```
From: "Martín Hötzel Escardó" <escardo.martin@gmail.com>
To: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'?
Date: Tue, 5 Nov 2019 15:14:07 -0800 (PST)
Message-ID: <cb153658-548a-4fe9-91ed-fc2e3db33723@googlegroups.com> (raw)
In-Reply-To: <CAH52Xb3s0+vweUaSQBMBNLa5mRc9F1jrsg2sSoFmcE_4=dAt1w@mail.gmail.com>
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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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```

next prev parent reply indexThread overview:18+ messages / expand[flat|nested] mbox.gz Atom feed top 2019-10-27 14:41 Nicolas Alexander Schmidt 2019-10-27 17:22 ` Bas 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. Grayson2019-11-05 23:14 ` Martín Hötzel Escardó [this message]2019-11-06 0:06 ` Stefan Monnier 2019-11-11 18:26 ` Licata, Dan 2019-11-03 7:29 ` Michael Shulman

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