From: Kevin Buzzard <kevin.m.buzzard@gmail.com> To: Noah Snyder <nsnyder@gmail.com> Cc: Juan Ospina <jospina65@gmail.com>, Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com> Subject: Re: [HoTT] doing "all of pure mathematics" in type theory Date: Sat, 25 May 2019 16:36:26 +0100 [thread overview] Message-ID: <CAH52Xb3k4fmuPfpDSO=Jmb3k19v3m6MjOLweQb_u8n0icdt0rQ@mail.gmail.com> (raw) In-Reply-To: <CAO0tDg4dyjZHueTPQEYKQaojSw6SDKNUC8y49JHMoR9oTQOmMw@mail.gmail.com> [-- Attachment #1.1: Type: text/plain, Size: 9733 bytes --] 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: >> >> [image: additivity.jpg] >> >> >> >> 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 >>> > >>> > -- >>> > You received this message because you are subscribed to the Google >>> Groups "Homotopy Type Theory" group. >>> > To unsubscribe from this group and stop receiving emails from it, send >>> an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. >>> > To view this discussion on the web visit >>> https://groups.google.com/d/msgid/HomotopyTypeTheory/a57315f6-cbd6-41a5-a3b7-b585e33375d4%40googlegroups.com. >>> >>> > For more options, visit https://groups.google.com/d/optout. >>> >>> -- >> You received this message because you are subscribed to the Google Groups >> "Homotopy Type Theory" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to HomotopyTypeTheory+unsubscribe@googlegroups.com. >> To view this discussion on the web visit >> https://groups.google.com/d/msgid/HomotopyTypeTheory/18681ec4-dc8d-42eb-b42b-b9a9f639d89e%40googlegroups.com >> <https://groups.google.com/d/msgid/HomotopyTypeTheory/18681ec4-dc8d-42eb-b42b-b9a9f639d89e%40googlegroups.com?utm_medium=email&utm_source=footer> >> . >> For more options, visit https://groups.google.com/d/optout. >> > -- > You received this message because you are subscribed to the Google Groups > "Homotopy Type Theory" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to HomotopyTypeTheory+unsubscribe@googlegroups.com. > To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/CAO0tDg4dyjZHueTPQEYKQaojSw6SDKNUC8y49JHMoR9oTQOmMw%40mail.gmail.com > <https://groups.google.com/d/msgid/HomotopyTypeTheory/CAO0tDg4dyjZHueTPQEYKQaojSw6SDKNUC8y49JHMoR9oTQOmMw%40mail.gmail.com?utm_medium=email&utm_source=footer> > . > For more options, visit https://groups.google.com/d/optout. > -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CAH52Xb3k4fmuPfpDSO%3DJmb3k19v3m6MjOLweQb_u8n0icdt0rQ%40mail.gmail.com. For more options, visit https://groups.google.com/d/optout. [-- Attachment #1.2: Type: text/html, Size: 12940 bytes --] [-- Attachment #2: additivity.jpg --] [-- Type: image/jpeg, Size: 18737 bytes --]

next prev parent reply other threads:[~2019-05-25 15:36 UTC|newest]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 Snyder2019-05-25 15:36 ` Kevin Buzzard [this message]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 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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