
Voevodsky’s Mathematical Revolution - juliangamble
http://blogs.scientificamerican.com/guest-blog/2013/10/01/voevodskys-mathematical-revolution/
======
auggierose
It is great that someone with a Fields medal is interested in interactive
theorem proving. And he is right that mathematics will completely change
because of it. Soon.

I doubt though that HoTT is the way this revolutionary change will happen. If
you don't believe me, try reading the first few pages of chapter 2 of the HoTT
book. It is basically total gibberish to me, and yes, this is because I am
neither well versed in constructive type theory, nor in homotopy theory. I DO
have though 10+ years of experience in interactive theorem proving, as well
have a diploma in math where I learnt the basic notions of topology. Now, if
this is gibberish to me, it will be gibberish to about 99% of all
mathematicians out there .

A revolution does not happen by excluding 99% of its targeted population. HoTT
will have to evolve into something much easier to understand with a clear
value proposition to non-homotopy / non-type-theorists before I would even
remotely consider it for providing the underpinnings of a system for
interactive proof.

~~~
pavelrub
I do not understand your criticism. HoTT's value proposition is about a
significant simplification of automated theorem proving in mathematics. As
such, the _value proposition_ itself has nothing to do with Homotopy Theory:
if what HoTT promises to deliver will indeed happen, mathematicians will learn
the necessary material _because_ of what it gives them.

Perhaps you are saying that it simply won't deliver, but you cannot possibly
reach such a conclusion by reading 1.5 chapters out of 11.

I do not understand how the fact that currently most people do not know the
relevant material is an argument against HoTT: of course nobody knows it: it's
new! Did you expect "revolutions" to happen by telling people something they
already know? Do you believe Grothendieck's work was a mistake because
99.9999% percent of mathematicians weren't Algebraic Geometers by the time EGA
was published? Your argument makes no sense, maybe because you are comparing
HoTT to Facebook, when in fact it's a mathematical achievement.

Also: _if this is gibberish to me, it will be gibberish to about 99% of all
mathematicians out there ._

I seriously doubt that.

~~~
auggierose
> HoTT's value proposition is about a significant simplification of automated
> theorem proving in mathematics.

Now that is utter and complete nonsense. And that is why I am criticising
HoTT. HoTT might be a beautiful mathematical theory. I am not competent in
judging that, to be so, I would have to invest more time. But there is not a
single shred of evidence that HoTT will improve automated, or for that matter
interactive theorem proving. And I am pretty competent to judge that.

~~~
pavelrub
I'm not sure what you are basing this on. How can you judge that if you do not
understand the theory or the motivation behind it? Because chapter 2 seemed
like gibberish to you?

Read for example: [http://homotopytypetheory.org/2013/03/08/homotopy-theory-
in-...](http://homotopytypetheory.org/2013/03/08/homotopy-theory-in-homotopy-
type-theory-introduction/)

~~~
auggierose
I know that we are already very well capable of doing succinct proofs in
interactive theorem proving systems like Isabelle that have a strong
resemblance to "human" proofs. None of the proofs displayed in HoTT or the
accompanying sources look shorter or more succinct than stuff I could do
without HoTT.

So HoTT might solve a problem for constructivists working with Coq. Problems
you can entirely avoid by not working constructively (or with Coq, for that
matter). Therefore, at this point, there is no evidence at all that HoTT might
improve the state of the art in interactive theorem proving.

That does not mean that this evidence might not come up at some point.

How can I be sure in my judgement? Well, I don't have to understand string
theory to be sure at this point that it is not relevant to interactive theorem
proving. I would change my mind if somebody points out evidence that string
theory is actually of major relevance to interactive theorem proving.

Edit: I would even start learning string theory then.

------
nilkn
I'm quite familiar with homotopy theory but had never, prior to this point,
taken a serious look at homotopy type theory. The following appears to be a
great (highly technical) introduction:

[http://hottheory.files.wordpress.com/2013/03/hott-
online-323...](http://hottheory.files.wordpress.com/2013/03/hott-
online-323-g28e4374.pdf)

~~~
ivan_ah
Interestingly, the entire source code of the HoTT book is available on github
and it is under active development:
[https://github.com/HoTT/book](https://github.com/HoTT/book)

~~~
Tyr42
Which is a great boon, as I've edited the latex and recompiled it to fit
perfectly on my Kindle.

I really wish more people would open source their books.

------
juliangamble
If you're interested in Agda Proofs - this is a well written post by Brian
McKenna demonstrating that the sum of two odd numbers is always even.

[http://brianmckenna.org/blog/plus_equals_even_take_2](http://brianmckenna.org/blog/plus_equals_even_take_2)

------
yetanotherphd
Everything the article says about the problems with set theory as foundations
of mathematics rings true. I haven't had time to investigate homotopy type
theory yet, but I hope that it lives up to the claims in the article.

~~~
jonsterling
HoTT is really fun. I'm particularly interested to see if the computational
behavior of univalent and higher inductive types gets sussed out (or else,
like Bob Harper, I shall have to pick up my toys and head home). Computing
with univalence and HITs sounds lovely, if only we could figure out how to do
it...

~~~
yetanotherphd
What is a good way to get started with this literature? And is it better to
read theory first, or read theory in conjunction with using languages like
Haskell and Coq?

I studied a lot of mathematics, and I find this fascinating, but I've found it
hard to read the literature on type theory, and mathematical logic also.

~~~
ek
It seems like I end up plugging the book really frequently here, but it's for
good reason -- it's exceptionally readable AND it's accompanied by a full Coq
development. That is, you can do basically every exercise in the book,
directly in Coq, if you want.

You can definitely just read the material and then come back and try the
exercises in Coq later, or more tightly couple the two. I think either
approach would work well, and it's up to personal preference.

The first chapter of HoTT is an introduction to Martin-Löf that I personally
find quite intuitive, to the point that I'd say it's clear than most other
expositions of the same material that I've tried to read.

It will be easier to understand dependent type theory if you're programmed in
a dependently typed language, but I wouldn't say it's strictly necessary to
understand dependent types theoretically. I have just a little bit of Coq
experience and found myself comfortable with the HoTT presentation of
dependent types.

~~~
tel
I have only managed to get through the first chapter myself and I really want
to reiterate that—the first chapter does a fantastic job presenting MLTT.

------
juliangamble
There is a question raised about the research outcomes of the event mentioned
in the blog post here:

[http://cs.stackexchange.com/questions/19001/what-were-the-
re...](http://cs.stackexchange.com/questions/19001/what-were-the-research-
outcomes-of-the-univalent-foundations-program-year-homot)

~~~
ek
Thanks! I tried to answer it.

------
continuations
> Soon, they won’t consider a theorem proven until a computer has verified it.

Is automated theorem proving really this close to being this powerful?

So soon we'd be able to feed the proof of Poincare Conjecture to a computer
and it'd be able to verify the proof? I was under the impression we were
nowhere close to being able to do that.

~~~
byerley
I was under the impression that even verification is Turing undecidable in
general - ie
[http://en.wikipedia.org/wiki/Entscheidungsproblem](http://en.wikipedia.org/wiki/Entscheidungsproblem)

~~~
joe_the_user
It's easy for impressions of undecidability to get out of hand. But actually,
verification is computable. Verification has to be computable for things like
Godel's theorem to work.

If you read your link carefully, you'll notice it's about provability via
axioms, not verification.

Peace

~~~
jonahx
joe, what is the difference between verification and provability?

~~~
lisper
It's the difference between showing that a candidate proof is correct, and
coming up with a correct proof in the first place. Once you have a proof,
showing it is correct is (easily) computable. But coming up with a correct
proof in the first place is not computable in general. (More generally,
deciding whether a given proposition is true -- i.e. whether a proof exists or
not -- is not computable. That is the "entsheidungsproblem".)

~~~
jonahx
thanks. after reading your explanation it's clear that the verification is
_verification of the proof_ , whereas i was interpreting it as verification of
the statement to be proved, hence my confusion.

~~~
joe_the_user
Wow,

Until I read your post, it never occurred to me you were confused by the
ambiguous implications of "verification".

You remind me how easy it is, once you get the jargon and the concepts, to not
actually write out everything into a form someone from the outside can
understand.

~~~
jonahx
Yeah, I personally file that under "Lessons You Never Learn Enough Times."
Glad my slowness could be of service :)

In retrospect, it seems like something I should have been able to deduce,
since it's kind of the only thing makes sense.

------
juliangamble
There are some great slides by Liam O'Connor that draw the link between first-
order logic and Martin-Löf type theory here:

[http://www.cse.unsw.edu.au/~liamoc/talk.pdf](http://www.cse.unsw.edu.au/~liamoc/talk.pdf)

They were presented at fpsyd.

------
ChristianMarks
Awodey and Voevodsky hit on connections between type theory and homotopy
theory about the same time. It isn't clear how to find normal forms for
dependent type theory with the univalence axiom added. Mathematicians of my
acquaintance find curious the attraction of what appears to be another
constructive type theory of interest primarily to (functional) programmers. To
them it looks like a calculus for doing synthetic homotopy theory.

------
menato
There is some rather interesting interview with Vladimir (in russian), which
shows a little of his underlying motivation:

[http://baaltii1.livejournal.com/198675.html](http://baaltii1.livejournal.com/198675.html)

[http://baaltii1.livejournal.com/200269.html](http://baaltii1.livejournal.com/200269.html)

------
tunesmith
I got lost on one basic thing. I thought Coq wasn't based on HoTT. I thought
Agda was based on Martin-Lof; Coq was based on calculus of constructions, and
that would mean that the HoTT people are looking to develop a new proof
assistant that is based on HoTT. Is that true?

~~~
chas
Coq isn't based on HoTT, however, the people working on HoTT use a modified
version of Coq in conjunction with a library that contains the essential
elements. [https://github.com/HoTT/HoTT](https://github.com/HoTT/HoTT)

~~~
nardi
The lightbulb moment for Voevodsky was apparently realizing that CoC (used in
Coq) is almost entirely equivalent to HoTT. So in a sense, Coq actually is
kind of based on HoTT, just using different words. (One could maybe say there
is a homotopy between CoC and HoTT?? IANAM.) The library they are developing
is basically a restatement of HoTT in Coq using more familiar terms/concepts.

------
jackmaney
I heard many of these claims--especially that there would soon be a Brave New
World of Mathematics where proofs via computer would be the norm--about a
decade ago, when I was in grad school. Let's just say that I'm not holding my
breath...

------
solomatov
Many people probably want to learn coq after reading the article and watching
the video.

Vladimir mentions that he had a course at Princeton where he learned Coq. Most
probably this is the course:
[http://www.cs.princeton.edu/courses/archive/fall10/cos441/in...](http://www.cs.princeton.edu/courses/archive/fall10/cos441/index.php)

And its based on on-line book Software Foundations
[http://www.cis.upenn.edu/~bcpierce/sf/toc.html](http://www.cis.upenn.edu/~bcpierce/sf/toc.html)
which is really accessible for about anyone with small programming experience.

------
aniijbod
For and on behalf of any genuine non-mathematicians, a couple of questions:
(1) what would have to be the case such that his latest advances in proof
assistance would not live up to his expectations (2) what things outside the
world of abstract mathematics (such as in, but not restricted to the world of
software) would change (such that anyone outside the field would notice) if
they did live up to his expectations?

------
DonGateley
I sure wish I could listen to his talk but the moron who recorded it wasn't
looking at his meters or listening to a monitor. The clipping distortion is
far beyond my ability to tolerate for more than about 10 seconds. Incredible
incompetence.

------
stefantalpalaru
>It turns out, however, that ArXiv isn’t yet up to the task — while it can
accept attached files, they can’t have any directory structure.

Why not archive and compress them in a tar.gz ?

