Re: [HoTT] "Identifications" ?

[Michael Shulman <shu...@sandiego.edu>]

I think "identification" is a nice middle-ground between "equality"
and "path/isomorphism".  On the one hand it signals that something
more is meant than the thing traditionally called "equality" (even
though I agree with Thorsten that inside of type theory this is the
thing called "equality") -- and, I think, does a good job of
suggesting precisely *what* more is meant.  On the other hand, it
still carries the meaning that two things are *the same* (in a
particular way) -- two things that have been "identified" are now
"identical" -- whereas "path" and "isomorphism" suggest their
traditional meanings of two things that are *not* fundamentally the
same but are related in some same-like-way.

On Mon, May 4, 2020 at 10:25 AM 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.
>
> 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.
>
> I noticed that many people use type theory always as something we talk about. It is just a formalism. Hence the equality type just expresses an identification of things which are actually different. In the real world. However, I think when we talk in type theory then this is our real world (at least metaphorically) and then the metatheoretic perspective is just a confusion.
>
> Does this make sense? Sorry, I realize it is a bit philosophical but then you are in the department of philosophy... __
>
> 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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