```
From: Bas Spitters <b.a.w.spitters@gmail.com>
To: Nicolas Alexander Schmidt <zero@fromzerotoinfinity.xyz>
Cc: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'?
Date: Sun, 3 Nov 2019 12:38:36 +0100
Message-ID: <CAOoPQuRQPMkCFKYtAbB+pNK90XtFk+aVT_aY59U_-9t17sBzeA@mail.gmail.com> (raw)
In-Reply-To: <CAOoPQuTfkp=PNeYE8bpO20APnTBdpzqJNfUekE5ECrr0vD5cww@mail.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
>>
>>
>> --
>> 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.
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>> https://groups.google.com/d/msgid/HomotopyTypeTheory/e491d38b-b50a-6172-dca9-40d45fe1b6d2%40fromzerotoinfinity.xyz
>> .
>>
>
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<div dir="ltr"><div>There's also VV homotopy lambda calculus, which he later abandoned for MLTT:</div><div><a href="https://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/Hlambda_short_current.pdf">https://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/Hlambda_short_current.pdf</a></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Sun, Oct 27, 2019 at 6:22 PM Bas Spitters <<a href="mailto:b.a.w.spitters@gmail.com">b.a.w.spitters@gmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="ltr"><div>I believe it refers to his 2-theories:<br></div><div><a href="https://www.ias.edu/ideas/2014/voevodsky-origins" target="_blank">https://www.ias.edu/ideas/2014/voevodsky-origins</a></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Sun, Oct 27, 2019 at 3:41 PM Nicolas Alexander Schmidt <<a href="mailto:zero@fromzerotoinfinity.xyz" target="_blank">zero@fromzerotoinfinity.xyz</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">In [this](<a href="https://www.youtube.com/watch?v=zw6NcwME7yI&t=1680" rel="noreferrer" target="_blank">https://www.youtube.com/watch?v=zw6NcwME7yI&t=1680</a>) 2014 talk<br>
at IAS, Voevodsky talks about the history of his project of "univalent<br>
mathematics" and his motivation for starting it. Namely, he mentions<br>
that he found existing proof assistants at that time (in 2000) to be<br>
impractical for the kinds of mathematics he was interested in.<br>
<br>
Unfortunately, he doesn't go into details of what mathematics he was<br>
exactly interested in (I'm guessing something to do with homotopy<br>
theory) or why exactly existing proof assistants weren't practical for<br>
formalizing them. Judging by the things he mentions in his talk, it<br>
seems that (roughly) his rejection of those proof assistants was based<br>
on the view that predicate logic + ZFC is not expressive enough. In<br>
other words, there is too much lossy encoding needed in order to<br>
translate from the platonic world of mathematical ideas to this formal<br>
language.<br>
<br>
Comparing the situation to computer programming languages, one might say<br>
that predicate logic is like Assembly in that even though everything can<br>
be encoded in that language, it is not expressive enough to directly<br>
talk about higher level concepts, diminishing its practical value for<br>
reasoning about mathematics. In particular, those systems are not<br>
adequate for *interactive* development of *new* mathematics (as opposed<br>
to formalization of existing theories).<br>
<br>
Perhaps I am just misinterpreting what Voevodsky said. In this case, I<br>
hope someone can correct me. However even if this wasn't *his* view, to<br>
me it seems to be a view held implicitly in the HoTT community. In any<br>
case, it's a view that one might reasonably hold.<br>
<br>
However I wonder how reasonable that view actually is, i.e. whether e.g.<br>
Mizar really is that much more impractical than HoTT-Coq or Agda, given<br>
that people have been happily formalizing mathematics in it for 46 years<br>
now. And, even though by browsing the contents of "Formalized<br>
Mathematics" one can get the impression that the work consists mostly of<br>
formalizing early 20th century mathematics, neither the UniMath nor the<br>
HoTT library for example contain a proof of Fubini's theorem.<br>
<br>
So, to put this into one concrete question, how (if at all) is HoTT-Coq<br>
more practical than Mizar for the purpose of formalizing mathematics,<br>
outside the specific realm of synthetic homotopy theory?<br>
<br>
<br>
--<br>
<br>
Nicolas<br>
<br>
<br>
-- <br>
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</blockquote></div>
</blockquote></div>
<p></p>
-- <br />
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```

next prev parent reply indexThread overview:18+ messages / expand[flat|nested] mbox.gz Atom feed top 2019-10-27 14:41 Nicolas Alexander Schmidt 2019-10-27 17:22 ` Bas Spitters2019-11-03 11:38 ` Bas Spitters [this message]2019-11-03 11:52 ` David Roberts 2019-11-03 19:13 ` Michael Shulman 2019-11-03 19:45 ` Valery Isaev 2019-11-03 22:23 ` Martín Hötzel Escardó 2019-11-04 23:20 ` Nicolas Alexander Schmidt 2019-11-04 18:42 ` Kevin Buzzard 2019-11-04 21:10 ` Michael Shulman 2019-11-04 23:26 ` David Roberts 2019-11-05 15:43 ` Daniel R. Grayson 2019-11-05 20:29 ` Yuhao Huang 2019-11-06 23:59 ` Daniel R. Grayson 2019-11-05 23:14 ` Martín Hötzel Escardó 2019-11-06 0:06 ` Stefan Monnier 2019-11-11 18:26 ` Licata, Dan 2019-11-03 7:29 ` Michael Shulman

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