| <<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\\ | |
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| <<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\\ | |
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| <<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\\ | |
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| <<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\\ | |
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| <<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\\ | |
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| <<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\\ | |
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| <<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\\ | |
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| <<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\\ | |
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| <<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\\ | |
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