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: [HoTT] Re: Why do we need judgmental equality?
Date: Fri, 8 Feb 2019 13:19:23 -0800 (PST)	[thread overview]
Message-ID: <1a3dfba4-816a-42c3-8eea-1a2906cb1cad@googlegroups.com> (raw)
In-Reply-To: <bcd56e20-1961-400b-91b4-2ca8c042d0e5@googlegroups.com>


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I would also like to know an answer to this question. It is true that 
dependent type theories have been designed using definitional equality.

But why would anybody say that there is a *need* for that? Is it impossible 
to define a sensible dependent type theory (say for the purpose of serving 
as a foundation for univalent mathematics) that doesn't mention anything 
like definitional equality? If not, why not? And notice that I am not 
talking about *usability* of a proof assistant such as the *existing* ones 
(say Coq, Agda, Lean) were definitional equalities to be removed. I don't 
care if such hypothetical proof assistants would be impossibly difficult to 
use for a dependent type theory lacking definitional equalities (if such a 
thing exists).

The question asked by Felix is a very sensible one: why is it claimed that 
definitional equalities are essential to dependent type theories?

(I do understand that they are used to compute, and so if you are 
interested in constructive mathematics (like I am) then they are useful. 
But, again, in principle we can think of a dependent type theory with no 
definitional equalities and instead an existence property like e.g. in 
Lambek and Scott's "introduction to higher-order categorical logic". And 
like was discussed in a relatively recent message by Thierry Coquand in 
this list,)

Martin 


On Wednesday, 30 January 2019 11:54:07 UTC, Felix Rech wrote:
>
> In section 1.1 of the HoTT book it says "In type theory there is also a 
> need for an equality judgment." Currently it seems to me like one could, in 
> principle, replace substitution along judgmental equality with explicit 
> transports if one added a few sensible rules to the type theory. Is there a 
> fundamental reason why the equality judgment is still necessary?
>
> Thanks,
> Felix Rech
>

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  parent reply	other threads:[~2019-02-08 21:19 UTC|newest]

Thread overview: 71+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-01-30 11:54 [HoTT] " Felix Rech
2019-02-05 23:00 ` [HoTT] " Matt Oliveri
2019-02-06  4:13   ` Anders Mörtberg
2019-02-09 11:55     ` Felix Rech
2019-02-16 15:59     ` Thorsten Altenkirch
2019-02-17  1:25       ` Michael Shulman
2019-02-17  7:56         ` Thorsten Altenkirch
2019-02-17  9:14           ` Matt Oliveri
2019-02-17  9:18           ` Michael Shulman
2019-02-17 10:52             ` Thorsten Altenkirch
2019-02-17 11:35               ` streicher
2019-02-17 11:44                 ` Thorsten Altenkirch
2019-02-17 14:24                   ` Bas Spitters
2019-02-17 19:36                   ` Thomas Streicher
2019-02-17 21:41                     ` Thorsten Altenkirch
2019-02-17 12:08             ` Matt Oliveri
2019-02-17 12:13               ` Matt Oliveri
2019-02-20  0:22               ` Michael Shulman
2019-02-17 14:22           ` [Agda] " Andreas Abel
2019-02-17  9:05         ` Matt Oliveri
2019-02-17 13:29         ` Nicolai Kraus
2019-02-08 21:19 ` Martín Hötzel Escardó [this message]
2019-02-08 23:31   ` Valery Isaev
2019-02-09  1:41     ` Nicolai Kraus
2019-02-09  8:04       ` Valery Isaev
2019-02-09  1:58     ` Jon Sterling
2019-02-09  8:16       ` Valery Isaev
2019-02-09  1:30   ` Nicolai Kraus
2019-02-09 11:38   ` Thomas Streicher
2019-02-09 13:29     ` Thorsten Altenkirch
2019-02-09 13:40       ` Théo Winterhalter
2019-02-09 11:57   ` Felix Rech
2019-02-09 12:39     ` Martín Hötzel Escardó
2019-02-11  6:58     ` Matt Oliveri
2019-02-18 17:37   ` Martín Hötzel Escardó
2019-02-18 19:22     ` Licata, Dan
2019-02-18 20:23       ` Martín Hötzel Escardó
2019-02-09 11:53 ` Felix Rech
2019-02-09 14:04   ` Nicolai Kraus
2019-02-09 14:26     ` Gabriel Scherer
2019-02-09 14:44     ` Jon Sterling
2019-02-09 20:34       ` Michael Shulman
2019-02-11 12:17         ` Matt Oliveri
2019-02-11 13:04           ` Michael Shulman
2019-02-11 15:09             ` Matt Oliveri
2019-02-11 17:20               ` Michael Shulman
2019-02-11 18:17                 ` Thorsten Altenkirch
2019-02-11 18:45                   ` Alexander Kurz
2019-02-11 22:58                     ` Thorsten Altenkirch
2019-02-12  2:09                       ` Jacques Carette
2019-02-12 11:03                   ` Matt Oliveri
2019-02-12 15:36                     ` Thorsten Altenkirch
2019-02-12 15:59                       ` Matt Oliveri
2019-02-11 19:27                 ` Matt Oliveri
2019-02-11 21:49                   ` Michael Shulman
2019-02-12  9:01                     ` Matt Oliveri
2019-02-12 17:54                       ` Michael Shulman
2019-02-13  6:37                         ` Matt Oliveri
2019-02-13 10:01                           ` Ansten Mørch Klev
2019-02-11 20:11                 ` Matt Oliveri
2019-02-11  8:23       ` Matt Oliveri
2019-02-11 13:03         ` Jon Sterling
2019-02-11 13:22           ` Matt Oliveri
2019-02-11 13:37             ` Jon Sterling
2019-02-11  6:51   ` Matt Oliveri
2019-02-09 12:30 ` [HoTT] " Thorsten Altenkirch
2019-02-11  7:01   ` Matt Oliveri
2019-02-11  8:04     ` Valery Isaev
2019-02-11  8:28       ` Matt Oliveri
2019-02-11  8:37         ` Matt Oliveri
2019-02-11  9:32           ` Rafaël Bocquet

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