Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'?

[Kevin Buzzard <kevin.m.buzzard@gmail.com>]

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On Sun, 3 Nov 2019 at 19:13, Michael Shulman <shulman@sandiego.edu> wrote:

> But does univalence a la Book HoTT *actually* make it easier to reason
> about such things?


I think this is a really interesting and important question.

I guess David was referring to my scheme fail of 2018. I wanted to
formalise the notion of a scheme a la Grothendieck and prove that if R was
a commutative ring then Spec(R) was a scheme [I know it's a definition, but
many mathematicians do seem to call it a theorem, in our ignorance]. I
showed an undergraduate a specific lemma in ring theory (
https://stacks.math.columbia.edu/tag/00EJ) and said "that's what I want"
and they formalised it for me. And then it turned out that I wanted
something else: I didn't have R_f, I had something "canonically isomorphic"
to it, a phrase we mathematicians like to pull out when the going gets
tough and we can't be bothered to check that any more diagrams commute. By
this point it was too late to turn back, and so I had to prove that 20
diagrams commuted and it wasn't much fun. I then got an MSc student to redo
everything using universal properties more carefully in Lean and it worked
like a dream https://github.com/ramonfmir/lean-scheme. A lot of people said
to me at the time "you wouldn't have had this problem if you'd been using
HoTT instead of DTT" and my response to this is still the (intentionally)
provocative "go ahead and define schemes and prove that Spec(R) is a scheme
in some HoTT system, and show me how it's better; note that we did have a
problem, but we solved it in DTT". I would be particularly interested to
see schemes done in Arend, because it always felt funny to me using UniMath
in Coq (and similarly it feels funny to me to do HoTT in Lean 3 -- in both
cases it could be argued that it's using a system to do something it wasn't
designed to do). I think it's easy to theorise about this sort of thing but
until it happens in practice in one or more of the HoTT systems I don't
think we will understand the issue properly (or, more precisely, I don't
think I will understand the issue properly). I have had extensive
discussions with Martin Escardo about HoTT and he has certainly given me
hope, but on the Lean chat I think people assumed schemes would be easy in
Lean (I certainly did) and then we ran into this unexpected problem (which
univalence is probably designed to solve), so the question is whether a
univalent type theory runs into a different unexpected problem -- you push
the carpet down somewhere and it pops up somewhere else.

I know this is a HoTT list but the challenge is also open to the HOL people
like the Isabelle/HOL experts. In contrast to HoTT theories, which I think
should handle schemes fine, I think that simple type theory will have
tremendous problems defining, for example, tensor products of sheaves of
modules on a scheme, because these are dependent types. On the other hand
my recent ArXiv paper with Commelin and Massot
https://arxiv.org/abs/1910.12320 goes much further and formalises
perfectoid spaces in dependent type theory. I would like the people on this
list to see this as a challenge. I think that this century will see the
rise of the theorem prover in mathematics and I am not naive enough to
think that the one I currently use now is the one which is guaranteed to be
the success story. Voevodsky was convinced that univalence was the right
way to do modern mathematics but I'm doing it just fine in dependent type
theory and now he's gone I really want to find someone who will take up the
challenge and do some scheme theory in HoTT, but convincing professional
mathematicians to get interested in this area is very difficult, and I
speak as someone who's been trying to do it for two years now [I recommend
you try the undergraduates instead, anyone who is interested in training
people up -- plenty of undergraduates are capable of reading the definition
of a scheme, if they know what rings and topological spaces are]

To get back to the original question, my understanding was that Voevodsky
had done a bunch of scheme theory and it had got him a Fields medal and it
was this mathematics which he was interested in at the time. He wanted to
formalise his big theorem, just like Hales did. Unfortunately he was
historically earlier, and his mathematics involved far more conceptual
objects than spheres in 3-space, so it was a much taller order. All the
evidence is there to suggest that over the next 15 or so years his
interests changed. The clearest evidence, in my mind, is that there is no
definition of a scheme in UniMath. Moreover his story in his Cambridge talk
https://www.newton.ac.uk/seminar/20170710113012301 about asking Suslin to
reprove one of his results without using the axiom of choice (46 minutes
in) kind of shocked me -- Suslin does not care about mathematics without
choice, and the vast majority of mathematicians employed in mathematics
departments feel the same, although I'm well aware that constructivism is
taken more seriously on this list. I think it is interesting that Voevodsky
failed to prove a constructive version of his theorem, because I think that
some mathematics is better off not being constructive. It is exactly the
interaction between constructivism and univalence which I do not understand
well, and I think that a very good way to investigate it would be to do
some highly non-constructive modern mathematics in a univalent type theory.

Kevin

PS many thanks to the people who have emailed me in the past telling me
about how in the past I have used "HoTT", "univalence", "UniMath",
interchangeably and incorrectly. Hopefully I am getting better but I am
still keen to hear anything which I'm saying which is imprecise or
incorrect.



> It allows us to write "=" rather than "\cong", but
> to construct such an equality we have to construct an isomorphism
> first, and to *use* such an equality we have to transport along it,
> and then we get lots of univalence-redexes that we have to manually
> reduce away.  My experience formalizing math in HoTT/Coq is that it's
> much easier if you *avoid* turning equivalences into equalities except
> when absolutely necessary.  (I don't have any experience formalizing
> math in a cubical proof assistant, but in that case at least you
> wouldn't have to manually reduce the univalence-redexes -- although it
> seems to me you'd still have to construct the isomorphism before you
> can apply univalence to make it an equality.)
>
> On Sun, Nov 3, 2019 at 3:57 AM David Roberts <droberts.65537@gmail.com>
> wrote:
> >
> > Forget even higher category theory. Kevin Buzzard now goes around
> telling the story of how even formally proving (using Lean) things in
> rather elementary commutative algebra from EGA that are stated as
> equalities was not obvious: the equality is really an isomorphism arising
> from a universal property. Forget trying to do anything motivic, when
> algebra is full of such equalities. This is not a problem with univalence,
> of course.
> >
> > David
> >
> > On Sun, 3 Nov 2019, 10:08 PM Bas Spitters <b.a.w.spitters@gmail.com>
> wrote:
> >>
> >> There's also VV homotopy lambda calculus, which he later abandoned for
> MLTT:
> >>
> https://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/Hlambda_short_current.pdf
> >>
> >> On Sun, Oct 27, 2019 at 6:22 PM Bas Spitters <b.a.w.spitters@gmail.com>
> wrote:
> >>>
> >>> I believe it refers to his 2-theories:
> >>> https://www.ias.edu/ideas/2014/voevodsky-origins
> >>>
> >>> On Sun, Oct 27, 2019 at 3:41 PM Nicolas Alexander Schmidt <
> zero@fromzerotoinfinity.xyz> wrote:
> >>>>
> >>>> In [this](https://www.youtube.com/watch?v=zw6NcwME7yI&t=1680) 2014
> talk
> >>>> at IAS, Voevodsky talks about the history of his project of "univalent
> >>>> mathematics" and his motivation for starting it. Namely, he mentions
> >>>> that he found existing proof assistants at that time (in 2000) to be
> >>>> impractical for the kinds of mathematics he was interested in.
> >>>>
> >>>> Unfortunately, he doesn't go into details of what mathematics he was
> >>>> exactly interested in (I'm guessing something to do with homotopy
> >>>> theory) or why exactly existing proof assistants weren't practical for
> >>>> formalizing them. Judging by the things he mentions in his talk, it
> >>>> seems that (roughly) his rejection of those proof assistants was based
> >>>> on the view that predicate logic + ZFC is not expressive enough. In
> >>>> other words, there is too much lossy encoding needed in order to
> >>>> translate from the platonic world of mathematical ideas to this formal
> >>>> language.
> >>>>
> >>>> Comparing the situation to computer programming languages, one might
> say
> >>>> that predicate logic is like Assembly in that even though everything
> can
> >>>> be encoded in that language, it is not expressive enough to directly
> >>>> talk about higher level concepts, diminishing its practical value for
> >>>> reasoning about mathematics. In particular, those systems are not
> >>>> adequate for *interactive* development of *new* mathematics (as
> opposed
> >>>> to formalization of existing theories).
> >>>>
> >>>> Perhaps I am just misinterpreting what Voevodsky said. In this case, I
> >>>> hope someone can correct me. However even if this wasn't *his* view,
> to
> >>>> me it seems to be a view held implicitly in the HoTT community. In any
> >>>> case, it's a view that one might reasonably hold.
> >>>>
> >>>> However I wonder how reasonable that view actually is, i.e. whether
> e.g.
> >>>> Mizar really is that much more impractical than HoTT-Coq or Agda,
> given
> >>>> that people have been happily formalizing mathematics in it for 46
> years
> >>>> now. And, even though by browsing the contents of "Formalized
> >>>> Mathematics" one can get the impression that the work consists mostly
> of
> >>>> formalizing early 20th century mathematics, neither the UniMath nor
> the
> >>>> HoTT library for example contain a proof of Fubini's theorem.
> >>>>
> >>>> So, to put this into one concrete question, how (if at all) is
> HoTT-Coq
> >>>> more practical than Mizar for the purpose of formalizing mathematics,
> >>>> outside the specific realm of synthetic homotopy theory?
> >>>>
> >>>>
> >>>> --
> >>>>
> >>>> Nicolas
> >>>>
> >>>>
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