Discussion of Homotopy Type Theory and Univalent Foundations
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From: Valery Isaev <valery.isaev@gmail.com>
To: Michael Shulman <shulman@sandiego.edu>
Cc: David Roberts <droberts.65537@gmail.com>,
	Bas Spitters <b.a.w.spitters@gmail.com>,
	 homotopytypetheory <homotopytypetheory@googlegroups.com>,
	 Nicolas Alexander Schmidt <zero@fromzerotoinfinity.xyz>
Subject: Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'?
Date: Sun, 3 Nov 2019 22:45:53 +0300
Message-ID: <CAA520ftr3b4cGi4vXdmJL-GbAVksV9ggrTAmqZ4E75P-ch1hVQ@mail.gmail.com> (raw)
In-Reply-To: <CAOvivQz47kSm9WbKDmUsndrpAJMkNwiWmVABqOFrVqyTOvSAbw@mail.gmail.com>

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The isomorphism invariance might be useful when you don't care about the
actual stuff you get after transporting. If you have two different
definitions of the same set, you can transport its *properties* along the
isomorphism (that you still need to construct). For example, we can define
rational numbers in two different ways: as arbitrary quotients and as
reduced quotients. Then you can prove that they are isomorphic and that one
of them gives you a field. Then you know that the other one is also a
field. You may be interested in the actual definitions of the operations,
but you dob't usually care about the actual proof of the properties. So,
you construct the addition and the multiplication explicitly, but you get
proofs that they determine a field "for free" from the other definition. If
the proofs for one of the definitions is easier than for the other one,
this might be a significant simplification.

Regards,
Valery Isaev


вс, 3 нояб. 2019 г. в 22:13, Michael Shulman <shulman@sandiego.edu>:

> But does univalence a la Book HoTT *actually* make it easier to reason
> about such things?  It allows us to write "=" rather than "\cong", but
> to construct such an equality we have to construct an isomorphism
> first, and to *use* such an equality we have to transport along it,
> and then we get lots of univalence-redexes that we have to manually
> reduce away.  My experience formalizing math in HoTT/Coq is that it's
> much easier if you *avoid* turning equivalences into equalities except
> when absolutely necessary.  (I don't have any experience formalizing
> math in a cubical proof assistant, but in that case at least you
> wouldn't have to manually reduce the univalence-redexes -- although it
> seems to me you'd still have to construct the isomorphism before you
> can apply univalence to make it an equality.)
>
> On Sun, Nov 3, 2019 at 3:57 AM David Roberts <droberts.65537@gmail.com>
> wrote:
> >
> > Forget even higher category theory. Kevin Buzzard now goes around
> telling the story of how even formally proving (using Lean) things in
> rather elementary commutative algebra from EGA that are stated as
> equalities was not obvious: the equality is really an isomorphism arising
> from a universal property. Forget trying to do anything motivic, when
> algebra is full of such equalities. This is not a problem with univalence,
> of course.
> >
> > David
> >
> > On Sun, 3 Nov 2019, 10:08 PM Bas Spitters <b.a.w.spitters@gmail.com>
> wrote:
> >>
> >> 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
> .
> >>
> >> --
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> .
> >
> > --
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> > To view this discussion on the web visit
> https://groups.google.com/d/msgid/HomotopyTypeTheory/CAFL%2BZM_%3D%3DiLS16Vy7sGiEqNkCxOMYL4j%2BZFqKv5uJ-ivkuemEg%40mail.gmail.com
> .
>
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<div dir="ltr"><div>The isomorphism invariance might be useful when you don&#39;t care about the actual stuff you get after transporting. If you have two different definitions of the same set, you can transport its <i>properties</i> along the isomorphism (that you still need to construct). For example, we can define rational numbers in two different ways: as arbitrary quotients and as reduced quotients. Then you can prove that they are isomorphic and that one of them gives you a field. Then you know that the other one is also a field. You may be interested in the actual definitions of the operations, but you dob&#39;t usually care about the actual proof of the properties. So, you construct the addition and the multiplication explicitly, but you get proofs that they determine a field &quot;for free&quot; from the other definition. If the proofs for one of the definitions is easier than for the other one, this might be a significant simplification.</div><br clear="all"><div><div dir="ltr" class="gmail_signature" data-smartmail="gmail_signature"><div dir="ltr"><div><div>Regards,</div><div dir="ltr">Valery Isaev<br></div></div></div></div></div><br></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">вс, 3 нояб. 2019 г. в 22:13, Michael Shulman &lt;<a href="mailto:shulman@sandiego.edu">shulman@sandiego.edu</a>&gt;:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">But does univalence a la Book HoTT *actually* make it easier to reason<br>
about such things?  It allows us to write &quot;=&quot; rather than &quot;\cong&quot;, but<br>
to construct such an equality we have to construct an isomorphism<br>
first, and to *use* such an equality we have to transport along it,<br>
and then we get lots of univalence-redexes that we have to manually<br>
reduce away.  My experience formalizing math in HoTT/Coq is that it&#39;s<br>
much easier if you *avoid* turning equivalences into equalities except<br>
when absolutely necessary.  (I don&#39;t have any experience formalizing<br>
math in a cubical proof assistant, but in that case at least you<br>
wouldn&#39;t have to manually reduce the univalence-redexes -- although it<br>
seems to me you&#39;d still have to construct the isomorphism before you<br>
can apply univalence to make it an equality.)<br>
<br>
On Sun, Nov 3, 2019 at 3:57 AM David Roberts &lt;<a href="mailto:droberts.65537@gmail.com" target="_blank">droberts.65537@gmail.com</a>&gt; wrote:<br>
&gt;<br>
&gt; Forget even higher category theory. Kevin Buzzard now goes around telling the story of how even formally proving (using Lean) things in rather elementary commutative algebra from EGA that are stated as equalities was not obvious: the equality is really an isomorphism arising from a universal property. Forget trying to do anything motivic, when algebra is full of such equalities. This is not a problem with univalence, of course.<br>
&gt;<br>
&gt; David<br>
&gt;<br>
&gt; On Sun, 3 Nov 2019, 10:08 PM Bas Spitters &lt;<a href="mailto:b.a.w.spitters@gmail.com" target="_blank">b.a.w.spitters@gmail.com</a>&gt; wrote:<br>
&gt;&gt;<br>
&gt;&gt; There&#39;s also VV homotopy lambda calculus, which he later abandoned for MLTT:<br>
&gt;&gt; <a href="https://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/Hlambda_short_current.pdf" rel="noreferrer" target="_blank">https://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/Hlambda_short_current.pdf</a><br>
&gt;&gt;<br>
&gt;&gt; On Sun, Oct 27, 2019 at 6:22 PM Bas Spitters &lt;<a href="mailto:b.a.w.spitters@gmail.com" target="_blank">b.a.w.spitters@gmail.com</a>&gt; wrote:<br>
&gt;&gt;&gt;<br>
&gt;&gt;&gt; I believe it refers to his 2-theories:<br>
&gt;&gt;&gt; <a href="https://www.ias.edu/ideas/2014/voevodsky-origins" rel="noreferrer" target="_blank">https://www.ias.edu/ideas/2014/voevodsky-origins</a><br>
&gt;&gt;&gt;<br>
&gt;&gt;&gt; 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>
&gt;&gt;&gt;&gt;<br>
&gt;&gt;&gt;&gt; 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>
&gt;&gt;&gt;&gt; at IAS, Voevodsky talks about the history of his project of &quot;univalent<br>
&gt;&gt;&gt;&gt; mathematics&quot; and his motivation for starting it. Namely, he mentions<br>
&gt;&gt;&gt;&gt; that he found existing proof assistants at that time (in 2000) to be<br>
&gt;&gt;&gt;&gt; impractical for the kinds of mathematics he was interested in.<br>
&gt;&gt;&gt;&gt;<br>
&gt;&gt;&gt;&gt; Unfortunately, he doesn&#39;t go into details of what mathematics he was<br>
&gt;&gt;&gt;&gt; exactly interested in (I&#39;m guessing something to do with homotopy<br>
&gt;&gt;&gt;&gt; theory) or why exactly existing proof assistants weren&#39;t practical for<br>
&gt;&gt;&gt;&gt; formalizing them. Judging by the things he mentions in his talk, it<br>
&gt;&gt;&gt;&gt; seems that (roughly) his rejection of those proof assistants was based<br>
&gt;&gt;&gt;&gt; on the view that predicate logic + ZFC is not expressive enough. In<br>
&gt;&gt;&gt;&gt; other words, there is too much lossy encoding needed in order to<br>
&gt;&gt;&gt;&gt; translate from the platonic world of mathematical ideas to this formal<br>
&gt;&gt;&gt;&gt; language.<br>
&gt;&gt;&gt;&gt;<br>
&gt;&gt;&gt;&gt; Comparing the situation to computer programming languages, one might say<br>
&gt;&gt;&gt;&gt; that predicate logic is like Assembly in that even though everything can<br>
&gt;&gt;&gt;&gt; be encoded in that language, it is not expressive enough to directly<br>
&gt;&gt;&gt;&gt; talk about higher level concepts, diminishing its practical value for<br>
&gt;&gt;&gt;&gt; reasoning about mathematics. In particular, those systems are not<br>
&gt;&gt;&gt;&gt; adequate for *interactive* development of *new* mathematics (as opposed<br>
&gt;&gt;&gt;&gt; to formalization of existing theories).<br>
&gt;&gt;&gt;&gt;<br>
&gt;&gt;&gt;&gt; Perhaps I am just misinterpreting what Voevodsky said. In this case, I<br>
&gt;&gt;&gt;&gt; hope someone can correct me. However even if this wasn&#39;t *his* view, to<br>
&gt;&gt;&gt;&gt; me it seems to be a view held implicitly in the HoTT community. In any<br>
&gt;&gt;&gt;&gt; case, it&#39;s a view that one might reasonably hold.<br>
&gt;&gt;&gt;&gt;<br>
&gt;&gt;&gt;&gt; However I wonder how reasonable that view actually is, i.e. whether e.g.<br>
&gt;&gt;&gt;&gt; Mizar really is that much more impractical than HoTT-Coq or Agda, given<br>
&gt;&gt;&gt;&gt; that people have been happily formalizing mathematics in it for 46 years<br>
&gt;&gt;&gt;&gt; now. And, even though by browsing the contents of &quot;Formalized<br>
&gt;&gt;&gt;&gt; Mathematics&quot; one can get the impression that the work consists mostly of<br>
&gt;&gt;&gt;&gt; formalizing early 20th century mathematics, neither the UniMath nor the<br>
&gt;&gt;&gt;&gt; HoTT library for example contain a proof of Fubini&#39;s theorem.<br>
&gt;&gt;&gt;&gt;<br>
&gt;&gt;&gt;&gt; So, to put this into one concrete question, how (if at all) is HoTT-Coq<br>
&gt;&gt;&gt;&gt; more practical than Mizar for the purpose of formalizing mathematics,<br>
&gt;&gt;&gt;&gt; outside the specific realm of synthetic homotopy theory?<br>
&gt;&gt;&gt;&gt;<br>
&gt;&gt;&gt;&gt;<br>
&gt;&gt;&gt;&gt; --<br>
&gt;&gt;&gt;&gt;<br>
&gt;&gt;&gt;&gt; Nicolas<br>
&gt;&gt;&gt;&gt;<br>
&gt;&gt;&gt;&gt;<br>
&gt;&gt;&gt;&gt; --<br>
&gt;&gt;&gt;&gt; You received this message because you are subscribed to the Google Groups &quot;Homotopy Type Theory&quot; group.<br>
&gt;&gt;&gt;&gt; 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>
&gt;&gt;&gt;&gt; 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>
&gt;&gt;<br>
&gt;&gt; --<br>
&gt;&gt; You received this message because you are subscribed to the Google Groups &quot;Homotopy Type Theory&quot; group.<br>
&gt;&gt; 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>
&gt;&gt; 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" rel="noreferrer" target="_blank">https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOoPQuRQPMkCFKYtAbB%2BpNK90XtFk%2BaVT_aY59U_-9t17sBzeA%40mail.gmail.com</a>.<br>
&gt;<br>
&gt; --<br>
&gt; You received this message because you are subscribed to the Google Groups &quot;Homotopy Type Theory&quot; group.<br>
&gt; 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>
&gt; To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/CAFL%2BZM_%3D%3DiLS16Vy7sGiEqNkCxOMYL4j%2BZFqKv5uJ-ivkuemEg%40mail.gmail.com" rel="noreferrer" target="_blank">https://groups.google.com/d/msgid/HomotopyTypeTheory/CAFL%2BZM_%3D%3DiLS16Vy7sGiEqNkCxOMYL4j%2BZFqKv5uJ-ivkuemEg%40mail.gmail.com</a>.<br>
<br>
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</blockquote></div>

<p></p>

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  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
2019-11-03 11:52     ` David Roberts
2019-11-03 19:13       ` Michael Shulman
2019-11-03 19:45         ` Valery Isaev [this message]
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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