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

[Bas Spitters <b.a.w.spitters@gmail.com>]

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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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>>
>

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