Re: [HoTT] doing "all of pure mathematics" in type theory

[Juan Ospina <jospina65@gmail.com>]

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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 
> <javascript:>> 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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