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Unimath.txt

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+HLF 2016, Sep. 22, 2016, Heidelberg.
+
+UniMath
+by Vladimir Voevodsky
+from the Institute for Advanced Study in Princeton, NJ.
+
+Part 1. Univalent foundations
+
+2
+
+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
+
+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
+
+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
+
+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
+
+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
+
+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
+
+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
+
+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
+
+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
+
+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
+
+Part 2. The UniMath library
+
+13
+
+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
+
+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
+
+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
+
+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
+
+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
+
+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
+
+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
+
+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
+
+