Discussion of Homotopy Type Theory and Univalent Foundations
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From: Steve Awodey <awo...@cmu.edu>
To: Thorsten Altenkirch <Thorsten....@nottingham.ac.uk>
Cc: Michael Shulman <shu...@sandiego.edu>,
	Stefan Monnier <mon...@iro.umontreal.ca>,
	"homotopyt...@googlegroups.com" <homotopyt...@googlegroups.com>
Subject: Re: [HoTT] "Identifications" ?
Date: Mon, 4 May 2020 13:55:15 -0400	[thread overview]
Message-ID: <4657A23E-2B56-4A87-8CF3-A3A1159BA044@cmu.edu> (raw)
In-Reply-To: <055F0AF8-C683-48CE-88A0-3BC9A0EEF28A@nottingham.ac.uk>



> On May 4, 2020, at 1:25 PM, Thorsten Altenkirch <Thorsten....@nottingham.ac.uk> wrote:
> 
> Hi Steve,
> 
>    I remember that conversation.
>    I think we decided to put the question “what does x=y mean?” aside,
>    until we had taken care of more important things.
> 
> I suppose this was just a way to move on without having to reach an agreement.

yup

> 
> I think it is more than a discussion about terms. What do we mean by equality? Does the equality type in HoTT is something fundamentally different? In a way yes, because it is proof relevant so some of the old terminology doesn't apply anymore. That is equality of structures is a structure not a proposition. But nevertheless I find it confusing to call it anything but equality. I would say two mathematical objects which share all the same properties, which behave the same, are equal. I don't like Leibniz's "equality of indiscernibles" because it uses a negative.

you seem to prefer the term “equality” over “identity” - even when (mis)quoting Leibniz!

maybe that’s part of the problem: if we talk about “identity types” instead of “equality types” then it’s easier to regard them as structures, 
the inhabitants of which are “identifications”, rather than “equality proofs” (which introduces an unwanted meta-perspective).

Frege once considered equality as a relation between symbols, rather than what they denote — he later rejected that view, but I don’t think it's so bad.  
Definitional equality can be thought of like this — it’s a meta-statement about (pseudo-) syntactic things.
The relation that we can reason about inside the system is called *identity*, and it’s a "proof relevant” relation 
— or better put, it’s a proper structure, not a mere proposition — 
its elements are identifications, and these we know admit an oo-groupoid structure. 

> 
> Does this make sense? Sorry, I realize it is a bit philosophical but then you are in the department of philosophy… __

to my everlasting regret — and thanks for not calling me a philosopher!

Steve

> 
> Thorsten
> 
> On 04/05/2020, 17:54, "Steven Awodey on behalf of Steve Awodey" <awo...@andrew.cmu.edu on behalf of awo...@cmu.edu> wrote:
> 
> 
> 
>> On May 4, 2020, at 12:17 PM, Thorsten Altenkirch <Thorsten....@nottingham.ac.uk> wrote:
>> 
>> 
>>   I’m afraid that someone may have hacked Thorsten’s email account. The real Thorsten went through all this with us many years ago. 
>>   : - )
>> 
>> One of our dogs gained access to my laptop - sorry. These animals can be very awkward.
>> 
>> However, even the real Thorsten had a friendly argument with Andre Joyal when we were writing the book about whether to use = for the equality type. 
> 
>    I remember that conversation. 
>    I think we decided to put the question “what does x=y mean?” aside, 
>    until we had taken care of more important things.
> 
>    So is it time now?
> 
>    Steve
> 
>> 
>> Thorsten
>> 
>> On 04/05/2020, 17:08, "Steve Awodey" <steve...@gmail.com> wrote:
>> 
>>   I’m afraid that someone may have hacked Thorsten’s email account. The real Thorsten went through all this with us many years ago. 
>>   : - )
>> 
>> 
>>> On May 4, 2020, at 12:00, Michael Shulman <shu...@sandiego.edu> wrote:
>>> 
>>> The word "path" is closely tied to the homotopy interpretation, and to
>>> the classical perspective of oo-groupoids presented via topological
>>> spaces, which has various problems.  This is particularly an issue in
>>> cohesive type theory, where there is a separate "point-set level"
>>> notion of path that it is important to distinguish from
>>> identifications.
>>> 
>>>> On Mon, May 4, 2020 at 7:48 AM Stefan Monnier <mon...@iro.umontreal.ca> wrote:
>>>> 
>>>>> I don't think using "identification" necessarily implies any
>>>>> difference between "identification" and "equality".  I don't think of
>>>>> it that way.  For me the point is just to have a word that refers to
>>>>> an *element* of an identity type.  Calling it "an equality" can have
>>>>> the wrong connotation because classically, an equality is just a
>>>>> proposition (or a true proposition), whereas an element of an identity
>>>>> type carries information.  Calling it "an identification" suggests
>>>>> exactly the information that it carries: a way of identifying two
>>>>> things.
>>>> 
>>>> I thought that's what "path" was for?
>>>> 
>>>> 
>>>>      Stefan "who really doesn't know what he's talking about"
>>>> 
>>> 
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>> 
>> 
>> 
>> 
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> 
> 
> 
> 
> 
> This message and any attachment are intended solely for the addressee
> and may contain confidential information. If you have received this
> message in error, please contact the sender and delete the email and
> attachment. 
> 
> Any views or opinions expressed by the author of this email do not
> necessarily reflect the views of the University of Nottingham. Email
> communications with the University of Nottingham may be monitored 
> where permitted by law.
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> 
> 
> 


  parent reply	other threads:[~2020-05-04 17:55 UTC|newest]

Thread overview: 26+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2020-05-04  9:35 Thorsten Altenkirch
2020-05-04 10:59 ` [HoTT] " stre...
2020-05-04 11:04   ` Steve Awodey
2020-05-04 11:17   ` Thorsten Altenkirch
2020-05-04 11:42     ` Nicolai Kraus
2020-05-04 12:04       ` Thorsten Altenkirch
2020-05-04 12:06     ` Thomas Streicher
2020-05-04 12:12       ` Thorsten Altenkirch
2020-05-04 12:39         ` Thomas Streicher
2020-05-04 13:16 ` Michael Shulman
2020-05-04 14:17   ` Thorsten Altenkirch
2020-05-04 14:48   ` Stefan Monnier
2020-05-04 15:46     ` Nicolai Kraus
2020-05-04 15:57       ` Thorsten Altenkirch
2020-05-04 15:59     ` Michael Shulman
2020-05-04 16:07       ` Steve Awodey
2020-05-04 16:17         ` Thorsten Altenkirch
2020-05-04 16:53           ` Steve Awodey
2020-05-04 17:25             ` Thorsten Altenkirch
2020-05-04 17:43               ` Michael Shulman
2020-05-04 17:55               ` Steve Awodey [this message]
2020-05-04 16:21         ` Peter LeFanu Lumsdaine
2020-05-04 16:16       ` Joyal, André
2020-05-04 20:38         ` Joyal, André
2020-05-07 19:43 ` Martín Hötzel Escardó
2020-05-08 10:41   ` [HoTT] " Thorsten Altenkirch

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