* Voevodsky obituary in Nature
@ 2017-11-07 21:40 Daniel R. Grayson
2017-11-07 21:51 ` [HoTT] " Dimitris Tsementzis
0 siblings, 1 reply; 4+ messages in thread
From: Daniel R. Grayson @ 2017-11-07 21:40 UTC (permalink / raw)
To: Homotopy Type Theory
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See https://www.nature.com/articles/d41586-017-05477-9
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* Re: [HoTT] Voevodsky obituary in Nature
2017-11-07 21:40 Voevodsky obituary in Nature Daniel R. Grayson
@ 2017-11-07 21:51 ` Dimitris Tsementzis
2017-11-07 22:49 ` Martín Hötzel Escardó
0 siblings, 1 reply; 4+ messages in thread
From: Dimitris Tsementzis @ 2017-11-07 21:51 UTC (permalink / raw)
To: Daniel R. Grayson; +Cc: Homotopy Type Theory
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Dan,
It is an interesting idea to refer to univalence as a “mechanism” rather than an “axiom”.
Even if you did it for expository reasons, I believe that in certain contexts it is a good phrase to adopt.
Dimitris
> On Nov 7, 2017, at 16:40, Daniel R. Grayson <danielrich...@gmail.com> wrote:
>
> See https://www.nature.com/articles/d41586-017-05477-9
>
> --
> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.
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* Re: [HoTT] Voevodsky obituary in Nature
2017-11-07 21:51 ` [HoTT] " Dimitris Tsementzis
@ 2017-11-07 22:49 ` Martín Hötzel Escardó
2017-11-07 23:11 ` Martín Hötzel Escardó
0 siblings, 1 reply; 4+ messages in thread
From: Martín Hötzel Escardó @ 2017-11-07 22:49 UTC (permalink / raw)
To: Homotopy Type Theory
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On Tuesday, 7 November 2017 21:50:24 UTC, Dimitris Tsementzis wrote:
>
> Dan,
>
> It is an interesting idea to refer to univalence as a “mechanism” rather
> than an “axiom”.
>
> Even if you did it for expository reasons, I believe that in certain
> contexts it is a good phrase to adopt.
>
I very much like this way of expression, too, even though univalence became
a mechanism for the first time with cubicaltt.
Nevertheless, although Vladimir certainly didn't have a mechanism for
univalence, he envisioned its possibility, and even formulated a conjecture
that is still unsettled, namely that if univalence implies that there is a
*number* n:N with a property P(n), then we can find, metatheoretically,
using an algorithm, a *numeral* n':N (a closed term of type N) and a proof
that univalence implies P(n'). No mechanism for univalence can currently
provide that, as far as I know.
Martin
>
> Dimitris
>
> On Nov 7, 2017, at 16:40, Daniel R. Grayson <daniel...@gmail.com
> <javascript:>> wrote:
>
> See https://www.nature.com/articles/d41586-017-05477-9
>
> --
> 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 HomotopyTypeThe...@googlegroups.com <javascript:>.
> For more options, visit https://groups.google.com/d/optout.
>
>
>
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* Re: [HoTT] Voevodsky obituary in Nature
2017-11-07 22:49 ` Martín Hötzel Escardó
@ 2017-11-07 23:11 ` Martín Hötzel Escardó
0 siblings, 0 replies; 4+ messages in thread
From: Martín Hötzel Escardó @ 2017-11-07 23:11 UTC (permalink / raw)
To: Homotopy Type Theory
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On Tuesday, 7 November 2017 22:49:30 UTC, Martín Hötzel Escardó wrote:
>
>
>
> On Tuesday, 7 November 2017 21:50:24 UTC, Dimitris Tsementzis wrote:
>>
>> Dan,
>>
>> It is an interesting idea to refer to univalence as a “mechanism” rather
>> than an “axiom”.
>>
>> Even if you did it for expository reasons, I believe that in certain
>> contexts it is a good phrase to adopt.
>>
>
> I very much like this way of expression, too, even though univalence
> became a mechanism for the first time with cubicaltt.
>
> Nevertheless, although Vladimir certainly didn't have a mechanism for
> univalence, he envisioned its possibility, and even formulated a conjecture
> that is still unsettled, namely that if univalence implies that there is a
> *number* n:N with a property P(n), then we can find, metatheoretically,
> using an algorithm, a *numeral* n':N (a closed term of type N) and a proof
> that univalence implies P(n').
>
The conjecture said that we can find a numeral n':N and a proof that
univalence implies n=n' .
Here is what he actually conjectured:
"
Conjecture 1 There is a terminating algorithm which for any term nn of type
nat constructed
with the use of the univalence axiom outputs a pair (nn 0 , pf ) where nn 0
is a term of type nat
constructed without the use of the univalence axiom and pf is a term of
type paths nat nn nn 0 (i.e.
a proof that nn 0 = nn). The term pf may use the univalence axiom.
"
http://www.math.ias.edu/vladimir/files/univalent_foundations_project.pdf
Martin
No mechanism for univalence can currently provide that, as far as I know.
>
> Martin
>
>
>>
>> Dimitris
>>
>> On Nov 7, 2017, at 16:40, Daniel R. Grayson <daniel...@gmail.com>
>> wrote:
>>
>> See https://www.nature.com/articles/d41586-017-05477-9
>>
>> --
>> 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 HomotopyTypeThe...@googlegroups.com.
>> For more options, visit https://groups.google.com/d/optout.
>>
>>
>>
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2017-11-07 21:51 ` [HoTT] " Dimitris Tsementzis
2017-11-07 22:49 ` Martín Hötzel Escardó
2017-11-07 23:11 ` Martín Hötzel Escardó
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