Re: [HoTT] "Identifications" ?

[Thorsten Altenkirch <Thorsten....@nottingham.ac.uk>]

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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