Discussion of Homotopy Type Theory and Univalent Foundations
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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>

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  parent reply index

Thread overview: 18+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-10-27 14:41 Nicolas Alexander Schmidt
2019-10-27 17:22 ` Bas Spitters
2019-11-03 11:38   ` Bas Spitters
2019-11-03 11:52     ` David Roberts
2019-11-03 19:13       ` Michael Shulman
2019-11-03 19:45         ` Valery Isaev
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 [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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