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) [thread overview]
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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next prev parent reply other threads:[~2019-11-05 23:14 UTC|newest]
Thread overview: 32+ 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
2019-11-24 18:11 ` Kevin Buzzard
2019-11-26 0:25 ` Michael Shulman
2019-11-26 8:08 ` Ulrik Buchholtz
2019-11-26 19:14 ` Martín Hötzel Escardó
2019-11-26 19:53 ` Kevin Buzzard
2019-11-26 20:40 ` Martín Hötzel Escardó
2019-11-26 22:18 ` Michael Shulman
2019-11-27 0:16 ` Joyal, André
2019-11-27 2:28 ` Stefan Monnier
2019-11-27 1:41 ` Daniel R. Grayson
2019-11-27 8:22 ` N. Raghavendra
2019-11-27 10:12 ` Thorsten Altenkirch
2019-11-27 16:37 ` Michael Shulman
2019-11-27 20:21 ` 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 [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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