Discussion of Homotopy Type Theory and Univalent Foundations
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From: Kevin Buzzard <kevin.m.buzzard@gmail.com>
To: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: [HoTT] doing "all of pure mathematics" in type theory
Date: Sat, 25 May 2019 03:12:04 -0700 (PDT)	[thread overview]
Message-ID: <a57315f6-cbd6-41a5-a3b7-b585e33375d4@googlegroups.com> (raw)

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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 

Kevin Buzzard

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             reply	other threads:[~2019-05-25 10:12 UTC|newest]

Thread overview: 31+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-05-25 10:12 Kevin Buzzard [this message]
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
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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