Unimath.org

<<1>>HLF 2016, Sep. 22, 2016, Heidelberg.  UniMath by Vladimir
Voevodsky from the Institute for Advanced Study in Princeton, NJ.

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<<2>>Part 1. Univalent foundations 2

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<<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.
#+begin_src interesting
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.
#+end_src
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

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<<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

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<<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

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<<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

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<<13>>Part 2. The UniMath library 13

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<<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

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<<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

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<<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

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<<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

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<<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

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<<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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