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* [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'?
@ 2019-10-27 14:41 Nicolas Alexander Schmidt
  2019-10-27 17:22 ` Bas Spitters
  2019-11-03  7:29 ` Michael Shulman
  0 siblings, 2 replies; 18+ messages in thread
From: Nicolas Alexander Schmidt @ 2019-10-27 14:41 UTC (permalink / raw)
  To: Homotopy Type Theory

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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^ permalink raw reply	[flat|nested] 18+ messages in thread

* Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'?
  2019-10-27 14:41 [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'? Nicolas Alexander Schmidt
@ 2019-10-27 17:22 ` Bas Spitters
  2019-11-03 11:38   ` Bas Spitters
  2019-11-03  7:29 ` Michael Shulman
  1 sibling, 1 reply; 18+ messages in thread
From: Bas Spitters @ 2019-10-27 17:22 UTC (permalink / raw)
  To: Nicolas Alexander Schmidt; +Cc: Homotopy Type Theory

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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>I believe it refers to his 2-theories:<br></div><div><a href="https://www.ias.edu/ideas/2014/voevodsky-origins">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">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>
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</blockquote></div>

<p></p>

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^ permalink raw reply	[flat|nested] 18+ messages in thread

* Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'?
  2019-10-27 14:41 [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'? Nicolas Alexander Schmidt
  2019-10-27 17:22 ` Bas Spitters
@ 2019-11-03  7:29 ` Michael Shulman
  1 sibling, 0 replies; 18+ messages in thread
From: Michael Shulman @ 2019-11-03  7:29 UTC (permalink / raw)
  To: Nicolas Alexander Schmidt; +Cc: Homotopy Type Theory

I know hardly anything about Mizar, so I can't comment on it.  And I
don't know for sure exactly what Voevodsky meant (and perhaps no one
now does).  But I am pretty sure that he was not thinking about
synthetic homotopy theory at the time, because the possibility of
synthetic homotopy theory wasn't really even imagined until later on
(specifically, with the advent of higher inductive types).

Instead, I expect he was thinking about category theory and higher
category theory.  The advantage of univalence in such contexts is that
it formalizes the isomorphism-invariant behavior of category theory,
incorporating it directly into the logical structure of the
foundations so that you don't have to worry about whether your
theorems are invariant under isomorphism, or prove explicitly that
they are.  Is this a substantial advantage over a ZFC-based system?
Maybe, maybe not.  (One might argue that it's not much of an advantage
at all until univalence becomes "computational", as with cubical type
theories, so that transporting along it can be done automatically by
the proof assistant.)  But there are other points to consider.

Firstly, before even getting to univalence, there is a difference
between membership-based set theories and type theories, which is more
closely analogous to the assembly language / high level programming
language divide.  Type-theoretic foundations for mathematics, like
strongly/statically typed programming languages, provide automatic
"error-checking" for the user, catching various kinds of mistakes at
"compile time", and allow more intelligent compiler optimization and
inference due to the more informative nature of types.  Dependent type
theories are even better at this.  And just as the advantages of
static typing are not belied by the fact that a lot of people happily
write code in dynamically typed languages, so the advantages of typed
mathematics for formalization are not belied by the fact that some
mathematicians are happy to do without them.  It's worth noting that
many of the recent high-profile formalizations of *recent*
mathematical results, such as the four-color theorem, the odd-order
theorem, and the Kepler conjecture, use type-theoretic proof
assistants like Coq and HOL rather than set-theoretic ones like Mizar.

From this perspective, the advantage of HoTT/UF is that it "fixes"
certain infelicities in previously existing type theories.  Now we
understand quotients better and have more tools for doing without
setoids; now we know what the equality type of the universe is; now we
are better at computing with function extensionality and propositional
extensionality; etc.  So HoTT/UF gives us the benefits of a
type-theoretic foundation while ameliorating some of the traditional
disadvantages of that approach.

But in my own opinion (which is, I believe, rather different from
Voevodsky's, at least in emphasis), all of this is kind of beside the
point.  It's like arguing about whether or not my laptop can play
movies better than an 80s-era VCR; it overlooks the main point that my
laptop does *so much more* than just play movies.  The real advantage
of type-theoretic, and homotopy-type-theoretic, foundations, is that
they have a plethora of categorical and higher-categorical models.
Proving a theorem once, in constructive homotopy type theory,
automatically entails the "internal" truth of the corresponding
theorem in all higher toposes.  I think this is a much more important
selling point than whether or not we get a more practical system for
formalizing plain old set-based mathematics.

Mike


On Sun, Oct 27, 2019 at 7:41 AM 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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^ permalink raw reply	[flat|nested] 18+ messages in thread

* Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'?
  2019-10-27 17:22 ` Bas Spitters
@ 2019-11-03 11:38   ` Bas Spitters
  2019-11-03 11:52     ` David Roberts
  0 siblings, 1 reply; 18+ messages in thread
From: Bas Spitters @ 2019-11-03 11:38 UTC (permalink / raw)
  To: Nicolas Alexander Schmidt; +Cc: Homotopy Type Theory

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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.
>> 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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<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>
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^ permalink raw reply	[flat|nested] 18+ messages in thread

* Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'?
  2019-11-03 11:38   ` Bas Spitters
@ 2019-11-03 11:52     ` David Roberts
  2019-11-03 19:13       ` Michael Shulman
  0 siblings, 1 reply; 18+ messages in thread
From: David Roberts @ 2019-11-03 11:52 UTC (permalink / raw)
  To: Bas Spitters, homotopytypetheory; +Cc: Nicolas Alexander Schmidt

[-- Attachment #1: Type: text/plain, Size: 4939 bytes --]

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.
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>>> .
>>>
>> --
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<div dir="auto">Forget even higher category theory. Kevin Buzzard now goes around telling the story of how even formally proving <span style="font-family:sans-serif">(using Lean)</span> 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.<div dir="auto"><br></div><div dir="auto">David</div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Sun, 3 Nov 2019, 10:08 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:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><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" target="_blank" rel="noreferrer">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" target="_blank" rel="noreferrer">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" rel="noreferrer">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" rel="noreferrer">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 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>
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^ permalink raw reply	[flat|nested] 18+ messages in thread

* Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'?
  2019-11-03 11:52     ` David Roberts
@ 2019-11-03 19:13       ` Michael Shulman
  2019-11-03 19:45         ` Valery Isaev
  2019-11-04 18:42         ` Kevin Buzzard
  0 siblings, 2 replies; 18+ messages in thread
From: Michael Shulman @ 2019-11-03 19:13 UTC (permalink / raw)
  To: David Roberts; +Cc: Bas Spitters, homotopytypetheory, Nicolas Alexander Schmidt

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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>> To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com.
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>
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^ permalink raw reply	[flat|nested] 18+ messages in thread

* Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'?
  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 18:42         ` Kevin Buzzard
  1 sibling, 1 reply; 18+ messages in thread
From: Valery Isaev @ 2019-11-03 19:45 UTC (permalink / raw)
  To: Michael Shulman
  Cc: David Roberts, Bas Spitters, homotopytypetheory,
	Nicolas Alexander Schmidt

[-- Attachment #1: Type: text/plain, Size: 7975 bytes --]

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.
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> .
> >>
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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>
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&gt;&gt;<br>
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&gt;<br>
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</blockquote></div>

<p></p>

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^ permalink raw reply	[flat|nested] 18+ messages in thread

* Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'?
  2019-11-03 19:45         ` Valery Isaev
@ 2019-11-03 22:23           ` Martín Hötzel Escardó
  2019-11-04 23:20             ` Nicolas Alexander Schmidt
  0 siblings, 1 reply; 18+ messages in thread
From: Martín Hötzel Escardó @ 2019-11-03 22:23 UTC (permalink / raw)
  To: Homotopy Type Theory

[-- Attachment #1.1: Type: text/plain, Size: 8618 bytes --]

This discussion reminds me of this 
question: https://mathoverflow.net/questions/336191/cauchy-reals-and-dedekind-reals-satisfy-the-same-mathematical-theorems/336233?noredirect=1#comment840649_336233

M.

On Sunday, 3 November 2019 19:46:32 UTC, Valery Isaev wrote:
>
> 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 <shu...@sandiego.edu 
> <javascript:>>:
>
>> 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 <drober...@gmail.com 
>> <javascript:>> 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...@gmail.com 
>> <javascript:>> 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...@gmail.com 
>> <javascript:>> 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 <
>> ze...@fromzerotoinfinity.xyz <javascript:>> 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 
>> <javascript:>.
>> >>>> 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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>> .
>>
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<div dir="ltr">This discussion reminds me of this question: https://mathoverflow.net/questions/336191/cauchy-reals-and-dedekind-reals-satisfy-the-same-mathematical-theorems/336233?noredirect=1#comment840649_336233<div><br></div><div>M.<br><br>On Sunday, 3 November 2019 19:46:32 UTC, Valery Isaev  wrote:<blockquote class="gmail_quote" style="margin: 0;margin-left: 0.8ex;border-left: 1px #ccc solid;padding-left: 1ex;"><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"><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">вс, 3 нояб. 2019 г. в 22:13, Michael Shulman &lt;<a href="javascript:" target="_blank" gdf-obfuscated-mailto="OuRI0sBOAgAJ" rel="nofollow" onmousedown="this.href=&#39;javascript:&#39;;return true;" onclick="this.href=&#39;javascript:&#39;;return true;">shu...@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="javascript:" target="_blank" gdf-obfuscated-mailto="OuRI0sBOAgAJ" rel="nofollow" onmousedown="this.href=&#39;javascript:&#39;;return true;" onclick="this.href=&#39;javascript:&#39;;return true;">drober...@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="javascript:" target="_blank" gdf-obfuscated-mailto="OuRI0sBOAgAJ" rel="nofollow" onmousedown="this.href=&#39;javascript:&#39;;return true;" onclick="this.href=&#39;javascript:&#39;;return true;">b.a.w...@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="nofollow" target="_blank" onmousedown="this.href=&#39;https://www.google.com/url?q\x3dhttps%3A%2F%2Fwww.math.ias.edu%2F~vladimir%2FSite3%2FUnivalent_Foundations_files%2FHlambda_short_current.pdf\x26sa\x3dD\x26sntz\x3d1\x26usg\x3dAFQjCNGgR_wCvme0B7Sh5s7fCKpUG45TAg&#39;;return true;" onclick="this.href=&#39;https://www.google.com/url?q\x3dhttps%3A%2F%2Fwww.math.ias.edu%2F~vladimir%2FSite3%2FUnivalent_Foundations_files%2FHlambda_short_current.pdf\x26sa\x3dD\x26sntz\x3d1\x26usg\x3dAFQjCNGgR_wCvme0B7Sh5s7fCKpUG45TAg&#39;;return true;">https://www.math.ias.edu/~<wbr>vladimir/Site3/Univalent_<wbr>Foundations_files/Hlambda_<wbr>short_current.pdf</a><br>
&gt;&gt;<br>
&gt;&gt; On Sun, Oct 27, 2019 at 6:22 PM Bas Spitters &lt;<a href="javascript:" target="_blank" gdf-obfuscated-mailto="OuRI0sBOAgAJ" rel="nofollow" onmousedown="this.href=&#39;javascript:&#39;;return true;" onclick="this.href=&#39;javascript:&#39;;return true;">b.a.w...@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="nofollow" target="_blank" onmousedown="this.href=&#39;https://www.google.com/url?q\x3dhttps%3A%2F%2Fwww.ias.edu%2Fideas%2F2014%2Fvoevodsky-origins\x26sa\x3dD\x26sntz\x3d1\x26usg\x3dAFQjCNEAo7GxDHRN5NR9FukMwKHtnnvj0g&#39;;return true;" onclick="this.href=&#39;https://www.google.com/url?q\x3dhttps%3A%2F%2Fwww.ias.edu%2Fideas%2F2014%2Fvoevodsky-origins\x26sa\x3dD\x26sntz\x3d1\x26usg\x3dAFQjCNEAo7GxDHRN5NR9FukMwKHtnnvj0g&#39;;return true;">https://www.ias.edu/ideas/<wbr>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="javascript:" target="_blank" gdf-obfuscated-mailto="OuRI0sBOAgAJ" rel="nofollow" onmousedown="this.href=&#39;javascript:&#39;;return true;" onclick="this.href=&#39;javascript:&#39;;return true;">ze...@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="nofollow" target="_blank" onmousedown="this.href=&#39;https://www.youtube.com/watch?v\x3dzw6NcwME7yI\x26t\x3d1680&#39;;return true;" onclick="this.href=&#39;https://www.youtube.com/watch?v\x3dzw6NcwME7yI\x26t\x3d1680&#39;;return true;">https://www.youtube.<wbr>com/watch?v=zw6NcwME7yI&amp;t=1680</a><wbr>) 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>
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^ permalink raw reply	[flat|nested] 18+ messages in thread

* Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'?
  2019-11-03 19:13       ` Michael Shulman
  2019-11-03 19:45         ` Valery Isaev
@ 2019-11-04 18:42         ` Kevin Buzzard
  2019-11-04 21:10           ` Michael Shulman
                             ` (3 more replies)
  1 sibling, 4 replies; 18+ messages in thread
From: Kevin Buzzard @ 2019-11-04 18:42 UTC (permalink / raw)
  To: Michael Shulman
  Cc: David Roberts, Bas Spitters, homotopytypetheory,
	Nicolas Alexander Schmidt

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On Sun, 3 Nov 2019 at 19:13, Michael Shulman <shulman@sandiego.edu> wrote:

> But does univalence a la Book HoTT *actually* make it easier to reason
> about such things?


I think this is a really interesting and important question.

I guess David was referring to my scheme fail of 2018. I wanted to
formalise the notion of a scheme a la Grothendieck and prove that if R was
a commutative ring then Spec(R) was a scheme [I know it's a definition, but
many mathematicians do seem to call it a theorem, in our ignorance]. I
showed an undergraduate a specific lemma in ring theory (
https://stacks.math.columbia.edu/tag/00EJ) and said "that's what I want"
and they formalised it for me. And then it turned out that I wanted
something else: I didn't have R_f, I had something "canonically isomorphic"
to it, a phrase we mathematicians like to pull out when the going gets
tough and we can't be bothered to check that any more diagrams commute. By
this point it was too late to turn back, and so I had to prove that 20
diagrams commuted and it wasn't much fun. I then got an MSc student to redo
everything using universal properties more carefully in Lean and it worked
like a dream https://github.com/ramonfmir/lean-scheme. A lot of people said
to me at the time "you wouldn't have had this problem if you'd been using
HoTT instead of DTT" and my response to this is still the (intentionally)
provocative "go ahead and define schemes and prove that Spec(R) is a scheme
in some HoTT system, and show me how it's better; note that we did have a
problem, but we solved it in DTT". I would be particularly interested to
see schemes done in Arend, because it always felt funny to me using UniMath
in Coq (and similarly it feels funny to me to do HoTT in Lean 3 -- in both
cases it could be argued that it's using a system to do something it wasn't
designed to do). I think it's easy to theorise about this sort of thing but
until it happens in practice in one or more of the HoTT systems I don't
think we will understand the issue properly (or, more precisely, I don't
think I will understand the issue properly). I have had extensive
discussions with Martin Escardo about HoTT and he has certainly given me
hope, but on the Lean chat I think people assumed schemes would be easy in
Lean (I certainly did) and then we ran into this unexpected problem (which
univalence is probably designed to solve), so the question is whether a
univalent type theory runs into a different unexpected problem -- you push
the carpet down somewhere and it pops up somewhere else.

I know this is a HoTT list but the challenge is also open to the HOL people
like the Isabelle/HOL experts. In contrast to HoTT theories, which I think
should handle schemes fine, I think that simple type theory will have
tremendous problems defining, for example, tensor products of sheaves of
modules on a scheme, because these are dependent types. On the other hand
my recent ArXiv paper with Commelin and Massot
https://arxiv.org/abs/1910.12320 goes much further and formalises
perfectoid spaces in dependent type theory. I would like the people on this
list to see this as a challenge. I think that this century will see the
rise of the theorem prover in mathematics and I am not naive enough to
think that the one I currently use now is the one which is guaranteed to be
the success story. Voevodsky was convinced that univalence was the right
way to do modern mathematics but I'm doing it just fine in dependent type
theory and now he's gone I really want to find someone who will take up the
challenge and do some scheme theory in HoTT, but convincing professional
mathematicians to get interested in this area is very difficult, and I
speak as someone who's been trying to do it for two years now [I recommend
you try the undergraduates instead, anyone who is interested in training
people up -- plenty of undergraduates are capable of reading the definition
of a scheme, if they know what rings and topological spaces are]

To get back to the original question, my understanding was that Voevodsky
had done a bunch of scheme theory and it had got him a Fields medal and it
was this mathematics which he was interested in at the time. He wanted to
formalise his big theorem, just like Hales did. Unfortunately he was
historically earlier, and his mathematics involved far more conceptual
objects than spheres in 3-space, so it was a much taller order. All the
evidence is there to suggest that over the next 15 or so years his
interests changed. The clearest evidence, in my mind, is that there is no
definition of a scheme in UniMath. Moreover his story in his Cambridge talk
https://www.newton.ac.uk/seminar/20170710113012301 about asking Suslin to
reprove one of his results without using the axiom of choice (46 minutes
in) kind of shocked me -- Suslin does not care about mathematics without
choice, and the vast majority of mathematicians employed in mathematics
departments feel the same, although I'm well aware that constructivism is
taken more seriously on this list. I think it is interesting that Voevodsky
failed to prove a constructive version of his theorem, because I think that
some mathematics is better off not being constructive. It is exactly the
interaction between constructivism and univalence which I do not understand
well, and I think that a very good way to investigate it would be to do
some highly non-constructive modern mathematics in a univalent type theory.

Kevin

PS many thanks to the people who have emailed me in the past telling me
about how in the past I have used "HoTT", "univalence", "UniMath",
interchangeably and incorrectly. Hopefully I am getting better but I am
still keen to hear anything which I'm saying which is imprecise or
incorrect.



> 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.
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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 dir="ltr"><br></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Sun, 3 Nov 2019 at 19:13, Michael Shulman &lt;<a href="mailto:shulman@sandiego.edu" target="_blank">shulman@sandiego.edu</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">But does univalence a la Book HoTT *actually* make it easier to reason<br>
about such things?  </blockquote><div><br></div><div>I think this is a really interesting and important question.</div><div><br></div><div>I guess David was referring to my scheme fail of 2018. I wanted to formalise the notion of a scheme a la Grothendieck and prove that if R was a commutative ring then Spec(R) was a scheme [I know it&#39;s a definition, but many mathematicians do seem to call it a theorem, in our ignorance]. I showed an undergraduate a specific lemma in ring theory (<a href="https://stacks.math.columbia.edu/tag/00EJ" target="_blank">https://stacks.math.columbia.edu/tag/00EJ</a>) and said &quot;that&#39;s what I want&quot; and they formalised it for me. And then it turned out that I wanted something else: I didn&#39;t have R_f, I had something &quot;canonically isomorphic&quot; to it, a phrase we mathematicians like to pull out when the going gets tough and we can&#39;t be bothered to check that any more diagrams commute. By this point it was too late to turn back, and so I had to prove that 20 diagrams commuted and it wasn&#39;t much fun. I then got an MSc student to redo everything using universal properties more carefully in Lean and it worked like a dream <a href="https://github.com/ramonfmir/lean-scheme">https://github.com/ramonfmir/lean-scheme</a>. A lot of people said to me at the time &quot;you wouldn&#39;t have had this problem if you&#39;d been using HoTT instead of DTT&quot; and my response to this is still the (intentionally) provocative &quot;go ahead and define schemes and prove that Spec(R) is a scheme in some HoTT system, and show me how it&#39;s better; note that we did have a problem, but we solved it in DTT&quot;. I would be particularly interested to see schemes done in Arend, because it always felt funny to me using UniMath in Coq (and similarly it feels funny to me to do HoTT in Lean 3 -- in both cases it could be argued that it&#39;s using a system to do something it wasn&#39;t designed to do). I think it&#39;s easy to theorise about this sort of thing but until it happens in practice in one or more of the HoTT systems I don&#39;t think we will understand the issue properly (or, more precisely, I don&#39;t think I will understand the issue properly). I have had extensive discussions with Martin Escardo about HoTT and he has certainly given me hope, but on the Lean chat I think people assumed schemes would be easy in Lean (I certainly did) and then we ran into this unexpected problem (which univalence is probably designed to solve), so the question is whether a univalent type theory runs into a different unexpected problem -- you push the carpet down somewhere and it pops up somewhere else.<br></div><div><br></div><div>I know this is a HoTT list but the challenge is also open to the HOL people like the Isabelle/HOL experts. In contrast to HoTT theories, which I think should handle schemes fine, I think that simple type theory will have tremendous problems defining, for example, tensor products of sheaves of modules on a scheme, because these are dependent types. On the other hand my recent ArXiv paper with Commelin and Massot <a href="https://arxiv.org/abs/1910.12320">https://arxiv.org/abs/1910.12320</a> goes much further and formalises perfectoid spaces in dependent type theory. I would like the people on this list to see this as a challenge. I think that this century will see the rise of the theorem prover in mathematics and I am not naive enough to think that the one I currently use now is the one which is guaranteed to be the success story. Voevodsky was convinced that univalence was the right way to do modern mathematics but I&#39;m doing it just fine in dependent type theory and now he&#39;s gone I really want to find someone who will take up the challenge and do some scheme theory in HoTT, but convincing professional mathematicians to get interested in this area is very difficult, and I speak as someone who&#39;s been trying to do it for two years now [I recommend you try the undergraduates instead, anyone who is interested in training people up -- plenty of undergraduates are capable of reading the definition of a scheme, if they know what rings and topological spaces are]<br></div><div><br></div><div>To get back to the original question, my understanding was that Voevodsky had done a bunch of scheme theory and it had got him a Fields medal and it was this mathematics which he was interested in at the time. He wanted to formalise his big theorem, just like Hales did. Unfortunately he was historically earlier, and his mathematics involved far more conceptual objects than spheres in 3-space, so it was a much taller order. All the evidence is there to suggest that over the next 15 or so years his interests changed. The clearest evidence, in my mind, is that there is no definition of a scheme in UniMath. Moreover his story in his Cambridge talk <a href="https://www.newton.ac.uk/seminar/20170710113012301">https://www.newton.ac.uk/seminar/20170710113012301</a> about asking Suslin to reprove one of his results without using the axiom of choice (46 minutes in) kind of shocked me -- Suslin does not care about mathematics without choice, and the vast majority of mathematicians employed in mathematics departments feel the same, although I&#39;m well aware that constructivism is taken more seriously on this list. I think it is interesting that Voevodsky failed to prove a constructive version of his theorem, because I think that some mathematics is better off not being constructive. It is exactly the interaction between constructivism and univalence which I do not understand well, and I think that a very good way to investigate it would be to do some highly non-constructive modern mathematics in a univalent type theory.<br></div><div><br></div><div>Kevin</div><div><br></div><div>PS many thanks to the people who have emailed me in the past telling me about how in the past I have used &quot;HoTT&quot;, &quot;univalence&quot;, &quot;UniMath&quot;, interchangeably and incorrectly. Hopefully I am getting better but I am still keen to hear anything which I&#39;m saying which is imprecise or incorrect.<br></div><div><br></div><div> </div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">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>
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</blockquote></div></div>

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^ permalink raw reply	[flat|nested] 18+ messages in thread

* Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'?
  2019-11-04 18:42         ` Kevin Buzzard
@ 2019-11-04 21:10           ` Michael Shulman
  2019-11-04 23:26           ` David Roberts
                             ` (2 subsequent siblings)
  3 siblings, 0 replies; 18+ messages in thread
From: Michael Shulman @ 2019-11-04 21:10 UTC (permalink / raw)
  To: Kevin Buzzard
  Cc: David Roberts, Bas Spitters, homotopytypetheory,
	Nicolas Alexander Schmidt

Valery has a good point that transporting properties along an
equivalence is definitely somewhere that Book HoTT could get you some
mileage.  But I suspect that a more significant advantage would come
from using a cubical type theory in which transport computes (as long
as the equivalence was defined constructively -- a good reason to care
about being constructive even putting aside philosophy and internal
languages).  I'd be curious to know what those 20 diagrams were.

FWIW, I think that nowadays Coq *is* designed for Book HoTT, certainly
more than Lean 3 is.  My understanding is that Lean 3 is actually
technically incompatible with univalence, whereas over the past decade
the Coq developers have incorporated various new features requested by
the HoTT community to improve compatibility, and the HoTT Coq library
is I believe one of the test suites that new Coq versions are tested
against to ensure that breakage is dealt with on one side or another.
I'm not sure how a proof assistant could be more designed for Book
HoTT than modern Coq and Agda are.  (Arend is not designed for Book
HoTT, but for a flavor of HoTT that's partway to a cubical theory,
with an interval type representing paths.)

On Mon, Nov 4, 2019 at 10:43 AM Kevin Buzzard <kevin.m.buzzard@gmail.com> wrote:
>
>
>
> On Sun, 3 Nov 2019 at 19:13, Michael Shulman <shulman@sandiego.edu> wrote:
>>
>> But does univalence a la Book HoTT *actually* make it easier to reason
>> about such things?
>
>
> I think this is a really interesting and important question.
>
> I guess David was referring to my scheme fail of 2018. I wanted to formalise the notion of a scheme a la Grothendieck and prove that if R was a commutative ring then Spec(R) was a scheme [I know it's a definition, but many mathematicians do seem to call it a theorem, in our ignorance]. I showed an undergraduate a specific lemma in ring theory (https://stacks.math.columbia.edu/tag/00EJ) and said "that's what I want" and they formalised it for me. And then it turned out that I wanted something else: I didn't have R_f, I had something "canonically isomorphic" to it, a phrase we mathematicians like to pull out when the going gets tough and we can't be bothered to check that any more diagrams commute. By this point it was too late to turn back, and so I had to prove that 20 diagrams commuted and it wasn't much fun. I then got an MSc student to redo everything using universal properties more carefully in Lean and it worked like a dream https://github.com/ramonfmir/lean-scheme. A lot of people said to me at the time "you wouldn't have had this problem if you'd been using HoTT instead of DTT" and my response to this is still the (intentionally) provocative "go ahead and define schemes and prove that Spec(R) is a scheme in some HoTT system, and show me how it's better; note that we did have a problem, but we solved it in DTT". I would be particularly interested to see schemes done in Arend, because it always felt funny to me using UniMath in Coq (and similarly it feels funny to me to do HoTT in Lean 3 -- in both cases it could be argued that it's using a system to do something it wasn't designed to do). I think it's easy to theorise about this sort of thing but until it happens in practice in one or more of the HoTT systems I don't think we will understand the issue properly (or, more precisely, I don't think I will understand the issue properly). I have had extensive discussions with Martin Escardo about HoTT and he has certainly given me hope, but on the Lean chat I think people assumed schemes would be easy in Lean (I certainly did) and then we ran into this unexpected problem (which univalence is probably designed to solve), so the question is whether a univalent type theory runs into a different unexpected problem -- you push the carpet down somewhere and it pops up somewhere else.
>
> I know this is a HoTT list but the challenge is also open to the HOL people like the Isabelle/HOL experts. In contrast to HoTT theories, which I think should handle schemes fine, I think that simple type theory will have tremendous problems defining, for example, tensor products of sheaves of modules on a scheme, because these are dependent types. On the other hand my recent ArXiv paper with Commelin and Massot https://arxiv.org/abs/1910.12320 goes much further and formalises perfectoid spaces in dependent type theory. I would like the people on this list to see this as a challenge. I think that this century will see the rise of the theorem prover in mathematics and I am not naive enough to think that the one I currently use now is the one which is guaranteed to be the success story. Voevodsky was convinced that univalence was the right way to do modern mathematics but I'm doing it just fine in dependent type theory and now he's gone I really want to find someone who will take up the challenge and do some scheme theory in HoTT, but convincing professional mathematicians to get interested in this area is very difficult, and I speak as someone who's been trying to do it for two years now [I recommend you try the undergraduates instead, anyone who is interested in training people up -- plenty of undergraduates are capable of reading the definition of a scheme, if they know what rings and topological spaces are]
>
> To get back to the original question, my understanding was that Voevodsky had done a bunch of scheme theory and it had got him a Fields medal and it was this mathematics which he was interested in at the time. He wanted to formalise his big theorem, just like Hales did. Unfortunately he was historically earlier, and his mathematics involved far more conceptual objects than spheres in 3-space, so it was a much taller order. All the evidence is there to suggest that over the next 15 or so years his interests changed. The clearest evidence, in my mind, is that there is no definition of a scheme in UniMath. Moreover his story in his Cambridge talk https://www.newton.ac.uk/seminar/20170710113012301 about asking Suslin to reprove one of his results without using the axiom of choice (46 minutes in) kind of shocked me -- Suslin does not care about mathematics without choice, and the vast majority of mathematicians employed in mathematics departments feel the same, although I'm well aware that constructivism is taken more seriously on this list. I think it is interesting that Voevodsky failed to prove a constructive version of his theorem, because I think that some mathematics is better off not being constructive. It is exactly the interaction between constructivism and univalence which I do not understand well, and I think that a very good way to investigate it would be to do some highly non-constructive modern mathematics in a univalent type theory.
>
> Kevin
>
> PS many thanks to the people who have emailed me in the past telling me about how in the past I have used "HoTT", "univalence", "UniMath", interchangeably and incorrectly. Hopefully I am getting better but I am still keen to hear anything which I'm saying which is imprecise or incorrect.
>
>
>>
>> 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.
>> >>
>> >> --
>> >> 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.
>> >
>> > --
>> > 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/CAFL%2BZM_%3D%3DiLS16Vy7sGiEqNkCxOMYL4j%2BZFqKv5uJ-ivkuemEg%40mail.gmail.com.
>>
>> --
>> 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/CAOvivQz47kSm9WbKDmUsndrpAJMkNwiWmVABqOFrVqyTOvSAbw%40mail.gmail.com.
>
> --
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^ permalink raw reply	[flat|nested] 18+ messages in thread

* Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'?
  2019-11-03 22:23           ` Martín Hötzel Escardó
@ 2019-11-04 23:20             ` Nicolas Alexander Schmidt
  0 siblings, 0 replies; 18+ messages in thread
From: Nicolas Alexander Schmidt @ 2019-11-04 23:20 UTC (permalink / raw)
  To: HomotopyTypeTheory

Dear Bas, Michael, David, Valery, Martín and Kevin,


thank you all for your replies. When I posed my question, I didn't
expect there to be so much room for debate. In particular, I am
surprised that it seems not at all clear how much univalence actually
helps the practicality of doing formal proofs (as opposed to any
theoretical benefits).

Kevin, I would like to second Michael's interest in seeing these 20
commutative diagrams. Moreover, I'd also be very interested in seeing
your "spaghetti code" (quote from the slides of your Big Proof talk): it
seems it should be informative to see where your initial approach went
wrong, and how much these problems and their solution had anything to do
at all with the formal system you were working in. Are your original
files perhaps available somewhere?


--

Nicolas




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^ permalink raw reply	[flat|nested] 18+ messages in thread

* Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'?
  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 23:14           ` Martín Hötzel Escardó
  3 siblings, 0 replies; 18+ messages in thread
From: David Roberts @ 2019-11-04 23:26 UTC (permalink / raw)
  To: Kevin Buzzard
  Cc: Michael Shulman, Bas Spitters, homotopytypetheory,
	Nicolas Alexander Schmidt

Hi,

Going back a step from DTT vs HoTT...

> I think that simple type theory will have tremendous problems defining, for example, tensor products of sheaves of modules on a scheme, because these are dependent types.

this is probably where VV found himself just under 20 years ago. I
don't know what the state of play was around 2000, but certainly if he
didn't know about Coq, then VV's options for formal proof systems were
looking more like Mizar, MetaMath and similar, which are probably even
worse than simple type-theory systems: imagine trying to formally
define a scheme in the language of ZFC!

Best regards,
David


David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts
Blog: https://thehighergeometer.wordpress.com

On Tue, 5 Nov 2019 at 05:13, Kevin Buzzard <kevin.m.buzzard@gmail.com> wrote:
>
>
>
> On Sun, 3 Nov 2019 at 19:13, Michael Shulman <shulman@sandiego.edu> wrote:
>>
>> But does univalence a la Book HoTT *actually* make it easier to reason
>> about such things?
>
>
> I think this is a really interesting and important question.
>
> I guess David was referring to my scheme fail of 2018. I wanted to formalise the notion of a scheme a la Grothendieck and prove that if R was a commutative ring then Spec(R) was a scheme [I know it's a definition, but many mathematicians do seem to call it a theorem, in our ignorance]. I showed an undergraduate a specific lemma in ring theory (https://stacks.math.columbia.edu/tag/00EJ) and said "that's what I want" and they formalised it for me. And then it turned out that I wanted something else: I didn't have R_f, I had something "canonically isomorphic" to it, a phrase we mathematicians like to pull out when the going gets tough and we can't be bothered to check that any more diagrams commute. By this point it was too late to turn back, and so I had to prove that 20 diagrams commuted and it wasn't much fun. I then got an MSc student to redo everything using universal properties more carefully in Lean and it worked like a dream https://github.com/ramonfmir/lean-scheme. A lot of people said to me at the time "you wouldn't have had this problem if you'd been using HoTT instead of DTT" and my response to this is still the (intentionally) provocative "go ahead and define schemes and prove that Spec(R) is a scheme in some HoTT system, and show me how it's better; note that we did have a problem, but we solved it in DTT". I would be particularly interested to see schemes done in Arend, because it always felt funny to me using UniMath in Coq (and similarly it feels funny to me to do HoTT in Lean 3 -- in both cases it could be argued that it's using a system to do something it wasn't designed to do). I think it's easy to theorise about this sort of thing but until it happens in practice in one or more of the HoTT systems I don't think we will understand the issue properly (or, more precisely, I don't think I will understand the issue properly). I have had extensive discussions with Martin Escardo about HoTT and he has certainly given me hope, but on the Lean chat I think people assumed schemes would be easy in Lean (I certainly did) and then we ran into this unexpected problem (which univalence is probably designed to solve), so the question is whether a univalent type theory runs into a different unexpected problem -- you push the carpet down somewhere and it pops up somewhere else.
>
> I know this is a HoTT list but the challenge is also open to the HOL people like the Isabelle/HOL experts. In contrast to HoTT theories, which I think should handle schemes fine, I think that simple type theory will have tremendous problems defining, for example, tensor products of sheaves of modules on a scheme, because these are dependent types. On the other hand my recent ArXiv paper with Commelin and Massot https://arxiv.org/abs/1910.12320 goes much further and formalises perfectoid spaces in dependent type theory. I would like the people on this list to see this as a challenge. I think that this century will see the rise of the theorem prover in mathematics and I am not naive enough to think that the one I currently use now is the one which is guaranteed to be the success story. Voevodsky was convinced that univalence was the right way to do modern mathematics but I'm doing it just fine in dependent type theory and now he's gone I really want to find someone who will take up the challenge and do some scheme theory in HoTT, but convincing professional mathematicians to get interested in this area is very difficult, and I speak as someone who's been trying to do it for two years now [I recommend you try the undergraduates instead, anyone who is interested in training people up -- plenty of undergraduates are capable of reading the definition of a scheme, if they know what rings and topological spaces are]
>
> To get back to the original question, my understanding was that Voevodsky had done a bunch of scheme theory and it had got him a Fields medal and it was this mathematics which he was interested in at the time. He wanted to formalise his big theorem, just like Hales did. Unfortunately he was historically earlier, and his mathematics involved far more conceptual objects than spheres in 3-space, so it was a much taller order. All the evidence is there to suggest that over the next 15 or so years his interests changed. The clearest evidence, in my mind, is that there is no definition of a scheme in UniMath. Moreover his story in his Cambridge talk https://www.newton.ac.uk/seminar/20170710113012301 about asking Suslin to reprove one of his results without using the axiom of choice (46 minutes in) kind of shocked me -- Suslin does not care about mathematics without choice, and the vast majority of mathematicians employed in mathematics departments feel the same, although I'm well aware that constructivism is taken more seriously on this list. I think it is interesting that Voevodsky failed to prove a constructive version of his theorem, because I think that some mathematics is better off not being constructive. It is exactly the interaction between constructivism and univalence which I do not understand well, and I think that a very good way to investigate it would be to do some highly non-constructive modern mathematics in a univalent type theory.
>
> Kevin
>
> PS many thanks to the people who have emailed me in the past telling me about how in the past I have used "HoTT", "univalence", "UniMath", interchangeably and incorrectly. Hopefully I am getting better but I am still keen to hear anything which I'm saying which is imprecise or incorrect.
>
>
>>
>> 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.
>> >>
>> >> --
>> >> 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.
>> >
>> > --
>> > 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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>>
>> --
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^ permalink raw reply	[flat|nested] 18+ messages in thread

* Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'?
  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
  2019-11-05 23:14           ` Martín Hötzel Escardó
  3 siblings, 1 reply; 18+ messages in thread
From: Daniel R. Grayson @ 2019-11-05 15:43 UTC (permalink / raw)
  To: Homotopy Type Theory

[-- Attachment #1.1: Type: text/plain, Size: 1554 bytes --]

Re: "To get back to the original question, my understanding was that 
Voevodsky had done a bunch of scheme theory and it had got him a Fields 
medal and it was this mathematics which he was interested in at the time. 
He wanted to formalise his big theorem, just like Hales did."

I think he was more interested in formalizing things like his early work 
with Kapranov on higher categories, which turned out to have a mistake in 
it.  He once told me that he wasn't interested in formalizing his proof of 
Bloch-Kato, because he was sure it was right.  (I should have asked him at 
the time how he could be so sure!)

Re: "The clearest evidence, in my mind, is that there is no definition of a 
scheme in UniMath."

That's sort of accidental.   In early 2014, expecting to speak at an 
algebraic geometry in the summer, he mentioned that one idea he had for his 
talk would be to formalize the definition of scheme in UniMath and speak 
about it.  I think he was distracted from that by thinking about 
C-systems.  The UniMath project aims at formalizing all of mathematics, so 
the definition of scheme is one of the next things that (still) needs to be 
done.

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<div dir="ltr">Re: &quot;To get back to the original question, my understanding was that Voevodsky had done a bunch of scheme theory and it had got him a Fields medal and it was this mathematics which he was interested in at the time. He wanted to formalise his big theorem, just like Hales did.&quot;<br><div><br></div><div>I think he was more interested in formalizing things like his early work with Kapranov on higher categories, which turned out to have a mistake in it.  He once told me that he wasn&#39;t interested in formalizing his proof of Bloch-Kato, because he was sure it was right.  (I should have asked him at the time how he could be so sure!)<br><br>Re: &quot;The clearest evidence, in my mind, is that there is no definition of a scheme in UniMath.&quot;<br><br>That&#39;s sort of accidental.   In early 2014, expecting to speak at an algebraic geometry in the summer, he mentioned that one idea he had for his talk would be to formalize the definition of scheme in UniMath and speak about it.  I think he was distracted from that by thinking about C-systems.  The UniMath project aims at formalizing all of mathematics, so the definition of scheme is one of the next things that (still) needs to be done.</div></div>

<p></p>

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^ permalink raw reply	[flat|nested] 18+ messages in thread

* Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'?
  2019-11-05 15:43           ` Daniel R. Grayson
@ 2019-11-05 20:29             ` Yuhao Huang
  2019-11-06 23:59               ` Daniel R. Grayson
  0 siblings, 1 reply; 18+ messages in thread
From: Yuhao Huang @ 2019-11-05 20:29 UTC (permalink / raw)
  To: Homotopy Type Theory

[-- Attachment #1.1: Type: text/plain, Size: 4927 bytes --]


>
> He once told me that he wasn't interested in formalizing his proof of 
> Bloch-Kato, because he was sure it was right.  (I should have asked him at 
> the time how he could be so sure!)
>

Oh this is interesting... do you remember when this conversation was 
happening? Because in these slides (
https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2014_09_Bernays_3%20presentation.pdf) 
he said "Next year I am starting a project of univalent formalization of my 
proof of Milnor’s Conjecture using this formalization of set theory as the 
starting point." (Page 11)

在 2019年11月5日星期二 UTC-8上午7:43:06,Daniel R. Grayson写道:
>
> Re: "To get back to the original question, my understanding was that 
> Voevodsky had done a bunch of scheme theory and it had got him a Fields 
> medal and it was this mathematics which he was interested in at the time. 
> He wanted to formalise his big theorem, just like Hales did."
>
> I think he was more interested in formalizing things like his early work 
> with Kapranov on higher categories, which turned out to have a mistake in 
> it.  He once told me that he wasn't interested in formalizing his proof of 
> Bloch-Kato, because he was sure it was right.  (I should have asked him at 
> the time how he could be so sure!)
>
> Re: "The clearest evidence, in my mind, is that there is no definition of 
> a scheme in UniMath."
>
> That's sort of accidental.   In early 2014, expecting to speak at an 
> algebraic geometry in the summer, he mentioned that one idea he had for his 
> talk would be to formalize the definition of scheme in UniMath and speak 
> about it.  I think he was distracted from that by thinking about 
> C-systems.  The UniMath project aims at formalizing all of mathematics, so 
> the definition of scheme is one of the next things that (still) needs to be 
> done.
>

在 2019年11月5日星期二 UTC-8上午7:43:06,Daniel R. Grayson写道:
>
> Re: "To get back to the original question, my understanding was that 
> Voevodsky had done a bunch of scheme theory and it had got him a Fields 
> medal and it was this mathematics which he was interested in at the time. 
> He wanted to formalise his big theorem, just like Hales did."
>
> I think he was more interested in formalizing things like his early work 
> with Kapranov on higher categories, which turned out to have a mistake in 
> it.  He once told me that he wasn't interested in formalizing his proof of 
> Bloch-Kato, because he was sure it was right.  (I should have asked him at 
> the time how he could be so sure!)
>
> Re: "The clearest evidence, in my mind, is that there is no definition of 
> a scheme in UniMath."
>
> That's sort of accidental.   In early 2014, expecting to speak at an 
> algebraic geometry in the summer, he mentioned that one idea he had for his 
> talk would be to formalize the definition of scheme in UniMath and speak 
> about it.  I think he was distracted from that by thinking about 
> C-systems.  The UniMath project aims at formalizing all of mathematics, so 
> the definition of scheme is one of the next things that (still) needs to be 
> done.
>

在 2019年11月5日星期二 UTC-8上午7:43:06,Daniel R. Grayson写道:
>
> Re: "To get back to the original question, my understanding was that 
> Voevodsky had done a bunch of scheme theory and it had got him a Fields 
> medal and it was this mathematics which he was interested in at the time. 
> He wanted to formalise his big theorem, just like Hales did."
>
> I think he was more interested in formalizing things like his early work 
> with Kapranov on higher categories, which turned out to have a mistake in 
> it.  He once told me that he wasn't interested in formalizing his proof of 
> Bloch-Kato, because he was sure it was right.  (I should have asked him at 
> the time how he could be so sure!)
>
> Re: "The clearest evidence, in my mind, is that there is no definition of 
> a scheme in UniMath."
>
> That's sort of accidental.   In early 2014, expecting to speak at an 
> algebraic geometry in the summer, he mentioned that one idea he had for his 
> talk would be to formalize the definition of scheme in UniMath and speak 
> about it.  I think he was distracted from that by thinking about 
> C-systems.  The UniMath project aims at formalizing all of mathematics, so 
> the definition of scheme is one of the next things that (still) needs to be 
> done.
>

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<div dir="ltr"><blockquote class="gmail_quote" style="margin: 0px 0px 0px 0.8ex; border-left: 1px solid rgb(204, 204, 204); padding-left: 1ex;">He once told me that he wasn&#39;t interested in formalizing his proof of 
Bloch-Kato, because he was sure it was right.  (I should have asked him 
at the time how he could be so sure!)<br></blockquote><div><br></div><div>Oh this is interesting... do you remember when this conversation was happening? Because in these slides (<a href="https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2014_09_Bernays_3%20presentation.pdf">https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2014_09_Bernays_3%20presentation.pdf</a>) he said &quot;Next year I am starting a project of univalent formalization of my proof of Milnor’s Conjecture using this formalization of set theory as the starting point.&quot; (Page 11)<br></div><div> <br></div>在 2019年11月5日星期二 UTC-8上午7:43:06,Daniel R. Grayson写道:<blockquote class="gmail_quote" style="margin: 0;margin-left: 0.8ex;border-left: 1px #ccc solid;padding-left: 1ex;"><div dir="ltr">Re: &quot;To get back to the original question, my understanding was that Voevodsky had done a bunch of scheme theory and it had got him a Fields medal and it was this mathematics which he was interested in at the time. He wanted to formalise his big theorem, just like Hales did.&quot;<br><div><br></div><div>I think he was more interested in formalizing things like his early work with Kapranov on higher categories, which turned out to have a mistake in it.  He once told me that he wasn&#39;t interested in formalizing his proof of Bloch-Kato, because he was sure it was right.  (I should have asked him at the time how he could be so sure!)<br><br>Re: &quot;The clearest evidence, in my mind, is that there is no definition of a scheme in UniMath.&quot;<br><br>That&#39;s sort of accidental.   In early 2014, expecting to speak at an algebraic geometry in the summer, he mentioned that one idea he had for his talk would be to formalize the definition of scheme in UniMath and speak about it.  I think he was distracted from that by thinking about C-systems.  The UniMath project aims at formalizing all of mathematics, so the definition of scheme is one of the next things that (still) needs to be done.</div></div></blockquote><br>在 2019年11月5日星期二 UTC-8上午7:43:06,Daniel R. Grayson写道:<blockquote class="gmail_quote" style="margin: 0;margin-left: 0.8ex;border-left: 1px #ccc solid;padding-left: 1ex;"><div dir="ltr">Re: &quot;To get back to the original question, my understanding was that Voevodsky had done a bunch of scheme theory and it had got him a Fields medal and it was this mathematics which he was interested in at the time. He wanted to formalise his big theorem, just like Hales did.&quot;<br><div><br></div><div>I think he was more interested in formalizing things like his early work with Kapranov on higher categories, which turned out to have a mistake in it.  He once told me that he wasn&#39;t interested in formalizing his proof of Bloch-Kato, because he was sure it was right.  (I should have asked him at the time how he could be so sure!)<br><br>Re: &quot;The clearest evidence, in my mind, is that there is no definition of a scheme in UniMath.&quot;<br><br>That&#39;s sort of accidental.   In early 2014, expecting to speak at an algebraic geometry in the summer, he mentioned that one idea he had for his talk would be to formalize the definition of scheme in UniMath and speak about it.  I think he was distracted from that by thinking about C-systems.  The UniMath project aims at formalizing all of mathematics, so the definition of scheme is one of the next things that (still) needs to be done.</div></div></blockquote><br>在 2019年11月5日星期二 UTC-8上午7:43:06,Daniel R. Grayson写道:<blockquote class="gmail_quote" style="margin: 0;margin-left: 0.8ex;border-left: 1px #ccc solid;padding-left: 1ex;"><div dir="ltr">Re: &quot;To get back to the original question, my understanding was that Voevodsky had done a bunch of scheme theory and it had got him a Fields medal and it was this mathematics which he was interested in at the time. He wanted to formalise his big theorem, just like Hales did.&quot;<br><div><br></div><div>I think he was more interested in formalizing things like his early work with Kapranov on higher categories, which turned out to have a mistake in it.  He once told me that he wasn&#39;t interested in formalizing his proof of Bloch-Kato, because he was sure it was right.  (I should have asked him at the time how he could be so sure!)<br><br>Re: &quot;The clearest evidence, in my mind, is that there is no definition of a scheme in UniMath.&quot;<br><br>That&#39;s sort of accidental.   In early 2014, expecting to speak at an algebraic geometry in the summer, he mentioned that one idea he had for his talk would be to formalize the definition of scheme in UniMath and speak about it.  I think he was distracted from that by thinking about C-systems.  The UniMath project aims at formalizing all of mathematics, so the definition of scheme is one of the next things that (still) needs to be done.</div></div></blockquote></div>

<p></p>

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^ permalink raw reply	[flat|nested] 18+ messages in thread

* Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'?
  2019-11-04 18:42         ` Kevin Buzzard
                             ` (2 preceding siblings ...)
  2019-11-05 15:43           ` Daniel R. Grayson
@ 2019-11-05 23:14           ` Martín Hötzel Escardó
  2019-11-06  0:06             ` Stefan Monnier
  3 siblings, 1 reply; 18+ messages in thread
From: Martín Hötzel Escardó @ 2019-11-05 23:14 UTC (permalink / raw)
  To: Homotopy Type Theory

[-- Attachment #1.1: Type: text/plain, Size: 1901 bytes --]



On Monday, 4 November 2019 18:43:08 UTC, Kevin Buzzard wrote:
>
>  It is exactly the interaction between constructivism and univalence which 
> I do not understand well, and I think that a very good way to investigate 
> it would be to do some highly non-constructive modern mathematics in a 
> univalent type theory
>

Regarding *old* mathematics, you have the well-ordering principle proved in 
UniMath (from the axiom of choice, of course). 

Regarding your doubt about the interaction, we have that univalence is 
orthogonal to constructivism. 

In fact, univalence is not *inherently* constructive. It was hard work to 
find a constructive interpretation of univalence (which happens to rely on 
cubical sets as in homotopy theory). In particular (even if I lam fond of 
constructive mathematics, as you know), I work with univalence 
axiomatically, as a black box, rather than as a construction, in my (formal 
and informal) mathematical developments. And I do prefer to work with 
univalence-as-a-specification rather than univalence-as-a-construction.

There is nothing inherently constructive about univalence. There is no a 
priori interaction between univalence and constructivism. There is only an 
a posteriori interaction, constructed by some of the constructively minded 
members of this list. The constructivity of univalence was an open problem 
for a number of year, and I would say that, even if it is solved via the 
cubical model, it is far from being fully understood. 

Best,
Martin


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<div dir="ltr"><br><br>On Monday, 4 November 2019 18:43:08 UTC, Kevin Buzzard  wrote:<blockquote class="gmail_quote" style="margin: 0;margin-left: 0.8ex;border-left: 1px #ccc solid;padding-left: 1ex;"><div dir="ltr"><div dir="ltr"> It is exactly the interaction between constructivism and univalence which I do not understand well, and I think that a very good way to investigate it would be to do some highly non-constructive modern mathematics in a univalent type theory</div></div></blockquote><div><br></div><div>Regarding *old* mathematics, you have the well-ordering principle proved in UniMath (from the axiom of choice, of course). </div><div><br></div><div>Regarding your doubt about the interaction, we have that univalence is orthogonal to constructivism. </div><div><br></div><div>In fact, univalence is not *inherently* constructive. It was hard work to find a constructive interpretation of univalence (which happens to rely on cubical sets as in homotopy theory). In particular (even if I lam fond of constructive mathematics, as you know), I work with univalence axiomatically, as a black box, rather than as a construction, in my (formal and informal) mathematical developments. And I do prefer to work with univalence-as-a-specification rather than univalence-as-a-construction.</div><div><br></div><div>There is nothing inherently constructive about univalence. There is no a priori interaction between univalence and constructivism. There is only an a posteriori interaction, constructed by some of the constructively minded members of this list. The constructivity of univalence was an open problem for a number of year, and I would say that, even if it is solved via the cubical model, it is far from being fully understood. </div><div><br></div><div>Best,</div><div>Martin</div><div><br></div><div><br></div></div>

<p></p>

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^ permalink raw reply	[flat|nested] 18+ messages in thread

* Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'?
  2019-11-05 23:14           ` Martín Hötzel Escardó
@ 2019-11-06  0:06             ` Stefan Monnier
  2019-11-11 18:26               ` Licata, Dan
  0 siblings, 1 reply; 18+ messages in thread
From: Stefan Monnier @ 2019-11-06  0:06 UTC (permalink / raw)
  To: Martín Hötzel Escardó; +Cc: Homotopy Type Theory

> members of this list. The constructivity of univalence was an open problem 
> for a number of year, and I would say that, even if it is solved via the 
> cubical model, it is far from being fully understood.

In my case, I still find it odd in a situation such as:

    data Foo : Set -> Set where
      bar : Foo UnaryNat

since transport supposedly allows us to take a proof of equivalence
between UnaryNat and BinaryNat and turn a `bar` into something of type
`Foo BinaryNat` although I can't see any way to directly construct an
object of this type.


        Stefan


^ permalink raw reply	[flat|nested] 18+ messages in thread

* Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'?
  2019-11-05 20:29             ` Yuhao Huang
@ 2019-11-06 23:59               ` Daniel R. Grayson
  0 siblings, 0 replies; 18+ messages in thread
From: Daniel R. Grayson @ 2019-11-06 23:59 UTC (permalink / raw)
  To: Homotopy Type Theory

[-- Attachment #1.1: Type: text/plain, Size: 1139 bytes --]

Sorry, at this point I don't remember precisely.

On Tuesday, November 5, 2019 at 2:29:56 PM UTC-6, Yuhao Huang wrote:
>
> He once told me that he wasn't interested in formalizing his proof of 
>> Bloch-Kato, because he was sure it was right.  (I should have asked him at 
>> the time how he could be so sure!)
>>
>
> Oh this is interesting... do you remember when this conversation was 
> happening? Because in these slides (
> https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2014_09_Bernays_3%20presentation.pdf) 
> he said "Next year I am starting a project of univalent formalization of my 
> proof of Milnor’s Conjecture using this formalization of set theory as the 
> starting point." (Page 11)
>

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<div dir="ltr">Sorry, at this point I don&#39;t remember precisely.<br><br>On Tuesday, November 5, 2019 at 2:29:56 PM UTC-6, Yuhao Huang wrote:<blockquote class="gmail_quote" style="margin: 0;margin-left: 0.8ex;border-left: 1px #ccc solid;padding-left: 1ex;"><div dir="ltr"><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">He once told me that he wasn&#39;t interested in formalizing his proof of 
Bloch-Kato, because he was sure it was right.  (I should have asked him 
at the time how he could be so sure!)<br></blockquote><div><br></div><div>Oh this is interesting... do you remember when this conversation was happening? Because in these slides (<a href="https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2014_09_Bernays_3%20presentation.pdf" target="_blank" rel="nofollow" onmousedown="this.href=&#39;https://www.google.com/url?q\x3dhttps%3A%2F%2Fwww.math.ias.edu%2Fvladimir%2Fsites%2Fmath.ias.edu.vladimir%2Ffiles%2F2014_09_Bernays_3%2520presentation.pdf\x26sa\x3dD\x26sntz\x3d1\x26usg\x3dAFQjCNEc51jJXwQodsO-qY-qDVfkTXkyCw&#39;;return true;" onclick="this.href=&#39;https://www.google.com/url?q\x3dhttps%3A%2F%2Fwww.math.ias.edu%2Fvladimir%2Fsites%2Fmath.ias.edu.vladimir%2Ffiles%2F2014_09_Bernays_3%2520presentation.pdf\x26sa\x3dD\x26sntz\x3d1\x26usg\x3dAFQjCNEc51jJXwQodsO-qY-qDVfkTXkyCw&#39;;return true;">https://www.math.ias.edu/<wbr>vladimir/sites/math.ias.edu.<wbr>vladimir/files/2014_09_<wbr>Bernays_3%20presentation.pdf</a>) he said &quot;Next year I am starting a project of univalent formalization of my proof of Milnor’s Conjecture using this formalization of set theory as the starting point.&quot; (Page 11)</div></div></blockquote></div>

<p></p>

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^ permalink raw reply	[flat|nested] 18+ messages in thread

* Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'?
  2019-11-06  0:06             ` Stefan Monnier
@ 2019-11-11 18:26               ` Licata, Dan
  0 siblings, 0 replies; 18+ messages in thread
From: Licata, Dan @ 2019-11-11 18:26 UTC (permalink / raw)
  To: Stefan Monnier; +Cc: Martín Hötzel Escardó, Homotopy Type Theory

For inductive families, one thing you can do is to think of them in terms of the translation to parametrized inductive types and identity types, so

>    data Foo (A : Set) : Set where
>      bar : Id U A UnaryNat -> Foo A 

in which case bar applied to the Id U BinaryNat UnaryNat that you get from univalence gives a Foo BinaryNat.

A related perspective is to think of some transports as additional constructors for inductive families; see e.g. this approach to inductive families in cubical type theory https://www.cs.cmu.edu/~ecavallo/works/popl19.pdf

-Dan

> On Nov 5, 2019, at 7:06 PM, Stefan Monnier <monnier@iro.umontreal.ca> wrote:
> 
>> members of this list. The constructivity of univalence was an open problem 
>> for a number of year, and I would say that, even if it is solved via the 
>> cubical model, it is far from being fully understood.
> 
> In my case, I still find it odd in a situation such as:
> 
>    data Foo : Set -> Set where
>      bar : Foo UnaryNat
> 
> since transport supposedly allows us to take a proof of equivalence
> between UnaryNat and BinaryNat and turn a `bar` into something of type
> `Foo BinaryNat` although I can't see any way to directly construct an
> object of this type.
> 
> 
>        Stefan
> 

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Thread overview: 18+ messages (download: mbox.gz / follow: Atom feed)
-- links below jump to the message on this page --
2019-10-27 14:41 [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'? 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
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

Discussion of Homotopy Type Theory and Univalent Foundations

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