Discussion of Homotopy Type Theory and Univalent Foundations
 help / color / Atom feed
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>

[-- Attachment #1: Type: text/plain, Size: 3752 bytes --]

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.
>> To view this discussion on the web visit
>> https://groups.google.com/d/msgid/HomotopyTypeTheory/e491d38b-b50a-6172-dca9-40d45fe1b6d2%40fromzerotoinfinity.xyz
>> .
>>
>

-- 
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/CAOoPQuRQPMkCFKYtAbB%2BpNK90XtFk%2BaVT_aY59U_-9t17sBzeA%40mail.gmail.com.

[-- Attachment #2: Type: text/html, Size: 5364 bytes --]

<div dir="ltr"><div>There&#39;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 &lt;<a href="mailto:b.a.w.spitters@gmail.com">b.a.w.spitters@gmail.com</a>&gt; 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 &lt;<a href="mailto:zero@fromzerotoinfinity.xyz" target="_blank">zero@fromzerotoinfinity.xyz</a>&gt; 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&amp;t=1680" rel="noreferrer" target="_blank">https://www.youtube.com/watch?v=zw6NcwME7yI&amp;t=1680</a>) 2014 talk<br>
at IAS, Voevodsky talks about the history of his project of &quot;univalent<br>
mathematics&quot; 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&#39;t go into details of what mathematics he was<br>
exactly interested in (I&#39;m guessing something to do with homotopy<br>
theory) or why exactly existing proof assistants weren&#39;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&#39;t *his* view, to<br>
me it seems to be a view held implicitly in the HoTT community. In any<br>
case, it&#39;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 &quot;Formalized<br>
Mathematics&quot; 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&#39;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>
You received this message because you are subscribed to the Google Groups &quot;Homotopy Type Theory&quot; group.<br>
To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory%2Bunsubscribe@googlegroups.com" target="_blank">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br>
To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/e491d38b-b50a-6172-dca9-40d45fe1b6d2%40fromzerotoinfinity.xyz" rel="noreferrer" target="_blank">https://groups.google.com/d/msgid/HomotopyTypeTheory/e491d38b-b50a-6172-dca9-40d45fe1b6d2%40fromzerotoinfinity.xyz</a>.<br>
</blockquote></div>
</blockquote></div>

<p></p>

-- <br />
You received this message because you are subscribed to the Google Groups &quot;Homotopy Type Theory&quot; group.<br />
To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory+unsubscribe@googlegroups.com">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br />
To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOoPQuRQPMkCFKYtAbB%2BpNK90XtFk%2BaVT_aY59U_-9t17sBzeA%40mail.gmail.com?utm_medium=email&utm_source=footer">https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOoPQuRQPMkCFKYtAbB%2BpNK90XtFk%2BaVT_aY59U_-9t17sBzeA%40mail.gmail.com</a>.<br />

  reply index

Thread 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 Spitters
2019-11-03 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

Reply instructions:

You may reply publically to this message via plain-text email
using any one of the following methods:

* Save the following mbox file, import it into your mail client,
  and reply-to-all from there: mbox

  Avoid top-posting and favor interleaved quoting:
  https://en.wikipedia.org/wiki/Posting_style#Interleaved_style

* Reply using the --to, --cc, and --in-reply-to
  switches of git-send-email(1):

  git send-email \
    --in-reply-to=CAOoPQuRQPMkCFKYtAbB+pNK90XtFk+aVT_aY59U_-9t17sBzeA@mail.gmail.com \
    --to=b.a.w.spitters@gmail.com \
    --cc=HomotopyTypeTheory@googlegroups.com \
    --cc=zero@fromzerotoinfinity.xyz \
    /path/to/YOUR_REPLY

  https://kernel.org/pub/software/scm/git/docs/git-send-email.html

* If your mail client supports setting the In-Reply-To header
  via mailto: links, try the mailto: link

Discussion of Homotopy Type Theory and Univalent Foundations

Archives are clonable: git clone --mirror http://inbox.vuxu.org/hott

Example config snippet for mirrors

Newsgroup available over NNTP:
	nntp://inbox.vuxu.org/vuxu.archive.hott


AGPL code for this site: git clone https://public-inbox.org/public-inbox.git