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Unimath.org

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+<<1>>HLF 2016, Sep. 22, 2016, Heidelberg.\\
+UniMath\\
+by Vladimir Voevodsky  \\
+from the Institute for Advanced Study in Princeton, NJ. \\
+
+--------------
+
+<<2>>Part 1. Univalent foundations\\
+2\\
+
+--------------
+
+<<3>>[[file:Unimath-3_1.png]]\\
+[[file:Unimath-3_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+Today we face a problem that involves two difficult to satisfy conditions. \\
+On the one hand we have to find a way for computer assisted verification of \\
+mathematical proofs.\\
+This is necessary, first of all, because we have to stop the dissolution of the \\
+concept of proof in mathematics.\\
+On the other hand  we have to preserve the intimate connection between \\
+mathematics and the world of human intuition.\\
+This connection is what moves mathematics forward and what we often \\
+experience as the beauty of mathematics. \\
+3\\
+
+--------------
+
+<<4>>[[file:Unimath-4_1.png]]\\
+[[file:Unimath-4_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+The Univalent Foundations (UF) is, a yet imperfect, solution to this problem.\\
+In their original form,  the UF combined three components:\\
+• the view of mathematics as the study of structures on sets and their higher \\
+analogs, \\
+• the idea that the higher analogs of sets are reflected in the set-based \\
+mathematics as homotopy types, \\
+• the idea that one can formalize our intuition about structures on these higher \\
+analogs using the Martin-Lof  Type Theory (MLTT) extended with the Law of \\
+Excluded Middle for propositions (LEM) , the Axiom of Choice for sets (AC), \\
+the Univalence Axiom (UA) and the Resizing Rules (RR).\\
+4\\
+
+--------------
+
+<<5>>[[file:Unimath-5_1.png]]\\
+[[file:Unimath-5_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+The main new concepts that were since added to these are the following: \\
+• the understanding that a lot of mathematics can be formalized in the MLTT \\
+without the LEM and the AC and that excluding these two axioms one \\
+obtains foundations for a /new form of constructive mathematics/,\\
+• the understanding that classical mathematics appears as a subset of this new \\
+constructive mathematics,\\
+• the understanding that the MLTT extended with the UA is an imperfect \\
+formalization system for this constructive mathematics and that it should be \\
+possible  to integrate the UA into the MLTT obtaining a new type theory \\
+with better computational properties.\\
+5\\
+
+--------------
+
+<<6>>[[file:Unimath-6_1.png]]\\
+[[file:Unimath-6_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+What does it mean for a formalization system to be constructive?\\
+Some expressions in type theory are said to be in normal form. Any \\
+expression can be automatically and deterministically “normalized”, that is, an \\
+equivalent expression in normal form can be computed. \\
+In type theory there are type expressions and element expressions. If “T” is a \\
+type (expression) and “o” is an element (expression) one writes “o:T” if the \\
+type of “o” is “T”. \\
+6\\
+
+--------------
+
+<<7>>[[file:Unimath-7_1.png]]\\
+[[file:Unimath-7_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+In most type systems there is the type of natural numbers. In the UniMath it is \\
+written as “nat”.\\
+There is the zero element “O:nat” and the successor function “S” from “nat” to \\
+“nat” that intuitively corresponds to the function that takes “n” to “1+n”. \\
+A constructive system satisfies the /canonicity property/ for “nat”, which asserts \\
+that the normal form of any expression “o:nat” has the form “S(S(....(SO)..))”.\\
+By counting how many “S” there is in the normal form one obtains an actual \\
+natural number from any element expression of type “nat”. \\
+7\\
+
+--------------
+
+<<8>>[[file:Unimath-8_1.png]]\\
+[[file:Unimath-8_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+This is a tremendously strong property. \\
+Consider the example: a set “X:hSet” is defined to be finite if there exists an \\
+isomorphism between it and the standard finite set “stn n”. Here “n” is an \\
+expression of type “nat”. It is well defined and one obtains a function “fincard”  \\
+from finite sets to “nat” called the cardinality - the number of elements of the \\
+set.\\
+Now suppose that I have proved, constructively, that “X” is finite. Then \\
+“(fincard X):nat” \\
+is defined. By normalizing “fincard X” I obtain an actual natural number.\\
+If I had a constructive proof of /Faltings's Theorem, /stating that the number of \\
+rational points on a curve of genus >1 is finite, I could find the actual number \\
+of points on any curve of genus >1.  \\
+8\\
+
+--------------
+
+<<9>>[[file:Unimath-9_1.png]]\\
+[[file:Unimath-9_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+We don't know whether such a proof exists. It is a very interesting and hard \\
+problem. \\
+The reason that the MLTT+UA is an imperfect system for constructive  \\
+formalization is that while MLTT itself has the canonicity property MLTT+UA \\
+does not.\\
+Therefore, formalizing the proof of Faltings's Theorem in the UniMath, which is \\
+based on MLTT+UA, would not immediately give us an algorithm to compute \\
+the number of rational points on a curve of genus >1.\\
+This is where a new type theory that integrates the UA into the MLTT in such \\
+a way as to preserve the canonicity would help. \\
+9\\
+
+--------------
+
+<<10>>[[file:Unimath-10_1.png]]\\
+[[file:Unimath-10_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+The search for such a type theory became one of the main driving forces in \\
+the development of the UF.\\
+Today several groups are working on the construction and  implementation in \\
+a proof assistant of candidate type theories.  \\
+The /cubical type theory/ and the prototype proof assistant /cubicaltt/ created by \\
+the group of Thierry Coquand with the help of many researchers from \\
+different parts of the world is at the most advanced stage of development \\
+today. \\
+A proof in the UniMath easily translates into a proof in the cubilatt.\\
+10\\
+
+--------------
+
+<<11>>[[file:Unimath-11_1.png]]\\
+[[file:Unimath-11_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+The new form of the UF that emerges can be seen as combining the following \\
+components:\\
+• the view of mathematics as the study of structures on sets and their higher \\
+analogs, \\
+• the view of mathematics as constructive with the classical mathematics being \\
+a subset consisting of the results that require LEM and/or AC among their \\
+assumptions,\\
+• the idea that the higher analogs of sets are reflected in the set-based \\
+mathematics as constructive homotopy types - objects of the new \\
+constructive homotopy theory that can so far be formulated only in terms of \\
+cubical sets,\\
+• the idea that one can formalize our intuition about structures on these higher \\
+analogs using Cubical Type Theory (CTT).\\
+11\\
+
+--------------
+
+<<12>>[[file:Unimath-12_1.png]]\\
+[[file:Unimath-12_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+In addition to the understanding that to obtain a formal system for the new \\
+constructive mathematics the UA needs to be integrated into the MLTT \\
+constructively, several more things are felt as lacking in the MLTT+UA:\\
+• higher inductive types, \\
+• resizing rules,\\
+• a possible strict extensional equality combined with the “fibrancy discipline”,\\
+• as yet unknown mechanism to construct the types of structures that involve \\
+infinite hierarchies of coherence conditions.  \\
+Surprisingly,  it might be easier to add these features to the CTT than to the \\
+MLTT. The work in these directions is ongoing. \\
+12\\
+
+--------------
+
+<<13>>Part 2. The UniMath library\\
+13\\
+
+--------------
+
+<<14>>[[file:Unimath-14_1.png]]\\
+[[file:Unimath-14_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+In the development of the UniMath library we attempt to do something that \\
+might be compared with the effort by the Bourbaki group to write a \\
+systematic exposition of mathematics based on the set theory and the view of \\
+mathematics as studying structures on sets.\\
+The effort by Bourbaki stalled at some point around the middle of the 20th \\
+century, in part, because it was very complicated to describe the emerging \\
+category-theoretic constructions in set-theoretic terms.\\
+14\\
+
+--------------
+
+<<15>>[[file:Unimath-15_1.png]]\\
+[[file:Unimath-15_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+One may however ask, is there any mathematical innovation in what we are \\
+doing? Is there a discovery of the unknown in the work on the UniMath?\\
+We have already seen how well-known problems in fields such as arithmetic \\
+algebraic geometry can be related to the search for a new foundation of \\
+constructive mathematics and for building proofs in the UniMath.\\
+Here is a different example.\\
+15\\
+
+--------------
+
+<<16>>[[file:Unimath-16_1.png]]\\
+[[file:Unimath-16_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+Some years ago, at the IAS, I had a conversation at lunch with Armand Borel. I \\
+mentioned how I like Bourbaki “Algebra” and how it helped me to become a \\
+mathematician. \\
+I then mentioned that some places there were really dense. For example, said I, \\
+the description of the tensor product was hard to follow. \\
+Of course, said Borel, /we have invented tensor product to get a systematic \\
+exposition of multi-linear maps/. \\
+It was new research, this is why it was not very smoothly written. \\
+16\\
+
+--------------
+
+<<17>>[[file:Unimath-17_1.png]]\\
+[[file:Unimath-17_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+I was amazed.\\
+It is hard to imagine today's mathematics without the concept of the tensor \\
+product. It would never occurred to me that it was invented by Bourbaki with \\
+the only purpose to obtain a more systematic exposition of multi-linear maps \\
+of vector spaces!\\
+This example shows how a major innovation can emerge from the work on \\
+systematization of knowledge. \\
+17\\
+
+--------------
+
+<<18>>[[file:Unimath-18_1.png]]\\
+[[file:Unimath-18_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+Finally, a few words to those mathematicians who will decide to understand \\
+UniMath and maybe to contribute to it. \\
+The UniMath library is being created using the proof assistant Coq. It is freely \\
+available on GitHub.\\
+The language of Coq is a very substantial extension of the MLTT and UniMath \\
+uses a very small subset of the full Coq language that approximately \\
+corresponds to the original MLTT.\\
+18\\
+
+--------------
+
+<<19>>[[file:Unimath-19_1.png]]\\
+[[file:Unimath-19_2.png]]\\
+[[file:Unimath-19_3.png]]\\
+Univalent Foundations\\
+UniMath library\\
+The first file in the UniMath after the /Basics/preamble.v/ is /Basics/PartA/.v.\\
+The first line in /Basics/PartA.v/ after the preamble section is as follows:\\
+    \\
+It should be understood as a declaration of intent to define a constant called \\
+/fromempty /whose type is described by the expression that is written to the \\
+right of the colon. \\
+Following this line there is a paragraph that starts with /Proof./ and ends with \\
+/Defined. /where the constant is actually defined using the little sub-programs of \\
+Coq called tactics which help to build complex expressions of the underlying \\
+type theory language in simple steps.  \\
+19\\
+
+--------------
+
+<<20>>[[file:Unimath-20_1.png]]\\
+[[file:Unimath-20_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+A mathematician who wants to understand UniMath should expect a very \\
+non-linear learning curve:\\
+• In the lectures that I gave in Oxford and in the similar lectures in the Hebrew \\
+University it took me the whole first lecture to explain what that first line \\
+and the following it paragraph really mean.\\
+• In the next lecture I was able to explain the next few hundred lines of PartA.\\
+• By the fourth lecture in Oxford, the video of which can be found on my \\
+website, I was explaining the invariant formalization of fibration sequences.\\
+20\\
+
+--------------
+
+<<21>>I hope that was able to show how important Univalent Foundations are and \\
+how important is the work on libraries such as UniMath.\\
+Thank you!\\
+21\\
+
+--------------