Discussion of Homotopy Type Theory and Univalent Foundations
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From: Bas Spitters <b.a.w.spitters@gmail.com>
To: Kevin Buzzard <kevin.m.buzzard@gmail.com>
Cc: Noah Snyder <nsnyder@gmail.com>,
	 Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>,
	Juan Ospina <jospina65@gmail.com>
Subject: Re: [HoTT] doing "all of pure mathematics" in type theory
Date: Sun, 2 Jun 2019 18:30:00 +0200
Message-ID: <CAOoPQuT4kJr4cr8rqS2kMjNkQyj6rj8CxChXx82rWcapXtRDxQ@mail.gmail.com> (raw)
In-Reply-To: <CAH52Xb1uvc=AA6Pe=Z98K-rJ+Q1xzLf4LDAwWQhJWW62bO5L3A@mail.gmail.com>

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Dear Kevin,

Looking at your slides from big proof, the transfer package you're asking
for seems to be very close to what is provided by HoTT.
https://xenaproject.wordpress.com/2019/06/02/equality-part-3-canonical-isomorphism/
This is explained in many places (e.g. the book). Here's an early article
explaining it for algebraic structures:
http://www.cse.chalmers.se/~nad/publications/coquand-danielsson-isomorphism-is-equality.pdf

Have you looked at any of this? Does it provide what you are looking for?

Best,

Bas

On Sun, May 26, 2019 at 7:00 PM Kevin Buzzard <kevin.m.buzzard@gmail.com>
wrote:

> I would like to again thank the people who have been responding to my
> posts this weekend with links and further reading. I know the Lean
> literature but I knew very little indeed about HoTT / UniMath at the
> start of this weekend; at least now I feel like I know where you guys
> are.
>
> On Sun, 26 May 2019 at 13:09, Bas Spitters <b.a.w.spitters@gmail.com>
> wrote:
>
> > It's a slow process, but I believe we are making progress.
>
> I agree that it's a slow process! I think that in any computer science
> department you can find people who know about these tools, indeed in
> the computer science department at my university there are people
> using things like Coq for program verification. I think that one
> measure of success would be when in most mathematics departments you
> can find someone who knows about this stuff. My personal experience is
> that we seem to be far from this, at this point. Bas points out the
> EU-funded project ForMath and I know that Paulson has an EU grant in
> Cambridge for Isabelle (my impression is that it is centred in the
> computer science department) and there is a Lean one based in
> Amsterdam which I know has mathematicians involved. For me the shock
> is that now I've seen what these things can do, I am kind of stunned
> that mathematicians don't know more about them.
>
> > You seem to be mixing at least two issues.
> > - HoTT/UF as a foundation
> > - Current implementations in proof assistants.
>
> Yes; when I started this thread I was very unclear about how
> everything fitted together. I asked a bit on the Lean chat but I guess
> many people are like me -- they know one system, and are not experts
> at all in what is going on with the other systems.
>
> I had forgotten about the mathcomp book! Someone pointed it out to me
> a while ago but I knew far less then about everything so it was a bit
> more intimidating. Thanks for reminding me.
>
> I think I have basically said all I had to say (and have managed to
> get my ideas un-muddled about several things). But here is a parting
> shot. Voevodsky was interested in formalising mathematics in a proof
> assistant. Before that, Voevodsky was a "traditional mathematician"
> and proved some great theorems and made some great constructions using
> mathematical objects called schemes. Theorems about schemes (his
> development of a homotopy theory for schemes) are what got him his
> Fields Medal. Schemes were clearly close to his heart. But looking at
> the things he formalised, he was doing things like the p-adic numbers,
> and lots and lots of category theory. I am surprised that he did not
> attempt to formalise the basic theory of schemes. Grothendieck's EGA
> is written in quite a "formal" way (although nowhere near as formal as
> what would be needed to formalise it easily in a proof assistant) and
> Johan de Jong's Stacks Project https://stacks.math.columbia.edu/ is
> another very solid attempt to lay down the foundations of the theory.
> I asked Johan whether he now considered his choice of "nice web pages"
> old-fashioned when it was now possible to formalise things in a proof
> assistant, and he said that he did not have time to learn how to use a
> proof assistant. But Voevodsky was surely aware of this work, and also
> how suitable it looks for formalisation.
>
> Thanks again to this community for all the comments and all the links
> and all the corrections. If anyone is going to Big Proof in Edinburgh
> this coming week I'd be happy to talk more.
>
> Kevin
>
> >
> > If you want to restrict to classical maths. Then please have a careful
> > look at how its done in mathematical components:
> > https://math-comp.github.io/mcb/
> > and the analysis library that is currently under development.
> > https://github.com/math-comp/analysis
> >
> > If you went to help connecting this to the HoTT library, it will be
> > much appreciated.
> > https://github.com/HoTT/HoTT
> >
> > Best wishes,
> >
> > Bas
> >
> > On Sun, May 26, 2019 at 1:41 PM Kevin Buzzard <kevin.m.buzzard@gmail.com>
> wrote:
> > >
> > > It seems to me, now I understand much better what is going on (many
> thanks to all the people who replied) that dependent type theory +
> impredicative prop "got lucky", in that Coq has been around for a long
> time, and Lean 3 is an attempt to model basically the same type theory but
> in a way more suited to mathematics, written by someone who knows what
> they're doing (de Moura). Using nonconstructive maths e.g. LEM or AC etc in
> Lean 3 is just a matter of writing some incantation at the top of a file
> and then not thinking about it any more. HoTT might be more appropriate for
> mathematics -- or at least for some kinds of mathematics -- but its
> implementation in an actual piece of software seems a bit more hacky at
> this point ("use Coq, but don't use these commands or these tactics"),
> which maybe raises the entry barrier for mathematicians a bit (and speaking
> from personal experience, already this entry barrier is quite high). High
> level tactics are absolutely crucial for mathematical Lean users. This is
> one of the reasons that the Lean documentation is not ideal for
> mathematicians -- mathematicians need very early on to be able to use
> tactics such as `ring` or `norm_num` to do calculations with real numbers
> or in commutative rings, and these tactics are not even mentioned in the
> standard Lean documentation.
> > >
> > > I am a working mathematician who two years ago knew nothing about this
> way of doing mathematics on a computer. Now I have seen what is possible I
> am becoming convinced that it can really change mathematics. In my
> experience the biggest obstruction to it changing mathematics is simply
> that mathematicians do not see the point of it, or what it has to offer a
> modern working mathematician; they can see no immediate benefits in
> learning how this stuff works. In short, I think type theory has an image
> problem. Sure there are category theorists who know about it, but how many
> category theorists are there in an average maths department? In the UK at
> least, the answer is "much less than 1", and I cannot see that changing any
> time soon. I would love to draw the mathematics and computer science
> communities closer together over ideas like this, but it's hard work. I am
> wondering whether developing accessible databases of undergraduate level
> mathematics would at least make mathematicians sit up and take notice, but
> when I look at what has been done in these various systems I do not see
> this occurring. This weekend I've learnt something about UniMath, but
> whilst it might do bicategories very well (which are not on our
> undergraduate curriculum), where is the basic analysis? Where is the stuff
> which draws mathematicians in? This by no means a criticism of unimath --
> it is in fact a far more broad criticism of all of the systems out there.
> Lean 3 might have schemes but they still can't prove that the derivative of
> sin is cos, and Isabelle/HOL might never have schemes. I know that Gonthier
> and his coauthors had to make a lot of undergraduate level maths (Galois
> theory, algebraic number theory, group theory) when formalising the odd
> order theorem in Coq, but it turns out that the odd order theorem is
> perhaps not a good "selling point" for mathematics formalisation when
> you're trying to sell to "normal research mathematicians", and I don't know
> what is. I'm now wondering making formalised undergraduate mathematics more
> accessible to untrained mathematicians is a better approach, but who knows.
> Obviously "AI which can solve the Riemann hypothesis" will work, but that
> looks to me like a complete fantasy at this point.
> > >
> > > One thing I have learnt over the last two years is that a whole bunch
> of extremely smart people, both mathematicians and computer scientists,
> have invested a lot of time in thinking about how to do mathematics with
> type theory. I find it very frustrating that mathematicians are not
> beginning to notice. Of course there are exceptions. One day though -- will
> there simply be a gigantic wave which crashes through mathematics and
> forces mathematicians to sit up and take notice? I guess we simply do not
> know, but if there is, I hope I'm still around.
> > >
> > > Kevin
> > >
> > >
> > >
> > > On Sun, 26 May 2019 at 06:50, Bas Spitters <b.a.w.spitters@gmail.com>
> wrote:
> > >>
> > >> There has been progress in making a cleaner interface with the
> standard Coq tactics. (Some abstractions were broken at the ocaml level)
> > >> I'm hopeful that this can be lead to a clean connection between the
> HoTT library and more of the Coq developments in the not too distant future.
> > >> As it exists in agda now.
> > >>
> > >> IIUC, UniMath does not allow any of the standard library or it's
> tactics, or even record types, since Vladimir wanted to have a very tight
> connection between type theory and it's semantics in simplicial sets. So, I
> don't expect them to connect to other developments, but I could be wrong.
> > >>
> > >> About the bundled/unbundled issue, which also exists in Coq, there's
> some recent progress "frame type theory" which should be applicable to both
> Coq and lean:
> > >>  http://www.ii.uib.no/~bezem/abstracts/TYPES_2019_paper_51
> > >>
> > >> Coming back to Kevin's question, yes, HoTT (plus classical logic for
> sets), seems to be the most natural foundation for mathematics as is
> currently published in the Annals.
> > >>
> > >> On Sat, May 25, 2019 at 6:42 PM Noah Snyder <nsnyder@gmail.com>
> wrote:
> > >>>
> > >>> UniMath vs HoTT wasn’t exactly my point, UniMath = Book-HoTT is of a
> category from “Coq with the indices-matter option plus the HoTT library."
>  “Coq with the indices-matter option plus the HoTT library" is of the same
> category as "Lean plus the math library" and then it makes sense to compare
> how practically useful they are for math.
> > >>>
> > >>> Here it's important to note that most advanced things that you can
> do in Coq are broken by using the "indices-matter" option and relatedly not
> using the built-in type Prop.  Quoting from
> https://arxiv.org/abs/1610.04591 "This small change makes the whole
> standard library unusable, and many tactics stop working, too.  The
> solution was rather drastic: we ripped out the standard library and
> replaced it with a minimal core that is sufficient for the basic tactics to
> work."
> > >>>
> > >>> (In particular, I was in error in my previous email, *some* tactics
> are available in Coq+indices-matter+HoTT, but not many of the more advanced
> ones, and to my knowledge, not tactics needed for complicated homotopical
> calculations.)
> > >>>
> > >>> I should say I've never used Coq, just Agda.  (When I was using Agda
> the situation was even worse, things like pattern matching secretly assumed
> k even if you used the without-k option, and HITs were put in by a hack
> that wasn't totally clear if it worked, etc.)  So I'm likely wrong in some
> places above.
> > >>>
> > >>> So I think from a practical point of view, “Coq with the
> indices-matter option plus the HoTT library" is well behind ordinary Coq
> (and also Lean) for doing ordinary mathematics.  However, if and when it
> does catch up some of the pain points involving transporting from my
> previous email should go away automatically.  (Side comment: once you start
> talking about transporting stuff related to categories across equivalences
> of categories it's only going to get more painful in ordinary type theory,
> but will remain easy in HoTT approaches.)
> > >>>
> > >>> Best,
> > >>>
> > >>> Noah
> > >>>
> > >>> p.s. Installed Lean last week.  Looking forward to using it next
> year when Scott and I are both at MSRI.
> > >>>
> > >>> On Sat, May 25, 2019 at 11:36 AM Kevin Buzzard <
> kevin.m.buzzard@gmail.com> wrote:
> > >>>>
> > >>>> Hi Noah. Thank you for pointing out the category error. It seems to
> me that sometimes when I say "HoTT" I should be saying, for example,
> "UniMath".
> > >>>>
> > >>>> Tactics in Lean are absolutely crucial for library development. Coq
> has some really powerful tactics, right? UniMath can use those tactics,
> presumably?
> > >>>>
> > >>>> I understand that UniMath, as implemented in Coq, takes Coq and
> adds some "rules" of the form "you can't use this functionality" and also
> adds at least one new axiom (univalence).
> > >>>>
> > >>>> On Sat, 25 May 2019 at 15:50, Noah Snyder <nsnyder@gmail.com>
> wrote:
> > >>>>>
> > >>>>> I’d also imagine that a “practical” implementation would likely
> have some kind of “two-level” type theory where you can use types that
> behave classically when that’s better and HoTT types when that’s better.
> > >>>>
> > >>>>
> > >>>> But plain Coq has such types, right?
> > >>>>
> > >>>> OK so this has all been extremely informative. There are other
> provers being developed which will implement some flavour of HoTT more
> "faithfully", and it might be easier to develop maths libraries in such
> provers.
> > >>>>
> > >>>>> For example, if G and H are isomorphic groups and you want to
> translate a theorem or construction across the isomorphism.  In ordinary
> type theory this is going to involve annoying book-keeping which it seems
> like you’d have to do separately for each kind of mathematical object.
> > >>>>
> > >>>>
> > >>>> Yes. This is a pain point in Lean. It's a particularly nasty one
> too, as far as mathematicians are concerned, because when you tell a
> mathematician "well this ring R is Cohen-Macauley, and here's a ring S
> which is isomorphic to R, but we cannot immediately deduce in Lean that S
> is Cohen-Macauley" then they begin to question what kind of crazy system
> you are using which cannot deduce this immediately. As an interesting
> experiment, find your favourite mathematician, preferably one who does not
> know what a Cohen-Macauley ring is, and ask them whether they think it will
> be true that if R and S are isomorphic rings and R is Cohen-Macauley then S
> is too. They will be very confident that this is true, even if they do not
> know the definition; standard mathematical definitions are
> isomorphism-invariant. This is part of our code of conduct, in fact.
> > >>>>
> > >>>> However in Lean I believe that the current plan is to try and make
> a tactic which will resolve this issue. This has not yet been done, and as
> far as I can see this is a place where UniMath is a more natural fit for
> "the way mathematicians think". However now I've looked over what has
> currently been formalised in UniMath I am wondering whether there are pain
> points for it, which Lean manages to get over more easily. That is somehow
> where I'm coming from.
> > >>>>
> > >>>>>
> > >>>>>  For example, say you have a theorem about bimodules over
> semisimple rings whose proof starts “wlog, by Artin-Wedderburn, we can
> assume both algebras are multimatrix algebras over division rings.”  Is
> that step something you’d be able to deal with easily in Lean?  If not,
> that’s somewhere that down the line HoTT might make things more practical.
> > >>>>
> > >>>>
> > >>>> This is a great example! To be honest I am slightly confused about
> why we are not running into this sort of thing already. As far as I can see
> this would be a great test case for the (still very much under development)
> transport tactic. Maybe we don't have enough classification theorems. I
> think that our hope in general is that this sort of issue can be solved
> with automation.
> > >>>>
> > >>>> Kevin
> > >>>>
> > >>>>
> > >>>>>
> > >>>>> But mostly I just want to say you’re making a category error in
> your question.  HoTT is an abstract type theory, not a proof assistant.
> > >>>>>
> > >>>>> Best,
> > >>>>>
> > >>>>> Noah
> > >>>>>
> > >>>>> On Sat, May 25, 2019 at 9:34 AM Juan Ospina <jospina65@gmail.com>
> wrote:
> > >>>>>>
> > >>>>>> On page 117 of https://arxiv.org/pdf/1808.10690.pdf appears the
> "additivity axiom".  Please let me know if the following formulation of the
> such axiom is correct:
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>>
> > >>>>>> On Saturday, May 25, 2019 at 5:22:41 AM UTC-5, awodey wrote:
> > >>>>>>>
> > >>>>>>> A useful example for you might be Floris van Doorn’s
> formalization of
> > >>>>>>> the Atiyah-Hirzebruch and Serre spectral sequences for cohomology
> > >>>>>>> in HoTT using Lean:
> > >>>>>>>
> > >>>>>>>  https://arxiv.org/abs/1808.10690
> > >>>>>>>
> > >>>>>>> Regards,
> > >>>>>>>
> > >>>>>>> Steve
> > >>>>>>>
> > >>>>>>> > On May 25, 2019, at 12:12 PM, Kevin Buzzard <
> kevin....@gmail.com> wrote:
> > >>>>>>> >
> > >>>>>>> > Hi from a Lean user.
> > >>>>>>> >
> > >>>>>>> > As many people here will know, Tom Hales' formal abstracts
> project https://formalabstracts.github.io/ wants to formalise many of the
> statements of modern pure mathematics in Lean. One could ask more generally
> about a project of formalising many of the statements of modern pure
> mathematics in an arbitrary system, such as HoTT. I know enough about the
> formalisation process to know that whatever system one chooses, there will
> be pain points, because some mathematical ideas fit more readily into some
> foundational systems than others.
> > >>>>>>> >
> > >>>>>>> > I have seen enough of Lean to become convinced that the pain
> points would be surmountable in Lean. I have seen enough of Isabelle/HOL to
> become skeptical about the idea that it would be suitable for all of modern
> pure mathematics, although it is clearly suitable for some of it; however
> it seems that simple type theory struggles to handle things like tensor
> products of sheaves of modules on a scheme, because sheaves are dependent
> types and it seems that one cannot use Isabelle's typeclass system to
> handle the rings showing up in a sheaf of rings.
> > >>>>>>> >
> > >>>>>>> > I have very little experience with HoTT. I have heard that the
> fact that "all constructions must be isomorphism-invariant" is both a
> blessing and a curse. However I would like to know more details. I am
> speaking at the Big Proof conference in Edinburgh this coming Wednesday on
> the pain points involved with formalising mathematical objects in dependent
> type theory and during the preparation of my talk I began to wonder what
> the analogous picture was with HoTT.
> > >>>>>>> >
> > >>>>>>> > Everyone will have a different interpretation of "modern pure
> mathematics" so to fix our ideas, let me say that for the purposes of this
> discussion, "modern pure mathematics" means the statements of the theorems
> publishsed by the Annals of Mathematics over the last few years, so for
> example I am talking about formalising statements of theorems involving
> L-functions of abelian varieties over number fields, Hodge theory,
> cohomology of algebraic varieties, Hecke algebras of symmetric groups,
> Ricci flow and the like; one can see titles and more at
> http://annals.math.princeton.edu/2019/189-3 . Classical logic and the
> axiom of choice are absolutely essential -- I am only interested in the
> hard-core "classical mathematician" stance of the way mathematics works,
> and what it is.
> > >>>>>>> >
> > >>>>>>> > If this is not the right forum for this question, I would be
> happily directed to somewhere more suitable. After spending 10 minutes
> failing to get onto ##hott on freenode ("you need to be identified with
> services") I decided it was easier just to ask here. If people want to chat
> directly I am usually around at https://leanprover.zulipchat.com/
> (registration required, full names are usually used, I'll start a HoTT
> thread in #mathematics).
> > >>>>>>> >
> > >>>>>>> > Kevin Buzzard
> > >>>>>>> >
> > >>>>>>> > --
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  parent reply index

Thread overview: 31+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-05-25 10:12 Kevin Buzzard
2019-05-25 10:22 ` Steve Awodey
2019-05-25 12:23   ` Kevin Buzzard
     [not found]   ` <B7D67BBA-5E0B-4438-908D-4EF316C8C1F1@chalmers.se>
     [not found]     ` <CAH52Xb1Y=Xq=012v_-KSDUuwgnKpEp5qjrxgtUJf+qc_0RWJUg@mail.gmail.com>
2019-05-25 13:13       ` Fwd: " Kevin Buzzard
2019-05-25 13:34   ` Juan Ospina
2019-05-25 14:50     ` Noah Snyder
2019-05-25 15:36       ` Kevin Buzzard
2019-05-25 16:41         ` Noah Snyder
2019-05-26  5:50           ` Bas Spitters
2019-05-26 11:41             ` Kevin Buzzard
2019-05-26 12:09               ` Bas Spitters
2019-05-26 17:00                 ` Kevin Buzzard
2019-05-27  2:33                   ` Daniel R. Grayson
2019-06-02 16:30                   ` Bas Spitters [this message]
2019-06-02 17:55                     ` Kevin Buzzard
2019-06-02 20:46                       ` Nicola Gambino
2019-06-02 20:59                         ` Valery Isaev
2019-06-04 20:32                       ` Michael Shulman
2019-06-04 20:58                         ` Kevin Buzzard
2019-06-06 16:30                         ` Matt Oliveri
2019-05-27 13:09                 ` Assia Mahboubi
2019-05-28  9:50                   ` Michael Shulman
2019-05-28 10:13                     ` Nils Anders Danielsson
2019-05-28 10:22                       ` Michael Shulman
2019-05-29 19:04                         ` Martín Hötzel Escardó
2019-05-30 17:14                           ` Michael Shulman
2019-06-02 17:49                             ` Kevin Buzzard
2019-06-04 20:50                               ` Martín Hötzel Escardó
2019-06-05 17:11                                 ` Thorsten Altenkirch
2019-05-28 15:20                     ` Joyal, André
2019-05-27  8:41           ` Nils Anders Danielsson

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Discussion of Homotopy Type Theory and Univalent Foundations

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