Among the most elemental aspects of a right has to be its nature as a valid claim or entitlement against others. That is what's implied by the thesis of logical correlativity after all. And it's not adding much to say that any such claim (noun) should also be one that can be meaningfully and justifiably claimed (verb) against its addressees.43 But there are different ways of interpreting the strength of this naturally consecutive thought, and we ought to carefully consider which one we mean to adopt. On one plausible interpretation, what we mean is the very thesis that we have been considering up to now: that claimability – and thus, the specificity of a right's holder(s), duty-bearer(s), content, and priority – is a conceptually necessary feature of rights. However, on another, equally eligible interpretation, what we mean is simply that claimability is an important and valuable feature of rights, but not a necessary one. Both of these interpretations present us with ways of building on the central motivating observation of O'Neill's critique: that a right consists in a valid claim against certain agents, and that its holders (or some third party) should be able to claim it against them. But the availability of the second interpretation is significant because it allows us to remain true to that observation without also saddling us with an unnaturally demanding conception of rights. Ordinarily understood, the existence or validity of a claim can be established in the absence of specific information about its addressees and content. For instance, consider how naturally we recognize the claim of a newborn infant to basic nourishing care and support; and this, without any specific information about who ought to deliver such care (e.g. are the parents of the child available and competent, or will these obligations fall on others?) and what concretely ought to be done. Such claims seem worthy of recognition on account of roughly three factors: (a) the existence of a sufficiently important need, status, interest, or vulnerability; (b) the feasibility of its satisfaction by the action and/or omission of some competent and available agent(s); and (c) the reasonableness of demanding the required action and/or omission from the agent(s) in question.44 Since these factors can be established even if all we know about (b) and (c) is that some worldly agent(s) is/are capable of fulfilling the relevant need or interest by some reasonable means, the question then becomes why we ought to abandon these intuitive criteria in favour of O'Neill's more stringent set. The answer is, upon reflection, not at all clear, especially when we consider that the difficulty of giving detailed or specific shape to a claim is usually an indication that there is simply more than one reasonable and effective way to fulfill it.45 Moreover, given that parallel forms of indeterminacy are expected and philosophically 43 See Feinberg (1973), pp. 64-67. 44 Shue (1996b), e.g. p. 165 rightly emphasizes the importance of this third set of considerations in establishing the validity of any right or claim. 45 Tasioulas (2008), p. 94. Also see Neil MacCormick, Legal Right and Social Democracy: Essays in Legal and Political Philosophy (Oxford, 1982), pp. 161-164, for a similar line of reasoning. tolerated in other areas of moral and legal reasoning – e.g. about the moral virtues, about the allthings-considered 'right thing to do', about the meaning of legislative statutes, etc.46 – we might ask why we shouldn't also expect and tolerate indeterminacy when we reason about claims or rights. The fact that O'Neill does not offer us any salient philosophical guidance at this critical juncture, beyond that of simply pointing us back to the original observation that rights are a kind of claim against others, is a serious weakness of her account, especially since her uncompromising endorsement of claimability imposes disturbing limits on the ascription of rights. Consider, for example, the case of the Murle of South Sudan, a people starved by drought and who live under a deeply fragile state. Since domestic remedies are (let us assume) unavailable, it is hard to say precisely who owes such people food relief. Nevertheless, given the patent feasibility of remedial action as well as the eminent reasonableness of holding at least some existing agent(s) bound to undertake it, it seems clear that help is rightfully owed to the Murle and that they would be wronged if such help were not somehow provided to them (we can even provide a rough account of who these agents are: affluent individuals and nations for whom providing life-sustaining aid would require comparatively minimal sacrifice, relief-oriented NGO's, international aid agencies, etc.).47 But these simple and intuitive conclusions are not available to us on O'Neill's account. On her view, a failure to provide aid to the Murle may constitute an injustice of some sort, perhaps even a deeply grave one, but not an injustice done unto them, i.e. one that can justify demands for further remedial action and that licenses special feelings of resentment and blame on their part. This ought to weigh heavily on us when we consider the merits of O'Neill's uncompromising position on the importance of claimability in rights claims. Particularly so in light of the fact there are, as mentioned, other ways of acknowledging claimability's importance that do not have an equally counter-intuitive effect. It is plainly open to O'Neill (and others) to straightforwardly deny the intuitive character of, say, the Murle's right to aid or a newborn child's right to nourishing care and support (at least in cases where confusion may exist about the identity of the relevant caretaker, and/or some other detail). Or she may reject the need to theoretically accommodate such intuitions in the first place, opting instead to correct, wherever necessary, our judgments about such matters. Given the already-noted lack of a deeper theoretical argument in favour of her strict position on claimability, however, the second strategy of reply isn't going to work; in the absence of such an argument, most (if not all) of what we have to work with to decide the issue at hand are our intuitions themselves. This leaves O'Neill with the first strategy. But if that is her method of reply, then it is fully open to each of us to consult our own intuitions and decide the matter accordingly. My suspicion is that most will not find themselves at ease in, for example, denying the Murle a valid claim or right to aid, and so the prospect of acknowledging the importance of claimability in a way that does not force them to do so will hold greater theoretical allure. What, then, would it mean to adopt the less stringent understanding of the importance of claimability that was outlined at the beginning of this section? At the very least, it would involve acknowledging the preferability of knowing just what demands are generated by a right as well as just who is bound by them, among other relevant details. As emphasized in the introduction, 46 Consider, for example, the widespread enthusiasm that has greeted Rawls' notion of the 'burdens of judgment,' according to which 'all our concepts... are vague and subject to hard cases.' John Rawls, Political Liberalism (New York, 1996), p. 56. 47 Shue's discussion of how to allocate duties of assistance is highly instructive in this regard. See Shue (1996b), Ch. 5. our understanding of a right is always imperfect and incomplete, and its meaning and practical relevance always diminished, until we possess a thorough comprehension of its content, weight or priority, holders, and duty-bearers, and of how these elements can change depending on the circumstances of application. This sort of comprehension is never there from the start, however. It is a product of continuous work and effort, much of which is taken on, as I stressed above in Section III, by legal and political institutions.48 Given the obvious importance of such work, moreover, it seems reasonable to affirm not just its value but also its obligatory nature. That is, on the sort of view we are contemplating here, abstract human rights can reasonably be seen as imposing widespread duties to help establish institutions that will undertake the arduous work of specifying, applying, and rendering them claimable.49 That the second interpretation allows us to affirm all of this without also forcing us to adopt an unnaturally demanding conception of rights is what accounts for its special dialectical appeal.50 With this plausible interpretation of the value of claimability in hand, the concerns raised by O'Neill's critique can be allayed.51 Not only can we make good philosophical sense of abstract rights that transcend and, indeed, motivate their own institutional implementation, we can also absolve human rights culture of any charge of naïveté, mockery, or cynicism that is premised on its heavy deployment of such rights. Moreover, lingering concerns about the realizability (if not falsity) of abstract rights can at least be contained by the plausible thought, noted above, that such rights impose widespread duties to help establish institutions that will perform the crucial work of determining their specific requirements. Now that all this has been established, I want to conclude by filling in the story from the other side, so to speak. That is, by illustrating some of the crucial functions of abstraction in rights discourse below, I hope to show that abstract rights are not only theoretically coherent but also useful and important. V. THE USES OF ABSTRACTION Human rights instruments may be replete with references to abstract rights that leave a host of important practical questions unanswered, but there are some ready explanations for these informational gaps. One is that the drafters of international declarations like the UDHR are under special pressure to produce documents that will be accessible to a wide audience that includes laypersons from a variety of cultural, linguistic, and national backgrounds. This means that a 48 This isn't to say that institutions are the sole means of specifying rights. For instance, Shue (1996b, pp. 111-131) is able to say many plausible things about what the affluent owe to the global poor without referring to any given institutional framework. 49 See Ashford (2008), p. 215 & Pablo Gilabert, 'Humanist and Political Perspectives on Human Rights' in Political Theory, Vol. 39, No. 4 (2011), p. 457, who make a similar point. 50 There is a third interpretation that, I should mention, also has this virtue. According to it, claimability is indeed a necessary feature of rights. However, it can be achieved with as little as a rough idea of a right's holder(s), duty-bearer(s), content, and priority. This corresponds, broadly, to the (weaker) sort of claimability requirement that we find endorsed by Nickel (2006), pp. 30-32, Griffin (2008), pp. 107-110, and Ashford (2008), pp. 215-217. But since it's unclear whether there is anything of real significance riding on the preservation of the term 'claimability' here, it's unclear how it differs from my own position, which is similarly tolerant of indeterminacy. I shall therefore leave this possibility aside. 51 The fact that the preceding observations are independently reinforced by the interest theory of rights – according to which the existence of a right can be known before one has answered the question of exactly which agents are bound by it and what is required of them [See Raz (1986), pp. 184-185; Tasioulas (2008), pp. 92-93] – speaks in favour of both. premium is placed on brevity and simplicity, rather than exhaustive treatment in the form of a huge multiplication of norms. 52 Another important factor is the nature of human rights declarations, covenants, and treaties as objects of international agreement. Since it is typically international committees that draft the human rights declarations and treaties adopted by global governance institutions such as the United Nations (UN), the content of such documents is inevitably constrained by the limits of what those committees are able to agree upon. And, of course, such constraints are especially intense when a drafting committee is made up of religiously, culturally, and politically diverse delegates, as is often the case. For instance, during the drafting of the UDHR, the Soviet delegation proposed an alternative formulation of Article 22 that was more explicit about the material and legislative duties of states with respect to the socio-economic rights of their citizens. But the Soviet formulation was shot down by a number of delegations (including that of the United States) that were ideologically uncomfortable with the idea of taking on clear and demanding obligations vis-à-vis socio-economic rights.53 The result is that we are now left with an Article 22 that is more opaque about state responsibilities than it might have been. And one can easily imagine how similar disagreements likely lead to the watering down of other proposed articles. A third factor favouring this sort of reticence stems from the intended universal scope of the human rights proclaimed in international declarations, treaties, and covenants. This is because the pressure to formulate a right in abstract or non-specific terms mounts in proportion to the range of contexts and cases it is intended to cover. If we look carefully at the human rights articulated even in a document as celebrated as the Universal Declaration, it quickly becomes apparent that many of its proclaimed rights are too specific to be universal. For instance, the right to work and to join trade unions clearly presuppose a modern market-based economy.54 This makes such rights inappropriate in the context of, say, remote hunter-gatherer societies (rare as these may be) that don't participate in any such economic system. The right to 'periodic and genuine elections'55 is equally likely to be inappropriate in some contexts. Without any detailed knowledge of how such societies function politically, for instance, the members of tribes buried deep in the Amazon jungle cannot necessarily be said to have a right to vote.56 Nor is it plausible to assign all persons a right to the institution of marriage understood as a union between spouses, i.e. 'husbands' and 'wives.'57 Among other peoples, the Mosuo of China famously do not practice marriage in any such conventional sense, favouring instead a custom of informal sexual visitations across matrilineal households.58 52 See Johannes Morsink, The Universal Declaration of Human Rights: Origins, Drafting, and Intent (Philadelphia, 2001), pp. 33-35. There are of course other reasons, like efficiency, for favoring generality and succinctness in the law. For instance, Jeremy Bentham famously complained of the senseless 'multitude' of laws prohibiting the stealing of vegetables in Georgian England, where instead of having generic statutes, one had 'one law for one sort of vegetable, another for another.' See 'Letter V' in The Collected Works of Jeremy Bentham: On the Liberty of the Press, and Public Discussion, and other Legal and Political Writings for Spain and Portugal (Oxford, 2012), eds. Catherine Pease-Watkin and Philip Schofield, p. 183. 53 See Morsink (2001), pp. 228-230. 54 Article 23, UDHR. 55 Article 21, UDHR. 56 This is, of course, providing we see such peoples as truly isolated and not, say, as citizens of Brazil. 57 Article 16, UDHR. 58 One question that is raised by these examples is that of what we are to make of these purported human rights, if they are not in fact universal. Human rights are, after all, supposed to be rights possessed Of course, the mere fact that some prescribed practice (e.g. elections, a market-based economy, etc.) has not taken hold in a given community is not in itself a reason to deny that it ought to. The ostensible non-universality of the rights described above seems to be due to at least two possible factors: (a) their fulfillment is drastically infeasible in the specified context; and (b) the values, interests, or considerations (e.g. in reproduction, intimacy, kinship, social standing, etc.) served by such rights seem adequately servable by other less conventional means that have already taken root in that context. Either way, the result is that a right can only attain full universality if it is formulated in such a way as to remain silent with respect to concrete questions of societal circumstance, deontic prescription, and institutional implementation, e.g. questions about what form of marriage one is entitled to, what type of economy one participates in, what kind of say one should have in one's government, etc. As a way of not pronouncing on such matters, then, abstraction is vital to the universal reach of rights. It allows us to avoid saddling rights with content that would render them inappropriate in certain contexts, objectionably partisan, and that would anchor them in a particular moral and institutional culture. And it also allows us to formulate rights in an open-ended manner that enables their practical content and implications to vary dynamically from context to context.59 All of this is surely important to addressing concerns about the ethnocentrism or 'Western' bias of human rights. Still, a reliance on abstraction also presents challenges of its own. Because it is a way of, essentially, not commenting on the nitty-gritty practical implications of a right, it leaves us with a great deal to sort out when the time comes to actually apply an abstract right to some specific case or range thereof. For any such application, an innumerable array of normative and empirical factors will have to be taken into account. And this raises the question of who can authoritatively undertake such work.60 In particular, I want to focus here on the political and legal side of this question, i.e. which institutions are best placed to determine the specific requirements of abstract human rights? I've already said a good deal (in Section IV) about the centrality of legal efforts to specify human rights, but this still leaves important jurisdictional questions wide open. So, for instance, should the authority to interpret abstract human rights rest in all cases with international legal institutions, or should regional and domestic legal institutions be able to decide for themselves how to implement and interpret such rights within their local jurisdiction? In deciding this question a number of epistemic considerations naturally come into view. In order to competently interpret the locally applicable content of a human right (e.g. the human right to adequate food), an interpreter must at least have (a) a basic grasp of the interests (or by all human beings, irrespective of nation, race, creed, tribe, or culture. One plausible solution is to accept the possibility of two varieties of human rights. On the one hand, there are abstract and consummately universal human rights: rights that apply to all individuals in all cases, even though their deontic implications are protean and may vary (e.g. the human right to political participation). And, on the other hand, there are derivative or specific human rights: rights that are generated by the application of a universal human right to a specific (cultural and/or institutional) context, taking into account the special features of that context, but that are not themselves universal in scope (e.g. the human right to periodic and genuine elections). According to this sort of picture, many of the rights posited by international and domestic law are indeed human rights, although they are so in light of being broadly applicable specifications of universal rights, not in virtue of being universal human rights themselves. For a similar view, see Gilabert (2011), p. 445; Griffin (2008), Sec. 2.8. 59 This is what Raz and Tasioulas refer to as the 'dynamicity of rights'. See Tasioulas (2008), pp. 9298, esp. p. 94; Raz (1986), pp. 185-186. 60 See Diane F. Orentlicher, 'Relativism and Religion' in Human Rights as Politics and Idolatry (Princeton, 2001), p. 144; Joseph Raz, 'Human Rights in the New World Order' (DRAFT), Sec. 3. other normative considerations) that ground the right, (b) a good sense of their moral weight or priority, and (c) a solid understanding of the relevant facts on the ground, e.g. facts about who is systematically deprived of the good protected by the right (and why), what local resources can be allocated towards its realization (and under what constraints), who can be fairly and feasibly saddled with duties to distribute those resources, and what social traditions, practices, and institutions are locally in play. If regional and domestic legal institutions are both responsible and cognitively competent (and I grant that this is a big if), primary interpretative authority would seem best placed in their hands, since local agents are likely to know the most about – as well as show the most sensitivity towards – (c) the relevant facts on the ground.61 The fact that, for obvious reasons, granting such interpretive license to adequate domestic institutions comports well with the ideal of national self-determination can only serve to reinforce this conclusion. And so the parallel but disjointed (i.e. national, regional, and international) legal specification of human rights, which we earlier lamented for its daunting complexity, now looks more like something to be celebrated on both moral and epistemic grounds. Moreover, that abstract rights are, in their dynamic open-endedness, uniquely suited to serve as the focal points of such an interpretive division of labour is one of their central advantages. This, then, constitutes a fourth function of abstraction: its facilitation of a delegative system of human rights interpretation, one that embraces something like the European Union's legal notion of a 'margin of appreciation,' according to which member states can be granted some interpretive leeway in their definition and implementation of human rights. This also completes the optimistic picture of abstract rights that I have been trying to defend in general. Not only are abstract rights philosophically coherent, they render the language of rights manageable, conciliatory, and serve as a conduit for its universalist aspirations. Last but not least, abstraction can help disarm worries about the culturally insensitive character of human rights, since it holds open the possibility that a single set of human rights (like the UDHR) can be implemented in contextually diverse ways and its local implications fleshed out by those best placed to understand them: often, local agents themselves.62 [email protected] 61 This parallels the justification for the allowance of a 'margin of appreciation' by the European Court of Human Rights in national interpretations of the ECHR among EU members. See Neuman (2003), pp. 1881-1886; Also Burleigh Wilkins, 'International Human Rights and National Discretion' in The Journal of Ethics, No. 6 (2002), pp. 373-382. 62 In writing this article, I benefited greatly from the constructive feedback of Elizabeth Ashford, Kimberley Brownlee, Roger Crisp, Pablo Gilabert, Peter Jones, Joshua Keton, James W. Nickel, Onora O'Neill, Adina Preda, Mauro Rossi, Jeremy Waldron, Daniel Weinstock, two anonymous reviewers at Utilitas, and editor Brad Hooker.

Wissenschaft ohne Wahrheit und Erkenntnis Das Problem epistemischer Verantwortung am Beispiel empirieferner Computersimulationen Eckhart Arnold erschienen in: Florian Steger / Rafaela Hillerbrand (Hrsg.): Praxisfelder angewandter Ethik. Ethische Orientierung in Medizin, Politik, Technik und Wirtschaft, mentis Verlag, Münster 2013, S. 309-326. Zusammenfassung Title: Science without truth and knowledge. The problem of epistemic responsibility as examplified with the case of empirically intractable computer simulations Abstract: Epistemic Responsibility means that scientists are responsible for their research being suitable to contribute to our understanding of the world, or at least some part of the world. As will be shown with the example of computer simulations in social sciences, this is unfortunately far from being understood as a matter of course. Rather, there exist whole research traditions in which the bulk of the contributions is quite free from any tangible purpose of enhancing our knowledge about anything. This essay is concerned with the causes of this phenomenon and pleads for taking epistemic responsibility as a scientific virtue serious. Science should be organized in such a way that it is possible and likely that scientists will assume epistemic responsibility for their research tasks. Zusammenfassung: Unter "Epistemischer Verantwortung" wird in diesem Artikel verstanden, dass Wissenschaftlerinnen1 dafür verantwortlich sind, dass ihre Forschung geeignet ist zur Erkenntnis der Welt oder eines Ausschnittes der Welt beizutragen. Wie am Beispiel sozialwissenschaftlicher Computersimulationen dargelegt wird, ist dies keineswegs so selbstverständlich, wie man meinen möchte. Vielmehr gibt es ganze Forschungstraditionen, innerhalb derer ein Grossteil der Beiträge frei von irgendeinem greifbaren Erkenntniszweck ist. Der Aufsatz fragt nach den Ursachen des Phänomens und plädiert dafür, die epistemische Verantwortung als wissenschaftlichen Wert ernst zu nehmen und Wissenschaft so zu organisieren, dass es der einzelnen Wissenschaftlerin möglich ist, die epistemische Verantwortung für ihre Forschungsaufgaben zu übernehmen. 1 Einleitung 1Von Professor Florian Steger noch einmal auf die Wichtigkeit des Problems der sprachlichen Geschlechtergerechtigkeit aufmerksam gemacht, verwende ich in diesem Artikel ausschliesslich die weibliche Form. Die wahrscheinlich beste Lösung dieses Problems wäre es nämlich, wenn alle Männer ausschliesslich die weibliche Form verwenden würden und alle Frauen nur und allein die männliche. Auf diese Weise lassen sich komplizierte Umschreibungen ebenso vermeiden wie artifiziell wirkende Spracherweiterungen (wie z.B. das Signal-"I"). Zugleich trägt bei dieser Lösung die gerade im Vorteil befindliche Gruppe ganz von selbst zur Herstellung ausgeglichener Verhältnisse bei. Und obendrein ist es auch ein Zeichen schöner Ritterlichkeit, es so zu halten. 1 Als Max Weber im Jahre 1919 in einem berühmt gewordenen Vortrag vor einer Studentenversammlung über den „inneren Berufe zur Wissenschaft"2 sprach, klärte er seine Zuhörerinnen auch über ein „Sinnproblem" der „Wissenschaft als Beruf" auf, das unter den Bedingungen der arbeitsteilig verfahrenden und rasch fortschreitenden modernen Wissenschaft entsteht. Dieses Sinnproblem bestand für Max Weber darin, dass die Einzelne nur als Spezialistin noch wissenschaftlich Belangvolles leisten könne. Will sie nicht hoffnungslose Dilettantin bleiben, so muss sie sich damit zufrieden geben, auf einem Spezialgebiet innerhalb seines Fachgebietes ein Detailproblem voranzutreiben – mit niemals sicherer Aussicht auf Erfolg und immer in dem Bewusstsein, dass sowohl die gefundene Lösung als auch das untersuchte Problem bald durch den Gang der Wissenschaft überholt sind und mitsamt der Wissenschaftlerin, die sie erforscht hat, in Vergessenheit geraten. Um einen Sinn in ihrem Tun finden zu können, müssen die Wissenschaftlerinnen also bereit sein, sich auf diese Bedingungen einzulassen und „rein der Sache" dienend eine Arbeit zu leisten, von der sie wissen, dass sie „in 10, 20, 50 Jahren veraltet ist."3 Denn nur unter dieser Bedingung ist fortschreitende wissenschaftliche Welterkenntnis möglich. Was Max Weber dabei aber stillschweigend und völlig selbstverständlich voraussetzte, war, dass die Wissenschaft der Erkenntnis dient, und dass sie dabei auf der Suche nach Wahrheit, und zwar nach wohlgeprüfter und wissenschaftlich begründeter Wahrheit ist. Unter dieser Bedingung ist es dann trotz dieser Einschränkung möglich, dass – wie Weber es sah – Wissenschaft Leidenschaft ist. „Denn nichts ist für den Menschen als Menschen etwas wert, was er nicht mit Leidenschaft tun kann."4 Glückliche Zeiten, in denen solches Pathos noch möglich war! In der heutigen Wissenschaft kann man nämlich zuweilen auf ein „Sinnproblem" ganz anderer Art treffen, das darin besteht, dass es wissenschaftliche Vorhaben gibt, die gar nicht mehr auf irgendeinen Erkenntniszweck bezogen sind, oder die zwar vorgeblich darauf bezogen sind, aber von vorn herein nicht geeignet sind ihn zu erfüllen. Derartige Phänomene werfen das Problem der epistemischen Verantwortung auf. Unter „epistemischer Verantwortung" ist in diesem Zusammenhang die Verantwortung dafür zu verstehen, dass das eigene Forschungsvorhaben – und sei es auch nur der Teillösung eines irgendeines Detailproblems gewidmet – einen Beitrag zu einem wissenschaftlichen Erkenntniszweck leistet. Wie in diesem Essay am Beispiel empirieferner Computersimulationen in den Sozialwissenschaften dargelegt werden soll, gelingt es dem Wissenschaftsbetrieb offenbar nicht immer zuverlässig, die Erfüllung epistemischer Verantwortung sicher zu stellen. Insbesondere ist es unter der Bedingung der „ »Trennung des Arbeiters von den Produktionsmitteln« "5 gerade den niederen Chargen des Wissenschaftsbetriebs, die sich ihre Forschungsaufgaben häufig nicht einmal selbst aussuchen können, sondern sie zugewiesen bekommen, oft gar nicht möglich, die epistemische Verantwortung für ihre Forschungstätigkeit in vollem Umfang zu übernehmen. Im Folgenden soll erläutert werden, was es mit der „epistemischen Verantwortung" auf sich hat, und wie es dazu kommt, dass sie vernachlässigt wird. Dazu wird zunächst der Begriff der epistemischen Verantwortung definiert und wissenschaftstheoretisch motiviert werden. Es wird die These vertreten, dass epistemische Verantwortung sich nicht schon im geregelten wissenschaftlichen Betrieb von allein ergibt, sondern als Wert 2 Weber, Wissenschaft als Beruf, S. 588. 3 Weber, Wissenschaft als Beruf, S. 592. 4 Weber, Wissenschaft als Beruf, S. 589. 5 Weber, Wissenschaft als Beruf, S. 584. 2 der wissenschaftlichen Tätigkeit zu Grunde liegen muss. Um zu zeigen, dass diesem Prinzip nicht immer Genüge geleistet wird, sollen einige Beispiele aus dem Bereich der Computersimulationen sozialen Verhaltens herangezogen werden, also aus einem Bereich, in dem die Wissenschaftlerinnen sich nicht wenig auf die Anlehnung an naturwissenschaftliche Methoden und damit eine vermeintlich besonders strenge Wissenschaftlichkeit zu Gute halten. Ich zeige im Folgenden, dass ein nicht unbeträchtlicher Teil der Computersimulationsforschung in den Sozialwissenschaften das Prinzip der epistemischen Verantwortlichkeit verletzt, indem er, ohne dass es die beteiligten Wissenschaftlerinnen besonders zu kümmern scheint, weder direkt noch indirekt in nachprüfbaren Erkenntnissen über den Untersuchungsgegenstand resultiert. Schliesslich wird gefragt werden, welche strukturellen und inhaltlichen Bedingungen die Übernahme epistemischer Verantwortung fördern oder behindern können. Und es werden einige Überlegungen dazu angestellt, wie epistemische Verantwortung ermöglicht bzw. begünstigt werden kann. Eine zentrale Forderung besteht darin, die individuelle Zuschreibung bzw. Zuschreibbarkeit von Forschungsleistungen sicherzustellen. Zugegebenermassen steht diese Forderung in einem Spannungsverhältnis zu den Bedingungen arbeitsteiliger Wissenschaft. 2 Der Begriff der epistemischen Verantwortung Unter „epistemischer Verantwortung" im wissenschaftlichen Kontext verstehe ich die Verantwortung einer Wissenschaftlerin dafür, dass die eigene Forschungsaufgabe einem wissenschaftlichen Erkenntniszweck dient. Als wissenschaftlicher Erkenntniszweck gilt dabei alles, was zur Erkenntnis eines als relevant angesehenen empirischen Untersuchungsgegenstandes dergestalt beiträgt, dass man die Richtigkeit oder Falschheit der gewonnen Erkenntnisse überprüfen kann. Ausserhalb der Betrachtung bleiben hierbei nicht-empirische Wissenschaften wie die Mathematik oder die Jurisprudenz, auf welche die gegebene Definition auch nicht anwendbar wäre. Um Missverständnisse zu vermeiden, sei darauf hingewiesen, dass der Begriff der "epistemic responsibility" in der Erkenntnistheorie in einem etwas anderen Sinn verstanden wird, nämlich als die Geneigtheit eines Subjekts etwas zu glauben, sofern es nur das glauben möchte, was wahr ist.6 Mir geht es hier aber nicht um Fragen der Erkenntnistheorie und damit der theoretischen Philosophie, sondern um eine Frage der praktischen Wissenschaftsethik. In diesem Zusammenhang ist es weniger von Bedeutung nach welchen Kriterien eine Überzeugung als gerechtfertigt und damit als "Wissen" betrachtet werden kann, als vielmehr von welchen Normen die Wissenschaft geleitet sein muss, damit die Wissenschaftlerinnen wohlbegründetes und relevantes Wissen hervorbringen. Das Problem der Relevanz soll an dieser Stelle allerdings nicht weiter vertieft werden. Fragen der Relevanz beinhalten immer auch Wertfragen, die noch einmal Schwierigkeiten ganz eigener Art aufwerfen. Eine gutwillige Anwendung der Definition epistemischer Verantwortung setzt voraus, dass man ihre Zielsetzung nicht durch freie Auslegung des Relevanzkriteriums aushebelt. Unter den Bedingungen arbeitsteiliger Wissenschaft stellt sich das zusätzliche Problem, dass die Tätigkeit einer einzelnen Wissenschaftlerin isoliert betrachtet noch überhaupt keine Erkenntnisse über irgend einen Untersuchungsgegenstand zu Tage fördern muss. Beispielsweise kann es vorkommen, dass die Überlegungen einer Theoretikerin erst im 6 Vgl. Greco und Turri, Virtue Epistemology. 3 Zusammenhang mit entsprechenden Experimenten, die andere sich ausdenken und durchführen müssen, zur Erkenntnis eines empirischen Gegenstandes beitragen. Erst recht gilt dies, wo Wissenschaftlerinnen sich rein heuristischer Methoden bedienen. Allerdings wird man auch in diesem Fall billigerweise fordern dürfen, dass dergleichen irgendwann einmal zur Formulierung einer Theorie oder eines Modells dient, dass dann empirisch prüfbar ist. Nun können zwischen der Formulierung einer Theorie und ihrer empirischen Überprüfung oft Jahre liegen, ja, es mag bei der Ausarbeitung einer Theorie noch gar nicht absehbar sein, wie sie empirisch geprüft werden kann. Aus diesem Grund ist es notwendig, das oben gegebene Kriterium zu verfeinern: Die Durchführung eines wissenschaftlichen Vorhabens ist dann epistemisch verantwortlich, wenn es geeignet ist, unter Voraussetzungen deren (künftige) Gegebenheit denkbar ist, zum Erkenntniszweck der Wissenschaft beizutragen. Die Formulierung ist bewusst vage gehalten, denn die Erfüllung des Kriteriums, dass die künftige Gegebenheit der weiteren, zum erfolgreichen Abschluss des Teilprojekts hinzukommenden Voraussetzungen für einen nachweisbaren Erkenntnisgewinn denkbar sein muss, lässt sich nur von Fall zu Fall bestimmen und bleibt auch dann noch in hohem Masse Einschätzungssache. Es gibt aber eine Reihe von Indizien, die im Einzelfall die Vermutung begründen können, dass eine bestimmte Methode auch in Zukunft zu keinerlei empirisch überprüfbarer Erkenntnis irgendeiner Art führen wird. Dazu gehören etwa ein lang anhaltendes Scheitern bei dem Versuch dies zu tun oder die Verwendung nicht messbarer Parameter wie z.B. die bei vielen ökonomischen Modellen vorausgesetzten kardinalen Nutzenwerte. Manchmal kommt es auch vor, dass in Folge der anhaltenden Erfolglosigkeit eines Forschungsansatzes der Erkenntnisanspruch schrittweise aufgegeben wird, die Forschung aber nichts desto trotz ungestört weiter läuft.7 Es gibt einen extremen Standpunkt, der sich auf eine relativistische Auslegung des Wissenschaftstheoretikers Paul Feyerabend8 stützt, und unter dem Schlagwort anything goes schlechthin jede wissenschaftliche Methode zulassen will. Denn, wie Feyerabend (vermutlich in ironischer Absicht) argumentiert, selbst aus dem abstrusesten Aberglauben können sich im Laufe der Geschichte bedeutende wissenschaftliche Theorien entwickeln.9 Dennoch, ungeachtet seltener historischer Glücksfälle ist es sicherlich ein schlechtes Forschungsprogramm, ohne eine halbwegs realistische Zielvorstellung irgendwelche Methoden auf gut Glück durchzuspielen. Epistemische Verantwortung fordert, dass die Wissenschaftlerinnen sich dafür verantwortlich fühlen, dass die eigene Forschung zum Erkenntniszweck der Wissenschaft beiträgt, und das dies nicht dem Glück und dem Zufall überlassen wird. 3 Beispiel mangelnder epistemischer Verantwortung im Bereich „Soziale Simulationen" Computersimulationen werden erfolgreich in der Wissenschaft eingesetzt, seit es Computer gibt. Dabei sind Computer im Grunde nichts weiter als besonders komplizierte Rechenmaschinen und Computersimulationen damit auch nichts anderes als besonders komplizierte Rechnungen.10 Daraus ergibt sich, dass man all das 7 Vgl. Green und Shapiro, Pathologies of Rational Choice Theory. 8 Feyerabend, Wider den Methodenzwang. 9 Vgl. Feyerabend, Wider den Methodenzwang, S. 385ff. 10 Zur Geschichte von Computersimulationen vgl. Grammelsberger, Computer Experimente. Zum Begriff und epistemischen Status von Simulationen u.a. Humphreys, Extending Ourselves, 4 Simulieren kann, was man auch berechnen kann, und dass Simulationen dort eine Grenze haben, wo sich nichts sinnvoll berechnen lässt. Sind Berechnungen aber möglich, dann können Simulationen unser Wissen über Naturvorgänge erheblich bereichern. So ist es z.B. möglich Naturvorgänge zu simulieren, die experimentell nicht zu oder nur schwer zugänglich sind. Ein Beispiel ist die Simulation chemischer Reaktionen im Bereich der Biochemie.11 Nicht zuletzt die Tatsache, dass Simulationen (in bestimmten Fällen) als Ersatz oder Alternative von Experimenten eingesetzt werden können, motiviert dazu Simulationen auch in den Sozialwissenschaften einzusetzen, denn in den Sozialwissenschaften sind Experimente manchmal nur schwierig durchzuführen und gleichzeitig oft wenig aussagekräftig, weil die Experimentalsituation sich von der Realweltsituation zu deutlich unterscheidet, als dass man zuverlässige Schlussfolgerungen ziehen könnte. Einer der Pioniere der sozialwissenschaftlichen Simulationen motiviert den Ansatz folgendermassen: Die experimentelle Literatur ist wenig hilfreich, weil sie praktisch vollständig auf der Analyse der Entscheidungen von Spielern beruht, die erstmals mit dem Spiel [dem sog. „Gefangenendilemma-Spiel", E.A.] in seiner formalen Fassung konfrontiert werden. Obwohl die Versuchspersonen möglicherweise über viel Erfahrung mit alltäglichen Vorkommnissen des Gefangenendilemmas verfügen, können ihre Fähigkeiten begrenzt sein, diese Erfahrungen in einem formalen Rahmen zu nutzen. [...] Um mehr über das iterierte Gefangenendilemma zu erfahren, ist ein neuer Ansatz erforderlich.12 Nun stimmt es zwar, dass es möglich ist, in einer Computersimulation Situationen durchzuspielen, die man niemals in einem sozialen Experiment nachstellen könnte. Aber erstens sind Computersimulationen sozialen Verhaltens deshalb nicht weniger artifiziell und zweitens besteht gerade dann, wenn man die simulierte Situation nicht in einem Experiment nachstellen kann, auch keine Möglichkeit mehr unmittelbar zu überprüfen, ob die Simulation das soziale Verhalten, das sie simuliert, tatsächlich richtig simuliert. Wie untauglich Axelrods Simulationsansatz der „Evolution der Kooperation" dazu ist, uns tiefere Erkenntnisse über seinen Gegenstand, die Evolution der Kooperation, zu liefern, demonstriert er unfreiwillig selbst mit seinem Paradebeispiel, dem „Leben und Leben-lassen System" zwischen verfeindeten Soldaten an der festgefahrenen Westfront im ersten Weltkrieg.13 Rein äusserlich scheint dieses Beispiel den Befund der Simulation, dass wechselseitige Kooperation nach dem Muster „Wie Du mir so ich Dir" evolutionär stabil ist14, auf dramatische Weise zu bestätigen. Wagt man jedoch einmal den Blick hinter die Kulissen, und blättert die historische Studie auf, die Axelrods Darstellung Winsberg, Science in the Age of Computer Simulations. 11 Vgl. Senn und Thiel, QM/MM Methods for Biomolecular Systems. 12 Axelrod, The Evolution of Cooperation, S. 26/27. 13 Axelrod, The Evolution of Cooperation, S. 67-79. 14 Axelrod, The Evolution of Cooperation, S. 50ff. – Axelrod spricht wörtlich nicht von "evolutionärer" sondern von "kollektiver" Stabilität. Aber sein Begriff der "kollektiven Stabilität" ist nur ein Sonderfall einer ganzen Klasse von evolutionären Stabilitätsbegriffen. Insofern ist es hier verständlicher von evolutionärer Stabilität zu reden, da es auch bei Axelrods Begriff der "kollektiven Stabilität" um die Stabilität bestimmter Verhaltensweisen im evolutionären Prozess geht. 5 zugrunde liegt, so stellt man fest, dass darin die vollständige Erklärung des ungewöhnlichen Vorgangs stillschweigender Kooperation zwischen Feinden bereits gegeben ist, und dass das einzige, was Axelrod leistet, darin besteht, sie noch einmal in vergleichsweise undifferenzierter Form in der Terminologie seiner Simulationsstudien zu wiederholen.15 Ein Erkenntnisgewinn irgendeiner Art ist damit nicht verbunden. Man kann auch nicht behaupten – ein oft unterstellter Vorzug abstrakter Modelle in den Sozialwissenschaften –, dass es Axelrod gelungen wäre, eine verallgemeinerungsfähige Erklärung für den historischen Sachverhalt zu geben, denn tatsächlich kommen in Axelrods Simulationsmodell viele wesentliche Faktoren gar nicht oder nur in den Parameterwerten versteckt vor. Und man wird eine hoffnungslos inadäquate Erklärung wohl kaum ihrer vermeintlichen Verallgemeinerbarkeit wegen rühmen wollen. Inwiefern berührt dies die epistemische Verantwortung? Insofern als hier offenbar ein Forschungsansatz gewählt wurde, der nicht geeignet ist, überprüfbare Erkenntnisse über empirische Gegenstände zu liefern, die über das Niveau dessen, was schon mit konventionellen und bekannten Methoden erreicht werden kann, hinaus gehen. In der Tat bleibt dieser Ansatz bei diesem Beispiel sogar weit dahinter zurück. Allerdings ist an dieser Stelle Vorsicht geboten. Denn erstens sind neue wissenschaftliche Methoden in ihren frühen Stadien naturgemäss unvollkommen und vielleicht nur anfangs noch weniger leistungsfähig als bestehende Methoden. Die Wissenschaft ist ein Wagnisunternehmen und oft ist erst, wenn man schon viel in eine Methode investiert hat, ein Erfolg oder Misserfolg überhaupt abzusehen. Und zweitens ist die Wissenschaft ein arbeitsteiliges Unternehmen, d.h. die Chance auf einen empirischen Erklärungserfolg, die ja den Massstab der epistemischen Verantwortung bildet, stellt sich bei einem zunächst nur spielerisch-theoretisch entwickelten Ansatz möglicherweise erst dann ein, wenn dieser Ansatz von der empirischen Forschung aufgegriffen wird. Das bedeutet jedoch nicht, dass das Prinzip epistemischer Verantwortung bei neuen wissenschaftlichen Ansätzen nicht schon wirksam werden sollte. (Würde man es ganz fallen lassen, wäre man wieder bei einer anything goes Epistemologie, die offensichtlich absurd ist.) Das Prinzip der epistemischen Verantwortung wirkt sich unter Berücksichtigung dieser Bedingungen für neue wissenschaftliche Ansätze lediglich anders aus, und kann für diesen Fall leicht in drei zentralen Forderungen konkretisiert werden: 1. Man muss sich der noch bestehenden Unvollkommenheiten des Ansatzes bewusst sein, und sie nicht kaschieren wollen. 2. Der Ansatz ist in der Richtung voranzutreiben, dass diese Unvollkommenheiten beseitigt werden und es ist stets die Frage zu stellen, ob die Chance besteht, dass sie beseitigt werden können. 3. Es ist dafür Sorge zu tragen, dass der Ansatz in der arbeitsteiligen Wissenschaft für die „Abnehmer" (in diesem Fall die empirische Laborund Feldforschung16) nützlich und fruchtbar werden kann. In der von Axelrod begründeten Forschungstradition geschah jedoch leider nichts dergleichen. Vielmehr setzte der Forschungsansatz Axelrods mitsamt seinen Defekten Standards für die nachfolgende Forschung. Noch heute dient der Verweis auf dieses „Vorbild" zur Legitimation ähnlich gelagerter Forschungsvorhaben.17 Empirische Erklärungserfolge blieben mit dem Simulationsansatz der Theorie der Evolution der Kooperation vollständig aus, ja es wurde – ein deutlicher Hinweis auf das mangelnde epistemische Verantwortungsgefühl der zugehörigen Fachgemeinschaft – nur äusserst 15 Vgl. Arnold, Explaining Altruism, S. 174-183. 16 Beispiele für Feldund Laborforschung aus diesem Bereich in: Poteete, Janssen und Ostrom: Working Together. Collective Action, the Commons, and Multiple Methods in Practice. 17 Vgl. Rendell et al., Why Copy Others?, S. 208f. 6 selten überhaupt versucht, diesen Ansatz empirisch auf die Probe zu stellen. Insgesamt war der Theorie der Evolution der Kooperation daher zwar ein durchschlagender sozialer Erfolg innerhalb der Fachwelt beschieden, der aber leider nicht durch einen durch empirische Überprüfung ausgewiesenen Erkenntnisgewinn gedeckt war. Deutlich wird dies an den Forschungsberichten zur „Evolution der Kooperation". So führt Hoffmann in seinem Forschungsbericht Twenty Years on: The Evolution of Cooperation Revisited18 als einzige empirische Studie eine Arbeit von Milinski (1987) über kooperatives Verhalten bei Schwarmfischen auf. Auch von anderen Autoren wird diese Studie – einer der ganz seltenen Versuche, irgendein Simulationsmodell der Evolution der Kooperation empirisch zu überprüfen, überhaupt – als vermeintlicher Beleg für die empirische Fruchtbarkeit dieses Ansatzes angeführt.19 In Wirklichkeit hatte die an diese Studie anschliessende wissenschaftliche Diskussion, die weder in dem eben zitierten Forschungsbericht noch in dem erwähnten Lehrbuch berücksichtigt wird, längst ergeben, dass das Simulationsmodell Axelrods zur Erklärung der fraglichen Verhaltensmuster bei Schwarmfischen inadäquat ist.20 Auf sehr viel breiterer empirischer Basis kommt Lee Allen Dugatkin (1997) in einer Meta-Studie über die empirische Forschung zu Kooperation und Altruismus im Tierreich zu dem Ergebnis, dass die Simulationsmodelle der Evolution der Kooperation nicht empirisch anwendbar sind. Von den ca. zwei Dutzend Simulationsund Modellstudien, die er zu Beginn seines Buches auflistet, taugt keine einzige dazu irgend eines der empirischen Beispiele kooperativen Verhaltens adäquat zu beschreiben. Und das obwohl der Simulationsansatz der Evolution der Kooperation zu der Zeit schon seit über 15 Jahren auf dem Markt war. In den meisten Fällen scheitert die Anwendung derartiger Simulationsmodelle an dem Problem, dass die Eingabe-Parameter der Modelle empirisch nicht bestimmbar sind. Das verdeutlicht einmal mehr, wie wenig Gedanken die Modellierer hier an die empirische Anwendung und Überprüfung ihrer Modelle verschwendet haben. Wenn Peter Hammerstein schliesslich das folgende Resümee zieht, so ist ihm nur zuzustimmen: Why is there such a discrepancy between theory and facts? A look at the best known examples of reciprocity shows that simple models of repeated games do not properly reflect the natural circumstances under which evolution takes place. Most repeated animal interactions do not even correspond to repeated games. [...] Most certainly, if we invested the same amount of energy in the resolution of all problems raised in this discourse, as we do in publishing of toy models with limited applicability, we would be further along in our understanding of cooperation.21 Dasselbe Fazit könnte man auch für die sozialwissenschaftliche Forschung zur „Evolution der Kooperation" ziehen. Der Versuch den Ansatz nun noch dadurch zu retten, dass man ihm einen – wie auch immer zu verstehenden – „theoretischen" Erkenntniswert zubilligt, scheitert an der durchschlagenden Kritik, die Axelrod von Seiten der mathematischen Spieltheorie erfahren hat.22 Eine Haltung weitgehender epistemischer Verantwortungslosigkeit ist nicht nur für die 18 Hoffmann, Twenty Years on. 19 Vgl. Osborne, An Introduction to Game Theory, S. 445/446. 20 Vgl. Dugatkin, Cooperation among Animals. 21 Hammerstein, Why Is Reciprocity So Rare in Social Animals?, S. 83, 92. 22 Vgl. Binmore, Game Theory and the Social Contract I+II. 7 speziellere Forschungstradition der Evolution der Kooperation charakteristisch gewesen. Sie findet sich in bemerkenswert grosser Verbreitung im Bereich der sozialen Simulationen überhaupt und zeigt sich beispielsweise an der geradezu verständnislosen Reaktion eines Simulationsforschers auf die naheliegende Frage eines Wissenschaftsjournalisten nach der empirischen Überprüfung der Simulationsmodelle: Keines der Modelle wurde bisher in psychologischen Experimenten bestätigt. Sollte einem das wirklich völlig egal sein? Rainer Hegselmann macht diese Frage fast ein wenig verlegen. 'Wissen Sie: In meinem Hinterkopf ist die Idee, dass eine bestimmte Sorte von Laborexperimenten uns gar nicht weiterhilft.23 Was an dieser Reaktion verblüfft ist, dass man doch eigentlich meinen sollte, dass ein Wissenschaftler ein natürliches Interesse daran haben müsste, zu erfahren, ob die eigenen Theorien oder Modelle denn nun wahr oder falsch sind. Der Fall wäre nicht weiter der Rede wert, wenn es ein Einzelfall wäre. Aber es gibt Indizien dafür, dass diese Einstellung im Bereich agentenbasierter Simulationen geradezu endemisch ist. So gelangen Heath, Hill und Ciarello (2009) in einer Meta-Studie über die Simulationsforschung im Zeitraum 1998-2008 zur der Feststellung, dass 2/3 aller untersuchten Simulationsstudien nicht hinreichend empirisch validiert sind, 1/3 sogar überhaupt nicht! Ein ernüchterndes Resultat, das von den Autoren der Studie auch mit angemessener Deutlichkeit bewertet wird: However, 65% of the surveyed articles were not completely validated. This is a practice that is not acceptable in other sciences and should no longer be acceptable in ABM [agent-based modelling, E.A.] practice and in publications associated with ABM.24 Immerhin zeigt sich auch, dass die Quote empirisch validierter Studien im Laufe des Untersuchungszeitraums angestiegen ist, was auf möglicherweise vorhandene Selbstheilungskräfte der Fachgemeinschaft hoffen lässt.25 Auch wenn die Autoren dieser Studie es nicht so formulieren, so entspricht ihre Kritik an der Praxis der Veröffentlichung unvalidierter Simulationsstudien der Einforderung dessen, was hier mit epistemischer Verantwortung bezeichnet wird. Zusammenfassend kann festgehalten werden: Das Beispiel führt vor Augen, dass mangelnde epistemische Verantwortung nicht nur als Einzelfall sondern als verbreitetes Phänomen innerhalb von ganzen Fachgemeinschaften vorkommen kann. Weiterhin zeigt es, dass mangelnde epistemische Verantwortung tatsächlich ein Problem ist. Zumindest im Bereich der Forschungstradition der „Evolution der Kooperation" hat sie zu einer Flut von weitgehend nutzlosen Computersimulationsstudien geführt, die auch dann nicht aufhörte als längst absehbar war, dass diese Simulationen nicht dazu geeignet waren zur wissenschaftlichen (d.h. empirisch überprüfbaren) Erkenntnis des Forschungsgegenstandes beizutragen. Beides spricht stark dafür „epistemische Verantwortung" als Wert zu etablieren. 23 Grötker, Reine Meinungsmache, S. 2. 24 Heath, Hill und Ciarello, A Survey of Agent-Based Modeling Practices. 25 Vgl. auch Poteete, Janssen und Ostrom: Working Together. Collective Action, the Commons, and Multiple Methods in Practice, S. 194ff., wo Möglichkeiten zur Verknüpfung agentenbasierter Modelle mit empirischer Forschung diskutiert werden. Bisher ist dergleichen, auch auf Grund mangelnden Problembewusstseins vieler Modellierer, leider eher noch die Ausnahme. 8 4 Bedingungen, unter denen epistemische Verantwortung gedeiht oder verkümmert Worauf ist es zurückzuführen, dass der Wert der epistemischen Verantwortung innerhalb der Wissenschaft (oder wenigstens einiger Wissenschaftszweige) unzureichend verankert ist, und wie kann dem abgeholfen werden? Zunächst einmal wäre es naiv anzunehmen, dass sich die epistemische Verantwortung im Rahmen der üblichen Qualitätssicherungsmechanismen der Wissenschaft von selbst ergibt. Die eben zitierte Meta-Studie hat sich fast ausschliesslich auf Artikel aus begutachteten Journalen gestützt, so dass es offensichtlich nicht stimmt, dass bestimmte Begutachtungsverfahren wie z.B. blind review -Verfahren26 automatisch garantieren, dass wissenschaftliche Publikationen einen Erkenntniswert haben. Ausserdem ist jede Begutachtung nur so gut, wie die angelegten Wertmassstäbe. Gehört die „epistemische Verantwortung" nicht zu den von einer Forschergemeinschaft als selbstverständlich geteilten Werten, dann wird sie sich auch nicht wie von selbst im wissenschaftlichen Begutachtungsverfahren einstellen. Grundsätzlich kann man vermuten, dass das Begutachtungsverfahren, da es sich tendenziell an den herrschenden Überzeugungen der Forschergemeinschaft ausrichtet,27 zwar sehr gut dazu geeignet ist, die Einhaltung wissenschaftlicher Standards und Sekundärtugenden zu fördern,28 dass es aber nicht gleichermassen erfolgreich dabei ist Paradigmenversagen abzufangen.29 Sofern dies stimmt, müsste man auch erwarten, dass dieses Verfahren für Wissenschaften mit herrschenden Paradigmen sehr viel besser geeignet ist, als für kontroverse Wissenschaften, die durch die 26 Vgl. zu den Vorzügen und Nachteilen unterschiedlicher Begutachtungsverfahren die jüngste Debatte dazu in nature: http://www.nature.com/nature/peerreview/debate/ doi:10.1038/nature04993 27 Vgl. dazu Stehbens, Basic philosophy and concepts underlying scientific peer review. Vgl. auch Campanario, Rejecting and resisting Nobel class discoveries. In dieselbe Kerbe schlägt auch der Autor des Editorials "Copying with peer rejection" der nature-Ausgabe vom 16. Oktober 2003, der zurückgewiesenen Autoren schlicht Hartnäckigkeit empfiehlt. Die Filterwirkung von Paradigmen und herrschenden Meinungen wird in der wissenschaftshistorischen und besonders der wissenssoziologischen Literatur immer wieder hervorgehoben: Vgl. Kuhn, Die Struktur wissenschaftlicher Revolutionen, S. 104ff. Vgl. Collins, Changing Order. Replication and Induction in Scientific Practice, S. 79ff., der sehr anschaulich vom "envelope of acceptable opinion" spricht. Kritisch zu Collins' Fallstudien allerdings: Franklin, No Easy Answers. Science and the Pursuit of Knowledge. Die Rolle sozialer Durchsetzungsmacht in der Wissenschaft stark übertreibend Latour und Woolgar, Laboratory Life. The Construction of Scientific Facts. Mit der wissenssoziologischen Diskussion nicht zu verwechseln ist die rein politisch motivierte Kritik an den wissenschaftlichen Begutachtungsverfahren, für die als Beispiel ein Vertreter des in dieser Hinsicht notorischen Cato Institute angeführt sei: Michaels, Peer Review And 'Pal Review' In Climate Science. Grundsätzlich zum Problem politisch motivierter Wissenschaftskritik: Oreskes und Conway, Merchants of Doubt. How a Handful of Scientists Obscured the Truth on Issues from Tobacco Soke to Global Warming. 28 Zumindest solange es sich nicht um bewussten Betrug handelt, der auch für die Gutachter oft nur schwer zu erkennen ist. 29 Ein interessantes Indiz dafür ist, dass inzwischen auch Kreationisten begonnen haben, ihre eigenen peer-reviewed journals heraus zu geben, wie z.B. das Answers Research Journal. http://www.answersingenesis.org/arj/about . Bisher gelingt die Imitation des Peer-Review Verfahrens den Kreationisten allerdings nur unvollkommen. Trotzdem führt das Beispiel vor Augen, dass man Wissenschaft im Sinne der Suche nach Wahrheit und Erkenntnis nicht mit der Kulisse von Wissenschaft, zu der Fachjournale, Fachtagungen, gelehrte Professoren und dicke Bücher gehören, verwechseln darf. Im günstigen Falle, aber eben nicht immer, trägt letzteres zu ersterem bei. 9 Dauerkonkurrenz rivalisierender Paradigmen gekennzeichnet sind. Tatsächlich hat es auch in den typischerweise durch Paradigmenvielfalt gekennzeichneten Geisteswissenschaften nach wie vor eine relativ geringere Bedeutung als in den Naturwissenschaften. Ohnehin darf der Erfolg innerhalb der institutionalisierten Verfahrensweisen der Wissenschaft nicht mit dem Erkenntniswert der Forschung gleichgesetzt werden. Erkenntniswert hat wissenschaftliche Forschung dann, wenn es gelingt, auf relevante Fragestellungen, die die Erfahrungswelt betreffen, nachweisbar richtige Antworten zu geben (d.h. solche Antworten, die ernsthaften Falsifizierungsversuchen stand halten). Das zentrale Problem der institutionellen Organisation von Wissenschaft kann man grob so beschreiben, dass es darin besteht, Regeln und Verfahrensweisen zu finden durch die für die Wissenschaftler und Wissenschaftlerinnen Anreize geschaffen werden, die sie dazu bewegen, den sachlichen Zweck der Wissenschaft (Erkenntnis) zu verfolgen und nach Kräften zu fördern. Hinsichtlich der epistemischen Verantwortung stellt sich die Frage unter welchen (institutionellen) Bedingungen epistemische Verantwortung gedeiht, und unter welchen Bedingungen sie verkümmert. Zu dieser Frage seien im Folgenden einige Überlegungen angestellt. Nachteilig wirkt sich auf die epistemische Verantwortung sicherlich die Universitätshierarchie und das damit einhergehende Hierarchiebewusstsein der Akteure aus, die gerade in Deutschland traditionell recht ausgeprägt ist und ihren intensivsten Grad im Verhältnis der Doktorandin zur Betreuerin erreicht. Eine Doktorandin ist der Betreuerin in der Regel über mehrere Jahre auf Gedeih und Verderb ausgeliefert. Häufig besteht eine Doppelabhängigkeit als Doktorandin und angestellter Mitarbeiterin. Von Seiten der Betreuerin wird das Verhältnis oft so ausgelegt, dass die Doktorandin Teil des eigenen Teams ist, was zu der Erwartung führt, sie müsste den eigenen Forschungsansatz, schlimmstenfalls sogar die wissenschaftlichen Meinungen der Betreuerin teilen. Sehr häufig bekommen Doktorandinnen ihr Thema, für das sie mehrere Jahre Lebenszeit opfern müssen, zugeteilt – bei nicht vorhandener oder nur geringer Wahlmöglichkeit. Das kann – ein keinesfalls seltener Fall – dazu führen, dass Doktorandinnen ein Forschungsvorhaben durchführen müssen, das sie selbst nicht überzeugt, und welches die Betreuerin nur einmal ausprobiert sehen wollte. Ohne einen gewissen Zynismus ist eine derartige Situation oft nur schwer zu ertragen. Es versteht sich von selbst, dass solche Bedingungen der Herausbildung intellektueller Gewissenhaftigkeit nicht zuträglich sind. Schon Max Weber hat die Situation, in der sich die Abhängigen in der Universitätshierarchie befinden, recht treffend charakterisiert: Der Arbeiter, der Assistent also, ist angewiesen auf die Arbeitsmittel, die vom Staat bereit gestellt werden; er ist infolgedessen vom Institutsdirektor ebenso abhängig wie ein Angestellter in einer Fabrik: – denn der Institutsdirektor stellt sich ganz gutgläubig vor, dass dies sein Institut sei, und schaltet darin –, und er steht häufig ähnlich prekär wie jede »proletaroide« Existenz...30 Es ist bemerkenswert, dass ein heutiger Beobachter beinahe hundert Jahre nach Max Weber zu einem ganz ähnlichen Befund gelangt: Die entscheidenden Fragen zu den strukturellen Bedingungen wissenschaftlicher 30 Weber, Wissenschaft als Beruf, S. 584. 10 Originalität und Produktivität werden erst gar nicht gestellt: Wie kann die hohe Unsicherheit und Abhängigkeit von jungen Wissenschaftlern an deutschen Universitäten einer originellen Forschung förderlich sein? Und: Wie kann in universitären Strukturen, in denen eine kleine Minderheit auf Lebensstellen über eine grosse Mehrheit auf Zeitstellen herrscht, dem Ideal einer wissenschaftlichen Kommunikation nachgelebt werden, in der Wahrheitsfragen so wenig wie möglich von Machtfragen kontaminiert sind?31 Wenn das hier beschrieben Phänomen tatsächlich einen neuralgischen Punkt trifft, dann müsste sich das im Vergleich zur Situation in anderen Ländern zeigen, in denen andere Verfahren gelten und das Hierarchiebewusstsein weniger ausgeprägt ist. Die im Auftrag des Bundesministeriums für Bildung Forschung erstellen Berichte der Hochschulinformationssystem GmbH deuten darauf hin, dass dies zumindest nach der subjektiven Einschätzung von Nachwuchswissenschaftlerinnen der Fall ist. Sowohl die USA als auch Grossbritannien schneiden im Vergleich zu Deutschland erheblich besser ab, wenn es um die Betreuung von Nachwuchswissenschaftlerinnen, die Offenheit für neuartige Forschungsansätze und den gleichberechtigten Umgang mit Wissenschaftlerinnen auf höheren Hierarchieebenen geht.32 Nun kann man sich zwar vorstellen, dass ein hoher Publikationsdruck (gegen den Nachwuchswissenschaftlerinnen in hierarchischen Systemen möglicherweise stärker abgeschirmt sind) eine nicht minder disziplinierende Wirkung ausübt, die sich – je nachdem wie stark dieser Gesichtspunkt innerhalb der Fachgemeinschaft berücksichtigt wird – ebenfalls nachteilig auf die epistemische Verantwortung auswirken könnte. Aber auch dann ist es ein anderes, ob man sich gegenüber einer anonymen Forschergemeinschaft bewähren muss oder in ein persönliches Machtund Autoritätsverhältnis gestellt ist, in dem die Versuchung stets gegeben ist, sich mehr durch Wohlverhalten und Botmässigkeit als durch Leistung zu qualifizieren. Gegen diese Kritik an der Universitätshierarchie sind aber auch Einwände möglich: Erstens einmal können viele wissenschaftliche Forschungsaufgaben nur arbeitsteilig durchgeführt werden. Das erfordert eine entsprechende Organisation, Aufgabenverteilung und wahrscheinlich auch Hierarchien. Und zweitens könnte man einwenden, dass die Tatsache, dass die Einzelne mangels Wahlmöglichkeiten nicht mehr die volle wissenschaftliche Verantwortung für ihre Forschung übernehmen kann, nicht bedeutet, dass sie verschwindet, sondern nur, dass die epistemische Verantwortung in dem Masse, wie Entscheidungen über das Forschungsvorhaben von der höheren Hierarchieebene getroffen werden, auch an die höhere Hierarchieebene delegiert wird. Aber auch wenn zuzugestehen ist, dass viele Bereiche der modernen Wissenschaften ohne Arbeitsteilung nicht denkbar sind, so bleibt dennoch das Problem bestehen, dass die einzelnen Wissenschaftlerinnen die Ergebnisse, die sie erarbeiten am Ende unter ihrem Namen veröffentlichen. Der Entstehungskontext lässt sich in den entsprechenden wissenschaftlichen Arbeiten aber nicht mehr unmittelbar nachvollziehen, und 31 Hirschi, Exportweltmeister beim akademischen Überschuss. – Wie Caspar Hirschi ebenfalls feststellt, hat die Exzellenzinitiative dieses Problem eher noch verschärft, was den Eindruck verstärkt, dass die Exzellenzinitiative die strukturellen Schwächen des deutschen Wissenschaftssystems jedenfalls nicht beseitigt. 32 Vgl. Jaksztat, Schindler, Briedis, Die internationale Ausrichtung des wissenschaftlichen Nachwuchses, S. 56ff. – Dem Bericht zu Folge schneiden die USA und Grossbritannien im Vergleich zu Deutschland in der Bewertung der Nachwuchswissenschaftlerinnen bei allen Fragen besser ab und bei manchen, wie den oben genannten, auch deutlich besser. Frankreich schneidet dagegen überwiegend schlechter ab. 11 dementsprechend auch nicht wie frei die Autorin war, bestimmte forschungsstrategische Entscheidungen zu treffen. Die Autorin ist daher wohl oder übel gezwungen, die so erzielten Ergebnisse zu vertreten. Das Problem dürfte wiederum umso gravierender sein, je „kontroverser" und weniger objektiv der Wissenschaftszweig ist, um den es geht. Es kommt durchaus vor, dass Wissenschaftlerinnen aus diesem Grund im persönlichen Gespräch ihre eigenen Veröffentlichungen dementieren und darauf hinweisen, dass sie es eigentlich ja ganz anders sehen – wiederum etwas, was nur mit einer gewissen professionellen Abgebrühtheit zu ertragen ist. Und noch aus einem anderen Grund funktioniert die Delegation von epistemischer Verantwortung nicht, oder zumindest nicht ohne Weiteres: Die Vorgesetzten können die Einzelheiten der Durchführung einer Forschungsaufgabe schon wegen des damit verbundenen Aufwandes nicht immer in aller Gänze überschauen und kontrollieren. Sie sind darauf angewiesen, dass ihre Mitarbeiterinnen eigenverantwortlich ordentliche Arbeit leisten. Daher kann man das Problem der Nicht-Verantwortbarkeit forschungsstrategischer Entscheidungen durch die abhängig arbeitenden Wissenschaftlerinnen auch nicht dadurch lösen, dass man Ergebnisse nominell dem Vorgesetzten in der Hierarchie zuschreibt. Die individuelle Zuschreibung wissenschaftlicher Leistungen ist für die Qualität wissenschaftlicher Arbeit vermutlich unerlässlich. Daher dürfte der Wissenschaft am besten gedient sein, wenn auch arbeitsteilige Forschungsprozesse so organisiert werden, dass die Möglichkeit zur Übernahme epistemischer Verantwortung durch die Einzelnen gegeben ist. Dies beinhaltet, dass Teilleistungen individuell zugeschrieben werden können, aber auch, dass den in der Hierarchie untergeordneten Forscherinnen möglichst grosse forschungsstrategische Entscheidungsfreiheiten verbleiben. 5 Zusammenfassung Epistemische Verantwortung bedeutet, dass Wissenschaftlerinnen dafür verantwortlich sind, dass ihre Forschung geeignet ist, zu einem wissenschaftlichen Erkenntniszweck (er mag selbst wie weltfremd auch immer sein), beizutragen. Sie bürdet der einzelnen Wissenschaftlerin auch unter Bedingungen der arbeitsteiligen Wissenschaft eine Mitverantwortung für den Sinn des Gesamtprojekts auf. Wie das Beispiel der sozialen Simulationen zeigt, ist epistemische Verantwortung in der Wissenschaftspraxis keineswegs eine Selbstverständlichkeit. Dieser Forschungszweig leidet bisher unter einer Flut von Simulationsstudien, deren Beitrag zur Erkenntnis des untersuchten Forschungsgegenstandes zumindest dubios bleibt. Es gibt immerhin Anzeichen, dass sich die Situation langsam bessert, und der relative Anteil von Simulationsstudien zunimmt, die ihre Erkenntnisqualität durch eine entsprechende Validierung ausweisen können. Insgesamt unterstreicht das Beispiel, dass epistemische Verantwortung innerhalb der Forschergemeinschaften als Wert begriffen und bewusst vertreten werden muss, damit sie in der Wissenschaftspraxis wirksam werden kann. Zu den Faktoren, die zur Erosion epistemischer Verantwortung führen können, gehören unter anderem die (unvermeidliche) Arbeitsteilung in der Wissenschaft und die besonders in Deutschland sehr ausgeprägte (aber schon eher vermeidbare) hierarchische Organisation wissenschaftlicher Institutionen und Forschungsprojekte. Literatur 12 Arnold, E.: Explaining Altruism. A Simulation-Based Approach and its Limits, Heusenstamm 2008. Axelrod, R.: The Evolution of Cooperation, New York. 1984. Binmore, K.: 1994. Game Theory and the Social Contract I. Playing Fair, Cambridge, Massachusetts / London, England: 2000 (1994). Binmore, K.: Game Theory and the Social Contract II. Just Playing, Cambridge, Massachusetts / London, England 1998. Campanario, J.: Rejecting and resisting Nobel class discoveries: accounts by Nobel Laureates. In: Scientometrics, Vol. 81, 2, 2009, S. 549-565. DOI: 10.1007/s11192-008-2141-5 http://www.mendeley.com/research/rejecting-and-resisting-nobel-class-discoveriesaccounts-by-nobel-laureates/ Collins, H.: Changing Order. Replication and Induction in Scientific Practice, Chicago 1992. Dugatkin, L. A.: Cooperation among Animals, Oxford 1997. Döscher, H.-J.: Seilschaften. Die verdrängte Vergangenheit des Auswärtigen Amtes, Berlin 2005. Feyerabend, P.:Wider den Methodenzwang, Berlin 1975/1983. Grammelsberger, G.: Computerexperimente. Zum Wandel der Wissenschaft im Zeitalter des Computers, Bielefeld 2010. Franklin, A.: No Easy Answers. Science and the Pursuit of Knowledge, Pittsburgh 2005. Greco, J.; Turri, J., Virtue Epistemology, in: Stanford Encyclopedia of Philosophy, Spring 2011 Edition (ed. N. Zalta). http://plato.stanford.edu/archives/spr2011/entries/epistemology-virtue/ Green, D. P.; Shapiro, I.: Pathologies of Rational Choice Theory. A Critique of Applications in Political Science, New Haven and London: 1994. Grötker, R.:. Reine Meinungsmache. 2005. http://www.heise.de/tr/artikel/Reine-Meinungsmache-277359.html Hammerstein, P.:. 2003. Why Is Reciprocity So Rare in Social Animals? A Protestant Appeal. In: Genetic and Cultural Evolution,2003, S. 83–94. Heath, B.; Hill, R.; Ciarello, F.: A Survey of Agent-Based Modeling Practices (January 1998 to July 2008). In: Journal of Artifical Societies and Social Simulation (JASSS) 12(4) 9, 2009. http://jasss.soc.surrey.ac.uk/12/4/9.html Hirschi, C.: Exportweltmeister beim akademischen Überschuss, in: Frankfurter Allgemeine Zeitung, 9. 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Republicanism Across Cultures Philip Pettit Introduction Every philosophy of the good society starts with an account of the central complaint that the state should help to put right: the evil that the society should drive out by means of political organization and initiative. If the philosophy is to be persuasive, then the complaint should attract widespread sympathy and support, being recognized as something that everyone will want to have remedied. And, of course, it should be a complaint that the state has the ability in principle to rectify, reducing or even eradicating the ill indicted. Republican philosophy identifies a complaint that is meant to be at once popularly motivating and politically implementable. It indicts the evil of subjection to another's will - particularly in important areas of personal choice - as an ill that we all recognize and recoil from and at the same time as an ill that the state is well placed to deal with. Such subjection can be effectively corralled and reduced, if not wholly eliminated, by means of political initiative. And yet it takes only a little imagination to realize just how repellent it can be. Think, by way of exercising such imagination, of how you would feel as a student if you depended for not failing a course on the whim of an instructor. Or as a wife if you had to rely on the mood of your husband for whether you could enjoy an unmolested day. Or as a worker if you hung on the favor of a manager for whether you retained your job. Or as someone destitute if you had to cast yourself on the mercy of others just to survive or maintain your family. Or think about how you would feel as the member of a cultural minority if you had to rely on the humor of majority groups for whether you escaped humiliation; or as an elderly person if you depended on escaping the notice of youth gangs for walking safely home; or as a citizen if you were dependent on winning the favor of some insider group for whether you or your kind ever caught the ear of government. 2 Already in classical, republican Rome, the evil of subjection to the will of others, whether or not such subjection led to actual interference, was identified and indicted as the iconic ill from which political organization should liberate people, in particular those in the fortunate position of citizens. It was described as the evil of being subject to a master or dominus - suffering dominatio (Lovett 2010, Appendix 1)- and was contrasted with the good of libertas or liberty. The accepted wisdom was that people could enjoy liberty, both in relation to one another and to the collectivity, only by being invested with the power and status of the civis or citizen. Being a free person became synonymous with not having anyone in the position of a dominus in your life. It was equated with being sufficiently empowered to stand on equal terms with others, as a citizen among citizens (Wirszubski 1968, Ch 1). The idea that citizens could enjoy this equal standing in their society, and not have to hang on the benevolence of their betters, became the signature theme in the long and powerful tradition of republican thought. Familiar from its instantiation in classical Rome, the idea was re--‐ignited in medieval and Renaissance Italy; spread throughout Europe in the modern era, sparking the English civil war and the French revolution; and inflamed the passions of England's American colonists in the late eighteenth century, leading to the foundation of the world's first modern democracy. With citizenship becoming more and more inclusive as a category, the idea was that the state could provide for all citizens in such a measure that they would each be able to walk tall, live without shame or indignity, and look one another in the eye without any reason for fear or deference. The recent revival of republican thought is built on this idea that there is an ideal for the state to promote - freedom understood as non--‐domination - that is both popularly motivating and politically implementable. Freedom in this sense is not meant to be the only value in life, or the only value that ultimately matters. The claim is merely that it is a gateway good, suited to guide the governments that people form and sustain. Let government look after the freedom of citizens in this sense, so the line goes, and it will also have to look after a plausible range of other goods and do so at a plausible level of provision. It will have to guard against 3 division and disorder, for example, and it will have to provide in a decent measure against misery and poverty, unfairness and inequality. Like a growing number of others I have embraced the research program, as I think of it, of exploring the implications of republican ideas for thinking about contemporary political problems.1 Republican theory offers a useful perspective on the three major issues in politics. In rough and ready slogans, it suggests that social justice requires citizens to be able to enjoy equal freedom as non--‐domination in their dealings with one another; that political justice requires them to enjoy equally shared control over the government that provides for social justice, thereby making government coercion non--‐dominating; and that international justice requires each people to enjoy equal freedom as non--‐domination in their dealings with other peoples - in effect, with other states - as well as with multi--‐national bodies and international agencies (Pettit 2012). In this paper I focus on how far the republican ideal of freedom as non--‐ domination can and should command allegiance across different cultures. Is the ideal bound to western culture, as its provenance may suggest? Or does it have a hold on the human imagination and sensibility that survives across various cultural and historical divides? I argue, in a deeply unfashionable vein, that it does command a form of universal allegiance. Or, to be more exact, I argue that freedom as non--‐ domination has this status in its role as an ideal of social justice. Reasons of space make it impossible to extend the argument to its role as an ideal of political and international justice - as an ideal of democracy and sovereignty - but the considerations I muster should make clear how that argument would go. The paper is in three sections. In section 1 I look at why the ideal of basic functioning capability, as advanced by Amartya Sen and Martha Nussbaum, counts as a universal ideal, albeit an ideal of a structural rather than substantive kind. In section 2 I argue that the ideal of freedom as non--‐domination should be considered as a structural ideal of a similar nature. And in section 3 I show that although it is more encompassing than basic functioning capability, freedom as non--‐domination ought to be taken to be an ideal with a universal appeal of the same kind. 4 1. Basic functioning capability In an article from over a quarter of a century ago the economist and philosopher, Amartya Sen (1983), took up the question of how we should define poverty for the purpose of comparing different societies and indeed for comparing the same society at different points in time. There were two salient approaches on offer in the literature. One was to define poverty in an absolute sense as the condition of having to live below the level required for subsistence. The other was to define it in a relative way as the condition of being in the economically lowest fifth or tenth or whatever percentile in the local society. These candidate conceptions of poverty each had a serious drawback. The absolute conception implied, outrageously, that in more affluent societies where few die of malnutrition or exposure there are no poor people and so no problem of poverty. The relative conception implied, equally outrageously, that no matter how developed or caring a society became, it could do nothing about poverty; there would always be some people nearer the economic bottom than others. In response to the difficulties with these two conceptions, Sen proposed that we should look for a middle way. He found a middle way implicit in Adam Smith's suggestion that people are poor when they lack the necessaries of life that enable them to live without shame among their fellows. 'By necessaries', Smith (1976, 351--‐ 2) says, 'I understand not only the commodities which are indispensably necessary for the support of life, but what ever the custom of the country renders it indecent for creditable people, even the lowest order, to be without'. In illustrating that idea for contemporaries, he argued that leather shoes count as a necessary in his own time and place: 'The poorest creditable person of either sex would be ashamed to appear in public without them'. Smith's observation suggests that we should define poverty in a universal, inter--‐cultural manner as the condition of being unable to live without a certain shame before compatriots. Yet it allows that the resources that would make someone relatively well--‐off in a less developed society, enabling them to live 5 without shame, might leave them below the poverty level in a comparatively more developed regime. Under this suggestion, poverty is defined inter--‐culturally but interpreted intra--‐culturally. It is a universal, structural evil that assumes a local content or substance as it is construed, now in the context of one society, now in the context of another. This approach to poverty led Sen (1985) to develop the well--‐known ideal according to which people should each enjoy a basic functioning capability in their society (see too Nussbaum 1992; 2006). Each should have the capacity to function up to a certain threshold of adequacy in their local world, having access to the resources required for being a functioning member of the community. This is a structurally universal ideal insofar as it is defined inter--‐culturally as an ideal for people in any society. But it is a substantively local ideal insofar as the demands it makes - the resources it requires people to have - vary greatly across cultural contexts. In order to be able to function properly within an indigenous community in highland Papua New Guinea you will certainly have to have access to a variety of locally valued resources. But the resources required vary greatly from those to which you must have access if you are to be able to function properly within an urban, industrialized society of the kind that most readers of this paper will inhabit. In order to function adequately within such a society, you must have access to shelter and sustenance but you must also have a degree of literacy, an ability to follow local and national news, a knowledge of your legal rights and duties, an address where you can be reached, an identity in the world of finance, employment and taxation, access to private or public transport, and so on through an open--‐ended range of material and institutional resources. The fact that the structural ideal of functioning capability assumes a different content as we move from society to society, culture to culture, ought to allay the worry that if we embrace such a universal ideal we may be insensitive to cultural difference. But does it remove the worry altogether? Ought we to think that no matter what the culture of their local society, people in every society should each 6 have the wherewithal for functioning there? Or are there any remaining grounds for retreating from that judgment, holding that still it imposes an ethnocentric ideal - a western ideal, as it will certainly be described - on others? To say that it is a universal ideal that people should each have a basic capability of functioning in their own society is just to say that it would be good if they had that capacity. It is not yet to say that regardless of collateral costs, for example, the local state should rely on coercion to ensure the capacity; that claim, however plausible, is bound to require further argument. And, again, it is not yet to say that the state should ensure this capacity in the unlikely event of a wholesale rejection of the ideal - say, its rejection under a system of unanimous, presumptively voluntary voting - by the members of the society. To uphold the ideal is merely to say that other things being equal it would be good - though good, period, not just good--‐by--‐our--‐lights - if in every society people had a basic functioning capability. I think that there are no compelling grounds for rejecting this claim and that we ought to have no fear that it would be ethnocentric to embrace the structurally universal ideal that Sen defends. In every society there are going to be many members who do have the resources to function adequately there. And in almost every society, so we may assume, it is going to be possible for everyone to have a capability for such functioning; that possibility is implicit in the description of the ideal as a basic or adequate level of capability. So why might we not think it good that in every society people should all enjoy this capability? Why might we not think that this is at least an ideal, even if collateral costs argue against the state realizing it? The fact that the capability is connected to living without shame as a creditable person of the society, as Smith expresses the ideal, makes it almost inconceivable that we should deny its universal appeal. Someone who denied the universal attraction of the ideal would have to think that there are some societies whose character ensures that it is not good that people there should all enjoy an equal functioning capability. But what might that character be? Presumably it must consist in the fact that members are committed in 7 the society to a form of hierarchy that presupposes inequality in functioning capability. Perhaps it is a patriarchal society that would be disrupted by any arrangement under which women would no longer depend on men for their basic welfare needs. Or perhaps it is a caste society that would be undone by any arrangement under which those in lower castes did not have to rely on the indulgence of those in higher. Ought we to shrink from the ideal of equal functioning capability in thinking about such a society? We might certainly shrink from encouraging the state to impose it coercively in the event, however unlikely, that there is a widespread, voluntary acceptance of such a hierarchy amongst the people. And perhaps we ought to shrink from such postures in the event that while the relatively deprived do not endorse the hierarchy, still the collateral costs of coercive imposition would be very high. But ought we to shrink from continuing to hold that still, other things being equal, it would be good if everyone were to enjoy an equal functioning capability in the society: that, in that sense, equal functioning capability remains an ideal? Surely not. We could shrink from embracing the ideal in this sense only if we reneged on the most fundamental commitment in political philosophy, which is endorsed in all but explicitly fascist or racist or similarly outrageous approaches. This is the commitment to the fundamental equality of human beings: the commitment to their each having a claim, however this is institutionally interpreted, to the same status as others in the eyes of the law and the state (Dworkin 1978). This discussion teaches two important lessons, then. The first is that there is a big difference, from the point of view of cultural sensitivity, between structurally universal and substantively universal ideals. And the second is that the barest commitment to equal human status argues for recognizing the universal validity of a structural ideal like that of adequate functioning capability. I build on the first of these lessons in the next section and on the second in the third. 2. The structural character of freedom as non--‐domination 8 The ideal of equal freedom as non--‐domination There are two aspects to the ideal of freedom as non--‐domination as it applies within a domestic polity. Whether or not there is a state in existence, it suggests that the members of the society - at the least, the adult, able--‐minded, more or less permanent residents - should enjoy equal freedom as non--‐domination in their dealings with one another, whether in individual--‐individual, individual--‐group, or group--‐group interaction. They should each enjoy private non--‐domination, as we may call it. And if there is a state that can be given the task of fostering such private non--‐domination - as, of course, there always will be - then it requires that in interfering coercively with its people, as in taxation, legislation and punishment, the state should be democratically controlled by them in a way that makes the interference non--‐dominating; it should be forced to operate on terms that the people share equally in imposing (Pettit 2012; 2014). As the people should enjoy private non--‐domination in their dealings with one another - for short, social justice - so they should enjoy public non--‐domination - political or democratic justice - in their dealings with the state. In this paper I am concerned with the issue of how plausible it is to take freedom as non--‐domination to be a universal ideal of social justice. I put aside the corresponding question that arises in its role as an ideal of political justice or indeed as an ideal of international justice. The issue is how far we ought to take it to be good that people everywhere should enjoy freedom as non--‐domination in their relationships with their fellow citizens; how far, other things being equal, we should want every state to implement that social ideal. I argue in affirmative response that freedom as non--‐domination is a universal ideal in the structural sense in which functioning capability is a universal ideal. Assume, in line with the fundamental commitment to equality, that people should have a certain equal status in society - that it would be good if they enjoyed such a status - whether this is established on the basis of received custom or coercive law. Each political philosophy will offer its own construal of what such a status requires, thereby putting forward a universal ideal of justice within a society. 9 It will draw on its preferred currency - utility, resources, capability or whatever - in order to give an account of what the ideal of equal status means. Since republican philosophy puts a premium on the value to people of freedom as non--‐domination, it is bound to argue that people everywhere should have an equal status in this regard: that it would be good if people in every society enjoyed equal freedom as non--‐ domination. The question before us now is whether this purportedly universal ideal should be taken to be universal in a substantive or structural sense. I argue in this section that it is a structural ideal that dictates a different content for different social or cultural contexts, resembling Sen's ideal of functioning capability in that regard. And then I argue in the next section that as a structural ideal it does indeed have universal appeal, as indeed the ideal of functioning capability has universal appeal. Short of rejecting the fundamental commitment to equality, we must agree that it would good if people in every society enjoyed equal freedom as non--‐domination in their dealings with one another, under the locally most suitable interpretation of that ideal. Interpreting equal freedom as non--‐domination Whatever equal freedom as non--‐domination is taken to mean, it has to require that people enjoy a certain equality in their freedom to exercise certain choices. In principle people could enjoy such equality in virtue of each enjoying the un--‐dominated exercise of such an individually customized range of choices that none has reason to be envious of others (Dworkin 2000). But in practice there is no prospect of identifying a range of customized option--‐menus, one for each individual, which would satisfy such an envy test. And even if we did identify a set of menus that proved to be satisfactory at a given moment, the chances are that it would soon cease to satisfy, losing its credentials with the changing preferences of members and, of course, with changes in the membership itself. The only scenario in which people might enjoy equal freedom as non--‐ domination is one in which they enjoy the un--‐dominated exercise of the same range 10 of basic choices. If we are to understand the ideal envisaged in republican theory, then, we have to inquire into the choices that ought to be available in this sense to every member or citizen of the society: as we understand citizenship, to every adult, able--‐minded, more or less permanent resident. In the republican tradition itself, especially as that came to be articulated in the seventeenth and eighteenth centuries, the choices that had to be available to every liber or freeman, to use the sexist language of the time, were described, in a phrase deployed by John Libourne (1646), as the fundamental or basic liberties. While these choices were often cast in his time and place as the ancient, historically sacred liberties of Englishmen, they gave institutional expression for Libourne to 'the freeman's freedom'; they reflected the fact that men and indeed women 'are, and were by nature all equal and alike in power, dignity, authority, and majesty - none of them having (by nature) any authority, dominion or magisterial power, one over or above another' (Sharp 1998). What then are the basic liberties? What are the choices to which people must all enjoy un--‐dominated access if they are to count equally as free citizens: if they are to enjoy equal freedom as non--‐domination? I suggest that they should be equated with the choices that it is possible for people each to exercise and enjoy fully - perhaps, to anticipate latter discussion, with plausible state support - consistently with others exercising and enjoying them at the same time. The ideal of the free citizen, which is central to the republican tradition, makes this the obvious way to proceed. If people generally failed to have un--‐dominated access to any co--‐ exercisable and co--‐enjoyable choices, then they would have less than the full range of free choice possible. And if people differed in their un--‐dominated access to those choices, then they would differ significantly in the range of free choice they enjoyed. It is common to think of the basic liberties as choices that are more or less stable across different societies and un--‐dominated access as a form of access that amounts to the same thing in all cultures. If the basic liberties were a stable, similarly accessible set of that kind, however, then the ideal of equal, un--‐dominated access to those choices would constitute a substantively rather than structurally 11 universal ideal. It would hold out the same desideratum for every society: viz, that the members should all be able to enjoy the same level of resourcing and protection in the exercise of the same set of choices. There are two reasons, however, why it would be a mistake to think of the demands of equal freedom as non--‐domination in this way. First, and most importantly, the basic liberties in any society - the choices that are co--‐exercisable and co--‐enjoyable there - are likely to be specific in important ways to that society. And second, the notion of un--‐dominated access is also likely to vary across social and cultural borders. I address these points in turn. The basic liberties When we ask about the choices that are likely to be co--‐exercisable and co--‐ enjoyable in any society, we naturally take account of the fact that two choices often relate to one another in such a way that if we enjoy access to the first we enjoy access to the second, but not vice versa. If you have the freedom to speak your mind to me, then you have the freedom to tell me about your holiday plans, but not vice versa. If you have the freedom to speak your mind to any audience, then you have the freedom to speak your mind to me, but not vice versa. If everyone has the freedom to speak their mind to any audience, then you have the freedom to speak your mind to any audience, but not vice versa. And so on. The freedoms in the first category of these examples are all more general than the freedoms in the second. In seeking after the co--‐exercisable, co--‐enjoyable choices to which equally free people should have the same un--‐dominated access, we naturally focus on the most general choices that fit those conditions. Look after the more general and the less general will look after themselves. This strategy makes it possible to identify at a very abstract level the sorts of choices that ought to be accessible to everyone. Plausible examples will include the following: • The freedom to think what you like • The freedom to express what you think 12 • The freedom to practice the religion of your choice • The freedom to associate with those willing to associate with you • The freedom to own certain goods and to trade in their exchange • The freedom to change occupation and employment • The freedom to travel within the society and settle where you will. But while it may be useful to construct a list like this, it is liable to have the unfortunate consequence of suggesting that the same range of choices has to be accessible to all in every society. That suggestion should be rejected, however. For it turns out on a little reflection that freedoms of the kind described in these abstract categories are certain to be interpreted very differently in different cultures and societies. Almost all of them have the feature we illustrated with Sen's equal capability for functioning. They may be capable of inter--‐cultural definition but they need intra--‐cultural interpretation. The requirements they impose vary from society to society. The basic liberties can vary in either of two ways. First, they can vary in a radical fashion that divides off nomadic from settled communities, and agricultural from industrial societies. And, second, they can vary in a less radical manner that appears in differences between otherwise quite similar dispensations, even between contemporary advanced democracies. The more radical divide shows up in the fact that the choices that would have called to be entrenched as basic liberties in classical Rome or medieval Italy or seventeenth century England - and the choices that would call to be secured in a extremely poor countries today - are very different from the choices that we would expect to be established in a contemporary liberal and democratic regime. The technology and affluence of advanced societies makes it possible for people to be able to co--‐exercise with enjoyment a much richer range of choices from those that would be available elsewhere. This is true for two reasons. First, everyone in such a society can do things, and can be secured in the choice of doing them, that would be unavailable to them elsewhere; think of how we can travel in an advanced society, or express ourselves, or win employment. And second, those who are limited in 13 their capacities can be enabled in an advanced society to exercise choices that would be unavailable elsewhere: think of the impact of eyeglasses or hearing aids or wheel chairs or various prosthetic devices. As functioning capability means something different in radically different cultures, then, so freedom as a person and the basic liberties with which it is associated is also bound to mean something very different across such divides. The idea of having freedom as non--‐domination in dealing with others may be a universally intelligible and attractive ideal - more on this in the next section - but it requires freedom in the exercise of quite different varieties of choice in radically different societies. The Roman commoner, the medieval burgher, the modern yeoman and a contemporary professional might each pass in their own society as enjoying the freedom of a person. But the choices we should expect to be established for each may differ quite deeply across those divides. Not only may the basic liberties vary radically insofar as different cultures offer different possibilities of action and relationship. They may also vary less radically across quite similar societies, even societies as similar as contemporary advanced democracies. There are three sources of this less radical variation, all of which turn on the fact that the basic liberties depend for their articulation on the introduction of conventions that are bound to vary across political divides. The first source of variation is that many basic liberties are co--‐exercisable only because they are defined by culturally variable rules. The outstanding example is provided by the liberties of ownership and exchange. No society can offer its members the choice of appropriating whatever natural or manufactured goods happen to appeal to them, since resources are scarce and in seeking to lay hold of what they want, people would be driven into violent conflict with one another. It makes sense therefore for the society to introduce rules of property that create a related co--‐exercisable liberty that enables people to appropriate whatever they like under the proviso that they abide by the local rules of ownership and exchange. Such rules would mean, in the words of the song, that the cowboy and the farmer can be friends, living under a regime that allows them each to pursue their different 14 uses of the land.2 But rules of property can vary dramatically, creating quite different liberties. They differ across societies in how far they establish public or communal as distinct from private ownership; in what restrictions they impose on the use and exchange of private property; and in the extent to which they create proportional or progressive taxation regimes on income or on wealth (Murphy and Nagel 2004). To argue that the rules of property may vary in this manner is to take a very different line from the traditional conservative and libertarian argument that there are natural property rights that ought to be respected in every society. It may be universally good, and in that sense a matter of natural right, that there should be a non--‐discriminating property regime in every society. But it is nonsense to suggest that, as a constraint of nature, there is just one legitimate way of distinguishing private, communal and public property; one legitimate set of titles to private property, whether titles of production, exchange or inheritance; one legitimate set of usage rights that go with owning something; and one legitimate form of taxation that ought to apply to property holdings and exchanges. Every society is liable to establish its own way of doing things and what republicans emphasize is just the need, as a matter of facilitating freedom as non--‐domination, to put some one regime in place. The cause of freedom as non--‐domination may argue against the formation of certain property conventions, of course, on the grounds that they allow a degree of poverty or inequality that fosters domination. But in all likelihood there are many different sets of conventions that are consistent with that cause. This being so, it should be clear that in arguing for a universal liberty in the sphere of property, we are not committed to a single style and size for all societies. What we are recommending is merely that each society should come up with some rules that allow for individuals to have the same freedom of choice in this domain. While the rules that ought to be implemented should not create imbalances of a potentially dominating sort, they may be as various as the local traditions and customs that they will naturally mirror. 15 What goes for freedom of ownership goes in other areas too. As there are differences in the conventions governing property rights, so there are great differences across societies in conventions bearing, for example, on rights of way, rights of residence, rights of privacy, rights of speech, rights of association, and so on. There may be considerations that argue against some such conventions on the basis of the needs of non--‐domination but it is very likely that most variations are not determinately better or worse than others. Thus we may expect to find differences in the conventions adopted, and the liberties established, even between quite similar and equally commendable societies. Not only are there going to be culturally variable rules needed to establish certain co--‐exercisable liberties, even across similar societies. A second source of variation across similar societies is that there are also going to be culturally variable rules required for making certain co--‐exercisable liberties co--‐enjoyable: making it possible for people to enjoy those liberties, even when all or many exercise them at once. Taking an example from Herbert Hart (1973, 543), consider the case where people each have a choice of addressing a group at will: say, a group comprising their fellow citizens. It will be clearly possible for them each to address the group and to do so at the same time, so that co--‐exercisability is not a problem. But if all or even just a number speak at once, no one will be heard and everyone will be frustrated. The example is artificial but it illustrates a general problem. Everyone may wish to address the group but if everyone addresses it at once then no one is going to be happy. In the same way, everyone may wish to own a gun but if everyone owns a gun then, plausibly, no one is defensively better off. And again, everyone may wish to drive into the city center but if everyone does this at the same time then the point of driving there may be undermined.3 Societies will be able to get over such problems by introducing rules that give people options that are close to the original, problematic options but that still meet the constraint of being co--‐enjoyable. Thus the problem illustrated by the case of speaking to a large group can be solved under rules such as Robert's rules of order. These allow people to take turns in speaking, dictating a pattern under which they 16 can each make proposals, suggest amendments to the proposals of others, and debate and vote on the various issues that arise in their discussion. As such rules might resolve the debating predicament, similar rules might resolve other problems too. For example, people might be given the rule--‐dependent option of owning guns on condition of passing certain, perhaps quite demanding tests; or of using free or cheap parking facilities and taking public transport into city centers; or of speaking their mind on any issue except when it amounts to hate speech and threatens public order. But the rules that get over such problems are very likely to vary between different societies, so that the basic liberties established in relevant areas will almost certainly assume quite different forms in different contexts. A third source of variation in basic liberties, even across broadly similar societies, derives from the fact that in many areas of complex choice there is bound to be a problem in definitively establishing, in the abstract, choices such that their exercise by some is not going to impact on the exercise of those choices by others or on the enjoyment that others derive from that exercise. There is bound to be a problem in finding a set that precludes all such negative interactions (Sen 1970; Dietrich and List 2008). Should people who live along a river have the basic liberty of using the water as they wish? In an agricultural society it may seem that they should each have this liberty. But the issue becomes more complex once there are possible ways of using the river - say, ways of using it in industrial production - that are going to impact negatively on the extent to which the downstream water remains usable for agricultural purposes. This sort of issue is typically resolved in contemporary societies by the courts. The law of torts allows plaintiffs to appeal to the courts for case--‐by--‐case judgments on whether someone should be allowed to exercise such a choice and, if allowed, whether they should be required to adopt precautions against damage to others. The Hand test, named after the U.S. Judge, Learned Hand, offers useful guidance in the area. The idea, roughly, is that if the expected cost of effective precautions to some agent, A, is less than the expected cost to others of A's making the choice without any particular care, then the choice should only be allowed when 17 relevant precautions are in place. Applied to the river case, it would require those upstream to make use of the river only under limitations or conditions that involve a lower expected cost for them than the cost to others of their not being subject to such constraints. The problem of complexity illustrated in this example argues that in general every society will need to establish in law - not just in the law of torts but also in criminal law, contract law and constitutional law - a way of handling different problems as they arise, leaving room for the sort of judicial intervention, or even legislative or constitutional amendment, that would interpret and revise the basic liberties so as to reduce unwanted interactions (Zucca 2007). The necessity of subjecting any system of basic liberties to such dynamic, case--‐driven adjustment means that two similar societies, even societies that begin from the same specification of basic liberties, are likely to come apart in the course of their development. The basic liberties will come to be interpreted differently in each. Not only do our three factors explain why variation in the interpretation of the basic liberties is possible across quite similar societies. The differences between otherwise quite close societies in the cultural expectations people bring to bear on public matters mean that variation is inevitable. In one society, there may be a very lax view of the background tests that gun owners should pass; in the other, the tests may be so strict that most citizens are given no access to guns. In one society people may take a very tolerant view, and in the other a very strict view, of how far offensive speech is damaging and should be restricted. And so on. The considerations rehearsed in this discussion show that equal freedom as non--‐domination is bound to mean significantly different things in different social contexts, even in contexts that differ in the relatively small ways that distinguish contemporary, advanced societies. But not only is it likely to mean different things as a result of variation in the basic liberties that it requires people to be able to access without domination. It is also going to mean different things as a result of differences in what is thought to be necessary for un--‐dominated access. 18 Un--‐dominated access People's access to certain choices will be un--‐dominated to the extent that they are able to make them without being exposed to a capacity for interference - in particular, of course, interference that is not licensed by the interferee - on the part of others. I take this to mean, first, that they must have the wherewithal to take any of the option in a relevant choice - they must have the personal capacity, perhaps with plausible state support, to make the choice; and, second, that they must not be exposed to a capacity for interference on the part of another. The first of these requirements, according to which people must have the wherewithal or capacity to choose any option in a relevant choice, runs counter to the weaker view suggested, for example, by Isaiah Berlin (1969) when he says: 'Mere incapacity to attain a goal is not lack of political liberty' (p.122). But the stronger view defended is more attractive, for two reasons. First, any serious incapacity is likely to expose you to domination, since it may require you to depend on others - and so to be subject to their power of interference - for obtaining the resources you need. And second, there is a paradox involved in holding that you can be free to do something, yet lack the resources and capacity required to do it. The paradox arises on the assumption, endorsed on all sides, that if you are free to do something or not to do it - say, to move elsewhere or stay put - then you are fit to be held responsible for how you choose. Suppose you self--‐ascribe un--‐ dominated access to the options in a certain choice. That means that you must self--‐ ascribe the freedom to take one or another option and the fitness to be held responsible for whatever option you choose. But you could not ascribe to yourself the fitness to be held responsible for whatever you choose, if you think that you do not have the personal wherewithal or capacity for realizing one or another option. Therefore in self--‐ascribing un--‐dominated access to the options in the choice, and so freedom to choose as you wish, you must take yourself to enjoy that wherewithal or capacity (Pettit 2012). 19 Assume that un--‐dominated access in a certain choice requires personal capacity, therefore, and that in a society where the range of the basic liberties is more or less fixed, people must have the wherewithal to make those choices. Turning now to the second requirement mentioned, how are we to determine whether or not people are exposed to a power of interference on the part of others that would make their access to the relevant options a dominated form of access? Exposure to a power of interference on another's part comes in degrees in at least two distinct dimensions. You may be exposed to a power on the part of another to interfere in a more serious or a less serious way, and in a wider or a narrower range of choice. And equally you may be exposed to a power on the part of others that itself comes in degrees: the others may be able to interfere with lesser or greater difficulty, for example, or at lesser or greater peril to themselves. How much exposure to a capacity for interference is going to mean that people are dominated? Or, putting the matter the other way around, how much protection is going to be required in order for them to enjoy the basic liberties without domination? The question is pressing because it is never going to be possible to eradicate altogether the power of others to interfere in someone's choices. I drew on the notion of the free citizen - in traditional language, the liber or freeman - in introducing the basic liberties as choices that all could co--‐exercise and at the same time co--‐enjoy. It was precisely such choices that the free citizen who is celebrated in the republican tradition was expected to be able to access. We can draw again on the image of the free citizen in addressing the question as to what level of protection is going to be enough to give people un--‐dominated access to such choices. In the received republican image, free persons can walk tall, and look others in the eye. They do not depend on anyone's grace or favor for being able to choose their mode of life. And they relate to one another in a shared, mutually reinforcing consciousness of enjoying this independence. Thus, in the established terms of 20 republican denigration, they do not have to bow or scrape, toady or kowtow, fawn or flatter; they do not have to placate any others with beguiling smiles or mincing steps. In short, they do not have to live on their wits, whether out of fear or deference. They are their own men and women and however deeply they bind themselves to one another, as in love or friendship or trust, they do so voluntarily, reaching out to one another from positions of relatively equal strength.4 The free--‐person image suggests that in order to enjoy freedom in the exercise of the basic liberties, people should have a publicly established and acknowledged status in relation to others; only this could enable them to walk tall and look others in the eye. Within the sphere of those liberties people ought to be protected on a public basis - and if necessary, resourced - against the incursions of others. They ought to enjoy objective safeguards that apply regardless of the will of others as to how they should choose in that domain. And it ought to be a matter of shared awareness in the society that they are so guarded. They should have an un--‐ dominated status both in the objective and the subjective or inter--‐subjective sense of status. The public safeguards required for the enjoyment of such status are traditionally taken to include the laws that provide in a saliently equal manner for the entrenchment of people's liberties. But given that universally beneficial laws are likely to be supported by attitudes of approval for compliance and disapproval for non--‐compliance, the safeguards are also bound to include associated norms or morals. Norms in this sense are rules of behavior such that, as a matter of public awareness, most members conform to them, most expect others to approve of conformity or disapprove of non--‐conformity, and most are policed into conformity by this expectation about what will attract approval and disapproval.5 Corresponding to the coercive effect of laws against fraud or violence or murder, we might expect to find norms that occasion a complementary, approbative effect, deterring potential offenders by holding out the prospect of communal disapproval. Machiavelli (1965) remarks in Discourses 1.18 on the importance of having norms available to support the laws in this way: 'just as good morals, if they are to be 21 maintained, have need of the laws, so the laws, if they are to be observed, have need of good morals'.6 But what is enough by way of public safeguards in order to ensure or help ensure that people are not dominated in the exercise of the basic liberties? The free--‐ person image suggests that they should be safeguarded to the point where people satisfy what we might call the eyeball test. They can look others in the eye without reason for the fear or deference that a power of interference might inspire; they can walk tall and assume the public status, objective and subjective, of being equal in this regard with the best. The eyeball test does not require that people should be able look one another in the eye, regardless of their personal lack of nerve. It says that they should have no reason - no good reason - for losing their nerve. What it requires, in other words, is that they have the capacity to look one another in the eye in the absence of factors like timidity or cowardice. The reference to such personal shortcomings is essential, since no public safeguards can compensate for differences between individual personalities and for variations in people's capacity to deal with the overbearing assumptions of others. But there is no cross--‐cultural standard of timidity or cowardice or anything of that kind. What counts as reasonable fearlessness, as distinct from excessive fearfulness or indeed excessive recklessness, is more or less bound to vary from society to society. People who live with different levels of vulnerability, natural or social, tend to be inured in different degrees to various threats and dangers. People who live under different regimes of honor and expectation will vary in how far it seems reasonable or rational to place trust in others, taking them to be sensitive enough to legal or normative sanctions to ensure that interference is not on the cards. And when people vary across societies in how far they are taken to be likely to offend against others, they will also vary in the habits of mind they develop and in the threshold at which they find timidity a prudent attitude to adopt. 22 The eyeball test for when people enjoy un--‐dominated access to their basic liberties implies, then, that what is needed for non--‐domination may vary across cultures and societies. Suppose that a society is one where the benchmark of security is low, the basis for mutual trust is high, and the general presumption is that others are unlikely to wish you harm. In that society it will be much easier to provide safeguards sufficient to enable people to pass the eyeball test than in a society where standards for security against risk are high, or the belief in trustworthiness low, or the presumption about others less flattering. Not only does the eyeball test serve to determine the level of resourcing and protection required for un--‐dominated access. In passing I should also note that it doubles in another role, which is to determine what I described earlier as plausible state support for those who otherwise would not have a personal capacity to exercise certain choices. Should those who require wheel chairs be enabled, if necessary, to purchase them, for example, and should they be facilitated by ramps in public and other buildings? Would such assistance count as plausible state aid? How such a question is to be answered will depend, in the spirit of the republican approach, on whether such provision is necessary in order for people to pass the eyeball test. Time to sum up. As the image of the free person offers direction on the range of choices that free persons ought to be able to access, so it provides guidance on how far those choices ought to secured if people are to enjoy un--‐dominated access to them. And as the direction it offers on the range of choices implies that the basic liberties are going to be interpreted differently in different cultures, so the guidance it provides on the security required for un--‐dominated access suggests that what is enough in one society for such security may not be enough in another. Not only does the interpretation of the basic liberties vary across societies, so does the interpretation of what un--‐dominated access to those liberties requires. 4. The universal appeal of this structural ideal The argument needed 23 In arguing for the universal appeal of the structural ideal of basic functional capability, we relied on two premises. One is the assumption that people are fundamentally equal in the sense that they each have a claim, however this is institutionally interpreted, to the same status as others in the eyes of the law and the state. The adult, able--‐minded and more or less permanent residents of any society deserve to be treated as equals by the collectively supported policies under which they live, where it is understood that being treated as equals can be differently interpreted in different cultures. They are each obliged, to put the assumption in another key, not to claim a special status in relation to their fellow citizens, whether on grounds of ethnicity or religion or gender or any such divide. No one is privileged in the way in which the nobility and the clergy claimed to be privileged in pre--‐revolutionary France. The second premise on which we relied in arguing for the universal appeal of functioning capability is that the ideal of equal status requires that people each have the resources needed for being able to function at a basic level in their society. There is room for interpretation of what exactly equal status requires, as we just mentioned, and there is also room for interpretation of what is required for functioning capability. But no matter how those interpretations go, so we assumed, the equal status of the members of any society demands that they each have the basic capability of functioning in their society. The very characterization of the capability as basic, as we mentioned, suggests that this has to be so. By the first of these two assumptions, the members of any society ought to have equal status in their relationships with one another: there is no denying that this is a good for all. By the second, equal status argues that it would be good if all could enjoy a basic functioning capability. And so the conclusion follows that the members of any society ought each to enjoy a basic functioning capability. It is good from any perspective - good, period - that in every society people should all be furnished with a basic capability of functioning there. The argument offers a model for arguing that not only should people enjoy an equal capability of this kind, they ought also to enjoy un--‐dominated access to the 24 basic liberties. If we are to be able to establish the universal appeal of our republican ideal, then we must be able to show that un--‐dominated access to the basic liberties is on a par in importance with basic functioning capability. Any argument in support of the universal value of freedom as non--‐ domination is entitled to make use of the first premise that appears in the other argument: the premise that equal status, however it should be interpreted, is a good for the people of every society. So the question is whether we can endorse a counterpart of the second premise, arguing that equal status should be interpreted in freedom terms. The question is whether equal status argues in any society for equal freedom as non--‐domination, ruling out variations in people's un--‐dominated access to the basic liberties. If we can argue that equal status does support equal freedom as non--‐domination, then we will be able to make a case for the republican ideal that is parallel to the argument for the universal appeal of the ideal of basic functioning capability. We will be able to re--‐use the first premise of that argument and we will be able to introduce a counterpart of the second. The argument outlined How to show that equal status supports equal freedom as non--‐domination? First we must show that such equal freedom is feasible or possible, at least under the assumption that scarcity is not extreme: there is no bar, under such circumstances of justice, as John Rawls (1971) describes them, to people's all enjoying such equal freedom. And then we must show that such equal status argues that that possibility should be realized. The possibility claim does not raise a problem. The ideal of freedom as non--‐ domination is defined in such a way that there is no bar to all the members of a society enjoying it, at least under the circumstances of justice. It requires universal access to liberties that, by stipulation, are such that all can exercise and enjoy them at the same time. And it requires a form of access that is un--‐dominated in such a sense that, short of great poverty or division, everyone can have un--‐dominated access to the basic liberties. Here the main point to register is that under the eyeball 25 test, differences of psychology or differences of personal power - wealth or fame or connection - need not undermine the un--‐dominated access of the less well off. That test implies that if your personal timidity affects your capacity to look others in the eye, still that doesn't mean you fail the eyeball test and suffer domination: the test requires only that you do not have good reason for fear or deference, not that you do not succumb out of weakness to such attitudes. And the test implies equally that having fewer resources and protections than others - say, as a result of enjoying below--‐average wealth - need not mean that you lack un--‐ dominated access. Excessive differences of wealth and power may jeopardize the freedom as non--‐domination of the less well off, as we noted, and be objectionable on that count. But, assuming that they are not allowed to be excessive, they are consistent with everyone's enjoying un--‐dominated access to the basic liberties. If freedom as non--‐domination, like a basic functioning capability, is feasible or possible for everyone in a society, then the question is whether their equal status argues for the value of everyone's enjoying it. We held in the case of basic functioning capability that insofar as such capability is required for individuals to live without shame as creditable people, it is inconceivable that we should ascribe equal status to persons and yet hold that in some societies it would not be good for everyone to have this capability. The only motive for thinking this would be the recognition that some societies are patriarchal or sectarian or whatever. But this would not be a good reason for maintaining that view, since patriarchy and sectarianism are inconsistent with ascribing an equal status to all human beings. This argument for ascribing universal appeal to the ideal of basic functioning capability suggests a parallel argument for ascribing universal appeal to the ideal of freedom as non--‐domination. Assuming that people are of equal status, it is not only hard to deny that it would be good in every society that all the members should enjoy a basic functioning capacity there; it is hard in the same measure to deny that it would be good that all of the members should enjoy un--‐dominated access to the same basic liberties. For just as the lack of basic functioning capability would 26 undermine people's capacity to live without shame as creditable people, so the lack of freedom as non--‐domination would have the same potentially demeaning effect. The ideal of freedom as non--‐domination is richer, it is true, than the ideal of basic functioning ability. People could hardly count as un--‐dominated - that is, free in the realm of co--‐exercisable and co--‐enjoyable choices - while not enjoying a basic functioning capability in their society. But people could strictly enjoy a basic functioning capability without enjoying such freedom as non--‐domination. They might enjoy that capability as a result of the beneficence of a patron and indeed all might enjoy it as a result of the beneficence of a single potentate or despot. But freedom as non--‐domination requires the absence of a power of interference on the part of others and would rule against any scenario in which some have to hang in this way on the goodwill of others. Where basic functioning capability might seem to be available just because the wealthy or powerful happen to be well--‐disposed towards the poor and powerless - just because the strong happen not to be disposed to interfere but rather to assist - freedom as non--‐domination requires a more robust provision against interference. But this difference of strength between the demands of the two ideals does not undermine the claim that the argument available with capability carries over to freedom as non--‐domination. To enjoy a basic functioning capability just by virtue of the indulgence of another would not be, in Smith's terms, to be able to live without shame as a creditable person. Thus the very consideration at the origin of the case for functioning capability argues equally for requiring that that capability should be robust enough to ensure that enjoying it means enjoying freedom as non--‐ domination. For this reason, indeed, I have argued elsewhere that a charitable interpretation of the ideal of basic functioning capability should lead us to take it as an ideal of a fundamentally republican kind (Pettit 2001).7 By this line of reasoning, then, the ideal of freedom as non--‐domination - the ideal of having an un--‐dominated status in relation to others - can claim a universal appeal on the same grounds as the ideal of basic functioning capability. Like that other ideal it is structural in character: it can be defined inter--‐culturally but is only 27 interpretable intra--‐culturally. And like that other ideal its appeal is as universally compelling as the claim that people in every society should be treated as equals by their state: the claim that in that sense they enjoy equal status. In each case it is supported by the fact that anyone who failed to enjoy the ideal would be unable to live without shame among their peers. It should not be surprising that the appeal of both values should be supported by a recognition of the horror of public shame. For just as having a basic functioning capacity is related to assuming a decent position in relation to others in your society, so the same is true of freedom as non--‐domination. Where the standard conception of freedom as non--‐interference suggests that you might be free in the absence of all other human beings - in a solitary state of nature - the conception of freedom as non--‐domination presupposes that there are other people around and identifies freedom with a publicly entrenched status in relation to those others. Freedom in this sense consists in having a level of public resourcing and protection sufficient by local standards to ensure that you do not have to depend on the goodwill of others to live your own life as you wish: to exercise the basic liberties available in the society. The ideal of having such a free status, like the ideal of having a capability of functioning adequately in your society, turns at bottom on the ideal of commanding the recognition and respect of your peers. Consistently with your enjoying the status of an un--‐dominated, free person, others may seek to influence you in various ways, trying to persuade you of their views, attempting to secure your assistance, or wanting to coopt you into a joint enterprise. But in seeking such influence they are constrained to acknowledge the inaccessibility - and, given the protective law and customs in place, the inadmissibility - of any modes of influence that would impose their will without your license. Thus they are forced to abjure or renounce the unlicensed removal, replacement or misrepresentation of your options in the sphere of the basic liberties. They have no choice but to put aside, and acknowledge putting aside, the use of force or penalty, coercion, deception or manipulation - but not of course teasing or joking or cajoling - in seeking to make you more amenable from 28 their point of view. They are so constrained by the protective field of law and norm that surrounds you that they can hope to influence you only by overt appeal to considerations that you are manifestly in a position to accept or reject: only by engaging you in conversation and co--‐reasoning. Cast in this way, the ideal of freedom as non--‐domination is a natural generalization of the ideal of functioning capability. It gives an interpersonal cast to the status associated with basic functioning capability, linking it with the capacity to command the respect of others. And it deepens the capability at the origin of such interpersonal status, arguing that it must be robust across variations in what you prefer to do and in what others prefer that you do; it must not depend on the indulgence of the powerful. Both forms of generalization are attractive and both argue for crediting republican freedom with the same universal appeal that functioning capability can claim. Conclusion It has become unfashionable to argue in political theory for the universal claims of any ideal, the suggestion being that to attempt such an argument is a sign of ethnocentric insensitivity or even a colonial or imperial mentality. But this is absurd. We are a young species with little to divide us at the genetic level and with a universally shared competence in language and in the exercise of conversation and co--‐reasoning that language facilitates. There is much diversity among human cultures, of course - that is what lends color to our civilizations - but it pales beside the fact that all of us, no matter what our tradition, are addressive and addressable agents. We are conversable creatures and each of us can savor the status we enjoy when, as a matter of shared recognition, others are constrained and committed to abjure any influence over our choices that is not mediated by persuasion and not premised on our acceptance. The appeal of non--‐domination as an ideal of social justice is precisely that, tailored to the conditions of our local culture and society, it would give each of us this great boon in relation to one another. Those with a special interest in the 29 positions of power their local culture gives them may wish to argue that non--‐ domination is not an indigenous value in their society but rather something imposed upon them from outside. But this should not deceive us. There may be variation over time and place in the artistic and religious, the intellectual and the philosophical, expressions of human experience. It is this sort of variation, after all, that requires the ideal of non--‐domination to assume a structural rather than a substantive cast. But there is neither a geography nor a history in our deepest, interpersonal needs. And nothing is deeper than our need to be able to command the respect of others, in particular the respect that ensures us a publicly acknowledged realm of ability and authority. This, by our argument, is nothing more or less than the need for freedom: the need to enjoy non--‐domination in the range of the basic liberties.8 30 References Bellamy, R. (2007). Political Constitutionalism: A Republican Defense of the Constitutionality of Democracy. Cambridge, Cambridge University Press. Berlin, I. (1969). Four Essays on Liberty. Oxford, Oxford University Press. Besson, S. and J. L. Marti (2008). Law and Republicanism. 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Oxford, Oxford University Press. Zucca, L. (2007). Constitutional Dilemmas: Conflicts of Fundamental Legal Rights in Europe and the USA. Oxford, Oxford University Press. 1 On the republican research program see (Lovett and Pettit 2009). The recent republican movement began from the historical work of Quentin Skinner (1978) on the medieval foundations of modern political thought, and from his subsequent articles in the 1980's on figures like Machiavelli who wrote within the republican tradition identified by John Pockock (1975). An up--‐to--‐date list of English works in contemporary republican thinking should include these books: (Pettit 1997; Skinner 1998; Brugger 1999; Honohan 2002; Viroli 2002; Maynor 2003; Lovett 2010; Marti and Pettit 2010; MacGilvray 2011); these collections of papers: (Van Gelderen and Skinner 2002; Weinstock and Nadeau 2004; Honohan and Jennings 2006; Laborde and Maynor 2007; Besson and Marti 2008; Niederbeger and Schink 2012); and a number of studies that deploy the conception of freedom as non--‐domination, broadly understood: (Braithwaite and Pettit 1990; Richardson 2002; Slaughter 2005; Bellamy 2007; Bohman 2007; Laborde 2008; White and Leighton 2008). 2 The farmer and the cowboy will be familiar from Rogers and Hammerstein's musical, Oklahoma, but the predicament they exemplify was already a matter of human experience in clashes between farming and foraging peoples, some as early as the fifth millenium BCE; see (Morris 2010, 112--‐14, 127--‐28, 271). 33 3 In these cases, as in the case of speaking to the group, it is generally true: first, that everyone has a reason to pursue the activity if no one else does; and second, that everyone prefers that no one pursue it to everyone's pursuing it. But it is worth noting that in some of the cases, unlike the group case, a third clause is satisfied too: everyone may have a reason-a new reason-to pursue the activity if others all do so. Setting virtue aside, no one will relish being the only person without a gun in a gun--‐toting society. This makes these particular examples into cases of a broadly free--‐riding character (Pettit 1986). In these cases people each have a reason for pursuing the activity even if all others do - they will not want to be made a sucker, as it is sometimes said - but that reason is not the consideration that originally gave the choice its appeal. 4 Presented in this way, the image of free persons may seem to be silent on the political front, implying nothing about people's political rights or responsibilities. But the presentation is adequate for our purposes, since we are abstracting in this discussion from the relation between citizens and government. 5 The definition follows (Pettit 1990; Brennan and Pettit 2004, Part III) with one amendment: it says that a norm is a regularity such that almost everyone expects others to approve of conformity rather than, in the older formulation, that it is a regularity such that almost everyone approves of conformity. The change allows us to recognize as norms regularities that, unbeknownst to people, do not actually attract general approval (Prentice and Miller 1993); for a fuller discussion of this possibility see (Pettit 2008). While I do not offer a defense for defining norms in this way, it should be noted that it fits extremely well, particularly in the way it connects norms and approval, with the understanding of norms in the larger literature (Hart 1961; Winch 1963; Coleman 1990; Sober and Wilson 1998; Elster 1999; Shapiro 2011). 6 See too (Tyler 1990). 7 For further important considerations on the nature of freedom as non--‐domination, and on its connection with Sen's ideal, see (List 2004; 2006) 34 8 I was enormously assisted in my thinking about the topics of this paper by participating in a workshop at the University of Korea in May 2012, which was organized by Professors Gil--‐Sung Park and Jun--‐Hyeok Kwak. I am indebted to the comments of the other participants, especially Professor Kwak, on a draft of this paper.

irit samet, Equity: Conscience Goes to Market (Oxford: Oxford University Press, 2018)* Irit Samet's Equity develops a novel and philosophically rich interpretation of the body of law originating in the English Court of Chancery, the body of law known as 'Equity.'1 Equity began as a response to particular cases where the rigid procedures of the common law led to substantive injustice. In such cases, the Court of Chancery could intervene to force individuals to act according to the dictates of 'conscience.' As Lord Ellesmere LC wrote, 'when a Judgment is obtained by Oppression, Wrong and a hard Conscience, the Chancellor will frustrate and set it aside, not for any error or Defect in the Judgment, but for the hard Conscience of the Party.'2 Over time, Equity was itself systematized, and in the 1870s the separate courts responsible for Equity and common law were fused: from then on, one court could apply both bodies of law. But many Equitable doctrines retained the marks of their origin: broad principles, stated in morally freighted language ('conscience,' 'clean hands,' 'loyalty,' and so on) and applied in a backwards-looking and fact-sensitive way (xv).3 The fusion of Equitable and common law courts raised the question whether the substantive bodies of law developed by these courts ought to be fused as well. Samet's aim, in this constructive and tightly argued book, is to defend Equity against the fusionists. We can reconstruct her argumentative strategy in two steps, which rebut different strands of the fusionist project (120). Some fusionists maintain that the separation of Equity from common law is a historical accident, so that little of substance would be lost by formalizing its rules in the style of property and contract.4 Others grant that Equity pursues a distinctive normative ideal but suggest that this ideal is unjustifiable in a modern legal order.5 Samet argues, first, that Equity is not just a grab bag of doctrines that happen to share an origin; rather, it is unified around a distinctive normative ideal. This ideal would be undermined in a fused legal system, because it requires the flexible and particularistic approach that is typical of Equity as opposed to common law (2). Second, Samet argues that the normative ideal served by Equity is still worth BOOK REVIEW * All parenthetical page references are to this text. I would like to thank Larissa Katz, Ngozi Okidegbe, Gurpreet Rattan, and Richa Sandill for their comments on an earlier draft of this review. 1 In this review, I will follow the convention of using 'Equity' to refer to the law (both procedural and substantive) originating in Courts of Chancery and 'equity' to refer to the concept of doing justice in particular cases. 2 Earl of Oxford's Case (1615), 21 ER 485. 3 On 'backwards-looking,' see Thorner v Major, [2009] UKHL 18 at para 101. 4 See Andrew Burrows, 'We Do This at Common Law But That in Equity' (2002) 22:1 Oxford J Leg Stud 1 [Burrows, 'We Do This']; Sarah Worthington, Equity, 2d ed (Oxford: Oxford University Press, 2006), ch 1. 5 For related discussion, see Peter Birks, 'Equity in the Modern Law: An Exercise in Taxonomy' (1996) 26:1 UWA L Rev 1 at sections 10, 53. 10.3138/utlj.2019-0084 e20190084 70 2 Season (Spring 2020) 70 UTLJ © UNIVERSITY OF TORONTO PRESS DOI: 10.3138/utlj.2019-0084 h ttp s: //u tp jo ur na ls .p re ss /d oi /p df /1 0. 31 38 /u tlj .2 01 900 84 F ri da y, M ar ch 1 3, 2 02 0 5: 49 :4 1 A M U ni ve rs ity o f T or on to I P A dd re ss :1 38 .5 1. 22 9. 2 pursuing in a modern legal order (30). Other things being equal, then, we ought to keep Equity separate from common law. The book's first chapter surveys the fusion debate, sets out the main lines of Samet's account of Equity and responds to fundamental objections. In the next three chapters, Samet offers detailed interpretations of three areas of Equitable doctrine: proprietary estoppel, fiduciary law, and the 'clean hands' principle. These interpretations serve both to articulate Samet's account of the purpose of Equity and to support her claim that this purpose would be undermined if Equitable doctrines were fused with the common law. Along the way, Samet engages with scholarship not only in law but also in moral philosophy, political theory, and economics. The range and detail of Samet's book is a strength, but it means that a review like this one will have to leave a lot out. I will focus on the core of Samet's account and raise two questions, corresponding to the first and second steps in my reconstruction of her argument. First, how far does Samet's proposed ideal fit with the doctrines of Equity? Second, does it justify retaining those doctrines against the fusionists? I will suggest that, though its immediate aim is to respond to fusionism, Samet's work also brings into the open a larger controversy about the purpose of private law. The ideal which, on Samet's account, Equity serves to promote is Accountability Correspondence (AC): 'When legal rules impose liability it should ideally correspond to the pattern of moral duty in the circumstances to which the rules apply' (28). AC is very general: it says nothing about what the patterns of moral duty in fact are, or in which circumstances legal rules should impose liability, but only specifies that liability ought to correspond to moral duty. Indeed, we might worry that AC is too general to be distinctive of Equity: what about other apparently morally laden bodies of law, such as tort or even criminal law? Surely Equity aims to enforce a special class of moral duties rather than moral duties as such? I will return to this worry below. For now, we can get a sense of the role that AC plays by contrasting it with another, more familiar legal ideal, the Rule of Law (ROL): 'an exemplary state of affairs wherein the government in all its actions is bound by legal norms fixed and announced beforehand so that people can foresee with fair certainty how the authority will use its coercive powers in given circumstances' (16). Samet suggests that much of private law, such as the law of property and contract, is composed of general, precise rules so as to promote the ROL (33). AC has a parallel relation to the particularistic, flexible rules of Equity. Samet does not deny that the ROL is a good thing. Nor does she deny that its demands can be in tension with those of AC (42). How, then, do the two ideals relate to each other? The answer is simple: AC is a value as well as the ROL, and sometimes the latter ought to give way to the former (74). A legal order might satisfy various ROL criteria, with rules that are general, clear, prospective, non-contradictory, and so on – and still be defective.6 This reflects the formality of the ROL, which specifies the form of a legal order while leaving open – at least to a large degree – what the content of the laws ought to be. Moreover, a body 6 For these and other rule of law criteria, see Lon Fuller, The Morality of Law (New Haven, CT: Yale University Press, 1964), ch 2. BOOK REVIEW 217 (Spring 2020) 70 UTLJ © UNIVERSITY OF TORONTO PRESS DOI: 10.3138/utlj.2019-0084 h ttp s: //u tp jo ur na ls .p re ss /d oi /p df /1 0. 31 38 /u tlj .2 01 900 84 F ri da y, M ar ch 1 3, 2 02 0 5: 49 :4 1 A M U ni ve rs ity o f T or on to I P A dd re ss :1 38 .5 1. 22 9. 2 of law composed of highly formal rules might allow for behaviour which, though technically legal, evades the law's point. When faced with such 'creative compliance,' regulators in tax and finance have responded using a 'substance over form' approach similar to that of AC (39). Samet suggests that AC is closely tied to the language of 'conscience' which permeates Equity: 'The concept of conscience is fundamental to Equity since it beautifully expresses its role as an advocate of Accountability Correspondence, as well as the mode of reasoning it employs in order to impel our legal system in the direction of this ideal' (43; emphasis in original). She develops a sophisticated account of conscience, building on Immanuel Kant's moral psychology. It combines a meta-ethical claim – that there are objective moral principles – with an epistemic one – that ordinary individuals are able to know these principles. The trouble comes in applying these principles to particular circumstances. Here we can be led astray by our self-interest, which leads us to speciously justify our conduct to ourselves. Conscience, on Samet's account, allows us to become aware of our self-serving rationalizations (56). A court of Equity, presupposing that individuals have a conscience, exists to hold them to what they know is right: to ' correct Mens Consciences for Frauds, Breach of Trusts, Wrongs and Oppressions, of what Nature soever they be.'7 This is why Equity will intervene 'only where sincere engagement with the moral aspects of [the individual's] behaviour would readily reveal its illicit nature' (61). To see how this works, consider Samet's interpretation of fiduciary law. Samet argues that fiduciary law creates a space for a distinctive kind of relationship centred on selflessness rather than self-interest (125); the language of 'loyalty' and 'conscience' serves as a reminder – both to the court and to the fiduciary – of the moral duties that the fiduciary owes to the principal (150). (Kantian moral psychology, in which self-interest distorts our application of moral rules to our circumstances, seems especially apt as a description of the fiduciary who has to decide whether they stand in a conflict of interest.) Supposing that the fiduciary does have a moral duty of selfless conduct in favour of the principal, a body of law that enforces this duty will thereby promote AC. This feature would be lost if, as fusionists propose, fiduciary law were simply folded into contract law as a set of implied terms for various types of contract, since parties to an ordinary contract are not in general obliged to behave selflessly (130). We might wonder, though, whether this sort of account, which places great emphasis on the duties of the fiduciary, and comparatively little on the rights of the principal, could be extended to the neighbouring area of Equitable property rights, such as the right of a beneficiary under a trust. For it is unclear how any amount of moral or legal duty on the part of the trustee alone could generate a proprietary right on the part of the beneficiary. Samet addresses this point in the introduction to her book, where she explains why the book does not discuss trust law (xvii). One reason for this omission is dialectical: the book is a defence of Equity against the fusionists, and trust law is not a popular candidate for fusion.8 7 Earl of Oxford's Case, supra note 2. 8 See e.g. Burrows, 'We Do This,' supra note 4 at 5. 218 UNIVERSITY OF TORONTO LAW JOURNAL (Spring 2020) 70 UTLJ © UNIVERSITY OF TORONTO PRESS DOI: 10.3138/utlj.2019-0084 h ttp s: //u tp jo ur na ls .p re ss /d oi /p df /1 0. 31 38 /u tlj .2 01 900 84 F ri da y, M ar ch 1 3, 2 02 0 5: 49 :4 1 A M U ni ve rs ity o f T or on to I P A dd re ss :1 38 .5 1. 22 9. 2 Another reason, however, is that trust law – with its precise rules and its strict liability for trustees – is simply a poor fit for the book's explanatory framework. Here as elsewhere, Samet shows an undogmatic willingness to allow that different bodies of law might need to be explained in different ways. Still, the resulting situation is not fully satisfactory. Not only is trust law a central part of the law historically developed by courts of Equity: it also has clear continuities with the more explicitly conscience-based bodies of law which Samet treats (xvii). For example, in Westdeutsche Landesbank Girozentrale v Islington LBC, Lord Browne-Wilkinson took it as 'uncontroversial' that 'the equitable jurisdiction to enforce trusts depends upon the conscience of the holder of the legal interest being affected.'9 It would be worth considering how much of trust law could, in fact, be explained by Samet's theory. This might require Samet to address a question on which her book appears to remain neutral. The book addresses itself to the normative question of whether Equity's separateness is justified, or whether it ought to be fused with the common law. It says little about the more conceptual question of what Equity is, or what distinguishes Equitable rights from rights at common law. For example, what is the nature of Equitable title, and how is it distinct from common law title? This is the sort of thing at issue between theorists like Frederic William Maitland, who argue that Equitable rights are always personal, and those like James Penner, who argue that Equitable title is a property right, just as it sounds.10 A version of Maitland's theory has recently been developed by Robert Stevens and Ben McFarlane, who suggest that Equitable rights are 'rights in rights' – for example, if A holds land in trust for B, then A has a right to the land and B has a right in A's right.11 My suspicion is that Samet's account is less neutral on this question than it might appear. Her focus on conscience as the animating idea of Equity could be taken to underpin the 'obligational' theory of Maitland, Stevens, and McFarlane, on the basis that it is precisely by operating on the conscience of rights-holders that Equity creates new rights in rights. If this is correct, the work of these theorists may offer a way of extending Samet's theory to explain at least the main principles of trust law. Moreover, to the extent that Samet provides a justification of Equity, we might wonder whether those principles that are not explicable by her theory appear defective in this light. Supposing that Equity is best understood as promoting AC, a different sort of fusionist may doubt that AC is worth promoting. Is it really acceptable to exercise the coercive power of the state to make people carry out their moral duties? If so, is it acceptable for moral duties as such, or only for those belonging to some 9 Westdeutsche Landesbank Girozentrale v Islington LBC, [1996] UKHL 12. On the role of conscience in the constructive trust, see Soulos v Korkontzilas, [1997] 2 SCR 217 at paras 29, 43. 10 Frederick W Maitland, Equity: A Course of Lectures (Cambridge, UK: Cambridge University Press, 1969), Lecture X [Maitland, Equity]; James E Penner, 'The (True) Nature of a Beneficiary's Equitable Property Interest under a Trust' (2014) 27:2 Can JL & Jur 473. 11 Robert Stevens & Ben McFarlane, 'The Nature of Equitable Property' (2010) 4:1 J Equity. See also Lionel D Smith, 'Trust and Patrimony' (2008) 38:2 Revue générale de droit 379. BOOK REVIEW 219 (Spring 2020) 70 UTLJ © UNIVERSITY OF TORONTO PRESS DOI: 10.3138/utlj.2019-0084 h ttp s: //u tp jo ur na ls .p re ss /d oi /p df /1 0. 31 38 /u tlj .2 01 900 84 F ri da y, M ar ch 1 3, 2 02 0 5: 49 :4 1 A M U ni ve rs ity o f T or on to I P A dd re ss :1 38 .5 1. 22 9. 2 special class? One reason that Samet gives for AC is that a legal order which satisfies it will be perceived to be legitimate and that this perception 'translates into willingness to obey the law' (31).12 But a deeper reason can be found in the broad tradition in which Samet is working. In this tradition, which originates with Joseph Raz and John Gardner, not only is it acceptable for law to serve morality: the authority of law depends on its doing so.13 A legal order which satisfies AC will not only appear more legitimate, but will be more legitimate. Relying on this view of the relation between law and morality is not a weakness in Samet's account, but I want to suggest that certain objections to the view emerge with particular force in the present context. We should distinguish two reasons for concern about the justifiability of AC. The first is based on doubts about moral knowledge: either there are no moral truths or, if such truths exist, we have no way of coming to know them as would be required by a court of Equity. While moral anti-realism and scepticism may have implications for general jurisprudence, it seems to me that they have little specific bearing on the theory of Equity. In any event, Samet defends her moral realism, making reference to relevant work in meta-ethics in an appendix (197). She does less to address a second reason for concern about AC: namely, that it is inconsistent with the neutrality that is required of a liberal state. As John Rawls argued in Political Liberalism, modern societies are characterized by 'a pluralism of incompatible yet reasonable comprehensive doctrines.'14 People are committed to deeply divergent world-views, and a modern legal order is constrained by this fact. Rawls suggested that laws should be justified only in terms of 'public reason,' which is neutral on certain fundamental questions.15 Two centuries earlier, Kant defended an even stronger version of liberal neutrality in which questions of morals are to be kept apart from questions of right. Lawmakers should not ask what people ought to do but, rather, what they can legitimately be coerced to do.16 Even if we can know moral truths, then, it might not be legitimate for the law to coerce people to act in accordance with them. Fusionists may argue that the very features that, in Samet's view, make Equity distinctive – its morally freighted language and flexible application – also make Equity distinctively problematic from the point of view of liberal neutrality. This issue becomes particularly pressing when we consider Samet's discussion of the 'clean hands' doctrine. When granting Equitable relief to a plaintiff would involve the court in some wrongdoing – for example, by allowing the plaintiff to carry out a fraud – the court may refuse its help on the principle that 'he who comes to Equity must come with clean hands.' Samet interprets this doctrine 12 For a similar argument, see Matthew Harding, 'Equity and the Rule of Law' (2016) 132 Law Q Rev 278 at 297 [Harding, 'Equity']. 13 See Joseph Raz, The Morality of Freedom (Oxford: Oxford University Press, 1988), ch 3; John Gardner, Law as a Leap of Faith (Oxford: Oxford University Press, 2012), ch 6. 14 John Rawls, Political Liberalism (New York: Columbia University Press, 1993) at xvi. 15 See John Rawls, 'The Idea of Public Reason Revisited' (1997) 63:4 U Chicago L Rev 765. 16 Arthur Ripstein, Force and Freedom (Cambridge, MA: Harvard University Press, 2009) at 14; c.f. Japa Palikkathayil, 'Neither Perfectionism nor Political Liberalism' (2016) 44:3 Philosophy & Public Affairs 171. 220 UNIVERSITY OF TORONTO LAW JOURNAL (Spring 2020) 70 UTLJ © UNIVERSITY OF TORONTO PRESS DOI: 10.3138/utlj.2019-0084 h ttp s: //u tp jo ur na ls .p re ss /d oi /p df /1 0. 31 38 /u tlj .2 01 900 84 F ri da y, M ar ch 1 3, 2 02 0 5: 49 :4 1 A M U ni ve rs ity o f T or on to I P A dd re ss :1 38 .5 1. 22 9. 2 by way of a creative analogy with Bernard Williams's notion of integrity.17 For Williams, this notion expresses our concern not only with what happens in the world, but also with our contribution to what happens, and with whether our contribution is consistent with our deepest ethical commitments. Samet suggests, similarly, that in certain cases the court must refuse its help to a plaintiff purely in order to safeguard its integrity as an institution committed to justice (180). This seems right, but can a court invoke its integrity in relation to just any kind of serious wrongdoing? Samet raises the possibility that courts might use the doctrine to exclude greedy company executives who seek to enforce contracts for 'overtly excessive remuneration' (155) or to reject claims 'made in the context of using premises for prostitution' (182). The only common factor seems to be that the court considers these things to be seriously wrong. In this way, the 'clean hands' doctrine threatens to allow for unconstrained moralism in the law. In response to this concern, we might try to draw on some commonplace ideas about Equity to narrow the range of relevant moral duties. Like many other Equitable doctrines, the 'clean hands' doctrine is second-order law, operating on top of a prior and complete set of legal rules.18 It responds to a specific kind of injustice – the sort which occurs when, as Georg Wilhelm Friedrich Hegel wrote, a legal process, in itself in any case a means, now begins to be something external to its end and contrasted with it. This long course of formalities is a right of the parties at law and they have the right to traverse it from beginning to end. Still, it may be turned into an evil, and even an instrument of wrong . . . .19 In other words, Equity responds to the way that the formalization of justice can itself work injustice.20 If this is right, then the moral duties enforced by Equity are of the same kind as those that the common law aims to enforce, and there is no prospect of unconstrained moralism. It is not clear whether Samet would, or could, agree with this suggestion. Her canonical statement of AC, which makes Equitable intervention conditional on legal rules which already impose liability, might be read in this way (28). But her accounts of proprietary estoppel and fiduciary relationships pull in the opposite direction. While property and contract rules provide a framework for equal freedom under law, these domains of Equity hold us to higher standards of interpersonal conduct. More importantly, then, reading Samet in line with commonplace 17 JJC Smart & Bernard Williams, Utilitarianism: For and Against (Cambridge, UK: Cambridge University Press, 1973), s 5. 18 Maitland, Equity, supra note 10, Lecture XII. The idea of Equity as 'second-order' law, responding to opportunistic uses of legal rules, is from Henry Smith. See Henry Smith, ' Equity as Second-Order Law: The Problem of Opportunism' (2015) [unpublished], online: <ssrn.com/abstract=2617413>. We might see the 'clean hands' doctrine as third-order law, responding to opportunistic uses of Equitable rules. 19 Georg WF Hegel, Hegel's Philosophy of Right, translated by Thomas M Knox (Oxford: Oxford University Press, 1968), s 223. 20 Harding, 'Equity,' supra note 12 at 301; Dennis Klimchuk, 'Equity and the Rule of Law' in Lisa M Austin & Dennis Klimchuk, eds, Private Law and the Rule of Law (Oxford: Oxford University Press, 2014) 247. BOOK REVIEW 221 (Spring 2020) 70 UTLJ © UNIVERSITY OF TORONTO PRESS DOI: 10.3138/utlj.2019-0084 h ttp s: //u tp jo ur na ls .p re ss /d oi /p df /1 0. 31 38 /u tlj .2 01 900 84 F ri da y, M ar ch 1 3, 2 02 0 5: 49 :4 1 A M U ni ve rs ity o f T or on to I P A dd re ss :1 38 .5 1. 22 9. 2 ideas about Equity would blunt the political edge of her account. The role of Equity, at least in part, is not to perfect the common law's pursuit of its own ideals, but to promote a different ideal. The debate with the fusionist is really 'a political controversy about the role of the state' (151). Samet's Equity brings this controversy into the open. Manish Oza University of Toronto Department of Philosophy  https://orcid.org/0000-0002-1807-5233 222 UNIVERSITY OF TORONTO LAW JOURNAL (Spring 2020) 70 UTLJ © UNIVERSITY OF TORONTO PRESS DOI: 10.3138/utlj.2019-0084 h ttp s: //u tp jo ur na ls .p re ss /d oi /p df /1 0. 31 38 /u tlj .2 01 900 84 F ri da y, M ar ch 1 3, 2 02 0 5: 49 :4 1 A M U ni ve rs ity o f T or on to I P A dd re ss :1 38 .5 1. 22 9.

DESIRING TO DESIRE: RUSSELL, LEWIS AND G.E MOORE1. By Charles R. Pigden 1. Moore's warning 'I am the person Moore warned you against,' joked David Lewis2 before reading his now famous paper, 'Dispositional Theories of Value', to the Aristotelian Society in 19893. The alleged warning occurs in §13 of Principia Ethica4, the crucial passage in which Moore expounds the Open Question Argument. 'To take, for instance one of the more plausible, because one of the more complicated of such [naturalistic] definitions, it may easily be thought, at first sight, that to be good may mean to be that which we desire to desire.' (PE: §13.) But plausible as it may be, Moore goes on to contend that this definition is false, since it is possible to wonder whether what we desire to desire is good. If 'good' meant 'what we desire to desire', the question 'Is what we desire to desire good?' would be a silly question since the answer would be very obvious – 'Yes'. Since the question is open and the answer is not obvious to every competent speaker, the definition cannot be correct. 1 1 This paper is dedicated to the memory of David Lewis 2 So I was told by an ear-witness, though Lewis himself had no recollection of the remark. However he certainly subscribed to the sentiment. 'The position defended is similar to the one that G. E. Moore chose as the target for his "naturalistic fallacy" argument.' See Lewis, David, Papers in Ethics and Social Philosophy (Cambridge: Cambridge University Press, 2000), 2. 3 Lewis, David, 'Dispositional Theories of Value', Proceedings of the Aristotelian Society Supplementary Volume, 63 (1989), 113-137. Reprinted in Lewis, David, Papers in Ethics and Social Philosophy (Cambridge: Cambridge University Press, 2000), 68-94. Henceforward DTV II with page references both to the original and to the reprint. 4 Moore, G. E. Principia Ethica, revised edn., ed. Thomas Baldwin, (Cambridge: Cambridge University Press, 1993). Henceforward PE. On the whole I give section rather than page references to accommodate those with earlier editions. Now Moore's 'warning' certainly applies to Lewis since he develops an analysis of value as what we are ideally disposed to desire to desire (an analysis which would fall foul of the Open Question Argument if that argument were sound). But Moore can hardly have had Lewis in mind when he penned this notorious passage in the early 1900s. Since he gives no citations it is tempting to suppose that Moore plucked his opponent out of thin air. But in fact, the person Moore warned us against was neither David Lewis nor Mr. Nobody but Bertrand Russell. For the definition of 'good' selected for dissection is precisely the definition suggested by Russell in 'Is Ethics a Branch of Empirical Psychology' a paper read to the Apostles in 18975. (RoE: 71-78/Papers 1: 100-104.) 'The criterion [of morality] must be supplied, therefore, by the contrast between ideal and actual desires, by the contrast between the desires we desire and desires we dislike.' Moore does not just criticize Russell's definition in 'Is Ethics a Branch of Empirical Psychology?' he criticizes the chief thesis of Russell's paper, namely that ethics is indeed a branch of empirical psychology. This thesis follows fairly obviously from the definition, since if 'good' means what we desire to desire, to find out what is good we need to ascertain what we desire to desire, which is a matter of psychological fact. And since [normative] ethics largely consists in the enquiry into what is good, ethics becomes a 'matter for purely psychological investigation' (RoE: 78/Papers 1: 104). The conclusion follows whether goodness consists in what we (the community, reasonable people or whatever) desire to desire (which is roughly Lewis's line), or whether goodness for each individual consists in what that person desires to desire (which is what Russell seems to suggest). Moore is very severe with this sort of thing. Naturalism, he says, consists in fixing on some natural property and then supposing that 'to be "good" means to 2 5 Most the works of Russell referred to in this paper are collected in Pigden, Charles R. (ed.) Russell on Ethics (London: Routledge, 1999), henceforward RoE. References are to RoE and to the relevant volumes of The Collected Papers of Bertrand Russell (abbreviated, for example, as Papers 1). possess the property in question ... thus replacing Ethics by some one of the natural sciences'. 'In general,' explains Moore, 'Psychology has been the science substituted, as by J.S. Mill' (PE: §26). Moore does not add 'and by Mill's secular godson, Bertrand Russell', presumably because the substituting was done in a confidential paper read to a secret Society (the Apostles). Moore was very scrupulous about keeping the Society secret, so much so that he was worried about discussing its doings by postcard. See Griffin (2002: 186). But the consequence was that until very recently nobody realized that at least one of Moore's targets in Principia Ethica was Bertrand Russell. I have two aims in this paper. In §§2-4 I contend that Moore has two arguments (not one) for the view that that 'good' denotes a non-natural property not to be identified with the naturalistic properties of science and common sense (or, for that matter, the more exotic properties posited by metaphysicians and theologians). The first argument, the Barren Tautology Argument (or the BTA), is derived, via Sidgwick, from a long tradition of antinaturalist polemic. But the second argument, the Open Question Argument proper (or the OQA), seems to have been Moore's own invention and was probably devised to deal with naturalistic theories, such as Russell's, which are immune to the Barren Tautology Argument. The OQA is valid and not (as Frankena (1939) has alleged) question-begging. Moreover, if its premises were true, it would have disposed of the desire-to-desire theory. But as I explain in §5, from 1970 onwards, two key premises of the OQA were successively called into question, the one because philosophers came to believe in synthetic identities between properties and the other because it led to the Paradox of Analysis. By 1989 a philosopher like Lewis could put forward precisely the kind of theory that Moore professed to have refuted with a clean intellectual conscience. However, in §§6-8 I shall argue that all is not lost for the OQA. I first press an objection to the desire-to-desire theory derived from Kripke's 3 famous epistemic argument. On reflection this argument looks uncannily like the OQA. But the premise on which it relies is weaker than the one that betrayed Moore by leading to the Paradox of Analysis. This suggests three conclusions: 1) that the desire-to-desire theory is false; 2) that the OQA can be revived, albeit in a modified form; and 3) that the revived OQA poses a serious threat to what might be called semantic naturalism. 2. Moore's two arguments Though Moore managed to convert Russell to non-naturalism (RoE: 73 & 75-104), there is reason to suspect that the desire-to-desire theory continued to be a worry. It is not always noticed that Moore has not one but two distinct arguments against naturalism, the Open Question Argument and the Barren Tautology Argument6. The first contends that 'good' cannot be synonymous with any naturalistic predicate 'X' since 'Are X things good?' is a significant or open question for every 'X'. The second contends that 'good' cannot be synonymous with any naturalistic 'X', if 'X things are good' is supposed to be a reason for action rather than a 'barren tautology'. The first is set forth at PE: §13, whilst the second crops up at PE: §11, though variants of it recur throughout the first four chapters (PE: §§14, 24 & 26). Russell (who was rather more succinct than Moore) summarizes it thus: Chapter II, on Naturalistic Ethics, discusses theories which hold that the only good things are certain natural objects, in so far as these theories are advocated as derivable from the very meaning of good. It is shown that such theories always confuse good, in its correct and indefinable sense, with the 4 6 It seems to me that the discussion of Moore in Darwall, Gibbard and Railton's justly famous 'Towards Fin de Siecle Ethics: Some Trends', The Philosophical Review, 101 (1992), 115-189, is vitiated by a failure to distinguish clearly between the Open Question Argument and the Barren Tautology Argument. sense which they assign to it by definition. For example, Evolutionist Ethics are apt to argue that good means more evolved, and on this to base practical recommendations. Yet, if their contention were correct, no practical consequences could follow. We ask: Why should I prefer this to that? And they reply: Because the more evolved is the better. But if they were right in the reason they give for thinking so, they have only said that the more evolved is the more evolved; and this barren tautology can be no basis for action. The meaning of two phrases cannot be the same, if it makes any difference whether we use one of them or the other; and, applying this test, it is easy to see that more evolved does not mean the same as better. (RoE: 100/Papers 4: 572.) More formally, we can restate the argument as follows: 1#) For any naturalistic or metaphysical 'X', if 'good' meant 'X', then (i) 'X things are good', would be a barren tautology equivalent to (ii) 'X things are X', or (iii) 'Good things are good'. 2#) For any naturalistic or metaphysical 'X', if (i) 'X things are good', were a barren tautology, it would not provide a reason for action (i.e. a reason to pursue or promote X-ness). 5 3#) So for any naturalistic or metaphysical 'X', either (i) 'X things are good', does not provide a reason for action (i.e. a reason to promote X-ness), or 'good' does not mean 'X'. To put the point another way: 3#') For any naturalistic or metaphysical 'X', if (i) 'X things are good', provides a reason for action (that is, a reason to promote X-ness), then 'good' does not mean 'X'. In other words, if you want the basic principles of your naturalistic ethic to be true by definition, they can't at the same time be action-guiding. (Note: this argument does not entail or presuppose that factual considerations cannot provide reasons for action.) This is, I think, the real argument for the naturalistic fallacy, since it suggests that most naturalists actually commit an intellectual mistake that can reasonably be described as a fallacy they propound as a reason for action some such principle as 'X things are good', or even 'Only X things are good', and then try to defend it by claiming that it is some sort of analyticity, 'the very meaning of the word', etc. (See Prior, A. N. (1949: chs. 1 & 9) and PE: 11, 24.) But this is to subvert the action-guiding power of their original pronouncement. It cannot both be that 'X things are good', is analytic (and thus secure from all shocks) and that it provides a reason for promoting X-ness. To suppose that it can, or to propound such an inconsistent view, is to make a mistake in reasoning that might reasonably be dubbed 'the naturalistic fallacy'. 6 But on this reading, not all forms of naturalism are fallacious. In some cases the suggested 'X' is not supposed to denote a property that its proponent wants to see promoted. Rather the 'X' constitutes an analysis of 'good' which is designed to explain why thinking something good provides (or might provide) some sort of motive to promote it. The analysis is supposed to forge a conceptual connection between moral belief and action. Moore's Russell-derived example of 'what we desire to desire' provides a case in point. When Russell and (later) Lewis claim that goodness (or value) is what we desire to desire, this is not because they have a special yen for what we desire to desire and think that by calling it 'good' they can get people to maximize it. Rather they think that if we construe 'good' as what we desire to desire, we can see why people have a rational motive to promote what they believe to be the good. If we desire what we desire to desire (which we don't always do), then we will have a desire (and hence a rational motive) to promote what we believe to be good. The aim of the proposed analysis is not action but understanding, specifically an understanding of the 'conceptual connection between value and motivation' (DTV II: 113/69). Thus Russell and Lewis would be willing to concede that 'What we desire to desire is good', is a barren tautology, in the sense that it is unlikely to beget anything very spectacular in the way of action. But though it is an analytic truth and hence, if you like, a tautology, it is fruitful rather than barren when it comes to understanding the action-guiding power of 'good'. Remember that the conclusion of the Barren Tautology Argument is a (quantified) conditional: 7 3#') For any naturalistic or metaphysical 'X', if (i) 'X things are good', provides a reason for action (i.e. a reason to promote X-ness), then 'good' does not mean 'X'. And it is quite consistent with 3#') that 'good' means 'X' for some naturalistic or metaphysical 'X' so long as (i) 'X things are good', (or 'Good things are X') does not provide a reason for action (in the sense of a reason to promote X-ness). But this means that the Barren Tautology Argument is a much less powerful engine against naturalism than is commonly supposed. It is not just the Russell/Lewis theory that escapes the net. The Hutcheson/Hume theory (that value consists in a disposition to excite the approbation of a suitably qualified spectator)7, the Michael Smith theory (that rightness is what we would desire ourselves to do if we were fully rational)8, even the Hobbes/Locke/ Paley theory (that rightness consists in obedience to some Authority – God, the Sovereign or even the Beau Monde)9 all of them are immune to the Barren Tautology Argument. This is most surprising in the case of Hobbes/Locke/Paley, but all three could concede that 'Obeying the Authority is right', is a barren and unmotivating tautology, since each supplies another motive for obeying the Authority – the fear of punishment or, in the case of the Beau Monde, the fear of ridicule and ostracism. At all events, we are a long way from a simple and unanalyzable property of goodness, which is what Moore wants to establish. 8 7 See the extracts from Hutcheson and Hume in Raphael, D.D, (ed.) The British Moralists 1650-1800, 2 vols., (Oxford, Oxford University Press, 1967). 8 See Smith, Michael, The Moral Problem (Oxford, Blackwell: 1991). 9 Again see Raphael (1967) for relevant extracts. 3. Why the Open Question Argument? Now, I am not sure how clearly Moore distinguished between his various arguments or if he was fully aware of how far they succeed. But the above analysis suggests an interesting speculation. We know that the BTA was developed before the OQA, since PE: §11 (which contains the Barren Tautology Argument) dates back to Moore's 1898 draft 'The Elements of Ethics' whereas PE: §13 (which contains the Open Question Argument) was written rather later. (See PE, revised edn.: 312-313.) There are indeed hints of the OQA in the Elements of Ethics, but so far as I can see they are only hints. In fact, the BTA is cribbed (with due acknowledgment) from Sidgwick's The Methods of Ethics. Now, it may be that Moore realized that the BTA does not do everything that he wanted. It points to a fallacy committed by many naturalists and it shows that you cannot found an action-guiding ethic upon a mere definition. But it does not exclude all forms of naturalism. In particular, it does not exclude the definition suggested by Russell, that 'good' means what we desire to desire. For this definition is not intended to provide a reason for action but to explain why goodness is a property which furnishes us with such reasons. If he realized this (and it is a pretty big 'if'), Moore may have been driven to invent the OQA in order to deal with naturalistic definitions such as this. For the OQA (if sound) would dispose of all brands of naturalism including the kind of theory propounded by Russell and Lewis. If this is correct, Russell's intervention may have forced Moore to move from the BTA to the OQA, which, despite one or two vague anticipations, seems to have been his own invention. (The final chapter of Prior (1949), 'The Naturalistic Fallacy – the History of Its Refutation', which deals at some length with anticipations of Moore, is exclusively concerned with the BTA.) Indeed, Russell's intervention might explain the long delay in the publication of Principia Ethica, which did not come out until 1903, even though Moore had a pretty good first draft by 1898. Perhaps 9 it took Moore a long time to come up with an adequate response. The flaw with this proposal is that Russell propounded his definition in 1897, before Moore wrote The Elements of Ethics. But it might have taken Moore a while to realize that the BTA could not deal with this particular threat. 4. The Open Question Argument stated It may be useful at this point to state the OQA a little more precisely. The OQA (PE: §13) rests on three premises. 1) 'Are X things good?' is a significant or open question for any naturalistic or metaphysical predicate 'X' (whether simple or complex). 'Every one does in fact understand the question'; it is 'intelligible', it can be 'asked with significance' and 'we can understand very well what is meant by doubting' the answer. (PE: §13) Such questions would not be 'significant' (in Moore's sense) if an understanding of the words involved were enough for an affirmative answer. This is the case with 'Are X things X?', 'Are good things good?' and 'Are bachelors unmarried?' where the questions posed are, in effect, interrogative tautologies But since 'Are X things good?' is 'significant' for any 'X' (indeed 'significant' for 'every one' by which Moore would appear to mean all competent speakers) it follows that an understanding of the words involved (which is shared by all competent speakers) does not suffice for an affirmative answer. 2) If two expressions (whether simple or complex) are synonymous this is evident on reflection to every competent speaker. 10 3) The meaning of a predicate or property word is the property for which it stands. Thus if two predicates or property words have distinct meanings they denote distinct properties. From 1) and 2) it follows that 4) 'Good' is not synonymous with any naturalistic or metaphysical predicate 'X' (or 'goodness' with any corresponding noun or noun-phrase 'X-ness'). If 'good' were synonymous with any predicate 'X', then this would be evident on reflection to every competent speaker and the question 'Are X things good?' would not be open or significant for that particular 'X'. Thus, the fact that 'Are X things good?' is significant or open for every 'X' shows that 'good' is not synonymous with any such predicate. But Subconclusion 4) does not give Moore everything he wants. It states that the word 'good' is not synonymous any natural predicate, not that goodness itself is not identical with any natural or non-moral property. It is tantamount to what I call the semantic autonomy of ethics, the thesis that moral words are not susceptible to a naturalistic definition. (See Pigden (1991).) It is therefore incompatible with semantic naturalism, which is precisely the thesis that the moral can be reduced to the non-moral by means of definitions, i.e. by establishing that 'good' (or whatever) means the same as some (presumably complex) naturalistic predicate. Moore however professes a lofty disdain for mere semantics. 'Verbal questions are properly left to the writers of dictionaries and other persons interested in literature; philosophy, as we shall see, has no concern with them' (PE: §2). He has bigger fish to fry. He wants to establish 11 what I call the ontological autonomy of ethics, the thesis that for moral judgments to be true there must be a realm of distinctively moral facts and properties, of which goodness is the chief. Nevertheless, Sub-conclusion 4) is not without importance, since if it is true, the Russell/Lewis theory is false. For Russell and his unwitting disciple David Lewis are both semantic naturalists. Lewis is quite explicit about this. His theory, he says is 'naturalistic [i.e. semantically naturalistic] since it advances an analytic definition of value' (DTV II: 113/68). But so too is Russell's. 'Unless, therefore, the good can be defined otherwise than in terms of desire, ethics, properly studied, must always remain ... purely a branch of empirical psychology' (RoE: 75/Papers I: 102). In Russell's view the good can't be defined otherwise than in terms of desire, which means that ethics is indeed a branch of empirical psychology. But the point is that it is a definition, a purported analysis of the concept 'good', that is supposed to do the trick. But important as it is for Moore to refute the likes of Bertrand Russell, he wants to go one better. He wants to go beyond the word 'good' to the property for which it stands. How does he get from semantic autonomy (a predicate, 'good', that cannot be defined in terms of the non-moral) to ontological autonomy (a non-natural property of goodness that cannot be identified with anything non-moral)? By appealing to Premise 3), the thesis that the meaning of a predicate is the property that it denotes and thus that if two predicates have distinct meanings they stand for distinct properties. Moore certainly believed in properties at the time he wrote Principia, and then and thereafter, he seems to have subscribed to a 'one-level' theory of meaning according to which the meaning of a word is the thing it denotes. (See Baldwin (1990: 39-50 and 203) and Hylton (1990: 140-141)). Premise 3) provides the bridge between semantics and ontology. From 3) and 4) it follows that 12 5) Goodness is not identical with any natural or metaphysical property of Xness. Since 'good' has a distinct meaning from every naturalistic or metaphysical predicate 'X', it denotes a distinct and non-natural property. And this is precisely what Moore set out to prove. 5. The OQA discredited Premises 1), 2) & 3) suffice to prove Moore's point. But Premise 3) is highly questionable. Bob Durrant (1970) was perhaps the first to point out a) that Moore's argument requires some such premise if it is to succeed but b) that the assumption depends upon a purely referential theory of meaning according to which there is nothing more to the meaning of a predicate than the property for which it stands. Once we admit that, non-synonymous predicates can refer to the same property (just as non-synonymous names can refer to the same thing), Moore's argument for 5) collapses and he is reduced to Sub-conclusion 4). 'Good' may not be synonymous with any naturalistic predicate 'X' (whether simple or complex) but this does not prove that goodness is not identical with some naturalistic property of X-ness. We can no longer proceed from an unanalysable and non-natural predicate 'good' to an unanalysable and non-natural property of goodness. Cornell realists rejoice in this fact and happily propound synthetic identities between moral properties and others analogous to the celebrated identity between water and H2O. We can have moral truth without either metaphysical spooks or implausible attempts to give a naturalistic definition of the word 'good'. Nevertheless, Sub-conclusion 4) is not without importance, since it suggests the semantic autonomy of ethics, the thesis that morals words are not susceptible to naturalistic 13 definition. It is therefore incompatible with semantic naturalism, which claims that the moral can be reduced to the non-moral by definition, i.e. by establishing that 'good' means the same as some naturalistic predicate 'X'. Thus if Sub-conclusion 4) is correct, the Russell/Lewis theory is false. But Sub-conclusion 4) depends upon Premise 2). And Premise 2) is false. For it leads straight to the Paradox of Analysis, a problem that Moore recognized but did not succeed in solving. The Paradox first appeared in a paper by Langford (1942) but was probably discovered by Moore himself (Baldwin (1990: 208)). The Paradox is that conceptual analysis (which was Moore's stock in trade) is either useless or productive of falsehoods. For suppose the analysandum (the expression to be analyzed) means the same as the analysans (or analyzing phrase). Then by 2) this will be evident to every competent speaker and the analysis will teach us nothing new. Suppose on the other hand that that the analysis teaches us something new, i.e. that it is not evident on reflection to every competent speaker. Then, again by 2), the analysis is false. For if it is not evident to every competent speaker that the analysans and the analysandum share the same meaning, then they won't share the same meaning and the analysis will be false. (Baldwin (1990: 210-211, Pigden (1990: 427) Darwall, Gibbard and Railton (1992: 115).) If conceptual analysis is to be a worthwhile enterprise, one capable of turning up new and interesting truths, Premise 2) which generates the Paradox had better be false. And it is false, since it presupposes that our concepts are transparent to us. This is a point now widely recognized. Baldwin, for instance states that what I call Premise 2) relies on 'the Cartesian conception of the content of thought as transparently available to the subject', whilst Darwall, Gibbard and Railton talk of 'assumptions about the transparency of concepts and the obviousness of analytic truth'. Moore therefore is in the embarrassing position of relying on an assumption which, if true, 14 would have sabotaged his philosophical career. It is an assumption that anyone who believes in the possibility of conceptual analysis from Moore and Russell through to David Lewis and Frank Jackson must reject. What is analytic isn't always obvious. Hence the fact (if it be a fact) that 'Are X things good?' is an open question for every naturalistic or metaphysical 'X' does not prove that 'good' is not synonymous with some such 'X'. Thus the Open Question Argument has collapsed. It relied on three premises, 1) 2) and 3). Premise 3) was shot down in the 1970s, first by Bob Durrant and subsequently by others such as Putnam (1981: 205-211.). This opened the way for brands of naturalism such as Cornell Realism which rely on synthetic identities. It was a liberating thought that you can have moral truths without resorting to non-natural properties or dubious conceptual analyses. But the deletion of Premise 3) still left the OQA able to limp along as a disproof of semantic naturalism. But somewhere around 1980 people began to realize that Premise 2) 'the publicity condition', leads to the Paradox of Analysis and therefore had to go. The first person to make this point was Casimir Lewy in a paper published as far back as 1964. But despite the most memorably bizarre set of lecturing mannerisms that I have ever encountered, Lewy was not a philosophical superstar, and his paper went largely unnoticed. I heard the point first from the lips of David Lewis in 1981 (though at the time I did not understand what he was getting at). Not surprising then, that Lewis went on to reinvent a version of the desire-to-desire theory that the OQA was devised to disprove. The wheel had come full circle. 6. G.E. Moore redux? At the moment Moore seems to be in a pretty bad way while Russell and Lewis are laughing. The BTA does not work against the desire-to-desire theory whilst the OQA, which was probably invented to dispose of it, relies on three premises, two of which are false. Moore 15 made his name with an argument, which, if it were sound, would have made mincemeat of much of his subsequent philosophy. But it was one of Lewis's pet theses that knock down refutations are rare to non-existent in philosophy. In the latter part of this paper I shall be illustrating this thesis by arguing that there is something to be said for the OQA though in a suitably amended form, of course. 7. Colors, values and the epistemic argument. As the title suggests, Lewis's version of the desire-to-desire theory is a dispositional theory of value. It stands in long a tradition which represents value properties as akin to secondary properties, and construes secondary properties themselves as dispositions to cause certain effects in us10. I want to consider an argument of Kripke's – the epistemic argument that can be deployed to show that neither colors nor values should be understood as dispositional properties. It is my contention that the argument fails with respect to colors but succeeds with respect to values. Thus this semi-successful argument tends to show that goodness unlike yellow is not a secondary property, and hence that Lewis's desire-to-desire theory is false. Furthermore, the successful argument turns out to be a variant of the OQA. The principal purpose of Kripke's Naming and Necessity is to argue that proper names are – at least typically – rigid designators and that they lack sense. But one of his subsidiary purposes is to argue against a dispositional account of secondary properties and, more specifically, colors. (Kripke (1980: 140n).) In Kripke's view, color terms such as 'yellow' are also rigid designators, and, like proper names, they too lack sense. 'Yellow' denotes yellowness, but not by abbreviating some such description as 'that (manifest) property of 16 10 There are hints of this in Shaftesbury, but Frances Hutcheson (1694-1746) was probably the first philosopher of modern times to produce a well-worked out account of moral properties as secondary qualities. He found a distinguished, if heterodox, disciple in Hume. See Raphael (1967) for relevant extracts. objects that causes them, under normal circumstances to be seen as yellow (i.e., to be sensed by certain visual impressions)'. That description fixes the reference but not the sense of 'yellow' since the word 'yellow' has no sense to fix. In my view Kripke is wrong about this. But I am not going to discuss the matter in detail. I am just going to focus on one of his arguments, the epistemic argument. We will first apply the argument to names, then to color terms and finally to 'good'. 1*) If 'Shakespeare' meant 'the actual11 author of Hamlet, Othello etc.', then it would be analytic that Shakespeare (if he existed) was the actual author of Hamlet, Othello etc. 2*) But it is not analytic that Shakespeare (if he existed) was the actual author Hamlet, Othello etc. 3*) So 'Shakespeare' does not mean 'the actual author of Hamlet, Othello etc.' (Salmon (1981: 27-29). I have no comment to make about this argument, which seems completely convincing at least with respect to the vast bulk of proper names. Can it be adapted to show that words like 'yellow' are senseless and should not be subjected to a dispositional analysis? 1**) If 'yellow' meant 'that property, if it exists, that given our actual optical propensities, excites yellow sensations under normal circumstances', then it 17 11 As is well known, Kripke's modal argument (to the effect that names and other terms lack descriptive content) can be neutralized if we rigidify the terms in question with the aid of some well-placed 'actual's and 'actually's. It seems to me that 'yellow' and 'good' are both used rigidly which means that an 'actually' must be read into them. For ease of exposition I have given 'Shakespeare' the same treatment. See Salmon, Nathan U., Reference and Essence (Oxford: Blackwell, 1982), 26ff and Lewis, DTV II: 132-133/88-89. would be analytic that given our actual optical propensities, yellow, if it exists, excites yellow sensations under normal circumstances. 2**) But it is synthetic not analytic that given our actual optical propensities, yellow, if it exists, excites yellow sensations under normal circumstances. 3**) So 'yellow' does not mean 'that property, if it exists, that given our actual optical propensities, excites yellow sensations under normal circumstances'. I deny the second premise. It is not a matter of synthetic fact that yellow (if it exists) actually excites yellow sensations, but something we learn with the language. What is synthetic is that things with a certain range of surface microstructures and reflexive propensities excite yellow sensations. But this I can happily admit. Indeed it paves the way for a synthetic identity between instances of yellow and the microstructural properties which excite the sensations. In this case the epistemic argument fails. So much for 'yellow', what about 'good'? 1***) If 'good' meant what Lewis thinks it means, then it would be analytic that what we are actually, ideally disposed to desire to desire is good. 2***) But this is, if true, a matter of synthetic fact. 3***) Accordingly the analysis is false and goodness is not a secondary property. In the case of yellow I denied the second premise that it is synthetic that if there is such a thing as yellow, it is what actually arouses yellow sensations in us. But this will not wash with good. For it is quite conceivable that what ideal human beings actually value is mostly 18 bad. We can imagine Luther's opinion of the Lewis's theory. The idea that the desires of unregenerate human beings should be a guide to the good would strike him as ridiculous. The whole project of converting oneself into an ideal desirer smacks of the impious vanity of the damned. For Luther our second order desires (whether idealized or not) would be like reason the Devil's whore. (See Luther (1957: 46).) The King of Brobdingnag, after hearing a somewhat slanted catalogue of human achievements, told his pet human that despite the high regard he felt for him it was obvious that mankind (well actually Englishmen) were 'the most pernicious race of little odious vermin that nature has ever suffered to crawl upon the face of the earth'. (Swift (1967: 173).) Is it likely that the desires of such contemptible creatures, even the best of them, will indicate what goodness is? Won't they rather give an absurdly high ranking to the interests of their own noxious species? Now I do not agree with Luther and the King of Brobdingnag (except in my most jaundiced moments). But their views embody no manifest contradiction. If it is analytic that 'What we humans ideally desire to desire is good.', it is certainly not an obvious analyticity. Isn't the epistemic argument just the Open Question Argument all over again? After all, the OQA was that 'good' cannot mean 'X' because we can conceive of something being X without it's being good. (That is why the question 'Are X things, good?' makes sense or can be sensibly asked). We cannot in the same way conceive of someone as a bachelor without conceiving of him as an unmarried man not if we know the meanings of the relevant words that is. Now the epistemic argument is very like this. It claims that 'good' cannot be analyzed as what we would be ideally disposed to desire to desire (i.e.: it is not analytic that 'What we are ideally disposed to desire to desire is good'). Why not? Because we can conceive that we are ideally disposed to desire to desire something which is not really good. Indeed we can imagine that it might be bad. 19 The candid answer to my question has to be yes the epistemic argument is pretty much a rehash of the OQA. Then shouldn't it be discarded? Once we admit that there are unobvious analyticities, the fact that we can conceive of X-s which are not Y-s does not demonstrate that 'X' is a mistaken analysis of 'Y'. It might be that we have not thought the matter through. Hypothetico-deductive methods and even empirical research are required to establish non-trivial analyticities. Linguistic intuitions are not enough. Nor are linguistic intuitions enough to disestablish an alleged analyticity. Our intuitions may not penetrate to the buried rules and presuppositions that govern our use of language. Nevertheless, our intuitions about what can and what cannot be conceived are not devoid of probative force. They reflect, albeit imperfectly, our understanding of the concepts we employ. Hence they can provide evidence for and against analytical hypotheses, though this evidence ceases to be decisive. Now it does seem to me clear that Luther and the King of Brobdingnag could be right, and that what we ideally desire to desire could be wrong. In which case we have evidence, though not conclusive evidence, against the desire-to-desire theory. Now Lewis (DTV II: 132/88) wants to argue that in this case the intuitive evidence is misleading. My intuitions reflect my superficial thinking not the deep structure of the relevant concepts. If we try to flesh out Luther's story or the story of the King, we see that the hypothesis collapses. We cannot really imagine what it would be like for what we ideally disposed to value to be wrong. To talk largely of human depravity is not enough. What we need is corroborative detail: a plausible disvalue ideal humans are inclined to value or a value they are inclined to disvalue. Although Swift was about as misanthropic as they come, and although Gulliver is from first to last a satire on human nature, he does not manage to provide this. Yahoos, of course, value all sorts of nasty things, but they are far from ideal. 20 Englishmen likewise value things of doubtful worth, such as money, titles and military renown. But Englishmen too can be improved upon; for a start, they could be more disinterested. But what about truly ideal people? Could ideal humans be wrongly inclined to value domination something Swift's repulsive puppets pursue? But although sub-ideal human beings in their fallen state may value domination, this is not something they need desire to desire, nor is it something they would desire to desire if the defects which make them less than ideal were removed. I think Lewis' challenge can be met and corroborative detail supplied. Indeed it has been supplied in numerous works of Science Fiction. It is the oldest trick in Science Fiction's book to present human beings through the eyes of the aliens, and to present them as ridiculous and repulsive. They are greedy, savage, selfish and stupid, destroying other species and the ecosystems on which they depend. Given space-travel they will spread through the galaxy like a noxious slime, blighting every planet they touch. They must be stopped! If I pile it on thick enough, I can get you to sympathize momentarily with the extermination of the human race. It is at least conceivable that the destruction of the human race would be a Good Thing from the point of view of the Universe as a whole. But could ideal human desirers be brought to desire this, not just in a moment of ecological frenzy, but as a settled policy? Surely not. Human Chauvinism, in the weak form of a tendency to desire our continued survival is surely part of our make-up and it is not a desire that we desire not to have – not even under ideal circumstances. So the extinction of the human race is a conceivable (if not a plausible) value that ideal human beings would be inclined to disvalue. Now if we can conceive of what is good and what the best of us are disposed to desire to desire coming apart, this is evidence (though not conclusive evidence) that the one does 21 not constitute an analysis of the other. The epistemic argument may be weaker than Kripke supposes, but in this context, it works after a fashion. 8. The OQA amended But if the epistemic argument works after a fashion, then there is something to be said for the OQA too since in this context they come to much the same thing. The OQA failed because Premise 2), the publicity condition had to be rejected. However, even if we reject 2), and with it the assumption that our concepts are transparent to us, it does not follow that Moore's argument is entirely worthless. For there are weaker and more plausible variants of 2) which would take us to probabilistic variants of Sub-conclusion 4). Consider for example: 2') If it is evident to some competent speakers that two expressions 'X' and 'Y' are not synonymous (since it is not analytic that X is Y), then this is evidence (though not conclusive evidence) that they are not, in fact, synonymous. This suggests that following reformulation of the OQA, though as reformulated it presupposes not an Open Question but a strong intuition on the part of some speakers that the two expressions do not mean the same. Nevertheless we shall call it the OQA*: 1') It is evident to some competent speakers that (so far as our understanding of the words is concerned) a thing could be X without being good. 22 2') If it is evident to some competent speakers that two expressions 'X' and 'Y' are not synonymous (since it is not analytic that X is Y), then this is evidence (though not conclusive evidence) that they are not, in fact, synonymous. 4') 'Good' is probably not synonymous with the predicate 'X' (or 'goodness' with the corresponding noun 'X-ness'). This argument, the OQA* can be deployed against most naturalistic definitions of the moral predicates, though whether it can be deployed against all such definitions is, to coin a phrase, an open question. Furthermore the argument has to be deployed piecemeal. We no longer have a blanket premise covering all naturalistic predicates. Instead we have a series of specific premises for specific naturalistic predicates 'X', saying that according to some competent speakers a thing could be X without being good. It is no longer enough not to believe that the two terms are synonymous (which is all that was required for the original OQA). Rather someone has to believe that they are not synonymous, because they can conceive of a thing's being X without being good. And we can't be sure that 1') will be satisfied for every naturalistic predicate 'X'. Thus the revised argument does not constitute a refutation of semantic naturalism but a series of potential refutations of specific versions of semantic naturalism And even when Premise 1') is true for some 'X', the refutation is far from conclusive. Nevertheless OQA* constitutes an argument schema which can be used to refute – or perhaps since that's a weaker word, to discredit – a wide variety of semantic naturalisms, among them the theory that to be good is to be what we are ideally disposed to 23 desire to desire. This is much less than Moore purported to prove, but it does indicate that he is not quite the undischarged intellectual bankrupt that he once appeared to be. 24 REFERENCES Baldwin, Thomas, G. E. Moore (London: Routledge, 1990).a Darwall, S., Gibbard, A. and Railton. P., 'Towards Fin de Siecle Ethics: Some Trends', The Philosophical Review, 101 (1992), 115-189. Durrant, R. G., 'Identity of Properties and the Definition of Good', Australasian Journal of Philosophy, Vol. 48 (1970), 360-361. Frankena, W. K., 'The Naturalistic Fallacy', Mind, 48, (1939) 464-477. Griffin, Nicholas, (ed.) The Selected Letters of Bertrand Russell, vol. 1: The Private Years 1884-1914 (London: Routledge, 2002). Hylton, Peter, Russell, Idealism and the Emergence of Analytic Philosophy (Oxford: Oxford University Press, 1990). Kripke, Saul, Naming and Necessity (Oxford: Blackwell, 1980). Langford, C.H., 'Moore's Notion of Analysis', in Schilpp, P.A. (ed.), The Philosophy of G. E. Moore (Lassalle: Open Court. 1942), 319-342. Lewis, David, 'Dispositional Theories of Value II' Proceedings of the Aristotelian Society Supplementary Volume, 63 (1989), 113-137. Lewis, David, 'Dispositional Theories of Value', in Lewis, David, Papers in Ethics and Social Philosophy (Cambridge: Cambridge University Press, 2000), 68-94. Lewy, Casimir, 'G.E. Moore on the Naturalistic Fallacy', Proceedings of the British Academy, 50 (1964), 251-262. Luther, Martin, On the Bondage of the Will, trans. J.I. Packer and O.R. Johnston, (Cambridge: James Clarke, 1957). Moore, G.E. The Elements of Ethics, ed. Tom Regan (Philadelphia: Temple University Press, 1991). Moore, G.E. Principia Ethica, revised edn,, ed. Thomas Baldwin (Cambridge: Cambridge University Press, 1993). Pigden, Charles R., 'Naturalism' in Singer, Peter (ed.), A Companion to Ethics, (Oxford; Blackwell, 1991). Pigden, Charles R. (ed.), Russell on Ethics, (London: Routledge. 1999). Prior, A. N., Logic and the Basis of Ethics, (Oxford: Oxford University Press, 1949). Putnam, Hilary, Reason, Truth and History, (Cambridge: Cambridge University Press, 1981). Raphael, D.D, (ed.) The British Moralists, 2 vols., (Oxford, Oxford University Press, 1967). 25 Russell, Bertrand, The Collected Papers of Bertrand Russell. Vol. 1, Cambridge Essays, 1888-1899, eds. Blackwell, K., Brink, A., Griffin, N., Rempel, R. and Slater, J. (London: Allen & Unwin. 1983) Russell, Bertrand, The Collected Papers of Bertrand Russell. Vol. 4, Foundations of Logic 1903-05, eds. Urquhart, A. and Lewis, A. C. (London: Routledge, 1994). Salmon, Nathan U., Reference and Essence (Oxford: Blackwell, 1982). Sidgwick, Henry, The Methods of Ethics, 7th edn. (London: Macmillan, 1907). Smith, Michael, The Moral Problem (Oxford, Blackwell: 1991). Swift, Jonathan, Gulliver's Travels (Harmondsworth: Penguin, 1967).

Bhaskar contra Kant: Why Critical Realism is not Transcendental Realism Let me start by thanking Dan Little for inviting me to write this guest-post. I'd like to take the opportunity to examine Roy Bhaskar's arguments for critical realism, in particular those presented in his A Realist Theory of Science (RTS). The aim of that work is remarkable: to establish by transcendental argument the mind-independence and structured nature of the objects of science. Bhaskar's views are explicitly grounded in Kantian arguments. But the rejection of Kantian transcendental idealism is a central feature of Bhaskar's critical realism. For Bhaskar, critical realism is also transcendental realism, a position he posits as an alternative to both Kantian and (neo-)Humean philosophy of science. Transcendental idealism is, at minimum, the idea that the conditions on human cognition – especially space and time, the forms of human intuition – in part determine the objects of knowledge. According to transcendental idealism, we cannot know things as they are 'in themselves', but rather only as they appear to beings like us. Kant thus distinguishes between things-in-themselves, the epistemically inaccessible noumena, and phenomena, things as they appear to us given the conditions on human cognition. The former are transcendentally real – unknowable but entirely mind-independent. The latter are empirically real – knowable, but in part dependent on the conditions on cognition. For Kant, science can study only the empirically real: to study the transcendentally real would require that we transcend the conditions on our own cognition – that we erase the distinction between the knower and the object of knowledge – a mystical feat of which we are evidently incapable. 2 Bhaskar makes a different distinction, between the intransitive and the transitive. Intransitive objects do not depend on human activity; they are entirely mind-independent (RTS 21). To say that some object is intransitive is therefore equivalent to saying that it is transcendentally real (this is clear throughout RTS; see also The Possibility of Naturalism 6). Hence, it is Bhaskar's aim to prove the transcendental reality (intransitivity) of the objects of science and perception. According to Bhaskar, we can know the objects of science as they are in themselves. Bhaskar defends this ambitious thesis by means of transcendental arguments. An argument is transcendental insofar as it shows that some commonly accepted claim x necessarily presupposes a controversial claim y; where y is the conclusion of the argument. Thus, a transcendental argument claims that its conclusion is the only possible way to account for the uncontroversial phenomenon which it takes as its premise. Unlike other arguments for scientific realism, then, Bhaskar's make a claim to necessity. Bhaskar's analysis of perception contains the first of his transcendental arguments: call it the argument from perception. It has roughly the following form: multiple agents can, at the same time, perceive the same object in different ways (x). This could be possible only given the mind-independence of the object (y). Therefore, given the occurrence of differential perception, the objects of perception must be transcendentally real. Here's Bhaskar himself making the argument: If changing experience of objects is to be possible, objects must have a distinct being in space and time from the experience of which they are the objects. For Kepler to see the 3 rim of the earth drop away, while Tycho Brahe watches the sun rise, we must suppose that there is something they both see. (RTS, 31) Earlier, he appears to be making the even stronger claim that perception simpliciter presupposes the intransitivity of the perceived: The intelligibility of sense-perception presupposes the intransitivity of the object perceived. For it is in the independent occurrence or existence of such objects that the meaning of 'perception', and the epistemic significance of perception, lies. (Ibid.) Let's take the argument from perception to involve the weaker claim that differential experience by different agents necessarily presupposes the intransitive nature of the object perceived. If the argument fails to ground this claim, we know a fortiori that it fails to ground the stronger conclusion. If it is possible for Brahe and Kepler to have different perceptions of the same object, there must be an object which they both see: this much seems clear. But the inference from this to the object's intransitivity is fallacious, for the presupposition that the objects of senseperception are empirically real is sufficient to explain differential perception. For the transcendental idealist, there is something which Brahe and Kepler both see: they both see the sun. The sun is empirically real, i.e., it partially depends on the conditions on human cognition. But Brahe and Kepler, being human, share the conditions on cognition and interact with the same mind-independent reality. Thus, there is nothing unintelligible about their different perceptions under the assumption that what they perceive is empirically real (partially minddependent). Bhaskar supposes that we must assume it is also transcendentally real (i.e., that Brahe and Kepler see the sun 'as it is in-itself') but does nothing to establish this. The argument 4 from perception does not show that the objects of knowledge must be intransitive given the occurrence of (differential) perception. It fails as a transcendental argument for critical realism. Bhaskar's second argument is much more central to the critical realist endeavor, and it is presented in his analysis of experimental activity. Call it the argument from experimentation. For Bhaskar, "two essential functions" are involved in an experiment: First, [the experimental scientist] must trigger the mechanism under study to ensure that it is active; and secondly he must prevent any interference with the operation of the mechanism. [...] Both involve changing or being prepared to change the 'course of nature', i.e. the sequence of events that would otherwise have occurred. [...] Only if the mechanism is active and the system in which it operates is closed can scientists in general record a unique relationship between the antecedent and consequent of a lawlike statement. (RTS, 53) Bhaskar notes that the experimenter who sets up a causally closed system thereby becomes causally responsible for a constant conjunction of events, but not for the underlying causal mechanism. Contra Humean accounts of law, Bhaskar's account of experimentation entails an ontological distinction between constant conjunctions and causal mechanisms. For Bhaskar, the intelligibility of such experimental activity can be used to transcendentally establish the intransitivity of the objects of science. "As a piece of philosophy," he claims, "we can say (given that science occurs) that some real things and generative mechanisms must exist (and act)," where by 'real' Bhaskar means 'intransitive' (RTS 52). In "Transcendental Realisms in the Philosophy of Science: On Bhaskar and Cartwright," Stephen Clarke provides the following helpful gloss on the argument: 5 Premise 1: Scientific explanatory practice (in particular the practice of exporting explanations from laboratory circumstances to general circumstances) is experienced by us as intelligible. Premise 2: Scientific explanatory practice could not be experienced by us as intelligible unless causal powers exist and those causal powers are governed by universal laws of nature. Conclusion: causal powers exist and are governed by universal laws of nature. (Clarke 302) Clarke calls this an "attack on idealism" (303) but Bhaskar explicitly frames it as an attack on transcendental idealism (RTS 27). Clarke's gloss is telling, for it is indeed unclear how the argument could work as an attack on the latter view. Bhaskar argues that we must suppose the world to be intransitively ordered if scientific explanatory practice is to be intelligible. But, he claims, "transcendental idealism maintains that this order is actually imposed by men in their cognitive activity" (RTS 27). And if order were imposed in cognitive activity, all experience would be ordered, eliminating the need for explanatory export from the closed causal systems of experimentation to the open causal systems of uncontrolled experience (RTS 27, Clarke 303). This argument is invalid. It does not follow from the premise that all experience is ordered that there is no need for explanatory export from closed to open causal systems. To the contrary: the very occurrence of such export presupposes that experience is ordered. After all, the aim of experimentation is to discover causal mechanisms and universal laws of nature. But to suppose that the causal mechanism discovered in a replicable scientific experiment generalizes to 6 open causal systems is to suppose that the same laws operate in open causal systems, even if other mechanisms sometimes obscure them. And to presuppose that there are such things as knowable universal laws of nature – operative in closed and open causal systems alike – just is to presuppose that all experience is ordered. The ordered nature of experience is, therefore, a necessary presupposition for experimentation. Now there are at least two ways in which experience could be thus ordered: because order is imposed on it in cognitive activity, or because the order is intransitive. Bhaskar supposes the former would render experimentation superfluous. This is a flummoxing claim to make. Surely Bhaskar does not mean to accuse the transcendental idealist of the view that the projection of order onto the world is somehow a conscious activity – that we already know every scientific truth. That would render experimentation superfluous, but I don't think it is a view anybody defends. Science is as much a process of gradual discovery for the Kantian as it is for everyone else. Maybe confusion arises from the fact that for Kantians genuinely universal scientific laws must be synthetic a-priori. Perhaps Bhaskar supposes that, because positing a universal law involves making a claim to synthetic a-priori knowledge, we should be able to derive the laws of nature by a-priori deduction, rendering experimentation superfluous. But this would be a misunderstanding of transcendental idealism. Suppose that because my perceptions of sparks and wood are frequently followed by perceptions of conflagration, I come to associate sparks and wood with fire. I can ask whether this association is subjective or objective. To claim that it is objective is, for the Kantian, to apply one of the Categories. For instance, one way of taking my association of sparks and dry wood with fire to be objective is to make a claim like "sparks and wood cause fire," applying the Category of causation. This claim is a-priori insofar as it involves 7 the application of an a-priori (pure) concept, a-posteriori insofar as it is about the objects of experience. Transcendental idealism entails we are entitled to make causal claims, but it does not entail the empirical truth of our claims. Experimentation with sparks and wood may lead me to modify my claim. For instance, I may discover that sparks and wet wood do not jointly give rise to fire, and adjust my claim to "sparks and dry wood cause fire." Further experimentation may lead to further refinements. I could not have deduced any of these conclusions about sparks and wood a-priori. The thesis that scientific claims have an a-priori component does not render experimentation either superfluous or unintelligible. As it turns out, Bhaskar supposes that, for the Kantian, causal mechanisms are mere "figment[s] of the imagination" (RTS 45). If true, this would provide an independent argument against the intelligibility of experimentation on a transcendentally idealist account. But, as should by now be clear, this is an incorrect characterization of transcendental idealism. It is only for skeptics and solipsistic idealists that causal mechanisms are figments of the imagination. Kantians and transcendental realists agree causal mechanisms exist: they disagree only about whether they are transcendentally or empirically real. Bhaskar's transcendental arguments for critical realism fail, and the Kantian view to which Bhaskar opposes his own is frequently misinterpreted. Most problematically, the meaning of the Kantian distinction between the transcendentally and empirically real is ignored, and the latter category is treated as if it contained only figments of our imagination. Bhaskar maintains that epistemic access to the transcendentally real is a necessary condition for science and perception. But, as we have seen, it is merely epistemic access to the empirically real that is 8 necessary. Bhaskar does not prove that we have knowledge of things as they are in-themselves. Critical realism is not transcendental realism.

Being Rational and Being Wrong Kevin Dorst [email protected] Draft-comments welcome! September 2020 Abstract Do people tend to be overconfident in their opinions? Many think so. They've run studies to test whether people are calibrated : whether their confidence in their opinions matches the proportion of those opinions that are true. Under certain conditions, people are systematically "over-calibrated"-for example, of the opinions they're 80% confident in, only 60% are true. From this observed over-calibration, it's inferred that people are irrationally overconfident. My question: When-and why-is this inference warranted? Answering this question requires articulating a general connection between being rational and being right-something extant studies have not done. I show how to do so using the notion of deference. This provides a theoretical foundation to calibration research, but also reveals a flaw: the connection between being rational and being right is much weaker than is commonly assumed; as a result, rational people can often be expected to be miscalibrated. Thus we can't test whether people are overconfident by simply testing whether they are over-calibrated; instead, we must first predict the expected rational deviations from calibration, and then compare those predictions to people's performance. I show how in principle this can be done-and that doing so has the potential to overturn the standard interpretation of robust empirical effects. In short: rational people can be expected to be wrong more often than you might think. 1 The Question Pencils ready! For each pair, circle the city that you think has a larger population (in the city proper), and then rate how confident you are in that guess on a 50−100% scale: 1) Denver or Phoenix? Confidence: % 2) San Jose or Seattle? Confidence: % 3) Indianapolis or Columbus? Confidence: % 1 1. THE QUESTION If you're like most people, this test will reveal two things. First, it's likely that only one or two of your answers is correct. Second-and perhaps more worryingly-it's likely that your confidence in your answers does not match this probability of being correct. Among 200 test-takers, the average confidence people had in their answers was 75%, while the proportion of correct answers to hard questions like this was only 45%.1 That rather striking result-the so-called "overconfidence effect"-is common: on a variety of tests, people's average confidence in their answers exceeds the proportion that are correct.2 Many have concluded from this result that people are often overconfident in their opinions-i.e. more confident than it is rational for them to be, given their evidence.3 Many have used these (and related) results to paint unflattering pictures of the human mind as prone to pervasive irrationality and bias.4 And many others have invoked overconfidence in particular to explain a variety of societal ills-from market crashes, to political polarization, to wars.5 Daniel Kahneman summed it up bluntly: 'What would I eliminate if I had a magic wand? Overconfidence' (Shariatmadari 2015). Okay. But how-exactly-did we reach this conclusion of pervasive overconfidence? The evidence comes in the form of calibration studies like the one you just took. We ask people a variety of questions, have them report their confidence in their answers, and then graph that confidence against the proportion of answers that are true.6 Say that a person is calibrated (at x) if exactly x% of the claims that they are x% confident in are true. They are over -calibrated (at x) if fewer than x% of such claims are true.7 And they are under -calibrated (at x) if more than x% of such claims are true. Focusing on binary-question ("2-alternative-forced-choice") formats-wherein people are asked to choose between two answers, and so are always at least 50% confident in their answer- schematic graphs of these different calibration curves are given on the left of Figure 1. Meanwhile, the right side of Figure 1 plots the results of my study (see §5.2 for details), replicating a well-known result that (on certain types of questions) people tend to be 1Answers: Phoenix, San Jose, Columbus. "Hard questions" means those with hit rates below 75%; see §5 for study details. 2 For summaries, see Lichtenstein et al. (1982), Harvey (1997), Hoffrage (2004), Glaser and Weber (2010), and Moore et al. (2015). 3Lichtenstein et al. (1982); Dunning et al. (1990); Vallone et al. (1990); Griffin and Tversky (1992); Kahneman and Tversky (1996); Budescu et al. (1997); Brenner (2000); Koehler et al. (2002); Brenner et al. (2005); Glaser and Weber (2010); Merkle and Weber (2011); Brenner et al. (2012); Moore et al. (2015); Ehrlinger et al. (2016); Magnus and Peresetsky (2018). 4 Plous (1993); Fine (2005); Ariely (2008); Hastie and Dawes (2009); Myers (2010); Kahneman (2011b); Thaler (2015); Lewis (2016); Tetlock and Gardner (2016). 5E.g. Howard (1984); Odean (1999); Glaser and Weber (2007); Johnson (2009); Myers (2010, 377); Johnson and Fowler (2011); Kahneman (2011a); Ortoleva and Snowberg (2015); van Prooijen and Krouwel (2019). 6 I'll focus on this type of calibration study-but see §6 for sketches of how the lessons may apply to both placement- (Kruger and Dunning 1999) and interval-estimation (Moore et al. 2015) methods. 7 "Over"-calibrated because their confidence in those opinions needs to be lower to be calibrated. In the graphs below, imagine the person controlling a left-right slider for their confidence; over-calibration is putting it too far to the right; under-calibration is putting it too far to the left. 2 1. THE QUESTION Under-calibrated opinions Calibrated opinions Over-calibrated opinions 0.5 0.6 0.7 0.8 0.9 1.0 0.5 0.6 0.7 0.8 0.9 1.0 Actual Confidence P ro po rt io n T ru e Calibrated My city-size test 0.5 0.6 0.7 0.8 0.9 1.0 0.5 0.6 0.7 0.8 0.9 1.0 Actual Confidence P ro po rt io n T ru e Figure 1: Left: Schematic calibration curves. Right: "Overconfidence effect" (i.e. overcalibration) in my study. (See §5.2 for study details.) substantially over-calibrated (Lichtenstein et al. 1982). That's the evidence-namely, that people are often over-calibrated. How do we get from there to the conclusion-namely, that people are often overconfident? Well, it's natural to think that if people's confidence is rationally placed, their opinions will be right about as often as they expect them to be. Conversely, if people are quite confident in their opinions and yet many (or most!) of those opinions are wrong, it's natural to infer that they are too confident-overconfident. This is natural. But it is also a substantive inference: it moves from an empirical observation-'you are (mis)calibrated'-to a normative conclusion-'you are (ir)rational.' Call it the rational-to-right inference since, stated bluntly, it relies on a general connection between your opinions being rational and your opinions being right. The Questions: What is the connection between being rational and being right? More specifically: When is the rational-to-right inference warranted? When is it not? And what does that tell us about how to interpret the results of calibration studies? The Plan: I'll first say what sort of connection the rational-to-right inference assumes, and explain why the existing literature has failed to articulate it (§2). I'll then go on to use the notion of deference to articulate this connection, and show how it vindicates the rational-to-right inference in certain simple cases (§3). However, it turns out that even the strongest (most contentious) version of this connection will break in predictable ways-meaning that often miscalibration is evidence for rationality (§4). I conclude by arguing that this result provides both a foundation for and a refinement to the standard calibration-study methodology: in testing whether people are rational, the null hypothesis should not be that they will be calibrated; rather, we must first predict the rational deviations from calibration on our test, and then compare people's performance 3 2. THE PROBLEM to those predictions. I show how in principle this can be done, and that doing so has the potential to overturn the interpretation of robust empirical effects (§§5–6). The Upshots: If all this is correct, it shows that certain philosophical and psychological literatures are much more intimately connected than has been realized. Contemporary philosophical debates about the formulation and tenability of deference principles have a direct and substantive bearing on the methodology and interpretation of empirical studies of confidence. Conversely, the methods developed by these studies show how we can make precise predictions about the relationship between rationality and truth in a variety of environments-and that being rational is not nearly the guide to being right that you might think. Regardless of whether you accept these particular conclusions, I hope to convince you that there are rich connections here-and thus that philosophers, psychologists, and behavioral economists can productively work together more closely in tackling the question of human (ir)rationality. 2 The Problem There is a problem here. The rational-to-right inference involves three quantities: (1) A person's actual degrees of confidence in some claims. (2) The proportion of those claims that are true. (3) The degrees of confidence it would be rational for them to have in those claims. The only quantities that are observed are (1) and (2). The rational-to-right inference uses these to infer something about (3): from the observation that (1) is higher than (2), it is inferred that (1) is higher than (3). Clearly this makes sense only if rational confidence-(3)-can be expected to align with proportion true-(2). The point can be made graphically. What would it mean to say that people tend to be overconfident (in a given domain8)? I'll take it to mean that they are (on average) more extreme in their opinions in that domain than they would be if they were rational. If we plot actual degrees of confidence against rational degrees of confidence (on 50− 100% scale), people tend to be rational if (averaging across opinions) rational confidence matches actual confidence-the curve is diagonal; they tend to be overconfident if rational confidence is less extreme than actual confidence-the curve is tilted. (See the left side of Figure 2.) That's the overconfidence hypothesis. What is the evidence offered in its favor? It's that in a variety of settings, people are over-calibrated : if we plot actual degree of confidence against proportion true, the curve is tilted-see the right side of Figure 2. 8 The "in a given domain" rider is important, as patterns of miscalibration vary widely across different sets of questions (see Koehler et al. 2002; Brenner et al. 2005). We'll introduce refinements to the empirical story in §4.1 and onwards-for now, I'll focus on tests for which the "overconfidence effect" is observed. 4 2. THE PROBLEM Rationality hypothesis Overconfidence hypothesis 0.5 0.6 0.7 0.8 0.9 1.0 0.5 0.6 0.7 0.8 0.9 1.0 Actual Confidence R at io n al C o n fi d en ce Calibration hypothesis Over-calibration hypothesis 0.5 0.6 0.7 0.8 0.9 1.0 0.5 0.6 0.7 0.8 0.9 1.0 Actual Confidence P ro p o rt io n T ru e Figure 2: Left: Rationality vs. Overconfidence hypotheses. Right: Calibration vs. Over-calibration hypothesis. Simple point: although the graphs look the same, the axes are different. It follows that the rational-to-right inference is warranted when and only when you should expect the two axes to align-i.e. when you should expect a rational person's judgment to be calibrated on the given test. More specifically, the inference works when and only when the following hold. Take all the claims that the person should be 50% confident in-you should expect roughly half of them to be true; take all the claims that the person should be 60% confident in-you should expect roughly 6 out of every 10 of them to be true; and so on. There's a point here worth emphasizing. To say that someone is overconfident in a set of opinions q1, ..., qn is to say that they are, on average, more confident than they should be-that is, that there is some number c that represents their average confidence, some other number r that represents the average confidence it would be rational for them to have, and that c > r. What this means is that calibration studies-and the rationalto-right inference they are based upon-presuppose that there are rational degrees of confidence (ri) that people ought to have in the claims they evaluate (qi), which may differ from their reported degrees of confidence (ci). Why am I banging on this drum? Because I know of no study that explicitly represents the rational degrees of confidence ri as variables to be investigated. None of the studies cited in this paper do so.9 As a result, none of these studies state the assumptions needed about rational confidence to derive the result that we should expect the 9Including those cited in footnotes 2, 3, and 5. Some studies model notions of probability distinct from subjects' reported confidence and observed frequencies-such as objective probabilities, true subjective confidence (as distinct from reported confidence), or the subjective confidence of differing agents (Gigerenzer et al. 1991; Erev et al. 1994; Juslin et al. 1997, 1999, 2000; Moore and Healy 2008). None of these quantities are treated as rational confidence needs to be-see below for more discussion. 5 2. THE PROBLEM rational opinions ri to be calibrated in their study. 10 In other words: I know of no study that states what assumptions it is making such that we should expect the two y-axes in Figure 2 to align. Yet over-calibration is evidence for overconfidence only if we should expect them these axes to align: observing that people's judgments are miscalibrated provides no evidence that they are irrational unless we have reason to think that the rational degrees of confidence would be calibrated. That is the problem.11 But how serious is this problem? Can we safely assume that-at least if the study is properly set up-rational confidence will on average be calibrated? No: there is no necessary connection between being rational and being right at any level of statistical generality. This is easy to see in extreme cases. Case 1: Take the philosopher's favorite rational brain-in-a-vat, Rajat. Rajat rationally uses all the information he gets. His information is a lot like yours or mine. As a result he's sure that he has hands, confident he's healthy, and suspects he'll soon grab lunch. But though rational, Rajat is wrong on all these fronts (and many others)-for, unbeknownst to him, he's a brain-in-a-vat being deceived by a mad scientist into thinking he's living a normal life. If we ran a calibration study on Rajat, he would be systematically over-calibrated-most of the things he's confident in are false. Yet, by stipulation, we know Rajat is perfectly rational. More mundane cases make the point as well. Case 2: Meet Georgie. She's quite confident-and quite wrong-in most of her geographical opinions. When she took a city-population test, her average confidence was 90% in her guesses, but the proportion she answers correctly was 50%. Does this provide evidence that she was irrationally overconfident? Not if we know that her geography teacher gave her an outdated textbook on city-sizes to memorize, for then we should chalk up her mistakes to bad information rather than irrationality. Obviously we can imagine scenarios in which the entire population of test-takers are the same position-we'd expect every student in Georgie's geography class to be rationally highly confident in their opinions, and yet also systematically wrong. Likewise, it's easy to construct cases in which we know the subject's have highquality evidence, and yet the rational-to-right inference fails. Case 3: I have a coin in my pocket that's 60% biased toward heads; I'm about to toss it 100 times. How 10 Some explicitly derive this result for a given Bayesian agent (Brenner et al. 2005; Moore and Healy 2008; Merkle and Weber 2011)-but to do so they all implicitly assume that the Bayesian's prior beliefs match the objective frequencies on the test. As we'll see, this cannot in general be assumed. 11 Lest you wonder if this suggests that psychologists are not interested in rationality, and instead are interested purely in the descriptive phenomenon of over-calibration, rest assured that the normative interpretation of these studies is clear. They are peppered with normative assessments of people's confidence: e.g. 'irrational" (Hoffrage 2004, 245; Magnus and Peresetsky 2018, 2), "unjustified" (Dunning et al. 1990, 579; Vallone et al. 1990, 588), "unreasonable" (Merkle and Weber 2011, 264), "biased" (Koehler et al. 2002, 686; Glaser and Weber 2010, 249; Moore et al. 2015, 182), etc. Kahneman and Tversky put it bluntly: "Our disagreement [with Gigerenzer (1991)] is normative, not descriptive. We believe that subjective probability judgments should be calibrated, whereas Gigerenzer appears unwilling to apply normative criteria to such judgments" (Kahneman and Tversky 1996, 589). 6 2. THE PROBLEM confident are you, of each toss, that it'll land heads on that toss? Write that number down-I'll look at it in a second. First to toss the coin (...done). Turns out it landed heads only 30 times. Now to compare that to your confidence.... Hm, 60%? You were 60% confident that each toss would land heads, but only 30% of those claims were true. Have I gained evidence that you are overconfident? Obviously not-your 60% confidence was perfectly rational, yet sometimes rational opinions turns out to be mistaken. Similarly, sometimes you can know that your rational opinions will be systematically mistaken. Case 4: I have an urn of mis-printed coins-60 of them are double-headed, and the remaining 40 are double-tailed. I'm about to pull a single coin from the urn and toss it 100 times. How confident are you, of each toss, that the coin I draw will land heads on that toss? 60% I take it. Yet you know that either I'll draw a double-headed or a double-tailed coin. If the former, all the tosses will land heads-100% of the things that you're 60% confident in will be true. And if the latter, then none of them will land heads-0% of the things that you're 60% confident in will be true. So you know that, either way, the rational opinions will be badly miscalibrated. Finally: in almost any conceivable scenario, a rational person will know that certain classes of their opinions will be systematically miscalibrated. Case 5: Suppose you're about to take a test drawn randomly from a representative set of your knowledge about geography. Suppose you know that the sources you've studied diligently and rationally are generally accurate. Nevertheless, your rational opinions aren't perfect-sometimes you'll be wrong. Consider the set of guesses W you'll be wrong about, and the set R you'll be right about. You won't know what these sets are until you finish the test and the answers are revealed. But you know you'll be miscalibrated on them-0% of the claims in W will be true, but your average confidence in them will be higher than that; and 100% of the claims in R will be true, but you average confidence in them will be lower than that. More generally, we should always expect that people will be over-calibrated on sets of opinions like "the set of answers they tend to get wrong" and under-calibrated on sets lke "the set of answers they tend to get right."12 Upshot: it is easy to imagine scenarios in which perfectly rational people are systematically miscalibrated.13 Thus when we run the rational-to-right inference-inferring 12 Cf. the discussion of "linear dependency" in Juslin et al. (2000) and elsewhere; we'll come back to this point as it relates to the "hard-easy effect" in §5. 13For those familiar with certain bits of theory, a clarification may be helpful here. Any Bayesian will expect any particular set of their own opinions to be calibrated (see below). But we are not them, and we know things that they do not. Therefore there is no theorem that we should expect them to be calibrated. Often we should not. (Why must they expect to be calibrated? Because a Bayesian's estimate of the proportion of truths amongst some particular set of claims will equal their average degree of confidence in them. Letting C be any probability function, E[X] be its expectation of a variable X (E[X] := ∑ t C(X = t) * t), and I(qi) be the indicator of qi (1 if qi is true, 0 if not), we have: E[ 1 n ∑n i=1I(qi)] = 1 n ∑n i=1E[I(qi)] = 1 n ∑n i=1C(qi). Thus our subject's estimate of the proportion of truths amongst the claims they are 80% confident in must be 80%. Moreover, so long as they treat the claims (relatively) independently, they will (by the weak law of large numbers) be confident that roughly 80% of those claims are true.) 7 2. THE PROBLEM from miscalibration to irrationality-we are somehow discounting concerns from scenarios like this. The question is what justifies us in doing so. To be clear: I am not claiming that these toy cases shed doubt on the rational-to-right inference in practice (nor that they will be of any surprise to the researchers conducting calibration studies!). What I'm claiming is that these cases make salient a conceptual question. As we've seen, whether we should expect the rational opinions to be calibrated on a given set of questions depends completely on the evidence that the test-taker has and how that set was determined. Surely, in some sense, we should expect that the opinions their evidence warrants will tend to be right-that's the point of evidence, after all-and thus that rational degrees of confidence will tend to be calibrated. The question is: When, why, and in what sense should we expect this? The answer to this question is not obvious. It requires formulating a systematic connection between being rational and being right. As discussed above, none of the calibration studies I know of have articulated such a connection, for none of them have represented the rational degrees of confidence as a variable to make assumptions and predictions about in their test. That is what I'm going to do. In §3 I'll articulate a general, probabilistic connection between being rational and being right-and on what this connection depends. This explains why the rational-to-right inference works in certain simple cases. But, as we'll see in §4, it also reveals that it'll fail in systematic ways whenever we have information that the test-takers don't-as we always will. §5 will turn to saying what this implies about the proper methodology of calibration studies. But before moving on, I should say more about how this project relates to an array of theoretical points that have been made in the calibration literature.14 "Ecological" approaches have made the point that subjects may well have misleading information about our test, and therefore that we must try to control for this by choosing representative questions from a natural domain (Gigerenzer 1991; Gigerenzer et al. 1991; Juslin 1994; Juslin et al. 2000; Hoffrage 2004). "Error-model" approaches have made the point that even if questions are chosen randomly from a natural domain, there will be stochastic errors ("noise") in both the selection of items and in the subject's reporting of their confidence that can naturally lead to them being miscalibrated on a given test-even if their true opinions are calibrated overall (Erev et al. 1994; Pfeifer 1994; Juslin et al. 1997, 1999, 2000). Similar points have been made using information asymmetries between subjects (Moore and Healy 2008; Jansen et al. 2018). Based on such considerations, 14What about precedents in the philosophical literature? To my knowledge no philosophers have directly addressed the rational-to-right inference, instead focusing on different questions about the epistemic significance of calibration. Some ask whether calibration can objectively vindicate a set of opinions (van Fraassen 1983; Dawid 1983; Seidenfeld 1985; Joyce 1998; Dunn 2015; Pettigrew 2016a); others ask whether a Bayesian agent's beliefs about their own long-run calibration are problematic (Dawid 1982; Belot 2013a,b; Elga 2016); and others ask how our your expectations about your (or your peer's) calibration should affect your confidence in your answers (Roush 2009, 2016, 2017; White 2009b; Christensen 2010a, 2016; Lam 2011, 2013; Sliwa and Horowitz 2015; Schoenfield 2015, 2016a; Isaacs 2019). 8 3. THE INSIGHT these researchers have built models of how people may form their degrees of confidence in an apparently rational way, and yet nonetheless we might expect to see the sorts of miscalibration that we in fact observe. I agree with these conceptual points, and some of my modeling choices in §5 are inspired by them. But the point I'm making is a broader one. These researchers have proposed particular, rational-seeming mechanisms for forming opinions15, and shown that they can lead to miscalibration. What I'm going to show is that no matter the mechanism used to form beliefs, rational opinions should be expected to be miscalibrated in systematic ways. Demonstrating this becomes possible once we explicitly represent the rational opinions as variables to be investigated. Interestingly, these rational deviations from calibration turn out to be broadly consistent with some of the core empirical trends (§5.2). But more importantly, they show that we've been using the wrong yardstick. In assessing whether people are overconfident, we should never simply compare their calibration curves to the diagonal calibrated line- rather, we must compare them to the predicted rational deviations from the calibrated line. I'll show how we can in principle predict these rational deviations from calibration without making any assumptions about mechanism. If this is right, the sorts of simulations for predicting miscalibration pioneered by Erev et al. (1994); Pfeifer (1994), and Juslin et al. (1997)-and which I will use in §5- are not the special purview of those trying to explain the empirical data with a rational model of confidence. Rather, they are a necessary precondition for figuring out what the null hypothesis should be when we aim to assess whether people's calibration curves provide evidence for overconfidence. 3 The Insight First things first, we need to delineate the connection between being rational and being right. When-and why-should we expect the rational degrees of confidence for a given person to be calibrated? When you learn about the results of a calibration study, you get a lot of evidence: how (mis)calibrated many subjects were across many levels of confidence; what sorts of test items were used, and how they were selected and presented; etc. All this evidence makes things complicated. Let's start by making things simple. Suppose you get very limited evidence. A single subject-Calvin-was given a calibration test; the questions were selected to be random and unrelated. Consider all the claims that Calvin was 80% confident in-call those his 80%-opinions. All you're told is which proportion of them were true. 15Which, in turn, have been criticized on a variety of grounds (Kahneman and Tversky 1996; Budescu et al. 1997; Brenner 2000; Koehler et al. 2002; Brenner et al. 2005; Merkle and Weber 2011). 9 3. THE INSIGHT I claim that in this simple scenario, the rational-to-right inference is warranted. If you learn that (roughly) 80% of Calvin's 80%-opinions were true, you get strong evidence that those opinions were rational; if you learn that far fewer (or far more) than 80% of these opinions are true-say, 60% (or 95%)-you get strong evidence that he was overconfident (underconfident). This, I claim, is the insight behind calibration studies. Why is it correct? Begin with a parable. Long ago, Magic Mary possessed a variety of magic coins- some were biased to come up heads almost every time; others to come up heads 90% of the time; others 80%, and so on. The magic coins had a variety of special markings on them-on some, George Washington has a large nose and small ears; on others, he has a thin neck and bushy eyebrows; etc. In principle, if one knew how to decipher the markings, one could tell what the bias of the coin was just by looking at it. Mary tossed the coins many, many times. She kept fastidious records: for each toss she wrote the details of the coin's markings on one side of a stone tablet, and the outcome of the toss (heads or tails) on the other. Alas, Magic Mary and her magic coins are long gone-but many of the tablets remain, stored in various historical archives. And alas, no one can decipher the markings to tell which bias a given tablet corresponds to. . . . or so we thought! But now bias-busting Bianca claims that she can decipher the markings and determine the coins' biases. How can we test her claim, given that we don't know how to decipher them? Here's a good strategy. Go to an archive that contains a representative sample of tablets; draw a tablet at random; show her the markings-side, having her announce her guess as to whether it landed heads or tails along with her confidence in that guess; write down whether she got it right (but don't tell her); then draw a new tablet and repeat. Suppose we do this with many, many tablets, and then I tell you this: "Of the guesses she was 80% confident in, 79% were correct!" How confident are you now that Bianca can reliably recognize the 80%-biased coins-i.e. those that are 80% biased toward heads and those that are 80%-biased toward tails? Quite confident, I take it. For-in brief-it is rather surprising that so nearly 80% of those coins landed the way she guessed; and if she can reliably decipher them, that would explain why this is so. Conversely, if I instead told you that only 60% of the judgments she was 80% confident in were correct, you should-for parallel reasons-suspect that she cannot reliably decipher the markings of the 80%-biased coins, and instead that she is likely over-estimating the strength of these coins' biases. Call this inference-from "Bianca was (mis)calibrated in her 80%-opinions" to "she probably can(not) reliably decipher the 80%-bias markings"-the deciphered-to-right inference, since it moves from her rates of being right to whether she has deciphered the markings. Clearly it is warranted in this simple scenario. And clearly there is an analogy between Bianca's test and Calvin's. If we can get clear on what exactly the 10 3. THE INSIGHT analogy is and why the deciphered-to-right inference works for Bianca, it'll show us what needs to be the case for the rational-to-right inference to work for Calvin. In fact, that's one of the main claims of this paper: if we want to know whether and to what extent we can expect the rational-to-right inference to work in a given scenario, imagine a parallel scenario for Bianca and her coins to see whether and to what extent the deciphered-to-right inference will work in that scenario. So: Why does the deciphered-to-right inference work in this scenario? I said that it is because the hypothesis that she can(not) decipher the coins would help explain her calibration if she is (mis)calibrated. But what does that mean more precisely, and why is it true? What it means more precisely is this. Before I tell you about Bianca's calibration, you should think to yourself: "If she can reliably recognize the 80%-biased coins, then the coins she says '80%' on will be (on average) around 80%-biased in the way she predicts- and conditional on that, I'm confident that roughly 80% of those tosses will land the way she predicts. Meanwhile, if she can't reliably recognize whether a coin is 80% biased, it's much more likely that a different proportion will land the way she predicts-for example, if she's over-estimating the bias, probably only 70% or 60% of the coins she says '80%' on will land the way she predicts." Thus the evidence you received-that 79% of her 80%-opinions were correct-is much more likely given that she can decipher the 80%-biased coins than it is given that she cannot; so it provides reason to think she can do so. Conversely, if you learn that only 60% of her 80%-opinions were correct, this is much more likely given that she's over-estimating the bias of the coins, so it provides reason to think that she is overestimating. The driving force of the deciphered-to-right inference, then, is that hypotheses about whether she is deciphering the coins' biases, over-estimating them, or under-estimating them, each have direct and strong implications for how many of the coins you should expect to land the way she guesses. Crucial question: why is this so? Answer: because hypotheses about the (average) biases of groups of coins have two very specific effects on how confident you should be in the outcomes of their tosses. First, you should defer to the average biases of the coins in setting your opinion for how a given coin will land: conditional on the coins corresponding to Bianca's 80%-opinions having an average bias of x% toward her predictions, you should be x%-confident that each of those predictions will be true. Second, this deference is independent: regardless of how her other predictions turn out, it is still the case that conditional on the coins having an average bias of x% toward 11 3. THE INSIGHT her prediction, you should be x%-confident that her next prediction will be true.16 Combined, these principles drive the deciphered-to-right inference by making it so that conditional on the coins having an average of x% bias toward Bianca's predictions, you're confident that roughly x% are true. Upshot: for the rational-to-right inference to work in Calvin's case, analogous deference and independence principles must hold. What does the analogy amount to? Bianca takes a bias-deciphering test in which she announces her best guesses about how coins with various markings landed, along with her confidence in those guesses. We want to use her resulting calibration score to draw conclusions about whether she is reliably deciphering the coins' biases, or over-estimating them, or under-estimating them. Meanwhile, Calvin takes a calibration test on which he announces his best guesses about the true answers to binary questions of various kinds, along with his confidence in those guesses. We want to use his resulting calibration score to draw conclusions about whether he is rational, overconfident, or underconfident. For each tablet Bianca is shown, there is a fact about what the corresponding coin's bias was. Likewise, for each question Calvin assesses, there is a fact about the rational degree of confidence he should have in the possible answers. We wanted to know whether Bianca can tell what the markings mean for the biases of the various coins. Likewise, we want to know whether Calvin can tell what his evidence means for the rational degree of confidence he should have in the various answers. In Bianca's case, the deciphered-to-right inference went though because we should defer to the biases of the coins, and do so independently of how her other predictions turn out. Likewise, then, in Calvin's case: the rational-to-right inference will go through when and because we should defer to the rational degrees of confidence for Calvin to have in his answers, and do so independently of whether his other answers turn out to be true or false. What does this mean more precisely? Consider all of the guesses Calvin assigns 80% confidence to-his 80%-opinions. Label them q1, ..., qn, so qi is the claim that the ith claim that Calvin was 80% confident in on this test (whatever it is) is true.17 We can entertain different hypotheses for what the average rational confidence is for Calvin to have in these claims. Let R be this quantity, whatever it is.18 Perhaps Calvin's 16 In standard setups of our case, these two principles follow from the well-known Principal Principle and its refinements (Lewis 1980, 1994; Hall 1994; Briggs 2009b). See below for formal statements of their analogues in Calvin's case. 17 For simplicity I assume you know that there are n such opinions. To generalize to the case where you don't know how many there are, we need to assume that learning how many there are would not affect our Deference and Independence principles below, and would not affect your confidence in what the (average) rational opinion for Calvin to have is. The inference will then go through by performing the reasoning described below, averaging over the various values n might take. 18 Formally, let R(qi) be the Rational confidence for Calvin to have in any given claim qi. Then R := 1 n ∑n i=1R(qi). I'll assume that the opinions that are rational for any given person can be modeled with a precise probability function. The same sort of reasoning may go through if the rational degrees of confidence were not unique (Schoenfield 2014) or not precise (Schoenfield 2012); for discussion of the 12 3. THE INSIGHT 80%-opinions are on average rational, in which case this quantity will be 80%: R = 0.8. Or perhaps they are on average overconfident (or underconfident), in which case it will be lower (or higher) than 80%: R < 0.8 (or R > 0.8). Let qi be any of Calvin's 80%-opinions. If you learn what the average rational opinion for Calvin to have in those opinions is, how does that affect your opinion in qi? For the case to be analogous to Bianca's, you must defer. Let P be a probability function representing your rational degrees of confidence. Then what we need is: Deference: Upon learning that the average rational confidence for Calvin to have in his 80%-opinions is x%, you should be x% confident in each of them. For all qi: P (qi|R = x) = x. How plausible is Deference? As I'll come back to in the conclusion, that depends heavily on the epistemological theory we accept.19 Importantly, no tenable epistemological theory will support a stronger deference principle than Deference-meaning that it's the tightest a connection between being rational and being right that we'll find. In particular, this means that all tenable epistemological theories will allow at least as much predictable rational deviations from calibration as those I illustrate in §§4–5. Why think Deference, holds, even in this simple scenario? It deserves far more discussion, but let me say two things in its defense. First, Deference tells you to defer to the opinions that are rational for Calvin to have, not the opinions he in fact has. Moreover, in our setup you don't know what claims are expressed by Calvin's 80%-opinions-qi is simply the claim that the ith claim on this test that Calvin was 80%-confident in (whatever that is) is true. Thus you have virtually no evidence about the qi. Meanwhile, Calvin has strictly more evidence than you about these claims-he knows all you do about the setup of the test, plus he knows which claims he was 80%-confident in, and therefore knows which facts bear on their truth. So conditional on Calvin's (more informed) evidence making it rational for him to be (on average) x% confident in these claims, it seems reasonable for you to be x% confident in it. Second, there is a strong intuition that the rational-to-right inference is sensible: it in principle makes sense to run calibration studies to test for overconfidence. As we'll see, whether this is so depends on whether a principle like Deference holds. Thus anyone (de)merits of such models, see White (2005, 2009a); Schultheis (2018); Carr (2019). 19 Deference is an interpersonal, rationalized, and "averaged" generalization of the well-known Reflection principle (van Fraassen 1984; Briggs 2009a; Christensen 2010b; Mahtani 2017). Appendix A.1 shows how this "averaged" version can be derived from a more familiar "point-wise" version. Whether interpersonal deference principles hold is highly dependent on the debate between uniqueness and permissivism (e.g. White 2005; Schoenfield 2014, 2019; Horowitz 2014b, 2019a; Greco and Hedden 2016; Schultheis 2018). Whether rationalized deference principles hold is highly dependent on debates around higher-order evidence (e.g. Williamson 2000, 2019; Christensen 2010b; Lasonen-Aarnio 2013, 2015, 2019; Elga 2013; Horowitz 2014a; Salow 2018; Dorst 2019a,b, 2020). Deference will be a theorem in our setup given uniqueness plus higher-order certainty; it'll be approximately true under a variety of weaker theories. 13 3. THE INSIGHT who thinks the rational-to-right inference makes sense in principle is under pressure to accept an epistemological theory that can support strong deference principles. Deference explains why the rational-to-right inference fails in many of our initial cases (§2). You shouldn't defer to Rajat (Case 1) because you something he doesn't-namely, that he's a brain in a vat. Likewise for Georgie-you know she had a bad geography teacher (Case 2). Similarly, when I saw that my 60%-biased coin landed heads only 30 of 100 times, I had evidence that you didn't when you formed your (rational) opinions about how it would land, so I shouldn't defer to them. Similarly for our final case-I shouldn't defer to your opinion about qi if I know that it's in the set W of guesses you were wrong about, since you (of course) didn't know you were wrong about them when you formed your guesses. However, Deference doesn't explain why the rational-to-right inference fails in our case of the misprinted coins. In that case, I haven't yet drawn the coin from the urn, so I defer to your rational opinions, yet I know you'll be miscalibrated. What's missing? This is where we need our second assumption to make Calvin's case analogous to Bianca's: independence. This says that once you learn the average rational confidence for Calvin to have in his 80%-opinions, learning about whether some of those opinions were true or false doesn't affect your confidence in the others. Precisely: Independence: Given that the average rational confidence for Calvin to have in his 80%-opinions is x%, further learning that certain of these opinions are true or false shouldn't affect your opinion in the others. For all qi0 , ..., qik : P (qi0 |R = x, qi1 , ..., qil ,¬qil+1 , ...,¬qik) = P (qi0 |R = x) How plausible is Independence? Again, there is much more to be said, but it is wellmotivated as a first approximation-after all, you know the test questions were selected randomly, so learning whether some are true or false shouldn't (significantly) affect your deference to information about Calvin's rational opinions on others.20 Independence explains why the rational-to-right inference fails in the case of the misprinted coins-in that case, we know that if the first toss lands heads, then the rest of them will as well. Deference and Independence imply that the rational-to-right inference is warranted in our simple scenario: learning that Calvin's 80%-opinions were (mis)calibrated provides strong evidence that they were (ir)rational. This is because the assumptions make the case analogous to Bianca's: "(average) rational confidence for Calvin" plays the same epistemic role for you as "(average) bias of Bianca's coins." Just as the deciphered-toright inference goes through in Bianca's case because you should defer (independently) to the biases of the coins, likewise the rational-to-right inference will go through in Calvin's 20 This is at best approximately true, as learning that all of Calvin's other 80%-opinions were false should make you suspect that the test is tricky. What's definitely true is that the qi are exchangeable (order doesn't matter) given R. Using this we could prove more general versions of the formula derived §A.2 by using beta-binomial distributions rather than binomial ones. The reasoning will be similar, and the closer the qi come to being independent, the stronger the rational-to-right inference will be. 14 4. THE LIMITS case when you should defer (independently) to the rational opinions for Calvin. In particular, conditional on Calvin's 80%-opinions being on average rational, you should be quite confident that roughly 80% of them will be true; and conditional on his 80%-opinions being on average overconfident (say, the average rational confidence is 60%), you should be quite confident that less than 80% (roughy 60%) of them will be true. Therefore when you learn that a given proportion of these opinions are true, that provides you with strong evidence about what the (average) rational confidence for Calvin to have is-i.e. about whether his actual opinions are (on average) rational. To give a simple example, suppose you are initially equally confident that the average rational confidence (R) for him to have in his 80%-opinions is any of 60%, 61%,..., or 99%. Suppose there are 50 such opinions. Let's say he is substantially overconfident if the average rational confidence in his 80%-opinions is less than 75% (R < 0.75). Then you are initially 37.5% ( 1540 ) confident that he is substantially over-confident. But if you were to learn that 70% of those opinions were true, then the rational-to-right inference is warranted: your confidence that he's substantially overconfident should jump to 78%.21 Upshot: despite a variety of concerns, the rational-to-right inference can be put on a firm theoretical foundation: when Deference and Independence hold, it is warranted. By the same token, however: when Deference fails, the exact same reasoning will show that the rational-to-right inference fails with it. For example, suppose that conditional on the average rational confidence being 80%, you should be 70% confident in each of Calvin's 80%-opinions: P (qi|R = 0.8) = 0.7. Then (if Independence holds) you should be confident that if Calvin's rational, 70% of his 80%-opinions will be true-and thus finding out that 70% of such opinions are true (he's slightly over-calibrated) will be evidence that he's rational, rather than overconfident! Thus we arrive at the key result: Deference is Key: Given Independence, the tenability of the rational-to-right inference in a given scenario stands or falls with the tenability of Deference. So the crucial question is: how robust is Deference to variations in our simple scenario? §4 argues that it is very fragile: there are common scenarios in which Deference systematically fails, and hence we should not expect rational people to be calibrated. However, §5 argues that these failures of Deference and the corresponding rational deviations from calibration are in principle predictable-meaning that a more nuanced type of calibration study is possible. 4 The Limits The real world isn't like the simple scenario, for you know a whole lot more about the test: its content, how it was constructed, what the experimenters were trying to 21 The general formula for this update is given in §A.2. 15 4. THE LIMITS show, what sorts of subjects were involved, and so on. Each of these bits of information threatens to undermine Deference and Independence in certain situations-and exploring the contours of these threats is important for having a full theory of the rational-to-right inference. Here I'll focus on just one type of information that a calibration study inevitably provides: our subject's full calibration curve-and, therefore, their overall proportion of true answers. Call that proportion their hit rate. Does knowing the hit rate cause a problem for the rational-to-right inference? Yes-the hit rate tells you which sorts of rational deviations from calibration to expect. To see why, start with a simple version of Bianca's case.22 Suppose in our archive all the tablets come from one of two coins-one that is 60% biased towards heads, the other that's 90%. Suppose we know that Bianca can decipher the coins, and that we'll randomly choose a couple dozen tablets from the archive. Should you expect her to be calibrated? Yes-but you should also expect her hit rate to be around 75%, because (1) she'll always guess heads (every coin is biased in favor of heads over tails), (2) we expect roughly 90% of the 90%-biased coins to land heads and 60% of the 60%-biased coins to do so, and (3) we expect they'll be roughly a 50-50 split between these coins (0.5 * 0.6 + 0.5 * 0.9 = 0.75). Now suppose it turns out that Bianca's hit rate is below 75%. Should you still expect her to be calibrated? Definitely not. This is easy to see in extreme cases: if the hit rate is very low (say, 50%), it's impossible for her to be calibrated-since the lowest credence she'll assign is 60%. Similarly if it's less extreme: whenever the hit-rate is below 75%, you should expect Bianca to be over-calibrated. The reason is that the hit-rate information breaks your deference to the biases of the coins. The connection between the biases of the coins and the frequency with which they land heads is probabilistic and therefore loose. Thus learning that the coins landed heads less often than you'd expect provides evidence that this is one of the cases where the biases and the frequencies came apart. That means upon learning that the bias of a given toss was 60% (90%), you should temper your deference downwards and be less than 60% (90%) confident that it landed heads. (Similar lessons apply if it turns out Bianca has a high hit rate: you should expect her to be under -calibrated.) The same lesson applies to Calvin: when you learn that his hit rate on some set of questions was low (or high), this provides evidence that it was one of the scenarios in which there is a gap between being rational and being right-that fewer (or more) of Calvin's guesses were correct than he'd be rational to expect. Thus you should temper your deference downwards (or upwards): conditional on the average rational confidence in his answers being x%, you should be less than (more than) x% confident in a given answer. Now let's state this line of reasoning more carefully. Let's assume that Calvin's hit 22 Thanks to Daniel Rothschild for putting his finger on this way of explaining the problem. 16 4. THE LIMITS rate, H, will be (approximately) equal to the rational hit rate, Hr-i.e. the hit rate he'd have if his degrees of confidence were rational. Since when faced with the question "A or B?" Calvin will guess the option he's more confident in, and he should guess the option he should be more confident in, this amounts to the assumption that in such binarychoice questions, Calvin will (usually) be more confident of A than B iff he should be. Grant this assumption for now-§4.1 explains why it's a reasonable one. Granting the hypothesis that H = Hr, we can see that the rational-to-right inference will break when we learn Calvin's hit rate because this will break Deference. Consider again whether Calvin's 80%-opinions are rational. Learning Calvin's hit rate does not itself significantly affect your opinion this question-after all, learning (merely) the rational hit rate shouldn't affect your opinion whether his 80% opinions are rational, and we're granting that his hit rate equals his rational hit rate.23 So learning his hit rate doesn't shift your opinions in his rationality. But it does shift your opinions in the truth-values of his answers-for example, if his hit rate is low, you know many of his answer are wrong. This gives you information that he couldn't have had when he formed his opinions (he can't know how many of his opinions are right when he's in the process of forming them). Therefore this shift in your opinions about the truth-values should temper your deference to his rational credences. For instance, suppose you learn that Calvin's hit rate is abnormally low-say, 50%. (75% is normal, since it's the average of 50−100%.) Now suppose you learn that Calvin's 80%-opinions were on average rational-should you be 80% confident in each of them? No! You should be less confident than that, since you know that more of them are false than he (rationally) expected. Thus although absent any information about the test you defer to his rational opinions, given hit-rate information you don't: P (qi|R = 0.8) = 0.8, but P (qi|R = 0.8, H = 0.5) < 0.8 Thus conditional on his 80% opinions being rational, you should only be (say) 70% confident in each one being true. And conditional on his 80% opinions being overconfident, you should be even less confident-say, 60%-in each one being true. If so, then-by exactly parallel reasoning to that at the end of §3-learning that only 70% of his 80%opinions are true (he's slightly over-calibrated) is evidence that he's rational. For it's evidence that his confidence matched his accuracy as as could be expected, given the difficulty of the questions. The rational-to-right inference is inverted. (Likewise, if you learn that Calvin's hit rate is abnormally high, the inference will be inverted in the other way-learning that he's slightly under -calibrated will be evidence that he's rational.) Here's a simple example. Again suppose you are initially equally confident that 23 Precisely: for any t, s, P (R = s|H = t) ≈ P (R = s|Hr = t) = P (R = s). 17 4. THE LIMITS the average rational confidence for him to have in his 80%-opinions (R) is any of 60%, 61%, ..., 99%, and there are 50 of them. Suppose that learning that his hit rate was 50% does not affect your confidence in any of these hypotheses, but it has the effect of tempering your deference in each downward by 10%: P (qi|R = x,H = 0.5) = x − 0.1 (so, for example, if his 80%-opinions are rational you should be 70% confident in each of them). Say that Calvin is approximately rational if 0.75 ≤ R ≤ 0.85. Then you are initially 27.5% (1140 ) confident that he's approximately rational, but upon learning that 70% of his 80%-opinions are true (he's slightly over-calibrated), you should increase this confidence to 61%. Meanwhile, you should decrease your confidence that Calvin is substantially overconfident (R < 0.75) from 37.5% to 22%, inverting the effect from the end of §3. In summary, we've arrived at the following result: Hit Rates are Key: The rational-to-right inference works only when (rational) hit rates are moderate-on any set of questions on which (rational) hit rates are high (or low), rational deviations from calibration should be expected. This qualitative claim raises a quantitative question: how much deviation from calibration should we expect, as hit rates vary? In §5 I'll show how we can answer this question under the assumption that people's actual hit rates match the rational ones; so first, we need to clarify why this is often a reasonable assumption. 4.1 (Rational) Hit Rates To do so, we need to get a bit clearer on what the overconfidence hypothesis and its alternatives might be (§2). We've been simplifying by focusing on the "overconfidence effect"-in fact, many studies find wildly different calibration curves for different types of questions. Sometimes people are over-calibrated at all levels of confidence; other times they are over-calibrated at high levels of confidence and under-calibrated at low levels of confidence; other times they are under-calibrated at all levels of confidence, and so on (more on this in §5; see Koehler et al. 2002; Brenner et al. 2005). Translating these calibration curves to corresponding (ir)rationality hypotheses, the varying types of possibilities are shown in Figure ??. In this figure, interpret the lines as averages: for example, the "over-extreme" hypothesis says that when a person's actual confidence is 80%, the confidence it is on average rational for them to have is merely 60% (as indicated by the red dot). With these options on the table, the live (ir)rationality hypotheses are claims of the form, "For questions of type X, people's confidence obeys (ir)rationality hypothesis Y ", where X is some specification of question-type, and Y is a curve having a shape like those in Figure ?? (Brenner et al. 2005). For example, ecological models have proposed that if X is "questions sampled randomly from a natural domain," then Y is the rational 18 4. THE LIMITS Rational Over-extreme Under-extreme Too confident Not confident enough 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 Actual Confidence R at io na lC on fid en ce Figure 3: The Various (Ir)rationality Hypotheses curve (Gigerenzer et al. 1991; Juslin 1994); meanwhile, case-based judgment models have proposed (among other things) that if X is "questions on which case-specific evidence is statistically weak and the base rate of truths is moderate," then Y is the over-extreme curve (Griffin and Tversky 1992; Koehler et al. 2002; Brenner et al. 2005). Here is an important prediction of any such (ir)rationality hypothesis: if the alternative claims someone is guessing between are from the same domain, then people's guesses will tend to be rational. Why? Note that all proposed (ir)rationality hypotheses have positive slopes, meaning that (on average) higher rational degrees of confidence correspond to higher actual degrees of confidence. Take any such (ir)rationality hypothesis, and consider a guess between a set of claims that it treats as in the same domain-say, "Which is bigger: Rome or Madrid?" The rational guess for Calvin is the one that he should assign higher confidence to: if R(Rome) > R(Madrid), it's rational for Calvin to guess Rome; and if R(Rome) < R(Madrid), it's rational for Calvin to guess Madrid. But since higher rational degrees of confidence correspond to higher actual degrees of confidence, the (ir)rationality hypothesis predicts that if the former, then Calvin's actual confidence will be higher in Rome than in Madrid-i.e. he'll guess Rome; and if the latter, Calvin's actual confidence will be higher in Madrid-i.e. he'll guess Madrid. Either way, the (ir)rationality hypothesis predicts that Calvin will guess rationally.24 24 Formally, let C(q) be Calvin's actual confidence in q, and let an (ir)rationality hypothesis be a function f : [0, 1] → [0, 1] mapping actual degrees of confidence to (average) rational degrees of confidence: R(q) = f(C(q)). Any such function that is monotonically increasing (f(x) > f(y) iff x > y) will be such that if R(q) > R(p), then f(C(q)) > f(C(p)), hence C(q) > C(p). Notably, since f is most plausibly interpreted as an average, there will be exceptions to this connection between rational and actual guesses. How common such exceptions will be depends on (1) how steep the slope of the (ir)rationality hypothesis is, and (2) how widely the deviations from f are distributed. Notably, if we 19 5. THE IMPLICATIONS Upshot: at least when people are guessing amongst claims that come from the same domain, all (ir)rationality hypotheses will agree that people's guesses will tend to be rational. People's hit rate is fully determined by their guesses-so if their hit rate is low, then (since their guesses will tend to be rational) this means that the rational hit rate is low as well. In other words, in many studies it is common ground amongst all (ir)rationality hypotheses that we are in a situation in which we know that Calvin's hit rate will be (close to) the rational hit rate. Since Hit Rates are Key, this means that when we learn the hit rate on our study, the rational-to-right inference will fail in predictable ways. 5 The Implications At this stage we've established that we should not expect rational opinions to be calibrated on sets of questions for which the hit rates turn out to be low (or high)-even if the questions were selected randomly from a domain that is representative of people's knowledge. This means that no matter how carefully we construct our test, we cannot evaluate the overconfidence hypothesis by simply checking whether people's opinions are calibrated-for we should often not expect rational opinions to be calibrated. What should we do, instead? My proposal is that we use the Bianca analogy to predict the rational deviations from calibration given our test setup, and then compare observed calibration curves to those predictions. We can do this in three steps: 1) Choose a test-construction procedure, along with a hypothesis about how this procedure will sample from rational opinions and right opinions. 2) Translate that hypothesis into the Bianca analogy and use it to build a simulation of the rational opinions. 3) Compare the predicted (mis)calibration of the rational opinions from this simulation to the actual calibration curves we observe. I'll spend the rest of this paper illustrating how this methodology can work and arguing that it calls into question the standard interpretation of certain empirical effects.25 5.1 The Hard-Easy Effect It turns out that the "overconfidence effect" is an overgeneralization: it is not the case that people are in general over-calibrated on binary-question tests. Rather, we can distinguish the tests that are hard from those that are easy based on the hit rate: an use the average hit rate (across subjects) on a test, and assume that subjects share similar evidence, such deviations from rationality should cancel out, and the average actual hit rate should be quite close to the average rational one. 25This methodology is a generalization of the simulation-based approaches found in, for example, Juslin et al. (1997, 1999); see §2 for more on the relation between the two. 20 5. THE IMPLICATIONS easy test is one with a hit rate of at least 75%; a hard test is one with a hit rate of less than 75%. The empirical generalization that subsumes the "overconfidence effect" is called the hard-easy effect: people tend to be over-calibrated on hard tests and under -calibrated on easy tests-see Figure 3. (The reason we see the "overconfidence effect" on general-knowledge trivia tests is simply that most such tests turn out to be hard.) The hard-easy effect has been called "fundamental bias in general-knowledge calibration" (Koehler et al. 2002, 687), and is widely cited as one of the core pieces of evidence in favor of the overconfidence hypothesis (e.g. Lichtenstein et al. 1982; Keren 1987; Gigerenzer et al. 1991; Griffin and Tversky 1992; Juslin 1994; Juslin et al. 2000; Koehler et al. 2002; Brenner et al. 2005; Hoffrage 2004; Moore and Healy 2008; Glaser and Weber 2010). The standard interpretation is that people do not make sufficient adjustments for task difficulty, leading them to be overconfident on hard tests and underconfident on easy ones.26 1.0 I *--Eas~subsetoFa test * D- ~ Esy;octual test 0-------Hard; subset of a test *1 Hard; actual test .4 a0 .4 .8 U 0 e # * Q. 00 A..6. Subjects' Response jFigure 3. Calibration for hard and easy tests and for I hard and easy subsets of a test. 19 7-7 Calibrated All questions Easy questions (hit rate ≥ 0.75) Hard questions (hit rate < 0.75) 0.5 0.6 0.7 0.8 0.9 1.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Actual Confidence P ro po rt io n T ru e Figure 4: The hard-easy effect. In both graphs, top curves are easy sets of questions; bottom curves are hard ones. Left: Lichtenstein et al. (1982). Right: My study. The hard-easy effect is one of the core pieces of evidence offered in favor of various versions of the irrationality hypothesis. However, we now know that systematic patterns of miscalibration should sometimes be expected of rational people when the hit rate varies. The question, then, is whether these empirical effects should be surprising given the null hypothesis that people are (approximately) rational. 26A closely related effect is the base rate effect: on tests in which subjects are simply presented with a series of claims and then rate their confidence from 0−100%, the overall proportion of truths (the base rate) has a dramatic effect on people's calibration curves (Lichtenstein et al. 1982; Koehler et al. 2002): low base-rates tend to lead to over-calibration and high base-rates tend to lead to undercalibration. Though for brevity I will omit simulations for this effect, the methodology and predictions are exactly the same, since learning the base rate on a set of questions breaks our Deference condition in the exact same way that learning the hit rate does (§4). 21 5. THE IMPLICATIONS 5.2 Testing for Rationality How can we know what to expect rational calibration curves to look like on tests of various types? The way we're going to answer this question is by returning to our coin analogy with Bianca. We are now going to assume that she can (at least usually) decipher the tablet markings-and thus set her confidence (at least approximately) equal to the biases of the coins-and go on to simulate what calibration curves we should expect from her as we vary the method of constructing the test and the difficulty of various sets of questions from it. Step 1 is to choose our test-construction procedure, and form a hypothesis about how this procedure will sample from the rational opinions and the right opinions. In particular: (i) how likely are we to include a question on which the rational credence in the answer is 50%? 60%? Etc. (ii) And on any given test we give, do we defer to the rational opinions? If so, how robust is that deference-does Independence hold, or would learning of false (true) answers temper our deference downwards (upwards), away from the rational credence? These questions matter because they affect (i) how often our simulations present Bianca with coins of various biases, and (ii) how robustly the bias of the coins lines up with our expectations about how many of them land heads. First focus on the simplest case: a test on which we can reasonably suppose that (i) the questions we pull are equally likely to have any level of rational confidence in their guess, between 50− 100%; and on which (ii) our deference is quite robust. One way to try to form such a test is to make one on which we pull questions randomly from a well-defined, representative domain on which we can expect that the accuracy of people's evidence will not be systematically correlated across questions. This turns out to be a difficult criterion to meet, but I'll take a standard paradigm from the literature (Gigerenzer et al. 1991), and pull pairs of American cities randomly from the top-20 most populous cities, and ask people to guess which they has a bigger population and to rate their confidence in that guess. On a representative-question test like this, it's reasonable to posit that the rational credences in answers will be fairly uniformly distributed between 50−100%. How robust your deference should be is a more vexed question-if we discover Calvin is wrong about whether San Francisco is larger than Phoenix, should that temper our deference to his rational opinion about whether San Jose is bigger than Austin? Perhaps-but let's ignore that for now (and come back to it in a moment). Given this, we can model perform Step 2: model (and then simulate) our test using the Bianca analogy. We toss a number of coins (equal to the number of questions on our test), selecting them uniformly at random from coins of varying biases between 50− 100%27, have her guess how they'll land and rate her confidence in that guess, and 27 I simplify by tossing coins of biases 50− 100% and having her always announce heads, rather than tossing coins of biases between 0 − 100% and having her first guess whether the coin lands heads or 22 5. THE IMPLICATIONS record her calibration curve. This is a single trial. Repeat this procedure thousands of times, and now look at the average results on trials (sets of questions) that have various hit rates. What do we expect to see for question-sets of various hit-rates? For all simulations, I'll display two versions. The perfection model assumes Bianca always gets the biases of the coins exactly right (analogy: Calvin is always perfectly rational). The noise model assumes that Bianca's announced confidence is a random perturbation of the bias of the coin-capturing the idea that she may be a reliable but imperfect at deciphering the coins' biases (analogy: Calvin's confidence may be a reliable but imperfect tracker of the degree of confidence his evidence warrants).28 The most plausible versions of the rationality hypotheses are ones in which there is some such error-though of course it's worth emphasizing that whenever their is such error, the person by hypothesis is not fully rational (cf. Brenner 2000). Nevertheless, since such deviations from rationality will be randomly distributed, there is still a good sense in which people who conform to such models are approximately rational. Calibrated All trials Hit Rate = 0.85 Hit Rate = 0.6 0.5 0.6 0.7 0.8 0.9 1.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Actual Confidence P ro po rt io n T ru e Calibrated Hit Rate = 0.85 All trials Hit Rate = 0.6 0.5 0.6 0.7 0.8 0.9 1.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Actual Confidence R at io na lC on fid en ce Figure 5: Random tests, restricted to various hit rates. Left: Perfection model. Right: Noise model (100,000 trials each). For illustration, the expected calibration curves for Bianca at various hit rates are displayed in Figure 4. When we consider all trials together, Bianca is calibrated- tails. The underlying statistics are the same. 28I assume the errors are normally distributed with mean 0; Figure 4 uses standard deviation 0.2. This model takes inspiration from "error models" (Erev et al. 1994; Pfeifer 1994; Juslin et al. 1997, 1999), but the interpretation is importantly different. Their models treats people's reported opinions as imperfect indicators of their true opinions (or, in some variations, the objective frequencies of truths), whereas mine treats people's reported opinions as imperfect indicators of the rational opinions. It is plausible that the latter errors will be larger than the former (it's harder to know what you should think than to know what you do think!). Moreover, while tests of the variance of people's reports have suggested that error in reporting their true confidence cannot account for the observed miscalibration (Budescu et al. 1997), these tests cannot test for error in matching their reported confidence to the rational confidence. 23 5. THE IMPLICATIONS perfectly so in the perfection model; slightly less so in the noise model due to "scale-end effects" (Juslin et al. 2000)-at the end-points of the confidence scale, errors can only go in one direction, resulting in the tilting of the curve. But amongst tests where the hit rate is low (high), Bianca tends to be over-(under-)calibrated-just as observed empirically with the hard-easy effect. Why? Consider a given trial on which the proportion of heads was lower than usual. Why was it lower? One explanation is that this trial had an abnormally large proportion of coins that were biased against landing heads. A different explanation is that this test was unusual in the sense that more of the coins landed tails than you'd usually expect, given their biases. Both are likely to play a role in any given trial with a low hit rate. Bianca will account for the first factor in setting her degrees of confidence, since she can recognize the coins and see that more of them than usual have a low bias-but it is impossible for her to account for the second factor. The result? As we consider cases with more extreme hit-rates, Bianca will become increasingly miscalibrated. For example, take the perfection model-where Bianca is as sensitive to the biases of the coins as she could possibly be. On the binary-question test, on trials with a hit-rate of 75%, Bianca's average confidence was 75%; on trials with a hit rate of 90%, her average confidence is 77% (becoming under-calibrated); and on trials with a hit rate of 60%, her average confidence is 72.7% (becoming over-calibrated). Upshot: even if the calibration tests contain questions that are random samples of the overall distribution of rational opinions (the best-case-scenario for the rational-to-right inference, as seen in §3), we would still expect some form of the hard-easy to emerge for rational subjects. Moreover, if they are merely approximately rational (the noise model), we should expect rational calibration curves that are qualitatively similar to the curves we observe empirically (compare the right side of Figure 4 to Figure 3). Let's now perform Step 3 and apply this model to my study (pre-registration available here). I generated all pairs from the 20 most-populous U.S. cities, and recruited 200 U.S. residents through Prolific (90 F, 107 M, 3 Other; mean age 34.7). After giving them standard instructions about how to use the 50–100% confidence scale ("Ideally, 8 out of 10 of the things you're '80%' confident in should be true", etc.), I presented each with 21 pairs-20 randomly selected from the 190 pairs of the top-20 U.S. cities, and 1 attention check. (Data from those who incorrectly answered this check were excluded; only 1 participant failed.) I pooled subjects' answers, and divided the questions into those that were easy (more than 75% of answers correct) and those that were hard (less than 75% correct). Figure 3 (page 20) above shows the calibration curves from my study-overall, amongst the hard questions, and amongst easy ones. The hard-easy effect was observed as expected- though it was especially stark. Amongst hard questions the average confidence was 24 5. THE IMPLICATIONS 75.1%, while the proportion true was only 45.2%.29 Meanwhile amongst easy questions, the average confidence was only 84.7%, while the proportion true was 92.1%.30 Somewhat unexpectedly, the test overall was slightly hard, with an average confidence of 79.8% and a proportion true of only 68.0%-accounting for the over-calibration observed across all questions in Figure 3.31 We can compare these results to both the perfection-model and noise-model predictions. As pre-registered, I generated these simulations by setting the number of questions Bianca faces to the size of the easy/hard/all-questions set, simulating millions of trials, and then removing trials with high/low hit rates until the mean hit rate matched the actual hit rate in the easy/hard/all-questions sets.32 The perfection model has no free parameters; its comparisons are displayed on the left of Figure 5. As can be seen, each predicted curve crosses the actual curve but the overalland hard-curves have significantly steeper slopes. Calibrated All questions (HR = 0.680) Simulated all questions Easy questions (hit rate = 0.921) Simulated easy questions Hard questions (hit rate = 0.451) Simulated hard questions 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Actual Confidence P ro po rt io n T ru e Calibrated All questions (HR = 0.680) Simulated all questions Easy questions (hit rate = 0.921) Simulated easy questions Hard questions (hit rate = 0.451) Simulated hard questions 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Actual Confidence P ro po rt io n T ru e Figure 6: Random tests run with the observed hit-rates in my study. Left: Perfection model (5 million trials). Right: Noise model (8 million trials, noise parameter = 0.3). The noise model has a free parameter for the standard deviation in the noise that leads people's credences to diverge from the rational credences. As pre-registered, to set 29The difference is significant: average confidence in hard questions (M = 0.751, SD = 0.165) was above proportion true of hard questions (M = 0.452, SD = 0.498), with a one-sided independent samples t-test of t(2487) = 25.82, p < 0.001, and d = 0.808. 30The difference is significant: average confidence in easy questions (M = 0.847, SD = 0.162) was below proportion true of hard questions (M = 0.921, SD = 0.270), with a one-sided independent samples t-test of t(3156) = 10.37, p < 0.001, and d = 0.333. 31 Since unexpected, this test was not pre-registered; but the difference was significant: mean of confidence across all questions (M = 0.798, SD = 0.170) differed from mean truth-value (M = 0.680, SD = 0.467), with a two-sided independent samples t-test of t(5013) = 14.97, p < 0.001, d = 0.336. 32The main limitation was that the observed hit rates were too extreme for me to obtain enough trials with the observed hit rate using the actual number of test questions in each set, so I had to use a lower number of 60 questions in each easy/hard/all category. The shape of the generated curves is quite robust to this parameter. 25 5. THE IMPLICATIONS this parameter I ran versions of the simulations with the parameter varying from 0−0.3, and chose the one with the resulting predicted calibration curves that minimized mean squared divergence between the model predictions (amongst hard and easy subsets) and the actual curves. This set the noise parameter to 0.3, and the resulting comparison of curves is displayed on the right of Figure 5.33 As can be seen, the predictions generated from the rational-credence-plus-noise model, though not a perfect fit, are generally close to the observed mis-calibration. But this isn't the end of the story. One puzzling thing about the above simulations is why it was so difficult to find instances with hit rates as extreme as we observed in the real study (of 8 million trials, only 183 had hit rates at or below 0.515, while my study's hard questions had a hit rate of 0.452). A natural answer is the following: in tests that share a common subject-matter-such as my city-comparison tests, and many others34-we need to revise our assumption of Independence, since the subject's evidence will be highly correlated across questions. In particular, though we should expect that the opinions warranted by their evidence will on the whole be calibrated, we should also expect that a there will be random fluctuations in how calibrated they are across subjectmatters. For instance, in my city-comparison test, some subjects will have evidence that warrants misleadingly strong opinions (only 70% of the opinions they should be 80% confident in are true), while others will have evidence that warrants misleadingly weak opinions (90% of the opinions they should be 80% confident in are true). Moreover, we expect these fluctuations in evidence to be correlated for a given person on a given subject-matter-if only 50% of the opinions Calvin ought to be 60% confident in on my test are true, we should expect that (say) only 60% of the ones he ought to be 70% confident in are true. Here's a natural way to model this. Again there is a random number of coins of varying biases that Bianca can recognize, but this time there is random variation across tablet archives in how representative they are of the broader distribution of tablets- some archives have higher proportions of heads from a coin of a given bias than would be expected; other have lower proportions. Thus for each trial (visit to an archive), we generate a random misleadingness parameter and add it to the coin biases to determine how far the proportions of heads in this archive deviates from the biases of the coins.35 Although I had constructed these models before running my city-calibration test, it only occurred to me that they were an apt model of it after running the test and seeing how extreme the variation in hit-rates were. As a result, these comparisons were not 33Obviously these are not the most rigorous statistical methods, but they suffice to illustrate the conceptual points of this paper. It should also be noted that this is a rather high noise parameter, corresponding to a fair amount of random deviation of actual confidence from rational confidence. 34(Dunning et al. 1990; Vallone et al. 1990; Brenner et al. 1996; Koehler et al. 2002; Brenner et al. 2005; Hoffrage 2004; Glaser and Weber 2007; Merkle and Weber 2011; Brenner et al. 2012). 35In the displayed simulations this parameter is normally distributed with mean 0 and (for illustration) standard deviation 0.2. In these simulations I assume that the variation in misleadingness is only in the magnitude-not the direction-of the evidence, so it never pushes below 50%. 26 6. THE OPEN QUESTIONS pre-registered, and therefore should be taken with a grain of salt. But it turns out to be much easier to find hit-rates as extreme as the ones we observed using this model, lending it some support. Running the same analysis as above yields find the optimal noise parameter at 0.15, and yields the comparisons in Figure 6. Calibrated All questions (HR = 0.680) Simulated all questions Easy questions (hit rate = 0.921) Simulated easy questions Hard questions (hit rate = 0.451) Simulated hard questions 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Actual Confidence P ro po rt io n T ru e Calibrated All questions (HR = 0.680) Simulated all questions Easy questions (hit rate = 0.921) Simulated easy questions Hard questions (hit rate = 0.451) Simulated hard questions 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Actual Confidence P ro po rt io n T ru e Figure 7: Tests with random misleadingness (parameter = 0.2), run using the observed hit rates in my study (20,000 trials). Left: Perfection model. Right: Noise model, parameter = 0.15. The point? The rational models are by no means a perfect fit. However, the largescale qualitative effects on this sort of study-such as the hard-easy effect, and the prediction that the subjects will be overall over-calibrated if the hit rate is below 75%-are predicted. Moreover, when we incorporate the possibility of random noise in subject's judgments or random misleadingness in subject's evidence, the observed calibration curves are close to what we should expect from rational people. The important takeaway is that it is the (much smaller) deviations from these predicted curves that we must study systematically, not the deviations between people's actual confidence and the perfectly-calibrated line. 6 The Open Questions I'll close by briefly considering a few open questions for the theory of rational (mis)calibration developed here: the philosophical tenability of Deference (§6.1), and the bearing of these results on the Dunning-Kruger (§6.2) and "over-precision" effects (§6.3). Readers uninterested in these issues may skip to §7. 27 6. THE OPEN QUESTIONS 6.1 The Tenability of Deference I have built a theory of rational (mis)calibration-of the connection between being rational and being right-based on Deference. But, as mentioned in §3, Deference is the strongest tenable interpersonal deference principle-and there are many philosophical reasons to be worried about it. First, if epistemic rationality is permissive, then you may have different epistemic standards than Calvin (White 2005; Schoenfield 2014, 2019). If so, the fact that his standards rationalize being 80%-confident in q doesn't imply that your standards do-perhaps your standards warrant being systematically more cautious (less extreme) in your opinions than Calvin's do. If so, Deference will fail-for instance, you may temper your deference downward from Calvin even without hit-rate information: P (qi|R = 0.8) < 0.8. This sheds doubt on whether permissive views of epistemic rationality can justify the rational-to-right inference even in our best-case scenario (§3). Similarly, if epistemic rationality is modest-meaning it can be rational to be unsure of what credences are rational-then it Deference must sometimes fail (Christensen 2010b; Elga 2013; Dorst 2019a). The only deference principle I know of that is both tenable in general in the case of modesty, and would warrant a variant of the reasoning from §3 is (a generalization of) the "Trust" principle in Dorst (2019a).36 So far as I know, every other proposed principle (Elga 2013; Pettigrew and Titelbaum 2014; Gallow 2019)-or argument that there can be no such general deference principle (LasonenAarnio 2015; Williamson 2019)-would allow wild and systematic deviations between what you learn about Calvin's rational credence and the resulting credence you should adopt. As a result, they would not undergird the rational-to-right inference even in ideal cases. These issues are pressing, since permissivism and modesty are both thought to be highly applicable to any notion of rationality that applies to human reasoners. As such, it's important to figure out whether such theories can explain the connection between being rational and being right in a way that could undergird the methodology of calibration studies-or whether such views are committed to the claim that none of these results provide us with evidence of overconfidence. 6.2 The Dunning-Kruger effect The Dunning-Kruger effect (Kruger and Dunning 1999) is the finding that those who are comparatively unskilled in a given domain are also unable to accurately assess how comparatively unskilled they are. Precisely: the gap between a person's relative performance on a test (which percentage of test-takers did they outperform?) and their 36 The "variant" reasoning requires looking not at whether the (average) rational credence is exactly t, but instead whether it is at least (or at most) t-which would require pooling people's judgments into categories "at least 50% confident", "at least 60% confident"; and so on. 28 6. THE OPEN QUESTIONS estimate of this number grows as relative performance decreases. For example, those in the 50th percentile may estimate that they are in the 60th percentile, while those in the 20th percentile may estimate that they are in the 50th percentile. This finding is routinely chalked up to a cognitive bias-a failure of the metacognitive ability to assess how competent one is (Dunning 2012). The theory developed here tells against this, and instead supports the theory from Moore and Healy (2008) and Jansen et al. (2018). As we've seen, for any set of rational opinions, there will be tests that are hard and easy for those opinions-in particular, that will lead to low or high hit rate (§2). We have also seen that in any such test, as the test gets harder for a rational person, they will become increasingly over-calibrated- meaning the gap between performance (actual hit rate) and estimated performance (average confidence, i.e. estimated hit rate) will grow (§5.2). Therefore, since even for rational people, the difficulty of a test will vary depending on their knowledge and skills, a straightforward consequence is that rational people who perform less well on a test will over-estimate their performance more than rational people who perform better. The Dunning-Krueger should be expected of them.37 6.3 Rational Over-precision? A different type of calibration test asks people to state various confidence intervals for the true value of some unknown parameter, such as the length of the Amazon. Empirically, there is a sense in which people tend to be systematically more over-calibrated on tests like this (Juslin et al. 1999; Moore and Healy 2008; Glaser and Weber 2010; Ortoleva and Snowberg 2015; Moore et al. 2015)-what has come to be known as "over-precision" (Moore and Healy 2008). The theory developed here may help explain this. First note that this test can be translated to our familiar format (Tversky and Kahneman 1974): your 90% confidence interval for the length of the Amazon is "1000 to 5000 miles" iff you are 95% confident in both "The Amazon is at Least 1000 miles long" (L), and "The Amazon is at Most 5000 miles long" (M); your confidence intervals tend to miss the true value too often iff you are over-calibrated on claims like L and M . Now, for someone to be calibrated in their interval estimates, items must fall outside the range of their 90%-confidence interval exactly 10% of the time. Yet studies regularly find "miss rates" as high as 50%, and almost never lower than 10% (Glaser and Weber 2010, 243). Is this evidence for a more robust form of overconfidence? Not obviously. Note that for hard binary-question tests, it is standard to see less than 75% of people's 95%-opinions being true-in fact, our noise models from §5 predict at least that much over-calibration when hit rates are low (Figures 4, 5, and 6). Now, Calvin's 90%-confidence interval for the Amazon's length in miles is "1000 to 5000" iff 37 Contra Merkle and Weber (2011)-whose response to Moore and Healy (2008) illicitly assumes that Bayesians will have priors that match the objective frequencies on the test. 29 7. THE CONCLUSION he is 95% confident in both"The Amazon is at Least 1000 miles" (L) and "The Amazon is at Most 5000 miles" (M). Our credence that both L and M are true (his interval covers the true value) is our credence in the former, multiplied by our credence in the latter given the former: P (L ∧M) = P (L) * P (M |L). As we've seen, given that Calvin's credence is 95% in each of these claims, if he's only approximately rational and the test his hard, we should be only 75% confident in each of them. That means that if they were independent, we'd expect his interval to cover the true value (L and M to both be true) about 0.75× 0.75 = 56.25% of the time, leading to a miss rate of around 44%. But note that they are (by definition) not independent: if L were false (the Amazon is less than 1000 miles), M would necessarily be true; hence learning that L is true necessarily lowers the probability of M : P (M |L) < P (M) = 0.75. Thus we should expect a hit rate for the conjunction L ∧ M of less than 56.25%, and hence a miss rate of greater than 44%; hence 50% does not seem especially surprising for hard tests. Moreover, by parallel reasoning we should expect less-than-10% miss rates only if more than 95% of a person's 95%-opinions are true. Yet we've seen that (due to scale-end effects) this is virtually never the case (none of our binary-question noise models-even with easy tests-see such high rates). Thus it seems an open question whether attention to rationality of over-calibration combined with the non-independence of the individual probability judgments that compose an interval estimate might shed new light on empirically observed over-precision. 7 The Conclusion Many have taken the results of calibration studies to demonstrate that people tend to be systematically overconfident in a way that is both dire and preventable. I've argued that the theoretical foundations of this inference are shaky (§2), but that we can secure them by articulating a probabilistic connection between being rational and being right (§3). Yet though this supplies a foundation to such studies, it also reveals a flaw: no matter how well-designed the study, rational people should be expected to be miscalibrated in systematic ways (§4). Using these systematic deviations, I proposed a modification of the standard methodology: we must use hit rates and other information about our study to predict the rational deviations from calibration, and then compare people's performance to those predictions. I illustrated how this can be done, and argued that it may overturn the standard interpretation of robust empirical effects (§§5–6). If even a portion of this discussion is correct, is suggests that certain debates in philosophy and psychology are much closer than has been realized. Psychologists have had a long, spirited debate about the bearing of empirical results (like those of calibration studies) on human (ir)rationality.38 Yet most contemporary philosophical debates 38 For classic statements of the "irrationalist" approach, see Tversky and Kahneman (1974, 1983); 30 7. THE CONCLUSION about rationality have been relatively isolated from these issues.39 As we've seen, these debates needn't-and arguably shouldn't-be isolated. Whether and the extent to which we have empirical evidence for overconfidence depends on the connection between being rational and being right. The proper formulation of such a connection is directly dependent on philosophical debates about the proper formulation of deference principles-and, relatedly, about permissivism and epistemic modesty (§6.1). Thus these philosophical debates have a direct bearing on the proper interpretation of these empirical studies. Conversely, the methods and results from calibration studies are directly relevant to ongoing the philosophical debate about how to understand the connection between being rational and being right.40 For instance, simulations like the ones I used in §5-based on the methods developed by psychologists (Erev et al. 1994; Pfeifer 1994; Juslin et al. 1997, 1999, 2000)-can be used to make precise predictions about the relation between rational confidence and accuracy. In short: both psychologists and philosophers have been investigating rationality- but often from radically different directions, and without substantial discussions. We've seen that the questions, methods, and tools from these investigations can be tied together in surprising and fruitful ways. That raises an exciting question: If we bring these investigations closer together, what other ties might we find?41 Kahneman et al. (1982); Kahneman and Tversky (1996); Fine (2005); Ariely (2008); Hastie and Dawes (2009); Kahneman (2011b); Thaler (2015). For defenses of "rational" approaches see Anderson (1990); Gigerenzer (1991); Oaksford and Chater (1994, 2007); Tenenbaum and Griffiths (2006); Hahn and Oaksford (2007); Hahn and Harris (2014); Harris and Hahn (2011); Tenenbaum et al. (2011); Griffiths et al. (2012); Cushman (2018). 39Though in recent years there are an increasing number of exceptions, e.g. Cohen (1981); Stich (1985); Kelly (2004, 2008); Crupi et al. (2008); Fitelson and Hawthorne (2010); Koralus and Mascarenhas (2013); Nebel (2015); Icard (2017); Hedden (2018); Mandelbaum (2018); O'Connor and Weatherall (2018); Singer et al. (2019); Doody (2020); Quilty-Dunn (2020). 40Joyce (1998); Littlejohn (2012); Pettigrew (2016b); Schoenfield (2016b); Horowitz (2014b, 2019b); Comesaña (2020); Staffel (2020). 41 Thanks to Lyle Brenner, Liam Kofi Bright, Thomas Byrne, Fiery Cushman, Chris Dorst, Dmitri Gallow, Cosmo Grant, Brian Hedden, Thomas Icard, Joshua Knobe, Harvey Lederman, Matt Mandelkern, Don Moore, Daniel Rothschild, Bernhard Salow, Miriam Schoenfield, Ginger Schultheis, James Shaw, and audiences at FEW 2020, MIT, and the Universities of Bristol, Pittsburgh, Oxford, and Sydney, for much helpful feedback and discussion. 31 7. THE CONCLUSION Appendix A.1 Deriving Deference Recall that q1, ..., qn are the claims that Calvin assign 80%-confidence to, that R is the rational probability function for him to have overall, and that R is the average rational confidence in the qi: R := ∑n i=1 R(qi) n . Recall Deference: Deference: Upon learning only that the average rational confidence for Calvin to have in his 80%-opinions is x%, become x% confident in each of them. For all qi: P (qi|R = x) = x. (For simplicity of notation I maintain focus on Calvin's 80%-opinions. Obviously, parallel principles and reasoning apply to the others thresholds.) Deference follows from two further principles: Point-wise Deference: Upon learning the rational credence function for Calvin is δ, become δ(qi)-confident in each qi. For all qi : P (qi|R = δ) = δ(qi).42 Equality: Upon learning only that the average rational confidence for Calvin to have in his 80% opinions is x%, be equally confident in each of them. For all qi, qj : P (qi|R = x) = P (qj|R = x). Since Equality is extremely plausible in the situations we're considering (where you don't know anything more about the qi than they they were claims that Calvin was 80% confident in), this shows that Deference follows from the more familiar Point-wise version. To prove this, for any random variable X (a function from possibilities to numbers), let E[X] := ∑ t P (X = t) * t be your rational expectation of X. (Assume a finite state space, for simplicity.) Note that R is a random variable; also note that if I(qi) is the indicator variable for qi (1 if qi is true, 0 otherwise), then E[I(qi)] = P (qi). Let Dx = {δ1, ..., δk} be the set of possible values of R such that ∑n i=1 δj(qi) n = x, so that R = x⇔ R ∈ Dx. First, focus on your expectations of the proportion of truths, conditional on R = x: E[ ∑ I(qi) n | R = x] = ∑ δ∈Dx P (R = δ| R = x) * E[∑ I(qi)n | R = δ] 42 Here 'δ' is a rigid designator for a particular probability function (an assignment of numbers to propositions), whereas R is a definite description for "the rational credence function for Calvin, whatever it is"-so R can vary across possibilities but δ cannot. 32 7. THE CONCLUSION By linearity of expectations, this equals = ∑ δ∈Dx P (R = δ| R = x) * 1 n n∑ i=1 E[I(qi)| R = δ] = ∑ δ∈Dx P (R = δ| R = x) * 1 n n∑ i=1 P (qi| R = δ) (Definition) = ∑ δ∈Dx P (R = δ| R = x) * 1 n n∑ i=1 δ(qi) (Point-wise Deference) = ∑ δ∈Dx P (R = δ| R = x) * x (Definition of Dx) = x. Therefore E[ ∑ I(qi) n | R = x] = x, so by linearity of expectations, your average rational credence in the qi equals x: 1 n ∑n i=1 P (qi|R = x) = x. By Equality, since each of the values in this sum is equal, they must all be equal to x-therefore for all qi: P (qi|R = x) = x, establishing Deference. A.2 The Rational-to-Right Formula Here I show how to calculate what your posterior confidence should be that Calvin is overconfident in his 80%-opinions when Deference and Independence hold, you know that there are n such opinions, and you learn how (mis)calibrated they are. Recall: Deference: For all qi: P (qi|R = x) = x. Independence: For all qi0 , ..., qik : P (qi0 |R = x, qi1 , ..., qil ,¬qil+1 , ...,¬qik) = P (qi0 |R = x) Suppose you initially leave open that R will be any of t1, ..., tm, with prior probabilities P (R = ti). Note that Deference and Independence imply that P (*|R = ti) treats the qi as independent, identically-distributed Bernoulli variables with success probability ti. 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A brief explanation of Kant's Enlightenment article Florian Millo* Abstract A presentation of Kant's idea for enlightenment process that was happening at that time. I try to be objective as it is needed to give a thorough explanation for what was the main subject in this process. Kant explains the main idea of enlightenment and describes it with examples for which stands descriptive and understandable for that period. *[email protected] Universite Paris-Saclay, Department of Physics 1 Introduction The problem of 18th-Century was deep inside man's mind, laziness, cowardice, immaturity. In Kant's paper, he starts the opening paragraph with a sentence. Enlightenment is man's emergence from his self-imposed immaturity. [1] What can we understand from this sentence? First let's say that we see enlightenment as man's emergence, then what is the next process to take place? Many philosopher prefer to call enlightenment as multiple enlightements, as there were a lot of contributions from others fields that illuminated generations. One of Kant's preferred philosopher was Jean Jacque Rousseau and his main work was Social Contract, which was a thorough study of political and moral philosophy that motivated Kant to write on enlightenment. 2 Understanding sentences One of keywords in Kant's paper is immaturity. When he says - "Immaturity is the inability to use one's understanding without guidance from another", he is talking about laziness. Is important the fact that immaturity is deeply related with laziness. One can say, no, for which none has an argument. The man has to use his mind to understand things by himself and by a guidance that does not requires any third parties. A good example of this was so called "Motto of Enlightenment" Dare to know!.....A line taken from Horace's Epode - 'Sapere Aude!' Next, I present a paragraph that is known to be a good example, it follows:-"If I have a book to serve as my understanding, a pastor to server as my consience, a physician to determine my diet for me, .... , I need not exert myself at all" In this example, Kant tells us that laziness had sorrounded everyone back then and everyone could not do anything without help from other parts of society. Thus, my understanding follows, if individual would work out to help theirselves, immaturity would become mature and never existed. Why immaturity existed? Is it even important? Rules and formulas, those mechanical aids to the rational use, or rather misuse, of his natural gifts, are the shackles of a permanent immaturity. Comment: One should free himself from immaturity and pursue a secure course. Kant argues that freedom of thinking is the key problem and not all had it. Freedom is the least harmful of all, and the freedom to use our reason publicly in all matters. Kant heard in all his sides: "Do not argue!" The officer says, "Do not argue, drill!" The tax man says, "Do not argue, pay!" The pastor says, "Do not argue, believe!. Comment: There were many restrictions back then, because of man's immaturity. All obeyed and with no protest. Checking all the chronology and what was the main problem, Kant deduced an answer for that:-"The use of man's reason must always be free, and it alone can brind out enlightenment among mans" Question: Which reason, private or public? Kant then defines two uses of reason: 1st: Public use of reason The use that everyone as a scholar makes of reason. 2nd: Private use of reason The use for which 1 a person may make in a civic post or office that has been trusted him. Comment: We have to understand, reason must be carefully used, because when it is used wrong in wrong places, one can call it a mistake but one can use it from his immaturity. Later on, Kant explains that we do not live in an enlightenment age but in an age of enlightenment. He then puts religion to present his main points, as for, Manner of thinking, Spiritual Freedom, Dignity. In the end Kant concludes that nature cares about religions principles applied to us, and thus they must be in accordance to principles of government, thus being in accord with people's dignity. Peoples must raise their spirit for freedom and the free of speach is one main thing, we should do. Dignity must agree with these principles, otherwise you need to change! As presenting these principles, we should have in mind Knowledge and Reason. Knowledge is important and as an archive of our brain, we must use it in different situations and to solve them. Theories of Knowledge explains that laziness and cowardice comes because one can not use his knowledge to finish what one wants and to do what one needs. Acknowledgement I thank Prof. Ted Humphrey for the good translation that he did, for which was a good source for me. References [1] An Answer to the Question: What is Enlightenment? IMMANUEL KANT. Translation by Ted Humphrey [2] An Answer to the Question: What is Enlightenment? 1784. Immanuel Kant [3] Image Other Works: [4] Observation of high-energy astrophysical neutrinos in three years of IceCube Data Florian Millo [5] High Velocity Cloud Analysis in HI4PI Data Florian Millo [6] γ − β Spectrometry of 207 Bi Florian Millo

© Dissertatio Volume Suplementar, Junho 2015 WITTGENSTEIN: UMA SOLUÇÃO FUNDACIONISTA AO PROBLEMA DO REGRESSO EPISTÊMICO Eduardo Ferreira das Neves Filho Juliano Santos do Carmo Universidade Federal de Pelotas RESUMO: As notas que compõem a obra Da Certeza (Über Gewissheit) expressam nitidamente a preocupação de Ludwig Wittgenstein com os problemas clássicos da epistemologia, em especial o uso dos termos epistêmicos tradicionais e os erros costumeiros dos filósofos que negligenciam suas profundas estruturas gramaticais. Em diversas passagens é fácil observar a tentativa de esclarecer os erros de realistas, idealistas e céticos no que diz respeito às nossas alegações ordinárias de conhecimento em contextos céticos moderados. A questão do ceticismo sobre a justificação é um tema recorrente na epistemologia analítica contemporânea e, de certo modo, as soluções ofertadas a este problema ainda não são completamente adequadas. Existem muitas passagens de Da Certeza que possuem o potencial de lançar luz sobre questões fundamentais que encontram-se imbricadas neste debate, cuja discussão contemporânea tem sido fomentada pela instigante análise de Daniele Moyal-Sharrock (2005, 2007). O objetivo deste artigo é justamente tentar esboçar uma reposta wittgensteiniana ao problema do regresso epistêmico. ABSTRACT: The notes that compose On Certainty (Über Gewissheit) clearly express the concern of Ludwig Wittgenstein with the classic problems of epistemology, in particular the use of traditional epistemic terms and the usual mistakes of philosophers who neglect their deep grammatical structures. In various passages is easy to observe the attempt to clarify errors of the realistic, idealistic and skeptical with regard to our common claims of knowledge in moderate skeptical contexts. The issue of skepticism about the justification is a recurring theme in Contemporary Analytical Epistemology and, in a sense, the solutions offered to this problem are not fully adequate. There are many passages in On Certainty that have the potential to shed light on fundamental questions that are intertwined in this debate, whose the contemporary discussion has been fostered by provocative analysis offered by Daniele Moyal-Sharrock (2005, 2007). This article is just trying to sketch a possible Wittgenstein's response to the epistemic regress problem. Eduardo das Neves Ferreira Filho Juliano do Carmo 104 1. Ceticismo: O Problema da Justificação Epistêmica O ceticismo sobre o conhecimento pode ser considerado a partir de uma dupla perspectiva1: (1) o Ceticismo Epistêmico diz que não existe tal coisa como o conhecimento, uma vez que o mundo é considerado como incomensurável e indeterminável (contexto cético exigente), e (2) o Ceticismo sobre a Justificação diz que não existe crença verdadeira "adequadamente justificada" (contexto cético moderado). Em se tratando do Ceticismo sobre a Justificação, a maioria dos epistemólogos costuma assumir que o conhecimento implica necessariamente em crença verdadeira adequadamente justificada, ainda que não exista consenso a respeito do que pode contar como "justificação adequada"2. A estratégia habitual dos céticos consiste em tentar mostrar que as condições impostas às teorias da justificação (em especial o problema do Regresso Epistêmico) não são ou não podem ser satisfeitas. O problema do Regresso Epistêmico remonta pelo menos a Pirro de Élis (365-270 a.C.), mas encontra muitos ecos em diversas posições filosóficas modernas e contemporâneas. Seja no caso das crenças que suportam outras crenças na cadeia inferencial de justificação, seja no caso das regras ou métodos aplicados no curso da justificação de uma crença, o problema do regresso epistêmico parece invalidar completamente as nossas alegações ordinárias de conhecimento. As teorias da justificação epistêmica que visam reabilitar nossas pretensas alegações de conhecimento podem ser divididas em "teorias fundacionais" e "teorias não-fundacionais", sendo que a diferença principal entre ambas as posições é justamente a resposta que oferecem à questão: "quais crenças e outros métodos inferenciais requerem justificação?". As teorias não-fundacionais costumam defender que todas as crenças e métodos inferenciais requerem justificação. As teorias fundacionais, por outro lado, assumem que algumas crenças e métodos 1 Ver: LUPER, S. The Skeptics: Contemporary Essays. Aldershot: Ashgate, 2003. 2 Existem diversos outros problemas com a definição clássica de conhecimento, especialmente os contraexemplos de Gettier e o "Dilema de Sellars" que, embora sejam fundamentais para a discussão contemporânea sobre epistemologia, não serão tratados aqui. Ver: LUZ, A. Conhecimento e Justificação. Pelotas: Dissertatio-Filosofia, 2013. SARTORI, C. O Dilema de Sellars: Desafio ao Fundacionismo Epistêmico. Porto Alegre: Veritas, 2009. Dossiê Wittgenstein, Dissertatio Volume Suplementar 01 | UFPel [2015] 105 (crenças e métodos fundacionais) não necessitam de justificação, enquanto todas as demais crenças e métodos de justificação estão em última análise assentadas em crenças e métodos fundacionais3. Na base do problema do Regresso Epistêmico, portanto, está a ideia de que para que uma crença seja adequadamente justificada ela precisa estar necessariamente vinculada a uma outra inferência ou a uma cadeia de inferências justificadas. O cético sugere, neste caso, que não é possível encontrar um modo adequado de garantir este vínculo, pois (i) ou bem a cadeia inferencial deve começar com inferências que não estão baseadas em outras inferências; (ii) ou bem a cadeia inferencial não possui um início fundacional e admite processos circulares de justificação; (iii) ou bem a cadeia de justificação inferencial deve seguir infinitamente. Uma vez que cadeias infinitas de justificação inferencial são impossíveis para seres humanos finitos, pois sequer poderíamos compreendê-las, a última alternativa (iii) fica imediatamente excluída. Uma vez que cadeias circulares não geram a justificação adequada e que cadeias de justificação inferencial não podem começar com suposições arbitrárias, então ficam excluídas também as duas primeiras possibilidades. Não parece restar outra alternativa senão concordar com o cético sobre a justificação epistêmica acerca do fato de que não existem crenças adequadamente justificadas ou fundamentadas. As três condições expostas acima formam o famoso Trilema de Agripa4: (1) A justificação não pode regredir infinitamente (Problema do Regresso ao Infinito); (2) A justificação não pode ser circular (Problema da Petição de Princípio) (3) A justificação não pode estar baseada em meras suposições (Problema da Arbitrariedade); A última condição às vezes é dividida em duas: (3) e (4) A justificação não pode estar baseada em proposições disputáveis (Problema da Autoevidência)5. 3 HARMAN. G. Skeptcism and Foundations. In. LUPER, S. The Skeptics: Contemporary Essays. Aldershot: Ashgate, 2003. 4 LAÉRTIUS, D. Vidas e Doutrinas dos Filósofos Ilustres. Brasília: UNB, 2008. Livro IX. 5 A formulação original encontra-se em: LUPER, S. Cartesian Skepticism. In: BERNECKER, S., PRITCHARD, D., Ed(s). The Routledge Companion to Epistemology. New York: Routledge, 2011, p. 415. Eduardo das Neves Ferreira Filho Juliano do Carmo 106 De acordo com a posição cética, contudo, as teorias candidatas a resolver o Trilema de Agripa fracassam na exata medida em que não parece possível encontrar uma crença adequadamente justificada sem fracassar ao mesmo tempo em satisfazer conjuntamente as condições (1)-(4). Basta notar que se conhecimento é identificado com crença adequadamente justificada e se a justificação depende da satisfação das condições (1)-(4), então qualquer justificação finita (1) e não circular (2) deveria começar com uma premissa que não está baseada em qualquer outra afirmação, o que tornará essa premissa uma mera suposição (3). Além disso, se levarmos em conta a condição (4), torna-se realmente difícil encontrar uma noção adequada de justificação, pois qualquer proposição é em alguma medida disputável. Se uma "proposição disputável" for compreendida como uma "proposição bivalente", então as coisas se tornam um pouco mais complicadas, pois a justificação não poderia estar baseada em qualquer proposição, uma vez que a bivalência pode ser considerada como uma característica básica de qualquer proposição genuína. As principais estratégias disponíveis para tentar resolver o Trilema de Agripa são as seguintes: (a) Mostrar que o conhecimento não requer justificação6 a partir da suposição de que nossas crenças sobre o mundo são produzidas através de um método confiável (Confiabilismo, Externalismo, Naturalismo e demais teorias nãofundacionais), mesmo que não estejamos justificados em pensar que elas tenham sido obtidas dessa forma7. (b) Recusar a condição (3), demonstrando que algumas crenças podem ser justificadas de um modo não-inferencial (ou ainda, não justificadas por outras crenças). Neste caso, seria preciso mostrar que as crenças justificadas de modo não-inferencial não implicam em deixar de satisfazer a condição (1). (c) Negar que a justificação adequada de uma crença deva satisfazer especialmente a condição (2), e procurar defender a legitimidade da justificação circular. (d) Rejeitar a quarta condição (4), pois ela supostamente inviabiliza qualquer 6 Ver: LUPER, S. The Causal Indicator Analysis of Knowledge. Philosophy and Phenomenological Research 47: 563–89. DRETSKE, F. Skepticism: What Perception Teaches. In: LUPER, S. The Skeptics. Aldershot: Ashgate Publishing, 2003, pp. 105-19; DRETSKE, Fred. Knowledge and the Flow of Information. MIT Press, 1982. 7 Ver: MILLIKAN, R. White Queen Psychology and Other Essays for Alice. MIT Press, 1993; Dossiê Wittgenstein, Dissertatio Volume Suplementar 01 | UFPel [2015] 107 reivindicação de conhecimento, já que seria aparentemente um absurdo pensar que não temos justificação ao crer em proposições que outros contestam. (e) Procurar satisfazer as quatro condições e mostrar o que há de errado nas considerações habituais do ceticismo sobre a justificação. Em Da Certeza, Wittgenstein procurou mostrar que certas proposições que superficialmente parecem proposições empíricas são, pelo contrário, proposições lógicas ou gramaticais8. Certas proposições mooreanas9 do tipo "Eis aqui uma mão", parecem dizer algo factual sobre o mundo e por isso poderiam erroneamente ser consideradas como estando sujeitas à dúvida e aos habituais recursos de verificação aplicados às proposições empíricas. As proposições gramaticais estão logicamente imunes à dúvida e servem como o próprio fundamento a partir do qual as proposições empíricas adquirem sentido. Ao que tudo indica, Wittgenstein não estava tão interessado em refutar as dúvidas céticas a respeito da existência de um mundo externo (ceticismo global), mas, antes, seu interesse estava voltado prioritariamente para o uso dos termos epistêmicos (tais como a utilização de verbos tais como "saber" e auto-atribuições de crença do tipo "eu sei"). Temos razões para crer que a obra Da Certeza tem o potencial de lançar luz sobre uma série de questões pertinentes ao Trilema de Agripa na epistemologia contemporânea (sobretudo no que se refere ao problema do Regresso Epistêmico), pois além de fomentar a ideia de que conhecimento e certeza pertencem a categorias distintas (satisfazendo a condição (2)), Wittgenstein defende que "certezas" são os próprios fundamentos de todo o nosso sistema conceitual (satisfazendo as condições (1) e (3)) e que "certezas" não são proposições (satisfazendo a condição (4)). Seguindo os insights de Daniele Moyal-Sharrock sobre a obra Da Certeza, vamos esboçar agora uma possível solução wittgensteiniana de cunho fundacionista ao problema do Regresso Epistêmico. 8 Ver: MCGUINN, M. Responding to the Sceptic: Therapeutic versus Theoretical Diagnosis. In: LUPER, S. The Skeptics: Contemporary Essays. Aldershot: Ashgate, 2003. 9 Ver: CHILD, W. Wittgenstein: uma introdução. Porto Alegre: Artmed, 2013. Eduardo das Neves Ferreira Filho Juliano do Carmo 108 2. Wittgenstein: Certeza Objetiva e Conhecimento É reconhecido por boa parte dos comentadores de Wittgenstein que as anotações que produziu em seus dois últimos anos de vida têm grande importância para a epistemologia, particularmente no debate com o ceticismo em suas versões mais radicais e abrangentes. A tarefa de organizar e interpretar os seiscentos e setenta e seis parágrafos, reunidos sob o título Da Certeza (OC), foi desenvolvida por G.E.M. Anscombe e G.H. Von Wright, está em edição bilíngue (alemão/inglês), e cujas traduções alcançam diversos idiomas. As recalcitrantes observações de Wittgenstein em OC têm provocado as mais variadas polêmicas quando se trata de compreender possíveis consequências da análise sobre os conceitos de conhecimento e certeza, mas também sobre o desenvolvimento de sua própria filosofia10. Ponto pacífico na fortuna crítica é atribuir o ensejo de Wittgenstein, ao produzir as referidas anotações, a seu interesse pela defesa que G. E. Moore tentou proporcionar ao senso comum, cujos resultados foram publicados por este último em dois artigos no início do século XX: Prova de um mundo exterior (MOORE, 1939) e Uma defesa do senso comum (MOORE, 1925). Na 'prova' da existência de um mundo exterior, Moore pretende derivar que 'há objetos exteriores' da premissa de que 'existem duas mãos humanas', e esta do fato de levantar cada uma de suas mãos à audiência, durante famosa conferência, dizendo duas vezes 'eis aqui uma mão'. Ora, segundo Wittgenstein, o 'erro' na 'prova' não seria decorrente do mau uso da lógica, já que realmente se pode inferir a conclusão das premissas sem dificuldade, mas de sua irrelevância filosófica, visto que as premissas não seriam mais certas do que a conclusão (CHILD, 2013, 201). Ainda, se o objetivo de Moore com a 'prova' era refutar idealistas e céticos, Wittgenstein vai lembrar a Moore que ainda lhe faltaria, para atingir esse fim, realizar "um diagnóstico de exatamente onde e como o 10 Por exemplo, D. MOYAL-SHARROCK (2007) defende que o Da Certeza seria o marco divisor de águas para uma 'terceira' fase do pensamento de Wittgenstein, ou, como sustenta, inaugura no pensamento do autor um Pragmatismo Lógico, ou um 'Terceiro' Wittgenstein. Para maiores informações, veja Moyal-Sharrock, 2007. Já CHILD (2013) sustenta que nas Investigações Filosóficas já há indícios de temas semelhantes, sobretudo nos parágrafos 324-6 e 466-486. Dossiê Wittgenstein, Dissertatio Volume Suplementar 01 | UFPel [2015] 109 idealista e o cético se enganam" (CHILD, 2013, 202). Moore não cumpre a tarefa trivializando a argumentação e permanecendo na vala comum em que os oponentes (céticos e idealistas) pretenderiam que ele permanecesse: não prova nem que realmente existe um mundo exterior independente de nós, tampouco que, se realmente este mundo existe, que possamos conhecê-lo, por razões óbvias. O erro de Moore ao asserir que "Sei com certeza (grifos nossos) que tenho duas mãos", se pode sustentar, está em atribuir à certeza um papel subjetivo, que indica, como Wittgenstein diversas vezes chama a atenção em OC, apenas uma espécie de 'convicção inabalável'. O ponto é que a convicção de Moore não poderia, como tal, ser dada como garantia de conhecimento, mas apenas indicaria uma simples atitude de crença. Quando se levanta sobre uma crença uma pretensão de conhecimento devem-se dar razões, de modo a 'demonstrar' a alguém que de fato estamos de posse de algo forte como 'conhecimento': Se diz 'Sei ...' quando se está em condições de dar razões apropriadas. 'Sei...' está vinculado a uma possibilidade de demonstrar a verdade. Se alguém sabe algo sempre que esteja convencido – pode manifestar isso. Mas se o que acredita é de tal tipo que as razões que possa dar não são mais seguras que sua asserção, não pode dizer que sabe o que acredita (OC 243). Moore teria, então, de responder como sabe que há objetos exteriores, do mesmo modo que responder como sabe que tem duas mãos, premissa usada por ele para derivar a conclusão de que há objetos exteriores. Para isso, seria insuficiente levantar suas mãos à audiência. As 'razões apropriadas' para uma alegação de saber, de que fala Wittgenstein, falham, no caso, tanto para a premissa, quanto para a conclusão da derivação na 'prova' de Moore, pois no uso correto da expressão 'Eu sei ...', somos capazes de oferecer "os fundamentos deste saber, ou pelo menos, eles podem ser dados" (OC 484). É importante observar que em contextos em que conhecimento é alegado "(i) há uma possibilidade lógica de erro ou ignorância, e (ii) tal possibilidade tenha sido afastada por meio de 'regras evidenciais claras'" (GLOCK, 1998, p. 74). Na ausência de condições adequadas para uma alegação de conhecimento, pois, amplia-se o erro de Moore: ele afirma 'saber com Eduardo das Neves Ferreira Filho Juliano do Carmo 110 certeza' que tem duas mãos. Ora, se não é adequado o uso da palavra 'saber' do modo como o fez Moore (como aponta Wittgenstein), não é adequado qualificar esse 'saber' com um caso de 'certeza', que é considerada, ao menos em parte da tradicional teoria do conhecimento, como o mais alto grau da justificação epistêmica. Ao contrário, 'certeza', do modo como Moore utiliza a expressão, apenas indica uma certeza subjetiva, uma forte convicção, e nada mais, além disso. A certeza de Moore, no máximo, ganha ares 'metafísicos', ideia recusada por Wittgenstein em vários parágrafos de OC11. Até aqui a análise se desenvolve de um ponto de vista negativo. No entanto, embora Wittgenstein rejeite que Moore possa 'saber com certeza' que possui duas mãos, 'Tenho duas mãos' e 'Há objetos exteriores' e diversos outros truísmos semelhantes em função, admite, fazem parte de uma classe de certezas objetivas, que pertencem a uma categoria diversa da categoria do conhecimento (MOYAL-SHARROCK, 2005, p. 77), possuindo determinado (e importante) papel em nossos jogos de linguagem: Com a palavra 'certo' expressamos a convicção absoluta, a ausência de qualquer tipo de dúvida, e tratamos de convencer aos demais. Isso é certeza subjetiva. Mas quando uma coisa é objetivamente certa? Quando o erro não é possível. Mas que tipo de possibilidade é esta? O erro não tem que ficar excluído logicamente? (OC 194) Wittgenstein menciona as certezas objetivas em OC de diferentes maneiras, no entanto, parece haver uma confluência de significados nas expressões que são utilizadas para caracteriza-las: por exemplo, elas aparecem como 'imagem de mundo' (OC 94), isto é, seriam 'proposições'12 que compõem nossa imagem de mundo, possuindo um 11 De Acordo com Glock (2008, p. 74), "a expressão 'Eu sei ...' não pode tolerar, no entanto, uma 'ênfase metafísica' dessa natureza (OC 21, 251, 425, 481-2, 533-4)" a saber, no uso atribuído por Moore. 12 Utilizamos o termo 'proposição' destacado nessa passagem, visto que elucida o modo standard de apresentação do pano de fundo das certezas objetivas em OC. Entretanto, adiante sustentaremos que, na verdade, as certezas objetivas não se caracterizam como proposições, e são melhor compreendidas como crenças básicas. Dossiê Wittgenstein, Dissertatio Volume Suplementar 01 | UFPel [2015] 111 'papel lógico determinado' (OC 136), como o que diz respeito às nossas 'crenças mais básicas' (OC 253), servindo como uma espécie de 'fundamento' aos nossos jogos de linguagem (OC 401-2), como aquilo que 'está firme' (OC 655). Esses 'adjetivos', que qualificam as certezas objetivas, fazem alusão a característica de estarem imunes à dúvida. Como todo o jogo de linguagem, o jogo de linguagem do duvidar possui suas regras. Não faz sentido duvidar das regras, dos próprios fundamentos dos nossos jogos de linguagem, daquilo que nos ajuda a compor nossa imagem de mundo, do quadro de referência sobre o qual erigimos nossas crenças: 'Desta proposição não posso duvidar sem renunciar a todo o juízo'. Mas, que tipo de proposição é esta? Isso lembra o que Frege disse sobre a identidade. Evidentemente, não se trata de uma proposição empírica. Não pertence à psicologia. Tem, melhor, o caráter de uma regra (OC 494). (...) É dizer, nos interessa, para que o juízo seja possível, que não possa haver dúvida alguma com respeito a certas proposições empíricas. Ou também: tendo a crer que nem tudo que tem a forma de uma proposição empírica é uma proposição empírica (OC 308). Wittgenstein afirma que não podemos duvidar dos fundamentos (ou regras) de nossos jogos de linguagem, pois dúvidas sobre eles paralisariam os jogos de linguagem. Essas 'regras' são certas, pois constituem a 'base' sob a qual erigimos nossos jogos de linguagem, e quaisquer dúvidas legítimas só poderão ser levantadas quando as tivermos sob nossa 'imagem de mundo': "Quem quer duvidar de tudo, nem sequer chegaria a duvidar. O mesmo jogo da dúvida pressupõe a certeza" (OC 115). Aqui temos uma 'pista' para tentar argumentar sobre a possibilidade de um fundacionismo em OC, e tentar mostrar como haveria, nesta perspectiva, uma solução ao problema do regresso epistêmico. A investigação, aqui, passa pela compreensão de MoyalSharrock (2005, 2007, 2013, 2014) tem das certezas objetivas como crenças básicas, o que passamos a expor a seguir. Eduardo das Neves Ferreira Filho Juliano do Carmo 112 2.1 Certezas Objetivas como Crenças Básicas que ocorrem em ação No parágrafo 204 de OC, Wittgenstein faz uma importante afirmação: No entanto, a fundamentação, a justificação da evidência tem um fim; mas o fim não está em que certas proposições nos pareçam verdadeiras de forma imediata, como se fosse uma espécie de ver de nossa parte; pelo contrário, é nossa ação que jaz no fundo do jogo de linguagem (OC 204). Essa passagem sugere, de acordo com Moyal-Sharrock, que Wittgenstein está traçando um limite 'lógico', e não sustentado graus, entre certeza objetiva e conhecimento. Conhecimento está, obviamente, baseado em graus. Certezas objetivas não. O 'erro não é possível', no caso das certezas objetivas, pois não se trata de acrescentar razões à crença: "a certeza que Wittgenstein está procurando definir como objetiva, é objetiva não meramente oposta à subjetiva, mas assim: ela não está baseada de modo algum em graus. Pois, quando razões são apresentadas, estamos no domínio do conhecimento" (MOYALSHARROCK, 2005, p. 77). Por seu turno, as certezas objetivas são mencionadas por Wittgenstein como não-fundadas, não-epistêmicas, e desempenham um papel lógico importante na 'armação de nossos pensamentos'. Decorre que os modos utilizados por Wittgenstein para destacar a certeza objetiva fazem com que, de acordo com MoyalSharrock (2005, p. 78), aparentemente Wittgenstein pareça sustentar inconsistências. A primeira delas: certeza objetiva é tomada tanto como regra da gramática, como quanto modo de ação (o que o parágrafo 204, citado logo acima, pode sugerir). Essa inconsistência é dissipada, segundo a autora, quando Wittgenstein acaba considerando, mesmo que isso não apareça de modo claro o bastante em OC, 'certeza objetiva' e 'certezas objetivas'; no primeiro caso, Wittgenstein identifica o aspecto fundacional da certeza objetiva, e, quando fala de certezas objetivas, os objetos dessa certeza, isto é, as hinges (dobradiças), tais como 'Aqui há uma mão', 'Nosso cérebro não está oco', 'A Terra existe há mais de cinco minutos', etc. Ao se tentar explicar certeza objetiva tomada em seu sentido Dossiê Wittgenstein, Dissertatio Volume Suplementar 01 | UFPel [2015] 113 fundacional aparentemente surge outra inconsistência. Isto é, dois 'ângulos' filosóficos parecem ser oferecidos na análise de Wittgenstein: um fenomenológico e o outro categorial. A aparente inconsistência é fruto apenas das imagens utilizadas para informa-los, mas, na verdade, eles são complementares (MOYAL-SHARROCK, 2005, p. 79). Como categoria doxástica (MOYAL-SHARROCK, 2005, pp. 789), certeza objetiva tem caráter fundacional, seria a rocha dura das nossas convicções, aquela em que se apoia todo o nosso sistema de crenças, o que nos permite distinguir o verdadeiro do falso (OC 94), compondo uma imagem de mundo (OC 162), assegurando normas de descrição (OC 167), sob as quais não se pode supor corretamente, visto que a suposição, nesse caso, dar-se-ia sobre o próprio fundamento de nossas ações (OC 411). A consequência de todos estes apontamentos nos sugere que o papel da certeza objetiva é o de regra da gramática: Se concebermos 'Eu sei, etc.' como uma proposição gramatical, é óbvio que o 'Eu' não pode ser importante. O que, no fundo, quer dizer: 'Não há, neste caso, nada como uma dúvida', ou 'A expressão 'Não sei' carece de sentido aqui'. Por suposto, disso se segue que 'Eu sei' tampouco tem sentido (OC 58). Por seu turno, como atitude doxástica, a análise é tomada, para MoyalSharrock, em um sentido fenomenológico, e atitude deve ser compreendida, nesse caso, tanto como disposição, quanto de ocorrência propriamente dita. Quer dizer, se está falando de nossas 'certezas objetivas' (no plural mesmo), aquilo que se mostra no que dizemos e fazemos. Mas em que relação a que se orientam nossas falas e ações? Moyal-Sharrock observa que este tipo de atitude doxástica, ao contrário daquilo que se poderia supor, não é direcionado a crenças que, mas orienta-se em relação a 'objetos' que fazem parte da gramática, tais como estados de coisas, indivíduos, etc., que pertencem a gramática, "objetos que são usados como paradigmas de nossos métodos de descrição (...). Tais objetos (estados de coisas, indivíduos, etc.) são, assim como amostras ou objetos usados em definições ostensivas, parte da gramática" (MOYAL-SHARROCK, 2005, p. 81). Como tais, não são apreendidos em palavras, mas constituem nossas crenças mais básicas, mostram-se em nossas atitudes, são know-how, e pertencem à gramática. Eduardo das Neves Ferreira Filho Juliano do Carmo 114 A criança aprende a acreditar em muitas coisas. Isso é, aprende, por exemplo, a agir de acordo com essas crenças pouco a pouco, se forma um sistema com as coisas que acredita e, em tal sistema, alguns elementos se mantêm imutáveis e firmes, enquanto outros são mais ou menos móveis. O que se mantém firme o faz não por que seja óbvio intrinsecamente ou convincente, senão porque se sustenta naquilo que o rodeia (OC 144). Certezas objetivas formam uma estrutura, uma fundação. Em OC 96, Wittgenstein nos pede para imaginar que 'proposições', "que tem a forma de proposições empíricas", e que funcionariam como os canais por onde fluem as proposições empíricas. Esse seria o "fundo me minhas convicções" (OC 248). Justamente aí se encontra o fim das razões13, o fundamento das crenças, ou seja, as crenças sem fundamentos (OC 251), o lugar em que a dúvida não faz sentido. Essa seria a rocha dura14 das nossas certezas, "constituem um sistema, um edifício" (OC 102). Moyal-Sharrock chama a atenção ao fato de que é esse um 'conjunto' que funciona com todas as nossas crenças básicas, e não cada uma delas isoladamente (OC 141); cada uma dessas crenças será fruto de aprendizado e experiências (OC 279). Mas vejamos. Estamos falando de crenças, porém, crenças que não possuem um conteúdo proposicional15. Por que chamar a esse 13 E, portanto, como ainda frisaremos, uma resposta ao problema do regresso epistêmico. 14 Aqui, também, tomada a obra de Wittgenstein como uma continuidade, posição que sustentamos, a rocha dura das nossas convicções de que Wittgenstein trata em OC seria uma apresentação não-enigmática da menção à rocha dura nas Investigações Filosóficas. Esse ponto, no entanto, não é tema de discussão aqui. 15 Moyal-Sharrock sustenta que o uso 'especializado' do termo proposição, em Wittgenstein, se mantém desde a perspectiva tractariana, e coloca, como condição à proposicionalidade, a bipolaridade. De acordo com a autora, boa parte da fortuna crítica corrobora essa posição, garantindo que há evidências para isso em todos os textos de Wittgenstein, particularmente no Da Certeza (OC 320-1); ou seja, o conteúdo de uma proposição é compreendido "como uma entidade abstrata à la Frege; como o 'pensamento' de Frege, o sentido de uma sentença" (MOYAL-SHARROCK, 2007, p. 34). As proposições empíricas, por isso, estariam sujeitas à bipolaridade. Ora, as certezas objetivas, apesar de aparentarem serem proposições empíricas, não estão sujeitas ao critério da bipolaridade, tendo antes a função de regras da gramática. Assim sendo, seriam casos degenerados de proposição, ou não seriam genuínas proposições; uma regra não é verdadeira, nem falsa, tampouco uma regra é candidata à dúvida, verificação ou falsificação. Assim, aquilo que se costumou chamar de 'proposições-dobradiças' (hinge propositions) seria, antes de qualquer coisa, apenas um modo heurístico de nos reportarmos às regras da gramática, que se mostram em ação. Dossiê Wittgenstein, Dissertatio Volume Suplementar 01 | UFPel [2015] 115 'fenômeno' de crença? Em primeiro lugar, pois "crença não demanda justificação, não implica que uma inferência tenha sido feita" (MOYALSHARROCK, 2007, p.24), e desse modo prescinde das demandas para uma alegação de conhecimento, como vimos acima. Em segundo lugar, pois parece um termo apropriado para destacar uma 'manifestação imediata', uma 'segurança' (OC 510), algo que não é nem justificado, nem injustificado, mas 'algo animal' (OC 359). Essa 'confiança', segundo a autora, desperta uma espécie de fé cega (certeza implica, pois, a completa ausência de dúvida), nos fazemos valer de todo o nosso background, e expressamos essas crenças mais básicas não em palavras, mas em ação, elas se expressam não em um saber-que, mas em know-how. Quando falamos de background, não nos referimos a um conjunto de observações teóricas sobre o mundo, mas ao nosso arcabouço de práticas, sobre os quais nenhuma atitude 'consciente' deve se fazer necessária quando agimos, quando não desenvolvemos nenhuma resistência a isso (a ausência de reflexão, aqui, é uma característica importante, pois não há 'atenção' presente nas ações, quando se trata de agir; ao contrário, a atenção não é associada a uma crença básica16). Por exemplo, ao acordamos pela manhã, não checaremos diariamente se há, ou não, um precipício ao lado de nossa cama, de modo que venhamos a cair ao nos levantarmos. Tampouco nos certificaremos se nossa escova de dente realmente é sólida, se todos os dias nós teremos dentes para escovar, etc. São justamente esses 'pontos de contato' com nossas crenças básicas, certezas objetivas, que fazem do know-how 'objetivo' para nós (MOYAL-SHARROCK, 2005, p. 87), seja construindo nossa imagem de mundo, seja compondo o quadro de referência sobre o qual erigimos nossas crenças: a fundação de todas as nossas ações (OC 414) (incluindo, é claro, nossos jogos de linguagem (OC 403, 411)) é (...) descrita em termos do agir. (...) Com isso, Wittgenstein quer dizer que nossa certeza fundacional é uma certeza prática (não é teorética, nem proposicional, nem uma certeza advinda de uma pressuposição) que se manifesta como modo de ação (OC 7, 284-5, 395) (...) (MOYAL-SHARROCK, 2005, p. 89). 16 Obviamente, a reflexão sobre o que se passa no caso das certezas objetivas é resultado na análise filosófica, não há incoerência aqui. Eduardo das Neves Ferreira Filho Juliano do Carmo 116 Em sua perspectiva fenomenológica, as certezas objetivas ocorrem em ação, isto é, nessa leitura, uma certeza objetiva, uma hinge (dobradiça), não pode ser dita, ela se mostra em ação (MOYALSHARROCK, 2005, p. 89); a assunção mais adequada para caracterizar os seus papéis, como hinges, é apresenta-las como regras de gramática, como vimos, e sublinhar que tais regras se mostram no fluxo dos nossos jogos de linguagem. Há casos, contudo, em que algumas 'proposições', que têm a forma de hinges, poderão ser faladas17 em nossos jogos de linguagem. Nessas 'circunstâncias', não funcionam como hinges, mas apenas como proposições empíricas. O termo empregado por Moyal-Sharrock para destacá-las é doppelgänger, uma espécie de representação idêntica de uma hinge, mas com função de proposição empírica propriamente dita. Por exemplo: suponha-se que a administração central de um campus universitário quer garantir que todas as pessoas com necessidades especiais possam ser atendidas com qualidade em todos os prédios dos diferentes campi. Então, uma checagem é realizada junto à comunidade universitária, mediante um conjunto de perguntas. Entre elas, pergunta-se, por telefone ou por e-mail, quantas mãos, etc., tem a pessoa, e a resposta poderia ser "Eu tenho duas mãos". Nesse caso, a função da 'resposta' é de proposição empírica. Em outros casos podemos proferir (falar) certas doppelgänger com o propósito de esclarecer o uso de regras, como o fazemos com crianças, por exemplo. Isso não quer dizer que hinges, como hinges, possam ser ditas com propósitos descritivos (OC 548)18 como tentou fazer Moore em sua 'prova': Uma regra gramatical não pode ser dita; ela apenas se mostra no que dizemos e fazemos [o que mostraria o errro no caso da 'prova' de Moore – acréscimo nosso]. Quando escreve que algo não pode ser dito, Wittgenstein quer dizer que isso não pode 17 Moyal-Sharrock mantém que a distinção tractariana entre dizer/mostrar também vale para OC. Desse modo, algumas 'proposições', embora não possam ser ditas, podem ser faladas. Para a autora, é possível dizer que Wittgenstein endossaria essa distinção: nem tudo o que é falado, é dito; por exemplo, uma 'regra' pode ser falada, mas não dita. Confira em Moyal-Sharrock (2007, pp. 43-7). 18 Isso significa dizer que hinges são inefáveis de um ponto de vista 'lógico' e 'prático'. Dossiê Wittgenstein, Dissertatio Volume Suplementar 01 | UFPel [2015] 117 ser parte corrente em um jogo de linguagem; não que isso não possa ser proferido com propósitos heurísticos (MOYALSHARROCK, 2005, p. 91). Por seu turno, é importante notar que um proferimento de uma hinge pode ser realizado na análise filosófica, e tem, nesse caso, um propósito bem delimitado, tem função de esclarecimento, uma função heurística: "me sento junto a um filósofo no jardim; ele diz repetidamente 'Sei que isso é uma árvore' enquanto aponta para uma árvore junto a nós. Uma terceira pessoa se aproxima e o escuta, e eu digo a ela: 'Este homem não está louco: apenas filosofamos" (OC 467). Moyal-Sharrock pede atenção para a formulação/articulação das hinges, por um lado, e sua ocorrência ou manifestação, por outro. Neste último caso, uma hinge deve ser compreendida, como já vimos, como uma certeza animal, como não-proposicional, como uma espécie de confiança animal em certas coisas, e que se mostra em nossa ação, o que podemos chamar de uma descrição fenomenológica da função da hinge. No entanto, como a leitura do parágrafo (OC 467) nos sugere, a elucidação da 'função' das hinges faz com que (ao exemplificarmos sua função 'mencionando' algumas delas) possamos pensar que elas possuem algum conteúdo, e que, portanto, ocupariam o 'lugar' de proposições, sendo estas o 'conteúdo' de nossas crenças. Aqui reside o erro. A formulação de uma hinge terá, apenas, a função de elucidar seu papel no jogo de linguagem, por meio de uma descrição categorial (por meio de exemplos, de certezas objetivas): "isso tenta elucidar o status dessas certezas promulgadas (enacted) – nossos modos ordinários de agir – em nosso sistema de crenças" (MOYAL-SHARROCK, 2005, p. 93). 2.2 Crenças básicas como fundação e como solução ao problema do regresso Na última seção desse artigo, gostaríamos de esclarecer duas questões. Em primeiro lugar, visto que a assunção de que nossas certezas objetivas, nossas crenças básicas, constitui uma espécie de 'fundação', tentar dar uma resposta ao problema do regresso epistêmico. Eduardo das Neves Ferreira Filho Juliano do Carmo 118 Em segundo lugar, mostrar como seria possível que crenças básicas, ainda que não tenham o papel de 'princípios', possam, ainda assim, garantir as nossas demais crenças, ou seja, realmente poder-se falar de uma 'fundação', apesar de a relação entre crenças básicas e crenças nãobásicas não seja construída mediante processos inferenciais. Quanto a uma resposta ao problema do regresso epistêmico, a partir da leitura wittgensteiniana de OC proporcionada por MoyalSharrock podemos afirmar que se pode cumprir as quatro condições elencadas no princípio deste artigo, a saber: a) que a justificação de nossas crenças não se siga de um argumento circular, b) que a solução não se baseie em suposições, c) que não se caia em um regresso infinito de razões e d) que a justificação não esteja baseada em proposições disputáveis. Em relação à primeira condição, observe-se que não há circularidade na argumentação, visto que nossas crenças básicas compõem a 'rocha dura' de nossas crenças, a imagem de mundo sobre a qual construímos nossas convicções e nossas crenças justificadas em geral (as certezas são os fundamentos de todo o nosso sistema conceitual). Para que funcionem nossos jogos de linguagem, o próprio jogo de linguagem do duvidar necessitará de regras apropriadas, isto é, não se pode duvidar daquilo que é fundamento de nossas crenças, visto que "dúvidas sempre se baseiam em razões" (GLOCK, 1998, p. 74): 'Sei que sou um homem'. Para nos darmos conta de quão pouco claro é o sentido desta proposição, consideremos sua negação. Quanto muito, poderíamos interpretá-la assim: 'Sei que tenho os órgãos próprios de um ser humano''. (Por exemplo, um cérebro que, de todos os modos, nunca ninguém viu). Mas, que acontece com uma proposição do tipo 'Sei que tenho um cérebro'? Posso colocá-la em dúvida? Faltam-me razões para a dúvida! Todo conta a seu favor, nada contra ela. No entanto, é possível imaginar que por meio de uma operação se comprovasse que meu crânio está vazio (OC 4). Por seu turno, nossas crenças básicas não são 'suposições', o que responde a segunda condição, elas não são nenhum tipo de construto intelectual destinado a cessar, de modo artificial, uma cadeia de razões. Nossas crenças básicas, nossas certezas objetivas, não são 'intelectuais', mas estão presentes em nossa vida cotidiana como know-how, são mais Dossiê Wittgenstein, Dissertatio Volume Suplementar 01 | UFPel [2015] 119 bem compreendidas como as regras que orientam nossa participação nos jogos de linguagem, compõem a rocha dura das nossas convicções, como vimos, e ocorrem em ação. Desse modo, a terceira condição, a saber, que se apresente um modo de evitar o regresso infinito de razões, também é cumprida: O que os filósofos têm chamado de 'crenças básicas ou tácitas' não podem, às custas do regresso infinito, serem elas mesmas crenças proposicionais, e a concepção wittgensteiniana de certeza dobradiça mostra que elas não são. Enquanto epistemólogos tem sempre pensado em nossas crenças básicas como proposições, Wittgenstein as vê como regras da gramática ou limites de sentido que se manifestam como modos de ação (MOYAL-SHARROCK, 2013, p. 6) A quarta condição pode ser respondida de modo evidente. A 'fundação' oferecida por Wittgenstein é construída sobre certezas promulgadas (enacted), e não está baseada em proposições disputáveis; o ponto de partida de Wittgenstein não se dá sobre proposições, nem está aberto ao domínio do que pode ser verdadeiro ou falso; a 'base' ou a 'fundação' está em ação, manifesta-se como know-how, e é alcançada 'diretamente'. Mas aqui surge um problema. Se for possível sustentar que a base ou fundação ao conhecimento é composta por nossas crenças básicas, elas deveriam servir de sustentação para nossas crenças não básicas, o que particularmente é negado por Wittgenstein: "a partir das proposições fulcrais [como Glock as denomina, acréscimo nosso] não deduzimos outras verdades; calcamo-nos nelas como um 'pano de fundo' para nossa argumentação racional" (GLOCK, 1998, p. 77). Essa questão surge como fio condutor da investigação de D. Pritchard (2012) em seu artigo Wittgenstein and the groundlessness of our believing. Ele observa que Wittgenstein pretende assegurar que as crenças básicas possuam duas características importantes: ser imunes à dúvida e ao suporte racional, no que em nada diverge do que foi apresentado aqui até agora. Por sua vez, toda a avaliação racional é sempre local, e sempre pressuporá as hinges (dobradiças), que são nãoracionais. A consequência do argumento, para o autor, é que Wittgenstein apresenta certa concepção das 'estruturas racionais', cuja 'base' é sempre composta pelas dobradiças que são imunes à avaliação racional, visto que identificam um tipo confiança 'primitiva' (OC 475), Eduardo das Neves Ferreira Filho Juliano do Carmo 120 evitando o regresso de razões. O ponto de divergência entre Pritchard e Moyal-Sharrock irá concentrar-se, contudo, justamente sobre a nãoproposicionalidade das hinges, e das consequências que uma posição como essa poderá trazer à epistemologia. Pritchard adverte que se segue um problema fundamental para a análise de Wittgenstein, se as coisas são colocadas dessa maneira: não é justamente isso que o cético quer que assumamos, que a base para nossas crenças seja não-racional? Assim colocada a questão, a pretensa rejeição ao ceticismo ficaria comprometida: para Wittgenstein, o cético serraria o galho em que está sentado, visto que não se pode duvidar dos próprios fundamentos que permitem com o que o jogo do duvidar seja praticado. Mas, se isso é assim, então deveríamos ser capazes de mostrar como nossas crenças básicas 'suportam, realmente, a casa toda'; como, em cenários locais, nosso suporte racional não estaria definitivamente marcado pela 'ausência de um suporte racional' (se Wittgenstein está certo! visto que uma cadeia de razões, para Wittgenstein, sempre tem um fim, e, se está correta a leitura apresentada até agora em nosso artigo, o fim é a ação). Uma segunda questão a responder, segundo Pritchard (2012, p.5), que é decorrente da primeira, é que parece possível, em alguns casos, que determinados raciocínios nos permitam acreditar, com razões, em alguma hinge, o que Wittgenstein não aceitaria, visto que, para ele, não seria possível dar razões às hinges. O raciocínio a que se reporta Pritchard é construído mediante o recurso do Princípio de Fechamento: Se S sabe que P e S competentemente deduz Q de P, assim formando sua crença de que Q sobre a base de sua competente dedução, enquanto mantém seu conhecimento de que P, então S sabe que Q (PRITCHARD, 2012, p. 7). Em um exemplo: Se S sabe que P, 'Napoleão venceu a batalha de Austerlitz em 1805 e disso deduz competentemente que Q, 'O universo não pode ter vindo a existir a cinco minutos atrás', então, então teria base para saber que Q, uma hinge. Disso se segue que seria possível adquirir suporte racional para uma hinge, o que não se encaixaria na visão wittgensteiniana em OC, usualmente defendida. Para Pritchard, aqui teríamos um dilema: ou negamos um princípio aparentemente plausível, o Princípio do Fechamento, ou negamos que crenças nãoDossiê Wittgenstein, Dissertatio Volume Suplementar 01 | UFPel [2015] 121 básicas como P tenham algum suporte racional. Essa última implica que o ceticismo não seria rejeitado nos cenários 'locais' pretendidos por Wittgenstein19. A objeção de Pritchard é inadequada e recebeu resposta de Moyal-Sharrock. Mas, se por um lado nos parece uma leitura equivocada, por outro nos proporcionará acrescentar uma possível (e importante) consequência ao debate. Primeiramente, deixem-nos reconstruir a estrutura da argumentação de Pritchard ao colocar o problema exposto acima: Esquema Pritchard 1. Alguém possui sustentação para saber que P, 'Napoleão venceu a batalha de Austerlitz em 1805'; 2. Uma hinge (dobradiça) relevante no cenário previsto em 1 seria Q: 'A Terra existe há mais tempo do que cinco minutos'; 3. Q não pode ser, como é uma hinge, racionalmente sustentada; 4. Q não pode ser racionalmente sustentada, pois é fundamental (em conjunção com outras hinges, em um cenário local) para a negação de um cenário cético radical; 5. Qualquer pessoa racional (S), razoavelmente reflexiva, é capaz de reconhecer que saber que P implica na rejeição da hipótese cética-alvo. 6. Mas, se é racional aceitar o Princípio do Fechamento, então, se S sabe que P, então S sabe que Q, pois pode inferir Q de P racionalmente, mediante dedução. 7. Mas, se o passo 6 está correto, então 3 não é satisfeito. 8. Se 3 não é satisfeita, então ou deve-se negar o Princípio do Fechamento ou negar que crenças não-hinge possam ser racionalmente sustentadas. 9. Caso se aceite a segunda metade da disjunção em 8, então não há suporte algum às nossas crenças não-hinges, e Wittgenstein, consequentemente, não pode oferecer uma solução ao ceticismo. Para Moyal-Sharrock (2014, pp. 13-4), o que incomoda Pritchard e outros epistemólogos é o fato de Wittgenstein ter colocado uma 19 Não nos interessa expor a solução de Pritchard ao desafio por ele lançado, embora o autor pretenda tentar 'resguardar' a posição original de Wittgenstein, visto que, a nosso ver, seus erros de interpretação estão na origem. O que tentaremos mostrar a seguir. Eduardo das Neves Ferreira Filho Juliano do Carmo 122 alternativa para o fim do regresso epistêmico em um tipo de certeza que é não-epistêmica, não-proposicional, indubitável, gramatical e promulgada (enacted), é isto que Pritchard considera algo 'misterioso'. A resposta da autora para as objeções sintetizadas em EP é bastante simples. Afirma que não fazemos uma inferência como aquela indicada no passo 6 do Esquema Pritchard, isso é, não inferirmos uma hinge de uma proposição sobre a qual aparentemente temos conhecimento: "Não acreditaríamos que a batalha de Austerlitz tivesse ocorrido em qualquer ano, se não estivéssemos alicerçados sobre a certeza de que o universo tenha surgido há mais do que cinco minutos" (MOYAL-SHARROCH, 2014, p. 14). O acarretamento, em função disso, seria apenas aparente, apenas 'invisivelmente' suportaria a proposição sobre a batalha de Austerlitz, e não poderia ser 'resultante' dessa: "o assim chamado acarretamento seria na melhor das hipóteses circular" (MOYAL-SHARROCK, 2014, p. 14). Moyal-Sharrock, pois, ao não aceitar o passo 6 do Esquema de Pritchard, consequentemente rejeita o passo 9. O 'erro' de Pritchard (PE), para Moyal-Sharrock, poderia ser colocado do seguinte modo (onde P1 seria uma proposição como 'Napoleão venceu a batalha de Auterlitz', e Q a hinge 'A Terra tem mais do que cinco minutos'): (PE): (Regra) – Proposição P1 – Princípio do Fechamento (ou da Transmissibilidade) – 'Proposição' Q (Hinge) Derivada. O que D. Moyal-Sharrock (MS) observa é o seguinte: (MS): Regra – Proposição P1 – (formulação da Regra – Q) Como se vê no esquema (PE), Pritchard desconsidera que a Proposição P1 já tem sua sustentação em uma regra, como Q, por exemplo ('A Terra tem mais do que cinco minutos'), e que a 'derivação', via Princípio do Fechamento ou suas variantes, apenas alcança uma formulação da regra, e não uma 'proposição' propriamente dita, sujeita ao escrutínio da verdade ou falsidade. Para Moyal-Sharrock, é justamente isso que faz das hinges não-proposicionais, como destacamos antes. O fato de as hinges serem não-proposicionais é o Dossiê Wittgenstein, Dissertatio Volume Suplementar 01 | UFPel [2015] 123 que garante que não haja comprometimento algum da 'viabilidade' de princípios epistêmicos, tais como o do Fechamento. Esse princípio funciona para atitudes proposicionais, afirma a autora, e as hinges, por obséquio, são não proposicionais. Para a ela, o Princípio do Fechamento e suas variantes tratam de conhecimento, visto que tratam de proposições, e as dobradiças não estão no escopo do conhecimento, não são proposições, mas, regras da gramática. Não aceitar a distinção seria, para Moyal-Sharrock, voltar ao ponto de partida na busca por uma solução ao problema do regresso. No entanto, por que não podemos realizar o tipo de inferência prescrita por Pritchard por meio do altamente plausível Princípio do Fechamento? As observações de Moyal-Sharrock não parecem garantir uma resposta satisfatória à questão, e a autora, ao não esclarecer corretamente o ponto, pode ser acusada de sustentar outra espécie de 'mistério'20, É possível, contudo, desfazer o imbróglio que a defesa de MoyalSharrock à leitura não-proposicional das dobradiças pode causar: podemos manter, apesar de não se constituírem em razões para nossas outras crenças, que as dobradiças estão presentes como os fundamentos de nossa ação e de nossas pretensões de sustentar conhecimento de proposições empíricas (e, com isso, mantendo que Wittgenstein responde corretamente ao problema do regresso epistêmico). Não é preciso descartar o procedimento que Pritchard utiliza, via o Princípio do Fechamento (e talvez nem de suas variantes mais sofisticadas, como o Princípio da Transmissibilidade), para deduzir o 'conhecimento' de uma 'proposição' derivada. Por um lado, 1) tem-se que mostrar como este princípio, em certos contextos, é insuficiente para derivar tanto uma proposição, quanto o conhecimento dessa proposição. Por outro, e em consequência, 2) que a pretensa 'derivação' de uma 'proposição' de que se teria 'conhecimento' (no caso de Q) nada mais é do que a formulação, via o princípio do fechamento, de uma regra, uma 20 Ou seja, o mistério não está em assegurar a não-proposicionalidade das dobradiças, crítica que geralmente ocorre na fortuna crítica – já que é bastante plausível a argumentação de que partimos de certezas para que outras coisas possam ser genuinamente duvidadas – mas, sim, em não mostrar como, ao tomar como ponto de partida a não-proposicionalidade das hinges, este percurso de fato não afeta princípios da argumentação racional, como o Princípio do Fechamento. Eduardo das Neves Ferreira Filho Juliano do Carmo 124 dobradiça (o que Moyal-Sharrock apenas esboça ao sustentar que sua leitura não-proposicional não afeta o Princípio do Fechamento, como Pritchard acusa21). Nesse contexto, o Princípio do Fechamento teria apenas a função heurística de apontar como, apesar de não se constituírem em 'razão' para sustentar conhecimento de proposições, as hinges de fato funcionam como fundações às nossas crenças (dito de outro modo, 'trazê-las à tona'). Para responder 1, considere o seguinte procedimento, que gostaríamos de chamar de Teste-Dobradiça (TD). Note-se que este é um teste informal, mas, quando tratamos de uma fundação que realmente sustenta-se em uma prática, o teste também é motivo prático para rejeitar a aplicação do Princípio do Fechamento em determinados contextos: TD: Dada certa proposição P, da qual aparentemente S (um sujeito qualquer) tem conhecimento, e considerando i) que S seja capaz de realizar competentemente deduções, ii) que uma dada proposição Q possa seguir-se, aparentemente, de S, iii) que S seja capaz de manter seu conhecimento de que P, então iv) S só poderá garantir seu 'conhecimento' de que Q, sobre a base de P, com auxílio do Princípio do Fechamento, se Q não estiver pressuposta no conhecimento de S de que P. Quando isto ocorrer, isto é, quando Q estiver pressuposta no conhecimento de S de que P, então a dedução é sem valor, e a utilização do Princípio do Fechamento apenas torna manifesta a formulação de uma hinge, e, como tal, indica um procedimento heurístico (e racional!) de manifestação da hinge, não garante uma 'proposição', apenas manifesta (formula) uma 'regra'. Ora, é evidente que 'A Terra tem mais do que cinco minutos', a regra Q formulada, é dobradiça para a proposição P, 'Napoleão venceu a batalha de Austerlitz em 1805'. Alguém só pode ter vencido uma batalha qualquer se a Terra existe há mais do que cinco minutos. A razão da circularidade, mencionada por Moyal-Sharrock, pois, pode ser explicada com o recurso de TD por meio de um procedimento que em nada afeta a aplicação do Princípio do Fechamento em muitos outros contextos, mas o restringe em contextos como nesse exemplo: como afirmou a autora, nós não fazemos esse tipo de inferência! 21 Veja Moyal-Sharrock (2014, p. 14). Dossiê Wittgenstein, Dissertatio Volume Suplementar 01 | UFPel [2015] 125 A conclusão do Esquema de Pritchard (a saber, o passo 9 de EP), assim, pode ser refutada acrescentando-se três passos ao raciocínio: Esquema de Pritchard Reformulado (EPR) 1. Alguém possui sustentação para saber que P, 'Napoleão venceu a batalha de Austerlitz em 1805'; 2. Uma hinge (dobradiça) relevante no cenário previsto em 1 seria Q: 'A Terra existe há mais tempo do que cinco minutos'; 3. Q não pode ser, como é uma hinge, racionalmente sustentada; 4. Q não pode ser racionalmente sustentada, pois é fundamental (em conjunção com outras hinges, em um cenário local) para a negação de um cenário cético radical; 5. Qualquer pessoa racional (S), razoavelmente reflexiva, é capaz de reconhecer que saber que P implica na rejeição da hipótese cética-alvo. 6. Mas, se é racional aceitar o Princípio do Fechamento, então, se S sabe que P, então S sabe que Q, pois pode inferir Q de P racionalmente, mediante dedução. 7. Mas, se o passo 6 está correto, então 3 não é satisfeito. 8. Mas, o passo 6 não se aplica, no caso, mediante o que está garantido por TD. 9. Assim o passo 3 pode ser mantido. 10. Para que o passo 3 possa ser mantido, não é preciso rejeitar o Princípio do Fechamento. 11. Logo, as crenças não-hinge podem ser racionalmente sustentadas mediante um conjunto de 'regras', que se manifestam em ação. 12. E, assim, Wittgenstein consequentemente pode oferecer uma solução ao ceticismo, bem como ao problema do Regresso Epistêmico. Em relação a resposta à questão 2, que levantamos acima, a saber, que a pretensa 'derivação' de uma 'proposição' de que se teria 'conhecimento' (no caso de Q) nada mais é do que a formulação, via o princípio do fechamento, de uma regra, uma dobradiça, O Princípio do Fechamento, se fosse utilizado em circunstâncias como a indicada em EPR, seria apenas mais um recurso (heurístico) para elucidar o papel importante que as hinges desempenham em nossos sistemas de crenças. Cabe a ressalva de que, ao formularmos uma hinge, essa peculiaridade da atividade filosófica, seja ela baseada ou não em procedimentos Eduardo das Neves Ferreira Filho Juliano do Carmo 126 racionais, tal como o uso de um princípio bastante plausível na teoria do conhecimento, não 'transforma' uma regra em proposição, não submete uma regra à genuína característica das proposições: estarem abertas ao crivo de sua verdade ou falsidade22. REFERÊNCIAS CHILD, W. Wittgenstein. Porto Alegre: Artmed, 2013. GLOCK, H. J. Dicionário Wittgenstein. Tradução de Helena Martins. Revisão de Luiz Carlos Pereira. Rio de Janeiro: Jorge Zahar Editor, 1998. MOYAL-SHARROCH, D. Unravelling certainty. In: MOYALSHARROCH, D. & BRENNER, W. In: Readings of Wittgenstein's On Certainty. Hampshire: Palgrave Macmillan, 2005. ___________________. Understanding Wittgenstein's On Certainty. Hampshire: Palgrave Macmillan, 2007. ___________________. Wittgenstein today. Talk delivered on the occasion of the International Conference on Wittgenstein and Contemporary Philosophy and the Inaugural Meeting of the Chinese Wittgenstein Society, Beijing Normal University 12-13 October 2013. Forthcoming in Conference Proceedings. Disponível em: http://www.academia.edu/5126735/Wittgenstein_Today. __________________. The Animal in Epistemology: Wittgenstein's Enactivist Solution to the Problem of Regress. This paper was read at the 6th Annual BWS Conference at the U. of Edinburgh (June 2014); to be published in Hinge Epistemology: Basic Beliefs after Wittgenstein & Moore, a special issue of the International Journal for the Study of Skepticism, eds A. Coliva & D. Moyal-Sharrock (2015). 5.7.14. 22 Isso não implica na infalibilidade das hinges, visto que algumas delas podem, no decorrer do fluxo da vida, serem alteradas. No entanto, isso não significa que, quando uma hinge específica for abandonada ou alterada, todo o 'edifício' das hinges tenha de ser abandonado. Se assim o fosse, inclusive nossas práticas de tentar garantir 'conhecimento' de proposições empíricas ficariam completamente comprometidas (GLOCK, 1998, p. 77). Dossiê Wittgenstein, Dissertatio Volume Suplementar 01 | UFPel [2015] 127 Disponível em: http://www.academia.edu/7563211/The_Animal_in_Epistemology_Wit tgensteins_Enactivist_Solution_to_the_Problem_of_Regress PRITCHARD, D. Wittgenstein and the groundlessness of our believing. Synthese, 189(2), 255-272doi: 10.1007/s11229-011-0057-8,

Pain, Perception, and the Appearance-Reality Distinction Thomas Park Publication Note This is a draft prior to publication. The final publication is available in Philosophical Analysis 38 (2017), pp. 205-237; http://www.analyticphilosophy.kr/category/%E3%80%8E%EC%B2%A0%ED%95%99%EC% A0%81%20%EB%B6%84%EC%84%9D%E3%80%8F. Abstract I argue that pain sensations are perceptual states, namely states that represent (actual or potential) damage. I defend this position against the objection that pains, unlike standard perceptual states, do not allow for an appearance-reality distinction by arguing that in the case of pain as well as in standard perceptual experiences, cognitive penetration or malfunctions of the underlying sensory systems can lead to a dissociation between the sensation on the one hand, and what is represented on the other hand. Moreover, I refute the objection that the allegedly weak correlation between pain and bodily damage forces intentionalist accounts of pain to postulate so many malfunctions (misrepresentations respectively) that such accounts become implausible. I also rebut Murat Aydede's objection that our linguistic practice supposedly shows that there is a conceptual difference between standard perceptual experiences and pain sensations by challenging Aydede's premise that we always withdraw standard perceptual reports in case of counterevidence, while we never do that with pain reports. At the end, I propose an explanation as to why we do not express perceptual reports of (potential) bodily damage in objectivist, but in mental terms. Key words pain, perception, appearance-reality distinction, nociception, Murat Aydede, sensation, intentionalism, representationalism, representation 1. Bodily damage as something we interoceptively perceive When we get hurt, sick or suffer from problems like cavities, we often feel an unpleasant sensation in a certain part of our body. When we thus feel bodily1 pain, this experience is on a par with perceptual experiences created by sense modalities such as vision or olfaction. Why should we accept such a perceptual account of pain sensations? Experiences of pain share with visual, olfactory and other standard perceptual experiences that what is perceived stimulates subject S's sensory receptors which cause electrical signals that are transmitted to S's central nervous system. Moreover, this enables S to discriminate what is perceived, and to adjust his/her behavior in an adequate way, that is, in a way that is beneficial to its reproduction and/or survival. I believe that something like this constitutes the necessary and sufficient conditions for perceptual experiences. But whether there are exceptions to these conditions is not important as long as the exceptions not only apply to pain sensations, but also to the experiences of our other senses. In detail, experiences of (potential) damage fulfill these conditions, given that, first, we have sensory receptors that are stimulated by (potential) bodily damage caused by heavy pressure on, heat close to, or tissue-damaging chemicals in contact with our bodies (Perl & Kruger 1996). The stimulation of such "nociceptors" causes electrical signals which are, when not inhibited, transmitted to the central nervous system and usually cause an experience with distinct qualitative properties, viz. pain.2 Second, this enables us to discriminate (potential) bodily damage, and to adjust our behavior in an adequate way, viz. such that we prevent harm or further 1 Here I focus on what I call "bodily" pain in contrast to "emotional" pain, the latter designating suffering due to disappointment, loss, and the like. 2 Even though Melzack and Wall (2008, pp. 155) as well as Jennifer Corns (2014) argue that we should not call nociceptors "pain receptors" – mostly because stimulation of nociceptors does not always elicit pain –, these authors do not deny that nociceptors are specialized in such a way that they respond only to particular stimuli, and that such a stimulation evokes characteristic patterns of neural signals that usually leads to pain sensations (Melzack & Wall 2008, pp. 81, 86, 154-5). Hence, the existence of nociceptors and their connection to the CNS is uncontroversial. harm to our bodies. That pain normally makes us react in a way that is beneficial to us is evident in those persons who are congenitally insensitive to pain. Such persons often die young because of injury, accident, or inflammation of joints due to their failure to change positions, in particular during sleep (Melzack and Wall 2008, pp. 4-5). According to the perceptual account, nociception, usually defined as the detection and neural processing of noxious stimuli which usually yields pain sensations, is the sense modality by means of which we interoceptively perceive (potential) damage to our own bodies, and pain is the phenomenal state which comes to represent such damage. 2. The appearance-reality objection Some philosophers argue that pain cannot be a perceptual state because it does not allow for a distinction between appearance and reality, whereas we can have illusionary or hallucinatory experiences with our standard senses. When subject to illusion or hallucination, the world appears in a different way than it actually is. For example, a stick half-immersed in water may appear to be bent even though it is straight. By contrast, if it appears to me that I am in pain, then I am in pain (M. R. Bennett & Hacker 2003, pp. 121-122, 126; Block 2005, pp. 140-141; O'Callaghan 2013). Critics like Hacker conclude that there is a fundamental difference between the awareness of pain and standard sense perception, so that we should not construe the former as a kind of the latter. I agree with the critics that it cannot appear to me that I am in pain, while not really being in pain, at least after careful consideration (Tye 1995b, pp. 192-193). However, even if we grant that there are no such pain illusions and hallucinations, the appearance-reality objection is based on a category mistake. It falsely assumes that pain is an object of perception, and thus on a par with objects seen, heard, smelled, etc. Pains are intentional states that represent certain states in the world, namely (potential) bodily damage. The intentional objects of standard perception (such as sticks or explosions) are on a par with bodily damage, not with the pain sensations which represent it. What is analog to pain sensations are the visual, auditory, etc. experiences (or sensations) which represent things like sticks. And in the same way that there is no discrepancy between how pain sensations appear to me and how they "really" are, likewise there is no discrepancy between how visual, auditory, olfactory sensations appear to me and how they "really" are. By contrast, in the same way that a seen stick might look bent while actually being straight, a part of my body may feel as if being (potentially) damaged, while actually being intact. Chronic pain or phantom limb pain are famous examples. Bodily damage and pain can dissociate also in the opposite way: some humans are congenitally unable to feel any pain in spite of their bodies getting harmed. Some do not feel pain in certain situations despite physical damage to their bodies due to cultural norms, to specific expectations, or to distraction (see Melzack and Wall 2008, pp. 4-5, 9-11, 15-17, 22-27, 75-16, 137, 171, 188). 3. The dissociation of bodily damage and pain These dissociation phenomena show that pain is not simply the passive noticing of noxious stimuli detected by nociceptors, but that pain sensations partly depend on one's cognitive states, including one's (conscious or subconscious) beliefs about what to expect in a certain situation. According to Melzack and Wall's gate control theory of pain, the central nervous system actively manipulates the signals coming from nociceptors, depending inter alia on one's memories of prior experiences and the meaning of the current situation.3 3 As M.D. Shelley A. Cross states in her review article from 1994, it is uncontroversial that that the transmission of nociceptive neural signals is inhibited both within the spinal cord ("segmental modulation") as well as from the cortex and the hypothalamus ("descending modulation") (1994, pp. 380-381). Stokes would argue that the existence of neural top-down mechanisms alone is not sufficient to prove that experiences of pain causally depend on cognitive states, because such "inference would require a relatively uncontroversial mapping from mental functions or states onto neural structures, and neuroscience is far from achieving this." (Stokes, 2013, p. 654). But even though neuroscience has not been able to establish such a mapping, I assume that the cognitive states that apparently influence our (pain) experiences as well as those experiences themselves are realized by neural states. Hardcastle (2015) claims that such cognitive influence militates against perceptual approaches to pain, assuming that perceptual states are cognitively impenetrable. In this section I will therefore defend my perceptual account of pain by arguing that both our standard perceptual experiences and our pain sensations are penetrated by cognition. Assessing the empirical research on cognitive penetration, Dustin Stokes concludes that even though the empirical evidence is not conclusive, the cognitive penetrability thesis according to which our perceptual experiences are sometimes penetrated by cognitive states is more plausible than the thesis that only perceptual judgments or beliefs are (sometimes) affected by (other) cognitive states (2013, pp. 657-658). Focusing on cognitive penetration of visual experience, Albert Newen and Petra Vetter (2017) likewise conclude that in the light of 1. cortical brain areas being heavily interconnected (not only to adjacent areas, but also to other processing areas further away), 2. higher level processes occurring much faster than previously thought, and therefore capable of influencing visual (or other) processing before a stable perceptual experience occurs4, 3. functional evidence for categorical top-down influences to the early vision cortex5, 4. studies that show that activated memorized visual templates change our perceptual experiences6, and 4 For example, there is empirical evidence for feedback loops from higher level processing in motion area V5 to early level processing in visual processing area V1 that occur in 80 ms or less (Newen & Vetter 2017, p. 30). 5 For example, the early visual cortex of blind folded test subjects who listened to sounds from naturalistic environment was active (depending on the semantic category. such as animate vs. inanimate sounds, the sound belongs to) even though the subjects have not been visually stimulated. The neural activity in the early visual cortex hence cannot stem from feedforward visual stimulation, but must stem from feedback loops from other parts of the brain (ibid. pp. 30-31). 6 Newen and Vetter refer inter alia to a study by Hansen, Olkkonen, Walter & Gegenfurtner (2006) which suggests that activation of abstract concepts or so-called visual templates such as the template of yellow bananas modifies how subjects experience the color of these objects (Newen & Vetter 2017, p. 32). In the case of seeing an impoverished black and white image as an image with a dog once one's concept of a (Dalmatian) dog is activated, Newen and Vetter further argue that this effect cannot be explained only by a change of attentional processing. Newen and Vetter refer to the neuroscientific study by Frith & Dolan (1997) which suggests that the effect is more plausibly explained as involving processes of cognitive integration of the black and white dots, that is, as involving 5. hypnosis studies that refute the attempt to explain away the modification of perceptual experiences (as suggested by (4)) as being merely the effect of long-term changes within the perceptual module due to associative learning7 the most plausible explanation of the relevant phenomena is to assume that higher cognitive processes (such as beliefs, desires or concepts) can change perceptual experiences. 8 Accordingly, it is reasonable to assume that what we see and hear does not depend merely on the stimulation of the relevant receptor cells, but also on so-called top-down processes of the brain. What we are interested in, what we desire (here construed as cognitive states) can apparently affect how we perceive the world. Applied to the case of pain, this means that higher cognitive processes will influence whether and how (actual or potential) damage to a part of one's body is experienced by the subject as pain. This is nicely illustrated by an experiment conducted by Leknes et al. (2013). Their test subjects saw a red screen with white text saying "Heat stimulus coming up ..." and then were exposed either to a thermal stimulus they have rated as moderately painful or to one they have rated as intensely painful. When exposed to the former, the participants rated the moderate thermal stimulation within the pleasant range of a higher-level processing. In the study, Frith and Dolan showed impoverished black and white image of a banana on some background to their test subjects. "Usually, hardly anyone recognizes the banana but perceives a pattern of black and white patches that cannot be integrated into any meaningful image. Later on, participants are presented with a clear image of the banana before viewing the impoverished image again. Contrasting the fMRI signal of the perception of the impoverished image before and after it was paired with the clear image resulted in a significant activation of the medial parietal lobe. The medial parietal lobe cannot be regarded as a candidate for early visual processes; it is thus most plausibly a candidate for higher-level processing. Therefore, changes of the perceptual experience of impoverished images are best explained as a result of cognitive penetration from a high-level area." (Newen & Vetter 2017, p. 33). 7 Newen and Vetter refer to Cohen Kadosh, Henik, Catena, Walsh & Fuentes (2009) whose experiment suggests that non-synaesthetic test-subjects can be influenced by posthypnotic suggestion so that they see digits as colored. Newen and Vetter claim that this study shows that the semantic content of a short-term and reversible posthypnotic suggestion, as opposed to the long-term processes involved in perceptual learning, caused a change in perceptual experience. 8 Newen and Vetter thus reject the thesis shared by Pylyshyn (1999) and Raftopoulos (2014) that there is an encapsulated and impenetrable visual module in the brain. Newen and Vetter stress that even though brain area "V4 [...] mainly processes color information [...,] it does not follow at all that V4 cannot be influenced by higher cognitive contents. Functional specialisation of brain areas by itself does not imply cognitive impenetrability. We suspect that this unjustified implication is based on the fact that Fodor (1983) defined modules with several main criteria combining domain-specificity, impenetrability and being innate. But this definition of combined module criteria should not mislead us. Without further evidence the definition is just not well chosen: domain-specificity and impenetrability need not go together." (Newen & Vetter 2017, p. 28). sensation hedonics scale. By contrast, the same moderate thermal stimulation was rated as unpleasant in the "control session" in which a visual cue – a green screen and white text ("Warm stimulus coming up ...") – was followed either by a thermal stimulus they have rated as moderately painful or one they have rated as non-painfully warm. Given that the test subjects were able to rate their experience on a scale ranging from "very painful" to "very pleasant", and given that they rated the experience as pleasant (in the proper session), they implicitly rated the experience as not painful (and hence as not unpleasant).9 This experiment thus shows that the subjects' expectations affect whether a (potentially) noxious stimulus is experienced as pain or not. Besides cognitive penetrability, malfunctioning of the nociceptive system too will account for certain cases in which pain and bodily damage dissociate. For example, subjects who suffer from congenital insensitivity to pain do not feel pain despite (potential) bodily damage because their nociceptive system is not working properly. As Nagasako, Oaklander, and Dworkin point out, in subjects with such a condition at least one kind of those peripheral nerve fibers which are sensitive to noxious stimuli are completely absent or heavily reduced (2003, p. 215). Conversely, headaches such as migraine or cluster headaches are cases in which pain occurs even though the bodily part where the pain is felt is not (potentially) damaged. Rather than correctly indicating bodily damage at some part of the head, such headaches occur due to some kind of neurovascular disorder, that is, malfunctioning of neuronal processing and of 9 It is for this reason that I disagree with Peter Carruthers who construes the experiment by Leknes et al. as demonstrating that "moderate pain that is lesser than expected can even be experienced as pleasant" (Carruthers 2017, pp. 2-3). This interpretation is false because the test subjects implicitly denied that their experiences were unpleasant in the non-control session. As written above, the test subjects were able to rate their experience on a scale ranging from "very painful" to "very pleasant", and given that they rated the experience as pleasant, they implicitly rated the experience as not painful (and hence as not unpleasant). It is therefore more appropriate to conclude that the moderate thermal stimuli (in the non-control session) did not result in pain experiences rather than to conclude that they resulted in pleasant pain experiences. This is in accordance with the scientists own verdict, who, unlike Carruthers, merely conclude that "in a context of intense pain, a moderately noxious stimulus can elicit positive hedonic feelings." (Leknes et al. 2013, p. 407). intracranial extracerebral vessels (Goadsby 2009, p. 860).10 This is in accordance with my intentionalist thesis that pain experiences interoceptively represent a certain part of one's body as actually or potentially damaged because headaches and other pains caused by malfunctions of the nociceptive system can be explained as misrepresentations of the nociceptive system. 4. Pain and bodily damage only weakly correlated? Jennifer Corns (2014, pp. 368-369) and Sabrina Coninx11 have recently objected that instances of pain which do not correlate with bodily damage are so common that proponents of intentionalist accounts of pain have to treat many cases of pain as misrepresentations of bodily damage. And assuming that misrepresentations of (standard) sensory systems are rare, Corns and Coninx conclude that intentionalist accounts of pain become implausible. Their argument rests on three assumptions that can be contested: 1. Pain and bodily damage are weakly correlated. 2. Misrepresentations of (standard) sensory systems are rare. 3. A weak correlation of pain and tissue damage excludes (potential) bodily damage as being represented in pain experiences. Even though I believe that (2) and (3) can be challenged too12, it will be sufficient to show that (1) is not justified in order to undermine the argument by Corns and Coninx. Regarding (1), 10 Even though the exact cause of such headaches are still unknown, various findings more precisely suggest that migraines "might be part of the spectrum of diseases known as channelopathies, or now ionopathies: disorders involving dysfunction of ion channel fluxes." (Goadsby 2009, p. 861). 11 Coninx presented this argument in her talk "Challenging the Representational Approach to Pain" held on June 9th 2017 at the Rudolf-Carnap-Lectures at the Ruhr-Universität Bochum (Germany). 12 I do not have any specific argument against (2), but it is worth noting that neither Corns nor Coninx refer to any empirical studies that support (2) either. Given that (2) is an empirical hypothesis, it should be corroborated by empirical results rather than intuitions. – As for (3), it becomes questionable because a teleological account of intentional content does not presuppose that a perceptual state perfectly covaries with the worldly property it represents. The crucial question is not what the perceptual state actually covaries with, but what it is supposed to covary with, that is, its function. Millikan, for example, illustrates with the eyeblink reflex that a mechanism (or "token") can have a proper function even though the mechanism may often be triggered by what turns out to be a false alarm (1995, p. 187). Intentionalists can likewise argue that some pain experiences which have not been caused by any kind of bodily damage (at the site of the felt pain) can be explained as false alarms of the nociceptive system, thus insisting that the proper function of pain experiences is to interoceptively represent actual or potential Corns (2014, p. 368) argues that it is supported by what we know inter alia about headaches, lower-back pains, the thermal grill illusion13, and chronic pains. Given that it is widely accepted in the medical as well as the philosophical community that chronic pains are disorders which prevent the nociceptive system from working properly (such that the resulting pains are plausibly construed as misrepresentations even if there are more and more patients who suffer from chronic pains), I will focus on acute pains here.14 The first thing to note is that leading figures in medical science state that acute pains are caused either by injury, disease or abnormal function. Invariably, acute pain and these associated responses are provoked by noxious stimulation produced by injury or disease of skin, deep somatic structures, or viscera or abnormal function of muscle or viscera. [...] That acute pain of peripheral origin is usually caused by injury or disease of the skin, subcutaneous tissue, or deep somatic structures, spasm of skeletal muscles or smooth muscles of the hollow viscera, or disease or abnormal function of the viscera is well known. Acute pain involving peripheral-central mechanisms is caused by injury, disease, or inflammation of the peripheral nervous system, whereas acute pain of central origin is caused by disease of the neuraxis. (Coda & Bonica 2001, p. 222; italics added) Intentionalists will take this as evidence that pain either represents a certain part of one's body as actually or potentially damaged (due to injury or disease) or that it mispresents such bodily damage due to some malfunction of the nociceptive system. John D. Loeser, a M.D. and neurologist to whom Corns (2014, p. 368) herself refers in order to make her point, makes the quantitative judgment that in most cases the cause of acute pain is some kind of bodily injury or some disease (rather than some malfunction of the nociceptive system): In most patients with acute pain, the region of tissue damage is obvious and the patient's complaints clearly stem from the injured region. In patients with pain associated with cancer, the search for an etiology of the patient's pain usually reveals disease, such as bodily damage in a part of one's body. A case which can be plausibly explained in this way is the pain resulting from the thermal grill illusion (see my next footnote) because it is plausible to assume that the nociceptive system misconstrues innocuous stimuli for noxious ones in such cases, analog to the false alarms of the eyeblink reflex. 13 The thermal grill illusion is produced by an interlaced grill of warm (~40°C) and cool (~20°C) bars. When you touch both kinds of bars at the same time, you experience burning pain in your hand. For a plausible explanation of this phenomenon, see my previous footnote. 14 Acute pains are, at least in the case of acute low back pain, defined as pains that have not persisted for longer than three months, or, when opposed to "subacute" pains, not longer than five to seven weeks (McGuirk & Bogduk 2009, p. 1094). metastasis to bone or tissue damage from the attempts to cure the malignancy, such as radiation-induced fibrosis or chemotherapy-induced neuropathy. (Loeser 2001, p. 265) This suggests that dissociative cases are, at least for acute pains, rare. If this is true, then (1) is false, and Corns's and Coninx's argument does not get off the ground. Unfortunately, Loeser does not present any exact quantified data. By contrast, in a talk given at a workshop Coninx has presented some figures in order to support her argument. She referred to review articles by Leadley, Armstrong, Lee, Allen, and Kleijnen (2012) and by Burch, Loder, Loder, and Smitherman (2015). The former write that "the general adult population [in the EU] reported an average chronic pain prevalence of 27%" (Leadley et al. 2012, p. 310). And the latter state that, according to publicly available US summary statistics from 2005-2012, on average 14.9% of US adults eighteen or older reported having had migraine or severe headache in the previous three months (Burch et al. 2015, p. 21). Do these figures corroborate (1)? In order to render a judgment we first have to define what counts as weakly correlated or, correspondingly, as well correlated. Neither Corns nor Coninx do that, but in medical science a rule of thumb says that a correlation coefficient of 0.70 to 0.90 (−0.70 to −0.90 respectively) counts as a high (or strong) correlation, while one of 0.30 to 0.50 (−0.30 to −0.50 respectively) counts as a low (or weak) correlation (Mukaka 2012, p. 71). Applied to pain and bodily damage, I suggest that we speak of a strong correlation if we find bodily damage in more than 70 % of the cases in which a subject feels pain, and of a weak correlation if we find bodily damage in less than 50 % of the cases in which a subject feels pain. If we stick to this definition, then the numbers presented by Coninx do not corroborate (1). (1) would require a representative survey which assesses how many pain experiences are (apparently) not caused by any (potential) damage to one's body. If it turned out that more than 30% of all pain experiences consist in such pain experiences, then pain and bodily damage would not strongly correlate. If 50% or more consist in such pain experiences, then pain and bodily damage would correlate weakly. But we do not have such figures. Accepting Burch et al.'s result that 15% of the population sometimes suffer from migraine or severe headaches (while assuming that such headaches are not caused by any bodily damage) does not tell us anything useful about the relationship of pain and bodily damage. For example, we cannot infer that 85% of the population have only pain experiences which are caused by some kind of actual or potential bodily damage. Nor does the figure tell us that 15% of all pain experiences are not caused by some kind of actual or potential bodily damage. The figures presented by Coninx are thus insufficient to support (but likewise insufficient to falsify) (1). Given the lack of relevant statistical data, we have to rely on such estimations as given by Loeser. And these rather militate against (1). Hence, even though pain and bodily damage can come apart, such disassociations are apparently seldom enough not to render intentionalists/perceptual accounts of pain implausible. 5. Murat Aydede's argument from focus Murat Aydede has offered another influential argument against intentionalist accounts of pain. The difference between pain and uncontroversial perceptual experiences is supposed to be manifest in our ways to talk about instances of illusion or hallucination. Whereas we withdraw perceptual reports when faced with evidence to the contrary, Aydede claims that we do not do that in the case of pain "reports". I will illustrate Aydede's argument by comparing cases of visual and auditory misrepresentations with an alleged case of nociceptive misrepresentation. Imagine you enter a bus and see what looks like a flag outside the window. You utter (i) I see a flag on my left side and you tell your friend to look at it. But he tells you that there is no flag. You look again and notice that the colors of the flag-like object are unusually faded. It turns out that this is only a reflection of a flag to your right. You withdraw (i), and might tell your friend that it appeared to you as if there was a flag to your left. Now imagine that you want to buy a pair of jeans. You ask an employee about the price and the employee replies "fifteen dollars". If somebody had asked you to describe your auditory experience, you might have said (ii) I heard the words "fifteen dollars". You cannot believe that the jeans are so cheap, and ask again. When the employee repeats his words, this time you hear "fifty dollars", and conclude that you have misheard what he had said the first time. You might explain your second request by saying that you thought you heard him saying "fifteen dollars". Given that you believe you understood him correctly the second time, you would withdraw (ii) if asked about your past experience. Let's compare these two dialogues with a scenario in which you suffer from phantom limb pain. If asked about your current state, you might utter (iii) I feel pain in my left foot. Aydede argues that if utterances such as (iii) were about the supposed intentional object of pain experiences, that is, about the (potential) damage in a certain part of one's body, then in the last scenario you would and should withdraw (iii) if, say, we reminded you that you do not have a left foot anymore (and hence cannot be damaged there). However, Aydede claims that most people hold that you can and will keep on claiming (iii). Accordingly, you will not (and do not have to) say "I thought I had pain in my foot" or "I made a mistake. It appeared to me as if my foot hurts." Aydede maintains that we do not withdraw our pain reports in cases like phantom limb pain because pain reports are not meant to be statements about whatever pain supposedly represents. They are about the experience itself. And this is supposed to be a fundamental difference to our perceptual reports (Aydede 2009, pp. 536-537). Put differently, Aydede's "argument from focus" proceeds in the following way. If feeling pain amounts to perceiving a physical disturbance in a certain part of one's body on a par with seeing a (material) flag or hearing "fifty dollars", then we would expect the grammatical objects in pain reports to refer to physical disturbances, and not to subjective pain experiences. But this is not the case. Whereas the grammatical objects in (i) I see a flag on my left side (ii) I hear the words "fifteen dollars" refer to non-mental, public objects or states of affairs, the grammatical object in (iii) I feel pain in my left foot must refer to an experience. Aydede takes this to be evinced by the difference in our linguistic practice. The fact that we do not withdraw our pain reports even in cases where it is absurd to claim that one suffers from a certain physical disturbance of one's body, as is the case in phantom limb pain, is construed as revealing that there is a conceptual difference between standard perceptual experiences and pain sensations. Aydede claims that when I look at a red tomato my experience of redness will (in normal conditions) prompt me to apply the concept RED to the tomato (ibid., p. 549). By contrast, pain experiences prompt me to apply the concept PAIN not to some kind of tissue damage, but to the experience itself (ibid., p. 553). In Aydede's words, the difference in our linguistic practice shows that the "semantic "focus" [of the concept pain] is different from the focus of genuine sensory concepts." (ibid.). This is why Aydede chose to label his reasoning "argument from focus". I am going to show that Aydede's argument rests on several false assumptions. As stated above, Aydede assumes that we always withdraw perceptual reports when faced with evidence to the contrary. But what counts as a perceptual report? Are all statements of the form "I perceive (see, hear, smell, etc.) ____" perceptual reports? It will be helpful to distinguish between four kinds of statements that might count as perceptual reports: (a) There is a red cube in front of me (b) I see a red cube (c) It seems to me that there is a red cube in front of me (d) The cube in front of me looks red. In statements such as (a), we presumably make claims about particular objects or objective states of affairs in the world. Reports such as (c) or (d) are presumably about our subjective experiences, at least if "seems" and "looks" are used in a phenomenological as opposed to an epistemic way. 15 We will deal with (b) below. Now, whereas philosophers such as Berit Brogaard count all four statements as perceptual reports16, Aydede does not. He believes that only (a) and (b), and these merely in most, not in all cases, amount to reports of one's perceptual states. His judgment is based on his requirement that the verb in perceptual reports is used as a success verb, that is, based on the requirement that the utterer thereby commits him-/herself to a factual claim about an objective property (characterizable in physical terms).17 According to Aydede, this requirement is manifest in our practice to withdraw perceptual reports in the face of counterevidence. Accordingly, those statements which we do not withdraw in spite of counterevidence are not perceptual reports. Hence, when (c) and (d) are used in a phenomenological sense, they are not perceptual reports. The problem with Aydede's narrow understanding of perceptual reports is that it is supposed to reflect how we use terms like "perceive" or "see" in statements like (b), not merely how we ought to use them. However, Aydede's understanding does not do justice to our actual linguistic 15 Berit Brogaard illustrates this distinction well by means of two examples. If I say "It looks like the road is wet" and mean it in a phenomenological, as opposed to an epistemic sense, "a defeater is not going to change how things look. If l am told that the city has painted the roads to make them look wet as part of their drive-safe campaign, it will still look to me as if the roads are wet." (2015, p. 239) Likewise, "[e]ven if I am told that the lines in the Müller-Lyer optical illusion have the same length, it will still look to me as if they have different lengths." (ibid., p. 242). 16 Berit Brogaard can be said to consider (a)-(d) to be perceptual reports, since she defines perceptual reports as "utterances of sentences that contain a perceptual verb" (Brogaard 2015, p. 237), and since she defines not only "sound", "feel", "taste", "smell", "see", "hear", and "perceive", but also "look" (in the phenomenological sense) as a perceptual verb (ibid.). 17 According to Aydede, "experiences that are intuitively of the same phenomenal kind [...] are genuinely perceptual only if their report normally/dominantly uses success verbs, that is, takes the form exhibited by the likes of (1) ["I see a dark discoloration on the back of my hand"] and (3) ["I see a red cube partially occluded by a green ball to its left"], read transparently." (2009, p. 545). By a "transparent reading" Aydede means that rendering such reports implies an existential (or factual) claim about an objective property characterizable in physical terms. For example, in the case of Aydede's example (1) it implies the claim that there is a dark discoloration in the back of my hand that I am seeing (ibid., pp. 543-4). practice. Some experimental philosophers show that Aydede's appeal to it as well as to "our ordinary concept of pain" (Aydede 2009, pp. 533, n. 2, 534, 535, n. 6, 545, 546 ...) is unwarranted.18 My criticisms consists in showing that statements such as (b) are not always withdrawn in the face of counterevidence, partly because they are not always meant as factual claims about objective properties. Statements like (b) can be construed either as being about particular objects in the world, viz. a particular red cube, or my subjective experience of it. This is supported by the way we talk about illusionary or hallucinatory experiences. For example, even though in the flag example above we will hesitate to say that I have seen a flag on my left side (given that there was no flag there), I might insist that I have seen some flag-like thing to my left. If used in a phenomenological sense, statements like (b) are thus not always withdrawn. The same applies even to ordinary non-illusionary cases. Consider a situation in which Sylvia experiences the smell of a lasagna. There is no lasagna nearby. The only potential source is a spaghetti restaurant next door. Now, even if her interlocutors tell her that she probably confuses the smell of spaghetti with the one of lasagna, she may still insist that it is lasagna which she smells. Breathing in and out, she keeps on having that particular smell of a carrot lasagna in her nose. She may admit that there is probably no actual carrot lasagna around, and still believe that what she now smells is nothing but such a lasagna. Which report is it exactly that she would not withdraw? Compare 18 Based on online surveys, Reuter, Dustin & Sytsma (2014) refute Aydede's claim that "our ordinary concept of pain" is such that pain cannot consist in a physical feature or condition of a part of one's body. Targeting certain statements they attribute to Aydede (2005, pp. 123-124), Reuter et al. use an empirical study to show that, pace Aydede, a majority of people think that the statement "I see a dark discoloration on the back of my hand" can be correct even if the subject in question hallucinates, viz. even if the subject sees a (phantom) hand that does not exist. Hence, "see" is not used as a success verb here, that is, not as picking out existing objects in the environment (Reuter, Dustin & Sytsma 2014, pp. 93-94). Conversely, more than half of the survey participants stated that a person who takes an antidepressant with potentially strange side effects can hallucinate a pain, which implies that that the participants distinguish between real pain and illusionary pain. When asked about the possibility of hallucinations merely in one sense modality (in a second study), even 64.5% of the participants endorsed the possibility of pain hallucinations (ibid., pp. 84-6). Of course, we may reject such folk intuitions as wrong, but the point of Reuter et al. is that Aydede cannot appeal to folk intuitions ("our ordinary concept of pain") in order to justify his claim. (a*) There is a carrot lasagna somewhere close by (b*) I smell a carrot lasagna (c*) It seems to me that there is a carrot lasagna somewhere close by.19 I think we will all agree that Sylvia would withdraw (a*), but not (c*). She can insist that she has the olfactory experience of a carrot lasagna smell even if she believes that there is actually no such lasagna around. Because (b*) can be interpreted as (a*) or (c*), she does not have to withdraw (b*). In the case of smell, there is another particular reason why we may allow that she keeps on asserting (b*), namely the ambiguity of "carrot lasagna" in this situation. She might take "carrot lasagna" to refer to the objective chemical compounds in the air around me (the odor) whereas my interlocutors might have thought that she refers to the source of the odor (here: an actual carrot lasagna). What is more, even if she makes explicit that she refers to an odor, it will be difficult for her interlocutors to make her withdraw (b*) because ordinary people do not have the possibility to falsify that no such chemical compounds are in the air nearby. Of course, they can use their own noses and sniff in order to verify whether there is such a smell in the air. But I believe that if Sylvia persists in her viewpoint, the others will simply comment that they do not smell any lasagna. We can find similar insistence in cases where we disagree about flavors. For example, if the flavor of a wine reminds me of strawberries whereas its flavor reminds you of peaches, it will be difficult for you to convince me that the wine tastes like peaches. Hence, I may not only insist on 19 An analog statement to (d) "The cube in front of me looks red" is omitted because we do not have an appearance verb for olfaction which has the same function as "looks". We might construe "The odor (or air) around me smells as of lasagna", but terms like "as of" are technical terms used in philosophical circles, not something nonphilosophers would say to describe their experiences. An alternative is "The odor (or air) around me smells like lasagna", but how "smell" is used in this statement will be as controversial as suggestions about the usage of "smell" in (b*). I therefore omit a forth way to express one's olfactory experience. (c**) It seems to me that this wine tastes like strawberries, but also to (b**) This wine tastes like strawberries. Aydede probably did not consider such examples because he compared nociception mainly to vision and audition. In case of seeing and hearing we readily correct statements in which we have mistakenly claimed that the content of our visual or auditory experience corresponds to some object or objective state of affairs in the world because such statements can often be easily falsified by others, and/or by our other senses.2021 By contrast, perceptual reports about smells or tastes cannot be easily falsified by others. To sum up, two assumptions in Aydede's argument are false, namely that (I) all our perceptual reports (taking the form of (b) from above) imply factual claims about non-mental objects and/or their objective properties, and (II) we always withdraw standard perceptual reports in case of disagreement or counterevidence. To be fair, at times Aydede writes as if he does not claim that all reports which take the form of (b) are perceptual reports, but only those in which the utterer factually uses the verb in question as a success verb, that is, those in which he/she would withdraw his/her report in the face of counterevidence. However, such a reading has the corollary that in cases where a person S is subject to an illusionary experience without knowing it, S will assume that his/her statements regarding this experience are perceptual reports whereas Aydede will deny that. 20 Another reason seems to be that smell, unlike vision, does not pick out the particular object that is responsible for the smell (Batty 2011, pp. 166-167). Clare Batty defends such a thesis, and argues that smells nevertheless represent objects as thus and so, even though not particular objects, but merely indefinite objects: there is something that smells thus and so (ibid., pp. 165-172). 21 This does not mean that all perceptual reports based on vision or audition are withdrawn in the case of counterevidence. As I argued above, I believe that we can retain such statements as (b) even when we ourselves believe that there is no flag to our left, because the object of "see" can also refer to the subjective experience itself. What counts as a perceptual report would thus not be determined by the content of the actual utterance, but only by appeal to counterfactuals. Aydede could insist that in such cases we should not use "perceive" and its cognates in order to describe our experiences. However, then his argument would be a normative one. But as I understand Aydede's argument from focus it is supposed to be an argument based on factual linguistic practice. Aydede claims that pain sensations are not perceptual states because reports about pains are (factually) used in a different way than reports about standard perceptual states, not that they should be used in such-and-such way. Hence, I reject this reading. Let's turn to Aydede's assumption that, unlike in standard sense modalities, there are no cases in which we withdraw pain reports. I will present two cases that are supposed to refute also this thesis of Aydede. Consider, first, a case of phantom limb pain. If S says (iii) I feel pain in my left foot, but does not have a left foot anymore, (iii) is falsified simply by the fact that S's left foot has been amputated. S cannot feel pain (and thus perceive some (potential) bodily damage) in his left foot if he does not have a left foot. For David Bain, such cases are examples of somatosensory hallucinations (Bain 2007, p. 178). Bain does not deny that S is in pain, but merely denies that S feels pain in his left foot (ibid., p. 173). Critics will object that only parts of the pain report have been withdrawn, not the report per se. So let's consider a case in which the whole pain report is plausibly taken back. The second example is based on a case imagined by philosopher Dorit Bar-On in which a subject believes that she is in pain even though she is not. I sit in the dentist's chair. Having a long history of dental work, I dread what is to come. The dentist puts a sharp-looking instrument in my mouth, and I wince, or grunt. If I could speak, I might say (something which I may in fact think to myself), "Ow, that tooth!" or, more explicitly, "My tooth hurts so much!" (Bar-On 2004, p. 322) If the dentist puts down the drill, and explains that he has not touched any tooth yet, the subject might realize that it was just her fearful expectation that made her think that her tooth hurts, and she might admit that she was not really in pain. It is sufficient that we can merely conceive of such a case. Since Aydede (2009, p. 563) argues that there is a conceptual difference between standard perception and feeling pain, his thesis is falsified by it being conceivable that we withdraw pain reports in such cases. I believe that rejecting assumptions (I), (II), as well as Aydede's denial that we could withdraw pain reports are sufficient to refute his argument from focus. 6. Reasons for the use of mentalistic terms in case of pain One question that remains though is why, if nociception is a sense modality, our perceptual reports are about pain, and not about the bodily harm pains are said to represent. For example, why do we say (iii) I feel pain in my left foot rather than (iii*) I feel harm (or damage, a cut, a disturbance, etc.) in my left foot? I think that this way of talking is partly due to the inability of laymen to identify the cause of their pains in many cases. If, for example, pain were always the result of easily identifiable cuts, then our pain reports might have looked like (iii*). This is supported by the fact that we do express pain reports in objective terms where we are confident about the cause of our current pain. For instance, we say that we feel our sore muscles after a long hike. However, there are many cases in which the specific cause of a pain is unknown or difficult to determine. If we suddenly feel a stabbing pain in our chest, we will often not know whether this is due to some cardiac problem, muscle overuse, or other reasons. The same goes for the myriad variants of headaches or back pains. We merely come to believe that there is (probably) something wrong with the bodily part in question, but we do not know what the problem is. Sometimes an appeal to a certain condition of one's body is therefore not possible.22 In his unpublished PhD thesis, Bain too ponders on the reason why we use mentalist concepts, viz. pain or hurt, in order to describe our pain experiences rather than ascribing an appearance to ourselves. Why do we say "It seems to me (visually) as though this apple is red", but not "It seems to me (somatosensorily) as though this foot is damaged"? Bain conjectures that the sentence "I am somatosensorily perceiving my left foot as disordered" is a bit of a mouthful; and since what is often most urgent when one has a pain experience is that one has the experience, rather than whether it is veridical, we have adopted "hurts" as a convenient shorthand for its self-ascription. (1999, pp. 152-153) In other words, pain talk might have evolved for practical reasons, that is, not in order to let others know that there is something wrong with a certain part of one's body, but in order to inform others that one is undergoing an unpleasant experience – one that the subject (normally) wants to get rid of – and thus either make them suddenly stop what they are doing (say touching a sensitive body part) or to ask them for help (say lifting a heavy load that jams one's hand). Hence, according to Bain, it is not so much our ignorance of the kind of damage that happened to us, but more the importance of quick appropriate behavior that led to the use of concepts such as pain or hurt. Another plausible explanation is inspired by what Manolo Martínez writes. Against Tye's strong representationalism, Martinez first takes sides with Aydede and argues that there seems to be an asymmetry between standard perceptual states and pain. For example, when we see something, "we are first and foremost interested in the features of the external world which the experience represents and only secondarily in the experience itself." (Martínez 2011, p. 69; italics added). By contrast, when we feel pain, "our main object of interest seems to be the experience itself." (ibid., p. 70; italics added). Martínez suggests that the reason for our interest 22 For a similar response, see Ritchie & Carruthers (2015, p. 358). in the pain experience (as evinced by our practice to use "pain" and thus refer to our experience rather than to bodily damage, or whatever it is that pain represents) is that "pain feels awful, regardless of its success at representing the external world. This is why we want, first and foremost, to avoid this awfulness, independently of whatever it may [...] represent." (ibid., p. 71). If we follow such a suggestion, then it is the unpleasantness of pains that lets us focus on and talk about "pain" rather than the intentional object pains represent. I assume that all three suggested reasons affected (and still affect) the way we talk and think about pain and the bodily damage it represents. 7. Conclusion I have argued that pain shares enough with standard perceptual experiences that is makes sense to construe them as perceptual states, too. I have proposed that pain sensations are states which interoceptively represent (actual or potential) damage. I defend this intentionalist account against the objection that pains, unlike standard perceptual states, do not allow for an appearance-reality distinction, and that our linguistic practice shows that there is a conceptual difference between standard perceptual experiences and pain sensations. I show that the objection consists in a category mistake, and that in the case of pain as well as in standard perceptual experiences, cognitive penetration or malfunctions of the underlying sensory systems can lead to a dissociation between the sensation on the one hand, and what is represented on the other hand. Moreover, I refute the objection that the allegedly weak correlation between pain and bodily damage forces intentionalist accounts of pain to postulate so many malfunctions (misrepresentations respectively) that such accounts become implausible. In the last section, I proposed that the difficulty to identify the causes of pain, our desire to get rid of pain given that it is unpleasant, and the importance of quick appropriate behavior is probably responsible for our linguistic practice to express perceptual reports of (potential) bodily damage not in objectivist, but in mental terms. 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Reason Papers Vol. 39, no. 2 Reason Papers 39, no. 2 (Winter 2017): 105-107. Copyright © 2017 Book Review Liebreich, Karen. The Black Page: Interviews with Nazi FilmMakers. UK: McHugh Publications, 2017. For those interested in propaganda and its use as a tool for political power, the paradigm case to examine is the German National Socialist regime (hereafter the Nazi Regime). Most other governments-even democratic ones-use propaganda. Perhaps some other totalitarian regimes have matched the power and deceitfulness of the Nazi Regime, but I daresay none has surpassed it. Karen Liebreich's The Black Page is a gem of a book that helps us to understand how many of the key players in Nazi cinema felt about their work in retrospect. The book is based upon her personal interviews (conducted in the 1990s) with a number of actors, directors, critics, cameramen, and so on. It is remarkable that of the Nazi film industry players alive fifty years after World War II ended so many agreed to talk with her. She includes sixteen of these interviews in the book. These include, most notably, ones with: Wilfred von Oven, press officer to Joseph Goebbels; Fritz Hippler, director of the Reich's Film Department; Hans-Otto Meissner, Diplomat; Hans Feld, film critic; and Kristina Soderbaum, actress (the heroine in Jew Suss [1940]). The Nazi Regime put special weight on cinema as a medium of propaganda. Goebbels, the Regime's propaganda minister-with his training in literature-held film to be second only to radio in its propagandistic potential. As he put it, "We are convinced that in general film is one of the most modern and far-reaching methods of influencing the masses. A regime must not allow film to go its own way" (pp. 8-9). Adolf Hitler-with his early interest in becoming an artist-wrote cynically in Mein Kampf, "The mass of the people as such is lazy. The picture in all its forms up to the film has greater possibilities. Here a man needs to use his brains even less. It suffices to look . . . and thus many will more readily accept a pictorial presentation than read an article of any length. The picture brings them in a much briefer time, I might say at one stroke" (pp. 7-8). It is no surprise, then, to find that the Nazi Regime produced during its twelveyear reign nearly 1,100 films, which is almost two releases per week. It exploited the cover of film to indoctrinate the young by installing film projectors in 70,000 schools in the first two years alone and making film showings mandatory at Hitler Youth meetings (p. 16). Reason Papers Vol. 39, no. 2 106 Two of the interviews warrant special discussion. First, von Oven's is a fascinating interview. He was Goebbels's press secretary for the last two years of the war, after which he (like so many other Nazis) immigrated to Argentina. He there set up a German-language newspaper and wrote for it as well as for a number of other extreme right-wing publications. Von Oven told Liebreich that Goebbels only informed him of the existence of the death camps shortly before the end of the war; while he didn't overtly deny the Holocaust, he scoffed at the claim that six million Jews were killed (p. 26). He also told Liebreich that the war started when Polish Jews massacred 5,800 German civilians in the city of Bromberg (p. 25). (In reality, the Germans had invaded Poland two days before. It is arguable that the German civilians were killed by "friendly fire," that is, accidentally killed by German troops firing upon retreating Polish troops.1) Regarding Goebbels, von Oven said that Goebbels was arrogant, but was intelligent and knew more about film than most people. Von Oven was able to shed some light on Goebbels's theory of propaganda. Goebbels held that propaganda should be kept simple and geared to the slowest people, and used the analogy of a convoy, which "must adjust its speed to suit the slowest ship" (p. 27). Moreover, Goebbels insisted that he didn't want "didactic" films, but rather, propaganda conveyed through entertainment films. Also illuminating is the brief interview with Hippler in his house overlooking Berchtesgaden. Besides being the head of the Nazi film department, Hippler was the director of the infamous anti-Semitic "documentary" The Eternal Jew (1940). The film attacks Jews in a number of ways and at the grossest of levels (as being physically repellant, culturally inferior, and dangerous in their alleged thirst for control). Hitler wanted the film to push the idea that Jews form a parasitic race. Goebbels approved the initial film takes, writing in his (Goebbels's) diary that the scenes were "horrific and brutal" and supported his view that Jewry must be "eliminated" (p. 66). Liebreich concludes the interview by noting that Hippler was still an ardent supporter of the Nazi Regime's ideology. Two main points emerge from reading this book. First, Goebbels greatly favored film that purveyed the Nazi Regime's message opaquely, that is, disguised as pure entertainment. As he put it 1 "Bloody Sunday (1939)," Wikipedia, accessed online at: http://nlp.cs.nyu.edu/meyers/controversial-wikipedia-corpus/englishhtml/main/main_0089.html. Reason Papers Vol. 39, no. 2 107 in 1942, the ideal film would be 80% entertainment and 20% propaganda (p. 8). Certainly, the most effective propaganda movies the Regime produced were entertainment features. In the case of the major Nazi anti-Semitic propaganda films, arguably the least effective was The Eternal Jew, which was the only one made as a documentary. It wasn't nearly as effective as Jew Suss.2 Second, it is astounding that after nearly a half-century since the Nazi Regime ended with Germany in total ruins and the revelation of the death camps that killed eleven million people, "Only one or two of our interviewees showed any sign of self-awareness or self-doubt about their contribution to the success of the regime" (p. 13). Indeed, some of the Nazi film-makers-including Hibbler, Meissner, and von Oven-were completely unrepentant. Such people are difficult to explain. Are they delusional? Is the narcissism that characterizes so many in the film industry just especially deep in them? Is this the ultimate in cognitive dissonance? This puzzle is outside of the realm of propaganda studies; it can be answered only by psychiatry. Gary James Jason California State University, Fullerton 2 For a review of both of these movies, see Gary James Jason, "Selling Genocide II: The Later Films," Reason Papers vol. 39, no. 1 (Summer 2017), pp. 97-123.

Journal of Analytic Theology, Vol. 3, May 2015 10.12978/jat.2015-3.180813100804 ©2015 Simon Kittle • © 2015 Journal of Analytic Theology Grace and Free Will: Quiescence and Control Simon Kittle University of Sheffield Abstract: Stump and Timpe have recently proposed Thomistic based solutions to the traditional problem in Christian theology of how to relate grace and free will. By taking a closer look at the notion of control, I subject Timpe's account – itself an extension of Stump's account – to extended critique. I argue that the centrepiece of Timpe's solution, his reliance on Dowe's notion of quasi-causation, is of no help in addressing the problem. As a result, Timpe's account fails to avoid Semi-Pelagianism. I canvass two alternatives, each of which adheres to the broad theological assumptions made by Stump and Timpe. I conclude that both proposals fail, although I argue that one comes as close as it is possible to get to a solution given the assumptions made. 1. Introduction There is a long standing problem in Christian theology which can be generated with two thoughts central to Christian thinking. The first is that people are responsible for failing to come to faith in Christ and the second is that saving faith is entirely a gift from God, a gift that is not earned nor given based on any merit. If one accepts the view that responsibility requires free will, then the first claim implies that the person is to some degree active in coming to faith – it is something that the person does, and does freely. The second claim was originally made against the Pelagians and Semi-Pelagians.1 Roughly, the Pelagians denied that the Fall damaged the moral capacities of humans and affirmed that post-Fall humans had the power to do all that was required of them by God's commandments (including turning to God). The Semi-Pelagians denied this, but they did insist that fallen humans could at least begin or try to seek after God. The second claim is the denial of these ideas: it affirms that God is the sole actor in the process of a human coming to faith. This implies that the person is passive in a way which is in tension with the first claim. Eleonore Stump has recently presented a Thomistic solution to this traditional theological problem. Her account has been criticised by Kevin Timpe, who has suggested how the account might be extended to meet the problem. Here I argue that Timpe's account is unsuccessful in avoiding Semi-Pelagianism. I canvass two possible ways forward, suggesting that one fails outright and the other has partial success if one understands the condemnations of the (Semi-)Pelagians in 1 For some useful background see Pohle (1917a; 1917b). Grace and free will: quiescence and control Simon Kittle 90 a particular manner. This partial solution, however, has some untoward consequences. 2. Stump's solution and its flaw Stump has recently claimed that Aquinas's theory of the human will provides a way to reconcile the two claims outlined above (Stump 2003, 389–404). The key is to recognise that the human will has not just two "positions" but three: with respect to some issue, the human will can assent to it, reject it, or be in a state of quiescence. In this latter state the will is neutral with respect to the issue at hand – it is simply "turned off" (Stump 2003, 394). This promises a solution along the following lines: God is constantly offering grace to everyone (Stump 2007, 103). The default state of normal, post-Fall human beings is to resist this offer; moreover, such post-Fall humans are unable to accept it. However, people in this state can become quiescent with respect to God's offer of grace. That is, they can move to a state of neither accepting God's grace nor rejecting God's grace. Once the person is in this state of quiescence, God infuses grace into that person. This infusion of grace reconfigures the person's will and enables her to assent to loving God. But the assent only comes after God has operated on the agent's will. According to Stump's Thomistic account of the will there are a number of ways that an agent might become quiescent with respect to some issue. The one that is relevant to the issue of grace and free will, however, concerns the agent's intellect. An agent moves to a state of quiescence with respect to some issue when there is a conflict in her intellect on that issue such that the agent is unable to form a "single, integrated judgement about it" (Stump 2003, 399). Being unable to form a judgement about what is best with respect to the topic at issue, the person is said to abstain from judgement. The result is a state of quiescence: the agent neither affirms the thing nor rejects the thing. On this account nothing external to the agent causes the state of quiescence: the quiescence is due ultimately to the operation of the person's intellect and this, Stump argues, secures the person's control over, and responsibility for, whether or not she fails to come to faith (Stump 2003, 401). However, the process of becoming quiescent is not something the agent does – it is not an action. After all, to stop doing something (e.g. resisting grace) need not mean the performance of some further action (e.g. accepting grace) (Stump 2003, 402). The agent who stops resisting grace is simply doing nothing, i.e., she is in a state of quiescence. Moreover, Stump concludes that because the state of quiescence is not a positive state but a lack – the agent's will, having ceased resisting God, isn't doing anything else (like accepting God) instead – the person still does not possess a good will. This means that the person neither performs a good action nor possesses a good will and so the account avoids both Pelagianism and Semi-Pelagianism. C. P. Ragland (2006) and Timpe (2007) have both raised a problem for Stump's account, namely, that her account does not secure for the human person enough control over the process of becoming quiescent and, ultimately, over her Grace and free will: quiescence and control Simon Kittle 91 coming to saving faith. On the Thomistic view that Stump endorses, "whether the will is turned off or not is always in the power of the will itself" (Stump 2003, 394). Despite this, however, Stump says that the process of becoming quiescent is not itself an action. As mentioned above, a person becomes quiescent when her intellect enters a state of conflict which it cannot resolve – the intellect cannot produce a "single, integrated judgement" about what is best. This indecision in the intellect produces inactivity in the will (Stump 2003, 399). On this account it is true that the ultimate source of the agent's quiescence is in the agent herself. This can be so because we can suppose that the operations of the intellect and will are indeterministic, such that the resulting state of quiescence cannot trace its origin back to something external to the agent. It is indeed the case then, as Stump says, that only the agent's "own intellect and will that determine which position her will is in" (Stump 2003, 402). The problem is that this is not enough for control. Ragland puts the worry like this (2006, §5): To avoid slipping into Semi-Pelagianism, Stump is forced to represent humans as altogether too passive with respect to whether they become quiescent: quiescence is not a state that they actively choose, but is rather a kind of paralysis of intellect and will that befalls humans. Many states of an agent, including psychological states, might be indeterministically caused by other states or events internal to the agent, and yet not under the agent's control. Ragland gives the example of an evil thought that pops into one's mind unbidden: "we have control over whether or not to consent to such a thought or to dwell on it, and we might have control over its recurrence ... but we don't have control over the initial occurrence of the thought" (2006, §5). That is so even if the occurrence of the evil thought is not deterministically caused, and traceable only to states or events internal to the agent. Examples are not limited to moral contexts. Emotional or physical events or states internal to the agent might cause in the agent a further event or state over which the agent has no control: an involuntary twitch in a person's muscle might cause her to shuffle in her seat, a general feeling of fear might result in a person walking home more quickly, and so on. The point is that although the ultimate source of the agent's quiescence is internal to the agent, this does nothing to produce control over the event or state. And this point seems decisive. While it's plausible to think this kind of sourcehood is a necessary condition on something's being under the agent's control, it is not a sufficient condition. The very thing, then, that Stump rules out in order to avoid (Semi- )Pelagianism – there being a prior act of will – undermines her claim that the agent has the right kind of control which is capable of securing her responsibility. 3. Timpe's proposed solution Timpe thinks that Stump's account can be augmented so as to address the worry just outlined. What we need, he says, is "an account ... which can maintain Grace and free will: quiescence and control Simon Kittle 92 both that grace is the sole non-instrumental efficient cause of saving faith and that human agents control whether or not they come to saving faith" (Timpe 2007, 289). In addressing the problem Timpe outlines a constraint which he thinks any account aiming to stick to traditional orthodoxy should aim to satisfy: (Anti-Pelagian Constraint) No fallen human individual is able to cause or will any good, including the will of her coming to saving faith, apart from a unique grace (Timpe 2007, 285). Timpe employs a strong reading of the Anti-Pelagian Constraint which precludes the agent being a cause (as opposed to the sole cause) of her coming to saving faith. According to Timpe, if this constraint can be satisfied we will have an account which is neither Pelagian nor Semi-Pelagian. To provide this account Timpe takes a closer look at the state of quiescence. He thinks that Stump is on the right lines when she notes that an agent's state of quiescence is not a positive thing – it is rather an absence or omission. Indeed, Timpe argues that this provides the key to the solution because (on at least some dominant theories) omissions cannot be causes. This means quiescence so conceived would not count as a cause. But because, on these same accounts, omissions can enter into causal explanations they might nevertheless be able to ground the agent's control over (and so responsibility for) her lack of faith. To develop this idea Timpe appeals to Dowe's account of "causation" of and by omissions. Dowe accepts the view that omissions, because they are absences, are not genuine causes. But they can be cited in causal explanations, and because of this we can treat them as "quasi-causes" (2001, 217). Dowe's account has two parts: first, quasi-causation by an omission (2001, 222): (Quasi-causation by omission) An omission (not-A) is the quasi-cause of some event B if B occurred and A did not, and there occurred an x such that (O1) x caused B, and (O1) had A occurred, then A would have prevented B from occurring by interacting with x. Second, the quasi-causation of omissions, otherwise known as prevention (Dowe 2001, 221): (Quasi-causation of omissions) A quasi-caused not-B if A occurred and B did not, and there occurred an x such that (P1) there is a causal interaction between A and the process due to x, and (P2) if A had not occurred, x would have caused B. How does this help with the issue of control? Timpe's idea is that being able to quasi-cause an omission bestows on the agent control over that omission. This idea is motivated by the following kind of example (which comes from Dowe): suppose that a child is about to run out into the street and his father grabs him just at the last minute. The father's grabbing the child prevented an accident. In terms of the Grace and free will: quiescence and control Simon Kittle 93 accounts above: the father's grabbing the child quasi-caused the lack of an accident because there was a process x – the child's running about – which the father interfered with, and which was such that had the father not interfered with it, the process would have caused an accident. Thus, the father has control over whether or not there was an accident even though, if there had been an accident, the father would not have caused the accident. Timpe thus suggests the following view of control (Timpe 2007, 292): An agent controls an event e when either (1) an action of the agent causes e to occur, or (2) an omission by that agent quasi-causes e to occur. Timpe thinks that this allows us to develop an account according to which only God's grace is causally efficacious but which puts the individual in control of whether or not they come to saving faith. His account has two components. First, individuals quasi-cause their being quiescent with respect to God's offer of grace by an act of will (i.e. we first have an instance of the quasi-causing of an omission). This positive act of will quasi-causes (as opposed to causes) the agent's quiescence because quiescence is an omission. The act of will interferes with a process – the agent's natural disposition to resist God – which is such that had it not been interfered with, it would have produced the state of resisting God. Thus we can say that the act of will quasi-causes the lack of the will's resisting God, or in other words, the agent's state of quiescence. It is the positing of this act of will which clearly distinguishes Timpe's account from Stump's: Stump wanted to avoid positing an act of will because she thought it would mean that the agent could take credit for their coming to saving faith. Timpe thinks this can be avoided once we recognise the role that quasi-causation is playing here.2 The second step of his account is the quasi-causing of the act of saving faith by the individual's state of quiescence. Here we have a case of quasi-causation (as opposed to causation), not because an omission is being produced, but because the omission is leading to something (so Timpe's second step involves quasi-causation by an omission). The state of quiescence quasi-causes the agent's coming to saving faith because there was a process – God's acting on (reconfiguring of) the will of the agent – which resulted in the agent's saving faith, but which was such that, had the agent not been quiescent, the process would not have succeeded. The two instances of quasi-causation mean that the agent has control over whether or not she comes 2 Given the centrality of this act of will to Timpe's account it is natural to ask what the content of this act of will is. Timpe characterises it only as "an act of will through which [the agent] becomes quiescent" (Timpe 2007, 294). There are serious questions to be asked about whether such an act of will is possible. After all, we cannot simply will ourselves to be in any state we wish. If I'm anxious about an upcoming event, it's unlikely that I can simply will myself into a state of contentment. If I'm sitting in front of a blue car I cannot simply will myself to see a red car; I could decide to bring it about that I see a red car, but that would involve more than willing it would involve getting up and searching for such a car. There is then a very real threat looming for Timpe's account. I will not push this threat, however, and will grant that there is such an act of will Timpe may appeal to. Grace and free will: quiescence and control Simon Kittle 94 to saving faith. Even so, on this account it is God alone who is causally efficacious in the individual's coming to faith, and this, Timpe thinks, means that his account is in no way Pelagian or Semi-Pelagian (Timpe 2007, 293–94). 4. Control and agency 4.1 Timpe's account of control: problems and repairs In what follows I will argue that Timpe's account fails to avoid SemiPelagianism, even though his account may well satisfy his Anti-Pelagian Constraint. The following is a brief overview of my argument. First, I will suggest that Timpe's account of control is inadequate. It is too broad (on a number of counts), affirming, for example, that agents control all kinds of things even when they are asleep. The account is fixable, I will suggest, but once we make the required fixes then it will become clear that we have control only when we have an instance of "full-blooded" human agency. That is, Timpe is right to suggest that control can be exercised via quasi-causation, but that is only because agency itself can be transmitted via quasicausation. Once we're clear, however, that we have an instance of agency with respect to the person's coming to saving faith, then it will not be at all clear why we should deny that the agent is able to bring about her coming to saving faith. This point leads us to a deficiency with Timpe's Anti-Pelagian Constraint: it does not cover the entire range of human agency. It affirms that the agent cannot positively cause something which will lead to her salvation ('cause' being used in the technical sense in which it stands opposed to quasi-causation), and also that the agent cannot positively will his coming to saving faith ('will' being used in a technical sense where it refers to the active forming of an intention to accept grace directly), but it does not preclude the agent's being able to (intentionally) bring it about that she comes to saving faith in a different way, namely, by acting in a way which involves quasi-causation. The Anti-Pelagian Constraint should, of course, preclude the agent's being able to bring about her salvation however that is done. Just as it is absurd to think that those who condemned (Semi-)Pelagianism meant to exclude the possibility of an agent being able to directly will her salvation whilst allowing that an agent might be able to bring herself to saving faith by, say, tapping an orange three times whilst crouching, so too it is absurd to think that those who condemned (Semi- )Pelagianism meant to exclude the possibility of the agent positively willing her salvation whilst allowing that an agent might be able to bring herself to saving faith by an action involving quasi-causation. I will close this section by addressing various objections, some of which were offered by Timpe in response to points similar to those I'm raising here. To begin, recall Timpe's account of control: An agent controls an event e when either (1) an action of the agent causes e to occur, or Grace and free will: quiescence and control Simon Kittle 95 (2) an omission by that agent quasi-causes e to occur. The first point I want to make is that clause (2) is too broad. Timpe has adopted Dowe's theory of causation which makes a distinction between causation, which applies only to positive events, and quasi-causation, which may involve omissions. In Dowe's theory, an omission is a lack or absence. The term 'omission' implies no sense of agency. Let's be clear about what this means: right now there is no one in my study. Given this, my lack of being in the study is, according to Timpe, quasicausing my study to be at its uninhabited temperature. Moreover, your absence from my study is also quasi-causing it to be at its uninhabited temperature. Indeed, every single person on the planet is quasi-causing my study to be at its uninhabited temperature. Quasi-causation is, to understate the issue, abundant. Every human agent quasi-causes all sorts of things, and not just when awake and active but also when asleep. It's plausible to think that, in my case, right now, there might be a sense in which I omitted to be in my study; that is, my omission might be an instance of my agency. For example, it might be that I consciously decided to go and sit in the living room instead of the study. If that is the case then I refrained from sitting in my study and my omission is imputable to me as something I did. But this will not be plausible for anyone but me. And we must be clear: even though some omissions are instances of agency in this stronger sense, Timpe's account is not employing this stronger sense. It is employing the notion of a mere lack or absence. No agency is implied. And for this reason, it should be doubted that quasi-causation suffices for control. While my quasi-causing my study to be at its uninhabited temperature might, in the manner just outlined, be something that is imputable to me, that will not be plausible for anyone else. Your quasi-causing my study to be at its uninhabited temperature is not an exercise of your agency – you haven't exercised control over the temperature of my study. If it isn't yet obvious that quasi-causation is not sufficient for control, note that Timpe's first clause does involve an instance of agency. An agent controls an event if it is caused by one of the agent's actions. So with respect to positive causation, Timpe thinks that in order for there to be control, there has to be an action, an instance of agency. We might wonder why there is no symmetry here. If Timpe doesn't think we need an instance of agency to have control with respect to omissions, why does he think we need one when it comes to positive causation? Why not say that an agent controls an event whenever she causes it to happen, regardless of whether she acts? After all, if an agent exercises control over the temperature of my study in virtue of quasi-causing its uninhabited temperature, which would be the case when that person is, say, asleep and on the other side of the world, doesn't someone who is currently asleep in some room, such that she is causing (and not just quasi-causing) its temperature to be slightly higher than it would otherwise be, exert at least as much control over the room's temperature? Timpe's account, I take it, is better revised in the other direction: rather than diluting clause (1) we should strengthen clause (2). Not any old instance of quasicausation is enough for control; rather, just those cases of quasi-causation where the omission counts as a refraining, as a genuine instance of agency. Anyone who Grace and free will: quiescence and control Simon Kittle 96 agrees that the person sleeping on the other side of the world is not exercising control over the temperature of my study will be led to amend clause (2) in this way. If Timpe resists this move, and insists that quasi-causation without agency is enough for control, then we need an account of why this kind of "control" deserves the name, for ordinarily we do not think one thing's causing (or quasi-causing) another counts as control. I will assume, then, the clause (2) needs to be revised to focus on just those omissions which are instances of agency, although I will not attempt to delineate the conditions under which that is so. Why would making this revision be problematic for Timpe? The point threatens to undermine the basis for his solution. Timpe distinguishes two ways that an agent might control an event: either by causation or by quasi-causation. And this distinction is central to his solution. When someone controls something via quasi-causation, Timpe thinks that that instance of controlling need not be subject to the Anti-Pelagian Constraint. But why is this? As we've just seen, the agent controls something via quasi-causation only when she acts, only when the omission is an instance of the person's agency. But if that is so, why think instances of agency which involve quasi-causation should be excluded from the Anti-Pelagian Constraint? After all, we ordinarily say that agents are able to do the things which they can do via quasi-causation. Moreover, it's clear that agents can bear responsibility both for acting in a way which involves quasi-causation and for the associated omissions. Consider some examples. I see someone slowly reversing out of a driveway who is about to run over his puppy. Having a strong dislike of dogs, I deliberately refrain from signalling the driver, who subsequently runs over the puppy. According to Dowe I quasi-cause the puppy's death for there was a process – the driver's slowly reversing his car – which eventually caused the puppy's death and which was such that, had I interrupted it by signalling the driver, would not have done so. I deserve (at least some) blame both for not signalling the driver and for the puppy's death, and this is so even though I "only" quasi-caused the puppy's death. Note that on Dowe's theory you too quasi-caused the puppy's death because you too omitted to signal the driver. Indeed, everyone quasi-caused the puppy's death. Yet you are not responsible for your omission nor the puppy's death: they are not imputable to you, because your omission was not an instance of agency in the way my omission was. Still, despite my agency involving quasi-causation, the result is imputable to me and I bear responsibility for it. This provides an example of an agent being blameworthy for the quasi-causing of an event by an omission. People can also be praiseworthy for quasi-causing something by an omission: suppose that Jim is holding forth on some topic and utters some untruth; Kim knows this, but recognising that little harm will come of it, refrains from pointing out his error, thus saving Jim much embarrassment. Kim's omission – her not saying anything – quasi-causes Jim's lack of embarrassment. And Kim is praiseworthy both for her omitting to say anything and for Jim's lack of embarrassment. Similar things apply to the quasi-causation of omissions. We've already encountered Dowe's example of the father who, in grabbing his child before the child runs into the road, prevents an accident. The father's grabbing the child quasicauses the lack of an accident because there was a process – the child's running into Grace and free will: quiescence and control Simon Kittle 97 the road – which would have caused an accident had the father not grabbed him. The father is praiseworthy for his action but also for there not being an accident. That is, he's praiseworthy for there being no accident. And we can easily imagine a situation involving blameworthiness. Suppose that the child is running about as part of a two-player game of tag. The father grabs his child, removing him from the game, and so quasi-causes the lack of there being any game. Under the right circumstances, the father might be blameworthy for the absence of there being a game. Thus we have examples of an agents being praiseworthy and blameworthy for both the quasi-causing of an event by an omission and for the quasi-causing of an omission by an event. In each case we have a "full-blooded" instance of agency; in each case the agent bears the same kind of moral responsibility that he or she would bear had the action involved only causation. But these instances of agency are excluded from Timpe's Anti-Pelagian Constraint which, recall, runs as follows: (Anti-Pelagian Constraint) No fallen human individual is able to cause or will any good, including the will of her coming to saving faith, apart from a unique grace. This constraint says that agents are unable to come to saving faith by causing or willing anything good. The term 'cause' here should be read in the technical sense that Timpe employs. And of course, Timpe is free to formulate the constraint using a technical sense of the term 'cause.' But if he does so, then the question must be faced: what justifies the exclusion of those things that people do via quasicausation? After all, it certainly seems as if such people are able to do those things. I, for example, was able to bring about the puppy's death; Kim was able to save Jim from embarrassment; the father was able to prevent the accident, and so on. So the ordinary language use of 'able' covers things done both via causation and via quasicausation. And as we saw, the agents bore full responsibility for the things they did. If, therefore, the intent behind the condemnation of the (Semi-)Pelagians was to say that agents were not able to do any salutary work no matter how it was done Timpe's account will be in trouble – not because it fails to satisfy the Anti-Pelagian Constraint but rather because the Anti-Pelagian Constraint isn't what needs to be satisfied. And there is good reason for thinking this is what they did intend. Consider canon 4 from the 418 A. D. Council of Carthage (the first major council to condemn Pelagianism): Whoever shall say that the grace of God through Jesus Christ our Lord helps us only in not sinning by revealing to us and opening to our understanding the commandments, so that we may know what to seek, what we ought to avoid, and also that we should love to do so, but that through it we are not helped so that we are able to do what we know we should do, let him be anathema (Schaff, 497–98). This canon discusses what we are able (or rather, unable) to do and appears to be precluding the ability to do those things that we should do. We've already seen that Grace and free will: quiescence and control Simon Kittle 98 ordinary language allows use of 'able' when referring to things we do which involve omissions. Note too that this canon explicitly mentions things we ought to avoid – things which on Timpe's account will necessarily involve quasi-causation. If God's grace is needed to avoid those things we should avoid, then the Anti-Pelagian Constraint should cover instances of agency involving quasi-causation. Similarly, canons 6 and 7 of the Second Synod of Orange (the major pronouncement condemning Semi-Pelagianism) say: (Canon 6) If anyone says that God has mercy upon us when, apart from his grace, we believe, will, desire, strive, labor, pray, watch, study, seek, ask, or knock, but does not confess that it is by the infusion and inspiration of the Holy Spirit within us that we have the faith, the will, or the strength to do all these things as we ought ... he contradicts the Apostle. (Canon 7) If anyone affirms that we can form any right opinion or make any right choice which relates to the salvation of eternal life ... or that we can be saved, that is, assent to the preaching of the gospel through our natural powers without the illumination and inspiration of the Holy Spirit ... he is led astray by a heretical spirit. Again the thought is that human agents cannot do any salutary work apart from a special gift of grace from God. But it's not as if the pronouncements say "you cannot will the good by causing such a volition in yourself, although you may bring it about that you come to saving faith by quasi-causing in yourself a state of quiescence." Nothing is being said about the different kinds of human action – rather, a blanket denial is being made. In other words, it is irrelevant to the pronouncements of the councils what the correct metaphysics and causal structure of human action is: if such a metaphysics involves quasi-causation, then quasi-causation is precluded. This point can be brought home by highlighting that quasi-causation is not implicated in only those instances of agency which obviously involve omissions, and which we typically describe negatively. As Schaffer has pointed out, because of the way our muscles work – they contract when two components of muscle cells bind together, which by default is inhibited from happening, but which does happen when an electrical impulse from the nervous system quasi-causes a lack of the relevant inhibitor – most if not all human action involves quasi-causation (Schaffer 2012, 407). Certainly, when we "strive, labor, pray, study, seek, ask or knock," we act in ways which involve quasi-causation. The basic problem, then, is that the Anti-Pelagian Constraint doesn't exclude enough because it doesn't exclude from the range of things fallen agents can do those omissions which count as instances of agency. There are two ways to press this point. First, quasi-causation appears to be implicated in many human actions that are usually described in positive terms and which would therefore be excluded by the Anti-Pelagian Constraint. But if the Anti-Pelagian Constraint is already excluding some actions which involve quasi-causation, the exclusion of other actions involving quasi-causation appears unmotivated. Second, as I've suggested above, it's clear that many omissions are "full-blooded" instances of human agency. They Grace and free will: quiescence and control Simon Kittle 99 are things that are imputable to the agent, and things for which the agent is responsible. There is therefore every reason to think that if an agent comes to saving faith via becoming quiescent, her coming to saving faith should, to at least some degree, be attributable to her. And to the degree that she was involved in bringing it about, she will be praiseworthy. 4.2. Objections and replies What might Timpe say in response to the above? Timpe envisages and addresses a number of complaints that are similar to the issues raised above, and it will be instructive to consider them now. His most pertinent set of responses comes in reply to the following objection: Suppose that what is most important about the [Anti-Pelagian Constraint] ... is that the agent not be able to take credit for her own salvation. ... But agents can be praised and blamed for the omissions when brought about by their own acts of will, whether or not one wants to bring such cases under the rubric of quasi-causation. So it's not clear, even if [the Anti-Pelagian Constraint] is 'technically' satisfied, ... that it isn't violated in a more important sense (Timpe 2007, n. 49). Timpe's discussion of this point is brief, the objection and his response appearing only in a footnote, but we may extract three claims that might be designed to address the issue (Timpe 2007, n. 49): (a) "One doesn't deserve credit for one's becoming quiescent precisely because quiescence isn't a positive or good act of the will – it is instead a lack of an act of will." (b) The prior act of will to become quiescent is distinct from willing to accept grace. (c) While willing to accept grace is a good act and something for which the agent would deserve credit, willing to become quiescent is not a good act. It is a better act than willing to resist grace, but being better does not make it good. Thus, willing to become quiescent is not something for which the agent can take credit. Consider point (a). I take it that Timpe intends the phrase 'one's becoming quiescent' to refer to the action – the prior act of will – which leads to the agent's state of quiescence. Reading the phrase like this – as opposed to reading it as referring to the resulting state – is supported by Timpe's treatment of the agent's responsibility for the resultant state later on in the footnote (extracted here as point (c)). So the idea is that the agent doesn't bear responsibility for this action "precisely because quiescence isn't a positive or good act of the will – it is instead a lack of an act of will." Grace and free will: quiescence and control Simon Kittle 100 Is Timpe's emphasis here on quiescence not being a good act of will, or is the emphasis on the fact that quiescence is a lack? The two are to some degree related: quiescence cannot be a good act of will if it is no act of will at all. Timpe is right about that much. Suppose the focus is on quiescence not being a good act of will – doesn't that mean that the agent deserves no credit here, for surely "one cannot deserve credit for X if X is not a morally good action"?3 The principle that a person cannot deserve credit for X if X is not a morally good action may be challenged if 'good' here is understood as a matter of value – moral goodness or badness – as opposed to moral rightness or wrongness. To anticipate something that I will elaborate on below, plausibly, the rightness or wrongness of an action depends on the range of options available. If an agent has two options available, neither of which has much value (i.e. neither of which is morally good), still the agent might do the right thing – and so deserve credit – if she brings about the better of the two. The crucial point for present purposes is that the action Timpe's opponent is pointing to here is not the agent's quiescence (which isn't even an action) but the agent's becoming quiescent. And the latter may be an action for which the agent deserves moral praise even if the resulting state is not intrinsically good. Perhaps, then, Timpe's focus is on the fact that quiescence is a lack or absence. It's not entirely clear how this thought is to be developed. One possibility is that in order to be responsible for an action which is essentially characterised in terms of an absence, one would have to think that there is such an absence. That is, one would have to reify the absence. This, it might be thought, is too high a metaphysical price to pay. Not only are we not given any reason for thinking this (or even any reason for thinking this is what Timpe has in mind), the position is implausible. We can make sense of the agent's responsibility for her act of will and her responsibility for her quiescence (an absence) without reifying absences. Her act of will to become quiescent is an action that cannot be understood apart from the lack of a certain state, but this can be explained in terms of states of affairs: what the agent intends to do is bring about a certain state of affairs, one characterisable only negatively. Moreover, we can say, without reifying absences, that the agent is responsible for the fact that she is quiescent. So it is unproblematic to think of the agent as responsible for the action by which she becomes quiescent and for the resulting state of quiescence. There is nothing unusual or mysterious about this. When I forget to water a neighbour's plant I'm responsible, not for some strange negative entity, but rather for the fact that the plant died. If I promise to get milk on the way home but forget I'm responsible for the subsequent lack of any milk in my kitchen. This doesn't require that there be a strange 'non-milk' object in my kitchen. I'm simply responsible for the fact that there is no milk in my kitchen. The father who grabs his child in time is responsible for the fact that there was no accident. Similarly, when the agent wills to become quiescent she is responsible for the fact that she is now in a state of quiescence. So this first reply fails to gain any traction. 3 I thank an anonymous referee for pushing this point. Grace and free will: quiescence and control Simon Kittle 101 Now consider point (b). Timpe argues that the act of will to become quiescent is distinct from the willing to accept grace. Timpe needs to affirm this in order to avoid having to say that the agent's willing to become quiescent is good. If willing to become quiescent just is willing to accept grace, then it would be hard to deny that the former is good because the latter is clearly a good act. With respect to this point, it is important to be clear about what we're referring to. I take it that as used by Timpe, the act of will is understood as something like the agent's active forming of a first-order volition with the content that I become quiescent (or something similar). Likewise, the agent's act of will to accept God's grace is to be thought of as the agent's active forming of a first-order volition with the content that I accept God's grace (this latter is, of course, something the agent cannot do). Conceived of like so, Timpe's point is straightforward: the two actions are distinct. But that is not the end of the story, for human agents are able to – indeed, they often have to – do one thing by doing another thing. This opens up the possibility that the agent might bring it about that she accepts grace by willing to become quiescent. Consider the following example which I think makes the point easier to grasp: (Arm raising) My right arm is attached to a hoist which is capable of lifting it up. The hoist is mechanically operated by a handle which I can turn with my left hand. I know how to use the hoist. Given this setup there are two ways I can raise my right arm. I can raise it in the normal way or I can raise it by turning the handle of the hoist. When I raise my hand in the normal way I form a first-order intention to raise my hand in an unmediated way: I will that I raise my arm ("directly"). This corresponds to the person's actively forming a first-order volition to accept God's offer of grace. But there is a second way I can raise my arm: I can use the hoist. Now suppose I'm operating the machine and someone comes in and asks me what I'm doing. I could say, 'I'm hoisting my arm up,' but I could also say (equally correctly) 'I'm raising my arm' and 'I'm bringing it about that my arm is raised.' Now the Arm raising story is not quite parallel to the process of coming to faith as envisaged by Timpe because in Arm raising I have both routes available to me: I can raise my arm in the normal way, or I can use the hoist. Suppose, however, that my right arm is paralysed. I cannot raise my right arm "in the normal way." We could suppose too that I have some strange phobia that precludes me from even willing to raise my right arm: that is, I cannot actively form a first-order intention with the content that I raise my right arm (in the unmediated way). Still, I can raise my arm. I can do it using the non-standard method of hoisting it up by operating the machine with my left arm. And in that case, all the statements above are still true of me: I'm raising my arm, I'm bringing it about that I raise my arm, I'm intending to raise my arm, and I'm willing that I raise my arm.4 4 Leigh Vicens has suggested that this example would not carry as much force if the machine were replaced by a person. For example, suppose that instead of pushing a button I had to ask a person to Grace and free will: quiescence and control Simon Kittle 102 Similar things are true of the person who cannot actively form a first-order volition to accept God's grace. If that person knows that were she to become quiescent God would infuse grace into her and she would come to saving faith, then she has a means by which she can bring it about that she comes to saving faith. So when she wills to become quiescent, she also brings it about that she acquires saving faith. Finally we come to (c). Timpe argues that the agent does not deserve any credit for willing to become quiescent because, although willing to be quiescent is better than willing to resisting God, it is still not good. Better does not make good. Thus, he is under no pressure to concede that the agent deserves any credit on the basis that she willed to be quiescent. Timpe offers an analogy to support this idea: Suppose that Joe has the opportunity to steal $100 from his boss, but steals instead only $20. Joe's action here is better than it could have been, but this does not mean that Joe deserves any sort of moral credit for the action that he did do. I agree that, on the most natural interpretation of this case, Joe does not deserve credit for only stealing $20. But notice a crucial difference: in the Joe story we ordinarily assume that Joe has other options available. We assume that Joe could refrain from stealing altogether. The availability of this options makes the cases disanalogous because with respect to coming to saving faith the agent has no other options. We could bring Timpe's example more into line with the quiescence case by attempting to remove the possibility of the agent's refraining altogether from stealing. Suppose, for example, that Joe is subject to the following threat: if he does not steal from his boss, his whole extended family and all his friends will be killed. In addition, let's say that at one end of his boss's table lies a crisp twenty dollar bill, at the other end lies a one hundred dollar bill. Joe has an opportunity to steal from his boss, but not much of an opportunity: he has time to steal only one of the notes. Joe now faces two and only two options. Refraining from stealing is ruled out by the threat, and Joe will satisfy the terms of the threat whichever note he steals. Both of these options still meet Timpe's criteria, inasmuch as both are (at least intuitively) objectively bad states of affairs. Whichever thing Joe does, there is a sense in which we have a bad result: a theft. Moreover, whichever option Joe picks, there is a sense in which his very action will be a morally bad thing. But moral badness is one thing, and moral wrongness is another. It is plausible in this amended case to think that Joe would not be doing anything morally wrong if he stole the twenty dollar bill. Joe would be making the best of a bad situation. And in doing that he is doing what he should be doing, namely, choosing the least bad option. Thus we can say that Joe raise my arm for me – it doesn't seem right to say here that I raise my arm. If our intuitions are different here I suggest it's because (a) it's hard to imagine an agent ever having another person around as a reliable arm-raiser and (b) the suggested revision was in terms of a question: I am to ask the helper to raise my arm, which implies that my helper might not comply. Consider, instead, the orders given by a CEO or Field marshal; in issuing commands to their subordinates, CEOs and Field marshals will and intend certain results, and we readily say that they brought them about. Grace and free will: quiescence and control Simon Kittle 103 does the right thing if he steals only the $20 bill, and if he does the right thing, then he is praiseworthy (although perhaps only minimally).5 This doesn't mean that Joe's action is intrinsically good, but it doesn't have to mean that in order for the point to count against Timpe's view. How it is with Joe is how it is with the person who becomes quiescent: the individual has two options, each of which is a bad state of affairs (compared to what it could be), but in choosing to become quiescent the agent does the right thing. Her becoming quiescent is attributable to her, and, being the right thing to do, she is praiseworthy for it. I conclude, then, that Timpe's account, if it secures the agent's control in the way designed, does not avoid Semi-Pelagianism. In the following section I will canvass two possible ways of moving forward. 5. Moving forward: two proposals 5.1 Reversing the defaults I have argued that if we posit a prior act of will we inevitably end up with something for which the agent deserves credit. One way of potentially avoiding this problem would be to alter the default state of the agent. Suppose that instead of starting out as resisting God's offer of grace, post-Fall humans start out as quiescent. They start out with their will "turned off" with respect to God's offer. Now, because God's offer of grace is effective whenever the human person is quiescent, we cannot say that God is constantly offering his grace to everyone. That would imply that as soon as someone is born (or has the capacities required for faith), God's offer of grace will become effective and that person will come to saving faith. I take it that that is not what happens. So this part of the story needs altering. Instead, we could say that God will indeed offer his grace to each person at some point in their life. Moreover, we could suggest that he will do this after they have had some opportunity to move to a state of resisting him. If God's offer is only ever given after the agent has been given that opportunity, then the agent seems to have the required control, but at the same time we have not posited any prior act of will. Thus we can affirm that God is the only actor in the person's coming to faith, despite the fact that each person retains control over whether or not they come to faith. Unfortunately, however, this proposal does not work. There are at least two significant theological objections. First, it is likely that someone who is motivated to provide an account which meets the Anti-Pelagian Constraint will also be motive to 5 It might be objected that this conclusion relies on denying the doctrine of double effect. Double effect, if accepted, shows that someone might bring something about, and be aware of this, but yet not be responsible for doing so because she did not intend it. However, neither the above example nor the case of becoming quiescence fits with the doctrine of double effect. One key feature of the doctrine states that the unintended (but known) effect for which the agent is not responsible must not be a means to the agent's end (Mangan 1949, 43). But in both the case above and in the case of the agent who becomes quiescent, the least bad action performed is a means to the end, so double effect does not apply. Grace and free will: quiescence and control Simon Kittle 104 retain a substantial doctrine of original sin. One worry is that this proposal does not secure such a thing. On the current proposal, the default state for a person is not a particularly good one: she does not have a will disposed towards God. But neither does she have a will configured for evil. This consequence might undermine the positive benefits of this account (were it to work). Another theological worry concerns fairness and the problem of evil.6 If God doesn't offer each person grace straightaway, but instead has to wait until the agent has had the opportunity to move to a state of resistance, then perhaps the accumulated effect of years of both performing bad actions (unrelated to coming to faith) and being on the receiving end of bad actions might bias an individual in an unfair manner. The idea is that the individual's remaining quiescent or deciding to move to a state of resistance would become too much a function of those aspects of a person's life history which are outside of her control. The real problem for this view, however, is that despite initial appearances it doesn't solve the problem it sets out to solve. I've talked of a person needing the opportunity to move to a state of resistance. The idea was that if the person has such an opportunity, the person is in control of and responsible for a state of affairs which obtains. There are two complications with this idea. First, not every sense of opportunity would be enough for control. There is a sense in which, in virtue of being a British citizen, Suzy has the opportunity to attend university. But circumstances might be such that Suzy cannot make use of this opportunity: she might not have the financial means, for example. Suzy could possess the one kind of opportunity, without yet being in control. Moreover, it's not that her control is undermined by some other kind of consideration – she lacks control because she lacks the right kind of opportunity. Spelling out the kind of opportunity that is required for control is difficult. But for the reasons considered above, and sticking to our current assumptions, it is plausible to think that having the right kind of opportunity will involve some kind of awareness of what one's current state is (i.e. quiescence) and that this is something one could change (i.e. one could move to a state of resistance). But then it seems that if the agent properly recognises those facts, and still continues in her state of quiescence, she has consented to her state of quiescent. This produces control, but it also re-introduces the prior act of will. So this proposal is in fact no improvement on Timpe's account whilst also incurring additional worries. 5.2. Control over resistance without control over coming to saving faith Timpe's target is to secure the agent's control over whether or not she comes to saving faith (2007, 284,285,289,293). One way forward might be to investigate whether the agent would bear the required responsibility for her failing to come to faith if she had control over her resistance and her quiescence but lacked control over whether or not she comes to saving faith. 6 I owe this point to Tasia Scrutton. Grace and free will: quiescence and control Simon Kittle 105 It is certainly possible for an agent to control whether or not she is quiescent without being in control of whether she comes to faith. To see this, let A be the proposition that the agent accepts God's grace, R be the proposition that the agent resists God's grace, and Q be the proposition that the agent is quiescent. Now it might be thought that the agent could control Q without controlling A because Q and A are contraries (not contradictories): both might be false. This point does play some role but in fact things are more complicated, because it is even possible to control P and yet not control ~P. Suppose, for example, that Kate is control of a light switch which is connected in parallel to one other switch, and that both switches are connected in series to a light and power source. Let P be the proposition that the light is on. And suppose that the second switch is connected to a randomiser device. Then Kate controls P but she does not control ~P.7 At most, she controls whether there is a possibility of ~P. So control over P can come apart from control over ~P. But note that Kate has this control because she has options: she can turn her switch on or she can turn it off. This doesn't translate into the light's being on or off, but it's plausible to think that Kate's control over P requires her to have those options available to her. What does this show? It suggests that an agent could control R even if she isn't able to bring about A, as long as she can bring about Q. That is, if the agent is in control of whether she resists God or whether she is quiescent, then whatever she does will be imputable to her and so she will be responsible for the state she realises in herself. The person's responsibility for failing to come to faith would be grounded in her ability to realise in herself a state of quiescence. We would have to insist that the person does not know her becoming quiescent would lead to her coming to faith, because if we add this knowledge then it seems to "automatically" produce in the agent the ability to come to saving faith (indirectly). The most significant problem this idea faces is connected to the problem which I raised above for Timpe's account, namely, that the agent's becoming quiescent is the morally right action to perform. This remains the case even if we remove any possibility of the agent having control over her coming to saving faith (by removing the agent's knowledge that her becoming quiescent will lead to her saving faith). Moreover, this remains the case even if we accept that the agent's state of quiescence is an objectively bad state of affairs. The agent has two options: if she chooses the least bad option then she has done the right thing. The agent who becomes quiescent has done something for which she is praiseworthy. There is no way around this. If we deny that the agent has any knowledge of the fact that her becoming quiescent will lead to her coming to saving faith, we can deny that the latter was under her control and we can deny that she was in any way the agent of her coming to saving faith. So she won't bear any responsibility for coming to saving faith. The agent is unable to intentionally bring herself to saving faith, although she is able to do something which in fact will bring her to saving faith (it's just that she doesn't know this). 7 Walton (1974) has a useful discussion of a variety of cases of this sort. Grace and free will: quiescence and control Simon Kittle 106 Is this enough to avoid (Semi-)Pelagianism? It depends on how we read the pronouncements. Do they mean to affirm that humans are unable to perform any action with the intention that it would lead to their saving faith? Or do they make the stronger claim, namely, that humans are unable to perform any action that would in fact lead to their saving faith, whether they know it or not? If the pronouncements make the former claim but not the latter claim then the position sketched above might escape (Semi-)Pelagianism. This solution differs from Timpe's in the following way. First, it has nothing to do with the causation/quasicausation distinction. It is, I suggest, highly implausible that the solution to the problem of grace and free will could depend on such a thing, because our ordinary ascriptions of agency cut across that distinction. Second, it relies on explicitly disconnecting the agent's control over her resistance and quiescence on the one hand from her control over coming to saving faith on the other. Third, it requires us to employ a weak understanding of the pronouncements of the councils to the effect that they preclude the agent performing an action which she knows will lead to her salvation. This last point renders irrelevant the issue of whether the action which the agent does perform is morally right or not: we can accept that the agent's becoming quiescent is an action for which the agent is praiseworthy, as long as the agent doesn't conceive of what she is doing intentionally as in any way bringing about her saving faith. She's praiseworthy for something, just not coming to saving faith. However, this proposal is not without its own problems. The weak reading of the pronouncements of the councils has some unintuitive results. Suppose that God resolves to bring to saving faith any person who picks up a purple cup at noon on the third day after a full moon. Presumably fallen human beings have the ability to pick up purple cups, and so also have the ability to pick up purple cups at noon on the 3rd day after a full moon. On the above account they have that ability, and would come to faith were they to exercise it, only as long as they don't know what results from picking up the cup. What's puzzling here is that we would have to say that if someone came to know that her being saved would follow from her picking up the cup would, then she would lose the ability to pick up purple cups at noon on the third day of the month. Letting the acquiring of saving faith depend on a bizarre and arbitrary action – the agent's picking up a purple cup at noon on a certain day – allows us to see how strange this is: the addition of knowledge has to somehow remove what looks like a predominantly physical ability. Something similar is true in the case of quiescence. On the above story, fallen humans could choose to become quiescent without any problems as long as they were blind to the fact that as a result of this they would come to faith. Any fallen human learning this would, on the current account, then lose the ability to become quiescent. This feature of the account is, at the very least, puzzling. For this reason I want to make clear that I do not mean to endorse this model, although it is, I think, the closest we can get to a solution while still positing only one unique grace. Grace and free will: quiescence and control Simon Kittle 107 6. Conclusion I have argued that Timpe's account of an agent's coming to faith does not avoid Semi-Pelagianism. I have suggested that Timpe's account of control is inadequate, and that fixing it will reveal that there is little basis for excluding those things agents do via quasi-causation from the scope of the Anti-Pelagian Constraint. Agents can indeed control things via quasi-causation, but that is because omissions can be imputable to them as agents, and they can, as a result, bear moral responsibility both for acting so as to bring about the omission and the omission itself. Recognising this leaves us with little reason to think that instances of agency which involve quasi-causation should be treated any differently from those which involve only causation. If we extend the Anti-Pelagian Constraint to cover such instances of agency then Timpe's account will not satisfy it. I canvassed two possible ways forward. I explored the possibility of reversing the defaults such that the agent starts out quiescent but has the opportunity to move to a state of resistance. This introduced two theological worries and didn't, in the end, solve the underlying philosophical problem. A more promising approach limited the agent's control such that she did not have control over her coming to saving faith but did control whether she resisted or was quiescent (this required denying that the agent knows her becoming quiescent would lead to her coming to faith). I suggested that this proposal is the closest we can get to a solution while positing only one unique grace (other ways forward, not discussed here, might involve positing more than one unique grace). But still problems remain. The person who becomes quiescent does the morally right thing and so is praiseworthy for that act. I suggested that whether or not this constitutes a serious objection depends on the particular understanding of the condemnations of (Semi-)Pelagianism that we employ. However, if we employ a reading according to which the proposal avoids Semi-Pelagianism, the resulting view has some unintuitive consequences concerning the agent's abilities. In closing, it seems that one grace solutions (two grace solutions, if we count the grace of creation) do not fare well when it comes to securing the agent's responsibility for lack of faith while at the same time avoiding (Semi-)Pelagianism.8 References Dowe, Phil. 2001. "A Counterfactual Theory of Prevention and 'Causation' by Omission." Australasian Journal of Philosophy 79 (2): 216–26. Mangan, Joseph. 1949. "An Historical Analysis of the Principle of Double Effect." Theological Studies 10: 41–61. 8 I would like to thank audiences at the Centre for Philosophy of Religion at the University of Leeds, and at the Analytic Theology Summer School at the Universität Innsbruck, for probing questions and helpful comments. Special thanks also go to Kevin Timpe and Leigh Vicens for very helpful discussion. Grace and free will: quiescence and control Simon Kittle 108 Pohle, J. 1917a. "Pelagius and Pelagianism." In The Catholic Encyclopedia. Vol. 11, ed. Kevin Knight. New York: Robert Appleton Company. http://www.newadvent.org/cathen/11604a.htm. ---. 1917b. "Semipelagianism." In The Catholic Encyclopedia. Vol. 13, ed. Kevin Knight. New York: Robert Appleton Company. Ragland, C. P. 2006. "The Trouble with Quiescence: Stump on Grace and Freedom." Philosophia Christi 8 (2): 343–62. Schaff, Philip. Nicene and Post-Nicene Fathers: Second Series. Volume 14: The Seven Ecumenical Councils. Grand Rapids, Michigan: William B. Eerdmans Publishing Company. Schaffer, Jonathan. 2012. "Disconnection and responsibility." Legal Theory 18 (04): 399–435. doi: 10.1017/S1352325212000092. Stump, Eleonore. 2003. Aquinas. London,, New York: Routledge. ---. 2007. "Justifying faith, free will, and the Atonement." In Freedom and the human person, 90–105. Studies in philosophy and the history of philosophy v. 48. Washington, D.C: Catholic University of America Press. Timpe, Kevin. 2007. "Grace and controlling what we do not cause." Faith and Philosophy 24 (3). Walton, Douglas N. 1974. "Control." Behaviorism 2: 162–71.

ar X iv :1 91 1. 11 83 3v 2 [ m at h. L O ] 1 D ec 2 01 9 Twist-Valued Models for Three-valued Paraconsistent Set Theory Walter Carnielli and Marcelo E. Coniglio Centre for Logic, Epistemology and the History of ScienceCLE and Department of Philosophy University of Campinas -Unicamp [email protected]; [email protected] Abstract Boolean-valued models of set theory were introduced by Scott and Solovay in 1965 (and independently by Vopěnka in the same year), offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, Löwe and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific threevalued model called PS3 which satisfies all the axioms of ZF, and can be expanded with a paraconsistent negation *, thus obtaining a paraconsistent model of ZF. We observe here that (PS3,*) coincides (up to language) with da Costa and D'Ottaviano logic J3, a three-valued paraconsistent logic that have been proposed independently in the literature by several authors and with different motivations: for instance, it was reintroduced as CLuNs, LFI1 and MPT, among others. We propose in this paper a family of algebraic models of ZFC based on a paraconsistent three-valued logic called LPT0, another linguistic variant of J3 and so of (PS3,*) introduced by us in 2016. The semantics of LPT0, as well as of its firstorder version QLPT0, is given by twist structures defined over arbitrary complete Boolean agebras. From this, it is possible to adapt the standard Boolean-valued models of (classical) ZFC to an expansion of ZFC by adding a paraconsistent negation. This paraconsistent set theory is based on QLPT0, hence it is a paraconsistent expansion of ZFC characterized by a class of twist-valued models. We argue that the implication operator of LPT0 considered in this paper is, in a sense, more suitable for a paraconsistent set theory than the implication of PS3: indeed, our implication allows for genuinely inconsistent sets (in a precise sense, [[(w ≈ w)]] = 1 2 for some w). It is to be remarked that our implication does not fall under the definition of the so-called 'reasonable implication algebras' of Löwe and Tarafder. This suggests that 'reasonable implication algebras' are just one way to define a paraconsistent set theory, perhaps not the most appropriate. Our twist-valued models for LPT0 can be easily adapted to provide twist-valued models for (PS3,*); in this way twist-valued models generalize Löwe and Tarafder's three-valued ZF model, showing that all of them (including (PS3,*)) are, in fact, models of ZFC (not only of ZF). This offers more options for investigating independence results in paraconsistent set theory. 1 1 On models of set theory: Gödel shrinks, Cohen expands The interest for – and the overall knowledge about – models for set theory changed dramatically after the famous invention (or discovery) of Paul Cohen's methods of forcing. Cohen was able to show that the notion of cardinal number is elastic and relative, in contrast with the methods of "inner models" that Gödel used. Gödel has shown that, by shrinking the totality of sets in a model, they would turn to be 'well-behaved'. As a consequence, the constructible sets could not be used to prove the relative consistency of the negation of the Axiom of Choice (AC) or of the Continuum Hypothesis (CH). Paul J. Cohen, on the contrary, had the idea of reverting the paradigm, and instead of cutting down the sets within models, found a way to expand a countable standard model into a standard model in which CH or AC can be false, doing this in a minimalist but controlled fashion. Cohen elements are 'bad-behaved', but finely guided so as to make 'logical space' for the independence of AC and CH, As Dana Scott puts in the forward of Bell's book [1], "Cohen's achievement lies in being able to expand models (countable, standard models) by adding new sets in a very economical fashion: they more or less have only the properties they are forced to have by the axioms (or by the truths of the given model)." Cohen's methods, however, are not easy, being regarded by some researchers as somewhat lengthy and tedious – but were the only tool available until the Boolean-valued models of set theory put forward by Scott and Solovay (and independently by Vopěnka) in 1965 offered a more natural and rich alternative for describing forcing. This does not discredit the brilliant idea of Cohen, who did not have the machinery of Boolean-valued models available at his time. What is a Boolean-valued model? The intuitive idea is to pick a suitable Boolean algebra A, and define the set of all A-valued sets in M, generalizing the familiar {0, 1} valued models. Then add to the language one constant symbol for each element of the model. After this, define a map φ 7→ [[φ]]A from the sentences in S to A which obey certain equations so that it should assign 1 to all the axioms of ZFC. The resulting structure MA will not be a standard model of ZFC, because it will consist of "relaxed sets" somehow similar to fuzzy sets, and not sets properly. If we take an arbitrary sentence about sets (for instance, "does Y is a member of X" ?) and ask whether it holds in MA, then the answer may be neither plain "yes" nor "no", but some element of the Boolean algebra A meaning the "degree" to which Y is a member of X . However, MA will satisfy ZFC, and to turn MA into an actual model of ZFC with certain desired properties it is sufficient to take a suitable quotient of MB that eliminates the elements of fuzziness. Boolean-valued models not only avoid tedious details of Cohen's original construction, but permit a great generalization by varying on any Boolean algebra. 2 Losing unnecessary weight: the role of alternative set theories It is a well-known historical fact that the discovery of the paradoxes in set theory and in the foundations of mathematics was the fuse that fired the revolution in contemporary set theory around its efforts to attempt to rescue Cantor's naive theory from triviality. The usual culprit was the Principle of (unrestricted) Abstraction, also known as the Principle of 2 Comprehension. Unrestricted abstraction allows sets to be defined by arbitrary conditions, and this freedom combined with the axiom of extensionality, leads to a contradiction, which by its turn leads to triviality in the sense that "everything goes", when the laws of the underlying logic obey the standard principles that comprise the so-called "classical" logic. But there is a way out from this maze. Paraconsistent set theory is the theoretical move to maintain the freedom of defining sets, while stripping the theory of unnecessary principles so as to avoid triviality, a disastrous consequences of contradictions involving sets in ZF. This philosophical maneuver is in frank opposition to traditional strategies, which deprive the freedom of set theory so appreciated by Cantor, by maintaining the underlying logic and weakening the Principle of Abstraction, An analogy may be instructive. The basic goal of reverse mathematics is to study the relative logical strengths of theorems from ordinary non-set theoretic mathematics. To this end, one tries to find the minimal natural axiom system A that is capable of proving a theorem T . In a perhaps vague, but illuminating analogy, paraconsistent logic tries to find the minimal natural principles that are capable of permitting us to reason in generic circumstances, even in the undesired circumstances of contradictions. This does not mean that contradictions are necessarily real: [4] gives a formal system and a corresponding intended interpretation, according to which true contradictions are not tolerated. Contradictions are, instead, epistemically understood as conflicting evidence. There are indeed many cases of contradictions in reasoning, but the classical principle Ex Contradictione Quodlibet, or Principle of Explosion, is neither used in mathematics in general; it is not, therefore, a characteristic of good reasoning, and has to be abandoned. Some people may be mislead by thinking that Reductio ad Absurdum, which is a useful and robust rule of inference, would be lost by abandoning the Principle of Explosion. This is not so: even if discarding such a principle, proofs by Reductio ad Absurdum get unaffected, as long as one can define a strong negation. This is achieved in many paraconsistent logics, in particular in all the logics of the family of the Logics of Formal Inconsistency (LFIs), see [9, 8, 7]. Reasoning does not necessarily require the full power of Ex Contradictione Quodlibet, because contradictions reached in a Reductio proof are not really used to cause any deductive explosion; what is used is the manipulation of negation. 3 Expanding Cohen's expansion: twist-valued models Boolean-valued models were adapted by Takeuti, Titani, Kozawa and Ozawa to latticevalued models of set theory, with applications to quantum set theory and fuzzy set theory (see [21, 23, 24, 19, 20]). The guidelines of these constructions were taken by Löwe and Tarafder in [18] in order to obtain a three-valued model (in the form of a latticevalued model) for a paraconsistent set theory based on ZF. They propose a class of algebras based on a certain kind of implication, called reasonable implication algebras (see Section 9) which satisfy several axioms of ZF. From this class, they found an especific three-valued model which satisfies all the axioms of ZF, and it can be expanded to an 3 algebra (PS3, *) with a paraconsistent negation *, obtaining so a paraconsistent model of ZF. As we discuss in Section 9, the logic (PS3, *) is the same as the logic MPT introduced in [13], and coincides up to language with the logic LPT0 adopted in the present paper. Here, we will introduce the notion of twist-valued models for a paraconsistent set theory ZFLPT0 based on QLPT0, a first-order version of LPT0. Our models, defined for any complete Boolean algebra A, constitute a generalization of the Boolean-valued models for set theory, at the same time generalizing Löwe and Tarafder's three-valued model. Indeed, in Section 9 the model of ZF based on (PS3, *) will be generalized to twist-valued models over an arbitrary complete Boolean algebra, obtaining so a class of models of ZFC. The structure over (PS3, *) will constitute a particular case, by considering the two-element complete Boolean algebra. As a consequence of this, it follows that Löwe and Tarafder's three-valued structure is, indeed, a model of ZFC. Twist-structure semantics have been independently proposed by M. Fidel [15] and D. Vakarelov [25], in order to semantically characterize the well-known Nelson logic. A twist structure consists of operations defined on the cartesian product of the universe of a lattice, L× L so that the negative and positive algebraic characteristics can be treated separately. In terms of logic, a pair (a, b) in L×L is such that a represents a truth-value for a formula φ while b corresponds to a truth-value for the negation of φ. That is, a is a positive value for φ while b is a negative value for it, thus justifying the name 'twist structures' given for this kind of algebras. This strategy is especially useful for obtaining semantical characterizations for non-standard logics. As a limiting case, a Boolean algebra turns out being a particular case of twist structures when there is no need to give separate attention to negative and positive algebraic characteristics, since the latter are uniquely obtained from the former by the dualizing Boolean complement ∼. In this case, every pair (a, b) is of the form (a,∼a), hence the second coordinate is redundant. Our proposal is based on models for ZF based on twist structures, thus the sentences of the language of ZF will be interpreted as pairs (a, b) in a suitable twist structure, such that the supremum a∨b is always 1, but the infimum a∧b is not necessarily equal to 0. This corresponds to the validity of the third-excluded middle for the non-classical negation of the underlying logic, while the explosion law φ∧¬φ → ψ is not valid in general in the underlying paraconsistent logic LPT0. A somewhat related approach was proposed by Libert in [16]: he proposes models for a naive set theory in which the truth-values are pairs of sets (A,B) of a universe U such that A ∪ B = U where A and B represent, respectively, the extension and the anti-extension of a set a. However, besides this similarity, our approach is quite different: we are interesting in giving paraconsistent models for ZFC and not in new models for Naive set theory. It is important to notice that there exists in the literature several approaches to paraconsistent set theory, under different perspectives. In particular, we propose in [6] a paraconsistent set theory based on several LFIs, but that approach differs from the one in the present paper. First, in the previous paper the systems were presented axiomatically, by means of suitable modifications of ZF. Moreover, in that logics a consistency predicate C(x) was considering, with the intuitive meaning that 'x is a consistent set'. On the other hand, in the present paper a model for standard ZFC will be presented instead of a Hilbert calculus for a modified version of ZF. We will return to this point in Section 10. As mentioned above, twist structures over a Boolean algebra generalize Boolean algebras, and are by their turn generalized by the swap structures introduced in [7, Chapter 6] (a previous notion of swap structures was given in [5]). Swap structures are non4 deterministic algebras defined over the three-fold Cartesian product A×A×A of a given Boolean algebra so that in a triple (a, b, c) the first component a represents the truthvalue of a given formula φ while b and c represent, respectively, possible values for the paraconsistent negation ¬φ of φ, and for the consistency ◦φ of φ. Swap structures are committed to semantics with a non deterministic character, while twist structures are used when the semantics are deterministic (or truth-functional). Definition 4.6 below shows how the definition of twist structures for the three-valued logic LFI1◦ introduced in [10, Definition 9.2] can be adapted to LPT0. As noted in Section 7, the three-valued logic (PS3, *) used in [22] already appears in [13] under the name MPT, and it is equivalent to LPT0 and also to LFI1◦. Variants of this logic have been independently proposed by different authors at with different motivations in several occasions (for instance, as the well-known da Costa and D'Ottaviano's logic J3). The naturalness of this logic is reflected by the fact that the three-valued algebra of LPT0 (see Definition 4.2 below) is equivalent, up to language, to the algebra underlying Lukasiewicz three-valued logic L3. The only difference is that in the former the set of distinguished (or designated) truth values is {1, 1 2 } instead of {1}, and this is why LPT0 is paraconsistent while L3 is paracomplete. Twist-valued models work beautifully as enjoying many properties similar to Booleanvalued models (when restricted to pure ZF-languages). Such similarities lead to a natural proof that ZFC is valid w.r.t. twist-valued models, as our central Theorem 8.21 shows. This paper deals with a paraconsistent set theory named ZFLPT0, defined by using as the underlying logic a first-order version of LPT0, called QLPT0, proposed in [12] under the form of QLFI1◦ (that is, by replacing the strong negation ∼ by the consistency operator ◦). The paraconsistent character of twist-valued models as regarding ZFLPT0 as rival of ZFC is emphasized. Despite having some limitative results, as much as Löwe and Tarafder's model, ZFLPT0 has a great potential as generator of models for paraconsistent set theory. A subtle, but critical advantage of our models is that the implication operator of LPT0 is much more suitable for a paraconsistent set theory than the one of PS3. Indeed, our models allow for inconsistent sets, and this is of paramount importance, as we argue below. Moreover, as pointed out above, our models generalize the three-valued model based on PS3, since they can be defined for any complete Boolean algebra. In this way, we have several models at our disposal, and in principle this can be used to investigate independence results in paraconsistency set theory. Albeit Boolean-valued models and their generalization in the form of twist-valued models are naturally devoted to study independence results, this paper does not tackle this big questions yet. The paper, instead, is dedicated to clarifying such models while establishing their basic properties. 4 The logic LPT0 In this section the logic LPT0 will be briefly discussed, including its twist structures semantics. From now on, if Σ′ is a propositional signature then, given a denumerable set V = {p1, p2, . . .} of propositional variables, the propositional language generated by Σ′ from V will be denoted by LΣ′. The paraconsistent logics considered in this paper belong to the class of logics known as logics of formal inconsistency, introduced in [9] (see also [8, 7]). 5 Definition 4.1. Let L = 〈Σ′,⊢〉 be a Tarskian, finitary and structural logic defined over a propositional signature Σ′, which contains a negation ¬, and let ◦ be a (primitive or defined) unary connective. The logic L is said to be a logic of formal inconsistency (LFI) with respect to ¬ and ◦ if the following holds: (i) φ,¬φ 0 ψ for some φ and ψ; (ii) there are two formulas φ and ψ such that (ii.a) ◦α, φ 0 ψ; (ii.b) ◦α,¬φ 0 ψ; (iii) ◦φ, φ,¬φ ⊢ ψ for every φ and ψ. Recall the logic MPT0 presented in [7] as a linguistic variant of the logic MPT introduced in [13]. Definition 4.2. (Modified Propositional logic of Pragmatic Truth MPT0, [7, Definition 4.4.51]) Let MPT0 = 〈M,D〉 be the three-valued logical matrix over Σ = {∧,∨,→ ,∼,¬} with domain M = {1, 1 2 , 0} and set of designated values D = {1, 1 2 } such that the operators are defined as follows: ∧ 1 1 2 0 1 1 1 2 0 1 2 1 2 1 2 0 0 0 0 0 ∨ 1 1 2 0 1 1 1 1 1 2 1 1 2 1 2 0 1 1 2 0 → 1 1 2 0 1 1 1 2 0 1 2 1 1 2 0 0 1 1 1 ∼ 1 0 1 2 0 0 1 ¬ 1 0 1 2 1 2 0 1 The logic associated to the logical matrix MPT0 is called MPT0. The three-valued algebra underlying MPT0 will be called APT0. Observe that x → y = ∼x ∨ y for every x, y. Recall that, by definition, the consequence relation MPT0 of MPT0 is given as follows: for every Γ ∪ {φ} ⊆ LΣ, Γ MPT0 φ iff, for every homomorphism v : LΣ →M of algebras over Σ, if v[Γ] ⊆ D then v(φ) ∈ D. From [7] a sound and complete Hilbert calculus for MPT0, called LPT0, can be defined. This calculus is an axiomatic extension of a Hilbert calculus for classical propositional logic CPL over the signature Σc = {∧,∨,→,∼}. From now on, φ ↔ ψ will be an abbreviation for the formula (φ→ ψ) ∧ (ψ → φ). Definition 4.3. (The calculus LPT0, [7, Definition 4.4.52]) The Hilbert calculus LPT0 over Σ is defined as follows:1 1To be rigorous, in [7, Theorem 4.4.56] an additional axiom schema is required: ¬∼φ → φ. However, it is easy to prove that this axiom is derivable from the others, by using MP. 6 Axiom Schemas: (Ax1) φ→ (ψ → φ) (Ax2) (φ→ (ψ → γ)) → ((φ→ ψ) → (φ→ γ)) (Ax3) φ→ (ψ → (φ ∧ ψ)) (Ax4) (φ ∧ ψ) → φ (Ax5) (φ ∧ ψ) → ψ (Ax6) φ→ (φ ∨ ψ) (Ax7) ψ → (φ ∨ ψ) (Ax8) (φ→ γ) → ((ψ → γ) → ((φ ∨ ψ) → γ)) (Ax9) φ ∨ (φ→ ψ) (TND) φ ∨ ∼φ (exp) φ→ ( ∼φ→ ψ ) (TND¬) φ ∨ ¬φ (dneg) ¬¬φ↔ φ (neg∨) ¬(φ ∨ ψ) ↔ (¬φ ∧ ¬ψ) (neg∧) ¬(φ ∧ ψ) ↔ (¬φ ∨ ¬ψ) (neg →) ¬(φ→ ψ) ↔ (φ ∧ ¬ψ) Inference rule: (MP) φ φ→ ψ ψ It is worth noting that axioms (Ax1)-(Ax9), (TND) and (exp), together with (MP), constitute an adequate Hilbert calculus for classical propositional logic CPL in the signature Σc = {∧,∨,→,∼}. Moreover, (Ax1)-(Ax9) plus (MP) is an adequate Hilbert calculus for classical positive popositional logic CPL+ in the signature Σcp = {∧,∨,→}. Theorem 4.4. ([7, Theorem 4.4.56]) The logic LPT0 is sound and complete w.r.t. the matrix logic of MPT0: Γ ⊢LPT0 φ iff Γ MPT0 φ, for every Γ ∪ {φ} ⊆ LΣ. The latter result can be extended to twist-structures semantics, as shown in [10]. Indeed, LPT0 coincides (up to signature) with LFI1◦, an LFI defined over the signature Σ◦ = {∧,∨,→,¬, ◦} such that the consistency operator ◦ is defined as ◦ 1 1 1 2 0 0 1 7 In LFI1◦ the strong negation ∼ is defined as ∼φ =def φ → ⊥φ such that ⊥φ =def (φ ∧ ¬φ) ∧ ◦φ. On the other hand, the consistency operator ◦ is defined in LPT0 as ◦φ =def ∼(φ ∧ ¬φ). The twist-structures semantics for LFI1◦ introduced in [10, Definition 9.2] can be adapted to LPT0 as follows: Definition 4.5. Let A = 〈A,∧,∨,→,∼, 0, 1〉 be a Boolean algebra.2 The twist domain generated by A is the set TA = {(z1, z2) ∈ A×A : z1 ∨ z2 = 1}. Definition 4.6. Let A be a Boolean algebra. The twist structure for LPT0 over A is the algebra TA = 〈TA, ∧, ∨, →, ∼, ¬〉 over Σ such that the operations are defined as follows, for every (z1, z2), (w1, w2) ∈ TA: (i) (z1, z2) ∧ (w1, w2) = (z1 ∧ w1, z2 ∨ w2); (ii) (z1, z2) ∨ (w1, w2) = (z1 ∨ w1, z2 ∧ w2); (iii) (z1, z2) → (w1, w2) = (z1 → w1, z1 ∧ w2); (iv) ∼(z1, z2) = (∼z1, z1); (v) ¬(z1, z2) = (z2, z1). By recalling that the consistency operator ◦ is defined in LPT0 as ◦φ =def ∼(φ ∧ ¬φ), it follows that ◦(z1, z2) = (∼(z1 ∧ z2), z1 ∧ z2). 3 Definition 4.7. The logical matrix associated to the twist structure TA is MT A = 〈TA, DA〉 where DA = {(z1, z2) ∈ TA : z1 = 1} = {(1, a) : a ∈ A}. The consequence relation associated to MT A will be denoted by TA. Let MLPT0 = {MT A : A is a Boolean algebra} be the class of twist models for LPT0. The twist-consequence relation for LPT0 is the consequence relation MLPT0 associated to MLPT0, namely: Γ MLPT0 φ iff Γ TA φ for every Boolean algebra A. Remark 4.8. In [10, Theorem 9.6] it was shown that LPT0 is sound and complete w.r.t. twist structures semantics, namely: Γ ⊢LPT0 φ iff Γ MLPT0 φ, for every set of formulas Γ ∪ {φ}. On the other hand, if A2 is the two-element Boolean algebra with domain {0, 1} then TA2 consists of three elements: (1, 0), (1, 1) and (0, 1). By identifying these elements with 1, 1 2 and 0, respectively, then TA2 coincides with the three-valued algebra APT0 underlying the matrix MPT0 (recall Definition 4.2). Moreover, MT A2 coincides with MPT0. Taking into consideration Theorem 4.4, this situation is analogous to the semantical characterization of CPL w.r.t. Boolean algebras: it is enough to consider the two-element Boolean algebra A2. 2In this paper the symbol ∼ will be used for denoting the strong negation of LPT0 as well as for denoting the classical negation and its semantical interpretation (the Boolean complement in a Boolean algebra). The context will avoid possible confusions 3This is why in [10, Definition 9.2] clause (v) was replaced by this clause defining ◦. 8 5 The logic QLPT0 A first-order version of LPT0, called QLPT0, was proposed in [12] under the equivalent (up to language) form of QLFI1◦. 4 For convenience, we reproduce here the main features of QLPT0. Definition 5.1. Let V ar = {v1, v2, . . .} be a denumerable set of individual variables. A first-order signature Θ for QLPT0 is given as follows: a set C of individual constants; for each n ≥ 1, a set Fn of function symbols of arity n, for each n ≥ 1, a nonempty set Pn of predicate symbols of arity n. The sets of terms and formulas generated by a signature Θ will be denoted by Ter(Θ) and For(Θ), respectively. The set of closed formulas (or sentences) and the set of closed terms (terms without variables) over Θ will be denoted by Sen(Θ) and CTer(Θ), respectively. The formula obtained from a given formula φ by substituting every free occurrence of a variable x by a term t will be denoted by φ[x/t]. Definition 5.2. Let Θ be a first-order signature. The logic QLPT0 is obtained from LPT0 by adding the following axioms and rules: Axioms Schemas: (Ax∃) φ[x/t] → ∃xφ, if t is a term free for x in φ (Ax∀) ∀xφ→ φ[x/t], if t is a term free for x in φ (Ax¬∃) ¬∃xφ ↔ ∀x¬φ (Ax¬∀) ¬∀xφ ↔ ∃x¬φ Inference rules: (∃-In) φ→ ψ ∃xφ→ ψ , where x does not occur free in ψ (∀-In) φ→ ψ φ→ ∀xψ , where x does not occur free in φ The consequence relation of QLPT0 will be denoted by ⊢QLPT0. 6 Twist structures semantics for QLPT0 In [12] a semantics of first-order structures based on twist structures for LFI1◦ was proposed for QLFI1◦. That semantics will be briefly recalled here, adapted to QLPT0. From now on, only complete Bolean algebras will be considered. 4That is, by taking ◦ instead of ∼ as a primitive connective. 9 Definition 6.1. let A be a complete Boolean algebra. Let MT A be the logical matrix associated to a twist structure TA for LPT0, and let Θ be a first-order signature (see Definition 5.1). A (first-order) structure over MT A and Θ (or a QLPT0-structure over Θ) is pair A = 〈U, IA〉 such that U is a nonempty set (the domain or universe of the structure) and IA is an interpretation function which assigns: an element IA(c) of U to each individual constant c ∈ C; a function IA(f) : U n → U to each function symbol f of arity n; a function IA(P ) : U n → TA to each predicate symbol P of arity n. Notation 6.2. From now on, we will write cA, fA and PA instead of IA(c), IA(f) and IA(P ) to denote the interpretation of an individual constant symbol c, a function symbol f and a predicate symbol P , respectively. Definition 6.3. Given a structure A over MT A and Θ, an assignment over A is any function μ : V ar → U . Definition 6.4. Given a structure A over MT A and Θ, and given an assignment μ : V ar → U we define recursively, for each term t, an element [[t]]Aμ in U as follows: - [[c]]Aμ = c A if c is an individual constant; - [[x]]Aμ = μ(x) if x is a variable; - [[f(t1, . . . , tn)]] A μ = f A([[t1]] A μ , . . . , [[tn]] A μ) if f is a function symbol of arity n and t1, . . . , tn are terms. Definition 6.5. Let A be a structure over MT A and Θ. The diagram language of A is the set of formulas For(ΘU), where ΘU is the signature obtained from Θ by adding, for each element a ∈ U , a new individual constant ā . Definition 6.6. The structure  = 〈U, I  〉 over ΘU is the structure A over Θ extended by I  (ā) = a for every a ∈ A. It is worth noting that s = sA whenever s is a symbol (individual constant, function symbol or predicate symbol) of Θ. Notation 6.7. The set of sentences or closed formulas (that is, formulas without free variables) of the diagram language For(ΘU) is denoted by Sen(ΘU), and the set of terms and of closed terms over ΘU will be denoted by Ter(ΘU) and CTer(ΘU), respectively. If t is a closed term we can write [[t]]A instead of [[t]]Aμ , for any assignment μ, since it does not depend on μ. Notation 6.8. From now on, if z ∈ TA then (z)1 and (z)2 (or simply z1 and z2) will denote the first and second coordinates of z, respectively. Definition 6.9 (QLPT0 interpretation maps). Let A be a complete Boolean algebra, and let A be a structure over MT A and Θ. The interpretation map for QLPT0 over A and MT A is a function [[*]] A : Sen(ΘU) → TA satisfying the following clauses (using Notation 6.8 in clauses (iv) and (v)): (i) [[P (t1, . . . , tn)]] A = PA([[t1]] Â, . . . , [[tn]] Â), if P (t1, . . . , tn) is atomic; 10 (ii) [[#φ]]A = #[[φ]]A, for every # ∈ {¬,∼}; (iii) [[φ#ψ]]A = [[φ]]A # [[ψ]]A, for every # ∈ {∧,∨,→}; (iv) [[∀xφ]]A = (∧ a∈U([[φ[x/ā]]] A)1, ∨ a∈U([[φ[x/ā]]] A)2 ) . (v) [[∃xφ]]A = (∨ a∈U([[φ[x/ā]]] A)1, ∧ a∈U([[φ[x/ā]]] A)2 ) . Remark 6.10. A partial order can be naturally introduced in TA as follows: z ≤ w iff z1 ≤ w1 and z2 ≥ w2. It is easy to see that, with this order, TA is a complete lattice (since A is a complete Boolean algebra), in which ∧ i∈I zi = (∧ i∈I(zi)1, ∨ i∈I(zi)2 ) , and ∨ i∈I zi = (∨ i∈I(zi)1, ∧ i∈I(zi)2 ) . Note that 1 =def (1, 0) and 0 =def (0, 1) are the top and bottom elements of TA, respectively. These considerations justify the definition of the interpretation of the quantifiers given in Definition 6.9(iv) and (v). Recall the notation stated in Definition 6.5. The interpretation map can be extended to arbitrary formulas as follows: Definition 6.11. Let A be a complete Boolean algebra, and let A be a structure over MT A and Θ. Given an assignment μ over A, the extended interpretation map [[*]] A μ : For(ΘU) → TA is given by [[φ]] A μ = [[φ[x1/μ(x1), . . . , xn/μ(xn)]]] A, provided that the free variables of φ occur in {x1, . . . , xn}. Definition 6.12. Let A be a complete Boolean algebra, and let A be a structure over MT A and Θ. Given a set of formulas Γ ∪ {φ} ⊆ For(ΘU), φ is said to be a semantical consequence of Γ w.r.t. (A,MT A), denoted by Γ |=(A,MT A) φ, if the following holds: if [[γ]]Aμ ∈ D, for every formula γ ∈ Γ and every assignment μ, then [[φ]] A μ ∈ D, for every assignment μ. Definition 6.13 (Semantical consequence relation in QLPT0 w.r.t. twist structures). Let Γ∪{φ} ⊆ For(Θ) be a set of formulas. Then φ is said to be a semantical consequence of Γ in QLPT0 w.r.t. first-order twist structures, denoted by Γ |=QLPT0 φ, if Γ |=(A,MT A) φ for every pair (A,MT A). Theorem 6.14 (Adequacy of QLPT0 w.r.t. first-order twist structures ([12])). For every set Γ ∪ {φ} ⊆ For(Θ): Γ ⊢QLPT0 φ if and only if Γ |=QLPT0 φ. 5 In Remark 4.8 was observed that TA2 , the twist structure for LPT0 defined over the twoelement Boolean algebra A2, coincides (up to names) with the three-valued algebra APT0 underlying the matrix MPT0 and, moreover, MT A2 coincides with the three-valued characteristic matrix MPT0 of LPT0. In [12] it was proven that QLPT0 can be characterized by first-order structures defined over MPT0. 6 5As observed above, in [12] the logic QLFI1◦ was analyzed instead of QLPT0. However, both logics are equivalent, the only difference being the use of ◦ instead of ∼ as primitive connective. The adaptation of the adequacy result for QLFI1◦ given in [12] to the logic QLPT0 is straightforward. 6Once again, it is worth observing that the result obtained in [12] concerns the logic QLFI1◦ instead of QLPT0. 11 Theorem 6.15 (Adequacy of QLPT0 w.r.t. first-order structures over MPT0 ([12])). For every set Γ ∪ {φ} ⊆ For(Θ): Γ ⊢QLPT0 φ iff Γ |=(A,MPT0) φ for every structure A over Θ and MPT0. Remark 6.16. It is worth observing that Theorem 6.15 constitutes a variant of the adequacy theorem of first-order J3 w.r.t. first-order structures given in [14]. Indeed, both logics are the same (up to language), and the semantic structures are the same, up to presentation. 7 Twist-valued models for set theory As mentioned before, a three-valued model for a paraconsistent set theory based on latticevalued models for ZF, as a non-classical variant of the well-known Scott-Solovay-Vopěnka Boolean-valued models for ZF, was proposed by Löwe and Tarafder in [18]. Specifically, they introduce a three-valued logic called PS3 which can be expanded with a paraconsistent negation ¬ (which they denote by ∗) and then a model for ZF is constructed over the three-valued algebra PS3, as well as over its expansion (PS3,¬), along the same lines as the traditional Boolean-valued models. It is known that the logic (PS3,¬), introduced in [13] as MPT, coincides up to language with LPT0. We will return to this point in Section 9. In this section, a twist-valued model for a paraconsistent set theory ZFLPT0 based on QLPT0 will be defined, for any complete Boolean algebra A. It will be shown that this models constitute a generalization of the Boolean-valued models for set theory, as well as of Löwe-Tarafder's three-valued model. Our constructions, as well as the proof of their formal properties, are entirely based on the exposition of Boolean-valued models given in the book [1], which constitutes a fundamental reference to this subject. Consider the first order signature ΘZF for set theory ZF which consists of two binary predicates ǫ (for membership) and ≈ (for identity). The logic ZFLPT0 will be defined over the first-order language L generated by ΘZF based on the signature of QLPT0, that is: the set of connectives is Σ = {∧,∨,→,∼,¬}, together with the quantifiers ∀ and ∃ and the set V ar = {v1, v2, . . .} of individual variables. As usual, dom(f) and ran(f) will thenote the domain and image (or rank) of a given function f . Definition 7.1. Let A be a complete Boolean algebra, and let α be an ordinal number. Define, by transfinite recursion on α, the following: VTAα = {x : x is a function and ran(x) ⊆ TA and dom(x) ⊆ V TA ξ for some ξ < α}; VTA = {x : x ∈ VTAα for some α}. The class VTA is called the twist-valued model over the complete Boolean algebra A. Definition 7.2. Expand the language L by adding a constant ū to each element u of VTA, obtaining a language denoted by L(TA). The fragments of L and L(TA) without the connective ¬ will be denoted by Lp and Lp(TA), respectively. They will be called the pure ZF-languages. Observe that L(TA) and Lp(TA) are proper classes. Finally, a formula φ in Lp is called restricted if every occurrence of a quantifier in φ is of the form ∀x(x ∈ y → . . .) or ∃x(x ∈ y∧ . . .), or if it is proved to be equivalent in ZFC to a formula of this kind. 12 Notation 7.3. By simplicity, and as it is done with Boolean-valued models, we will identify the element u of VTA with its name ū in L(TA), simply writting u. Moreover, if φ is a formula in which x is the unique variable (possibly) occurring free, we will write φ(u) instead of φ[x/u] or φ[x/ū]. Remark 7.4 (Induction principles). Recall that, from the regularity axiom of ZF, the sets Vα = {x : x ⊆ Vξ for some ξ < α} are definable for every ordinal α. Moreover, in ZF every set x belongs to some Vα. This induces a function rank(x) =def least α such that x ∈ Vα. Since rank(x) < rank(y) is well-founded, it induces a principle of induction on rank: Let Ψ be a property over sets. Assume, for every set x, the following: if Ψ(y) holds for every y such that rank(y) < rank(x) then Ψ(x) holds. Hence, Ψ(x) holds for every x. From this, the following Induction Principle (IP) holds in VTA (similar to the one for Boolean-valued models): Let Ψ be a property over individuals in VTA. Assume, for every x ∈ VTA, the following: if Ψ(y) holds for every y ∈ dom(x) then Ψ(x) holds. Hence, Ψ(x) holds for every x ∈ VTA. Both induction principles are fundamental tools in order to prove properties in VTA. Definition 7.5. Define by induction on the complexity in L(TA) a mapping [[*]] VTA (or simply [[*]]) assigning to each closed formula in L(TA) a value in TA as follows: [[u ǫ v]] = ∨ x∈dom(v) (v(x) ∧ [[x ≈ u]]) = ( ∨ x∈dom(v) ((v(x))1 ∧ [[x ≈ u]]1), ∧ x∈dom(v) ((v(x))2 ∨ [[x ≈ u]]2) ) [[u ≈ v]] = ∧ x∈dom(u) (u(x) → [[x ǫ v]]) ∧ ∧ x∈dom(v) (v(x) → [[x ǫ u]]) = ( ∧ x∈dom(u) ((u(x))1 → [[x ǫ v]]1), ∨ x∈dom(u) ((u(x))1 ∧ [[x ǫ v]]2) ) ∧ ( ∧ x∈dom(v) ((v(x))1 → [[x ǫ u]]1), ∨ x∈dom(v) ((v(x))1 ∧ [[x ǫ u]]2) ) [[φ#ψ]] = [[φ]]#[[ψ]] for # ∈ {∧,∨,→} [[#ψ]] = #[[ψ]] for # ∈ {∼,¬} [[∀xφ(x)]] = ∧ u∈VTA [[φ(u)]] = ( ∧ u∈VTA [[φ(u)]]1, ∨ u∈VTA [[φ(u)]]2 ) [[∃xφ(x)]] = ∨ u∈VTA [[φ(u)]] = ( ∨ u∈VTA [[φ(u)]]1, ∧ u∈VTA [[φ(u)]]2 ) . [[φ]]V TA is called the twist truth-value of the sentence φ ∈ L(TA) in the twist-valued model VTA over the complete Boolean algebra A. 13 Remark 7.6. Observe that VTA can be seen as a structure for QLPT0 over MT A and ΘZF in a wide sense, given that its domain is a proper class. Under this identification, the twist truth-value [[φ]]V TA of the sentence φ in VTA is exactly the value assigned to φ by the interpretation map for QLPT0 over VTA and MT A (recall Definition 6.9). In this case we assume that the mappings (* ǫ *)V TA and (* ≈ *)V TA are as in Definition 7.5. Recall the notion of semantical consequence relation in QLPT0 (see Definitions 6.12 and 6.13). This motivates the following: Definition 7.7. A sentence φ in L(TA) is said to be valid in V TA, which is denoted by VTA |= φ, if [[φ]]V TA ∈ DA. The semantical notions introduced above can easily be generalized to formulas with free variables. Recall from Notation 7.3 that u is identified with u in VTA. Then: Definition 7.8. Let φ be a formula in L whose free variables occur in {x1, . . . , xn}. Given a twist-valued model VTA and an assignment μ : V ar → VTA, the twist truth-value of φ in VTA and μ is defined as follows: [[φ]]V TA μ =def [[φ[x1/μ(x1), . . . , xn/μ(xn)]]] VTA . The formula φ is valid in VTA if [[φ]]V TA μ ∈ DA for every μ. Definition 7.9. ZFLPT0 is the logic of the class of twist-valued models, seen as QLPT0structures over the signature ΘZF. That is, ZFLPT0 is the set of formulas of L which are valid in every twist-valued model VTA. 8 Boolean-valued models versus twist-valued models In this section, the relationship between twist-valued models and Boolean-valued models will be briefly analized. It will be shown that these models enjoy similar properties than the Boolean-valued models (when restricted to pure ZF-languages). These similarities will be fundamental in order to prove that ZFC is valid w.r.t. twist-valued models (see Theorem 8.21 below). The following basic results for twist-valued models are analogous to the corresponding ones for Boolean-valued models obtained in [1, Theorem 1.17]. All these results will be proven by using the Induction Principle (IP) (recall Remark 7.4). From now on we assume that the reader is familiar with the book [1]. Lemma 8.1. Let A be a complete Boolean algebra, and let u ∈ VTA. Then [[u ∈ u]]1 = 0. Proof. Assume the inductive hypothesis [[y ∈ y]]1 = 0 for every y ∈ dom(u). Note that [[u ǫ u]]1 = ∨ y∈dom(u) ((u(y))1 ∧ [[y ≈ u]]1). Let y ∈ dom(u). Then (u(y))1 ∧ [[y ≈ u]]1 ≤ (u(y))1 ∧ ∧ x∈dom(u) ((u(x))1 → [[x ǫ y]]1) ≤ (u(y))1 ∧ ((u(y))1 → [[y ǫ y]]1) ≤ [[y ǫ y]]1 = 0. Then u(y)1 ∧ [[y ≈ u]]1 = 0 for every y ∈ dom(u), hence [[u ∈ u]]1 = 0. 14 Theorem 8.2. Let A be a complete Boolean algebra, and let u, v, w ∈ VTA. Then: (i) [[u ≈ u]]1 = 1. (ii) u(x)1 ≤ [[x ǫ u]]1, for every x ∈ dom(u). (iii) [[u ≈ v]]1 = [[v ≈ u]]1. (iv) [[u ≈ v]]1 ∧ [[v ≈ w]]1 ≤ [[u ≈ w]]1. (v) [[u ≈ v]]1 ∧ [[u ǫw]]1 ≤ [[v ǫw]]1. (vi) [[v ≈ w]]1 ∧ [[u ǫ v]]1 ≤ [[u ǫw]]1. (vii) [[u ≈ v]]1 ∧ [[φ(u)]]1 ≤ [[φ(v)]]1 for every formula φ(x) in Lp(TA). Proof. The proof of items (i)-(vi) is analogous to the proof of the corresponding items found in [1, Theorem 1.17]. The proof of item (vii) is easily done by induction on the complexity of φ(x) by observing that: the proof when φ is atomic uses Lemma 8.1, for φ = (x ǫ x), and items (i)-(vi) for the other cases. For complex formulas the result follows easily by induction hypothesis. Lemma 8.3. Let A be a complete Boolean algebra. Then, for every formula φ(x) in Lp(TA) and every u ∈ V TA: [[∃y((u ≈ y) ∧ φ(y))]]1 = [[φ(u)]]1. Proof. It follows from Theorem 8.2 items (i), (iii) and (viii). Indeed, [[∃y((u ≈ y) ∧ φ(y))]]1 = ∨ x∈dom(u) ([[u ≈ y]]1 ∧ [[φ(y)]]1) ≤ [[φ(u)]]1 = [[u ≈ u]]1 ∧ [[φ(u)]]1 ≤ [[∃y((u ≈ y) ∧ φ(y))]]1. Notation 8.4. The following notation from [1] will be adopted from now on: ∃x ǫ u φ(x) =def ∃x(x ǫ u ∧ φ(x)); ∀x ǫ u φ(x) =def ∀x(x ǫ u → φ(x)). Theorem 8.5. Let A be a complete Boolean algebra. Then, for every formula φ(x) in Lp(TA) and every u ∈ V TA: [[∃x ǫ u φ(x)]]1 = ∨ x∈dom(u) ((u(x))1 ∧ [[φ(x)]]1) and [[∀x ǫ u φ(x)]]1 = ∧ x∈dom(u) ((u(x))1 → [[φ(x)]]1). Proof. The proof is similar to that for [1, Corollary 1.18], taking into account Theorem 8.2 and Lemma 8.3 Recall that a complete Boolean algebra A' is a complete subalgebra of the complete Boolean algebra A provided that A' is a subalgebra of A and ∨ A′ X = ∨ A X and ∧ A′ X =∧ A X for every X ⊆ |A′|. Analogously, we say that a twist-structure TA′ is a complete subalgebra of the twist-structure TA if TA′ is a subalgebra of TA and ∨ TA′ X = ∨ TA X and∧ TA′ X = ∧ TA X for every X ⊆ |TA′ |, recalling Remark 6.10. 15 Proposition 8.6. If A' is a complete subalgebra of A then TA′ is a complete subalgebra of TA. Proof. If follows from Definition 4.6 and Remark 6.10. Theorem 8.7. Let A' be a complete subalgebra of the complete Boolean algebra A. Then: (i) VTA′ ⊆ VTA. (ii) for every u, v ∈ VTA′ : [[u ǫw]]V T A′ = [[u ǫw]]V TA , and [[u ≈ w]]V T A′ = [[u ≈ w]]V TA . Corollary 8.8. Suppose that A' is a complete subalgebra of A. Then, for any restricted formula φ(x1, . . . , xn) in Lp (recall Definition 7.2) and for every u1, . . . , un ∈ TA′: [[φ(u1, . . . , un)]] V T A′ = [[φ(u1, . . . , un)]] VTA . Proof. The proof is analogous to that for [1, Corollary 1.21]. Remark 8.9. Recall from Remark 4.8 that TA2 , the twist structure for LPT0 defined over the two-element Boolean algebra A2, coincides (up to names) with the three-valued algebra APT0 underlying the matrix MPT0, where 1, 1 2 and 0 are identified with (1, 0), (1, 1) and (0, 1), respectively. Hence, the twist-valued structure VTA2 will be denoted by VAPT0 Since A2 is a complete subalgebra of any complete Boolean algebra A then V APT0 is a complete subalgebra of VTA, for any TA. By Theorem 8.7, [[u ǫ v]] VAPT0 = [[u ǫ v]]V TA and [[u ≈ v]]V APT0 = [[u ≈ v]]V TA for every u, v ∈ VAPT0 and every TA. As happens with the Boolean-valued model VA2, the twist-valued model VAPT0 is, in some sense, isomorphic to the standard universe V, as it will be shown in Theorem 8.13 below. Definition 8.10. Define by transfinite recursion on the well-founded relation y ∈ x the following, for each x ∈ V: x =def {〈ŷ, 1〉 : y ∈ x}. It is clear that x ∈ VAPT0 and so x ∈ VTA for every TA. Hence, if φ(v1, . . . , vn) is a restricted formula in Lp and x1, . . . , xn ∈ V then [[φ(x1, . . . , xn)]] VAPT0 = [[φ(x1, . . . , xn)]] VTA for every TA, by Corollary 8.8. Lemma 8.11. Let φ(v1, . . . , vn) be a formula in Lp, and let x1, . . . , xn ∈ V. Then, [[φ(x1, . . . , xn)]] VAPT0 ∈ {0, 1}. Proof. The result is proven by induction on the complexity of φ. Corollary 8.12. Let φ(v1, . . . , vn) be a restricted formula in Lp, and let x1, . . . , xn ∈ V. Then, [[φ(x1, . . . , xn)]] VTA ∈ {0, 1} for every A. Proof. It follows by Lemma 8.11 and by Corollary 8.8. Theorem 8.13. (i) For every x ∈ V and u ∈ VTA: [[u ǫ x]] = ∨ y∈x [[u ≈ ŷ]]. (ii) For x, y ∈ V: x ∈ y holds in ZFC iff VTA |= (x ǫ ŷ) for every A; x = y holds in ZFC iff VTA |= (x ≈ ŷ) for every A. 16 (iii) The function x 7→ x is one-to-one from V to VAPT0. (iv) For every u ∈ VAPT0 there is a (unique) x ∈ V such that VTA |= (u ≈ x) for all A. (v) For every formula φ(v1, . . . , vn) in Lp and every x1, . . . , xn ∈ V: φ(x1, . . . , xn) holds in ZFC iff V APT0 |= φ(x1, . . . , xn). In addition if φ is restricted (recall Definition 7.2) then, for every x1, . . . , xn ∈ V: φ(x1, . . . , xn) holds in ZFC iff V TA |= φ(x1, . . . , xn), for every A. Proof. It follows by an easy adaptation of the proof of [1, Theorem 1.23]. The only points to be considered are the following: (i) Note that 1 ∧ a = a for every a ∈ |TA|. Then, the adaptation of the proof of this item is immediate. (ii) Both assertions are simultaneously proven by induction on rank(y) (see Remark 7.4), where the induction hypothesis is: for every z with rank(z) < rank(y), x ∈ z iff VTA |= (x ǫ ẑ) for every x and A; x = z iff VTA |= (x ≈ ẑ) for every x and A; and z ∈ x iff VTA |= (ẑ ǫ x) for every x and A. For the first assertion, Corollary 8.12 should be used. For the second assertion, note that 1 → a = a for every a ∈ |A|. Hence ([[x ≈ ẑ]]V TA )1 =∧ y∈x ([[ŷ ǫ ẑ]]V TA )1 ∧ ∧ y∈z ([[ŷ ǫ x]]V TA )1. Use then the first assertion, induction hypothesis and the axiom of extensionality. (iii) It follows from (ii). (iv) By adapting the proof of [1, Theorem 1.23(iv)], at some point of the proof the set v = {y ∈ V : u(x) = 1 and ([[x ≈ ŷ]]V TA )1 = 1, for some x ∈ dom(u)} of V must be considered. (v) In order to adapt the proof of [1, Theorem 1.23(v)] it should be noted that, if ∅ 6= X ⊆ |APT0| is such that ∨ APT0 X = 1, then 1 ∈ X . From this, the inductive step φ = ∃xψ can be treated analogously to the proof of [1, Theorem 1.23(v)]. In addition, the use of the Leibniz rule (see [1, Theorem 1.17(vii)]) at this point of the proof can be adapted here to an application of Theorem 8.2(vii) as follows: 1 = ([[ψ(x, x1, . . . , xn]] VAPT0 )1 ∧ ([[x ≈ ŷ]] VAPT0 )1 ≤ ([[ψ(ŷ, x1, . . . , xn]] VAPT0 )1. Hence ([[ψ(ŷ, x1, . . . , xn]] VAPT0 )1 = 1, and the rest of the proof follows from here. Now it will be shown the Maximum Principle of Boolean-valued models (see [1, Lemma 1.27]) is also valid in twist-valued models. The adaptation to our framework of the proof of this result found in [1] is straightfoward. Definition 8.14. Let A be a complete Boolean algebra. Given sets E = {ai : i ∈ I} ⊆ |A| and F = {ui : i ∈ I} ⊆ V TA, the twist mixture of F with respect to E is the element u = ∑ i∈I ai ⊙ ui of V TA defined as follows:7 dom(u) = ⋃ i∈I dom(ui), and 7It is worth observing that the definition of the second coordinate of u(z) will be irrelevant. 17 u(z) = (∨ i∈I (ai ∧ [[z ǫ ui]]1),∼ ∨ i∈I (ai ∧ [[z ǫ ui]]1) ) , for every z ∈ dom(u). Lemma 8.15 (Mixing Lemma). Let {ai : i ∈ I} ⊆ |A| and {ui : i ∈ I} ⊆ V TA, and let u = ∑ i∈I ai ⊙ ui. Suppose that, for every i, j ∈ I, ai ∧ aj ≤ [[ui ≈ uj]]1. Then ai ≤ [[u ≈ ui]]1 for every i ∈ I. Proof. It can be proved by a straightforward adaptation of the proof of [1, Lemma 1.25], taking into account Theorem 8.2 items (ii), (iii) and (vi). The next fundamental result shows that the set of pure ZF-sentences validated by each twist-valued structure VTA is a Henkin theory: Lemma 8.16 (The Maximum Principle). Let A be a complete Boolean algebra. Then, for every formula φ(x) in Lp(TA), there is u ∈ V TA such that [[∃xφ(x)]]1 = [[φ(u)]]1. In particular, if VTA |= ∃xφ(x) then VTA |= φ(u) for some u ∈ VTA. Proof. The proof is obtained by a straightforward adaptation of the proof of [1, Lemma 1.27]. The collection X = {[[φ(u)]] : u ∈ VTA} is a set, since TA is a set. By the Axiom of Choice, there is an ordinal α and a set {uξ : ξ < α} ⊆ V TA} such that X = {[[φ(uξ)]] : ξ < α}, hence [[∃xφ(x)]]1 = ∨ ξ<α[[φ(uξ)]]1. For each ξ < α let aξ = [[φ(uξ)]]1∧∼ ∨ η<ξ[[φ(uη)]]1, and let u = ∑ ξ<α aξ⊙uξ. By the Mixing Lemma 8.15 and by Theorem 8.2 items (ii) and (vii) it follows that [[∃xφ(x)]]1 = [[φ(u)]]1. Corollary 8.17. Let φ(x) be a formula in Lp(TA) such that V TA |= ∃xφ(x). Then: (i) For any v ∈ VTA there exists u ∈ VTA such that [[φ(u)]]1 = 1 and [[φ(v)]]1 = [[u ≈ v]]1. (ii) Let ψ(x) be a formula in Lp(TA) such that V TA |= φ(u) implies that VTA |= ψ(u), for every u ∈ VTA. Then VTA |= ∀x(φ(x) → ψ(x)). Proof. Is an easy adaptation of the proof of [1, Corollary 1.28], taking into account Lemma 8.16 and Theorem 8.2 items (ii) and (vii). The notion of core for a Boolean-valued set (see [1]) can be easily adapted to twist-valued sets: Definition 8.18. Let u ∈ VTA. A core for u is a set v ⊆ VTA such that: (i) [[x ǫ u]]1 = 1 for every x ∈ v; and (ii) for every y ∈ VTA such that [[y ǫ u]]1 = 1, there is a unique x ∈ v such that [[x ≈ y]]1 = 1. Lemma 8.19. Any u ∈ VTA has a core. Proof. Is an easy adaptation of the proof of [1, Lemma 1.31]. Let ∅ be the empty element of VTA. As happens with Boolean-valued models, if u ∈ VTA is such that VTA |= ∼(u ≈ ∅) then, by the Maximum Principle, any core of u is nonempty. Corollary 8.20. Let u ∈ VTA such that VTA |= ∼(u ≈ ∅), and let v be a core for u. Then, for any x ∈ VTA there exists y ∈ v such that [[x ≈ y]]1 = [[x ǫ u]]1. 18 Proof. Is follows from Corollary 8.17. From the results obtained above, one of the main results of the paper can be established: Theorem 8.21. All the axioms (hence all the theorems) of ZFC, when restricted to pure ZF-languages Lp(TA) (recall Definition 7.2), are valid in V TA, for every A. Proof. It is a relatively easy (but arduous) adaptation of the proof of [1, Theorem 1.33], taking into account the auxiliary results obtained within this section, which are similar to the ones required in [1]. 9 Twist-valued models for (PS3,¬) In this section the three-valued model for set theory introduced by Löwe and Tarafder in [18] will be extended to a class of twist-valued models. As observed in Section 7, the three-valued logic (PS3,¬) (denoted as (PS3, ∗) in [18]) was already considered in [13] under the name MPT. Indeed, this logic has been independenly proposed by different authors at several times, and with different motivations.8 For instance, the same logic was proposed in 1970 by da Costa and D'Ottaviano's as J3. It was reintroduced in 2000 by Carnielli, Marcos and de Amo as LFI1 and by Batens and De Clerq as the propositional fragment of the first-order logic CLuNs, in 2014. As observed by Batens, this logic was firstly proposed by Karl Scütte in 1960 under the name Φv (see [7] for details and specific references). Each of the three-valued algebras above is equivalent, up to language, to the three-valued algebra of Lukasiewicz three-valued logic L3. Hence, these logics are equivalent to L3 with {1, 1 2 } as designated values. Moreover, as it was shown by Blok and Pigozzi in [2], the class of algebraic models of J3 (and so the class of twist structures for LPT0) coincides with the agebraic models of Lukasiewicz's three-valued logic L3. More remarks about these three-valued equivalent logics can be found in [7], Chapters 4 and 7. As shown in [13, p. 407], the implication ⇒ given by ⇒ 1 1 2 0 1 1 1 0 1 2 1 1 0 0 1 1 1 (which is the same implication ⇒ of PS3 and the primitive implication of MPT) can be defined in the language of LFI1 (hence in the language of LPT0) as follows: φ⇒ ψ =def ¬∼(φ→ ψ). From this, it is easy to adapt Definition 4.6 of twist-structures for LPT0 to (PS3,¬) (see Definition 9.1 below). Hence, the logic (PS3,¬) will be considered as defined over the signature Σ⇒ = {∧,∨,⇒,¬}. As observed in [13, pp. 395 and 407], the strong negation ∼ can be defined as ∼φ =def φ⇒ ¬(φ⇒ φ), while φ→ ψ =def ∼φ ∨ ψ. Definition 9.1. Let A be a complete Boolean algebra, and let TA as in Definition 4.5. The twist structure for (PS3,¬) over A is the algebra TA∗ = 〈TA, ∧, ∨, ⇒, ¬〉 over Σ⇒ such that the operations ∧, ∨ and ¬ are defined as in Definition 4.6, and ⇒ is defined as follows, for every (z1, z2), (w1, w2) ∈ TA: 8As mentioned in Section 3, LFI1◦ is another presentation of this logic. 19 (z1, z2) ⇒ (w1, w2) = (z1 → w1, z1 ∧ ∼w1). By considering (as mentioned above) ∼ and → as derived connectives in TA∗ , it is clear that ∼(z1, z2) = (∼z1, z1) and (z1, z2) → (w1, w2) = (z1 → w1, z1∧w2). Hence, the original operations of Definition 4.6 can be recovered in TA∗ . As it will be discussed below, we will adopt a technique different to the one used in [18] in order to show the satisfaction of ZFC in the twist-valued models based on TA∗ . However, it is interesting to observe that a nice property of (PS3,¬) is preserved by any TA∗ . Indeed, in [18] the following notion of reasonable implication algebras was proposed in order to provide suitable lattice-valued for ZF: Definition 9.2. An algebra A = 〈A,∧,∨,⇒, 0, 1〉 is an reasonable implication algebra if the reduct 〈A,∧,∨, 0, 1〉 is a complete lattice with bottom 0 and top 1, and ⇒ is a binary operator satisfying the following, for every z, w, u ∈ A: (P1) z ∧ w ≤ u implies that z ≤ (w ⇒ u); (P2) z ≤ w implies that (u⇒ z) ≤ (u⇒ w); (P3) z ≤ w implies that (w ⇒ u) ≤ (z ⇒ u). Proposition 9.3. For every complete Boolean algebra A, the twist structure TA∗ for (PS3,¬) is a reasonable implication algebra such that 0 = (0, 1) and 1 = (1, 0). 9 Proof. Let (z1, z2), (w1, w2), (u1, u2) ∈ TA. (P1): Assume that (z1, z2) ∧ (w1, w2) ≤ (u1, u2). That is, (z1 ∧ w1, z2 ∨ w2) ≤ (u1, u2). Then z1 ∧ w1 ≤ u1 and z2 ∨ w2 ≥ u2. From z1 ∧ w1 ≤ u1 it follows that z1 ≤ w1 → u1. Besides, since z1∨z2 = 1 then ∼z2 ≤ z1 ≤ w1 → u1. Hence z2 ≥ ∼(w1 → u1) = w1∧∼u1. From this, (z1, z2) ≤ (w1 → u1, w1 ∧ ∼u1) = (w1, w2) ⇒ (u1, u2). (P2): Assume that (z1, z2) ≤ (w1, w2). Then z1 ≤ w1, hence u1 → z1 ≤ u1 → w1 and so u1 ∧∼z1 = ∼(u1 → z1) ≥ ∼(u1 → w1) = u1 ∧∼w1. This means that (u1, u2) ⇒ (z1, z2) ≤ (u1, u2) ⇒ (w1, w2). (P3): It is proved analogously, but now taking into account that z1 ≤ w1 implies that w1 → u1 ≤ z1 → u1. Now, the three-valued model of set theory presented in [18] will be generalized to twistvalued models over any complete Boolean algebra. The structure VTA∗ is defined as the structure VTA given in Definition 7.1. This does not come as a surprise, given that the domain of TA and TA∗ is the same, the set TA. However, V TA and VTA∗ are different as first-order structures, namely, the way in which the formulas are interpreted. The only difference, besides using different implications in the underlying logics, will be in the form in which the predicates ǫ and ≈ are interpreted. Thus, the twist truth-value [[φ]]V T A∗ of a sentence φ in VTA∗ will be defined according to the recursive clauses in Definition 7.5, with the following difference: any occurrence of the operator → must be replaced by the operator ⇒ Note that the clause interpreting ∼φ is now derived from the others, taking into account the observation after Definition 9.1. 9To be rigorous, the ¬-less reduct of TA∗ expanded with 0 and 1 is a reasonable implication algebra. 20 In Theorem 9.4 below it is stated that every twist-valued structure VTA∗ is a model of ZFC. This constitutes a generalization of [18, Corollary 11]. Indeed, instead of taking just a three-valued model (generated by the two-element Boolean algebra), we obtain a class of models, one for each complete Boolean algebra. Moreover, we also prove that these generalized models (including, of course, the original Löwe-Tarafder model) satisfy, in addition, the Axiom of Choice. The proof of validity of ZF given in [18, Corollary 11] is strongly based on the particularities of the three-valued algebra of (PS3,¬). 10 This forces us to adapt, to this setting, the proof for twist-valued models over TA given in the previous sections (which, by its turn, is adapted from the proof for Boolean-valued sets). Such adaptations from TA to TA∗ are immediate, and all the results and definitions proposed in the previous sections work fine for TA∗ . Hence, we obtain the second main result of the paper: Theorem 9.4. All the axioms (hence all the theorems) of ZFC, when restricted to pure ZF-languages Lp(TA), are valid in V TA∗ , for every A. Remark 9.5. Oberve that, in [18, Corollary 11], it was proved that PS3 is a model of ZF, not of ZFC. Thus, Theorem 9.4 improves the above mentioned result in two ways: it is generalized to arbitary Boolean algebras and, in addition, it proves that the Axiom of Choice AC is also satisfied by all that models, including the original three-valued structure PS3. 10 ZFLPT0 as a paraconsistent set theory After proving that the two classes of twist-valued models proposed here are models of ZFC, in this section the paraconsistent character of both classes of models will be investigated. It will be shown that twist-valued models over TA (that is, over the logic LPT0) are "more paraconsistent" that the ones over TA∗ (that is, defined over (PS3,¬)). Recall from Theorem 8.2(i) that [[u ≈ u]] ∈ DA for every u in every twist-valued model VTA. The interesting fact of ZFLPT0 is that it allows "inconsistent" sets, that is, elements of VTA such that the value of (u 6≈ u) is also designated. Observe that 1 = (1, 0), 1 2 = (1, 1) and 0 = (0, 1) are defined in every TA. Since z ∈ DA iff z = (1, a) for some a ∈ A it follows that 1 2 ≤ z for every z ∈ DA (recalling the partial order for TA considered in Remark 6.10). Proposition 10.1. There exists u ∈ VTA such that [[u ≈ u]] = 1 2 . Proof. Let w be any element of VTA, and let u = {〈w, 1 2 〉}. Since [[w ≈ w]] ∈ DA then [[w ǫ u]] = u(w) ∧ [[w ≈ w]] = 1 2 ∧ [[w ≈ w]] = 1 2 . From this, [[u ≈ u]] = u(w) → [[w ǫ u]] = 1 2 → 1 2 = 1 2 . From the last result it can be proven that ZFLPT0 is strongly paraconsistent, in the sense that there is a contradiction which is valid in the logic: Corollary 10.2. Let σ = ∀x(x ≈ x). Then VTA |= σ ∧ ¬σ. 10For instance, the fact that expressions like [[u ≈ v]] ⇒ [[u ǫw]] can only take either the value 0 or 1 is used several times in [18]. Observe that, in TA∗ , the value of z ⇒w is always of the form (a,∼a) for some a ∈ |A|. Hence [[u ≈ v]]V T A∗ is always of the form (a,∼a) for some a ∈ |A|. 21 Proof. Let VTA be a twist-valued model for ZFLPT0. As observed above, 1 2 ≤ z for every z ∈ DA. By Theorem 8.2(i), [[v ≈ v]] ∈ DA for every v in V TA and so 1 2 ≤ [[v ≈ v]] for every v, that is, 1 2 ≤ [[∀x(x ≈ x)]], by Definition 7.5. On the other hand, [[∀x(x ≈ x)]] ≤ [[u ≈ u]] = 1 2 for u as in Proposition 10.1. This shows that [[σ]] = [[∀x(x ≈ x)]] = 1 2 and so [[¬σ]] = ¬ [[σ]] = 1 2 . Hence [[σ ∧ ¬σ]] = [[σ]] ∧ [[¬σ]] = 1 2 , a designated value. Since the extensionality axiom of ZF is satisfied by every twist-valued model VTA for ZFLPT0, [[u ≈ v]] ∈ DA iff u and v have the same elements, that is: for every w in VTA, [[w ǫ u]] ∈ DA iff [[w ǫ v]] ∈ DA. However, nothing guarantees that u and v will have the same 'non-elements', namely: it could be possible that [[¬(w ǫ u)]] ∈ DA but [[¬(w ǫ v)]] /∈ DA, for some w in V TA, even when [[u ≈ v]] ∈ DA. Given such w, consider the property φ(x) := ¬(w ǫ x), meaning that "w is a non-element of x". Then, this situation shows that VTA 6|= ((u ≈ v)∧φ(u)) → φ(v), which constitutes a violation of the Leibniz rule for the equality predicate ≈ in ZFLPT0. Theorem 10.3. The formula φ(x) := ¬(w ǫ x) is such that the Leibniz rule fails for it in every VTA, namely: VTA 6|= ∀x∀y((x ≈ y) ∧ φ(x) → φ(y)). Proof. Let VTA be a twist-valued model for ZFLPT0, and let ∅ be the empty element of VTA. Observe that w = {〈∅, 1〉}, u = {〈w, 1 2 〉} and v = {〈w, 1〉} belong to every model VTA. Now, [[∅ ǫ w]] = w(∅) ∧ [[∅ ≈ ∅]] = 1 ∧1 = 1. From this, [[w ≈ w]] = w(∅) → [[∅ ǫ w]] = 1 →1 = 1 and so [[w ǫ u]] = u(w) ∧ [[w ≈ w]] = 1 2 ∧1 = 1 2 . On the other hand, [[w ǫ v]] = v(w) ∧ [[w ≈ w]] = 1 ∧1 = 1. This implies that [[u ≈ v]] = (u(w) → [[w ǫ v]]) ∧ (v(w) → [[w ǫ u]]) = (1 2 →1) ∧ (1 → 1 2 ) = 1 2 . But [[φ(u)]] = [[¬(w ǫ u)]] = ¬ [[w ǫ u]] = ¬ 1 2 = 1 2 and [[φ(v)]] = [[¬(w ǫ v)]] = ¬ [[w ǫ v]] = ¬1 = 0. Thus, [[((u ≈ v) ∧ φ(u)) → φ(v)]] = (1 2 ∧ 1 2 ) →0 = 0, which implies that VTA 6|= ∀x∀y((x ≈ y) ∧ φ(x) → φ(y)). It is important to observe that the failure of the Leiniz rule in VTA shown in Theorem 10.3 does not contradict Theorem 8.2(viii): indeed, what Theorem 8.2(viii) states is the validity of the Leibniz rule in VTA for every formula φ(x) in the pure ZF-language Lp(TA). On the other hand, the formula φ(x) found in Theorem 10.3 which violates the Leibniz rule in VTA contains an occurrence of the paraconsistent negation ¬, that is, it does not belong to Lp(TA). In that example, two sets which are equal have different 'non-elements', where 'non' refers to the paraconsistent negation ¬. Besides the failure of the Leibniz rule for the full language, ZFLPT0 does not validate the so-called bounded quantification properties. Definition 10.4. For any formula φ and every u ∈ VTA, the universal bounded quantification property UBQuφ and the existential bounded quantification property EBQ u φ are defined as follows: (UBQuφ) [[∀x(x ǫ u → φ(x))]]1 = ∧ x∈dom(u)((u(x))1 → φ(x)) (EBQuφ) [[∃x(x ǫ u ∧ φ(x))]]1 = ∨ x∈dom(u)((u(x))1 ∧ [[φ(x)]]1) By simplicity, formulas on the left-hand size of UBQuψ and EBQ u φ will be written as [[∀x ǫ u φ(x)]]1 and [[∃x ǫ u φ(x)]]1, respectively. By adapting the proof of [1, Corollary 1.18] it can be proven the following: 22 Theorem 10.5. For any negation-free formula φ (i.e., φ ∈ Lp(TA)) and every u ∈ V TA, the bounded quantification properties UBQuφ and EBQ u φ hold in V TA. However, for formulas containing the paraconsistent negation the latter result does not holds in general: Proposition 10.6. There is u ∈ VTA and formulas φ(x) and ψ(x) such that the bounded quantification properties UBQuψ and EBQ u φ fail in V TA. Proof. It is enough to prove the falure of EBQuφ given that the failure of UBQ u ψ is obtained from it by using ψ(x) := ∼φ(x) and the duality between infimum and supremum through the Boolean complement ∼. Thus, let VTA and let w = {〈∅, 1〉}, v = {〈w, 1 2 〉}, y = {〈w, 1〉} and u = {〈y, 1〉}. Let φ(x) := ¬(w ǫ x). As in the proof of Theorem 10.3 it can be proven that [[v ≈ y]] = [[φ(v)]] = 1 2 and [[φ(y)]] = 0. Hence ∨ x∈dom(u)((u(x))1 ∧ [[φ(x)]]1) = (u(y))1 ∧ [[φ(y)]]1 = 0 while [[∃x ǫ u φ(x)]]1 = [[∃x(x ǫ u ∧ φ(x))]]1 = ∨ v′∈VTA ∨ x∈dom(u)((u(x))1 ∧ [[v ′ ≈ x]]1 ∧ [[φ(v′)]]1) = ∨ v′∈VTA ((u(y))1 ∧ [[v ′ ≈ y]]1 ∧ [[φ(v ′)]]1) ≥ (u(y))1 ∧ [[v ≈ y]]1 ∧ [[φ(v)]]1 = 1. This means that [[∃x ǫ u φ(x)]]1 = 1 6= 0 = ∨ x∈dom(u)((u(x))1 ∧ [[φ(x)]]1). It is worth noting that the limitations of ZFLPT0 pointed out above (namely, the Leibniz rule and the bounded quantification property for formulas containing the paraconsistent negation) are also present in Löwe-Tarafder's model [18]. As mentioned in Section 3, in [6] was presented a family of paraconsistent set theories based on diverse LFIs, such that the original ZF axioms were slightly modified in order to deal with a unary predicate C(x) representing that 'the set x is consistent'. The consistency connective ◦ is primitive in mbC, but it is definable as ◦φ := ∼(φ ∧ ¬φ) in any axiomatic extension of mbC which proves the schema (ciw): ◦φ ∨ (φ ∧ ¬φ) such as LPT0. In the same way, the consistency predicate C(x) can be expressed, in extensions of ZFmbC, in terms of a formula of ZFmbC without using the predicate C, and the same happens with the inconsistency predicate ¬C(x). For instance, ZFmCi is based on mCi, an extension of mbC in which ¬◦φ is equivalent to φ∧¬φ. Thus, ¬C(x) was defined to be equivalent to (x ≈ x)∧¬(x ≈ x) in ZFmCi. From this, ¬C(x) is equivalent to ¬◦(x ≈ x) in ZFmCi. Given that LPT0 is an extension of mCi, if a consistency predicate for sets were added to the language of ZFLPT0 then it seems reasonable to require the equivalence between ¬C(x) and ¬◦(x ≈ x) in ZFLPT0. But ◦C(x) is derivable ZFmCi, so it would be valid in ZFLPT0 (indeed, the proof in ZFmCi of ◦C(x) given in [6, Proposition 3.10] holds in QLPT0, assuming the axioms for C from ZFmCi). From this C(x) ↔ ◦(x ≈ x) would be also derivable in QLPT0 and so it would be valid in ZFLPT0 expanded with a suitable predicate C denoting 'consistency for sets'. This motivates the following: Definition 10.7. Define in ZFLPT0 the consistency predicate for sets, C(x), as follows: C(x) =def ∼¬(x ≈ x). According to the previous discussion, C(x) should be equivalent to ◦(x ≈ x) in ZFLPT0. But ◦φ is equivalent to ∼(φ∧¬φ) in LPT0, and (x ≈ x) is valid in ZFLPT0, hence C(x) should be equivalent to ∼¬(x ≈ x) in ZFLPT0, which justifies Definition 10.7. Proposition 10.8. The consistency predicate C(x) is non-trivial: there exist v, w ∈ VTA such that [[C(v)]] = 1 and [[C(w)]] = 0. Moreover, [[C(u)]] 6= 1 2 for every u in VTA. 23 Proof. Let VTA be a twist-valued model for ZFLPT0, and consider v = {〈∅, 1〉} and w = {〈∅, 1 2 〉} in VTA. It is easy to see that [[C(v)]] = 1 and [[C(w)]] = 0. On the other hand, for every u in VTA it is the case that [[C(u)]] = ∼z for z = [[¬(u ≈ u)]]. Hence [[C(u)]] = (∼z1, z1) 6= 1 2 , for every u. Finally, we can show now that twist-valued models over TA (that is, over the logic LPT0) are "more paraconsistent" than the ones over TA∗ (that is, defined over (PS3,¬)). Indeed, as we have seen, ZFLPT0 allow us to define in every twist-valued model V TA an "inconsistent set", namely u, such that (u ≈ u)∧¬(u ≈ u) holds. In fact, any u = {〈w, 1 2 〉} is such that [[u ≈ u]] = 1 2 →1 2 = 1 2 . The difference, of course, rests on the nature of the implication operator considered in each case: in (PS3,¬) the value of (u ≈ u) is always 1, since 1 2 ⇒1 2 = 1. Hence, ¬(u ≈ u) always gets the value 0. The same holds in any model over reasonable implicative algebras considered by Löwe and Tarafder (see [18, Proposition 1]). 10.1 Discussion: ZFLPT0 and the failure of the Leibniz rule At first sigth, having a (paraconsistent) set theory as ZFLPT0 in which the Leibniz rule is not satisfied for every formula φ(x) that represents a property could seem to be a bit disappointing. After all, ZF is defined as a first-order theory with equality, which pressuposes the validity of the Leibniz rule. The Leibniz rule states that the equality predicate preserves logical equivalence, namely: (a ≈ b) → (φ(a) ↔ φ(b) for every formula φ(x) (clearly this can be generalized to formulas with n ≥ 1 free variables, assuming ∧n i=1(ai ≈ bi)). In first-order theories based on classical logic, such as ZF, it is enough to require that this property holds for every atomic formula, and so the general case is proven by induction on the complexity of φ. Of course this proof cannot be reproduced in QLPT0, since ¬ is not congruential: φ(a) ↔ φ(b) does not imply ¬φ(a) ↔ ¬φ(b) in general (and this is the key step in the proof by induction). The solution is requiring the validity of the Leibniz rule for every φ from the beginning, adjusting accordingly the class of interpretations for QLPT0 expanded with equality (see [12]). However, the situation for ZFLPT0 is quite different: because of the extensionality axiom, the definition of the interpretation of the equality predicate depends strongly on the interpretation of the membership predicate. In fact, the interpretation of both predicates is simultaneously defined by transfinite recursion, according to Definition 7.5. The validity of the Leibniz rule, in the case of Boolean-set models for ZFC, is proven as a theorem. The simultaneous definition of the equality and membership predicates is designed to fit exactly the requirements of the extensionality axiom: two individuals (sets) are identical provided that they have the same elements. From this, it is proven by induction of the complexity of φ(x) that [[u ≈ v]] ∧ [[φ(u)]] ≤ [[φ(v)]] in every Booleanvalued model. As we have seen in Theorem 8.2(vii), the same holds in twist-valued models w.r.t. the first coordinate, namely: [[u ≈ v]]1 ∧ [[φ(u)]]1 ≤ [[φ(v)]]1. But then, it is required that this property just holds for 'classical' formulas, that is, formulas φ without occurrences of the paraconsistent negation ¬. The explanation for this fact is simple, from the technical point of view: assuming that the property above holds for φ then, when considering ¬φ, the value of [[¬φ(u)]]1 is [[φ(u)]]2, and we don't have enough information about the relationship between [[φ(u)]]2, and [[φ(v)]]2. The example given in the proof of Theorem 10.3 shows that it is impossible to satisfy the Leibniz rule in ZFLPT0 24 for formulas containing the paraconsistent negation, hence this is an unsolvable problem with the current definitions. Within the present approach, paraconsistent situations such as the existence of 'inconsistent' sets u satisfying ¬(u ≈ u) or the existence of a set being simultaneously an element and a non-element of another set seems to be irreconcilable with the fullfillment of the Leibniz rule for formulas behind the 'classical' language. Because of this, the predicate ≈ in ZFLPT0 should be considered as representing 'indiscernibility by pure ZF-properties', exactly as happens with Boolean-valued models for ZF. In this manner (u ≈ v) implies that, besides having the same elements, u and v have, for instance, the same 'non∗-elements', where 'non∗' stands for the classical negation ∼. That is, ∀w(∼(w ǫ u) ↔ ∼(w ǫ v)) is a consequence of (u ≈ v). On the other hand, as it was shown in Theorem 10.3, (u ≈ v) does not imply (in general) that u and v have the same 'nonelements', where 'non' stands for the paraconsistent negation ¬: ∀w(¬(w ǫ u) ↔ ¬(w ǫ v)) is not a consequence of (u ≈ v). Instead of being regarded as discouraging, the fact that (u ≈ v) does not necessarily imply that u and v have the same 'non-elements' (for 'non' the paraconsistent negation ¬) can be seen as an auspicious property, because it can be a way to circumvent undesirable consequences of 'non-elements', as it happens with the well-known Hempel's Ravens Paradox: evidence, differently from proof, for instance, has its own idiosyncratic properties. This point, however, will be left for further discussion. 11 Concluding remarks In this paper, we introduce a generalization of Boolean-valued models of set theory to a class of algebras represented as twist-structures, defining a class of models for ZFC that we called twist-valued models. This class of algebras characterizes a three-valued paraconsistent logic called LPT, which was extensively studied in the literature of paraconsistent logics under different names and signatures as, for example, as the well-known da Costa and D'Ottaviano's logic J3 and as the logic LFI1 (cf. [3]) . As it was shown by Blok and Pigozzi in [2], the class of algebraic models of J3 (hence, the class of twist structures for LPT0) coincides with the agebraic models of Lukasiewicz three-valued logic L3. With small changes, in Section 9 the twist-valued models for LPT0 were adapted in order to obtain twist-valued for (PS3,¬), the three-valued paraconsistent logic studied by Löwe and Tarafder in [18] as a basis for paraconsistent set theory. Thus, their three-valued algebraic model of ZF was extended to a class of twist-valued models of ZF, each of them defined over a complete Boolean algebra. In addition, it was proved that these models (including the three-valued model over (PS3,¬)) satisfy, in addition, the Axiom of Choice. Moreover, it was shown that the implication operator → of LPT0 is, in a sense, more suitable for a paraconsistent set theory than the one ⇒ of PS3: it allows inconsistent sets (i.e., [[(w ≈ w)]] = 1 2 for some w, see Proposition 10.1). It is worth noting that → does not characterize a 'reasonable implication algebra' (recall Definition 9.2): indeed, 1 ∧ 1 2 ≤ 1 2 but 1 6≤ 1 2 → 1 2 = 1 2 . This shows that reasonable implication algebras are just one way to define a paraconsistent set theory, not the best. Despite having the same limitative results than Löwe-Tarafder's model (that is, the debatable failure of Leibniz rule and the bounded quantification property for formulas containing the paraconsistent negation, recall Section 10) we believe that ZFLPT0 has a great potential as a paraconsistent set theory. 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Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 1 Taking empathy online Lucy Osler Department of Sociology, Philosophy and Anthropology, University of Exeter, Exeter, UK [email protected] Abstract: Despite its long history of investigating sociality, phenomenology has, to date, said little about online sociality. The phenomenological tradition typically claims that empathy is the fundamental way in which we experience others and their experiences. While empathy is discussed almost exclusively in the context of face-to-face interaction, I claim that we can empathetically perceive others and their experiences in certain online situations. Drawing upon the phenomenological distinction between the physical, objective body and the expressive, lived body, I: (i) highlight that empathy involves perceiving the other's expressive, lived body, (ii) show that the lived body is not tied to the physical body and that empathy can take place outside of face-to-face interactions, and (iii) argue that the lived body can enter online space and is empathetically available to others there. I explore two ways in which the other's lived body enters online space and can be empathetically perceived: first, in cases where our face-to-face encounter is technologically-mediated over video link and, second, by showing how the other's texts, as speech, can form part of the other's lived body. Investigating empathy online not only furthers our understanding of online encounters but also leads to a refined conception of empathy more generally. Keywords: phenomenology, empathy, online, sociality, lived body, expressivity Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 2 Introduction A huge number of my interpersonal encounters take place online. I spend a lot of time on WhatsApp chatting to friends (both individually and in groups); I use Messenger in a similar manner; I am an avid sharer of photos on Instagram; I sometimes peruse Facebook; I regularly Skype, Zoom and Houseparty with friends; I email colleagues; I've been known to play World of Warcraft via an avatar; and, very occasionally, I even use my phone to call people over WiFi. While I may be betraying just how often I encounter people online, I am certainly not an outlier here. Indeed, as I am making the final edits to this paper during a Covid-19 lockdown in the UK, this is currently the norm for many of us. As Miller puts it, 'the spaces of networked digital technologies are no longer liminal since they are now part-and-parcel of the experience of everyday life' (2012, 266; also see Miller 2016) (and this is particularly the experience of everyday life in lockdown). After an initial wave of optimism that praised the internet for creating a utopian hub for social encounters (e.g. Benkler 2006; Rainie and Wellman 2012; Rheingold 2000), there has been a pessimistic turn in the way that many people think of online sociality (e.g. Dotson 2017; Turkle 2015, 2017). Whether one remains optimistic about online sociality or embraces this more pessimistic outlook, what underlies this debate is a more fundamental question about how we encounter others online in the first place. What does it mean to encounter the other when we are no longer face-to-face but are mediated by our screens? Despite a long history of interest in intersubjectivity and sociality, phenomenology has, to date, said relatively little about interpersonal relations on the internet. Recent compendiums on the phenomenology of sociality, for example, do not include any chapters on the matter (e.g. Dolezal and Petherbridge 2017; Szanto and Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 3 Moran 2015; Salice and Schmid 2016).1 This paper serves not only to show that empathy (a notion at the heart of the phenomenology of sociality) takes place online but opens the door to a rich array of phenomenological investigations in relation to our experiences online. Moreover, by considering whether empathy takes place online, we are forced to refine and further our understanding of empathy more generally. The phenomenological tradition typically claims that empathy is the fundamental way in which we experience others and their experiences. 'Empathy' is used as a technical term to refer to the way that others' experiences can be directly perceptually available to me through their expressive behaviour (e.g. Husserl 1993, 2006; Fuchs 2014; Krueger and Overgaard 2012; Scheler 2008; Schutz 1967; Stein 1989; Zahavi 2001, 2014). This is not to imply that our bodies merely give inferential clues to others about our hidden inner mental life. Rather, it is the claim that our bodily expressions are, properly-speaking, constitutive parts of our experience and, thus, render experience something that, at least sometimes, can be directly perceived by others. Given the emphasis on the role of the expressive body, it is perhaps not surprising that empathy theorists2 within phenomenology almost exclusively discuss 1 For an interesting article bucking this trend, see Kekki's recent phenomenological exploration of how we authentically encounter the other in the context of online learning. While Kekki (2020) does not specifically consider whether we empathetically encounter the other online, she presents a persuasive account of how we encounter the expressive other in the online sphere. 2 Note that, throughout this article, when I refer to 'empathy theorists' or 'empathy proponents', I am specifically referring to those working with the phenomenological concept of empathy, rather than to empathy more broadly or colloquially construed. The word empathy is used Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 4 empathy in the context of face-to-face encounters. As such, it might seem that the phenomenological concept of empathy has little to contribute to discussions of online sociality. Yet, the assumption that empathy only occurs face-to-face rests on an unjustified restriction of expressivity to the physical body. Drawing upon the classic phenomenological distinction between the objective, physical body and the expressive, lived body, I argue that the expressive, lived body is not bound to our skin and, therefore, allows for empathy to occur 'at a distance'. Having liberated empathy from a strictly face-to-face interaction, I turn to the question of whether empathy can happen online. I claim that even though online sociality is often depicted as disembodied communication (e.g. Fuchs 2014), our lived bodies can and do enter online space. I, therefore, present the novel position that we have direct empathetic access to others and their experiences even when their bodies are technologically-mediated. In section 1, I provide an overview of the phenomenological concept of empathy. In section 2, I discuss how the lived body is not tied to the physical, objective body. I argue that, even in our offline lives, we can empathetically perceive others 'at a distance' from their physical bodies. In section 3, I outline why it might seem that empathy drops out of the picture when we go online. In section 4, I explore two ways in which we find the lived body online. First, where our face-to-face encounters are technologically-mediated, for instance via a video call. Second, where we experience the other's lived, expressive body in text. In section 5, I explore and respond to some initial challenges to my claim that empathy occurs online. I conclude that not only do within social cognition in a variety of (sometimes contradictory) ways (see Zahavi and Overgaard 2012) and I do not intend to refer to all theorists who think that some kind of empathy might be involved in social cognition. Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 5 concepts from the phenomenological tradition shed light on online sociality but that these very concepts are, in turn, enriched by such considerations. 1. The phenomenological notion of empathy The term empathy is used by both classic and contemporary phenomenologists to pick out a special form of other-directed act (e.g. Husserl 1993, 2006; Jardine 2015; Magrì and Moran 2018; Overgaard 2018; Scheler 2008; Schutz 1967; Stein 1989; Szanto 2015; Zahavi 2001, among many others). Empathy is typically taken to be a perceptionbased experience, where the other and their experience are given to me through the 'field of expression' that is their lived body (Schutz 1967, 22). Contra other theories of social cognition, phenomenologists do not take their starting point to be that others' experiences are essentially unobservable. Such a starting point rests on the assumption that experiences are something that happen inside a person, hidden behind the veil of our bodies; thus, rendering another's experience inaccessible to us. This 'Unobservability Thesis' is frequently taken as the very motivation behind the problem of other minds (e.g. Goldman 2012, 402). Two prominent social cognition theories that attempt to respond to the Unobservability Thesis are Theory Theory and Simulation Theory: Theory Theory asserts that our other-understanding is rooted in our ability to construct theories of mind about other people based upon their behaviour, which we then employ in order to infer what they might be experiencing (e.g. Gopnik and Wellman 1992). Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 6 Simulation Theory asserts that our other-understanding is rooted in our ability to simulate the experiences or feelings of another and project them onto them (e.g. Goldman 2012). Empathy proponents, however, rebut the Unobservability Thesis on the grounds that we are essentially embodied subjects. Empathy theorists claim that at least some aspects of another's experience are perceptually available to us, as these experiences are embodied in their expressive behaviour.3 This is not meant to imply that when I encounter another person that I see bodily changes which merely indicate that they are undergoing some kind of inner experience. Rather, the claim is that when I experience another person, I encounter them as an embodied subject. Their bodily expressivity is a constitutive part of their experience, not just a behavioural cue (Krueger and Overgaard 2012). When I see you smile, I do not infer from this that you are happy. I see your happiness in your smile; your happiness is perceptually given to me 'directly, unmediated, and non-inferentially' (Zahavi 2014, 125). Phenomenologists often point out that social cognition theories that rest on inference or simulation actually presuppose the very experiences they are meant to explain (e.g. Scheler 2008, 7; Stein 1989, 12). For, in order for us to know when to employ Theory Theory or Simulation Theory, we must already have recognised the other as an experiencing subject to whom we ascribe – or onto whom we project – experiences in the first place; must have recognised the other as a subject and their bodily actions as expressive. Consequently, empathy is presented not simply as a form 3 Although the most common examples of empathy refer to emotional experiences (such as seeing someone's happiness in their smile), other experiences can be empathetically perceived (such as seeing someone's intention to pick up their cup when seeing their hand move towards it). I, therefore, use 'experience' here broadly. Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 7 of other-understanding but the fundamental form of other-understanding (Scheler 2008; Stein 1989; Zahavi 2014). The question I pursue below is whether this fundamental form of other-understanding is also found in our online encounters. Importantly, while empathy gives us access to the other's experience, this is not to say that we have full access to their experience. If I see your smile and empathetically perceive your happiness, that is not to say that I perceive your entire experience of being happy. That I do not have full first-personal access to your happiness is precisely what makes it an experience of your happiness and not of my happiness (to which I do have full first-personal access). As such, empathy is understood as my experience of your experience; a structure that preserves the asymmetry between first-personal experience and a secondor third-personal experience of another's experience. Nor do I need to share your happiness, I can perceive your happiness while remaining grumpy myself (Scheler 2008; Stein 1989; Zahavi and Rochat 2015). While proposing empathy as the fundamental way of encountering others and their experiences, we should not mistake empathy theorists as claiming that empathy is our only way of grasping the experience of others (Zahavi 2001, 2014). I might empathetically grasp your disappointment when perceiving your furrowed brow and slouched shoulders but infer that this disappointment relates to the publication of a joblist on which your name did not appear. Nor is the empathy theorist claiming that all of another's experiences are perceptually available. It is possible that you are disappointed about the job-list but that your experience is not expressed in your bodily comportment and is, therefore, not empathetically available to me. Finally, it should be noted that what is being claimed is that empathy is how we experience others. This should not be mistaken for a claim about how well we do this. I can be entirely wrong in my empathetic grasp of you. For instance, I might mistake your grimace for a smile. Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 8 Empathy is like other perceptual experiences in this regard; just because I see something in a particular way does not guarantee its veridicality. Consequently, empathy is meant to capture the fundamental way that we encounter others as embodied and experiencing subjects, but it is not infallible nor is it the only instrument in our social cognition toolkit. 2. Empathy, expressivity and the body 2.1. The objective body and the lived body Before exploring whether empathy can happen online, it is worth expanding upon the relationship between empathy, expressivity and the body. Often not much more is said about this beyond the claim that empathy involves perceiving the other's experiences through the expressive bodily behaviour of the other. Due to this emphasis on the body, we can understand why empathy is usually taken to require a face-to-face interaction. However, by digging deeper into how empathy, expressivity and the body relate, the door is opened for unsettling this assumption. To empathetically perceive another person involves my having an experience of their experience which is perceptually given to me through their expressive body. Common examples of empathy are of seeing someone's happiness in their smile, their anger in their clenched fists, their sadness in their tears (e.g. Scheler 2008; Stein 1989; Zahavi 2014). What is important to note is that while being face-to-face with another can give rise to an empathetic experience of them, it need not do. I can, at least potentially, perceive another's body without having an empathetic experience of them. To understand this, we need to introduce a classic distinction made in phenomenology between the objective body and the lived body. This distinction highlights that there are Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 9 two ways in which we can experience the body, not a claim that we have two different types of body (Gallagher and Zahavi 2013, 136).4 The body, in one sense, is just an object in the world like other objects. It has extension, mass, colour, is subject to the laws of physics, and so on. This is the objective, physical body. I experience my body as an object if, for instance, I examine the texture of the skin on my arm or measure it in some way. However, I also experience my body as a subject. It is not just an object of experience; I am a bodily subject of experience. It is this lived body that phenomenology brings to the fore. When, for example, I reach out to grab my mug, I do not pick up my arm and move it towards the mug. I simply extend my arm without thinking about it. Due to processes such as proprioception and kinesthesis, I am always aware of where my limbs are and what I can do with them without needing to explicitly locate them. This is because my lived body is always given to me experientially from a first-person perspective. As MerleauPonty puts it, 'I am my body' (2012, 151) and I experience it as the centre of my agency and experience. Indeed, that I am a lived body is what enables me to experience my body as an object in the first place; for me to perceive my body as an object, I also must be the subject perceiving it as such. Crucial for our purposes is that I can experience the other's body either as an objective body or a lived body. Think of a surgeon cutting open a body to take out the appendix, a tailor measuring someone's waistband, an artist examining the bright green of someone's eyes in order to create a certain shade of paint. In these cases, the surgeon, the tailor and the painter are all attending to the objective body. 4 For an interesting discussion of whether we should conceive of the objective body and the lived body as wholly distinct, see Legrand 2010. Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 10 Empathy takes place when perceiving someone's lived body. For example, when I look into my sister's face and see tears falling down her cheeks, I experience her sadness through her expressive lived body. Remember, this is not an inferential claim – that I see her tears and infer from them that she is sad – but is a claim about how her sadness is given to me directly. I am not turned to her body as a physical object here. Rather, I have perceptual access to my sister's pain through her lived body. It is her lived body that I experience as a field of expressivity. When I see my sister's tears, I see her sadness; I am not turned to an objective, material body that is secreting water out of tear ducts. Empathy, then, is intimately related to the lived body as something expressive. On the face of it, this accounts for why it is commonly supposed that empathy requires a face-to-face interaction. It might seem obvious that we must be able to see someone's physical body in order to see their expressive, lived body, e.g. to see smiles, tears, clenched fists. However, the lived body is not strictly tied to the objective, biological body (i.e. to the limits of skin and skull). Think, for example, of the voice. The voice, while issuing from their physical body, is not itself merely a distortion of bodily mass. It is part of the lived body but is not skin-bound; we do not hear someone speak by attending to the movement of their voice box. We can hear someone's voice even when we cannot see them. We should not interpret the idea of the lived body as a field of expression, then, as referring to expressivity that only plays across someone's skin. That we can perceive someone's expressive, lived body while not being in the same physical place as their objective body will prove particularly important when exploring the idea of online empathy. That the lived body extends beyond skin and bone is also well-recognised in the phenomenological tradition where a part of the world is incorporated into someone's Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 11 lived body. The archetypal example of this is Merleau-Ponty's description of the blind man and his cane. Merleau-Ponty describes how a blind man uses a cane to navigate his way down a cobbled street, using the cane as way of feeling and navigating the stones in front of him. Merleau-Ponty notes how: [t]he blind man's cane has ceased to be an object for him, it is no longer perceived for itself; rather, the cane's furthest point is transformed into a sensitive zone, it increases the scope and the radius of the act of touching and has become analogous to a gaze. (Merleau-Ponty 2012, 144) The blind man's lived body extends beyond his fingers to the tip of the cane, thus allowing the man to feel the cobbled street where the tip of the cane meets the stones. The cane is no longer experienced as an object that the blind man holds but as part of the experiential field of his lived body. Thus, the lived body, in certain circumstances, can include aspects of the world. This, as we will see, adds an additional layer of complexity to how we understand the lived body as something that could be perceptually available to another outside of the face-to-face encounter. 2.2. Beyond the face-to-face Despite empathy and the lived body being two canonical concepts in the phenomenological tradition, that the lived body extends beyond the skin seems to have been forgotten by empathy literature. This has led to an overemphasis on the importance of being face-to-face with another in order for empathy to take place. Why has the question of whether empathy can take place outside of face-to-face interaction been typically overlooked? In large part, this can be attributed to the trend of talking about being able to 'see' someone's experiences. Examples of empathy abound with descriptions of seeing someone crying, blushing, smiling and scowling (e.g. Krueger and Overgaard 2012; Scheler 2008; Schutz 1967; Stein 1989; Zahavi 2001, 2014). Not Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 12 surprisingly, then, the framing of empathy in visual terms has resulted in a close connection between empathy and face-to-face encounters. That empathy is perception-based does not restrict us to visual perception. Take my hearing someone's anger in their frustrated tone of voice. Expressivity is not something that just plays out across the surface of the skin. Indeed, we can empathetically perceive the other through their tone of voice when we cannot see them at all. We would not want to deny that a blind individual can empathetically experience others. We also want to allow that I can empathetically experience another's anger when they yell at me down the stairs. Being face-to-face with someone is not a necessary condition for empathy, even in these everyday contexts. That the lived body can extend through incorporation also needs to be considered. If we can perceive incorporated objects as part of someone's lived body, they can also form part of the field of expression that is relevant for empathy. While this has not been discussed within empathy literature, it seems clear that we can empathetically perceive incorporated aspects of another's lived body. Imagine that I have been in an accident and lost my left arm. In its place, I have been fitted with a state-of-the-art prosthetic limb which I have become adept at using. Now suppose that you have just walked past my desk, knocked over my mug and spilt coffee all over my lecture notes. I jump up from my chair and shake my fists, both my flesh-and-blood fist and my prosthetic one, at you in annoyance. It seems appropriate to say that you empathetically perceive my annoyance in the shaking of my fists, both of them. It seems artificial to say that your empathy stops where my prosthetic limb starts. Rather, the field of expression that you are empathetically directed to encompasses my whole lived body, including both my physical body and my incorporated prosthetic limb. Aspects of Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 13 the world can, therefore, constitute part of someone's lived body and can be empathetically perceived by another. The lived body, then, can be empathetically perceived 'at a distance' from the objective, physical body. This frees empathetic perception from taking place only within face-to-face interactions and opens the door to the question of whether empathy can take place online. 3. Empathy online: a dead-end? That there has been next-to-no discussion of online empathy might be taken as an indication that this is simply a dead-end endeavour. Indeed, in one of the few phenomenological considerations of empathy online, Thomas Fuchs, in his article The Virtual Other (2014), rules out the possibility that empathy proper takes place online. According to Fuchs, when we go online, we lose our direct empathetic access to others as we no longer have perceptual access to the other's body. He states that online, '[i]nstead of interacting with embodied persons, we interact more and more with pictures and symbols' (2014, 167). As empathy involves perceiving the other's lived body, empathy is deemed impossible in this online world of 'disembodied communication' (ibid.). How, then, can we encounter others online when we leave our bodies behind? Fuchs recognises that virtual communication can still be suffused with emotion. However, what he says is lacking is 'the direct feedback from the embodied contact, based on emotional cues and expressive gestures by which we perceive one another empathetically' (ibid.). Fuchs suggests that we make the mistake of thinking we encounter the other online as we engage in a kind of imaginative form of otherunderstanding, a quasi-empathy 'as if' we were really encountering the other. Instead of Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 14 really encountering the other online, Fuchs states that '[t]he other has become a projection surface, a product of my imagination' (2014, 168). He claims that we do not empathetically perceive the other's experience but project our expectations onto their communication. He contrasts this to face-to-face interactions, where one frequently is surprised by the other's experience (also see Staehler 2014). This is quite a radical stance to adopt. For one, it seems at odds with our lived experience of online encounters. I suspect anyone who has had the misfortune of having an argument with someone over text or the fortune of having a romantic exchange, would strongly resist the idea that we only encounter an imagined other online; such interactions seem permeated with the emotions of the other person, emotions that can take us aback, that can teach us something about the other's experience. Yet, for Fuchs, these are nothing more than fictional emotions; emotions that I imagine and project onto the other based upon mere informational prompts (ibid.). While we might want to resist Fuchs' conclusion that we only encounter ourselves online, the idea that in online space what takes place is 'disembodied communication' has some intuitive appeal. It seems a banality to point out that when I pick up my phone and open WhatsApp that I do not take my body online. This is exactly the point that Dreyfus makes when he states that when we enter online space 'we leave behind our emotional, intuitive, situated, vulnerable, embodied selves' (2008, 6). This abandonment of the body might seem like a knock-down argument against the idea of online empathy, pushing us to look for another method for analysing how we experience others online. However, before throwing ourselves into formulating an alternative approach to online sociality, we need to re-examine what Fuchs and Dreyfus appear to be saying. The claim that these authors seem to be making is that because we cannot take our Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 15 physical bodies into online space, we cannot, therefore, encounter the other as an embodied subject online. Fuchs states that in 'virtual worlds there is a suspension of immediate bodily experience, a disembodiment' (Fuchs 2014, 165); while Dreyfus' entire book is motivated by a concern about what happens when we enter online space and 'the body goes' (Dreyfus 2008, 7). They both seem to assume that because we leave behind our physical bodies when we enter online space, we become disembodied. But, as we have discussed, the physical, objective body is not the same thing as the lived body and the lived body is not tied to skin and bone. More needs to be said to justify the move from saying that the physical, objective body cannot enter online space to saying that the lived body cannot enter online space. Simply put, there is a conflation of the objective body with the lived body in both Fuchs' and Dreyfus' accounts. As empathy involves perceiving the other as an embodied subject through the field of expression of their lived body, pointing out that the objective body is not online is not sufficient to rule out that empathy can occur online. The question that we need to return to, then, is whether the lived body can go online. When pursuing this question, we must also be careful not to refer to online space as if it were one homogenous realm in which we have one style of interpersonal interaction (Krueger and Osler forthcoming). The internet houses numerous platforms offering different forms and styles of encounter: whether this is on live video, on textbased platforms, through avatars and so on. We are not talking about the online interpersonal encounter but various types of online encounters. While I cannot offer a full taxonomy of online encounters here, in the following, I focus on specific types of online encounter to motivate the case for online empathy. Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 16 4. Empathy online: two case studies 4.1. Skype: direct but technologically-mediated perception I am skyping with Frida. When we are on a video call, rather than being physically faceto-face, our encounter is mediated by a pair of screens.5 Frida is not literally in front of me in a physical, flesh-and-blood sense. Yet, it seems strange to claim that we are not 'face-to-face'. There is her face on my screen. Can I empathetically perceive Frida's lived body here? It does not seem that Frida has incorporated her laptop's camera into her lived body; she does not experience the camera as part of her body in the same way as the blind man and his cane. So, when I perceive her face on my screen, I am not empathetically perceiving an incorporated part of her lived body. We might think that there is an analogy here to be made with empathetically hearing someone's anger while being in a different room. However, this doesn't straightforwardly fit either. Frida's body is not simply something that I am perceiving from far away. My perceptual access to her body is mediated by the screens, microphones and speakers between us. The question, then, is whether we can empathetically perceive someone's lived body when our perceptual access to their body is technologically-mediated. I think the answer to this is clearly yes. I still experience Frida's expressive behaviour: I can see her happiness play out across her lips, her enthusiasm in her gesticulating hands, hear her amusement in her chuckling. Although there are screens mediating us, it is not to my screen that I am perceptually directed; rather, I am perceptually directed at Frida (Osler, 2019, 14). The screen is transparent in the sense that I am not attending to it as a 5 I take it that this example also applies to video link platforms such as Zoom, WhatsApp video, FaceTime, Houseparty, and so on. Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 17 screen but attending to Frida. While Frida's body in all its full physical corporeality is not in front of me, I experience her lived expressive body as in front of me. There might be a concern here, though, that I do not really have access to Frida's lived body here but to an image or representation of Frida given to me via the video link. In Husserl's work Phantasy, Image-Consciousness, and Memory (2005), we find an in-depth analysis of image-consciousness that might be considered a challenge to my claim that I directly perceive Frida's expressive bodily gestures over Skype.6 Here, Husserl provides a detailed discussion of our perception of images, such as drawings, paintings, and photographs. In short, Husserl argues that when we look at an image of an individual we do not directly perceive the subject depicted in the image (what he calls the 'image subject') nor the material object that depicts the image (unless we are inspecting this material as an object), e.g. the photographic paper or canvas (what he calls the 'physical object'). Instead, Husserl claims that we are turned to the object depicted by the image (what he calls the 'image object'). He states that when we look at a photograph of a child, what we perceive 'is not the child itself but a photographic image' (2005, 20). Based on this analysis, one might claim that what I have access to on Skype is the image of Frida which, although mapping onto Frida's actual bodily expressions, is not in fact a perception of Frida. This seems to suggest that I cannot have direct empathetic perceptual access to Frida's lived body over Skype.7 The analysis that Husserl offers of image-consciousness is a complex and layered one, which deserves more careful attention that I can do justice to in this paper. 6 Thank you to the reviewer who helpfully summarized this particular concern and suggested including a more developed response to this challenge in this paper. 7 For an interesting discussion of how empathetic perception itself is akin to imageconsciousness, see Luo (2017). Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 18 Nevertheless, I want to provide an initial response to a challenge posed along these lines. There are a number of contrasts that Husserl draws between imageconsciousness and ordinary perception.8 I want to highlight three of these contrasts here. First, Husserl claims that in ordinary perception we see the object as real and actual. In contrast, he states that when we look at an image, we do not posit the image object as real, as actually existing. When we are looking at a photograph of a child, we are not 'deceived' into thinking that we are perceiving a real child (Husserl 2005, 155). Second, Husserl points out how there is a conflict between what appears in the image and the surroundings of the image. In the image, we see, say, a child in a garden. But the photo itself is pinned to a wall, and the internal world of the photograph is at odds with my orange wall, the staircase in my house, and so on. The image 'has its own space and time set apart from the space time of reality. It embraces a world internal to it and is discontinuous to the reality around it' (Brough 2012a, 549). Finally, the image object, unlike objects of ordinary perception, is exhausted by its appearance. When looking at the child, I can only see what is presented to me (e.g. their face). It lacks the spatial horizons that ordinary objects have; I do not experience the image child as something that I can explore the back side of. Image objects are, as it were, characterized by a 'what you see is what you get' staticism. We do not, for instance, look at a photograph and have 'expectations of movement and speech' (Eldridge 2018, 576). Importantly, I think that the three contrasts specified above do not (at least straightforwardly) apply to the case of Skype. First, unlike when I look at a photograph, 8 For a more detailed discussion of Husserl's account of image-consciousness see: Brough 2012a, 2012b; Eldridge 2018; Luo 2017. Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 19 I do experience Frida as existing, as real. This is not just because I know that the image refers to or resembles a real person out there in the world. Rather, I experience myself as having access to the real Frida, existing in the present moment. Indeed, what our Skype example complicates is what we mean when we say that in ordinary perception we experience objects as real and actually existing. Unlike photographs, which capture the 'That-has-been' (Barthes 2000, 73) or the 'not now' (Husserl 2005, 155), when we talk on Skype, I experience Frida as she is in the present moment. This leads us to the second point. While a photograph has its own internal time and space, this again is not clearly the case on Skype. First, as mentioned, Skype does not present me with an image of a past-Frida, but with Frida as she currently is. Frida and I share the same temporal present. To borrow Schutz's phrase, Frida and I 'grow older together' in a shared, lived time (1967, 103). Interestingly, I think this also applies to space. On Skype, I see Frida in her bedroom. Certainly, I am not in Frida's bedroom with her, I am not sharing the same physical space as her. Yet, I am not presented with a flat, frozen space that lacks horizonal possibilities. Frida herself is not hemmed in or constrained by the space that I can see through my screen. The space that is framed on my screen does not stop at the edges of the screen. Frida can get up and walk out of frame or Frida can turn the screen so I might see another aspect of her room. This is more analogous to the experience of looking out of a window. When I look out into my garden, my view of the garden is framed by the window and the garden is, in some sense, discontinuous with the grey curtains and beige walls of my study. Yet, I do not see the garden as having its own 'internal space'. What I see is a space of possibilities that I, currently, cannot physically access. What this suggests to me is that rather than seeing the screen as equivalent to a photograph that we see an image in it is more akin to a window that I see Frida through. This is also supported by the final point that I perceive Frida, unlike the Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 20 photographic image of the child, as having spatial horizons that can be explored. Part of my experience of Frida here on Skype is that she does have a back side that I can explore further (even if I cannot physically walk around her in actuality); I see a Frida who is not reduced to a mere appearance. What I think this analysis reveals is that my screen operates more like a window than an image. I do not have an internally-referring space represented on my screen, rather I am given access to Frida's lived space, the space that she is currently occupying; a space that is within the horizonal possibilities of our shared world. This renders my perception of Frida on Skype of a different sort to image-consciousness as Husserl characterises it. How, then, should we think about the experience of perceiving someone over video link? I think that Husserl's own work contains a potential pointer here. As noted, Husserl states that when we perceive images, we are not fooled into thinking that they are real. Yet, he does acknowledge that certain technologies do deceive us: Deception and sensory illusion of the sort belonging to panorama images, cinematographic images, and the like, depend on the fact that the appearing objects in their whole appearing state are slightly or imperceptibly different from the objects appearing in normal perception. One can know in these cases that they are mere image objects, though one cannot vitally sense this. (Husserl 2005, 146, my emphasis) Sadly, Husserl does not provide a detailed analysis of cinematic perception. However, what I take him to be saying here is that when we see cinematic images that so perfectly resemble normal perception, that the structure of our experience here is not that of image-consciousness but is better spoken of as analogous to illusion or hallucination. This, I think, fits our experience of Frida over Skype; although I am seeing Frida in a mediated manner, I do experience her as really there (albeit that she is not in the same Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 21 physical room as me). This may be some kind of illusion but it has the same structure as ordinary perceptual experience. Note that this is not to say that there are no differences between technologically-mediated perception and ordinary perception. As I will explore in more detail below, certain sensory possibilities are denied to me on Skype, for instance, I cannot touch Frida. However, I think these differences, while interesting, do not warrant the claim that I only perceive an image of Frida. Having defended the claim that I perceive Frida over Skype, the question now is whether we are justified in saying that this amounts to the kind of direct perceptual access that we require for empathy? As discussed, empathy is often described as a direct and unmediated experience of the other (e.g. Stein 1989, 24; Zahavi 2014, 167). Here, although I am turned towards Frida's lived body and not to the screen, her lived body is technologically-mediated. Does this mean that we are prevented from saying that I empathetically perceive Frida's happiness over Skype? I argue not. When we talk about directly perceiving someone's lived body in the context of empathy, what is direct is my experience of their experience. 'Direct' is meant to pick out that we do not use indirect methods such as inference or simulation to grasp the other's experience. I perceive their experiences in a direct and immediate manner. Contra Fuchs, I still directly perceive Frida's bodily happiness in her smiles and laughter. I do not suddenly infer or simulate her happiness because her lived body happens to be technologically-mediated. While empathy is always direct, in the sense of not being inferential, I argue that it need not always be unmediated, in the sense that the other's lived body is not given to me in an unmediated fashion. Online, then, we can have a direct but mediated empathetic encounter of the other's lived body. Remember, moreover, that according to empathy theorists we only use tools such as inference or simulation when we have already empathetically experienced the other as an embodied and expressive subject. In Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 22 order to start inferring or simulating Frida's experience, I must have first empathetically grasped that her smiles and gestures are expressive movements of an embodied subject. This case-study, I think, establishes that there is at least one instance where it makes sense to speak about empathically perceiving the other online. I now want to turn to a more controversial case of empathetically encountering the other online: texting in the form of instant messaging. 4.2. WhatsApp: text incorporated I am messaging Diego on WhatsApp. We are using text interspersed with emojis to chat about his love life. WhatsApp also indicates when Diego is online ('online'), that he has read my messages (two blue ticks), when he is typing ('typing...'), and when he was last active ('last seen today at 11:03'). I cannot, however, either see or hear Diego's body or voice. Yet, I can 'hear' the excitement in his messages, the over-enthusiastic tone, his desire to talk about his new partner Carl. But does it make sense to talk about my having an empathetic perception of Diego's happiness here? On the face of it, this seems like a starkly different case to skyping with Frida. On Skype, Frida's expressive behaviour is empathetically available because I can see her face, hear her voice; her lived body is there on my screen. With Diego, what is on my screen are letters and basic pictures. How could this amount to having access to Diego's lived body? Although he does not specify, it is likely that it is these kinds of interactions that Fuchs has in mind when he talks about how online we encounter one another via pictures and symbols. According to Fuchs, we do not have access to Diego as an embodied subject. I want to challenge this. To do so we need to understand how words as speech, while not a part of our physical body, are incorporated into the field of expression of our lived body. Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 23 Merleau-Ponty famously claims that 'speech accomplishes thought' (2012, 183). In stating this, he distances himself from the idea that speech is merely the externalization of inner thought. He argues that we should not think of speech as a sign of thought, the way that smoke is a sign of fire. Rather, it is through speaking that we think and feel. For example, think of the occasions where we talk through our thoughts and feelings in order to work them out. This accounts for how 'my words can surprise me and teach me my own thought' (Merleau-Ponty 1964, 111); they are not signs of something ready-formed but are part of the thought itself. Already we can see a resemblance between how Merleau-Ponty depicts speech and how we have discussed bodily expressivity as a constitutive part of experience. Indeed, Merleau-Ponty makes precisely this analogy when stating that speech itself is gesture (2012, 187). Not only is speech constitutive of thought but speech is itself embodied and expressive: 'Speaking is (also) something a human organism does with her body, like dancing, gesturing, grimacing, screaming, singing, etc.' (Colombetti 2009, 9). It is through speaking that we express our thoughts and feelings and through speaking that our experience is intersubjectively available: For the speaker, then, speech does not translate a ready-made thought; rather, speech accomplishes thought. Even more so, it must be acknowledged that the person listening receives the thought from the speech itself. (Merleau-Ponty 2012, 183) Speech, as something embodied, is accessible to others; the speaking subject is always an embodied subject. As Husserl notes: 'the hearer perceives the speaker as manifesting certain inner experiences, and to that extent he also perceives these experiences themselves' (1970, 278). Speech, then, forms part of the field of expression of our lived body that can be empathetically perceived. As a listener, I perceive your experience Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 24 through your speech. Thus, when I hear your anger, I not only hear the tone, I also hear your anger through your words. I do not need to infer what your words mean; I do not need to endow them with sense. I directly hear your words saturated with your experience, with your meaning. I am not directed to your words as objects, as a script, but at what you are expressing. I hear what you are expressing because I am attending to your subjective, lived, expressive body not to a physical body emitting noises. Words do not form part of our lived body simply by virtue of being part of our physical body.9 However, as discussed in section 2, tools can be incorporated into our lived body when they are used in a way that they become transparent, come to shape and mould our experiences. The blind man's cane is incorporated into his lived body as it extends his field of perception, experiences the cane not as a separable object but part of his being in the world. Moreover, incorporated objects can be perceived by others as part of the expressive field of someone's lived body. Language and words are also 'part of our equipment' (Merleau-Ponty 2012, 185) which we incorporate into our lived body. Just as the blind man's experience extends to the tip of the cane, so I experience my words as part of my experience not a mere container for it. Indeed, language is so deeply incorporated into our subjective experience that we literally struggle to think of what it would be like to be without it. How, though, does this help us in relation to texting? Texting involves written words, not spoken ones.10 Diego's texting also accomplishes his thoughts and feelings, 9 Note that simply being part of the physical body is not sufficient for something to be part of our expressive, lived body. My liver, for example, is part of my physical body but does not form part of the field of expression of my lived body. 10 Note that I will not deal with all forms of writing here, though this obviously opens up questions not only about empathy in other cases of writing such as letters or emails. I think, however, there are interesting divergences across these forms of writing that affect Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 25 he discovers his own feelings for Carl as he tells me about him, realises his own excitement in his fast-paced messaging. Texting, like speaking, can be a constitutive part of his experience, is incorporated into his lived, expressive body. What is more, his texts are perceptually available to others. Just as when I listen to someone speaking, when I am reading Diego's messages, I am not (usually) directed at the words but at what Diego is saying to me. What is special about texting is that we typically text in a style that is very close to face-to-face conversational style. We take turns, use informal language, engage in synchronous and reciprocal interactions (Ben-Ze'ev 2004; Baym 2015; Garde-Hansen and Gorton 2013). Moreover, 'text-based media afford many ways to express emotion. We use emoticons to signal friendliness, we use punctuation and capitalization to insert feeling, we use informal language and talk-like phonetics spellings to create an air of conversationality' (Baym 2015, 13). I experience Diego's voice given to me in his effusive words, his excited tone, the rapid style of responding, and so on. While I do not see his excitement play across his face in smiles, I experience it in his texting. The pace of Diego's messages, the patter of his speech, his choice of words, his use of emojis and wild punctuation all form part of the field of expression I directly perceive. The style of his texting has a certain 'vitality' (Stern 2010) to it that is not contained in the texts but unfolds through the texting itself, giving his messages a certain expressive tone. Like the hearer, I, as a reader, perceive Diego's excitement through his texting, which forms part of his lived, expressive body. As Kekki puts it: whether we can empathetically grasp the writer (e.g. time, reciprocity between writer and reader, the dynamics of the interaction). I restrict myself here to texting, which, in many ways, most closely resembles spoken, face-to-face communication as it often unfolds in the present in a dynamic and interactive manner, in informal language. Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 26 'the words we read or listen to become 'lived' for us...[and] we perceive the words as expressions of their producers' (2020, 8). Contra Fuchs, Diego's texts are full of emotional cues and expressive gestures which I perceive as his messages unfold on my screen. This motivates the claim that not only do we empathetically perceive the speaking subject but also the texting subject. To reduce Diego's messages to disembodied signs and symbols misses the way we experience speech (either spoken or texted) as expressive. Indeed, that we do experience Diego's messages as expressive at all should itself prompt us to understand that empathy is at play here. Again, recall that empathy theorists argue that empathy is the fundamental form of other-understanding, for unless we recognise someone as a subject, recognise their behaviour as expressive, we would not know to whom to project experience onto. If Fuchs is right that we project emotions onto the text of the other online, we must have first empathetically experienced their words as expressive in the first place. In Husserl's words: 'the possibility of sociality, the possibility of comprehension, presupposes a certain lived-bodily intersubjectivity' (Husserl 1970, 297) and this applies just as much to the online world as the offline one. Consequently, even in cases of texting, I argue that we empathetically perceive the other. 5. Exploring the limits While I have argued that we empathetically perceive others online, there are differences between online encounters and face-to-face ones. The reduced perceptual richness of these interactions, as well as their temporal structure, may be grounds for challenging my claim that empathy occurs online. I consider these challenges below and argue that while these considerations help us explore the limit cases for empathy, they do not undermine online empathy per se. Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 27 5.1. Perceptual richness One might be concerned that when we go online and encounter the other's lived body in a technologically-mediated manner, the perceptual richness of the face-to-face interaction is lost, thus impacting one's ability to empathetically perceive the other online. A badly connected Skype call renders the image of the other pixelated and blurry. Even when the connection is good, it does not have the same perceptual clarity as a face-to-face interaction. Also, while we have visual and auditory access to the other over Skype, we cannot apprehend them with all our sensorial capacities; for example, I cannot (yet) touch Frida through the screen. Our perceptual grasp of Diego is even less rich, as we are limited to perceiving him via text. There might be a certain amount of perceptual richness that is needed for empathy. Perhaps this threshold is not met in online encounters. Note that strictly speaking neither of these concerns rule out the idea of online empathy. If we lived in a world where our perceptual access to the other over Skype was as perceptually rich and multi-sensorial as face-to-face, this would allow for online empathy. Nevertheless, these concerns might suggest that online empathy is not currently possible. While we cannot grasp people using our full suite of sensorial capacities over Skype or WhatsApp, this full suite is not necessary for empathy. I can hear someone's anger in their voice if I am blind or see their anger if I am deaf. To empathetically perceive someone does not require that I perceive them with all my sensory capacities. Indeed, such a position would have worryingly ableist implications. This challenge does not, therefore, find its mark as we should not apply a higher standard of perceptual richness for online cases compared to offline ones. What about the claim that the quality of the other's technologically-mediated body may not be sufficient for empathy? For me to see your happiness in your smile, I Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 28 must be able to see your expressivity with sufficient detail. While this is doubtless the case, again we must be careful not to apply a standard to online encounters above offline ones. Point-light display experiments suggest that even with minimal perceptual input, we perceive certain actions, emotions, intentions of others (e.g. Pavelova et al. 2001; Runeson 1985). Such experiments indicate that we can empathetically perceive another's experiences even when our perceptual access is pretty sparse. It seems unlikely that we will be able to provide an exact cut-off for when someone's lived body is no longer available to me with enough perceptual richness for empathy. Factors such as how well I know someone, what the empathized person is doing, and so on, are likely to all play a role in this. Intense anger might be perceived with relatively low perceptual richness, while pity might require greater perceptual richness. I might be able to detect my partner's disgruntlement just by catching a glance of them out of the corner of my eye in a way I could not do with a stranger. It is helpful to think of the required perceptual richness as being on a spectrum; where, typically, as I experience someone with less perceptual richness, the less likely I am to empathetically perceive them. There might be a transition when the quality of the Skype call is so blurry that I can no longer empathetically perceive Frida. Perhaps at this point I perceive the pixelated screen and not Frida's lived, expressive body. As we move down the scale of perceptual richness, it also seems likely that I am going to be more prone to 'getting it wrong'; where the screen is blurry I seem more likely to mistake Frida's grimace for a smile. One last thing should be said here. I have implied that more perceptual richness is always better for empathy. There are a number of reasons to question this. One such reason is that what counts as perceptual richness is not particularly clear. If perceptual richness just means 'more', in some cases, it may well inhibit our empathetic Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 29 capabilities. If you are shouting in my ear, any subtly of tone is likely lost, thus impeding my empathetic perception of you. Likewise, getting extremely close to your mouth is not going to give me a better empathetic grip of your smile. People, like paintings, might have an optimal distance from which they are viewed. Using MerleauPonty's (2012) terminology, it seems possible that there is an 'optimal grip' that we can get of people that is not just a question of as much perceptual intensity as possible. Furthermore, what someone's optimal grip of another is, is going to be different for different people. This can be brought out, in particular, when we consider individuals who find a lot of perceptual stimulus distracting or overwhelming (e.g. individuals with autism). For some people, perceptual access to someone's lived, expressive body that is not too perceptually rich may well aid empathy, rather than inhibit it. For some, it might actually be easier to empathetically perceive others online than offline. 5.2. Temporality On Skype, the connection can stutter and glitch. Frida's face can freeze, waiting for the connection to re-establish and for her to be given to me in live time once again. Over WhatsApp, there are delays in the messages sent, received and read by me and Diego, from seconds, to minutes, to hours. In classic phenomenological discussions, empathy is characterized as my present experience of someone else's present experience (e.g. Stein 1989, 2). Could time-lag jeopardize my empathetic experience of Frida and Diego as I am not perceiving them with the same immediacy as face-to-face? Technically, I never perceive anyone's expressive lived body instantaneously; there is always a time-lag between someone smiling and the time it takes for the light to reach me, allowing me to perceive their smile. Nevertheless, in our Skype and WhatsApp examples, the time-lag is more significant than this. Could a temporal delay Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 30 prevent or disrupt empathy because we are not perceiving someone's present experience? Discussions of online encounters put pressure on the idea that empathy is directed at the other's present experience. We are forced to ask the rather odd-sounding question of 'how long is the present?'. Face-to-face, we allow that I perceive your present experience despite the micro-delay in light and sound waves reaching me. Can this micro-delay be stretched somewhat while still allowing us to experience the other's present experience? When I ask Frida how her day was and her reply reaches me with a slight delay created through the technological-mediation, my empathetic perception of her lived body is slightly out-of-sync with her embodied experience of happiness. I still perceive her bodily enthusiasm and it seems that I still experience her enthusiasm as her present enthusiasm. How far, though, can we push this temporal delay? What about Diego's texts, which I might read minutes or hours after he sent them? What if Frida sent me a video-recording of her chatting that I viewed the next day? We are faced with three possible routes with respect to this challenge. One: we accept that where we find temporal delay that is longer online than face-to-face, this does not count as empathy, as we are no longer experiencing the other's present experience. Such time-lag occurs on Skype where the connection is not fast enough and likely threatens all text-based communication. However, in adopting this position, one would need to defend why face-to-face interaction sets the limit on what counts as the 'present' in a non-arbitrary way. It also needs to account for why it seems like I am directly perceiving Frida and Diego's happiness online, despite their happiness potentially being in the past. Two: we do away with the idea that we can only empathise with someone's present experiences. This would allow us to say that if I watched a video-recording of Frida, I still empathetically grasp her happiness in her smile, even Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 31 though it is her past happiness. This leaves us facing questions about whether it makes sense to talk about directly seeing a constitutive part of someone's experience, where the rest of their experience has been-and-gone. Three: we phenomenologically investigate what we mean by 'present experience' in more detail. Note that routes two and three are not mutually exclusive. I cannot provide an answer to this here. However, I want to raise two points in favour of pursuing route three. First, experience is not static. Experience does not happen in freeze-frames but temporally unfolds. Happiness does not happen in a snapshot moment but is temporally extended. The window for my empathetically perceiving someone's present happiness may, then, be longer than we first suppose. Second, our experience of sharing an interpersonal-temporal-present with others may be shaped by our normative expectations. In conversations between neurotypical individuals, response times are usually quite fast. Individuals with autism typically respond with a longer-than-average time-delay (Leary and Donnellan 2012). This can lead to neurotypical individuals experiencing conversations with those with autism as stilted, as not unfolding smoothly in the present moment. Yet, neurotypical individuals can adjust their expectations and no longer experience the time-delay as awkward or disruptive (Krueger 2019). I suggest that one way to interpret this is that a time-delay is no longer experienced because the neurotypical's experience of what falls within their shared temporal-present is extended based upon modulated expectations of response times. Our perceptual experience of what is present might, then, be shaped by our expectations. When engaging in online communication we might also have altered expectations of what constitutes the present moment. If this is the case, even though I read Diego's messages minutes after they were sent, I might perceive his experience as being part of our shared temporal-present. These considerations, I think, motivate a more extensive Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 32 phenomenological analysis of how we experience the temporal-present online and at least staves off the challenge that empathy cannot occur online until such research is done. Conclusion This paper started with the question of how we can encounter others online when we are not face-to-face. I have shown that concepts found at the heart of the phenomenological tradition can be applied to the online sphere to answer this question. I have clarified that as empathy involves perceiving the other's expressive lived body, and the lived body is not tied to our physical body, we can empathetically encounter others outside of face-toface interactions. Having decoupled empathy from face-to-face interactions, I rebuffed the idea that we engage in disembodied communication online by drawing on the phenomenological distinction between the objective and the lived body. Although the objective body cannot enter online space, I have presented two ways in which the other's lived body enters online space and can be empathetically perceived: first, in cases where our face-to-face encounter is technologically-mediated over video link and, second, by showing how the other's texts, as speech, can be part of the other's lived body. By establishing that empathy takes place online, we can defend the idea that we really do encounter one another online, and not merely imagined others. Moreover, through phenomenological analysis of online encounters, we enrich our concept of empathy by feeling out its limit cases. In online interactions, issues such as perceptual richness and the temporal structure of other-experience, that have passed under the radar in face-to-face interactions, come to the fore. This is, though, not to say that we experience others online in exactly the same way as we do face-to-face. There might be Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 33 multiple other factors at play that make the face-to-face encounter special is some way. Indeed, if we think about empathy as something that happens on a spectrum, where I can have a better or worse empathetic grasp of the other – perhaps with simply recognising someone as an embodied subject on one end of the spectrum and empathetically perceiving a close friend and grasping a range of subtle emotions and experiences through their personal style of gestures, tics, expressions, and vitality enriched by my intimate knowledge of them at the other end – it should be noted that I have not suggested that this full empathetic range is available online. I am making the more restrained claim that at least some level of empathy is available in certain interpersonal encounters online. Having made this initial move, we are now in a position to pursue further considerations about the differences and similarities of empathetic encounters online and offline. As emphasised above, the online realm is not a homogenous space and contains many different styles of interpersonal interaction. I have, however, claimed that our foundational form of other experience, empathy, takes place in certain online settings. This opens the door for a plethora of other online interpersonal experiences that presuppose empathy; e.g., inference, simulation, projection, sympathy or shared experiences.11 Phenomenology, with its extensive research on intersubjectivity and sociality, along with its sophisticated analyses of embodiment, temporality, and spatiality, has much to add to our philosophical considerations of online experiences. Acknowledgements: I would like to thank Joel Krueger and Tom Roberts for their helpful comments on an earlier draft of this paper and for the many conversations they have endured 11 See Kekki (2020) for a discussion of online learning and Osler (2019) for an exploration of how we can have shared experiences and a sense of togetherness with others online. Osler, L. Forthcoming. Taking Empathy Online. Inquiry. 34 about Skype, WhatsApp, images and perception. I would also like to thank the blind reviewer who took the time to review my paper and provided extremely thoughtful feedback. Funding details: This work was supported by the AHRC South, West and Wales Doctoral Training Partnership under Grant Number AH/L503939/1. Disclosure of interest: The author reports no conflict of interest. References Barthes, Roland. 2000. Camera Lucida, trans. R. Howard. London: Vintage Books. Baym, Nancy. 2015. Personal connections in the digital age (2nd ed.). Cambridge: Polity Press. 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The Romanian Journal of Analytic Philosophy Vol. VIII, 1°, 2014, pp. 21‐31 POLITICAL IMPLICATIONS OF HUMOR Dan PANAET* Abstract : This paper discusses some political implications of humor, using as a point of departure the mechanisms that explain the sources of the comical. First, I briefly present the main explanations offered for why we laugh. I then focus on the cognitive view proposed Hurley, Dennett and Adams, according to which humor carries out the epistemic function of eliminating the errors that covert‐ ly entered a mental space. In the second section of the paper, I present two ac‐ counts of how liberalism continues to extend the scope of individual liberties. I use these views on liberalism as a background for my analysis of the politi‐ cal implications of humor, advancing the claim that, as a result of its epistemic function, humor has a strong conservative bias. Keywords : humor, liberalism, conservatism, oppression, redescriptions. I. INTRODUCTION Humor is usually considered an effective strategy of fighting off polit‐ ical tyranny. Because it sparks a visceral reaction, it is often seen as a di‐ rect and honest way of conveying a political message. However, humor has a rather dual condition, as it can be employed to support both progressive and conservative causes ; it can be used to propose new individual rights in the same measure in which the establishment may employ it to prevent emancipating measures from being carried out. The aim of this paper is to explore some moral and political implications of using humor as a rhetori‐ cal weapon. Looking more closely at how humor functions, the paper aims to give an answer to the following interrelated questions : How does humor defend individual rights and when does it forestall their implementation ? Considering the mechanisms of humor, what does it mean to make an edgy or an over the line joke ? * Dan PANAET, Faculty of Philosophy, University of Bucharest. Contact address : danpanaet@ gmail.com. 22 Dan Panaet The argument of this paper will unfold as follows. First, the paper brief‐ ly presents the main theories that attempt to explain the sources of humor, using these accounts of how a joke works as a point of departure in the en‐ deavor concerning the ethics of humor. More exactly, the aim is to bring to light what the internal mechanisms of humor can tell us about the political implications of using humor as a rhetorical weapon. In the second part, the problem of political change is tackled, focusing on how it relates to the modification of moral perceptions. First, the article brief‐ ly discusses two accounts of how liberalism continues to extend the scope of individual liberties. Secondly, it emphasizes how the moral problems of hu‐ mor pose a threat to the advent of liberalism. II. THE SOURCES OF HUMOR The Incongruity Theory is arguably the most popular explanation of why we laugh. According to this account, mirth ensues when something violates our expectations and an incongruity is perceived. Humor arises from the sud‐ den encounter with an unexpected and innocuous element. Kant, one of the supporters of this approach to humor, explained the internal mechanism of a joke in the following way : „In everything that is to excite a lively convulsive laugh there must be something absurd (in which the understanding, therefore, can find no satisfaction). Laughter is an affection arising from the sud‐ den transformation of a strained expectation into nothing. This transformation, which is certainly not enjoyable to the understand‐ ing, yet indirectly gives it very active enjoyment for a moment." (Kant 2007, 133) According to Schopenhauer (Morreall 2012), another proponent of the in‐ congruity theory, humor arises when we discover an incongruence between an abstract concept and an object that was supposed to fall under it. A dis‐ crepancy is thus suddenly found between what is thought and what is per‐ ceived. Another popular theory of humor emphasizes its unpleasant social im‐ plications. This approach is known as The Superiority Theory and focuses on the existence of a butt of the joke whose inferiority is found comical by those who suddenly discover it. The most famous articulation of this view belongs to Hobbes, according to whom „'sudden glory' is the passion which maketh those 'grimaces' called 'laughter' ; and is caused either by some sudden act of their own that pleaseth them, or by the apprehension of some deformed thing in another by comparison whereof they suddenly applaud them‐ selves." (Hobbes 1997, 36) 23 Political Implications of Humor According to this view, humor is malicious and undemocratic, as it tends to establish hierarchies. However, Hobbes finds humor blameworthy for stra‐ tegic reasons, claiming that when one indulges himself in the pleasures of humor, he will tend to surround himself with weak people, whose inferior‐ ity will make him laugh. (Hobbes 1997, 36) Yet another approach to humor is The Release Theory. Freud, maybe its most prominent defender, claimed that through laughter we release the nerv‐ ous energy that has been suddenly rendered useless, as we have stopped per‐ forming the psychological task for which this energy was summoned in the first place. A clown's clumsy actions spark laughter because we suddenly realize that we don't need the nervous energy we have initially summoned for understanding his behavior. (Morreall 2012) A more recent strategy of explaining humor consists in uncovering its evolutionary roots. These approaches are radically different from the afore‐ mentioned theories, as an evolutionary explanation tries to identify not the essence of humor, but a particular form of humor that has enhanced the re‐ productive success of individuals. For example, R. D. Alexander (Polimeni, Reiss 2006, 351) claims that humor generated an advantage in reproduction by raising the social status of the individuals who tell jokes. However, it is obvious that not all forms of humor can be explained in terms of raising the social status of the person who produces it. It would be very difficult to un‐ derstand how a clown raises his social status by being clumsy. Evolutionary approaches to humor tend to see the forms of humor that do not fit their ex‐ planation as mere exaptations. For this reason, an explanation like the one offered by Alexander is immune to counterexamples and hence logically im‐ possible to disprove. A more empirically laden theory of humor is the one advanced by Ramachandran, according to whom the laughter is the sign of a false alarm (Polimeni, Reiss 2006, 351). Ramachandran claims that „the main purpose of laughter is for the individual to alert others in the social group that the anomaly detected by the individual is of trivial consequences" (Polimeni, Reiss 2006, 351). He cites examples in which people burst into laughing when they suddenly realize that what they initially thought was a threat is in fact an innocuous presence. Polimeni and Reiss (Polimeni, Reiss 2006, 351) notice that theories of hu‐ mor are not mutually exclusive, as they focus on different aspects of this is‐ sue. The aforementioned theories offer a rather fragmentary view of the sub‐ ject, each of them focusing on a single feature of humor and utterly ignoring the others. For example, the incongruity theory emphasizes the cognitive fea‐ tures of humor, while the superiority theory brings into focus its social use. However, none of the two theories does explore the link between incongru‐ ity and the tendency to create social hierarchies, so as to put these two fea‐ tures in the same picture. 24 Dan Panaet For this reason, I will now turn my attention to a more comprehensive theory of humor. Hurley, Dennett and Adams advanced a cognitive‐evolu‐ tionary explanation for why we laugh, emphasizing how enjoying a joke is akin to problem solving (Hurley, Dennett, Adams 2011). According to them, humor evolved because of its epistemological use. Thus, the pleasant sensa‐ tion of mirth is the reward we get for identifying an error that covertly en‐ tered our conscious mind. In other words, humor evolved because the cog‐ nitive mechanism underlying it served for debugging the mind. Mirth may occur, for example, when, after frantically looking for our glasses, we suddenly discover we had them on top of our head the entire time. The sensation of mirth is the reward we get for discovering the incon‐ gruity and eliminating the false belief. Without such a reward, it would prob‐ ably be more difficult to make sense of what just happened. According to Hurley, Dennett and Adams, „humor occurs, when 1.an active element in a mental space that has 2.covertly entered that space and is 3.taken to be true within that space 4.is diagnosed as false in that space – simply in the sense that it is the loser in an epistemic reconciliation process ; 5.and the discovery is not accompanied by any (strong) negative emotional valence." (Hurley, Dennett, Adams 2011, 121) For example, if we reduce a joke to a setup and a punch line, the former will covertly comprise an error, which will be suddenly disclosed by the punch line. The setup of the joke usually suggests a direction that the punch line will unexpectedly prove wrong. We should notice that this explanation of humor also encompasses some of the theories I mentioned earlier. The reaction of laughter we have when suddenly realizing that what we thought was a threat is in fact innocuous can be explained in terms of the scheme proposed by Hurley, Dennett and Adams. Thus, the belief according to which we are in danger is suddenly identified as mistaken. Their view can also be seen as a more specific variant of incongruity theory. For this reason, I will take it as a point of departure in my attempt to analyze the moral consequences of humor. At this point, the article aims to give an answer to the following question : What does Hurley, Dennett and Adams' theory of humor tell us about its political implications ? It is not the object of this paper to assess the accuracy of the view Hurley et al. proposed for how humor works. I take their theory as a point of depar‐ ture because, as I already mentioned, it encompasses the main traits of the prevailing views on humor, only adding an evolutionary twist. It advances an explanation based on the resolution of an incongruity, but it also suggests the „sudden glory" generated by the lack of any tormenting element. Besides, 25 Political Implications of Humor the epistemological implications of the view proposed by Hurley et al. give us a glance as to why the incongruency and the „sudden glory" go hand in hand. The punch line of the joke does not bring with it a mere incongruence, but also the triumph over something we didn't understand and now we do. III. SOME POLITICAL IMPLICATIONS OF HUMOR A political implication of humor derives from its epistemological limita‐ tions. More exactly, the problem is posed by the lack of epistemological ac‐ curacy of the scheme I presented. Humor emerges when errors are identified and eliminated from a mental space, but this process of epistemic reconcilia‐ tion always occurs against a background that is not questioned as well. The scheme proposed by Hurley, Dennett and Adams depends on a shared lan‐ guage that may contain verbal remnants of old discriminations. The language is a living organism that changes slower than our conception of justice and usually has biases we are not always aware of. We cannot control all the im‐ plications of the words we are using as we are immersed in a language in which meanings are interconnected in ways that escape our perception. For example, a word like „manly" continues to have meanings that are not ap‐ propriate for a society in which men and women are considered equal and no gender should have a monopoly over the idea of bravery. However, words evoking an oppressive past are still in use, being a part of that background against which errors are identified as such. It follows that progressive poli‐ cies that purport to further individual rights are easy to ridicule, as the hu‐ morous discourse will sanction any departure from the entrenched language. A possible objection against this interpretation is that the error discovered at the conclusion of the joke is only relative to the setup. In many cases, the humor derives from exposing the absurdity or the double meaning of some common expressions, as it is the case with the following joke : „I went to the corner shop. I bought four corners." The element whose lack of sense is de‐ nounced here is part of the common language. However, it is hard to believe that people will stop talking this way as a result of the phrase being ridiculed. Two observations should be made in relation to this objection. First, the claim supported by the epistemological limitations of humor is not that hu‐ mor is always conservative, but that it has a strong conservative bias. In other words, the main contention is that we run the risk of mocking a progressive view without realizing how unfair we are. An entrenched point of view re‐ garding social issues can also be mocked, as long as the premise of the joke is set up so as to support the progressive leaning punchline. However, it is eas‐ ier to use the common view as a setup than to carefully devise a wholly dif‐ ferent one in hopes of leading to the liberal conclusion you want to support. Sometimes supporters of rather progressive causes have to go great lengths to mock an entrenched view. In the following example, an attack on 26 Dan Panaet the conservative view according to which art is immoral and useless had to be attached to a more progressive Dadaist critique of art in order to become comical : „Republicans, Dadaists, Declare War On Art Citing the „proliferation of immoral and offensive material through‐ out America's museums and schools," and waving placards em‐ blazoned with agit‐prop fotocollage reading, „diE KUnst ISt tOT, DadA ubEr aLLes" ("Art is dead, dada over all"), a coalition of lead‐ ing Republican congressional conservatives and early 20th‐century Dadaists declared war on art in a joint press conference Monday. Calling for the elimination of federal funding for the National Endowment for the Arts ; the banning of offensive art from museums and schools ; and the destruction of the „hoax of reason" in our in‐ creasingly random, irrational and meaningless age, the Republicans and Dadaists were unified in their condemnation of the role of the artist in society today." (The Onion, 1997) The second observation pertains to the moral bearing of the abovemen‐ tioned joke, as there is no controversy whether the language should be de‐ void of any equivocal meanings. In other words, the joke is not part of an ongoing debate between two quarreling sides. Everybody agrees language has breeches that allow double meanings and virtually nobody wants to re‐ form the language so as to exclude any equivocal use of words. The play on words type of jokes take aim at the way we usually use language, but the point of departure, the position from which the critique is launched, is also part of the common language and does not pertain to a progressive view someone wishes to impose. According to the scheme I mentioned, errors are always diagnosed as such against the background of a common language that is not always a re‐ liable guide. A similar objection is usually brought against custom (Posner 2003, 151‐152), as it changes slower than our needs in terms of legislation. According to Posner, „custom, being acephalous, tends to change very slow‐ ly. If economic or other social practices are changing rapidly, custom will of‐ ten fail to keep up and will become a drag on progress" (Posner 2003, 152). Like custom, language is a spontaneous growth and any verdict it gives will lack epistemological accuracy, as it will be marred by the slowness of its de‐ velopment. In the same way in which custom is not a reliable guide for leg‐ islation, the connotations words happen to have may not follow the lines of the discourse about rights we are currently supporting. This problem is similar with the objection Hayek brings against conserv‐ atism. According to him, conservatism lacks a proper critique of the present, as it sanctions any departure from the status quo using the present as a yard‐ stick. According to Hayek, „one of the fundamental traits of the conservative 27 Political Implications of Humor attitude is a fear of change, a timid distrust of the new as such." (Hayek 2011, 522) A conservative critique of the present is possible only to the extent to which the present has not acquired legitimacy in the light of the tradition‐ al institutions, only inasmuch recent changes have not became entrenched. To put it briefly, conservatism is not able to deliver a proper critique of the present because it lacks a set of general principles in relation to which the present can be judged. In the same way, humor will have a strong conserv‐ ative bias as its attitude is based on a language already entrenched. In oth‐ er words, the truth humor conveys is ascertained against the background of the language that is currently in use. For this reason, any progressive at‐ tempt to reveal a form of oppression is prone to be sanctioned as ridiculous. A possible objection against this argument is that humor does not claim to convey any truth. What jokes deliver is the rather pleasant sensation of mirth, not a political argument. However, according to Hurley, Dennett and Adams, humor originally had an epistemological function, as people usual‐ ly equate the message of the joke with a surprising insight. A simple act of introspection will reveal the fact that, when people tell a joke, they tend to agree with the punch line, not with the claim placed under attack. More than that, humor is effective when used in political debates, as it puts the person who employs it in a good light. In other words, not only do people agree with the conclusion of the joke they tell, but they also tend to persuade the others that their point is sound. Another Hayekian argument concerning the morality of humor is based on Hayek's mixture of conservatism and liberalism. Hayek complements his rather conservative view on the spontaneous development of liberal in‐ stitutions with a Kantian test of universalizability, meant to decide wheth‐ er a rule is just. According to him, „the test of the justice of a rule is usually (since Kant) described as that if its universalizability, i.e. of the possibility of willing that the rules should be applied to all instances that correspond to the conditions stated in it (the 'categorical imperative')" (Hayek 1969, 168). In order for a rule to pass the test, it has to be general and avoid any ref‐ erences to a particular group of people. The only forms of discrimination that are accepted are the ones on which we have the agreement of majorities from both outside and inside the class of people which is singled out. (Hayek 2011, 222‐223) However, there is a close dependency between the two ele‐ ments I mentioned above (i.e., the process of cultural evolution and the uni‐ versalizability test), as we are constrained to apply the universalizability test within the conceptual confines of what our contingent language allows us to think. In other words, in order to rationally decide whether a rule is just, we have to work with morally laden concepts that are the product of a con‐ tingent path of societal evolution. In order for this limitation to become clearer, let us consider the interpre‐ tation Gray offers for the Hayekian universalizability test. He claims that 28 Dan Panaet Hayek's test is able to ensure a protected domain of the individual, thus dis‐ proving the critique formulated by Hamowy (Hamowy 1978) and Raz (Raz 1979). According to Gray, the process of universalizability has three stages (Gray 1998, 60‐62). The first stage aims at the consistence between similar cas‐ es. The second stage of the process of universalization tests whether we are ready to accept that the rule will govern the conduct of others toward our‐ selves. The final stage aims at the impartiality between preferences and ide‐ as about the good life, irrespective of whether they are ours or they belong to other persons and are contrary to what we think about how life should be lived. However, we have no rational criteria to decide whether a preference or an idea about the good life is legitimate or the externalities it carries with it make it unacceptable. The only guide we have is the moral attitude of the society in which we belong. It results that we always apply the universaliz‐ ability test within the confines of a contingent language, thus being prone to refuse liberties because an entrenched vocabulary limits our perception. In this context, humor functions as an additional hindrance, as it sanc‐ tions any departure from the official discourse, from the prevailing rhetoric. In other words, humor can be employed as a rhetorical means to reduce the already scanty leeway reason has in evaluating a rule against a contingent background of concepts. In the cases in which the two sources of rules – the societal evolution and the application of the universalizability test – suggest different outcomes, humor employed as a rhetorical weapon is prone to have a bias toward the former, as it usually reinforces the common sense. For ex‐ ample, critics of the liberal tendency to come up with less offensive phrases mocked this penchant for reforming the language by proposing further re‐ designations : vertically deployed antipersonnel device instead of bomb and sex‐ ually focused, chronologically gifted individual for dirty old man.1 A possible objection to this critique of the conservative bias of humor con‐ sists in the fact that we cannot know for sure which inequalities are justified and which should be discarded. Some forms of discriminations (e.g. against lazy people) may function for the advantage of everyone, as other discrim‐ inations, like the one against people of a certain religion, are utterly illegiti‐ mate. However, it is not clear whether the legal bias against some categories of people should be discarded or maintained. Humor provides a good way of dealing with this uncertainty, being part of a debate in which the preten‐ tions of equality are appraised. Without such a mechanism, any claim to a more egalitarian scheme would be accepted unconditionally, without being sure whether the inequality plays any role. As I already mentioned, some forms of discriminations the law makes are legitimate according to Hayek as long as they are accepted by majorities from both inside and outside of the class of people singled out by the rule. 1 According the the web page http ://www.funny2.com/dictionary.htm. 29 Political Implications of Humor Rorty's view on the rise of liberalism highlights another problem of hu‐ mor. According to him, liberalism has developed as the solidarity between people deepened and individuals gradually accepted more and more groups into the sphere of what they defined as „we" (Rorty 1995). This kind of change was propelled not by rational arguments, but by powerful redescrip‐ tions, which made people sympathize with the less fortunate (Rorty 1995). However, humor seems to go in the opposite direction. It reinforces the connection with the other members of the group as it relies on the language and the references we share. We should notice that even the liberal humor does exclude others, as it is based on drawing a clear line between us and them, conservatives being seen as backward for not agreeing with more pro‐ gressive policies. In the terms of the scheme proposed by Hurley et al., peo‐ ple are able to take part in the identification of the error only as long as they share the language against which the anomalous element is identified as such. If liberalism developed through gradually extending the limits of a group, so as to include new categories of people, humor reinforces the lim‐ its the group already has. Moreover, humor blocks our empathy, given that in order to see the humor in other people's unusual behavior you should not try to understand them. This conclusion is also supported by Bergson's view on humor, as he claimed that laughter is accompanied by insensitivity. According to Bergson, „it is enough for us to stop our ears to the sound of music in a room, where danc‐ ing is going, for the dancers at once to appear ridiculous." (Bergson 2003, 8) To summarize, the main moral problem posed by humor can be described using the following conjunction : 1) In a liberal democracy, an important danger consists in not being able to perceive all the forms of oppression that surround us, because of the com‐ mon language in which we are immersed. 2) Humor is based on an epistemological mechanism of identifying the errors using as a benchmark a common language that is not similarly questioned. However, we should notice that in a dictatorship, humor is more effec‐ tive in supporting individual rights. In a tyranny, the main problem is not that we cannot perceive some forms of oppression, as the presence of the op‐ pression will be obvious for everyone. In these cases, the rulers usually try to devise an artificial legitimation discourse that can easily be questioned in the terms of the common language. For this reason, humor will prove useful in denouncing the transgressions of a dictatorial government. In communist Romania, for example, the most common form of humor was the one directed at the political situation, as the official language of the rulers was obviously spurious and could be easily identified as such. The following joke is a clev‐ er take on the fact that grocery stores were insufficiently supplied : 30 Dan Panaet A man walks into a store and asks the clerk : „Do you have bread ?" „I am sorry, sir", answers the clerk, „but this is the store where we don't have meat. The store where they don't have bread is just across the street." The punch line reveals the despairing reality lying behind the façade of the official discourse. What is denounced are the pretenses of the communist government, according to which you can find bread and meat in any store that purports to sell these items. The background against which the debug‐ ging is carried out includes the observation that the dearth of the commu‐ nist economy makes the grocery stores to be insufficiently supplied. The er‐ ror that has been identified is stated in a clever manner : if a store purports to sell bread, what the storefront sign really says is that you can be sure you won't find any bread in there. IV. CONCLUSION Humor is usually used as a rhetorical means by both conservative and liberals. However, I argued in this paper that humor has a strong conserv‐ ative bias, as jokes are made against the background of an established dis‐ course that is not questioned as well. For this reason, more progressive caus‐ es are easier to ridicule, which makes humor prone to perpetuate stereotypes. I used as a point of departure the cognitive interpretation of humor offered by Hurley, Dennett and Adams. According to their view, humor has been se‐ lected during the process of evolution because of its epistemological use, as it helps at eliminating the errors that covertly entered our minds. However, this process of debugging the mind is carried out against a contingent back‐ ground that escapes a similar evaluation. After briefly presenting two view on liberalism, I argued that the epistemic limitations of humor function as hindrances in the way of denouncing covert forms of oppression. REFERENCES Bergson, Henri. 2003. Laughter : An Essay on the Meaning of the Comic. The Project Gutenberg Etext. Gray, John. 1998. Hayek on Liberty, London : Routledge. Hamowy, Ronald. 1978. „Law and the Liberal Society : F.A. Hayek's Constitution of Liberty", Journal of libertarian Studies, Vol. 2, No. 4 : 287‐297. Hayek, F.A. 2011. The Constitution of Liberty, Chicago : The University of Chicago Press. Hobbes, Thomas. 1997 – Leviathan. Touchstone. Political Implications of Humor Hurley, Matthew M., Daniel Dennet, Reginald Adams. 2011. Inside Jokes : Using Humor to Reverse‐engineer the Mind. The MIT Press. Kant, Immanuel. 2007. Critique of Judgement. Cosimo Inc. Morreall, John. 2012. „Philosophy of Humor", Stanford Encyclopedia of Philosophy, http ://plato.stanford.edu/entries/humor/. Polimeni, Joseph and Jeffrey Reiss. 2006. „The First Joke : Exploring the Evolutionary Origins of Humor", Evolutionary Psychology, 4 :347‐366. Posner, Richard A. 2003. „Hayek, Law, and Cognition", NYU Journal of Law and Liberty, Vol. 1, No. 0 : 147‐165, pp. 152‐153. Raz, Joseph. 1979. „The Rule of Law and Its Virtue", in Joseph Raz – The Authority of Law : Essays on Law and Morality. Oxford : Clarendon Press. Rorty, Richard. 1995. Contingency, Irony, and Solidarity. Cambridge University Press. The Onion. 1997. „Republicans, Dadaists, Declare War On Art", The Onion, http ://www.theonion.com/articles/republicans‐dadaists‐de‐ clare‐war‐on‐art,858/

Penultimate Draft. For the published version, see Oxford Studies in Normative Ethics, Volume 5. Fairness, Participation and the Real Problem of Collective Harm1 1. Introduction In a wide range of cases, people collectively cause a morally significant outcome but no individual act seems to make a difference. For example, while climate change is caused in part by millions of people acting in certain ways, any one such act does not seem to make a difference. Take a single driver off the road and the problem of climate change will surely be just as bad. Similarly, while large-scale consumer patterns can have a major impact on the lives of people across the globe, it's hard to believe that a single purchase will itself make a difference. Will things be any better for anyone if, for instance, the cup of coffee I buy this morning is fair trade rather than conventional? Or take the case of voting in a national election, where – let us suppose – it matters a great deal for the welfare of the population which candidate wins. While our votes collectively determine who wins, a single vote won't make a difference: the winner will be the same give or take any individual vote. The problem in cases of this sort is that it seems each person can argue, "it won't make a difference whether or not I do X, so I have no reason to do it." The challenge is to explain where this argument goes wrong. If we cannot, this would be very troubling; morality would be powerless in wide range of cases in which it should have force. I call this "the Problem of Collective Harm", and the cases in which it arises "Collective Harm Cases".2 1 I am very grateful for helpful comments from Markus Kohl, Niko Kolodny, Mark Lebar, Alex Rennet, Abraham Roth, Carolina Sartorio, Sergio Tenenbaum, R. Jay Wallace, and two anonymous referees for Oxford University Press. I also owe much gratitude to the participants of the Fifth Annual Arizona Normative Ethics Workshop, and to audiences at Carleton University and the 2013 Toronto-Montreal Joint Ethics Centre Conference at the University of Montreal. 2 This follows my terminology in Julia Nefsky, "Consequentialism and the Problem of Collective Harm: A Reply to Kagan." Philosophy & Public Affairs 39 (2011): 364-395 and in my "How You Can Help, Without Making a Difference", draft. I think these names, on balance, do a nice job of giving a sense of the relevant phenomenon, but they are in one way misleading. While in many Collective Harm Cases it is harm that is at stake, this need not be so. In a voting case, for instance, it could matter a lot morally which candidate wins, but not because there will be more harm if one candidate wins than if the other does. This would still count as a "Collective Harm Case" on my Julia Nefsky 2 There are two ways of trying to solve the problem.3 The first is to try to deny the description of the cases: to argue that, contrary to appearances, one's individual act actually does, or at least might, make a difference with respect to the morally significant outcome in question and that there is, thus, no problem after all. The second is to accept that there are cases in which one's individual act cannot itself make a difference, but to argue that there is still reason for action – a reason that doesn't have to do with the difference you make in outcome. Those who take this second type of approach, in other words, respond by rejecting the implication from 'it won't make a difference' to 'there is no reason to do it.' In this paper I am interested in the possibility of taking the second kind of approach – a "Rejecting the Implication" approach.4 I centre the discussion around three such proposals. The first two – which I call "Weak Participation" and "Strong Participation" – both claim that we can solve the problem by attending to the fact that individuals in Collective Harm Cases are acting as part of a group that makes a difference. The third proposal – the "Fairness Approach" – attempts to solve the problem by appealing to considerations of fairness. After raising some initial questions about each proposal individually, I argue that there is a common, basic problem with all three. While they do identify considerations other than the difference you make, it's not clear that these considerations can provide the reasons for action we are looking for as long as it is true that your act won't make a difference. More specifically, I argue that unless your act could play a significant instrumental role in bringing about the outcome in question, the explanations these views are offering as to why you have reason to do it cannot get off the ground. And yet not playing any such role seems to be just what we are definition. We can think of "Collective Harm Cases" as naming a wider class of cases after a subset, namely the subset in which what is at stake is harm. 3 This excludes biting the bullet and accepting that there is no reason for action in Collective Harm Cases. 4 The "Denying the Description" approach will briefly come back into the discussion at the end of this paper. I discuss the plausibility of such an approach elsewhere (in "Consequentialism and the Problem of Collective Harm"). Julia Nefsky 3 accepting when we accept that your act won't make a difference. I bring out this problem through a discussion of the Fairness Approach in Section 6. In Section 7 we see how the same problem arises for the other two approaches. The problem, I then argue, extends beyond these three proposals. In particular, if we try to apply certain general non-consequentialist moral theories to Collective Harm Cases, we encounter the same basic problem. This clarifies why it is that the Problem of Collective Harm poses a genuine challenge not just for consequentialists but also for non-consequentialists. None of this will be to say that notions of fairness or participation will not be important in understanding the morality of Collective Harm Cases, or that non-consequentialist theories cannot ultimately handle these cases effectively. Rather, the aim of the discussion is to obtain a sharper understanding of what the core challenge in the Problem of Collective Harm really is. This, in turn, helps us identify what the central task needs to be if we are to properly address it. 2. Two Examples I have given a few quick examples of real-world Collective Harm Cases, but I will largely focus on the following two imaginary cases. Both are from Derek Parfit. Drops of Water: Ten thousand men lie in the desert, suffering from intensely painful thirst. We are ten thousand people near the desert, and each of us has a pint of water. We cannot go into the desert ourselves, but we can pour our pints into a water cart. The cart will be driven into the desert, and any water in it will be evenly distributed amongst the men. If most of us add our pints, this will be enough to alleviate their suffering. But while our acts of adding our pints would collectively alleviate their suffering, any individual such act does not seem to make a difference. A single pint added to the cart only allows each man to drink an extra tenthousandth of a pint of water. This is no more than a single drop. And, however much water they are receiving, one drop more or less is simply too miniscule an amount to make any difference to how they feel.5 Harmless Torturers: A victim is hooked up to an electric shock machine that has a thousand settings, controlled by a dial. There are a thousand "torturers", each of whom turns the dial up a single notch. At the first setting there is no electric current. At the second setting 5 This is adapted from Derek Parfit, Reasons and Persons (Oxford: Oxford University Press, 1984), p. 76 and follows my presentation in "How You Can Help, Without Making a Difference." Julia Nefsky 4 there is a tiny electric current, but it is too tiny to be perceived by the victim. In general, the settings increase the voltage by increments so tiny that the difference between any two adjacent settings is too miniscule to make a difference to how the victim feels. But, while any two adjacent settings are indistinguishable, after many increases the victim feels pain, and by the thousandth setting the victim is in excruciating pain. While the torturers' dial-turns together result in a lot of pain for the victim, it seems that no individual makes a difference for the worse. Take any given torturer's dial-turn away, and the pain would be just as bad.6 The Problem of Collective Harm arises in both cases. In Harmless Torturers, each torturer can argue, "my dial-turn did not make a difference; the victim would have been in just as much pain had I not done it. So I did not do anything wrong." In Drops of Water, each of us can say, "it won't make a difference whether or not I add my pint, so there's no reason to add it." True, the dehydrated men will not be relieved from their pain unless enough of us add our pints. But this does not change the fact that things will be the same for them give or take any single pint. 3. Weak Participation Weak Participation and Strong Participation both claim that we can solve the Problem of Collective Harm by attending to the fact that, even if you do not yourself make a difference, you may be part of a group that does. We together make a great deal of difference in Collective Harm Cases, and this can explain why each of us has acted wrongly or rightly. Weak Participation essentially starts and stops with this claim. Parfit tries this simple appeal to what people together do in Reasons and Persons, giving the following principle, which he calls "C7": (C7) Even if an act harms no one, this act may be wrong because it is one of a set of acts that together harm other people. Similarly, even if some act benefits no one, it can be what someone ought to do, because it is one of a set of acts that together benefit other people.7 6 This is adapted from Reasons and Persons, p. 80 and follows my presentation in "Consequentialism and the Problem of Collective Harm". 7 Reasons and Persons, p. 70. Parfit intends this to be a consequentialist proposal. For a discussion of whether it should be accepted by consequentialists see Frank Jackson, "Which Effects," Reading Parfit, Ed. Jonathan Dancy (Oxford: Blackwell, 1997), 42-53, and Ben Eggleston, "Should Consequentialists Make Parfit's Second Mistake: A Reply to Jackson," Australasian Journal of Philosophy 78 (2000): 1-15. In this paper, though, I am interested in the proposal independently of whether it would be attractive to a consequentialist. Julia Nefsky 5 In Harmless Torturers, for instance, while turning up the dial one notch makes no difference to the harm suffered by the victim, it is one of a set of acts that together cause him a lot of pain. The torturers together make a difference for the worse, and so each of them has acted wrongly. Does this response work? Not as it stands. After all, what determines who is in the group?8 Why can't each individual say: my act makes no difference, and so it's not part of the set of acts that together makes a difference for the worse? It is unclear, in other words, why I should count as part of the group that harms if what I do makes no difference with respect to that harm. The point is not (or not just) that we need some criteria that will tell us which acts are in the group and which are not. The point is that if we want to derive normative implications directly from my being part of a group, we need an understanding of what this sort of participation consists in that explains why it has these normative implications. Still, Weak Participation might seem to be pointing down the right path. Weak Participation says: even if you do not by yourself make a difference, if you together with others do, this can explain why you have acted wrongly or rightly. Strong Participation can be thought of as an attempt to substantiate this idea.9 4. Strong Participation Strong Participation, unlike Weak Participation, employs a robust conception of acting with others – of participating in a collective action. Using this conception of what it is to participate, it attempts to explain why mere participation, regardless of whether you can make a difference, has normative implications. 8 Parfit attempts to address this question. He sets out various principles that attempt to further elaborate on C7 and, if they succeed, would specify who is in the group. But, as he admits in a later paper, they do not succeed: the further principles exclude the very acts he is trying to capture, acts that cannot themselves make a difference. See his "Comments," Ethics 96 (1986), pp. 846-848. 9 When substantiated in this way, though, it is clearly not consequentialist. (i.e. See note 7 above.) Julia Nefsky 6 Christopher Kutz develops such a proposal. On Kutz's account, you participate in a collective action when you act with a "participatory intention", an intention to do your part in a collective project.10 Suppose we are building a sandcastle. Your act of filling the bucket with water is part of our collective action because it is performed with the intention of contributing to our collective end, a sandcastle. You fill the bucket so that we can mix water with sand, which will allow us to sculpt the castle. If you had instead filled it with the intention of pouring it on my head, this would not have been part of our collective action. "Jointly acting groups," Kutz writes, "consist of individuals who intend to contribute to a collective end."11 Next Kutz argues that (1) you are accountable for actions that are ascribable to you, and for the consequences (whether intended or not) of those actions, and that (2) when you participate in a collective action, that action is ascribable to you. When you take part in a collective action, you are an "inclusive" author of that action.12 You are one of a number of people who can say, "We did it." It is, in this sense, your action, and so you can be held accountable for it and its consequences.13 Kutz claims that this holds regardless of whether you individually make a difference. He gives examples like the following. Suppose that you and I are having a picnic. While I am getting the drinks from the car, you put the blanket down on a bed of flowers and ruin it. Even though it is your individual act and not mine that causes this damage, you put the blanket down 10 Christopher Kutz, Complicity (Cambridge: Cambridge University Press, 2000), Chapter 3. There is a further condition, namely that your conception of the joint project sufficiently overlaps with that of the other members of the group. (Complicity, pp. 94-95.) 11 Complicity, p. 89. There is disagreement about how to understand the phenomena of collective action. Kutz's account – as he explains – is minimalistic. Other accounts bring in stronger conditions, conditions requiring, for instance, mutual knowledge or mutual responsiveness. And they often invoke not just intentions to do your part but intentions that we do the group action. In part because Kutz's account is already a minimalist one, it is not as though we could resolve the core problem that I will raise (in section 7) by substituting in another available account of collective action. 12 Where this is contrasted with being the "exclusive" author of an action. (Complicity, pp. 105-106.) 13 Complicity, pp. 137-139. Julia Nefsky 7 as part of our collective action of picnicking together. So, while none of my individual acts make a difference for the worse, the damage is a result of a collective action that is ascribable to me. Thus, I can be to some extent accountable for it.14 A similar thing can be said in Collective Harm Cases: even if one's individual act makes no difference with respect to the outcome of concern, one can be accountable if one is a participant in a collective action that results in this outcome. One issue that arises in trying to solve the Problem of Collective Harm in this way is that there are, as Kutz points out, many Collective Harm Cases that simply do not involve collective action. These are cases of "unstructured collective harms": cases in which harm results from many people acting in certain ways, without there being any joint project that they are all intentionally participating in.15 Much environmental damage is unstructured harm; drivers, for instance, are not intending to do their part in some collective project whenever they drive. Similarly, people who buy conventional rather than fair trade goods are not intending to do their part in a joint project. There might be, however, a simple way of extending the picture to cover unstructured cases. Typically when there is a risk of unstructured collective harm, there is also an opportunity to participate in a collective action of preventing that harm. In Drops of Water, if none of us add our pints and the men continue to suffer, we would not be intentionally participating in a collective project. So, this would be a case of unstructured collective harm. But if we do add our pints, it seems we would be engaged in a joint project of alleviating their suffering. Similarly, while drivers are not engaged in a collective project when they drive, they could participate in a collective action aimed at mitigating environmental damage if they each reduce the amount they drive with the aim of doing their part in such a project. If Strong Participation explains why there 14 Complicity, p. 154. 15 Complicity, p. 166. Julia Nefsky 8 is reason to refrain from participating in harmful collective actions even if you make no difference, it should also be able to explain why there is reason to participate in beneficial collective actions even if you make no difference. The next question to ask, though, is: how exactly does Strong Participation explain why there is reason to do either? So far, what it has explained is why participants in a collective action can be accountable, even if they make no difference. But, at least on Kutz's notion of accountability, asserting that someone can be accountable for a harm is not equivalent to saying that this person did something that she ought not to have done. Accountability, for Kutz, concerns "warranted response", where this is not just a matter of responses owed "because of the rights and wrongs" one has done.16 Think of the picnic example. Perhaps it is true that I am partially accountable for the damage caused to the flowers: perhaps I ought to express apology or help compensate the gardener because of the way in which I am connected to the damage through our collective action. But one thing that is clear is that I did not do anything I ought not to have done. Everything I did was perfectly fine. As Kutz writes: I am not at fault with regard to the gardener, but it seems I do nonetheless owe some form of response ... Because I have made our picnic mine by my intentional involvement, I have also made its consequences mine. My accountability is therefore intermediate, between what one owes for faultless harms (for I was not at fault) and what one owes for faulty harms (for you, and hence we, were). Your fault sticks to me but not with its full force.17 Are things similar in Collective Harm Cases? In Harmless Torturers, my individual act of turning the dial makes no difference to the suffering the victim undergoes. Perhaps I am to some extent accountable because of the way in which I am connected to the harm through my participation. But if what I did couldn't have made any difference with respect to the harm, how does Strong Participation give us grounds for thinking that I ought not to have done it? 16 Complicity, p. 18. That is, accountability for Kutz is specifically not a retributivist or desert-based notion. 17 Complicity, p. 154. Julia Nefsky 9 I take it the reply is: in the picnic case I have no reason to expect the bad outcome; this is why we cannot say that I ought to have acted otherwise. In Harmless Torturers, on the other hand, the collective project is a harmful one and so I can expect that the collective action will result in harm. If you expect that a collective action will result in harm, then you ought to refrain from participating in it, and – crucially – you ought to refrain from participating regardless of whether you could make any difference for the better by refraining. This is because if you participate, you will be an "inclusive author" of the harm. The harmful collective action will be, in a sense, yours – something that we, and thus inclusively you, did. Intuitions will likely diverge here. Some might find this picture appealing. Others might think: while I can expect that the collective action will result in harm, I also know (in advance) that my "inclusive authorship" won't make any difference with respect to that harm. Why should that sort of inclusive authorship make my individual act wrong? Why not think that the most it can do is make me accountable in a faultless, associative sense?18 5. The Fairness Approach A common idea is that the reason one ought to act in the relevant way in Collective Harm Cases is that not to do so would be unfair. Garrett Cullity advances such a proposal, and it is his version that I will make use of here.19 18 Of course, I might not want to be even faultlessly accountable; I might not want to owe or be owed any response. In that case, the risk of being accountable could itself give me reason to refrain from turning the dial. But, divorced from any claim that I would be doing something I ought (or, at least, have independent reason) not to do, this appeal to the risk of accountability seems to simply give a reason concerning how turning the dial could affect me. This seems to be the wrong sort of reason to respond to the Problem of Collective Harm that arises in Harmless Torturers. 19 The appeal to fairness is common for those concerned with Collective Harm Cases in which not acting in the relevant way is considered free-riding. These are cases in which the relevant agents are also the beneficiaries. For example, why should I pay my subway token if I can sneak on? $3 more or less surely won't make a difference to subway operations. A common answer is: I am only able to gain the benefits of the subway because others pay their tokens; so, to not pay would be unfair. Not all Collective Harm Cases are potential free-rider cases. In Drops of Water, if I keep my pint, I am not gaining a benefit that is only possible because others have given up their pints. (Note also that not all free-rider cases are Collective Harm Cases.) But while the two kinds of cases only overlap, and are not identical, it seems plausible that not acting in the relevant way is wrong for the same reason in both Julia Nefsky 10 The Fairness Approach first claims that in Collective Harm Cases there is a collective obligation to prevent (or to avoid bringing about) the relevant bad outcome. Unless enough people act in a certain way, bad consequences will result, and so there is an obligation on the group as a whole to ensure that this does not happen. But if we have a collective obligation – the argument goes – it is unfair for some but not others to do the work of satisfying it. If I stand by while others do their part in fulfilling the obligation, this is not fair to the others in the group and is therefore wrong. Thus, regardless of whether or not one could make a difference by doing so, there is reason to act: "it is a matter of pulling one's weight in what we all ought to be doing."20 Suppose that each of the potential torturers would receive a benefit for turning up the dial – say, fifty dollars. Each knows that he won't make a difference to the victim's pain by doing so. So, why should he refrain? According to the Fairness Approach, without making any assumptions about the torturers' individual obligations, we can say that there is an obligation on the group as a whole not to cause the victim to suffer. Most people in the group must refrain from turning the dial if this collective obligation is to be met, and if most people refrain while you go ahead and claim the fifty dollars, this is unfair. So, the right thing to do is to refrain. Now, the point is not that from a collective obligation we can always derive individual obligations. If no one is trying to satisfy the collective obligation, I am not being unfair by refraining from acting myself. The idea is, rather, that when there is a collective obligation and people are trying to fulfill it, it can be unfair for me not to do my part too. Cullity explains: I have not been arguing that individual imperatives can be derived from every collective imperative... Before the collective action is started, none of us is arrogating any special privileges in refusing to get it started. We are collectively acting wrongly, but no individual is taking advantage of others' propensity to contribute to doing what we ought to be doing.21 kinds of cases. This is the idea that Cullity pursues in "Pooled Beneficence", Imperceptible Harms and Benefits, Ed. M. Almeida (Dordrecht: Kluwer Academic Publishers, 2000), 1-23. 20 "Pooled Beneficence," p. 17. 21 "Pooled Beneficence," p.18. Julia Nefsky 11 This, however, seems to be a problem for the Fairness Approach. This means that as long as all the torturers continue turning up the dials, or as long as no one has given a pint to the cart, the Fairness Approach has nothing to say about why one ought to act otherwise. But the Problem of Collective Harm is that it's not clear why any individual has reason to give up her pint, or to refrain from turning the dial, and so on. Each individual such act seems to make no difference, and thus it is hard to see why anyone has reason to do it. If the Fairness Approach only kicks in once people have already started acting in the relevant way, it can't count as a full solution to the problem. In response, perhaps we can say that while the collective obligation does not itself entail individual obligations, its presence does give individuals, right from the start, at least some morally relevant reason for action. While I may not have an obligation to add my pint if others have not yet done so, perhaps the fact that we are collectively obligated to alleviate the suffering explains why I have at least some reason to add my pint. Once others, or enough others, have added their pints, I then become obligated to do so because not doing so would be unfair. I do not think this reply succeeds. But this is a consequence of a more basic and important problem – one that arises for all three of the views under discussion. Let's turn to that problem. 6. Fairness and Difference-Making Jonathan Glover dismisses the Fairness Approach, claiming that it invokes a notion of fairness that we do not or, at least, should not care about. He considers a case in which a car needs to be pushed up a hill; only six people are necessary for the job, but there are eight altogether. Even though six could manage the task on their own, it would be unfair for two to sit by while others do all the work. This sounds much like the point that the Fairness Approach tries Julia Nefsky 12 to make in Collective Harm Cases. In the voting case, for instance, the Fairness Approach says: even though we do not all need to vote, it would be unfair for some but not others to do so. But, Glover replies: We are free to accept the argument from injustice in the car pushing case without accepting it in the voting case. If I do not push the car, the others will have to push a bit harder. Many of us are against the kind of injustice that involves giving benefits to some at the cost of additional hardships to others. But no-one has to vote harder because I do not vote. It seems a dog-in-the-manger version of justice that objects to one person benefiting because others are left unchanged.22 Glover is right that there is an important disanalogy between the car-pushing case and the voting case.23 But it misunderstands the Fairness Approach to say that it invokes a "dog-in-themanger version of justice" or that it "objects to one person benefiting because others are left unchanged." Take Drops of Water. Unlike a "dog in the manger" situation, each of us would benefit from keeping our pints; we would each have it as a refreshing drink. The point the Fairness Approach makes is that it would be unfair for some to keep their pints, receiving the benefits that each of us would have enjoyed, while others give theirs up in order to achieve what we as a group are obligated to achieve. The type of unfairness involved in keeping my pint is not that doing so benefits only myself and not others – that it benefits one person while leaving others unchanged. It is, rather, that if I keep my water, while others give theirs up, I am not – as Cullity writes – "pulling my weight in what we all ought to be doing."24 I am "relying on others to satisfy ... [the collective] imperative, while excepting myself from doing so."25 It seems reasonable to propose that this sort of unfairness is morally objectionable. And someone who 22 Jonathan Glover, "It Makes No Difference Whether or Not I Do It," Proceedings of the Aristotelian Society 49 (1975), p. 182. 23 Here is the disanalogy. In the car pushing case, my act is not necessary for achieving the desired outcome – there are ways of bringing about the outcome without it. But in Collective Harm Cases, it's not just that my act is not necessary; it's that it simply can't make any difference. We can take my act away, and not replace it with anything (e.g. not replace it with more effort on the part of the others), and the outcome of concern will be the same in all relevant respects. Because of this disanalogy, while appealing to fairness might work to explain why I ought to push in the car-pushing case, it won't work as a solution to the Problem of Collective Harm, as we will see shortly. 24 "Pooled Beneficence", p. 17. 25 "Pooled Beneficence", p. 17. Julia Nefsky 13 finds this view compelling will not be moved by the reply that we should only care about unfairness if it is matter of one person gaining a benefit by making things worse for others. Such a reply seems to simply contradict the proposal, rather than identifying a problem with it. Glover's objection is, thus, not effective as it stands. But there is an important issue in the vicinity. The Fairness Approach says: if we collectively ought to bring about some outcome, it is unfair to let others do the work of satisfying this obligation without doing so yourself. You ought to pull your weight in what we collectively ought to do, regardless of whether in doing so you will make any difference. Glover's reply is: we shouldn't care about the unfairness in not pulling one's weight if it won't make any difference. But the problem is more basic. If acting in the relevant way won't make any difference then it does not seem that it pulls any weight at all. It doesn't seem to count as doing the work of satisfying the collective obligation. Consider this revised version of the car-pushing example. We need to get the car up the hill, but one of us – call him "Small" – has a rare physical condition: while he is otherwise healthy, he has the muscles of a two-year-old. He is, let's say, an adult trapped in a two-yearold's body. Because of this, it makes no difference whether or not Small pushes. If he pushes with all his might, this will exert a tiny amount of force on the car, but this tiny amount of force is too tiny to make any difference at all with respect to the task at hand. The car won't progress up the hill any faster if he pushes than if he doesn't, nor will it be any easier for the others involved. And this is so regardless of how many others push. Whether or not enough other people push to get the car up the hill, adding or removing Small from the equation simply won't change things in any relevant respect. In this situation, does it make sense to say that Small ought to push because this is his fair share of our collective burden? His act of pushing wouldn't help us achieve the goal. It would be Julia Nefsky 14 merely superfluous. It is true that there is a burden on the collective to get the car up the hill. But if his pushing won't help discharge this burden, how could it be his fair share of the burden? It doesn't seem to be any real share at all. Instead it seems like a mere waste of his efforts. Perhaps Small should do something out of fairness. If there is some useful task he could perform, he might have a duty of fairness to do that. But it doesn't make much sense to say that what he ought to do is something merely superfluous and unhelpful. In Collective Harm Cases each individual act seems to be like Small's act of car-pushing. It appears that it can't make any difference with respect to the outcome of concern, and so seems to be merely superfluous. But what we are seeing is that the Fairness Approach cannot get off the ground if this is so. The Fairness Approach appeals to (a) the presence of a collective obligation to bring about some outcome, and (b) the idea that it is unfair to let others take on the task of satisfying this obligation while excepting yourself from doing so. But as long as my act won't help satisfy the collective obligation – as long as it would be merely superfluous with respect to what we are collectively obligated to do – (a) and (b) cannot explain why I should do it. Even if there is a sense in which we are collectively obligated to relieve the men's suffering in Drops of Water, if adding my pint to the cart won't help to relieve their suffering, this collective obligation wouldn't give me reason to add my pint. And bringing in a requirement to do one's fair share does not fill the gap. How could doing something merely superfluous with respect to the alleviation of their suffering count as doing my fair share of our obligation to alleviate their suffering? It seems to be no real share at all.26 26 Cullity considers and tries to respond to a similar objection. Part of his response is to argue that one's individual act counts as making a real contribution toward meeting the collective obligation because "the contribution is perceptible" even though "it has no perceptible effect on what is achieved by the group" ("Pooled Beneficence", p. 16). In Drops of Water, for instance, my contribution is perceptible: a pint is a perceptible amount of water. This does not work. The perceptibility of a pint of water is irrelevant to the question of whether it actually contributes in any morally significant sense to the task of alleviating their suffering. The problem here is similar to a problem I discuss in "Consequentialism and the Problem of Collective Harm"; there is a tendency in writing about this topic to Julia Nefsky 15 I am not claiming that the idea that I ought to do my fair share has no force in this case. As with Small's case, if there is some useful task I could perform I might have that sort of duty to do it. The point is that as long as it's true that pouring my pint into the cart wouldn't do anything useful toward alleviating their suffering – that it will not help matters in any real way – it does not seem that that would be the thing to do in a response to a requirement to do my part in the task of alleviating their suffering. One might reply as follows: "putting the idea of 'doing one's fair share of the collective obligation' aside, there is a simple explanation as to why fairness gives one reason to add one's pint. If we really think that fairness matters morally in itself, then it does not need to be that adding a pint would be useful with respect to satisfying this collective obligation. It is unfair for some of us to keep our pints if others have given theirs up just because this would mean having something that those others no longer have." Now, this kind of fairness – unlike the one that occurs in the Fairness Approach – is the "dog-in-the-manger" type of fairness that Glover finds objectionable. It asks us to "level down": to give something up, lowering our own position, simply for the sake of making things more equal. Still, while many object to the idea that there is any reason to level down, some do think that there is moral reason to do so and that it only seems objectionable if we do not attend to the fact that this reason can often be outweighed. Regardless, this is really beside the point. Even if you do have reason to give up your pint coming simply from the fact that this would make things more equal amongst the original pintholders – that is, even if there is a reason to "level down" – this is not a reason that can count as a solution to the Problem of Collective Harm. We can see this by noting that it does not explain confuse perceptible with morally significant. Still, while Cullity's point about perceptibility does not work, some of his other comments hint in a potentially promising direction. Julia Nefsky 16 why I have reason to add my pint to the cart, rather than – for example – to pour it onto the ground. As far as bare equality in the resources of the original pint-holders is concerned, it would be just as effective if I pour my pint onto the ground. To respond to the Problem of Collective Harm we need an explanation of why I have reason specifically to add my pint to the cart. That is, we need an explanation of why each of us ought, or at least has reason, specifically to do the sort of thing that if enough of us do it, will alleviate the suffering. A reason that is just as much a reason to, instead, pour one's pint onto the ground is no such reason. Notice, moreover, that if I have reason to add my pint coming simply from the fact that this would make the resources more equal amongst the group of original pint-holders, this is a reason that has nothing particularly to do with the suffering men in the desert. It only concerns how my resources compare to the others in this group. Any such reason cannot count as a solution to the Problem of Collective Harm. We need a reason for action that connects appropriately to the fact that the men in the desert are suffering. Otherwise, we are essentially just changing the topic. The Fairness Approach tries to provide this connection by invoking a specific kind of fairness – that of pulling one's weight in what we collectively ought to do. But, as we have seen, it's not clear that this can work as long as we are accepting the claim that one's individual act won't make any difference.27 Let's call this "the Superfluity Problem". 27 Some might object to the comparison between Small's case and Drops of Water. In Small's case, everyone but Small is able to do something instrumentally significant; Small is an exception. In Drops of Water, on the other hand, everyone is symmetrically situated: each can add at most a pint, and so no one can make a significant difference, though we together can get the job done. If we made Small's case like that it would have to be that everyone was like Small. There could be thousands of people with Small's condition who need to get a car up a hill; they can do so if most of them push, but any one of them won't make a difference. When we move to this "symmetrical" version it no longer seems unintuitive that Small is doing his fair share by pushing. My response is largely to agree. I agree that there is an important difference between the original Small case, on the one hand, and Drops of Water and this "symmetrical" Small case on the other. I agree that, while there might be no (morally relevant) reason for Small to push in the original case, Small does have reason to push in the symmetrical version, and each of us has reason to add our pints in Drops of Water. This is because I think there is a Julia Nefsky 17 7. A Common Problem The Superfluity Problem arises just as much with the other two views. We have essentially already seen this for Weak Participation. Weak Participation says that even if you don't harm anyone yourself, if you together with others do this explains why you have acted wrongly. But in Collective Harm Cases, it's not just that you don't cause the harm in question all by yourself; it's that you make no difference with respect to that harm. Your act doesn't seem to play any significant role in bringing it about. So, why should you count as part of the group to begin with? Why is your act part of the set of acts that harms, if it does not contribute in any nonsuperfluous way to bringing about the harm? To come at the point another way: as long as your act doesn't play any significant role in what the group does, we cannot simply point to what the group does to explain why you have acted wrongly (or rightly). And yet this is just what Weak Participation tries to do. Strong Participation does not do any better. Whether or not we find appealing the idea that there is something wrong with participating in a harmful collective action even when one's doing so can't make a difference, there is a more basic issue. If your act can't make any difference, we may not be able to count it as an act of participation to begin with. In Harmless Torturers, Strong Participation says: 'even if each torturer makes no difference to the harm, the torturers would be engaged in a collective action of harming the victim. By turning the dial, you are participating in this collective action. This makes the collective action in a sense yours, and solution to the Problem of Collective Harm. (Note that the symmetrical Small case is a Collective Harm Case.) Moreover, I do not deny that it ultimately will make sense to think of adding a pint as one's fair share of the collective obligation. The point about symmetry is certainly not enough to establish this, but it does – I think – help refocus our attention in the right direction: namely, one of trying to identify features that will allow us to explain why in Collective Harm Cases it is not actually true that one's individual act is instrumentally superfluous. What our discussion in this section is meant to reveal is that we cannot deal with the "it makes no difference" argument by invoking the idea of a duty to "pull one's weight", or to do one's "fair share." Instead, before we can make use of these ideas in understanding the morality of Collective Harm Cases, we first need to be able to debunk the claim that my acting in the relevant way would be instrumentally superfluous and unhelpful. Julia Nefsky 18 so you are accountable for it.' But, on closer inspection, it's not clear this story can get off the ground. For any given torturer, how can we say that she is participating in this collective action? To participate in a collective action is to intentionally do your part in a joint project – in this case, presumably, a project of torturing the victim.28 But for any given "torturer", her act of turning up the dial a single notch makes no difference to the pain of the victim. It doesn't seem to play any non-superfluous role in causing the harm. So, how can we say that in turning it she is intending to do her part in this joint project? Her intention in turning up the dial could just be to acquire fifty dollars at no cost to anyone. Indeed, if she decided to turn the dial only after realizing that doing so won't make any difference, that probably is her intention. Kutz explains that the intentions that make an act participatory need not be "explicit in deliberation"; they can be "functionally implicit" in one's behavior.29 But this does not help here. We cannot say that while an intention to do her part in a collective project of torturing the victim may not be explicit in her deliberation, it is functionally implicit in her behavior. Her behavior, after all, is behavior that won't make any difference to the torture the victim undergoes. The extent to which the victim is tortured will be exactly the same give or take her turn of the dial. So, we can't simply appeal to that action to reveal that she is implicitly trying to do her part in a joint project of torturing the victim. Nor can we say that there are other aspects of her behavior that reveal her implicit intention to do her part in the collective project; that would be stipulating extra features that need not be present in the case. We have said that to handle cases like Drops of Water or environmental damage – cases of "unstructured collective harm" – Strong Participation can talk about reason to participate in a collective action aimed at preventing or mitigating the harm. But the same problem arises here. 28 There is no other potential joint aim specified in the example. 29 Complicity, p. 82. Julia Nefsky 19 Again, participating in a collective action is to intentionally do your part in a collective project. In Drops of Water it would be a project of alleviating the men's suffering; in the case of environmental damage it would be a project of mitigating environmental damage. But how can doing something that would be merely superfluous with respect to our bringing about these outcomes count as doing my part in our bringing them about? Kutz writes, "my part is defined as the task I ought to perform if we are to be successful in realizing our shared goal."30 A task that can't make any difference with respect to our shared goal seems to be no such task. Indeed, if we could have said that adding my pint is the task that I ought to perform if we are to relieve the men's suffering, there would not have been a Problem of Collective Harm there to begin with. Advocates of Strong Participation might object that I am presupposing an instrumental conception of "doing my part". Why can't Strong Participation invoke a broader conception – a conception under which my act need not do anything useful toward the achievement of the shared goal in order to count as doing my part? Indeed, while in some places Kutz writes things like "participatory intentions can, thus, be seen as merely a species of ordinary, instrumental intentions, differentiated by the group-oriented content of the goal they specify"31, in other (nearby) places, Kutz explicitly endorses a broader conception. For example: Contributory relations ["my part" relations between individual act and collective end] might take an instrumental form if what the agent does helps cause the collective outcome (my pushing helps to move the car), or if the agent's part is a constitutive element of the group act (stepping this way is part of dancing a tango). The relation might be expressive if by doing one's part, one thereby exemplifies one's membership in a group or participation in an activity, as when by voting I express my membership in a political community. And the relation might be normative if one performs one's part because of norms internal to some group or institution that demand certain behavior (I wear a dark suit as an IBM employee.) ... What makes my behavior participatory is nothing more (and nothing less) than my conception of what I do as related to the group act.32 30 Complicity, p. 81. 31 Complicity, p. 84. 32 Complicity, p. 82, my emphasis. Julia Nefsky 20 If Strong Participation invokes this broad understanding of doing one's part, can't it avoid the Superfluity Problem by saying that it is non-instrumental contributory relations that are at play in Collective Harm Cases? For instance, while adding a pint will not do not anything instrumentally significant, it can still count as doing my part in our collective project of alleviating the suffering, because it expresses my support for the project or my solidarity with those who have donated.33 The problem is, though, that on this proposal, Strong Participation cannot explain why I have reason specifically to add my pint to the cart, rather than – for instance – to wear a T-shirt that says, "I support the rehydration project!" Especially insofar as we are accepting that neither amounts to doing anything instrumentally significant, wearing the T-shirt could just as well express my support, or my solidarity, as adding my pint could. Strong Participation, in other words, would be unable to differentiate between an act of pouring in one's pint and any other instrumentally insignificant act that could similarly express support or solidarity: neither makes any significant causal contribution to the achievement of the collective goal, and either can make you a participant in the collective project because of its expressive features. The problem here is similar to the one we saw with the attempt to appeal to a broader notion of fairness. Turning to non-instrumental ways of doing one's part in the joint project avoids the Superfluity Problem only by deflating Strong Participation's potential to address what is at issue in the Problem of Collective Harm. As we've said, to address what is at issue, we need an explanation of why each of us specifically has reason to do an act that is of the sort that if 33 Why not say that adding a pint counts as "doing my part" in the constitutive instrumental sense? We could say that my adding a pint is a constitutive element of the collective act of together adding our pints to the cart. The problem with this is that, at least on Kutz's account, a collective act is defined in terms of the collective end. The collective end could be either (a) a state of affairs, or (b) an activity (e.g. dancing the tango), or (c) an institution of some kind. (Kutz, p. 82) In Drops of Water, the most plausible understanding of what our collective end would be is the alleviation of the men's suffering (a state of affairs), not the activity of together adding our pints to the cart. We would not be adding our pints for its own sake, with the alleviation of the suffering as a mere foreseeable consequence. (When we dance the tango, on the other hand, we may be doing this for its own sake, rather than to bring about some further state of affairs.) Julia Nefsky 21 enough of us do it, this will collectively cause (or, depending on the case, prevent) the morally significant outcome of concern. Finding a reason to add a pint to the cart that is just as much a reason to instead wear a T-shirt that expresses one's support or solidarity does not do this. To summarize: Rejecting the Implication approaches try to solve the Problem of Collective Harm by accepting that your individual act can't make any difference, and explaining why nonetheless there is still reason to do it (or, depending on the case, reason to refrain). But the problem is that – at least for the three views we have seen – the explanations they offer do not seem to work as long as your act would be instrumentally superfluous. And yet that seems to be exactly what we are accepting when we accept that your act can't make any difference. 8. Can't We Just Apply a Non-Consequentialist Theory and Call it a Day? It is tempting to think that it is only consequentialist theories that will have trouble with Collective Harm Cases.34 After all, non-consequentialists do not think that all that matters morally is the difference you make in outcome. So, the suggestion goes, their theories will do just fine. Our discussion of the Superfluity Problem helps reveal why this view about nonconsequentialist theories is mistaken. Recall, we cannot solve the Problem of Collective Harm by giving a reason for action that has nothing particularly to do with the morally significant outcome in question. I may have all sorts of reasons to refrain from driving: to get more exercise, to save money, to reduce the risk of causing or being in an accident, to avoid stressful traffic jams. But, of course, none of these can provide a solution to the Problem of Collective Harm that arises with respect to car driving and environmental damage. We need a reason for action that connects in some appropriate way to the fact that widespread car use causes environmental damage. 34 This is a common reaction I have received from philosophers in conversation. Julia Nefsky 22 The problem for non-consequentialists is that, even though they do not think that all that matters morally is the difference you make in outcome, it's not clear that there can be any reason to X, which connects appropriately to outcome Y, if X-ing cannot make any difference with respect to Y. If acting in the relevant way would be merely superfluous with respect to the outcome in question, it's not clear that we can get any relevant story going as to why you have reason to do it. Suppose, for instance, that you subscribe to a theory of Virtue Ethics. Suppose you think that the right thing to do is what the virtuous person would do. Can this view explain why you ought to add your pint to the cart, even if doing so won't make a difference? It's not clear that it can. If it really won't make a difference to the well-being of the suffering men, then it's not clear that the virtuous person would add her pint. Surely, the virtuous person does not act wastefully, and insofar as it is true that adding a pint won't make any difference, doing so appears to be a waste. Unless we can say that adding a single pint could genuinely help to alleviate the men's suffering, it does not seem we can say that this is what the virtuous person would do. Or, consider the virtues and vices that a Virtue Ethicist might appeal to in order to explain why one ought to add one's pint. They might want to say that keeping your pint when these men are suffering would be greedy or selfish, and that to donate it would be generous or compassionate. But if adding your pint would be superfluous with respect to the alleviation of the men's suffering, it doesn't seem any of these would apply. Making a 'donation' that will not do anything useful does not seem generous or compassionate; it seems foolish and wasteful. And Julia Nefsky 23 if you have considered whether it would be of any use to pour in your pint and have correctly determined that it would not be, then keeping this water does not seem greedy or selfish.35 Now, suppose you are a Kantian. Suppose you think that Kant's Formula of Humanity gets things right. You agree that you should "act that you use humanity, whether in your own person or in the person of any other, always at the same time as an end, never merely as a means."36 Can this explain why you ought to refrain from turning up the dial a notch in Harmless Torturers? If turning the dial won't make a difference for the worse to anyone, it's not clear why it would count as treating someone as a mere means. If you are not contributing in any nonsuperfluous way to bringing about the victim's suffering by turning the dial, then how can we say that in doing so you are treating him as a mere means? You don't seem to be treating him in any way at all by doing that particular act. Unless your act would play some significant role in causing the outcome, it does not seem we can appeal to the Formula of Humanity to explain why you ought not to do it.37 'Okay', you might say, 'but surely the more promising place to look is Kant's Formula of Universal Law.' That formulation might seem to be perfectly suited to avoid the Superfluity Problem. Let's see if this is so. The Formula of Universal Law says that an act is right if and only if you can will that the maxim of the action become a universal law of nature without contradiction.38 To determine whether this condition is met, we must imagine a world in which, by your will, everyone acts on 35 Walter Sinnott-Armstrong makes a similar point in "It's Not My Fault: Global Warming and Individual Moral Obligations," Perspectives on Climate Change: Science, Economics, Politics, Ethics (Advances in the Economics of Environmental Research), Volume 5 (2005), pp. 303-304. 36 Immanuel Kant, Groundwork of the Metaphysic of Morals, Trans. and Ed. Mary Gregor (Cambridge University Press: Cambridge, 1997), 4: 429. 37 The Formula of Humanity might imply that you have other duties with respect to that victim: for example, to do what you can – if anything – to alleviate the pain, or to help him get back on his feet afterward. 38 Groundwork of the Metaphysic of Morals, 4: 421. Julia Nefsky 24 the maxim in question, and we must look for, first, a contradiction in conception39 and, second – if there is no contradiction in conception – a contradiction of the will.40 If there is neither, acting on the maxim is permissible. There is a difficult question concerning what should go into the description of the maxim. But it seems that for many Collective Harm Cases, however we describe the maxim, we are not going to get a contradiction in conception.41 In Drops of Water, whether we think of the maxim as 'refrain from adding my pint of water to the cart, so as to keep it for myself', or as simply 'keep my water supply for myself', or as 'don't give something away when doing so won't make a difference', there is no contradiction in conception. For each of these, it is possible to act successfully on the maxim in a world in which everyone has the same maxim. What about the 'contradiction of the will' test? Here we might get different results depending on how we describe the maxim. Suppose, first, that we describe the maxim with a fair amount of specificity. For example, let's take the maxim to be 'refrain from adding my pint of water to the cart, so as to keep it for myself'. We can universalize this maxim without a contradiction of the will. For there to be a contradiction of the will it must be that my willing this maxim as a universal law is in contradiction with something else that I must will as a rational agent. There is no such contradiction. I will not be negatively affected if no one puts water into this cart. The maxim described in this way passes the test. But the problem might be that we are describing the maxim in an overly narrow way. Perhaps if we give a more general description, we can obtain the contradiction of the will that we 39 As I'm understanding it, there is a contradiction in conception if it is not possible to act successfully on the maxim in the world in which everyone has that maxim. (But note that the point I will make remains if we go with a stronger interpretation under which there is a contradiction in conception only if the world in which everyone has the maxim is logically impossible.) 40 As I'm understanding it, there is a contradiction of the will if willing the universalization of the maxim contradicts something else that you must will as a rational agent. 41 I say 'many' rather than 'all', because in free-rider Collective Harm Cases (see note 19 above), we may get a contradiction in conception under some descriptions of the maxim. Julia Nefsky 25 are looking for. How can we describe the maxim of refraining from adding a pint more broadly so that the universalization test yields a contradiction of the will? Here is what seems to me to be the best (if not the only good) answer: we should describe the maxim as, 'refrain from helping those in need.' Willing the universalization of this maxim produces a contradiction of the will. This, after all, is the maxim of non-beneficence, one of Kant's own examples of a maxim that fails the contradiction of the will test. But here – just when it seems we have found the contradiction we were looking for – we hit the Superfluity Problem. We can only describe my maxim of refraining from adding my pint as a maxim of 'refraining from helping those in need' if my adding the pint would help those in need. But the claim that my act won't make any difference to those in need seems to be precisely a claim that it will not help them – that it would, instead, be instrumentally superfluous when it comes to alleviating their suffering. In any case, we can at least conclude that it is far from clear how the Formula of Universal Law would yield the result that you ought to add your pint to the cart, unless we can first debunk the claim that doing so will not help.42 9. The Problem of Collective Harm (and How to Solve It) In Collective Harm Cases, one's individual act (or omission) doesn't seem to make any difference. It appears that things will be the same in all relevant respects, whether or not it is performed. And what this seems to mean is that acting in this way would be instrumentally superfluous – that it would not itself play any significant role with respect to bringing about the 42 Perhaps there is a (correct) way of describing the maxim that avoids this problem. Even if that is so, there is a different level at which the Superfluity Problem might show up. If we are going to appeal to the Formula of Universal Law, we need an understanding of why this test tells us what we ought to do. We need to know, in other words, why universalizability matters. It could very well be that the explanation we want to give about why universalizability matters will not work in Collective Harm Cases unless we can first show that one's individual act is not instrumentally superfluous. For instance, a common idea is that acting in a way that is not universalizable is wrong because it is exhibits a certain kind of unfairness: it amounts to relying on others to do something, while excepting yourself. For reasons similar to those given in our discussion of the Fairness Approach, this story might not work in Collective Harm Cases unless we can first debunk the impression that one's act would be instrumentally superfluous. Many thanks to Sergio Tenenbaum for this point. Julia Nefsky 26 outcome in question. This is why it is hard to see how there could be any point in acting. While one knows that unless enough people act in this way bad consequences will result, one is faced with the apparent uselessness of acting in this way oneself. The three Rejecting the Implication views that we discussed approach this problem by trying to show that, even if acting in the relevant way can't make a difference, there is still reason to do so. They do this by turning the focus away from the concern about the apparent instrumental superfluity of your individual act, and onto other sorts of considerations: being part of a group that makes a difference, participation in a collective action, fairness in the context of a collective obligation. But these considerations can't do the work they are supposed to do as long as your act would be merely superfluous. Unless your act could play a significant, non-trivial role with respect to bringing about the outcome in question, the explanations these views are offering as to why you have reason to do it cannot properly address what is at issue in the Problem of Collective Harm. The same problem seems to arise for various non-consequentialist theories, if we think we can turn to them to provide a solution to the Problem of Collective Harm. In general, to address what is at issue in the Problem of Collective Harm we need to be able to explain how it is that the fact that the men in the desert are suffering from painful thirst means that each of us has reason to add our pints, and that the fact that we are collectively causing climate change means that each of us has reason to take a bicycle to work instead of a car, and so on. That is, we need to give a reason for action that has to do with – in a central way – the morally relevant outcome of concern, and that tells us specifically to do the sort of acts that could collectively cause (or, depending on the case, prevent) that outcome. It is doubtful we can Julia Nefsky 27 do this adequately while at the same time accepting that acting in this way would be merely superfluous with respect to bringing about that outcome. What all of this reveals is that to solve the Problem of Collective Harm, we need to show that one's individual act isn't merely superfluous. We need to show that we are mistaken when it seems to us that an individual act of the relevant sort won't itself do anything instrumentally significant. Of course, it's not clear that we can do this. But unless we can, a satisfactory answer to the problem seems unlikely. If we can show that an individual act of the relevant sort isn't merely superfluous, and rather could significantly help to bring about the outcome in question, this would address the Superfluity Problem that Weak Participation, Strong Participation and the Fairness Approach face. So, that core problem with those views would be resolved. Similarly, the general nonconsequentialist theories we considered would then, it seems to me, be able to handle Collective Harm Cases. For instance, if one's turning up the dial on the shock machine (in order to get $50) might play a significant role in harming the victim, it makes sense to say – as the Kantian would want to – that in doing so one is treating him as a mere means. But the point isn't just that this would allow these views about our moral reasons in Collective Harm Cases to get off the ground. Rather than just being a matter of filling in the hole that we've identified with these views, showing that an individual act can do something significant toward bringing about the relevant outcome is the central task of solving the Problem of Collective Harm. I've argued that it is doubtful that we can solve the problem in a satisfactory way without showing this. But, importantly, if we can explain how it is that your individual act can do something instrumentally significant, this would itself solve the core issue of the Problem of Collective Harm. What leads us to fail to see how we could have reason for action in Julia Nefsky 28 Collective Harm Cases is the impression that doing so would be useless – that it wouldn't do anything significant toward bringing about the relevant good outcome, or preventing the relevant bad one. If we could show that this were not true, that core issue would be addressed. Now, what this might seem to mean is that if we want to solve the problem, we need to take a Denying the Description approach. That is, we must try to argue that in Collective Harm Cases, contrary to appearances, individual acts do, or at least might, make a difference with respect to the morally relevant outcome of concern. This line of response has largely been taken by those seeking a consequentialist solution to the problem. But, since it would address the Superfluity Problem if it worked, one implication of our discussion is that non-consequentialists also have a stake in whether such a view can succeed. I discuss the Denying the Description strategy elsewhere. Unfortunately, if my conclusions there are correct, such an approach does not work.43 I don't doubt that in some cases we can and should deny the "it won't make a difference" claim: we are surely sometimes making a mistake when we think that one act won't make a difference. But, while that will surely be true in some cases, I argue that we cannot draw that conclusion in general. We are not, I argue, in a position to deny that there are Collective Harm Cases in which one's individual act simply can't make a difference. Suppose, for the sake of argument, that I am right about this. It might seem, then, that we have hit a standstill. On the one hand, the upshot of the present paper is that we cannot solve the Problem of Collective Harm unless we can maintain that one's individual act can play a significant instrumental role with respect to the outcome of concern. But on the other hand, if my conclusions elsewhere are correct, we cannot deny that there are Collective Harm Cases in which one's individual act simply can't make a difference. 43 "Consequentialism and the Problem of Collective Harm: A Reply to Kagan". Julia Nefsky 29 But rather than a standstill, these conclusions together carve out a new space in which a solution to the problem might be found. Instead of taking a Denying the Description approach, we could try to show that an act can play a significant instrumental role in bringing about an outcome, even if it cannot make a difference with respect to that outcome. If an act cannot make a difference, we take this to mean that it would be instrumentally superfluous. A way of solving the Problem of Collective Harm would be to show that this assumption is mistaken. What would it mean for this assumption to be mistaken? Whether one's act makes a difference is a particular, counterfactual matter. It is a matter of whether had one not done it, things would have been (relevantly) different. The assumption would be mistaken if this were not the correct test for determining whether an act is instrumentally useful. Perhaps my adding a pint to the cart could play a significant, non-superfluous role in alleviating the men's suffering even though the extent to which their suffering is alleviated will not be different depending on whether or not I do so.44 In summary: the upshot of our investigation of Rejecting the Implication approaches is that to respond to the "it makes no difference" argument, we need to be able to show that one's individual act in Collective Harm Cases is not instrumentally superfluous. This is the real challenge of the Problem of Collective Harm. Attempting to address the problem by turning our attention to other sorts of reasons for action will not work. One way of trying to meet the challenge is to take a Denying the Description approach: to try to show that an individual act actually can make a difference in Collective Harm Cases. But that is not the only way. There is a 44 I am not claiming that this strategy will work. For the purposes of this paper, the point is just that this strategy exists as a second way – besides denying the "it makes no difference" description – of trying to address the superfluity challenge. I attempt to develop a solution of this sort in "How You Can Help, Without Making a Difference." Julia Nefsky 30 second strategy that remains open: we could question whether an act needs to be a differencemaker in order to be instrumentally useful.

SEE ALSO EPISTEMOLOGY; KNOWLEDGE, THEORIES OF. BIBLIOGRAPHY Aristotle. The Basic Works of Aristotle (1941). Edited by Richard McKeon. New York: Modern Library, 2001. Augustine. On the Trinity [De trinitate]. Edited by Gareth B. Matthews. Translated by Stephen McKenna. Cambridge: Cambridge University Press, 2002. Augustine. On the Profit of Believing [De utilitate credendi]. Translated by C. L. Cornish. In Nicene and Post-Nicene Fathers, First Series, edited by Philip Schaff. Vol. 3: Augustine: On the Holy Trinity, Doctrinal Treatises, Moral Treatises, 347–366. Buffalo, NY: Christian Literature Publishing, 1887. Copleston, Frederick. A History of Philosophy. 9 vols. Westminster, MD: Newman Press, 1946–1975. Descartes, René. The Philosophical Writings of Descartes. Translated by John Cottingham, Robert Stoothoff, and Dugald Murdoch. Cambridge: Cambridge University Press, 1985. Gerson, Lloyd. Ancient Epistemology. Cambridge: Cambridge University Press, 2009. Kant, Immanuel. Critique of Pure Reason. Translated and edited by Paul Guyer and Allen Wood. Cambridge: Cambridge University Press, 1998. Kenny, Anthony. A New History of Western Philosophy. Vol. 4: Philosophy in the Modern World. Oxford: Clarendon Press, 2007. Kirk, G. S., J. E. Raven, and M. Schofield. The Presocratic Philosophers: A Critical History with a Selection of Texts. 2nd ed. Cambridge: Cambridge University Press, 1983. Miner, Robert. Truth in the Making: Creative Knowledge in Theology and Philosophy. New York: Routledge, 2004. O'Callaghan, John P. Thomist Realism and the Linguistic Turn: Toward a More Perfect Form of Existence. Notre Dame, IN: University of Notre Dame Press, 2003. Pasnau, Robert. Theories of Cognition in the Later Middle Ages. Cambridge: Cambridge University Press, 1997. Plato. Complete Works. Edited by John M. Cooper. Indianapolis, IN: Hackett, 1997. Sokolowski, Robert. Phenomenology of the Human Person. New York: Cambridge University Press, 2008. Tachau, Katherine. Vision and Certitude in the Age of Ockham: Optics, Epistemology, and the Foundations of Semantics. Leiden, Netherlands: Brill, 1988. Chad Engelland Assistant Professor, Department of Philosophy Borromeo College Seminary and John Carroll University Cleveland, OH (2013) EPOCHÉ Epoché designates for phenomenologists an entryway into philosophical contemplation and self-reflection in which the meditating "I," the thinker, gains distance from the concerns of everyday life. From the Greek d  (suspension of disbelief ), the term was used by ancient skeptics. The word was revived by Edmund HUSSERL (1859–1938), who taught Martin HEIDEGGER (1889–1976) and Edith STEIN (1891–1942) and who was an important influence on Max SCHELER (1874– 1928), Dietrich von HILDEBRAND (1889–1977), Karol Wojtyła (Pope JOHN PAUL II; 1920–2005), Robert Sokolowski, and many others. By performing the epoché, I (the thinking person) disengage from belief or disbelief by holding off on assertions or questions about the existence of things that extend beyond (transcend) their appearance in my conscious life. That is, I suspend my belief in such things as cats, atoms, houses, psychological states, scientific theories, and so forth-things that are "transcendent" in the sense that they appear to be what they are independent of any ideas that I may have about them. Philosophical disinterestedness is thus made possible. For Husserl, philosophy is marked off from the "natural attitude" of human life in which, without much ado, we accept the existence of this or that object. In the natural attitude we occasionally doubt or deny the reality of something (the tooth fairy) or suspend judgment on a possible truth (the existence of extraterrestrial life). When we do, our intellectual step back happens within our overall acceptance of the world, the whole in which every real thing has its causes, effects, and meaning. That is, in normal living, we always believe that the world's reality goes beyond our experience of it. Such world-belief is naïve because lacking in self-understanding-we are not self-aware of our motives for positing transcendent reality. Usually we are carried along by the evidence, but do not reflect on the conscious acts by which we accept-and thereby "constitute" for ourselves-the world we believe in. Husserl's epoché "neutralizes" this basic belief and "brackets" the existence of the world, so that we can trace any reality's being-for-us back to conscious acts that establish its validity for us. This tracing-back is called the phenomenological reduction. (If we do not practice the epoché consistently while philosophizing, Husserl claims, the reduction would become a false psychological idealism, a relativism claiming that reality depends on our psychological states.) The epoché primarily appears as an ascetic act: while engaging in theoria, we refuse to participate in the natural attitude and deny ourselves any appeal to transcendent facts. Husserl emphasizes what is gained from this philosophical entryway. In overcoming the naïveté of normal belief, we do not reject its truths or deny its realities but deepen our understanding of them as accomplished by persons. The things appearing as real are parenthesized, not denied; we contemplate the phenomena but do not turn them into mere phantoms Epoché N E W C A T H O L I C E N C Y C L O P E D I A S U P P L E M E N T 2 0 1 2 − 1 3 : E T H I C S A N D P H I L O S O P H Y , V O L U M E 2 487 (c) 2013 Cengage Learning. All Rights Reserved. or meanings. For Husserl, the epoché conceals nothing. It reveals what straightforward life overlooks: (1) the spiritual and intellectual life by which reality is real for us and truth is achieved by us, and (2) the presenceto-us of the world and its objects, a presence that we usually live through without appreciating. Phenomenology seeks to follow the ancient command "Know Thyself," while also clarifying the objectivity of the world. Finally, the epoché reveals that persons have a transcendental (spiritual and intellectual, not just psychological or thingly) mode of being. SEE ALSO EIDETIC VARIATION; PHENOMENOLOGY. BIBLIOGRAPHY Husserl, Edmund. The Crisis of European Sciences and Transcendental Phenomenology: An Introduction to Phenomenological Philosophy. [1954.] Translated by David Carr. Evanston, IL: Northwestern University Press, 1970. Husserl, Edmund. Ideas: General Introduction to Pure Phenomenology. [1913.] Translated by W. R. Boyce Gibson. London and New York: Routlege, 2012. Sokolowski, Robert. Introduction to Phenomenology. Cambridge, UK, and New York: Cambridge University Press, 2000. Zahavi, Dan. Husserl's Phenomenology. Stanford, CA: Stanford University Press, 2003. Molly Brigid Flynn Associate Professor of Philosophy Assumption College, Worcester, MA (2013) EQUALITY The concept of equality is an important aspect of political and moral philosophy and also of theology. It is a bedrock principle of economic and political justice and of efforts to advance human dignity and human rights. Thus it is not surprising that meditations on the notion of equality have formed an essential ingredient of the earliest works of ancient Greek and Roman philosophers, the Hebrew Scriptures, and the preaching of Jesus as recorded in the New Testament. In turn, for two millennia the Catholic Church, together with its philosophers, theologians, and doctors, has preserved its own understanding of this concept in its teachings about humanity and in its charitable practices. What Is Equality? Equality is usually understood as sameness in regard to some specific kind of measure when comparing entities of the same species or kind. The idea can apply to the size, weight, stature, worth, extent, or nature of an entity. Applied to human beings it can refer to their inherent being or to their material condition or status, and thus it could apply to their nature, moral worth, dignity, station, rank, status, economic class, social condition, or legal standing. Equality need not always imply exact identity but rather similarity, recognizing that individual differences among a class of beings can subsist in the midst of the equality of kind. Equality aims at the essence of the nature of a thing, even while accidents, such as shape, size, color, age, and position, may differ. Some kinds of equality are easily measurable, such as the distribution of wealth or material goods. Other kinds of equality refer less to outcomes than to opportunities. Equality of income, for instance, is unlikely for human beings, so long as they are left free to use their talents, to work and to innovate in ways that could readily lead to economic disparity. But legal equality would presume that all individuals have the same access to justice and opportunities. In democratic systems equality applies to the principle of one-man, one-vote, and to equal treatment before the law. The essential equality of human beings does not presuppose equal virtue, talent, or effort in individual human beings. But it does imply that persons are not discriminated against as persons. In terms of moral or essential equality, JudeoChristian teaching insists that all persons have equal dignity as members of the one human family, all of which are created in God's divine image and likeness (see, for example, Gen 1:26–27 and the Catechism of the Catholic Church [CCC], 356–360). In the history of Western Civilization, this fundamental teaching would prove extremely influential in the ultimate development of the idea of equality in Western thought. Equality in the History of Philosophy. The issue of equality played an important role in Aristotle's ethical (Nicomachean Ethics) and political philosophy (Politics). He showed that justice demands that equals be treated as such, and that unequals should be treated unequally. In the Politics, Aristotle (384–322 BC) noted that unequal distributions of wealth in a city could lead to conflict, whereas a large middle class could serve as a check on rebellious tendencies and their consequent injustice. In his On Duties, Cicero (106–43 BC) included all humanity within the orbit of duty, thereby implying human equality. Hebrew thought warned that aliens and slaves should be treated with consideration because all men spring from the same divine creative activity. Greek and Roman Stoics regarded all human beings as part of a universal brotherhood of men. In the New Testament, Jesus treated everyone, the mighty and the lowly, with respect. Christ's particular care for the poor, and his charge that they be charitably served, became a clarion call in Christian social development, giving rise to the modern expression of the "preferential option" or "preferential love" for the poor in the work of the Equality 488 N E W C A T H O L I C E N C Y C L O P E D I A S U P P L E M E N T 2 0 1 2 − 1 3 : E T H I C S A N D P H I L O S O P H Y , V O L U M E 2 (c) 2013 Cengage Learning. All Rights Reserved.

The Logicality of Language: A new take on Triviality, "Ungrammaticality", and Logical Form ∗ Guillermo Del Pinal Leibniz-ZAS & University of Michigan January 9, 2018. forthcoming in Noûs (DOI: 10.1111/nous.12235) Abstract Recent work in formal semantics suggests that the language system includes not only a structure building device, as standardly assumed, but also a natural deductive system which can determine when expressions have trivial truthconditions (e.g., are logically true/false) and mark them as unacceptable. This hypothesis, called the 'logicality of language', accounts for many acceptability patterns, including systematic restrictions on the distribution of quantifiers. To deal with apparent counter-examples consisting of acceptable tautologies and contradictions, the logicality of language is often paired with an additional assumption according to which logical forms are radically underspecified: i.e., the language system can see functional terms but is 'blind' to open class terms to the extent that different tokens of the same term are treated as if independent. This conception of logical form has profound implications: it suggests an extreme version of the modularity of language, and can only be paired with non-classical-indeed quite exotic-kinds of deductive systems. The aim of this paper is to show that we can pair the logicality of language with a different and ultimately more traditional account of logical form. This framework accounts for the basic acceptability patterns which motivated the logicality of language, can explain why some tautologies and contradictions ∗ For many helpful discussions on this project, I am especially grateful to Gennaro Chierchia, Daniel Rothschild and Uli Sauerland. This paper also benefited from comments and discussions with Alexander Dinges, Gabriel Dupre, Patrick Elliot, Ezra Keshet, Clemens Mayr, Marie-Christine Meyer, Eleonore Neufeld, Carlotta Pavese, Hazel Pearson, Francois Recanati, Eric Swanson, Emmanuel Viebahn and Julia Zakkou. Two anonymous reviewers for Noûs provided me with extremely helpful and constructive comments that led to several key improvements of the paper. I am also grateful to audiences at the University of the Basque Country, Leibniz-ZAS Berlin, University of Cologne and the University of Turin. Finally, thanks to the Alexander von Humboldt Foundation for supporting this project. 1 Del Pinal are acceptable, and makes better predictions in key cases. As a result, we can pursue versions of the logicality of language in frameworks compatible with the view that the language system is not radically modular vis-á-vis its open class terms and employs a deductive system that is basically classical. Keywords: logical form, triviality, quantifiers, contradictions, tautologies, natural logic, modularity. Words: 11,773 1 Introduction One of the most important recent hypothesis about the computational architecture of language is that it consists not only of (i) a structure building device (e.g., 'Merge' + the corresponding semantic operations), but also of (ii) a 'natural logic' or automatic deductive system. We shall call the view that (i) and (ii) work together to determine the set of acceptable expressions of natural languages, the 'logicality of language' (Fox 2000, Fox & Hackl 2007, Chierchia 2006, 2013, Abrusán 2011a, 2014). The question explored in this paper is: What notion of logical form should we pair with the logicality of language? The term 'logical form' is used here in its broad descriptive sense. As a starting point, we can say that the 'logical form' of an expression is the underlying representation which is the input to its semantic interpretation. At this level, ambiguities are resolved and semantic values can be assigned to complex expressions as a function of those assigned to their constituents (Heim & Kratzer 1998, Fox 2003). Accordingly, our main question can be reformulated as follows: to develop a defensible version of the logicality of language, what kinds of revisions do we need to make to the standard conception of logical form? The answer to this question will determine not only the ultimate viability of the logicality of language hypothesis, but also its implications to various foundational issues such as the degree of modularity of the language system and the nature of its interface with our general reasoning capacities.1 1 Terminological note. Most philosophers distinguish between two broad notions of 'logical form' (see e.g., Stanley 2000, Szabó 2012, Iacona 2017). In its 'descriptive' sense-which is the primary focus of this paper-the 'logical form' of an expression is a level of representation that is the input to semantic interpretation. In its second, 'revisionary' sense, 'logical forms' are formulae in artificial languages which, for scientific or other investigations, we can assign to expressions of natural languages. Depending on the goals, we can choose different regimentations. We are not directly concerned with 'logical forms' in this second sense. Still, since a major project of linguistic semantics is to model why native speakers find certain inferences bad or compelling, the two notions can be intimately connected. Indeed, some philosophers have searched for a unified conception (for critical discussion see Szabó 2012 and Iacona 2017). 2 Triviality and Logical Form To appreciate the importance of this issue, we begin by briefly describing why theorists have proposed the logicality of language. Two observation are crucial. First, some robust acceptability patterns cannot be explained purely syntactically. Important examples include systematic restrictions on the kinds of quantifiers that can occur with exceptive phrases, illustrated in (1), and there-existentials, illustrated in (2). Second, given independently plausible logical forms and interpretations for the functional terms, the unacceptable examples in each pair can be shown to be trivial: i.e., in all worlds or situations, (1-a) is false and (2-a) is true. (1) a. *Few students but Sue passed the exam. b. All students but Sue passed the exam. (2) a. *There is every red apple. b. There is a red apple. As we review in §2, this kind of triviality underwrites various general semantic restrictions on the distribution of quantificational determiners, among other systematic acceptability patterns (Gajewski 2002, 2009, Fox & Hackl 2007, Chierchia 2006, 2013, Abrusán 2014). It follows that we can explain these patterns if we accept the logicality of language, i.e., the hypothesis that the language system works with a deductive system that can automatically compute whether, and filter-out when, an expression is trivial (true/false in all worlds or situations). As it stands, however, the logicality of language raises an obvious worry (see e.g., Fox & Hackl 2007, Gajewski 2002, 2009, Chierchia 2013). If the deductive system marks as ungrammatical or unacceptable trivial sentences such as (1-a)-(2-a), why are apparently simpler examples of triviality, such as the contradictions and tautologies in (3), perfectly acceptable? (3) a. It is raining and it is not raining. b. If John is wrong, then John is wrong. Indeed, communication with such superficially trivial expressions is not uncommon. Furthermore, we can imagine a language device which, being paired with an automatic and unforgiving deductive system, would force us to rescue expressions like those in (3) by overtly modifying at least one of the problematic predicates, as in it is raining and it is not raining hard and if John is somewhat wrong, then John is totally wrong. How, then, can we reconcile the presumed 3 Del Pinal logicality of language, so as to account for the patterns in (1)-(2), with the ubiquity of acceptable trivial sentences such as those in (3)?2 In response to this challenge, most proponents of the logicality of language add to the basic framework the following hypothesis: the deductive system operates on logical forms which are radically underspecified with respect to the content of their non-logical terms (Gajewski 2002, Fox & Hackl 2007, Chierchia 2006, 2013). This hypothesis is often attributed to Gajewski (2002, 2008b, 2009), who proposed one of its most simple and elegant formulations: (4) Logical skeletons a. The subset of the trivial sentences which are unacceptable is formally definable by the configuration of their functional terms at logical form. Call these 'L-trivial'. b. Language and its deductive system 'see' only 'logical skeletons': representations that are underspecified with respect to the content of their non-logical expressions. c. Logical skeletons treat all tokens of non-logical expressions as independent-even tokens of the same expression. The basic idea is that if we accept Logical skeletons, as defined in (4), we can explain key patterns of semantic restrictions on quantifiers, such as those illustrated in (1)-(2), without incorrectly predicting that trivial sentences such as those in (3) are also marked. For example, we will show later that in a case like (1-a), the triviality can be traced solely to the configuration of logical terms-the interpretation and identity of the non-logical terms is irrelevant. In a case like (3-a), however, the contradiction is due also to the identity of the non-logical terms. If the deductive system cannot 'see' that the two tokens of rain are the same, it cannot determine that there is a contradiction in (3-a) and so doesn't mark that expression as unacceptable. Assuming the details can be worked out, postulating that the Grammar and its deductive system see only logical skeletons seems to account for the difference between acceptable and unacceptable trivial sentences, a considerable feat.3 2 Terminological note. In this paper, I follow the standard convention in linguistics of using the terms 'unacceptable' and '*' to mark expressions that are bad in a strong sense, i.e., indistinguishable or quite similar to the phenomenology of ungrammaticality. The flip side of this is that, in the sense used here, an expression can be strictly 'acceptable' and still be somewhat odd. There are of course many borderline cases, but in most of the cases explored in this paper the classifications are relatively uncontroversial. 3 I should clarify at the outset that none of the competing views for how to develop the logicality of language identify logical triviality with ungrammaticality. These views are compatible with the standard position that some non-trivial sentences are unacceptable for 4 Triviality and Logical Form Treating logical forms as logical skeletons has far reaching implications for our conception of language, natural logic, and their interface with general knowledge and reasoning. First, it entails a division between semantics and pragmatics according to which what is delivered by the compositional processes to pragmatics are not even characters, as standardly conceived (cf. Chomsky 2005, 2013). Secondly, logical skeletons are best paired with a view of the Grammar as radically modular: i.e., as insulated not only from conceptual systems and general knowledge, but even from information that is standardly taken to be encoded in the lexicon. Thirdly, a deductive system which can only see logical skeletons-such that every predicate hence sentential token is treated as if independent-is one for which most classical formulas and rules of inference are not valid (cf. Williamson 1994). To be sure, proponents of the logicality of language such as Chierchia (2013) and Fox & Hackl (2007) explicitly embrace versions of the first two implications. At the same time, the full effect of the third implication is less appreciated, and arguably problematic for some accounts based on the logicality of language, as I argue in §5.2-5.3. At any rate, it is obviously worth exploring whether we can wed the logicality of language with different assumptions about logical form. The main task of this paper is to show that we can maintain the logicality of language, as outlined in (i) and (ii) above, and account for the difference between L-trivial and acceptable trivial sentences, without assuming that the deductive system operates on logical skeletons. We adopt instead a standard view of logical forms, such that they represent when different tokens are of the same non-logical terms; but assume, following Sauerland (2014) and related proposals by Martı (2006), Pagin & Pelletier (2007), Recanati (2010), Stanley (2000), Szabó & Stanley (2000), Kamp & Partee (1995), among others, that at LF non-logical or open class terms-e.g., nouns and verbs-can be arguments to operators which modulate their meaning.4 purely grammatical reasons. They are also compatible with the view that some expressions which ultimately have trivial truth conditions are acceptable. With respect to the latter point, however, there are some subtle differences between the views (see §3.3 below). 4 For our purposes, the key characteristic of this family of views is that open class terms are represented in a way that allows for modulation. The implementation I present-using an optional higher-order covert operator-is most directly inspired by Sauerland (2014). For reasons that will emerge, I think this implementation has key advantages. Still, I expect that other implementations could, with some refinements, be adopted. For example, we could use a constrained version of the system defended in Pagin & Pelletier (2007) and Recanati (2010), where the interpretation function is defined in terms of a modulation function. We could also pursue a version of Stanley (2000, 2007), and assume that open class terms are restricted via intersective combinations with covert syntactic elements. In all these cases, the hypothesized operations that perform modulation can be heavily constrained-which is 5 Del Pinal (5) Logical forms + Rescale a. The subset of the trivial sentences which are unacceptable is formally definable by the configuration of their functional terms at logical form. Call these 'L-trivial'. b. Language and its deductive system see representations whose constituents, including non-logical terms, have been assigned their semantic values, and see when different tokens are of the same open class term. c. Non-logical items can be arguments of an optional Rescale operator. Different tokens of the same expression can differ with respect to whether/how each token is modified by Rescale. In §4 I show that we can adopt LF+Rescale, as defined in (5), and still account for the target semantic restrictions on the distribution of quantifiers. That is, given standard logical forms and an optional Rescale of open class terms, only L-trivial sentences can be proven to be trivial. In addition, superficially trivial sentences which are perceived as acceptable, such as those in (3), are correctly predicted to be acceptable. It follows that accounting for L-triviality is not a strong reason to accept logical skeletons over LF+Rescale. Furthermore, I argue in §5 that in some key cases only LF+Rescale makes the right predictions. If this is correct, we can maintain the logicality of language without assuming its radical modularity, and in particular that language does not see information encoded in the lexicon that is unique to particular open class terms. We can also maintain, or at least explore the hypothesis, that the natural deductive system of language follows classical inference rules. 2 Restrictions on quantificational determiners The acceptability patterns which are the main focus of this paper concern three well-known restrictions on the distribution of quantifiers. The aim of this section is to see (i) that the key generalizations can best (and arguably only) be captured semantically, and (ii) that in each case we can systematically show that the unacceptable cases have trivial truth-conditions. Our presentation follows closely that of Gajewski (2009).5 required to avoid over-generation of meanings-without affecting any of the points I make here. These options are compared in more detail in §3. 5 There are other acceptability patterns which have been explained as arising from trivial truth-conditions, some of which we will discuss in §5.2-§5.3. These include constrains on adverbial modification (Dowty 1979), polarity items (Chierchia 2006, 2013), modified numerals (Fox & Hackl 2007), and weak islands (Abrusán 2011a, 2014). We focus primarily on the cases involving quantifiers in there-existentials, connected exceptives and comparatives 6 Triviality and Logical Form 2.1 Definiteness effect in there-existential sentences The first case we explore involves a definiteness effect in there-existential sentences, which we already encountered above in (2-a). The basic contrast is captured by examples such as (6-a)-(6-b) below. As noted in the generalization in (7), quantifiers such as many and few are acceptable in there-existentials, whereas quantifiers such as all and most are unacceptable. (6) a. There are some curious students b. *There is every curious student. (7) Generalization: a. Acceptable: some, three, a, many, few, exactly two, no b. Unacceptable: every, all, neither, both, the, most Crucially, the target generalization in (7) can be captured in semantic terms. Specifically, the determiners that can occur in there-existentials are the weak determiners (Barwise & Cooper 1981), defined as in (8): (8) a. Determiner D is positive strong iff for every model M =< J K, De > and every A ⊆ De, if JDK(A) is defined, then JDK(A)(A) = 1 b. D is negative strong iff for every model M =< J K, De > and every A ⊆ De, if JDK(A) is defined, then JDK(A)(A) = 0 c. D is weak if D is not strong. A paradigmatic example of a positive strong determiner is every, as illustrated by the observation that (9-a) is always true, regardless of what interpretation we assign to JstudentK. A paradigmatic example of a weak determiner is some, as illustrated in (10-a), which is false in a world where there are no students, and true in a world in which there is at least one student. (9) JeveryK(A)(B) = 1 iff A ⊆ B [strong] a. Every student is a student. (10) JsomeK(A)(B) = 1 iff A ∩B 6= ∅ [weak] for two reasons. First, they have played a central role in the argument for logical skeletons; secondly, the corresponding generalizations and accounts are less controversial than other generalizations and accounts which appeal to trivial truth-conditions. Of course, whether the logicality of language should be paired with logical skeletons or LFs+Rescale will remain an interesting open issue as long as new cases of unacceptability due to triviality (or counter-examples thereof) continue to be discovered. 7 Del Pinal a. Some student is a student. It is easy to see why the semantic property which captures the acceptability patterns of quantifiers in there-existentials-namely, that positive/negative strong quantifiers result in unacceptability-gives rise to trivial truth-conditions: (i) Assume, as is independently plausible, that Jthere is/areK simply denotes De, i.e., the set of entities in the model. (ii) Given Conservativity,6 if D is (positive) strong, then for all A ⊆ De, JDK(A)(De) = JDK(A)(De ∩ A) = JDK(A)(A) = 1.7 From (i) and (ii) it follows that (6-b), given the logical form in (11-a) (where S = the set of curious students), is trivially true, whereas (6-a), given the logical form in (11-b), is true or false depending on whether there are any curious students. (11) a. JeveryK(S)(S ∩De) = 1 iff S ⊆ S b. JsomeK(S)(S ∩De) = 1 iff S ∩ S 6= ∅ If we assume that trivial sentences are filtered out by the deductive system, we can explain why only weak determiners can occur in there-existential sentences. 2.2 Selection properties of Connected Exceptive Phrases The second case we consider involves the selection properties of connected exceptive phrases. The basic pattern is illustrated by the contrast in (12-a)- (12-b). As specified in (13), the target generalization is quite simple: only determiners such as every, all, none can host connected exceptives. (12) a. *Some student but Sue passed the exam. b. Every student but Sue passed the exam. (13) Generalization: a. Acceptable: every, all, none b. Unacceptable: the rest Note that the class of determiners that can host connected exceptives is semantically definable: they are the universal (negative/positive) quantifiers. 6 Conservativity: For all M and all A,B ⊆M , JDK(A)(B) iff JDK(A)(A ∩B). 7 We focus on the case of positive strong quantifiers for simplicity. If D is negative strong (e.g., neither), then we should change the right hand side of the equation to 0, and instead of a trivially true we get a trivially false sentence (i.e., false in all words/situations). 8 Triviality and Logical Form As before, this suggests that the semantic property which captures the target generalization will play a key role in its explanation. The basic account, due to von Fintel (1993), is based on two observations: (i) The complement of but is the least that you have to take out of the restrictor of the host quantifier to make the statement true. (ii) Universal determiners-e.g. every and no-are the only determiners that systematically allow such minimal exceptions. Other quantifiers yield logical trivialities. Let us see how this works. Initially, one could be tempted to assign but the entry in (14-a). The problem, however, is that then (12-b), for example, would not entail that Sue did not pass the exam. To avoid this problem, von Fintel argues that we need an entry closer to (14-b): (14) EveryD [studentA but MaryC ] smokesP a. JbutK(C)(A) = A− C b. JbutK(C)(A)(D)(P ) = 1 iff C 6= ∅ andD(A−C)(P ) = 1 and ∀S[D(A− S)(P ) = 1→ C ⊆ S]} {{ } captures 'the least you have to take out' constraint To see the difference between the two entries for but, consider a world w were every student, including Mary, smokes. According to (14-a), Every student but Mary smokes would be true in w, which is an incorrect prediction. In contrast, (14-b) correctly predicts that the statement is false in w. Now, consider the interaction between but and different kinds of quantifiers. Focusing on examples such as (12-a), we can show that any left upward entailing quantifier (e.g., some, many, three), when hosting a connected exceptive phrase, will result in a trivially false statement:8 (15) D is a left upward entailing quantifier iff ∀A,B,C s.t. JDK(A)(C) = 1 & A ⊆ B, JDK(B)(C) = 1 Why? If D is left upward entailing and you have removed some individuals from D's restrictor and the statement is true, then you could always have removed fewer and still be left with a true statement. To see this: suppose 8 Note that universal positive/negative quantifiers are left downward entailing. So what we have to show is why the other left monotonic quantifiers, specifically, the left upward entailing, cannot host connected exceptives. For simplicity, we ignore for now the left non-monotonic quantifiers such as exactly 3, which also cannot host connected excepitives, as is illustrated by *Exactly three students but/except Mary smoked. 9 Del Pinal A = B − s, then A ⊆ B. In other words, the set A denoted by 'B but s' ⊆ B. Given a left upward entailing quantifier, you can thus replace A with B (its superset) in its restrictor. It follows that s is not the least you have to take out to make statement true, since you can simply take out nothing. 2.3 Negative islands in comparatives The third and final acceptability pattern concerns negative islands in comparatives (Gajewski 2008b). The basic observation, illustrated in (16-a)-(16-b), concerns constraints on the kinds of quantifiers that can appear inside a comparative clause. (16) a. *Mary is taller than no other student is. b. Mary is taller than every other student is. (17) Generalization a. Acceptable: the rest. b. Unacceptable: no, few, fewer than 4, at most 7, not every The target generalization can again be captured in semantic terms. The problematic quantifiers are the downward entailing (generalized) quantifiers. As before, this suggests that this semantic property is essentially involved in the explanation of the basic acceptability pattern. Moving to the explanation, we begin by specifying the truth-conditions of comparative statements such as (16-a) and (16-b). Following Gajewski (2008b)'s account, the truth-conditions of the target comparatives are as in (18).9 We assume, in addition, that gradable adjectives, represented by P in (18), are monotonic, as defined in (19): (18) A is P-er than Q is = 1 iff ∃d [A is d-P and Q is not d-P ] (19) A gradable adjective P is monotonic iff P (d)(x) = 1 and d′ < d, then P (d′)(x) = 1 Given these assumptions, if Q in (18) is downward entailing, we get tautologies. 9 The precise logical form and truth-conditions of comparatives is an area of lively debate. For overviews see Schwarzchild & Wilkinson (2002), Schwarzschild (2008), Morzycki (2016). Gajewski (2008b) argues that a key point in favor of his theory-called the 'existential' theory for reasons that will become clear below-is precisely that it can account for the acceptability pattern captured in (17). Morzycki (2016: ch.4), however, argues that other standard theories, in particular the 'maximality theory', can also capture the target pattern (via undefinedness of the maximality operator in the unacceptable cases). 10 Triviality and Logical Form (20) A (generalized) quantifier Q is downward entailing iff for all A,B s.t. A ⊆ B and JQK(B) = 1, then JQK(A) = 1 To see why, consider two worlds, w1 and w2, and a domain which consists of the students Mary, Susan, and Bill. In w1, Mary = 1.6m, Susan = 1.5m, Bill = 1.4m. In w2, Bill = 1.6m, Mary = 1.5m, and Susan = 1.4m. We can easily see that (21) makes a contingent statement, whereas (22), where Q is replaced with a downward entailing generalized quantifier, is trivially true. (21) Mary is taller than every other student is. a. ∃d [Mary is d-tall and every other student is not d-tall] = 1 @ w1, for let d = 1.6m, then Mary is d-tall but no one else is d-tall. = 0 @ w2 since any d ∈ Mary's height (0, 1.5] is in Bill's height (0, 1.6] (22) *Mary is taller than no other student is. a. ∃d [Mary is d-tall and no other student is not d-tall] = ∃d [Mary is d-tall and every student is d-tall] = 1 in w1/w2 since d can be between (0, shortest student's height] To sum up, if we assume that the deductive system marks as ungrammatical trivial statements (true/false in all worlds or situations), we can explain the basic acceptability pattern concerning negative islands in comparatives. 3 The Glitch: Acceptable trivialities, logical skeletons and enriched logical forms We have seen that if we assume the logicality of language-i.e., that the Grammar works with a deductive system which can determine whether, and filter out when, sentences are trivial-we can account for various systematic restrictions on the distribution of quantifiers. However, as mentioned in §1, this otherwise powerful account has a glitch: tautologies and contradictions are not, in general, marked as unacceptable. Examples such as those in (23), which superficially seem like the most obvious cases of trivial sentences, are perfectly grammatical, even if in some contexts they feel somewhat odd: (23) a. It is raining and it isn't raining b. If Fred is wrong, then he is wrong. c. Every square is a square. d. My brother is an only child 11 Del Pinal How can we maintain the logicality of language accounts for the semantic restrictions on the distribution of quantifiers examined in §2, without incorrectly predicting that superficially trivial expression such as those in (23) are ungrammatical? This section presents and develops two competing solutions to this problem, Logical skeletons and LF+Rescale. 3.1 Logical skeletons As noted in §1, Gajewski (2002, 2009) argues that we can solve the glitch by adopting Logical skeletons (see also Chierchia 2006, 2013, Fox & Hackl 2007). The basic idea, captured in (4) above, is that there is a formally specifiable subset of the trivial sentences, called 'L-trivial', whose members are unacceptable. Gajewski argues that unacceptable trivial expressions such as those discussed in §2 are L-trivial, whereas acceptable trivial expressions such as those in (23) are not L-trivial. This proposal rests on two assumptions about the architecture of language: A1. Terms can be sorted into two classes, roughly corresponding to the traditional dichotomies of logical vs. non-logical terms, functional vs. non-functional terms, or closed-class vs. open-class words. A2. The deductive system does not 'see' the non-logical terms. Specifically, their semantic type is represented, so that compositionality can proceed, but the language system does not encode different tokens of the same non-logical terms as the same. The suggestion, then, is that the language system sees only 'logical skeletons'. (24) Logical skeleton To obtain the logical skeleton of a standard logical form α: a. Identify the maximal constituents of α containing no logical terms. b. Replace each such constituent with a new constant of the same semantic type. We can now formulate precisely which trivial sentences are unacceptable: (25) (i) A sentence S is L-trivial iff S's logical skeleton = 1 (or 0) in all its interpretations (in which S is defined). (ii) S is ungrammatical if its logical form contains an L-trivial sentence. In what follows, we assume A1 (for discussion, see van Benthem 1989, 2002, Gajewski 2002, 2009, Chierchia 2013, Abrusán 2014). In most of the cases 12 Triviality and Logical Form we consider, the target terms clearly fall on either the logical/closed-class or non-logical/open-class side of this dichotomy.10 In addition, we will see in §4 that, as Gajewski has shown, the triviality of our target cases concerning the distribution of quantifiers can indeed be proven from their logical skeletons. What is crucial, at this point, is to see why acceptable trivial sentences such as those in (23) cannot be proven as trivial from their logical skeletons, which is the desired result. If we apply the procedure in (24) to (23-a), repeated in (26-a), we get as its logical skeleton (26-b). Note that (26-b) is indistinguishable from the logical skeleton of a structurally equivalent yet informative contingent statement such as (27-a). This can be seen by comparing (26-b) and (27-b). It follows that the deductive system cannot determine (even if we enrich it with heuristics) that superficial contradictions such as (26-a), which it sees as (26-b), are trivial. As a result, such expressions are not filtered out. (26) a. It is raining and it isn't raining. b. It is P1,<e,t>-ing and it isn't P2,<e,t>-ing (27) a. It is raining and it isn't snowing. b. It is P1,<e,t>-ing and it isn't P2,<e,t>-ing The same argument can be easily extended to show that the other superficially trivial examples in (23) are not L-trivial. As mentioned above, we will see in §4 that the target acceptability patterns concerning the distribution of quantifiers examined in §2 can be predicted from logical skeletons. Taken together, these results suggest that endorsing Logical skeletons is the key to maintain the logicality of language, i.e., the hypothesis that the Grammar has an automatic deductive system which can identify and filter out trivial sentences. Indeed, most proponents of the logicality of language accept this approach. From now on, I shall refer to this view using the shorter 'Skeletons'. 10 This claim should be qualified. As a reviewer pointed out, there is currently no fool-proof method for distinguishing between functional/logical and content/open-class terms. Indeed, this is explicitly acknowledged by most proponents of the logicality of language (see, e.g., Gajewski 2009, Abrusán 2014). One proposal for picking out the logical terms is by appealing to the property of permutation invariance (e.g., van Benthem 1989). However, as Gajewski (2009) and Abrusán (2014) show, this leaves out some terms whose identity is crucial to prove some L-trivialities, and arguably allows some terms that are intuitively open-class. von Fintel (1995) presents a more promising proposal which singles out the functional terms by appealing to a cluster of properties including permutation invariance, having high-types, and being subject to universal constraints. 13 Del Pinal 3.2 Logical forms + Rescale The main contention of this paper is that there is a better way to maintain the logicality of language in light of the acceptability of (superficially) trivial examples such as those in (23). As a starting point, note that there are various independently motivated views about logical form which entail that (most) cases in (23) are not trivial (e.g., Sauerland 2014, Alxatib et al. 2013, Recanati 2010, Stanley 2007, Kamp & Partee 1995). These views share the idea that the meaning of open-class terms can often/always be modulated, either because of the presence of covert (optional) operators (cf. Sauerland 2014, Martı 2006, Jacobson 2005, Stanley 2007), or because the interpretation function is defined in such a way that it modulates the meaning of those expressions as a function of their fine-grained utterance context (cf. Kamp & Partee 1995, Pagin & Pelletier 2007, Recanati 2010, Lasersohn 2012).11 The specific proposal I defend-which is most directly inspired by Sauerland (2014)-was introduced in (5) as LF+Rescale. On this view, we assume that the Grammar and its deductive system see standard logical forms, in particular whether different tokens are of the same open class term. To capture the idea that the interpretation of open class terms can be modulated as a function of context, we assume that logical forms include an optional polymorphic type Rescale operator which can take non-logical terms and fine tune (e.g, intersect) their meaning in certain constrained ways. (28) Logical Form + Rescale To obtain an LF+Rescale a. Identify the minimal projections of open class heads (adjectives, nouns, adverbs, verbs). b. You may optionally add Rescale as a sister. 11 Abrusán (2014: ch. 6) makes a similar observation in her discussion of logical skeletons and the logicality of language. Her overall position is broadly congenial to the position developed here, and most of the objections she raises against logical skeletons complement the arguments presented in §5. Still, I should mention a key difference between the positive proposal made by Abrusán (2014) and my proposal. To successfully pair views which allow modulation with the logicality of language, it is, in my view, crucial that the mechanism which modulates open class terms be implemented as part of the compositional semantics rather than as a post-compositional, pragmatic processes. The basic argument for this will be presented in §3.3 and further developed in §5 (cf. Alxatib et al. 2013). Abrusán (2014), however, follows Kamp & Partee (1995) in treating the relevant modulations as pragmatic processes. In contrast, the account I defend is closer to the syntactic/semantic accounts defended by Sauerland (2014) and Stanley (2007). 14 Triviality and Logical Form Overt operators similar to Rescale are terms such as typical and good. Like those modifiers, Rescale is technically a character, hence its precise effect depends on the context of utterance. At the most general level, we assume that, for any open class term P , argument of suitable type x and context c, {x : Rescalec(P )(x)} ⊆ {x : P (x)}. That is, the meaning modulation is constrained to specialize meanings, where the precise refinement depends on the context parameter c. Like its overt counterparts, Rescale can appear at various positions in an expression, and its context sensitive parameter can be fixed differently at each position. To model this, we assume that each token of Rescale is interpreted in its dynamically updated local context.12 LF+Rescale can be refined and developed in various ways. Still, we can already begin to see how it can account for acceptable trivial sentences. On this view, acceptable trivial sentences such as (29) and (30) can be assigned the LFs in (29-a) and (30-a), which are non-trivial and potentially informative expressions. To be clear, (29) and (30) can also have LFs without Rescale, as in (29-b) and (30-b); but since these are formally trivial, they are marked as unacceptable and are dispreferred relative to the alternative disambiguations which are acceptable and potentially informative. (29) It is raining and it isn't raining. a. It is raining and it isn't Rescalec(raining). ≈ It is raining and it isn't raining hard. b. *It is raining and it isn't raining. (30) If Fred is wrong, then he is wrong. a. If Fred is Rescalec(wrong), then he is Rescalec′(wrong). ≈ If Fred is slightly wrong, then he is totally wrong. b. *If Fred is wrong, then he is wrong. The basic idea can be generalized: when Rescale is inserted the resulting logical forms can 'rescue' acceptable tautologies/contradictions such as those in (23). As a result, LF+Rescale can also explain why superficially trivial sentences such as those in (23) have acceptable readings.13 12 The notion of a 'local context' is here used broadly, and is in principle compatible with different implementations. Since this paper uses a standard static semantics, it is easier to opt for implementations designed for static systems, such as Schlenker (2009) and Stalnaker (2014). But we could also opt for more dynamic implementations, such as Heim (1983, 1982), Barker (2002) and Rothschild (2011). 13 Some clarifications are in order. First, we will see later that we can adopt less constrained accounts of the expressive power of Rescale without affecting any of the points made in this paper. Put in terms familiar from discussions of contextualism, although we assume that the effect of Rescale is intuitively that of 'enriching' meanings, everything we say here is 15 Del Pinal 3.3 Rescale is independently needed: Embedded trivialities Skeletons and LF+Rescale both have the resources to explain why, even if we assume the logicality of language, superficial trivial statements such as those in (23) can be acceptable. However, I now want to suggest that something like Rescale of open class terms is independently needed to account for certain simple variations of acceptable trivial sentences. A crucial difference between Skeletons and LFs+Rescale is that only the latter has the resources to fully explain the default intuitive readings of acceptable superficial trivialities. To see why, recall that, according to Skeletons, an acceptable contradiction such as (31), is seen by the deductive system as (31-a), which is the reason why it is not ruled out. (31) It is raining and not raining. a. It is P1 and not P2. b. It is P and not P . At a later post-compositional stage of processing, namely, when (31-a) is sent to pragmatics for the assignment of full truth-conditions (and further inferential processes), something like the information represented in (31-b) is recovered, namely, that P1 = P2. At this point, the contradiction can be identified. What happens at this stage? As far as I know, neither Gajewski nor others who endorse Skeletons directly address this question. Still, this account is compatible with familiar pragmatic stories. Namely, when (31) is recovered at the (post-compositional) pragmatic stage, we can use Gricean reasoning to derive, from its assertion, an informative implicature, such as that it is raining but not that hard. The problem for this kind of pragmatic account is that there are cases of embedded uses of acceptable contradictions in non-asserted positions, as illustrated in (32)-(34). As is well known, such enriched embedded readings challenge post-compositional pragmatic stories which work on asserted contents as inputs (Chierchia et al. 2012, Recanati 2003, 2010). compatible with assuming that it can also 'loosen' meanings (cf. Recanati 2010). In this case, we say that for any open class term P , argument of suitable type x and context c, either {x : Rescalec(P )(x)} ⊆ {x : P (x)} or {x : P (x)} ⊆ {x : Rescalec(P )(x)}. Secondly, like typical, Rescale is defined relative to a contextual parameter, and is only felicitous if the context provides the required information. This predicts that some rescued superficial contradictions, uttered out of context, can feel somewhat zeugmatic, analogous to an out of context utterance of that is not like that. Finally, we assume that Rescale can apply recursively, which is desirable given its status as a modifier similar to typical, and is in any case required to deal with examples like a typical gun is not a typical gun and it is raining hard and not raining hard. 16 Triviality and Logical Form (32) If it is raining and not raining, I am willing to go out and play. a. If it is raining and not Rescalec(raining), I am willing to go out and play. ≈ If its raining but not that hard, I am willing to go out and play (33) If John is tall and not tall, I bet he won't make it into the basketball team a. If John is tall and not Rescalec(tall), I bet he won't make it into the basketball team ≈ If John is borderline tall, I bet he won't make it into the basketball team (34) Peter is either smart but not smart, or he has no experience running a tough business. a. Peter is either Rescalec(smart) but not Rescalec′(smart), or he has no experience running a tough business. ≈ John is either book smart but not street smart, or he has no experience running a tough business. Although out of the blue cases like (32)-(34) can feel a bit odd, it is clear that they are strictly acceptable, and it is easy to imagine contexts in which they get the suggested readings. Furthermore, embedded enrichments in non-asserted clauses call for a treatment within the compositional semantics. Accordingly, (32)-(34) point to the availability of a semantic rescue mechanism along the lines of Rescale: i.e., a way of generating, within the compositional semantics, formally non-contradictory/non-tautologous logical forms for superficially trivial sentences in embedded positions, as illustrated in (32-a), (33-a) and (34-a). As so far presented, LF+Rescale seems to entail that, when using natural languages, we can't really assert trivialities: for it works as if the language system always generates logical forms which make superficially trivial statements of the kind in (23) potentially informative. However, there are cases in which we use such trivial statements to convey precisely the trivial readings. This is illustrated by a salient reading of the antecedent in (35) below. To see why LF+Rescale is not in tension with examples like this, recall that Rescale, just like similar overt modifiers such as typical, is technically a character. Its full modulatory effect is determined only once certain contextual parameters are provided. (35) If John believes that it is raining and not raining, then he has inconsistent beliefs. a. *If John believes that it is raining and not raining, . . . 17 Del Pinal b. If John believes that it is raining and not Rescalec(raining), . . . c. If John believes that it is Rescalec(raining) and not Rescalec′(raining), . . . To be clear, this account does predict a default preference for logical forms such as (35-b)/(35-c) over (35-a). Crucially, however, this technically allows that, for some c, c′, Rescalec(P ) = P , or that Rescalec(P ) = Rescalec′(P ) (see the definition of Rescale in §3.2). Hence this account allows for the eventual assignment of trivial readings to embedded statements such as it is raining and not raining. Still, since the decision to fix the context sensitive parameter of Rescale is a post-semantic, pragmatic process, these trivial readings are not seen as such by the deductive system, hence are not filtered out.14 To sum up, we have seen that although Skeletons and LF+Rescale can both account for the acceptability of superficially trivial statements such as those in (23), cases of embedded superficial trivial statements such as (32)-(34) provide independent motivation for something like Rescale. Obviously, this is not yet to suggest that we should abandon Skeletons for LF+Rescale. For what Gajewski (2002, 2009) has crucially shown is that by adopting Skeletons we can rescue just the acceptable trivial sentences. Only if the same result can be achieved by adopting LF+Rescale can we say that we have really undermined, by a kind of Occam's razor argument, the need to posit logical skeletons. Establishing that result is the aim of §4. I will then argue, in §5, that there are various cases in which LF+Rescale makes different and better predications than Skeletons. 4 Back to restrictions on quantificational determiners: Logical skeletons and LF+Rescale Gajewski (2002, 2009) argues that the triviality patterns involving thereexistentials, connected exceptives and negative islands in comparatives presented in §2 can be proven from their logical skeletons. Our main task now is to show that the target trivialities can also be proven from their standard LFs+Rescale. If correct, this means that, despite common assumptions to 14 One could argue that this account also predicts that there should be a strong default tendency to interpret superficial tautologies and contradictions non-trivially. Although there are ways of blocking or hedging this prediction for proponents of LF+Rescale, it is perhaps best for now to simply accept it. For there is empirical work which suggests that ordinary speakers tend to interpret superficial tautologies and contradictions informatively (in contrast to, say, trained philosophers, logicians and formal semanticists). For relevant empirical and theoretical work, see Osherson & Markman (1975), Wierzbicka (1987), Ward & Hirschberg (1991), Sauerland (2011), Cobreros et al. (2012), Alxatib et al. (2013), Snider (2015) 18 Triviality and Logical Form the contrary (see e.g., Chierchia 2006, 2013, Fox & Hackl 2007, Gajewski 2009), the account of L-triviality used to separate acceptable from unacceptable trivial sentences does not depend on pairing the logicality of language with Skeletons. Accounts that use standard logical forms but also allow constrained modulation of open class terms via something like Rescale work just as well. 4.1 There-existential sentences Consider first the acceptability patterns involving there-existentials, focusing again on examples (6-a) and (6-b), repeated here as (36) and (37). It is easy to see that the original explanation, as spelled out in §2.1, goes through given their logical skeletons, specified in (36-a) and (37-a). Since some is weak, (36-b) may get 1 or 0 depending on whether the semantic value assigned to P1-i.e., I(P1)-is ultimately empty. On the other hand, since every is strong, (37-b) will always be true regardless of the interpretation assigned to P1, since for any assignment, I(P1) ⊆ (I(P1) ∩De) ⊆ De (note: P1 is of type < e, t >). It follows that only (37) is L-trivial, and predicted to be marked as unacceptable. (36) There are some curious students. a. Logical skeleton: [there [are [some P1,<e,t> ]] b. Interpretation: JsomeK(I(P1))(De) = 1 iff I(P1) ∩De 6= ∅ (37) *There is every curious student. a. Logical skeleton: [there [is [every P1,<e,t> ]] b. Interpretation: JeveryK(I(P1))(De) = 1 iff I(P1) ⊆ De Note that essentially the same story holds even if the deductive system can see LFs+Rescale. We have already shown this in §2.1 for standard LFs, so what we have to show now is just that modifying the open class terms by Rescale does not change the acceptability patterns: • In (36), even if the deductive system can see that P1 = Jcurious studentsK, information about the evaluation world is required to determine whether anything falls under it. Introducing Rescale(Jcurious studentsK) (e.g., to mean something like 'there are some extremely curious students') doesn't change that fact.15 15 Although here I briefly consider a case in which Rescale potentially downgrades an LF, it is in general unnecessary to check this option. For suppose that adding Rescale to an otherwise acceptable LF would result in unacceptability. The original LF without Rescale 19 Del Pinal • In (37), even if we introduce Rescale(Jcurious studentsK) to its LF, including recursive applications, the expression will still be tautologous, since it will always hold that: Rescale(Rescale(Jcurious studentsK)) ⊆ Rescale(Jcurious studentsK) ⊆ De As with logical skeletons, then, only (37) comes out as L-trivial. The reason why the target explanation holds with LFs+Rescale if it holds with logical skeletons is obvious. Suppose sentence S comes out as L-trivial on the basis of its logical skeleton. This means that the particular semantic values of its open class terms do not make a difference to S's truth value. A fortiori, it does not make a difference whether its open class terms are (recursively) modified by Rescale. 4.2 Connected exceptive phrases Consider next the acceptability patterns on quantifiers imposed by connected exceptives, illustrated by (12-a) and (12-b), repeated here as (38) and (39). Some interpretations of P1 . . . P3 will make (38) true, and others will make it false. However, all interpretations (which do not result in presupposition failure, hence where I(P2) is not empty) will map (39) to false. Since some is left upward entailing, we can always subtract less than I(P2), whatever that is, namely, we can subtract nothing (let S = ∅). (38) Every student but Mary smokes a. Logical skeleton: [every [P1 but P2] P3] b. Interpretation: JbutK(I(P1))(I(P2))JeveryK(I(P3)) = 1 iff I(P2) 6= ∅ and JeveryK(I(P1)− I(P2))(I(P3)) = 1 and ∀S[JeveryK(I(P1)− S)(I(P3)) = 1→ I(P2) ⊆ S] (39) *Some student but Mary smokes. a. Logical skeleton: [some [P1 but P2] P3] b. Interpretation: JbutK(I(P1))(I(P2))JsomeK(I(P3)) = 1 iff I(P2) 6= ∅ and JsomeK(I(P1)− I(P2))(I(P3)) = 1 and ∀S[JsomeK(I(P1)− S)(I(P3)) = 1→ I(P2) ⊆ S] would then be selected as the preferred disambiguation. In subsequent sections I discuss only cases in which adding Rescale could potentially rescue an otherwise unacceptable LF. 20 Triviality and Logical Form As in the case of there-existentials, what is important to note is that essentially the same story holds if we assume instead LFs+Rescale. In §2.2, we have already shown how we capture the acceptability patterns for standard LFs; so what we now have to show is that introducing Rescale does not lead to incorrectly rescuing (39). The only position where the modification could be problematic is Rescale(I(P2)), since this could potentially narrow the extension of the complement of but. In other words, we know that Rescale(I(P2))⊆ I(P2); however, to avoid presupposition failure, Rescale(I(P2)) 6= ∅. At this point, its easy to see that, since some is left-upward entailing, we can still choose S = ∅, hence (39) will (when defined) always be false.16 The following intuitive examples illustrate the basic point. The problem with (40-a) is that we could always substract less and keep truth, as in (40-b). But the same holds if we try to rescue (40-a) by adding Rescale as in (41-a): although this can take us to a subset of the complement of but, it still holds that we can substract less, namely nothing, and keep truth, as in (41-b) (40) a. *Some [students but the smart ones] smoke. b. Some [students − ∅] smoke. (41) a. *Some [students but the Rescale(smart) ones] smoke. ≈ *Some [students but the very smart ones] smoke. b. Some [students − ∅] smoke 4.3 Negative islands in comparatives Consider finally the restrictions on quantifiers in comparatives, illustrated by (42) and (43). Their logical skeletons and corresponding interpretations are given in (42-a)-(42-b) and (43-a)-(43-b) respectively. (42) Mary is taller than every student is tall. a. Logical skeleton: [A is P1,<d,<e,t>>-er [than [every P2,<e,t>] is P3,<d,<e,t>>]] b. Interpretation: ∃d [I(A) is d-I(P1) and every I(P2) is not d-I(P3)] (43) *Mary is taller than no student is tall. a. Logical skeleton: [A is P1,<d,<e,t>>-er [than [no P2,<e,t>] is P3,<d,<e,t>>]] 16 Note that, if we let Rescale 'widen' interpretations (as some modulation-friendly theorists would certainly insist we should), then, a fortiori, it cannot rescue (39). Indeed, none of the points I make here depend on defining Rescale as a narrowing operation. 21 Del Pinal b. Interpretation: ∃d [I(A) is d-I(P1) and no I(P2) is not d-I(P3)] Note that, by the definition of logical skeletons, the tokens of tall are treated as if they could be different predicates (since gradable adjectives such as tall are open class terms). Despite this, Gajewski argues that L-triviality can still be proven for (43) if we place the following independently plausible constraints on D<d,<e,t>>, the class of gradable predicates: (44) a. Gradable adjectives are monotonic, as defined in (19) above. b. The domains of gradable adjectives are restricted to scales, as illustrated in (45) for tall and old. (45) a. JtallK = λd ∈ Sheight. λx : ∃d ∈ Sheight[height(x) = d]. d ≤ height(x) b. JoldK = λd ∈ Sage. λx : ∃d ∈ Sage[age(x) = d]. d ≤ age(x) Given these assumptions, if I(P1) and I(P3) do not match, as in tall and old, then both (42-b) and (43-b) come out as undefined. If I(P1) and I(P3) match, then (42-b) is contingent. For as we saw in §2.3, for any gradable adjective I(P1), there are some worlds in which I(A) is the I(P1)-est, compared to all I(P2)s, and some in which I(A) is not the I(P1)-est. In contrast, (43-b) comes out as trivially true when I(P1) and I(P3) match. To see this, recall the basic argument from §2.3. We need to find a degree d such that I(A) (e.g., Mary) has d, and in addition each entity in the comparison class I(P2) has d. Given monotonicity, we can choose a d between (0, dsmallest], where 'dsmallest' is the degree assigned to the smallest member of the union of I(A) and the comparison class I(P2). Since there is always such a degree, (43) is, when defined, always true, and is therefore marked as unacceptable. Accounting for the difference between expressions such as (42) and (43) given LF+Rescale instead of logical skeletons is quite simple. We have already shown, in §2.3, that we can generate the basic pattern of restrictions on negative generalized quantifiers in comparatives from standard logical forms. So what we have to show now is that we cannot use Rescale to turn, say, (43), repeated below as (46), into a contingent statement. This is easy to see. We can use Rescale to restrict the interpretation of student, and get a contextual reading like (46-a). We can also use it to restrict either or both tokens of tall, and get a contextual reading like (46-b). Modulate any way you want, there will always be a degree d between (0, ds], were ds is the height of the smallest member of the union of Mary and the (restricted) set of students. In short, 22 Triviality and Logical Form (46) is trivially true, even if we sprinkle Rescale wherever it's allowed, and is therefore marked as unnacceptable by the deductive system. (46) *Mary is taller than no student is tall. a. ∃d [Mary is d-tall and no smart student is not d-tall] b. ∃d [Mary is roughly d-tall and no student is not roughly d-tall] 5 Logical skeletons vs LFs+Rescale Where do we stand? We have seen that, as Gajewski (2002, 2009) and others (e.g., Chierchia 2013, Fox & Hackl 2007) have argued, if we assume that the deductive system operates on logical skeletons, we can generate just the right class of L-trivial statements, i.e., the subset of the trivial sentences which are unacceptable. Specifically, if the representations that feed into the compositional semantics are logical skeletons, we cannot prove triviality for acceptable tautologies and contradictions such as those in (23), yet can still prove triviality for the unacceptable patterns involving quantifiers in thereexistentials, connected exceptives and negative islands in comparatives. We have shown, however, that essentially the same results are obtained if we assume LFs+Rescale. On this view, the logicality of language is paired with a relatively standard conception of logical form, except that it is enriched with the hypothesis that open class terms can be optionally modulated via a covert Rescale operator, which I argued is independently needed (see §3.3). I now try to show that, in key cases where their predictions differ, LF+Rescale makes better predictions. §5.1 presents cases in which Skeletons systematically overgenerates assignments of unacceptability, and in §5.2-§5.3 cases in which it undergenerates assignments of unacceptability. Crucially, the cases explored in §5.2-§5.3 constitute-together with there-existentials, connected exceptives, and negative islands in comparatives-core accounts within the overall argument for the logicality of language. Taken together, these results strongly suggest that the logicality of language should be paired with a view of logical form akin to LF+Rescale. 5.1 Contradictions with variable co-binding of predicates We begin by examining cases involving co-binding of predicative variables in which the two views of logical form make different predictions, and only LF+Rescale generates the correct ones. Take a sentence such as (47), which is superficially trivial but not L-trivial, and consider the variant in (48), which involves co-binding of the tokens of tall, as represented by its logical form in 23 Del Pinal (48-a). Gajewski (2009) admits that this variant, which can be systematically generated for other acceptable trivial sentences, presents a problem for the notion of L-triviality.17 Assume, as seems required, that these binding relations are seen by the Grammar, hence must be encoded by logical skeletons. It follows that L-triviality can be easily proven for (48-a), which is therefore (by the logicality of language) predicted to be marked as unacceptable. However, (48) is, even if odd, clearly not ungrammatical. In addition, this is precisely what is predicted by LF+Rescale. For on this view, (48) can have the logical form in (48-b), which is preferred over the contradictory and hence marked logical form in (48-a). (47) Bill is tall and Bill isn't tall. (48) Tall is what Bill is and isn't. a. Tall is [what1 Bill is t1 and is not t1] b. Tall is [what1 Bill is t1 and is not Rescalec(t1)] c. Tall is [what1 Bill is posc(t1) and is not posc′(t1)] Gajewski (2009) concludes that cases such as (48) present an open problem for the pairing of the logicality of language with logical skeletons. Still, it might be tempting to respond on his behalf as follows. Assume that gradable adjectives are degree functions of type < e, d >, which can occur with a covert degree morpheme pos of type << e, d >,< e, t >> (Kennedy & McNally 2005, Kennedy 2007). Pos determines the relevant standard, as a function of context, for an object to fall under the adjective which is its argument. If we can leave pos in situ when we move the adjective, one possible logical form for (48) would be (48-c). Recall that various frameworks allow that contexts are updated as information is processed, such that c 6= c′. It follows that since each token of pos could determine a different standard, the deductive system cannot treat (48-c) as a contradiction. The problem with this response is that it doesn't generalize to simple variations of the original example. Specifically, we can construct examples with explicitly scoped out overt degree morphemes, such as (49). Furthermore, the proposed response cannot be applied to variations with non-gradable adjectives, such as (50). In short, the simple variations in (49) and (50) are still incorrectly predicted to be L-trivial, hence marked. In contrast, LF+Rescale again makes the correct predictions: it generates for (49) the logical form in (49-a) and for (50) the logical form in (50-b), neither of which is L-trivial. (49) 2 meters tall is what Bill is and isn't. 17 Gajewski (2009) attributes this observation to Danny Fox. 24 Triviality and Logical Form a. 2m tall is [what1 Bill is t1 and isn't Rescalec(t1)]. ≈ Bill is around 2m tall but is not exactly 2m tall. (50) Raining is what is and isn't happening. a. Raining is [what1 is t1 and is not t1 happening] b. Raining is [what1 is t1 and is not Rescalec(t1) happening] These examples might feel odd, especially when considered out of the blue. Indeed, they might be in tension with other principles, such as principles of manner or economy. The point here is just that they are clearly not marked as ungrammatical, which is the prediction made by LFs+Rescale, but not by Logical skeletons. One could of course hold that logical skeletons can include a mechanism of nominal/verbal restriction, perhaps along the lines of Rescale. If a solution along these lines is accepted, however, the need to distinguish between acceptable trivialities such as those in (23), and the unacceptable trivialities involving the distributions of quantifiers examined in §2, can no longer be a reason to adopt Logical skeletons, as demonstrated in §4. Importantly, cases of acceptable trivialities involving co-bound variables present a systematic problem for Logical skeletons. In contrast, most of these cases can be easily handled by LF+Rescale.18 To consider a different kind of case, take (51), suggested by a reviewer. Note that (51) contains a trivial embedded question. Suppose Mary and Peter are discussing John's strange recent behavior. Exasperated, Peter utters (51). Although the embedded question is a superficial contradiction, (51) is not ungrammatical, and can be easily rescued in this context. Assuming Hamblin (1973)'s account of questions, the logical form and meaning of the embedded question in (51) would be (51-a) and (51-b) respectively. Since the predicative variables in the embedded question are co-bound by what, the logical skeleton cannot generate independent variables. As a result, each proposition in the set of possible answers will be seen as a contradiction, as shown in (51-b). In contrast, LF+Rescale also generates the logical form in (51-c), which in this context can denote the set of non-contradictory propositions in (51-d). This view correctly predicts that, in this scenario, the discourse in (52), where Mary's assertion entails that John is a friend but not a good friend, is perfectly coherent. (51) I wonder what John is and is not. a. what1 John is t1 and John is not t1 b. {p: ∃Q[p = John is Q and John is not Q]} 18 I'm grateful to an anonymous reviewer for emphasizing this point, and for providing me with a range of insightful examples and initial analyses. The rest of the discussion in this section is greatly indebted to the reviewer's constructive comments. 25 Del Pinal c. what1 John is t1 and John is not Rescalec(t1) d. ≈ {John is a cousin and not a good cousin, John is a friend and not a good friend, John is a partner and not a good partner, . . . } (52) a. Peter: I wonder what John is and is not. b. Mary: A friend . . . Another manifestation of the problem of co-bound variables for Skeletons involves trivial sentences with reflexive pronouns (Gajewski 2009). Presenting a full account of the target acceptability patterns is a difficult and controversial task which I cannot embark here. Still, let us briefly consider two representative examples which illustrate the prima facie advantage of LF+Rescale over Skeletons in this domain. The first example consists of comparatives, such as (53), in which the clausal comparative contains a reflexive pronoun. Although acceptability intuitions are in this case a bit fuzzy, it seems that (53) is strictly acceptable. To nudge your intuitions, suppose (53) is uttered by Peter in a situation where John is outshining his usual self, as in Look at John debate today! He is simply smarter than himself !, used to say that John is smarter than he typically is. Assuming the account of comparatives presented in §2.3, and a bound variable account of reflexives, Skeletons incorrectly predicts that (53) is, when defined, trivially false, hence strictly unacceptable. This can be seen from its logical skeleton in (53-b), where to avoid presupposition failure P1 and P2 are required to take a degree on the same scale (see §4.3). In contrast, LF+Rescale generates a logical form for (53)-spelled out in (53-c)-which captures the target reading. In this case, Rescale works as an adverbial modifier, and the sentence could be resolved in context to say that there is a degree d such that John is at some salient time/location d-smart, although he is not typically d-smart. (53) John is smarter than himself. a. John λ1 ∃d [t1 is d-smart and t1 is not d-smart] b. John λ1 ∃d [t1 is d-P1,<d,<et>> and t1 is not d-P2,<d,<et>>] c. John λ1 ∃d [t1 is Rescalec d-smart and t1 not is Rescalec′ d-smart] The second example of acceptable trivialities with reflexives is illustrated by the superficially simple sentence in (54). To nudge intuitions, consider again the scenario in which Mary and Peter are dismayed by John's recent uncharacteristic behavior. Clearly, they can discuss John's behavior using stylistic variants of John is/is not himself, meaning that John is/is not behaving in characteristic ways. LF+Rescale generates an appropriate logical form for this kind of 26 Triviality and Logical Form acceptable target reading. To see this, let us assume, plausibly, that the target reading involves predication and not directly identity. Accordingly, himself has to be type-shifted to a predicate, which can be done via the ident operator (Partee 1986b,a), as illustrated in (54-b). The output of this operation is 'λx. x = t1'. This resolves the type mismatch but results, in this environment, in a clear triviality, namely, '[λx. x = t1](t1)'-which is still trivial given its logical skeleton. However, since 'ident(himself1)' is a predicate, it can be modified by Rescale, as shown in (54-c). The output of this intersective modification can be represented as 'λx. x = t1 ∧ Pc(x)'; which is not trivial in this environment-i.e., in '[λx. x = t1 ∧ Pc(x)](t1)'-and captures the target reading. For example, suppose that, in c, P is assigned a property of John at his best, e.g., generosity; we then get [λx. x = t1 ∧ generous(x)](t1).19 (54) John is himself. a. John λ1 [t1 is himself1] b. John λ1 [t1 is ident(himself1)] c. John λ1 [t1 is Rescalec ident(himself1)] 5.2 Polarity items and contradiction A key difference between Skeletons and LF+Rescale is that only for the latter are classical formulas and inference rules, such as the Law of Non-contradiction (LNC) and Modus Ponens (MP), valid at the level of representation where acceptability/grammaticality is determined.20 Here we focus on the LNC. To adopt Skeletons is to assume that the deductive system of language doesn't 'see' the identity of open class terms, to the extent that different tokens of the same term are treated as if independent. This entails that superficial contradictions such as (31) above are not seen as such, as captured in (31-a). The underlying generalization is simply that LNC doesn't hold given logical skeletons, since this formula is valid only if the dependency between non-logical terms is preserved, 19 This account also captures the target reading of John is not himself. Consider again the scenario and assignments described above. Since 'John = John' is necessarily true, the statement 'not [John = John and John is generous]' is resolved to the claim that John is not generous, which corresponds to the intended reading. To be clear, I haven't provided an independent justification for this account of expressions like (54). Indeed, I am not sure whether they really present a problem for Skeletons, although Gajewski (2009) seems to think that they do. My aim here is just to show that LF+Rescale provides us with key tools to explain why superficially trivial expressions with reflexives can be acceptable. 20 Relative to the familiar entries for the connectives and other relevant functional terms. 27 Del Pinal as in (31-b).21 In contrast, according to LF+Rescale logical forms such as (31-b) can be seen by the language system, but since they are trivially false they are disfavored relative to potentially informative disambiguations where Rescale modifies (at least) one of the tokens of raining. The question, then, is this: Should the result that the LNC is valid if we adopt LF+Rescale but not if we adopt Skeletons be taken as support for the former view? There is no simple move to that conclusion. For the issue here concerns the properties of a natural deductive system, which could be radically different from classical systems. Furthermore, even if we adopt Skeletons, there is still a level of representation where the dependency between tokens of open class terms is recovered. Accordingly, one could hold that it is at this post-compositional level that classical formulas/rules of inference, including the LNC, apply (cf. Chierchia 2013).22 Still, this relative neutrality (with respect to the classicality of the natural deductive system) is available to proponents of the logicality of language only if the corresponding accounts of acceptability based on logical triviality do not depend on the validity of any of the formulas/rules of inference which are allegedly suspended at the level of representation where grammaticality is determined. Can this be maintained relative to the LNC? Admittedly, in the case of there-existentials, connected exceptives, and negative islands in comparatives we proved triviality-i.e., truth or falsity under all models-without assuming the LNC, and still captured the target acceptability patterns.23 However, this is not the case for other key accounts based on the logicality of language, including Chierchia's (2006, 2013) influential account of the distribution of polarity sensitive items. Specifically, I will now argue that this account requires that the LNC be valid at the level of representation where the deductive system 21 Gajewski is aware of this consequence of Skeletons. He notes that the basic idea has a precursor in Körner's (three-valued) logic of inexact concepts (1955, 1960), which provides truth tables that, as shown by Williamson (1994), effectively treat each token of a propositional variable as independent. 22 From this perspective, it is only at this level that we can engage in, e.g., pragmatic reasoning from Gricean maxims, since this presumably depends on respecting the LNC and MP. To be sure, one could also hold that pragmatic reasoning is determined or constrained by a kind of natural logic; but this system should then be strictly distinguished from the automatic deductive system of the language module. For related discussions, see Szabó (2012) and Iacona (2017). 23 To be fair, since those are the main acceptability patterns examined by Gajewski (2002, 2009), he could consistently adopt a deductive system which treats all non-logical terms as independent. 28 Triviality and Logical Form determines grammaticality, as entailed by LF+Rescale but not-at least without additional ad hoc stipulations-by Skeletons.24 Consider the distribution of the NPI any, illustrated in (55)-(56). According to Chierchia (2013), any is an indefinite with existential force which, unlike its plain counterpart a/an, obligatorily activates alternatives. This, in turn, triggers a process of automatic exhaustification which is the key component to explain the difference in the distribution of any and its plain counterparts. (55) a. John doesn't have an egg. b. John doesn't have any eggs. (56) a. John has an egg. b. *John has any eggs. The mode of exhaustification and set of alternatives relevant to assertions with any is defined in (57). This definition will become clear once we apply it to some examples below. For now, note that (i) the second conjunct of (57-a) guarantees the negation of alternatives which are logically stronger than the prejacent, and (ii) the set of alternatives, as defined in (57-b), is just the domain restricted versions of the prejacent. (57) a. JODA φKg,w = JφKg,w ∧ ∀p ∈ JφKDA[p→ λw′JφKg,w ′ ⊆ p] b. JφKDA = {JφK : D′ ⊆ g(D)} To explain the distribution of any, two observations are crucial. The first is that, in downward entailing environments, the obligatory exhaustification triggered by any is empty and thus unproblematic. Consider (58). The prejacent- namely, that John doesn't have an egg-entails all the alternatives. To see this, suppose the relevant domain D (for brevity, we use 'D' below instead of 'g(D)') concerns the eggs in John's house. That John doesn't have an egg in his house entails that John doesn't have an egg in any of its rooms. Recall that the second conjunct in (57-a) guarantees that alternatives which are entailed by the prejacent are not negated. So exhaustification is in this case vacuous and simply returns the prejacent. (58) John doesn't have any eggs. a. ODA(¬[∃x ∈ D[eggw(x) ∧ havew(j, x)]]) 24 Abrusán (2014: ch. 6) raises a similar point to argue that Logical skeletons, as the view is standardly conceived, must be substantially revised to allow for accounts of acceptability patterns which appeal to contradictions. This includes not only Chierchia's account of the distribution of polarity-sensitive items, but also various logicality based accounts-including Abrusán's own-of presuppositional, negative and other weak island effects. See §5.3 below. 29 Del Pinal b. DA = {¬[∃x ∈ D′[eggw(x) ∧ havew(j, x)]] : D′ ⊆ D} The second observation is that, in upward entailing environments, the obligatory exhaustification triggered by any generates contradictions. The key difference is that, in this case, the prejacent-namely, that John has an egg-doesn't entail any of the alternatives. Suppose John's house has a living room and a kitchen. That John has an egg doesn't entail that John has an egg in the living room, and it doesn't entail that John has an egg in the kitchen. Exhaustification will therefore negate these stronger alternatives and generate the proposition in (59-c), which is a contradiction. (59) *John has any eggs. a. ODA(∃x ∈ D[eggw(x) ∧ havew(j, x)]]) b. DA = {∃x ∈ D′[eggw(x) ∧ havew(j, x)] : D′ ⊆ D} c. ∃x ∈ D[eggw(x) ∧ havew(j, x)] ∧ ∀D′ ⊆ D[¬∃x ∈ D′[eggw(x) ∧ havew(j, x)]] ≈ John has an egg ∈ Dhouse ∧¬John has an egg ∈ Dkitchen ∧¬ John has an egg ∈ Dliving room As in the cases in §2, the presence of trivial truth-conditions explains, given the logicality of language, why any is unacceptable in upward entailing environments. The key point, however, is that unlike for there-existentials, connected exceptives, and comparatives, in this case the triviality can be traced to a violation of the LNC. The problem is that this formula is not valid given logical skeletons: for its validity requires that the system respect the dependency between-i.e., the uniform substitutions of semantic values for-tokens of the same open class terms, as in (59-c). If we generate a logical skeleton for (59-c), shown in (60), we can immediately see that the result is not L-trivial: (60) John has a P1 ∈ Dhouse ∧¬John has a P2 ∈ Dkitchen ∧¬ John has a P3 ∈ Dliving room In contrast to Skeletons, LF+Rescale doesn't affect the account of NPIs such as any. The reason is simple. First, on this view violations of LNC can be identified. Second, application of Rescale doesn't affect the monotonicity of the relevant environments, hence it doesn't change the basic outcomes of each case of obligatory exhaustification. To illustrate, consider (61), which is like (59) except that we introduced Rescale to (try to) rescue it. Applied to this logical form, ODA as defined in (57) has two key implications. First, ODA is sensitive only to domain (and not syntactically simpler) alternatives, as captured in the formulation in (57-b). Since in (61) Rescale is a constituent 30 Triviality and Logical Form of the logical form of the prejacent, it must be present in all the domain alternatives DA. Second, an assignment function determines the value of the context parameter of Rescale in (61). Since alternatives don't have a context of their own, but instead inherit their context from the utterance context of the source, it follows from (57) that this assignment must be uniform-i.e., the same parameter value assigned to the token of Rescale in the prejacent is assigned to each token in each alternative. Again, the only variation allowed for each item in DA is in the domain of the existential quantifier.25 This generates the contradiction in (61-c): (61) *John has any Rescalec(eggs). a. ODA(∃x ∈ D[Rescalec(eggw)(x) ∧ havew(j, x)]]) b. DA = {∃x ∈ D′[Rescalec(eggw)(x) ∧ havew(j, x)] : D′ ⊆ D} c. ∃x ∈ D[Rescalec(eggw)(x) ∧ havew(j, x)] ∧ ∀D′ ⊆ D[¬∃x ∈ D′[Rescalec(eggw)(x) ∧ havew(j, x)]] In short, LF+Rescale doesn't affect Chierchia's basic account of the distribution of any. In particular, we need not make any additional or ad hoc stipulations to preserve the basic explanation of why, in upward entailing environments, any generates contradictions and is thus unacceptable. In light of this, consider the following response on behalf of Skeletons. Instead of taking blindness to open class terms as a kind of general property of the language system at the relevant level, think of the algorithm for logical skeletons, specified in (24) above, as a kind of rule. This rule can be applied to generate a logical skeleton for (59) either before or after we get the relevant alternatives. The previous objection holds only if we apply the rule after we generate the relevant alternatives. If, however, we apply the rule before we generate alternatives, we would get the alternatives to 'copy' the skeleton, as in (62-b), and thus get a contradiction, specified in (62-c). 25 The requirement that there be a uniform assignment for the open parameter for each token of Rescale in the prejacent and its alternatives is independently motivated. This is how, in general, we must treat the context sensitive parameters of (non-focused) characters in the prejacent and their corresponding alternatives. For example, an assertion of someF of the walls in [that house]1 are red in response to the question do you know any house which is totally red?, would invoke the alternative all of the walls in [that house]1 are red, where the index/value of that house is fixed across alternatives. For another kind of example, consider assignments of comparison classes. Suppose John wants to buy a soccer team with only tall players. Knowing this, Peter tells John that someF Man U players are tall ; a relevant alternative in this case is all Man U players are tall, where the same comparison class is used in the assertion and its alternatives (e.g., 'tall for a soccer player'). 31 Del Pinal (62) *John has any eggs. a. ODA(∃x ∈ D[Pw(x) ∧ havew(J, x)]]) b. DA = {∃x ∈ D′[Pw(x) ∧ havew(J, x)] : D′ ⊆ D} c. ∃x ∈ D[Pw(x) ∧ havew(J, x)] ∧ ∀D′ ⊆ D[¬∃x ∈ D′[Pw(x) ∧ havew(J, x)]] ≈ J has a P ∈ Dhouse ∧¬J has a P ∈ Dkitchen ∧¬ J has a P ∈ Dliving room This response is technically available to proponents of Skeletons, but it comes with a price. Once it is adopted, logical skeletons can no longer be understood as resulting from a general property of the language system at the level where grammaticality is determined (cf. Chierchia 2013, Fox & Hackl 2007). The theoretical assumptions which underlie the view that the language system doesn't 'see' or 'care' about open class terms would have to be revised. In its place, proponents would have to hold that the deductive system of language can sometimes see and sometimes not whether different tokens are of the same open class term. In contrast, LF+Rescale, which preserves the validity of LNC, can accommodate the basic account of NPIs without making any ad hoc/additional stipulations. Most importantly, note that the problem posed to Skeletons by the logicality-based account of polarity distribution, when taken together with that posed by acceptable contradictions with co-binding (see §5.1), suggests that any potential reformulation of Skeletons will be rather ad hoc. For, on one hand, to account for the unacceptability of NPIs in upward entailing environments, we would need to stipulate that the deductive system can see contradictions when induced by conjoining the prejacent and its negated domain alternatives. This is possible only if logical skeletons do encode when tokens are copies of particular open class terms. On the other hand, to allow for the acceptability of (superficial) contradictions involving co-bound predicates, we would need to say that tokens of open class variables, formally co-bound by an open class term, are somehow not encoded as such by logical skeletons. 5.3 Weak presuppositional islands in manner questions The last case in support of LF+Rescale that we'll discuss centers around Abrusán's (2011a, 2014) logicality-based account of weak presuppositional islands in manner questions. Abrusán's account of this and other weak island constraints together constitute a key piece of evidence for the logicality of language. Hence supporters of this hypothesis should adopt a notion of logical form that is compatible with Abrusán's account, even if there is disagreement on the details. In addition, this discussion will allow us to explore some central 32 Triviality and Logical Form issues, relevant to various accounts in this tradition, concerning the logicality of language and the formal status of attitude verbs. In recent work (2014: ch. 4), Abrusán has already argued that her logicality-based account of weak islands is not compatible with Skeletons. In this section, I develop a reasonable response on behalf of Skeletons, and then present some novel considerations for Abrusán's conclusion. Finally, I show that LF+Rescale coheres perfectly with Abrusán's account of weak presuppositional islands. The target generalization is that wh-words that range over manners can escape weak islands when embedded under a non-factive attitude verb (e.g., hope, desire), but not when embedded under a factive one (e.g., regret, know). This basic pattern is illustrated in (63), and can be contrasted with the pattern displayed by the identity questions in (64): (63) a. *How does John regret that Peter fixed the car? b. How does John hope that Peter fixed the car? (64) a. Who does John hope fixed the car? b. Who does John regret fixed the car? Abrusán (2011a, 2014) argues that the best explanation for this pattern is that, in these manner questions, factive verbs generate contradictory presuppositions. This logicality-style account is based on two premises. The first is that, in this environment, the presence of a variable in the scope of a factive verb results in a universal presupposition. This is illustrated in (65): (65) Who among these ten people does Mary regret that Bill invited? a. λp.∃x[x ∈ {these ten people} ∧ p = λw′: Mary believesw′ that Bill invited x. Mary regretsw′ that Bill invited x] b. Presupposition: ∀x ∈ {these ten people} : Mary believes that Bill invited x In light of this, consider again the unacceptable example (63-a), repeated in (66), of a weak island violation. At first glance, the universal presupposition of the question, captured in (66-b), seems innocent: it just says that Mary believes that Peter fixed the car in each of a given set of ways. (66) *How does John regret that Peter fixed the car? a. λp.∃α[α ∈ DM ∧ p = λw′ : John believesw′ that Peter fixed the car in way α. John regretsw′ that Peter fixed the car in way α] b. Presupposition: ∀α ∈ DM : John believes that Peter fixed the car (i.e., Peter's car fixing event e) in way α 33 Del Pinal The second premise of Abrusán's account, however, is that domains of manners always contain contraries. Contrary manners-e.g., carefully vs. carelessly, slowly vs. quickly-cannot both be true of the same event, although they can both be false. On this view, any contextual assignment of a set of relevant manners, g(DM), has the following property: for every manner predicate P ∈ g(DM) there is another P ′ ∈ g(DM) such that P ∩ P ′ = ∅. Given these two premises, we can now identify the problem with (66-b): if, say, {careful, erratic}∈ g(DM) the presupposition of (66) is that John has the incoherent belief that Peter's car fixing event e was careful and erratic.26 The question that concerns us here is whether Abrusán's logicality-style account can be maintained in Skeletons and in LF+Rescale. According to Abrusán (2014), the target difference between (63-a) and (63-b) cannot be computed from their corresponding logical skeletons. To see why, consider their (partial) logical skeletons in (67-a) and (67-b). Since attitude verbs like regret and hope are (on most accounts) open class terms of the same semantic type, Skeletons entails that they should be replaced with different variables of the same type. As a result, a deductive system operating on these skeletons is blind to the original difference in their presuppositions. Since factivity is not a property of all verbal predicates that take propositional complements, we cannot stipulate it as a general constrain on D<<s,t>,<e,<s,t>>>. In short, a deductive system operating on logical skeletons cannot distinguish between (67-a) and (67-b), and thus fails to predict the contrast in their acceptability. (67) a. *How does John regret that Peter fixed the car? {John V1,<<s,t>,<e,<s,t>>> that Peter VP3 in way α : α ∈ DM} b. How does John hope that Peter fixed the car? {John V2,<<s,t>,<e,<s,t>>> that Peter VP3 in way α : α ∈ DM} In contrast, Abrusán's account can be maintained in LF+Rescale without making any ad hoc stipulations. We have seen that the difference between unacceptable cases with factives like (67-a) and acceptable cases with nonfactives like (67-b) can be proven given standard logical forms, so we only 26 Note that the source of unacceptability, in the example with regret, is the attribution of a contradictory/incoherent belief. This is quite different from generating a contradiction. The latter cannot be entailed by any context, but presumably there are contexts that entail/admit that some agent has contradictory beliefs. Parallel manner questions with knows presumably generate a stronger presupposition. Still this account might ultimately need an additional stipulation such that contradictions/tautologies are banned in every environment, including as complements of attitude verbs (this is compatible with allowing logical forms with acceptable superficial trivialities that, after contextual saturation, can be resolved to contradictions/tautologies, as I argued in §3.3). 34 Triviality and Logical Form need to show that Rescale cannot rescue the cases with factives. Two observations are crucial. First, attitude verbs like regret can be modified, as in John superficially regrets what he did, but he doesn't truly regret it. Secondly, intersective modifiers do not in general change the presuppositions of their arguments or sisters: e.g., John partially regrets that φ and John regrets that φ have the same presupposition (on this account, that John believes that φ). The simplest assumption about Rescale, in this domain, is that it behaves like its overt counterparts. Accordingly, for any context c, John Rescalec knows that φ entails φ, and John Rescalec hopes that φ does not entail φ. It follows that Rescale cannot rescue the unacceptable manner questions: (68) a. *How does John Rescalec regret that Peter fixed the car? b. *How does John really regret that Peter fixed the car? (69) a. How does John Rescalec hope that Peter fixed the car? b. How does John really hope that Peter fixed the car? Now, we can also try to rescue (66)/(67-a) by adding Rescale as an adjunct to the trace of how. The effect of this modification would be a relevance restriction on the set DM . However, Abrusán's account is based on the stipulation that any domain DM of manners, independently of how it is contextually restricted, comes with pairs of contraries. A fortiori, any restriction by Rescale would output a subdomain of manners that still includes pairs of contraries. There is, however, a way of revising Skeletons which can be used to capture the target acceptability patterns for manner questions (cf. Fox & Hackl 2007, Mayr 2017). The basic idea is this. Attitude verbs have a quantificational component, and in this respect are like some logical terms. We can thus decompose each of these items into a logical and a non-logical component. Each items factivity, or lack thereof, can be encoded in its logical component. This is illustrated in the (simplified) lexical entries (70) and (71)-note that what is erased in the corresponding skeletons for know, in (70-b), and hope, in (71-b), is just the specific accessibility relation. With this revision, the contrast between the unacceptable manner question in (67-a) and the acceptable one in (67-b) can be computed from their logical skeletons. (70) a. JknowK = λp.λx.λw : p(w) = 1. ∀w′[w′ ∈ Doxx,w → p(w′) = 1] b. Skeleton: λp.λx.λw : p(w) = 1. ∀w′[w′ ∈ Rx,w → p(w′) = 1] (71) a. JhopeK = λp.λx.λw. ∀w′[w′ ∈ Hopx,w → p(w′) = 1] b. Skeleton: λp.λx.λw. ∀w′[w′ ∈ Qx,w → p(w′) = 1] This revision of Skeletons, however, gives rise to a version of the glitch (see §3) for attitude verbs, the solution of which arguably requires that we 35 Del Pinal independently appeal to Rescale. Consider the examples in (72) and (73), all of which are strictly acceptable. Assuming that the factivity of knows is encoded in logical skeletons, the examples in (73) each seem to entail a contradiction which can be seen by the deductive system. Now, we can still account for the acceptability of (73-a) by noting that the complement of each token of knows is assigned a different logical skeleton. However, this solution is not available for (73-b), for reasons familiar from the discussion of variable co-binding in §5.1. Specifically, (72-b) and (73-b) involve the co-binding of propositional variables. Assuming factivity can be read from logical skeletons, (73-b) entails a contradiction, and is thus incorrectly predicted to be marked as unacceptable. (72) a. John believes that God exists and he also believes that God does not exist. b. That God exists is what John believes is the case and also believes is not the case. (73) a. John knows that God exists and he also knows that God does not exist. b. That God exists is what John knows is the case and also knows is not the case. In contrast, LF+Rescale correctly predicts the acceptability of these examples. At first, one might be tempted to explain this by appealing to logical forms in which the attitude verbs are modified, as in (74-a)-(74-b). However, since modification with Rescale cannot affect the presuppositions of the corresponding attitude verb, these logical forms still predict a contrast, such that only (74-b) entails a contradiction and is thus marked as unacceptable. (74) a. That God exists is what John Rescalec believes is the case and also Rescalec′ believes is not the case. b. *That God exists is what John Rescalec knows is the case and also Rescalec′ knows is not the case. Recall from the discussion of superficial contradictions in §3.2 that LF+Rescale generates a logical form for sentences like God exists as God [Rescalec exists ]. So LF+Rescale generates a logical form with covert Rescale for the topicalized clause, as in (75-a) and (76-a). In this case, (76-a) no longer entails a contradiction, and is correctly predicted to be strictly acceptable, just like (75-a).27 This logical form accounts for the default reading in contexts such as 27 Note that the value of the contextually sensitive parameter of Rescale can be different at each cite. This assumption, which also holds of other overt characters in similar constructions, 36 Triviality and Logical Form this: 'That God exists is what John knows is and is not the case. He knows God exists in each of us, but also knows that God isn't anything beyond that.' (75) a. That God Rescalec exists is what John believes is the case and also believes is not the case. b. John believes [God Rescalec′ exists] and believes not [God Rescalec′′ exists] (76) a. That God Rescalec exists is what John knows is the case and also knows is not the case. b. John knows [God Rescalec′ exists] and knows not [God Rescalec′′ exists] I should add that, even if some of the details of Abrusán's account of weak presuppositional islands in manner questions are rejected, our main results still bear on various logicality-based accounts in which the non/factivity of attitude verbs plays a key explanatory role by generating trivialities. This includes Abrusán's (2011b) account of wh-islands and Mayr's (2017) recent account of interrogative embeddings. Although I can't defend that claim here, I hope this discussion suggests that these sorts of accounts are best pursued within a version of the logicality of language that is paired with LF+Rescale rather than with Skeletons. 6 Conclusion The logicality of language is one of the most important hypotheses about the computational architecture of language to come out of recent work in formal semantics. It issues in elegant accounts of general acceptability patterns involving the distribution of not only quantificational determiners (§2), but also scalar implicatures, polarity sensitive items (§5.2), adverbs, verbs, and attitude verbs in presuppositional (§5.3) and negative islands (Dowty 1979, Fox 2000, Fox & Hackl 2007, Gajewski 2008b,a, Chierchia 2006, 2013, Abrusán 2011a,b, 2014, Mayr 2017). This paper explored which notion of logical form should be paired with the logicality of language. Minimally, any plausible candidate should have the result that, when combined with the assumption that the deductive system automatically filters out trivial expressions, it does is independently motivated. For example, in the default interpretation of (i), the comparison class of tall can vary at each position. (i) Distinctively tall is [what1 Mary was t1 in kindergarten and also t1 in college]. 37 Del Pinal not also incorrectly predict that acceptable trivial sentences, such as those in (23), are marked as unacceptable. The dominant response, we have seen, is to wed the logicality of language to the view that, at the level of representation where grammaticality is determined, logical forms are logical skeletons. The implications of this move should not be underestimated. It entails that compositional operations are blind to the content, and even the character, of all open class words. This results in a particular division between semantics and pragmatics which opens a considerable gap between the outputs of the compositional semantics and intuitions about what is said. Not only are truth-conditions the product of post-compositional pragmatic processes, but even the characters of complex expressions are not fully determined by linguistic compositional processes. The resulting view, in which semantics plays a minimal role within the language system, is reminiscent of positions advocated by Chomsky (2005, 2013). Adopting logical skeletons also has substantial implications for our conception of the automatic deductive system. The end-result of seeing every token of an open class term, hence every sentential clause, as if it was independent is that the deductive system cannot see/follow classically valid formulas and rules such as LNC and MP, or indeed almost any inference rules at all (see Williamson 1994: ch.4 on Körner's 1955, 1960 three-valued logic). The main contention of this paper is that distinguishing between L-trivial and acceptable trivial sentences within the basic framework of the logicality of language doesn't require endorsing Skeletons. There is at least one more live option, LF+Rescale, which is based on more standard assumptions about the information encoded in logical forms. In particular, on this view language can see when different tokens are of the same open class terms, and as a consequence can be paired with a deductive system which follows classical rules such as MP. Indeed, if we are correct about the comparative advantages of this view, then its ability to adequately distinguish between L-trivial and acceptable trivial sentences is a good reason for including something like Rescale-i.e., some optional item for modulation of open class terms-in our accounts of logical form. This constitutes a significant piece of evidence, so far overlooked, in favor of families of 'contextualist' frameworks which allow, albeit in different ways, some fine tuning of nouns and other open class terms within the compositional semantics (e.g., Pagin & Pelletier 2007, Recanati 2010, Stanley 2000, Szabó & Stanley 2000, Sauerland 2014, Martı 2006, Lasersohn 2012, Dekker 2014). At the same time, the argument presented here issues in a substantial constraint on all modulation-friendly frameworks that accept the logicality of language. In our implementation, Rescale does not apply to any logical or functional terms. Otherwise, we would risk-and in some cases certainly 38 Triviality and Logical Form lose-the accounts of why some L-trivial sentences are marked as unacceptable and cannot be rescued. For if we could modify the meaning of logical terms, some L-trivial sentences would arguably be rescued. For example, consider the entry for JbutK in (14-b) above, and suppose that, in some cases, we could modulate its meaning by dropping the conjunct which specifies the 'least you have to take out' condition. Given this flexibility, we should be able to rescue some students but Mary passed the exam, and incorrectly predict that it has an acceptable (and potentially quite informative) reading along the lines of 'some students passed the exam, and Mary need not be in that group'. In short, if logical/functional terms could be modulated, some L-trivial expressions should be odd but not strictly unacceptable. Since this is not the case, we conclude that functional terms are not modulated by the language system. 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Forthcoming in Ergo OBLIGATION, PERMISSION, AND BAYESIAN ORGULITY MICHAEL NIELSEN AND RUSH T. STEWART Abstract. This essay has two aims. The first is to correct an increasingly popular way to misunderstand Belot's Orgulity Argument. The Orgulity Argument charges Bayesianism with defect as a normative epistemology. For concreteness, our argument focuses on Cisewski et al.'s recent rejoinder to Belot. The conditions that underwrite their version of the argument are too strong and Belot does not endorse them on our reading. A more compelling version of the Orgulity Argument than Cisewski et al. present is available, however-a point that we make by drawing an analogy with de Finetti's argument against mandating countable additivity. Having presented the best version of the Orgulity Argument, our second aim is to develop a reply to it. We extend Elga's idea of appealing to finitely additive probability to show that the challenge posed by the Orgulity Argument can be met. Keywords. Finite additivity; orgulity; probability; topology; modesty 1. Introduction How much confidence does rationality demand in one's ability to get at the truth? A natural thought is that it depends on the nature of the inquiry at hand. A certain amount of humility might be understandable in the face of sufficiently complex problems. But, for Bayesians, such humility is out of the question. Coherence compels Bayesians to always be fully confident. Based on this fact, Gordon Belot advances an argument against Bayesianism that we call the Orgulity Argument. The upshot of the Orgulity Argument is that Bayesianism-one of the most compelling, popular, powerful, and principled epistemological frameworks ever studied-is defective as a normative theory. The argument has already proven to be influential and has produced a growing literature (Huttegger, 2015a; Weatherson, 2015; Elga, 2016; Belot, 2017; Cisewski et al., 2018; Pomatto and Sandroni, 2018). This literature raises several interesting and important points about the foundations of Bayesian learning. But we have found that many contributors to this literature have misread Belot and have failed to identify the best version of the Orgulity Argument. This is important because, if we are correct, the challenge posed to Bayesianism by the Orgulity Argument is not being addressed let alone met by most contributions to this debate. In order for the debate to advance, it is necessary to correct these mistakes. That is the first aim of this essay. Below, we focus on the most recent contribution to the debate from Cisewski et al. (2018), though we indicate also how our argument applies to other readers of Belot (Huttegger, 2015a; Weatherson, 2015; Pomatto and Sandroni, 2018). We argue that Cisewski et al.'s version of the Orgulity Argument relies on two premises that Belot does not endorse and, interpretative issues aside, are too strong to underwrite a plausible challenge to Bayesianism. The main point is simple. There is a clear difference between mandating certain behavior and insisting that the behavior be permissible. On Cisewski et al.'s reading of the Orgulity Argument, a certain kind of epistemological modesty (to be explained below) is rationally mandatory. On our reading, the argument relies on a considerably weaker premise: such modesty is permissible. Having articulated the correct way to understand the Orgulity Argument, the second aim of our essay is to defend a line of response to the argument, initiated by Elga (2016), 1 2 NIELSEN AND STEWART that involves relaxing countable additivity. This approach was initiated by Elga (2016) and involves relaxing countable additivity. Here, the first part of the essay proves useful. The tools that we develop for thinking about the correct version of the Orgulity Argument also help us to see that criticisms of Elga's reply are flawed. We then extend Elga's argument, showing that replies to the Orgulity Argument based on finitely additive probability theory are surprisingly robust. After introducing the framework and the relevant concepts for the debate in Section 2, we present and discuss the Orgulity Argument as Cisewski et al. understand it (Section 3). By our lights, their interpretation exemplifies a common way of misunderstanding Belots point. We then present a better version of the Orgulity Argument and clarify the normative confusion at stake by drawing an analogy to de Finetti's argument against mandating countable additivity (Sections 4 and 5). Finally, in Section 6, we develop the finitely additive approach to answering the Orgulity Argument. 2. Preliminaries All versions of the Orgulity Argument rely on the mathematical concepts that we introduce in this section. Let C " t0, 1uN be the set of all countably infinite binary sequences. We could interpret C as the set of possible outcomes of an experiment in which a coin is tossed infinitely many times, for example. In order to produce a probability space, we pair C with a particular sigma-algebra of its subsets, which we denote B.1 Elements of B are called hypotheses or events. Bayesianism is the view that a rational agent's state of uncertainty regarding some set of live possibilities is represented by a probability measure.2 This probability measure is subject to revision as the agent learns. In particular, agents learn by conditionalization, forming posterior probabilities from prior conditional probabilities. The Orgulity Argument is a critique of the epistemological significance often attributed to the Bayesian convergenceto-the-truth theorem. To present the theorem, we introduce some more notation. Let P be a probability measure on pC,Bq. If ω P C, let ωn be the set of sequences that agree with ω up to the nth digit. If H is an event in B, let 1H be the indicator function for H. 3 The convergence-to-the-truth theorem says that P ptω : 1Hpωq " lim nÑ8 P pH|ωnquq " 1 (1) for all hypotheses H.4 Suppose that ω is the actual sequence of 0s and 1s, and that this sequence is to be revealed one bit at a time to an agent with personal probabilities given by P . After observing the nth bit, the agent updates her probabilities by conditionalizing on ωn. Then, (1) says that for any hypothesis H, the agent believes (with probability 1) that her posterior probabilities for H will converge to 1 or 0 according to whether H is true or false, respectively. For any hypothesis H, the agent believes that she will converge 1We endow C with the infinite product topology of discrete topologies on t0, 1u. The resulting topological space is called Cantor space. B is the (Borel) sigma-algebra that is generated by the open subsets of Cantor space. 2With Belot, we will assume for the moment that probabilities are countably additive. The probability P satisfies countable additivity if, for any countable collection tAiuiPN of pairwise disjoint hypotheses in B, we have P ` Ť8 i"1 Ai " ř8 i"1 P pAiq. We return to countable additivity below. 31Hpωq " # 1, ω P H 0, ω R H 4See, for example, Billingsley (2008, Theorem 35.6). OBLIGATION, PERMISSION, AND BAYESIAN ORGULITY 3 to the truth about H. We call the complement of the set that appears in (1), namely tω : 1Hpωq ‰ limnÑ8 P pH|ω nqu, the failure set for H (with respect to P ). If ω is in the failure set for H, then conditionalizing on ωn will not lead P to the truth about H. Another way of summarizing (1) is to say that, for any hypothesis H, a Bayesian agent assigns probability 0 to the failure set for H. The failure set for H is probabilistically "negligible" or "small" because it is assigned probability 0. The Orgulity Argument appeals to a distinct notion of smallness that uses topological concepts. A subset A of a topological space is nowhere dense if its closure has empty interior, i.e. the smallest closed set containing A does not contain any open sets. Nowhere dense sets are topologically small. For example, any finite subset of R is nowhere dense in the standard topology. A set is called meager if it is the countable union of nowhere dense sets.5 Since countable unions of probability 0 sets have probability 0, it makes sense to say that countable unions of probabilistically small sets are probabilistically small. Similarly, meager sets, like the nowhere dense sets that compose them, are often said to be small or "atypical." The complement of a meager set is a comeager (or residual) set and is topologically "large" or "typical." We can now state the mathematical fact that is the driving force behind the Orgulity Argument. Topological and probabilistic notions of smallness need not coincide. In particular, there are probabilities P on pC,Bq and hypotheses H in B such that the failure set for H with respect to P is comeager.6 But applying (1) to the failure set for H, we see that it is assigned probability 0. Although the failure set for H is topologically large, it is probabilistically small. An agent with personal probabilities given by P assigns probability 0 to failing to converge to the truth about H even though that event is "typical." 3. Cisewski et al.'s Version of the Orgulity Argument As Cisewski et al. (2018) read Belot, assigning probability 0 to a topologically large set is a sign of epistemological immodesty. An agent with such probabilities is "practically certain" that a (topologically) typical event will not occur. The main premise of the Orgulity Argument, on Cisewski et al.'s reading, is that epistemological immodesty in this sense is rationally forbidden. More precisely, Cisewski et al. claim that the following conditions underwrite the Orgulity Argument. Topological Condition #1: Do not assign probability 1 to a meager set. Topological Condition #2: Assign probability 0 to each hypothesis that is a meager set.7 5Some further examples: the Cantor set is an uncountable set but is nowhere dense, and so meager. Q is a countable and dense subset of R, but is also meager. 6Belot shows that if a probability is "open-minded" in a technical sense (2013, p. 496), then its failure set for a hypothesis that is countable and dense is comeager (2013, p. 499). Treating this sense of openmindedness will not be essential for our purposes since we can focus directly on the existence of comeager failure sets. 7Cisewski et al. qualify these conditions to apply only to meager sets of observables. It isn't entirely clear what they intend with this qualification. One possibility is that it indicates that the above conditions do not apply to hypotheses whose truth values are not determined by (even infinitely many) observations. Bayesian convergence to the truth, at any rate, does not hold for such hypotheses (Huttegger, 2015b, p. 590). (Our claim in the introduction that coherence compels Bayesians to "always be fully confident" likewise fails for such hypotheses.) If this is what Cisewski et al. intend, then our omission is harmless since we have restricted ourselves to the space of observables. 4 NIELSEN AND STEWART We will focus on Condition #1, but our criticisms carry over straightforwardly to the "more demanding" Condition #2. As pointed out in the previous section, there exist probability spaces pC,B, P q and meager subsets of C that are assigned P -probability 1. Such a P violates Topological Condition # 1. The conclusion of the Orgulity Argument, on Cisewski et al.'s reading, is that Bayesianism should be rejected as a theory of rationality. Cisewski et al. deny the Orgulity Argument's conclusion and go on to argue convincingly that the above topological conditions have some untenable consequences. Others seem to share Cisewski et al.'s view that the above topological conditions underwrite the Orgulity Argument. For example, according to Huttegger, Belot's argument "appears to be along the lines of 'if a set is residual in the topology, then it should have positive probability'" (2015b, p. 594). The injunction that Huttegger attributes to Belot is equivalent to Topological Condition #1. If residual sets should have positive probability, then their complements should not have probability 1. And, by definition, the complement of a residual set is meager. So, Huttegger is attributing to Belot the view that meager sets should not have probability 1. Similarly, Weatherson presents the Orgulity Argument as introducing the "requirement" of modesty for residual sets (2015, p. 532). For Weatherson, a probability P is modest with respect to a hypothesis H if P assigns positive probability to the failure set for H.8 To say that modesty is required for residual sets, then, is tantamount to an endorsement of Topological Condition #1. Although neither Weatherson nor Huttegger is as explicit in their presentation of the Orgulity Argument as Cisewski et al., it is clear that the above topological conditions underwrite their interpretations of Belot. We agree with Cisewski et al. that the conclusion of the Orgulity Argument is that Bayesianism should be rejected. But as we interpret Belot, the above topological conditions do not in fact occur as premises in the Orgulity Argument. Belot ends his paper with the following summary: "The truth concerning Bayesian convergence-to-the-truth results is significantly worse than has been generally allowed-they constitute a real liability for Bayesianism by forbidding a reasonable epistemological modesty" (2013, emphasis ours). His indictment of Bayesianism is based on the claim that modest probability judgments, which ought to be permissible, are deemed impermissible (forbidden) on the Bayesian account. Cisewski et al. reply instead to a version of the Orgulity Argument according to which such modesty is mandatory. But the Orgulity Argument need not (and it appears that Belot does not) mandate modesty to arrive at its conclusion. We develop this alternative way to construe the Orgulity Argument in Sections 4 and 5. But it is worth pointing out already that the Cisewski et al. topological conditions are not even plausible on their face. Consider the probability according to which each digit in ω is determined by the outcome of a fair coin toss. By the strong law of large numbers, the hypothesis that the limiting relative frequency of 1s is 1{2 is assigned probability 1. But, it turns out, this hypothesis is a meager subset of C.9 Topological Condition #1 implies that this probability measure is rationally impermissible. If one thinks that there are some contexts in which it is rationally permissible to adopt probabilities represented by the fair coin measure, then one should reject Topological Condition #1. Whether or not Belot in fact 8We agree with Weatherson that this is the correct way to understand modesty in the context of the Orgulity Argument (cf. Section 5). 9Belot suggested this example to us in personal communication. For details, see Oxtoby (2013, p. 85). OBLIGATION, PERMISSION, AND BAYESIAN ORGULITY 5 endorses the topological conditions that Cisewski et al. attribute to him, a better version of the Orgulity Argument is available. This form of the argument relies on neither of the above topological conditions. We turn to this version of the Orgulity Argument now. 4. An Analogy: de Finetti's Lottery We understand the Orgulity Argument in analogy to de Finetti's criticisms of countable additivity. Since we expect the latter to be familiar to most participants in the orgulity debate, we find it helpful to point out that the Orgulity Argument has the same structure as the argument based on de Finetti's lottery. So far, we have assumed that all rational probability judgments are countably additive, but de Finetti denies this. On his view, if we take countable additivity to be a universal constraint on rational probability judgments, eminently reasonable credal states are ruled out. De Finetti does not insist that personal probability must always violate countable additivity. Rather, he insists that violating countable additivity is permissible (at least under certain circumstances). Insistence on countable additivity is heavy-handedness. Consider a lottery consisting of countably many tickets, one for each integer. According to de Finetti, it should be open to a rational agent to consider such a lottery fair (de Finetti, 1990). To do so, each ticket must be assigned equal probability. But if probabilities are countably additive, this is not possible. If each ticket is assigned the same positive probability, then the probability of the union of the tickets, being equal to the sum of the probabilities of the individual tickets, is infinite, and not 1 as the probability axioms demand. If, on the other hand, each ticket is assigned probability 0, then so is the probability of their union. Again, this contradicts the probability axioms. The example is meant to show that countable additivity is heavy-handed as a constraint on rational probabilities. It forbids an opinion that is reasonable, at least in some circumstances, namely that a lottery on the integers is fair. If one is willing, as de Finetti is, to relax countable additivity in response to these considerations, then one can adopt a merely finitely additive probability that assigns each ticket in the integer lottery probability 0, thus reflecting the opinion that the lottery is fair.10 There are various ways to counter de Finetti's argument. One could contest its conclusion- that violations of countable additivity are permissible-by denying the premise that it is sometimes reasonable to believe that a lottery on the integers is fair. Or, one could admit that de Finetti's argument counts as a consideration against mandating countable additivity but claim that this consideration is outweighed by arguments that support mandatory countable additivity. For example, one could appeal to extended Dutch book arguments based on countable collections of bets or to countable additivity's theoretical fruitfulness. Rejoinders to these reservations about de Finetti's position have been voiced a number of times in the literature (e.g., Kelly, 1996; Howson, 2014). Our aim in this section is not to adjudicate debates about the integer lottery. We have rehearsed de Finetti's argument in order to show that the best version of the Orgulity Argument has an analogous structure. As we will explain at greater length in the next section, according to proponents of the Orgulity Argument, Bayesianism forbids an opinion that is reasonable, at least in some circumstances, namely, having less than full confidence in converging to the truth. Later in the paper (Section 6), we will push the analogy further when 10For a relatively recent study of uniform distributions on the natural numbers, see (Schirokauer and Kadane, 2007). 6 NIELSEN AND STEWART we turn to evaluating a particular reply to the Orgulity Argument based on finitely additive probability. 5. A Modest Orgulity Argument As we have just explained, the inconsistency of countable additivity with a fair lottery on the integers is taken by some as grounds to reject countable additivity as a requirement of rationality. Similarly, we suggest, the correct version of the Orgulity Argument sees the inconsistency of Bayesianism with a "reasonable epistemological modesty" as grounds to reject Bayesianism. We call this version of the argument modest because it relies on a premise that is strictly weaker than the topological conditions that appear in Cisewski et al.'s version. According to the modest Orgulity Argument, a probability P is modest concerning some hypothesis H P B if P assigns the failure set for H positive probability. Such modesty is reasonable if the failure set is comeager (topologically "typical"). "Reasonable" here is interpreted as it was above for the de Finetti lottery, that is, as synonymous with "permissible" not with "required." Some agent may adopt a modest opinion regarding H without thereby surrendering his rationality. In order to accuse Bayesianism of orgulity, it is enough to think that it should be rationally permissible for an agent to assign positive probability to a comeager failure set. Such modesty need not be required as Cisewski et al.'s first topological condition puts it. The following topological condition underwrites the modest Orgulity Argument. Topological Condition #3: It is permissible (in some circumstances) to assign positive probability to a comeager failure set. What is desired, on our reading of Belot, is the existence of some probability P on pC,Bq that assigns a comeager failure set positive probability. Some such P should be rationally permissible. In other words, for some H, among the rational probability measures for which the failure set for H is comeager, there should be some P such that P ` tω : 1Hpωq ‰ lim nÑ8 P pH|ωnqu ą 0. But Bayesianism forbids this, by (1). There is no such probability. So Bayesianism, the argument concludes, should be rejected.11 On our view, attacks on Topological Conditions #1 and #2 are analogous to attacks on mandatory violations of countable additivity. They do not bear on the modest Orgulity Argument, just as attacks on the position that mere finite additivity is mandatory do not bear on de Finetti's position as we presented it in the last section. Propounders of the Orgulity Argument, as we understand them, are pleading for less stringent epistemological demands, not proposing new ones. We find that this modest version of the Orgulity Argument, even if ultimately wrong, is more compelling than the version that forbids assigning probability 1 to meager sets. It is the modest Orgulity Argument that should be the focus of the debate. 11Some seem to endorse something akin to our reading of the argument. For example, regarding the assignment of positive probability to a comeager failure set, Elga asks, "Is such humility rationally permissible? According to Gordon Belot's orgulity argument: the answer is yes, but long-run convergence-to-the-truth theorems force Bayesians to answer no" (2016, p. 305, emphasis ours). OBLIGATION, PERMISSION, AND BAYESIAN ORGULITY 7 6. More on Comeager Failure Sets and Finite Additivity Having identified the best version of the Orgulity argument, we are now in a position to respond to it. Here, the analogy to de Finetti's criticism of countable additivity shows itself to be doubly apt. Just as in the integer lottery case, the epistemic state desired by proponents of the Orgulity Argument is permissible if countable additivity is relaxed (Elga, 2016; Cisewski et al., 2018). Elga shows that there exists a merely finitely additive probability measure that assigns positive probability to a comeager failure set.12 This probability assignment exhibits the sort of epistemological modesty that Belot thinks is reasonable. While a Bayesianism that assumes countable additivity is inconsistent with Topological Condition #3, Elga's reply shows that finitely additive Bayesianism is not, just as some finitely additive probabilities are consistent with the judgment that the integer lottery is fair. It seems the relevance of Elga's point has been misunderstood along the same lines as Belot's original argument has. For example, Pomatto and Sandroni claim that concerns about Bayesian orgulity extend to merely finitely additive probabilities because some merely finitely additive probabilities converge to the truth in the context of a certain "hard" kind of inference problem (2018, p. 15). By our lights, the important point in the context of Pomatto and Sandroni's framework is that, unlike countably additive probabilities, not all merely finitely additive probabilities are fully confident in solving the problem. Cisewski et al. find Elga's reply unsatisfactory for similar reasons. They complain that his example shows modesty about only one event. They write, "Hence, though Elga argues that P is modest with respect to one limiting frequency hypothesis, according to Condition # 1 P is immodest for a different but related hypothesis about existence of well-defined limiting frequencies" (Cisewski et al., 2018, p. 61). In view of our articulation of the modest Orgulity Argument, it is now easy to see that these criticisms are misguided. Recall that, as we have shown above, Topological Condition #1 does not occur as a premise in the Orgulity Argument. Rather, the main premise of the Orgulity Argument is Topological Condition #3. With this point firmly in mind it is clear that a reply to the Orgulity Argument based on relaxing countable additivity is legitimate. Since it can be shown that there exist merely finitely additive probabilities that assign positive measure to some comeager failure set, it follows that finitely additive Bayesianism is consistent with Topological Condition #3. This should go some way towards answering Belot's intended challenge. While we do not find their criticisms of Elga compelling, Cisewski et al.'s remarks do raise the possibility of strengthening the Orgulity Argument's main premise. A premise intermediate to Belot's and Cisewski et al.'s asks that modesty concerning every comeager failure set be permissible under some circumstances. Although not explicitly endorsed by Belot, we find the following condition worthy of further investigation. Topological Condition #4: It is permissible (in some circumstances) to assign positive probability to every comeager failure set. Topological Condition #4 is clearly stronger than Topological Condition #3, but it is weaker than Topological Condition #1 on two counts. First, in keeping with the main theme of this 12Cisewski et al. dispute the mathematical details of the particular example Elga offers, arguing that his proofs involve conditioning on null events. They then offer a slightly modified example, which they claim achieves what Elga intends. But even if a merely finitely additive probability assigns positive probability to some comeager failure set, it may assign probability 0 to some other comeager set. 8 NIELSEN AND STEWART paper, it claims that certain probability assignments are permissible. It does not mandate any particular assignments. Second, Topological Condition #4 applies only to comeager failure sets, not to every comeager set. Only the former are relevant to the Orgulity Argument. Cisewski et al.'s criticism raises the question, Can Elga's reply be extended by showing that finitely additive Bayesianism is consistent with Topological Condition #4? In fact it can be, as we show with the following proposition. Proposition 1. There exist uncountably many finitely additive probabilities that assign positive probability to all their comeager failure sets. Proof. To see that the proposition holds, note that the collection of comeager subsets of C forms a filter on the powerset of C. That is, the collection of comeager sets contains C and is closed under supersets and finite intersections. The filter of comeager subsets of C can be extended to an ultrafilter U , a filter that contains exactly one of A or Ac for every A Ď C. The set function U defined on the powerset of C by UpAq " 1 if A P U and UpAq " 0 if A R U is a finitely additive probability measure that assigns probability 1 to every comeager subset of C. Hence, the restriction of U to the Borel sigma-algebra B, which we will also denote by U , is a finitely additive probability measure that assigns probability 1 to every comeager, Borel-measurable subset of C. A fortiori, U assigns positive probability to all of its comeager failure sets. This establishes that U satisfies Topological Condition #4. To see that there are uncountably many other such finitely additive probabilities, fix λ P p0, 1q and observe that the function P ÞÑ λP ` p1 λqU injects the space of probability measures on pC,Bq into the space of probability measures that satisfy Topological Condition #4. This proves the proposition. We remark that the argument just given actually establishes the following result. For any topological space X, and any measurable space pX,Fq, let P be the collection of finitely additive probabilities P on pX,Fq satisfying: P pAq ą 0 for every measurable, comeager subset of X. Then the cardinality of P is equal to the cardinality of the space of all probabilities on pX,Fq.  Although the argument just given shows that finitely additive Bayesianism is consistent with Topological Condition #4, there is an objection to consider. One might regard the probabilities constructed above as mere mathematical curiosities. It's not clear that they bear any resemblance to the more familiar probability constructions that Belot uses to advance his argument. In addition to answering this objection, the next proposition, as we will see, also improves on Proposition 1. To begin, we must first recall how probabilities are typically constructed in the countably additive setting. The standard approach is to first define P on a collection of "simple" or "low-complexity" subsets. One then appeals to an extension theorem to establish that P extends to the entire sigma-algebra generated by the simple subsets. In our setting, the simple subsets are all those of the form ωn, as well as finite unions of these sets. This collection forms an algebra, which we denote by F , and F generates B. For example, to construct the fair coin measure that was mentioned before, one first defines P by P pωnq " 2 n and then shows that P is countably additive on F . By the Carathédory extension theorem, P extends to a unique countably additive probability whose domain is all of B. This is how more or less every familiar and interesting countably additive measure on pC,Bq is constructed.13 The next result shows that any P defined on F can be extended to 13There are ways to define P directly, without appeal to extension theorems, for example, by letting P be point mass concentrated on some ω P C. OBLIGATION, PERMISSION, AND BAYESIAN ORGULITY 9 a finitely additive probability that is consistent with Topological Condition #4. This means that for every countably additive probability, there exists a finitely additive probability that "resembles" it in the sense that the two agree on F , and the finitely additive probability is modest in the strong sense of condition #4. Proposition 2. Let P be a finitely additive probability with domain F . Let G be any algebra containing F . There exists an extension P 1 of P to G such that P 1pCq " 1 for every comeager subset C of C in G.14 Proof. The proof follows closely the proof of Theorem 4 in Pomatto et al. (2014). Let M be the collection of meager subsets of C, and let A " pF X Cq YM : F P F , Cc PMX G,M PMX G ( . Observe that F Ď A Ď G. One can now verify that A is an algebra, and that the function P 1 defined on A by P 1ppF X Cq YMq " P pF q is a well-defined, finitely additive probability measure (see the reference above for details). By construction, with F " H, P 1pMq " 0 for every meager subset M in G. With a harmless abuse of notation, P 1 now extends to G, and the result is proved.  Our propositions show that replies to the Orgulity Argument based on finitely additive probability are robust. By relaxing countable additivity, various notions of modesty akin to the one in which Belot is interested are easily accommodated.15 As with de Finetti's lottery argument, there are responses to the Orgulity Argument besides relaxing countable additivity. One could deny the basic assumption that topology is of any relevance to epistemology, for example (Huttegger, 2015b).16 In fact, we are quite sympathetic to this line of response. In order for the Orgulity Argument to be truly convincing, an explanation of when and why confidence should coincide with topological typicality is needed. An effort to fill this gap would be an interesting line of research. However, as a reply to the Orgulity Argument, appealing to finitely additive probabilities has the rhetorical advantage of allowing us to grant both the general relevance of topology to epistemology and the various specific topological conditions for the sake of the argument. Even with these assumptions granted for the sake of the argument, finitely additive Bayesianism need not be abandoned. Alternatively, one may find the assumptions of the Orgulity Argument plausible or genuinely compelling. In that case, Elga's point and our propositions contribute to the case against mandating countable additivity. 7. Conclusion The modest Orgulity Argument takes as a premise that it is permissible, at least sometimes, to assign probability greater than 0 to a comeager failure set. It does not appeal to a universal injunction against assigning probability 1 to meager sets à la Topological Condition #1. To do so would be analogous to claiming that countably additive probabilities are rationally forbidden. We hope that future contributors to the debate focus their efforts on the modest version of the Orgulity Argument that we have articulated here. 14Thanks to Belot for pointing out in conversation that Proposition 2 strengthens Proposition 1. 15For what it's worth, one can glean from the above arguments that even Cisewski et al.'s demanding conditions are satisfiable by moving to finitely additive probabilities. 16Pollard observes, "Many probabilists seem to regard topology as completely extraneous to any discussion of conditioning, or even to any discussion of abstract probability theory" (2002, p. 117). 10 NIELSEN AND STEWART Following Elga, we have also defended a reply to the modest Orgulity Argument that appeals to finitely additive probability. We have argued that this reply is robust by considering topological conditions that are stronger than those that Belot appeals to but weaker and more plausible than those of Cisewski et al. In particular, Topological Condition #4 claims that it is permissible in some circumstances to assign positive probability to all comeager failure sets. We have shown that finitely additive probabilities can meet the even more demanding challenges raised by this new condition. There are uncountably many finitely additive probabilities on pC,Bq that assign positive probability to all their comeager failure sets. Just as in de Finetti's lottery, finitely additive probability permits agents to adopt seemingly reasonable states of probability judgment that are ruled out by countable additivity. OBLIGATION, PERMISSION, AND BAYESIAN ORGULITY 11 References Belot, G. (2013). Bayesian orgulity. Philosophy of Science 80 (4), 483–503. Belot, G. (2017). Objectivity and bias. Mind 126 (503), 655–695. Billingsley, P. (2008). Probability and Measure. John Wiley & Sons. Cisewski, J., J. B. Kadane, M. J. Schervish, T. Seidenfeld, and R. Stern (2018). Standards for modest Bayesian credences. 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© 2010. Epoché, Volume 14, Issue 2 (Spring 2010). ISSN 1085-1968. 399–415 A Phenomenological (Husserlian) Defense of Bergson's "Idealistic Concession" MICHAEL KELLY Boston College Abstract: When summarizing the fi ndings of his 1896 Matter and Memory, Bergson claims: "That every reality has . . . a relation with consciousness-this is what we concede to idealism." Yet Bergson's 1896 text presents the theory of "pure perception," which, since it accounts for perception according to the brain's mechanical transmissions, apparently leaves no room for subjective consciousness. Bergson's theory of pure perception would appear to render his idealistic concession absurd. In this paper, I attempt to defend Bergson's idealistic concession. I argue that Bergson's account of cerebral transmissions at the level of pure perception necessarily entails a theory of temporality, an appeal to a theory of time-consciousness that justifi es his idealistic concession. When summarizing the fi ndings of his 1896 Matter and Memory,1 Bergson makes a rather strong and-perhaps to some-peculiar concession to idealism; he writes, "That every reality has . . . a relation with consciousness-this is what we concede to idealism. . . . No philosophical doctrine, moreover, provided that it is consistent with itself, can escape from this conclusion" (360/229). One only need refl ect briefl y upon the title of the work in question, with its prioritized order of terms and dualistic assumption, to discern the tension created by Bergson's idealistic concession (IC). And if one penetrates the pages of this rich and complex work and examines the theory of pure perception (PP) therein, then the tension heightens, for Bergson's theory accounts for perception according to the brain's mechanical transmissions. Unsurprisingly, phenomenologists from the second to the present generation, as well as contemporary Bergsonists, have concluded that Bergson's theory of PP renders his IC absurd.2 In this paper, I attempt to defend Bergson's IC by appealing to the view of temporality at the base of his theory of PP. Such a defense comes by way of examining Bergson's account of the relation between memory and PP. While Bergson confusedly expresses this 400 Michael Kelly distinction between memory and PP, his account of the preservation of cerebral stimuli invites us to clarify this distinction by appealing to Husserl's view of inner time-consciousness and thereby justify his IC. In section one, I explain the tension between Bergson's theory of PP and his IC. In section two, I consider the possibility of reconciling PP and IC by examining Bergson's account of bodily consciousness, which reveals two central claims in Bergson's account of PP: (1) PP entails duration because "every sensation translates a . . . succession . . . of vibrations," and (2) duration at the level of PP entails durée, i.e., consciousness' "memory of . . . immediate experience . . . not memory of former experience" (280, 377/138, 249). These claims, I argue in section three, reveal that Bergson's account of PP implies a consciousness of temporal distention, which view amounts to something quite similar to Husserl's distinction between retention and memory. In short, we shall see that Bergson's theory of PP entails a theory of time-consciousness that underscores the sensibility of IC. 1. The Idealistic Concession and Pure Perception: A Hypothetical Concession to Materialism Bergson begins Matter and Memory with an assumption designed to overcome philosophy's persistent "problem of appearances." Bergson's assumption amounts to a methodological move designed to circumvent the ontological opposition of inner to outer. He writes, We . . . assume for the moment that we know nothing of the theories of matter and . . . the theories of spirit, nothing of the discussions concerning the reality or ideality of the external world. (169/17) Like Husserl's epoché 3 Bergson's methodological suspension of dogmatic beliefs about the world, does not amount to a Cartesian denial of the external world. Rather, Bergson "ask[s] . . . of the reader . . . to forget . . . the disputes between philosophers" (162/10). This epoché, argues Bergson, makes possible the discovery of images in "the vaguest sense of the word, images perceived when my senses are opened to them, unperceived when they are closed" (169/17). In opposition to realism and idealism, the image denotes "a certain existence that is more than what the idealist calls a representation, but less than what the realist calls a thing" (161/9). For Bergson, the "object exists in itself and . . . is pictorial as we perceive it: image it is, but an image which exists in itself " (162/10). In place of Cartesian idealism's separation of thing from thought and Cartesian materialism's reduction of idea to cerebral transmissions, Bergson proposes to consider the common denominator of these positions, namely the image in its specifi c relation to the perceiver. One may even venture to say, in phenomenological parlance, that Bergson understands the image as the phenomenon as it appears to the perceiver independently of traditional scientifi c and philosophical theories. A Phenomenological Defense of Bergson's "Idealistic Concession" 401 The complicated status of Bergson's image aside for the moment,4 his choice of term implies an innovative sense of idealism: That every reality has a kinship, an analogy-in short, a relation with consciousness, this is what we concede to idealism by the very fact that we term things "images." No philosophical doctrine, moreover, provided that it is consistent with itself, can escape from this conclusion. (360/229-my italics) The apparent expansiveness of the realm of images in Bergson's philosophy is undeniable. Everything, argues Bergson, is image after the epoché, including my body. Indeed, "the afferent nerves, . . . the brain, . . . the disturbance traveling through the sensory nerves and propagated in the brain"-i.e., all the signifi cations with which neuroscience operates-amount to "images" (170/19). The letter of Bergson's text establishes his IC indisputably. But it is not even clear that IC is coherent within Bergson's broader account. Bergson himself qualifi es both his notion of the image and his IC. Taking the latter qualifi cation fi rst, Bergson insists contra idealism that we must restrict perception to its "true offi ce . . . [which] is to prepare actions" (360/229). And the foundation for this offi ce of action, i.e., perception, Bergson argues, rests in the body's rudimentary preparation of actions. Here, the former qualifi cation becomes important for understanding the tension created by IC. Though an image among images, my body constitutes a part of the whole of images, which Bergson terms matter (173, 176/22, 25). Having obfuscated his concept of the image and tied perception to the body as an image among images, or a part of matter, Bergson's IC begins to seem absurd (173–4/22). Moreover, to explain exactly how perception prepares actions, Bergson proposes to "simplify the conditions under which perception takes place" (183–4/33). This simplifi cation, which amounts to a schematic rendering of human perception that "exists in theory rather than fact" (185/34), is Bergson's hypothesis of PP. Here is the difference: Since the hypothesis of PP forms the basis on which we perceive images, and since images form the basis of Bergson's IC, these theories collide. The relation between the theories thus requires a closer look. PP requires the philosopher to bracket all activities contributing to everyday concrete perception, e.g., memory, recognition, etc., and consider perception "confi ned to the present" cerebral interval (183–5, 212/33–4, 65). At this "lowest degree of the mind" (356/222), the brain connects inseparably to the whole of images, i.e., matter. Specifi cally, the brain functions as the switchboard (180, 194/30, 45)5 through which centripetal and centrifugal forces transfer at the impersonal, non-subjective level of PP.6 The body's interaction with the world in PP parallels that of an amoeba's (180, 182, 203/30, 32, 55). But unlike the amoeba that reacts immediately to touch, the higher functioning organism enjoys a minimal freedom of response, a zone of indetermination, as Bergson calls it (184/34). In the higher functioning organism, the brain absorbs stimuli from which it prepares 402 Michael Kelly possible actions and "chooses" (179/29) the most effi cacious neural pathway to satisfy its basic needs, e.g., hunger, thirst, sex etc (359–60/229). Perception thus arises for advanced organisms in a moment of hesitation between the stimulus and the response. On Bergson's account, the brain contracts and narrows the whole of images, selects from this whole the partial image that suits its interest, and initiates bodily action (356–9/225–8). This material and impersonal cerebral instant in PP, the rudimentary stage in the preparation of actions, Bergson terms the "poverty of consciousness" (188/38). And he deems this poverty of consciousness in PP "necessary" because concrete perception-that is, common sense or mundane perception before the hypothetical construct of PP-could not arise apart from this "material" substrate (183, 188–9/34, 38–9). The "absurdity" of IC now emerges. If perception arises from the 'telephonic exchange' between the brain and the world, then "no need" exists "for a subjective consciousness;"7 and if no need exists for a subjective consciousness, then, the phenomenologist retorts, Bergson's IC appears absurd.8 Since bodily consciousness connects to the world via the brain (359/228) on Bergson's account,9 the coherence of his IC depends upon establishing a place for conscious perception within this vast network of image-relations (the totality of which, as we have seen, Bergson terms matter). But it is precisely a place for conscious perception within PP that phenomenologists think Bergson's very theory eliminates. In his typically laconic way, Sartre dismisses the possibility of Bergson's IC because he (Sartre) cannot see how this "impersonal consciousness" in the material body "becomes the conscious consciousness of an individual subject."10 Unlike Sartre's sweeping charge, Merleau-Ponty specifi cally targeted Bergson's theory of PP as the specifi c source of psychic blindness in Bergson's philosophy.11 The phenomenologist of the lived-body believed that PP advanced a theory of the body's perception based upon the brain's mechanistic relation to the material world. Unlike his own phenomenological account of embodied consciousness, the early Merleau-Ponty considers Bergson's account of the body in PP as an objective, scientifi c account that precludes a proper investigation into the intentional structures of bodily consciousness. Hence, Merleau-Ponty concluded that PP constituted at best another form of psychologism, at worst a reductionistic materialism.12 Perhaps underscoring the phenomenological critique of Bergson, moreover, the latter of these charges does not ring as a criticism in the ears of recent poststructural readings of Bergson developed out of Deleuze's infl uential Bergsonism.13 My point is not so much that Deleuze and those reading Bergson with Deleuze read Bergson as a thoroughgoing materialist; a reading of Deleuze's What is Philosophy? is enough to see that he does not.14 What is interesting in this juxtaposition of readings of Bergson is that phenomenology's supposed allergy to any materialist concessions makes phenomenologists dismissive of Bergson's IC A Phenomenological Defense of Bergson's "Idealistic Concession" 403 on the very score on which Deleuze and Bergsonists appreciate Bergson, namely the privileging of neither idealism nor materialism. On such Deleuzean inspired Bergsonist accounts, the redeeming quality of Bergson's thought is the materialist concession that explains conscious perception with "a feature of matter in its most immanent mode"15 as "impersonal" (195/46). My point in carving out this opposition between phenomenologists and Bergsonists, then, is that from very different philosophical starting points and thus very different readings of Bergson, both camps share the same conclusion regarding Bergson's IC. And the conclusion can be stated as follows according to the Deleuzean interpretation: How can one interpret Bergson's IC as anything more than a "so-called concession" when his theory of PP admittedly deals "hardly . . . with the spirit" (365/235)?16 Indeed, as Merleau-Ponty put it, "the action of which Bergson is thinking is always virtual action, that by which the organism maintains itself in existence."17 2. The Poverty of Consciousness: Foretelling but not yet Establishing the Idealistic Concession Bergson's theory of PP certainly appears to render his IC absurd. Bergson's IC certainly requires a place for consciousness, but this place seems unlikely in an account perception that attributes perception to "the work of the brain."18 To address this issue, one might argue for a place for consciousness in PP by examining Bergson's description of impoverished consciousness as the "impersonal" and "physical basis of my personality" (195, 209/46, 61). In his account of the body at the level of PP in that state of impoverished consciousness, Bergson describes the body as having a "double faculty" of awareness (209/61). He writes: [T]his image [my body] always occupies the center of representation, so that the other images range themselves round it in the . . . order in which they might be subject to its actions; on the other hand, I know it from within, by sensations which I term affective, instead of knowing only, as in the case of other images, its outer skin. There is, then, in the aggregate of images, a privileged image. . . . [It] is this particular image which I adopt as the center of my universe . . . the physical basis of my personality. (209/61) Not merely an image among the whole of images, not merely a part of matter, Bergson thus articulates a sense of bodily self-givenness in which the self is aware of itself not as an object, i.e., not as it is aware of "other images." In this case, the reader of Bergson can identify a descriptive account of the distinction between fi rst-personal, non-objective, tacit bodily self-givenness and third-personal, objective, conceptual self-consciousness.19 That the impersonal founds the personal descriptively implies and necessarily entails that the former includes some sense of consciousness' ownership of its body's rudimentary functions such that this impoverished consciousness can assimilate these experiences into the life of the self, i.e., its personality. 404 Michael Kelly The value of this approach for assessing Bergson's idealistic concession lies in its readily accessible phenomenological descriptions of bodily motility in the mode of fi rst-person givenness. These ready examples that might elucidate the nature of this impersonal, fi rst-person bodily self-givenness-and thereby locate a place for consciousness or an IC within PP-come at the expense of an argumentative leap from Bergson's account of PP, which he restricts to matter, to his account of concrete-perception, which he admits to include memory. At best, a refl ection on Bergson's account of the body's double function merely reports Bergson's assertion of two separate claims about the body as lived-body and corporeal-body. But since "Bergson . . . attributes [perception] to the action of the body" where the "brain receives the messages coming to it from the senses and transforms it into a corporal movement,"20 a defense of Bergson's IC must explain the interaction of these two functions of the body. Indeed, if this mechanistic account of perception in PP presents the fi nal word on Bergson's theory of perception, then the large issue remains: Does Bergson's materialism in PP drive out anything resembling a self-aware consciousness since there must be some space between that which is (pre-refl exively) aware of the self (not as an object but precisely as self) and the self of which it is aware (non-objectively)? As diffi cult questions often do, this one points precisely to the issue upon which rests a defi nitive resolution to the interpretive dilemma of IC. If Bergson's theory of PP precludes a sense of consciousness, then it also drives out the sensibility of his IC. But, when scrutinized closely, the materialism of PP becomes less pronounced. In fact, Bergson argues, PP marks the "poverty of consciousness" that "foretells of spirit" (188, 194/38, 45). Specifi cally, as we shall see, even the brain's activity at the level of PP depends upon a minimal form of consciousness because PP entails duration insofar as "every sensation translates a . . . succession of elementary vibrations" (280/138). If PP involves a temporal span, as Bergson seems to suggest that it does, then the interpretive tension surrounding IC ultimately concerns the place and coherence of durée in Matter and Memory. 3. Translation as Transition-Synthesis: Establishing Bergson's Idealistic Concesssion Let us return for a moment to the brain's relation to images in PP, that moment when the brain contracts and narrows the whole of images to select the particular image that suits its needs. In this instance, if the interest of the organism infl uences the choice of one image from the mind's assessment of the whole of images, can this selection reduce to a mere neurological process? Asked differently, can this process of selection, depending as it does on a hesitation characterizing the higher organism's response to external stimuli received from the senses, reduce to a mere neurological process? It seems that it cannot. Though Bernet, for example, has A Phenomenological Defense of Bergson's "Idealistic Concession" 405 argued against the possibility of fi nding a place on a phenomenological panel for Bergson, he also seems to suggest, as I would like to here, that Bergson's account of the cerebral transmissions at that material level cannot amount merely to a neurological process. As Bernet writes, In our view this "contraction" can no longer be attributed to a simple neurological process-even if it is not yet a question of . . . recollection or . . . explicit foresight. Properly speaking, this comes down to neither memory nor expectation, but to what Husserl calls a "retention" and a "protention" and which he furthermore qualifi es as being a "perception" of the past and the future. There seems to be no way of getting around the fact that such a primitive consciousness of temporal duration already belongs to "pure" perception."21 Bergson's text seems to support Bernet's observation, for (1) perception does not occur in the brain and (2) this contraction entails a 'perception' of the past along with a perception of the present, or the perception of a succession of elementary vibrations that the brain cannot secure. In the section entitled "Of the Survival of Images: Memory and Mind," Bergson qualifi es his examination of the brain's function in perception. Although "physicochemical phenomena take place in the brain," an image itself within the whole of images (matter), the brain "never occupies more than the present moment" and thus "cannot store up images" (290–1, 292/148–9, 151). The brain's function may process elementary vibrations, but such processing occurs over time, which the ever-present brain cannot traverse. As such, the brain itself cannot account for the apprehension of successive vibrations in PP. Hence, Bergson concludes, it is "a chimerical enterprise to seek to localize past or even present perceptions in the brain: they [past or present perceptions] are not in it [the brain]; it is the brain that is in them" (292/151). This is the poverty of consciousness, a temporal awareness that foretells of spirit. Bergson will traverse the gap between past and present vibrations by correcting two classical problem concerning philosophy's understanding of time. First, "nothing is less than the present moment, if you understand by that the indivisible limit which divides the past from the future;" second, the past itself cannot provide the substrate for my present, for how can that which no longer is "preserve itself " (291–8/149–58)? By locating the brain in perceptions both past and present, Bergson means to locate the brain in "my present," which "consists in the consciousness I have of my body" (280/138). The question, then, concerns whether Bergson attaches to the present vibration in the brain my memory of past vibrations as an addendum? Or, does Bergson describe a more primordial form of time-consciousness that distinguishes the perception of a succession of elementary vibrations that makes possible memory? On a Deleuzean reading of Bergson's resolution to this classical paradox, one emphasizes the distinction between PP and memory, or matter and memory, for 406 Michael Kelly these denote ontologically separate realms somehow integrated in the higher organism.22 Cerebral transmissions fl eetingly pass through the switchboard brain and memory's storehouse, which Bergson does not think parasitic upon the brain, preserves them. Memory not only can preserve the past, Deleuze contends, but thanks to Bergson's discovery of a new notion of the past it also marks the condition for the possibility of the passage of time. Bergson's distinctive discovery in Matter and Memory, an advance beyond his psychological theory of time from Time and Free Will, holds that the past constitutes a separate realm of ontological existence that makes possible the passage of the present into the past, for without considering the past as a separate ontological realm of existence the present moment would have nothing into which it could pass. On this reading, we cannot understand the past as the mere residue of a no longer existing present, for such residue needs a place to reside. The past and the present, then, must coexist, and this "past which never was present" constitutes the passage of time.23 That Deleuze emphasizes the importance of this discovery seems most reasonable, for Bergson's observes that the past continues to exist just as unperceived objects in space do (284, 286–7/142, 145).24 Deleuze seizes on Bergson's notion of the "past in general" (275–6/133–4) treating it as something of an ontological reservoir.25 Here, the present and past, PP and memory, butt up against one another like Aristophanes' mythical lovers just at the point when Zeus scornfully splits them.26 Concerning their interaction, Deleuze's Bergson reconstructs temporal passage by maintaining (i) that each present cerebral instant's nature is to pass, (ii) that such passage occurs thanks to the leap consciousness makes into the past in general, and (iii) that memory constitutes the perception of succession by recuperating the deceased moments of (i) relevant for the immediate experience.27 Deleuze's Bergson thus argues for a "contemporaneity of the present and the past"28 that understands "my present's" extension beyond the now to consist in memory plus the absolutely new (cerebral transmission).29 But this Deleuzean reading of Bergson, which holds in abeyance PP and memory, cannot remove the absurdity of Bergson's IC, since it does not (wish to) acknowledge any such concession to idealism. Rather, this Deleuzean reading insists on a fundamental dualism, a mixture of matter and memory, present and past, materialism and idealism. It does seem reasonable to me to deny that Bergson's notion of a past in general can contribute to an understanding of the perception of successive states insofar as it believes itself to explain the sense of the past upon which time's passage depends. It does not follow as reasonable, however, to countenance Deleuze's resultant dichotomous reading. To be sure, Deleuze's reading preserves the difference in kind between memory and PP upon which Bergson insists, even in the title of Matter and Memory (279/137). At the same time, however, Deleuze's theory seems to retain the "arbitrary" defi nition of the present as "that which is . . . the indivisible limit which divides the past from A Phenomenological Defense of Bergson's "Idealistic Concession" 407 the future" (291/149–50). As such, Deleuze's reading struggles to accommodate Bergson's belief that the present is "that which is being made" (291/150). To preserve the difference in kind between memory and perception upon which Bergson insists, then, one must fl esh out the more fundamental distinction in kind that Bergson articulates between memory of former and memory of immediate experience, a distinction Bergson confl ated in his psychological account of durée in Time and Free Will.30 Bergson resolves the problem of the apprehension of successive vibrations in PP- and thus secures the sensibility of IC-through a proto-phenomenological turn to the living-present, durée, that which is constantly fl owing and being made.31 According to Bergson, the metaphysical paradox of the preservation of time-past dissolves when one considers the "present such as it is actually lived by consciousness," even at the impoverished level of PP (150). He writes, In the fraction of a second which covers the briefest possible perception of light, billions of vibrations have taken place, of which the fi rst is separated from the last by an interval which is enormously divided. Your perception, however instantaneous, consists then in an incalculable multitude of remembered elements; in truth, every perception is already memory. Practically, we perceive only the past. . . . Consciousness, then, illumines . . . that immediate part of the past which, impending over the future, seeks to realize and associate with it. (150) We shall return in the conclusion to this important Bergsonian notion of perceiving the past. For now, we should note that Bergson explains the preservation of time to consist in the perception of the immediate past referred to above as memory of the immediate (not former) experience. Conscious perception consists not in the memorial "revival" of past vibrations brought into the present but the "survival" of these past impressions in the fl ow of conscious life, durée. And the fl ow of conscious life, durée, enables a consciousness of the past of the immediate experience relevant for a present action soon to be completed, e.g., listening to a sentence. The vibrancy and self-apprehension of the fl ow of conscious life enables the survival of those successive ("material") vibrations in a consciousness of the past of the relevant series.32 As Bergson put it in Time and Free Will, durée "is the form which the succession of our conscious states assumes when our ego lets itself live. . . . It is because I endure . . . that I picture to myself what I call the past" of an object.33 Durée, then, understood as the consciousness of consciousness's past (a dimension of its fl ow in general), makes possible the apprehension of the occurring and deceased moments of the successive vibrations in PP relevant for consciousness's present concerns, thereby underscoring the function of PP and vindicating the sensibility of IC.34 Unfortunately, Bergson's account in itself appears slightly confused. Is the extension that is the fl ow of conscious life constituted by a multitude of remem408 Michael Kelly bered elements? Or, does the fl ow of conscious life constitute a perception of the past rather than remembrance of these successively passing vibrations? To make clear the diffi culty facing Bergson's account of the relation between durée and PP, consider Husserl's critical self-refl ection on his early theory of time-consciousness, where he concluded that primary memory cannot explain the consciousness of succession.35 Merely attaching memory to the present fl ow, argued Husserl, renders the consciousness fl ow of life nonsensical, for such an account has the effect of confi ning the fl ow and the past of consciousness to the now that always is no longer. What's more, for Husserl at least, appealing to memory to explain the fl ow's apprehension of a successive object implies that "at any given moment I perceive only the actually present phase of the tone and the objectivity of the whole enduring tone is constituted in an act-continuum that is in part memory, in smallest part perception."36 Memory and perception, then, mark acts different in kind, as Bergson seemingly agrees despite his equivocation in using these terms (279–80/137).37 Indeed, something quite different occurs when I remember my tenth birthday and when I listen to a sentence or perceive a passing train. In order to explain this experience of the fl ow that constitutes the consciousness of succession, Husserl distinguishes retention from primary memory, much as Bergson distinguished memory of an immediate from memory of a former experience.38 For both, it seems, we cannot differentiate memory from retention merely as a matter of temporal distance. Memory and retention are structurally opposed: The former is an active, mediated, objectifying awareness of a past object, the latter a passive, immediate, nonobjectifying awareness of the elapsing phase of conscious experience.39 In short, memory presents the object as past, whereas retention presents a consciousness of the past phase of experience and thereby the object as past, a consciousness of the past of the object rather than a consciousness of the object as past.40 Were retention thought as a re-production, re-petition or re-cognition of past states, it would not differ from the thematizing activity of primary memory-or memory of a former experience, as Bergson expresses it. Given these differences, Husserl concludes, that retention-as an inseparable though distinguishable moment of the living-present-founds memory, and this is because of the role that the retentional moment of the living-present plays in constituting the perception of an temporal object, which object only can become a memory once fully constituted, i.e., after it has become a completed perception. To return to Bergson's account, just as Husserl says that retention perceives rather than remembers the past, Bergson says that "practically we perceive only the past," and this sense of perception is not, for Bergson, memory in the traditional sense of the revival of a former experience. Indeed, Bergson insists, "to picture is not to remember" (135). This perception of the past, then, according to Bergson, is a consciousness of the elapsed phase of consciousness's fl ow, its durée, by A Phenomenological Defense of Bergson's "Idealistic Concession" 409 which it apprehend the successive vibrations in PP. It is this insight that we have clarifi ed by our appeal to Husserl's distinction between memory and retention despite Bergson's unhappy description of the perception of the past as memory of the immediate experience. 4. Retaining Bergson's Idealistic Concession Even if the present account provides reasonable textual support for and validation of Bergson's concession to idealism, it may not persuade either phenomenologists or Bergsonists. The important point to note, it seems to me, however, is that Bergson's notion of a memory of immediate experience entails a "poverty of consciousness" that amounts to an activity that provides a transition-synthesis-a retentional mode of intentionality-that translates cerebral vibrations into images, phenomena.41 If my reading of Bergson's idealistic concession is persuasive, then since it entails a theory of the life of consciousness in its retentional-intentional relation to the world, we are left to choose between clarifying or dismissing his IC. That is, we argue for a mechanical, cerebral relation that precludes subjectivity's awareness of itself and objects, or we see the self apprehending itself and therefore phenomena in the world by way of understanding the apprehension of successive cerebral vibrations as a pre-refl ective, "impersonal" self-consolidating that makes possible 'memory' of immediate experiences and thereby memory of former experiences. The former approach cannot reconcile Bergson's IC, while the latter approach can accommodate Bergson's IC. It seems that the letter and spirit of Bergson's text calls us to reject the former option in favor of the latter. As I noted at the close of section two of this essay, a defense of Bergson's IC must explain the interaction of the "double-awareness" enjoyed by the body. Indeed, if this mechanistic account of perception in PP presents the fi nal word on Bergson's theory of perception, then Bergson's materialism in PP drives out anything resembling a self-aware consciousness since there must be some space between that which is (pre-refl exively) aware of the self (not as an object but precisely as self) and the self of which it is aware (non-objectively)? But the materialism underscoring the hypothesis of PP is not the end of the story. Indeed, Bergson identifi es the brain only as a "necessary condition" for experience, for "the cerebral mechanism does indeed in some sort condition memories but is in no way suffi cient to ensure their survival" (222/75). That the brain constitutes a necessary but not suffi cient condition for experience is not anathema to phenomenology and thus on this reading does not undermine Bergson's concession to idealism; Husserl himself maintained that the neurophysiologic events in a human organism constitute the "'turning point' where causal relations are transformed into conditional relations between external world and the Bodily-psychic subject."42 Such a synthesis of these cerebral moments cannot stem from a reduction of 410 Michael Kelly conscious perception to the brain because the brain is a "thing like others" and "it would be absurd that the container should issue the content" (190/41). That is, under Bergson's hypothesis of PP, moreover, the body considered as the brain can be nothing otherwise than a necessary condition because it has no time, i.e., it "never occupies more than the present moment" and thus appears anew each moment (291/149). Since perception arises only when consciousness brings these cerebral moments together, Bergson maintains that "pure perception, . . . however rapid we suppose it to be, occupies a certain depth of duration, so that our successive perceptions are never the real moments of things . . . but the moments of our consciousness" (216/69). In the fi nal analysis, since cerebral activity itself cannot experience the time delay that it creates, Bergson concludes that experience must result from timeconsciousness and its intersection with the body. Summarizing his fi ndings from Matter and Memory, Bergson writes, [T]he humblest function of spirit is to bind together successive moments of the duration of things. . . . We note that [the nervous system's] increasing complexity appears to allow an ever greater latitude to the activity of the living being, the faculty of waiting before reacting. . . . The more complex organization of the nervous system . . . is only the material symbol of that independence itself . . . the symbol of the inner energy which allows the being to free itself from the rhythm of the fl ow of things and to retain in an ever higher degree the past in order to infl uence ever more deeply the future. (352/221–2) And this is why, Bergson, although he maintains that perception includes a motor phenomenon, consistently describes the human body as an ambiguous center characterized by a "double faculty" that I "know . . . from within, by sensations which I term affective, instead of knowing only, as in the case of other [objects], its outer skin" (209/61). Rather than examining the intrinsic properties of neural systems as a positivist or vitalist, Bergson establishes mind and matter as necessary conditions for living and experiencing by focusing on the dynamic interplay of the body, its neural activity and the world. Bergson's insights into the 'turning point' from matter to 'memory' (of immediate experience, or retention) at least place his philosophy on a parallel rather than perpendicular plane to phenomenology. But, Bergson's parallel plane, which itself includes a concession to idealism, might benefi t from and benefi t phenomenology, for as Husserl writes, It can be said that, if [the] psychology of cognition had ever gone to work with a consciousness of its aim and had consequently been successful, its results would also have been work accomplished directly for the philosophic theory of cognition. All insights into structure that had been acquired for the psychology of cognition would also have benefi ted transcendental philosophy.43 A Phenomenological Defense of Bergson's "Idealistic Concession" 411 Read in this way, Husserl's distinction between memory and retention makes explicit Bergson's implicit distinction between memory and perception. And this distinction between memory and retention allows the sensibility of Bergson's IC to shine forth, a glimmer that, in turn, shines new light on phenomenologists' dismissal of Bergson's thought, as well as the contemporary Deleuzean inspired Bergsonist revivals of his thought. NOTES 1. H. Bergson, Matter and Memory, trans. N. M. Paul and W. S. Palmer (New York: Zone Books, 1991). All references to Bergson's work with the pagination of the English translation following the pagination of H. Bergson, OEuvres. Édition du Centenaire (Paris: Presses Universitaires de France, 1959). Henceforth cited parenthetically by page number alone; no other work will receive parenthetical citation. 2. For a few examples of phenomenological thinkers who express reservations about Bergson's philosophical project and his idealistic concession see R. Barbaras, "Le problème de l'expérience: proximité ou corrélation? Bergson et la phenomenology" in Vie et intentionnalité Recherches phénoménologiques (Paris: Vrin 2004); R. Barbaras, Introduction à une phénoménologie de la vie (Paris: Vrin 2008), 141–56; R. Bernet, "A Present Folded Back on the Past (Bergson)," Research in Phenomenology 35 (2005): 55–76; J. P. Sartre, Imagination, trans. F. Williams (Ann Arbor: University of Michigan Press, 1962); M. Merleau-Ponty, The Structure of Behavior, trans. A. L. Fisher (Pittsburgh: Duquense University Press, 1983); D. Zahavi, "Life, Thinking and Phenomenology in the Early Bergson," in Bergson and Phenomenology, ed. M. Kelly (London: Palgrave, forthcoming). On the Bergsonist side, Keith Ansell-Pearson, Philosophy and the Adventure of the Virtual: Bergson and the Time of Life (New York: Routledge, 2002); G. Deleuze, Bergsonism, trans. H. Tomlinson and B. Habberjam (New York: Zone Books, 1999); L. Lawlor, The Challenge of Bergsonism (London: Continuum Press, 2003); Frédéric Worms, "La conscience ou la vie? Bergson entre phénomenologie et métaphysique" in Annales Bergsoniennes II: Bergson, Deleuze, La Phénoménologie, ed. Frédéric Worms (Presses Universitaires de France, 2004), 191–206. Other works expressing similar reservations along these party lines will emerge throughout the course of this essay. 3. Cf. Lawlor, The Challenge of Bergsonism, chap. 1, where Lawlor compares the methodological assumption orienting Matter and Memory with Husserl's notion of the epoché. 4. The notion of the image in Bergson's thought is one of his most perplexing. A quick glance at the concept may make it appear to resemble Kant's conception of phenomena. But Bergson quite clearly argues against such a reading. Sartre has provided the most sustained refl ections on this issue. See Sartre, Imagination. 5. Cf. Lawlor, The Challenge of Bergsonism, 16. 6. Ansell-Pearson, Philosophy and the Adventure of the Virtual, 151. We have in the theory of pure perception Bergson's reminder that we have left behind not only the 412 Michael Kelly Cartesian material-realism that sequesters the theater of the mind on which images of the world play, but also the traditional al Cartesian idealism that would "fain go without [matter]" (MM 34, 39). 7. Bernet, "A Present Folded Back on the Past," 60ff. Cf. Sartre, Imagination, 39–40. 8. Sartre, Imagination, 40. As Bernet has recently put it, it seems impossible to fi nd space for Bergson on a "phenomenological panel" given his theory of PP; Bernet, "A Present Folded Back on the Past," 61. Indeed, Bernet writes, since "Bergson attributes [perception] to the action of the body . . . there is no need for a subjective consciousness to introduce a break into the machinery of the universe." 9. Bergson writes, "Here is my body with its 'perceptive centers.' . . . Where is it? I cannot hesitate as to the answer: positing my body, I posit a certain image, but with it also the aggregate of other images, since there is no material image which does not owe its qualities, its determinations, in short, its existence to the place which it occupies in the totality of the universe." 10. Sartre, Imagination, 40. 11. Merleau-Ponty, The Structure of Behavior, 162. Cf. M. Merleau-Ponty, The Incarnate Subject: Malebranche, Biran, and Bergson on the Union of Body and Soul, trans. P. Milan (Amherst, N.Y.: Humanity Books, 2001), 89. In that text, Merleau-Ponty claims that Bergon's theory of pure perception suffers from a "psychic blindness," a materialism "blind" to the issue of intentional consciousness and the body's directedness toward self and other. 12. "When Bergson stresses the unity of perception and action and invents . . . the term 'sensory motor process,' he is clearly seeking to involve consciousness in the world. . . . Generally speaking, Bergson saw that the body and the mind try to communicate with each other. . . . But the body remains for him what we have called the objective body . . . and one cannot see why . . . consciousness becomes in involved in a body and a world." M. Merleau-Ponty, Phenomenology of Perception, trans. C. Smith (New York: Routledge, 1995), 78n2. 13. Deleuze, Bergsonism. 14. G. Deleuze and F. Guattari, What is Philosophy?, trans. H. Tomlinson and G. Burchell (New York: Columbia University Press, 1994), 48–9. 15. Ansell-Pearson, Philosophy and the Adventure of the Virtual, 141, 156. 16. Lawlor, The Challenge of Bergsonism, 9. 17. Merleau-Ponty, The Structure of Behavior, 176. Cf. Barbaras, Introduction à une phénoménologie de la vie, 148. Although it goes beyond the scope of this essay, the reader should note that Merleau-Ponty later in his career ultimately comes to a greater appreciation of Bergson's thought. As he writes in "The Philosophy of Existence" (1959), "If we had been careful readers of Bergson, and if more thought had been given to him, we would have been drawn to a much more concrete philosophy, to a philosophy much less refl exive than Brunschvicg's. But since Bergson was hardly read by my contemporaries, it is certain that we had to wait for the philosophies of existence in order to be able to learn much of what he would have been able to teach us. It is quite certain-as we realize more and more today-that Bergson, has we read him carefully, would have taught us things that ten or fi fteen years later we believed to be A Phenomenological Defense of Bergson's "Idealistic Concession" 413 discoveries made by the philosophy of existence itself." M. Merelau-Ponty, Texts and Dialogues, ed. H. Silverman and J. Barry, Jr., trans. M. Smith et al. (Atlantic Highlands, N.J.: Humanities Press, 1992), 132. 18. Bernet, "A Present Folded Back on the Past," 56. 19. For a discussion of the distinction between firstand third-person modes of bodily self-givenness, see D. Zahavi, Self-awareness and Alterity: A Phenomenological Investigation (Evanston, Ill.: Northwestern University Press, 1999); as well as his Subjectivity and Self-hood: Investigating the First-Person Perspective (Cambridge, Mass.: The MIT Press, 2005). 20. Bernet, "A Present Folded Back on the Past," 60. 21. Cf. ibid., 61. Despite this claim, the reader should note that Bernet still resists a phenomenological reading of Bergson. See note vii herein. 22. Deleuze, Bergsonism, chap. 3. 23. G. Deleuze, Difference and Repetition, trans. P. Patton (New York: Columbia University Press, 1994), 82. 24. Deleuze, Bergsonism, 58–9. Cf. A. Al-Saji, "The Memory of Another Past: Bergson, Deleuze and a New Theory of Time," Continental Philosophy Review 37 (2004): 203–39. 25. Deleuze, Bergsonism, 59; Deleuze, Difference and Repetition, 81. 26. Plato, "Symposium," trans. M. Joyce, in The Collected Dialogues of Plato, ed. E. Hamilton and H. Cairns (Princeton, N.J.: Princeton University Press, 1989), 542–3, 190a–e. 27. Deleuze, Difference and Repetition, chap. 2, "Repetition in Itself." 28. Deleuze, Bergsonism, 59 29. Lawlor, The Challenge of Bergsonism, ix. 30. For an account of this confl ation, see M. Kelly, "Husserl, Deleuzean Bergsonism and the Sense of the Past in General," Husserl Studies 24 (2008) 15–30. 31. That a similarity exists between Bergson's notion of duration and Husserl's notion of inner time-consciousness is a matter of great dispute. In addition to Deleuze's clear appreciation for Bergson's account of time-consciousness over Husserl's as noted in Chapter II, "Repetition in Itself " of Difference and Repetition (London: Athlone Press, 1994), two papers inspired by the tradition of Deleuzean-Bergsonism recently defended Bergson's theory of time at the expense of Husserl's account of time: S. Crocker, "The Past is to Time What the Idea is to Thought or, What is General in the Past in General," Journal of the British Society for Phenomenology, vol. 35, no. 1: 42–53; Al-Saji, "The Memory of Another Past," 204. Nevertheless, without dismissing Deleuze's contributions to Bergson's theory of time, Rudolf Bernet recently has argued that a similarity exists between Bergson's theory of the duration associated with pure perception and Husserl's theory of the living-present with its notions of retention and protention: Bernet, "A Present Folded Back on the Past," 61. For a fuller account that pursues a line similar to Bernet's suggestion, see Kelly, "Husserl, Deleuzean Bergsonism and the Sense of the Past in General." 414 Michael Kelly 32. This is perhaps why Bergson entitles the chapter dealing with PP and time-consciousness "Of the Survival of Images. Memory and Mind" rather than on the revival of images. 33. H. Bergson, Time and Free Will, trans. F. L. Pogson (Mineola, N.Y.: Dover Publications, 2001), 100, 108. 34. Bernet, "A Present Folded Back on the Past," 61. 35. E. Husserl, On the Phenomenology of the Consciousness of Internal Time (1893–1917), trans. J. Brough (Dordrecht: Kluwer Academic Publishers, 1991), 23, 130. 36. Ibid., 25. 37. Bergson writes, "the illusion which consists in establishing only a difference of degree between memory and perception is more than a mere consequence of associationism, more than an accident in the history of philosophy. It roots lie deep. It rests, in the last analysis, on a false idea of the nature and of the object of external perception. . . . But there is much more between past and present than a mere difference of degree. My present is that which interests me, which lives for me, and in a word, that which summons me to action; in contrast, my past is essentially powerless." 38. While this is not the place to engage in a sustained defense of Husserl's theory of the living-present against possible Bergsonian objections, I feel compelled to make the following brief remark. Husserl certainly describes the living-present according to the distinguishable yet inseparable moments of retention, primal impression and protention. Such a description, however, does not render these moments divorced from one another, thereby requiring a synthesis to rejoin them; against this potential Bergsonian worry, the reader of Husserl's time-consciousness writings must keep it in mind that Husserl regards the fl ow as a non-temporal temporalizing, an identity in a manifold that in no way resembles a series of morsels of lived-experience that require re-connection (for such a view implies that these moments are spread out in temporal order, which Husserl maintains they are not). 39. R. Cobb-Stevens, "James and Husserl: Time-Consciousness and the Intentionality of Presence and Absence," in Self-Awareness, Temporality and Alterity, ed. D. Zahavi (Dordrecht: Kluwer Academic Publishers, 1999), 45. 40. Husserl, On the Phenomenology of the Consciousness of Internal Time (1893–1917), 31, 324. 41. One may justifi ably wonder at this stage if Bergson has not traded in one Husserlian concern for another. On the one hand, the reader of Bergson can explain away the confl ation of memory and retention upon which rests the confl ation between memory and perception. On the other hand, when Bergson claims that to picture is not to remember, one might ask whether or not this expression distinguishes perception from memory or imagination from memory. The notion of picturing rightfully tempts one to conclude the latter. This presents a certain diffi culty for a phenomenological interpretation of Bergson's account of time-consciousness. It is well-known that Husserl rejected Brentano's account of originary association because it accounted for the extension of perception beyond the now by making recourse to the imagination, a move that reduced perception to imagination rather than explaining its extension beyond the now. To picture may not be to remember, but to perceive is not merely to A Phenomenological Defense of Bergson's "Idealistic Concession" 415 image. At any rate, an attempt to address this diffi culty must reckon with Bergson's account of the image mentioned in section one and tease out its relation to perception, a task well beyond the scope of this brief inquiry. 42. E. Husserl, Ideas Pertaining to a Pure Phenomenology and to a Phenomenological Philosophy, Second Book, trans. R. Rojcewicz and A. Schuwer (Dordrecht: Kluwer Academic Publishers, 1989), 169. 43. E. Husserl, Formal and Transcendental Logic, trans. D. Cairns, (The Hague: Martinus Nijhoff, 1969), 254.