From: Thorsten Altenkirch <Thorsten....@nottingham.ac.uk> To: Steve Awodey <awo...@cmu.edu> 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 17:25:03 +0000 [thread overview] Message-ID: <055F0AF8-C683-48CE-88A0-3BC9A0EEF28A@nottingham.ac.uk> (raw) In-Reply-To: <14AEC162-00A7-41E5-88EF-11501EF7C2AB@gmail.com> 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" >>> >> >> -- >> 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 HomotopyT...@googlegroups.com. >> To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOvivQx_2TinRHBrmOAZFnmFp8VVQ-yMcPvtKFtX-d9wGsD%2B2Q%40mail.gmail.com. > > > > > 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. > > > > > -- > 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 HomotopyT...@googlegroups.com. > To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/D83C2B3E-AF61-409B-BE3A-A98839A00CF6%40nottingham.ac.uk. 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. 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next prev parent reply other threads:[~2020-05-04 17:25 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 [this message] 2020-05-04 17:43 ` Michael Shulman 2020-05-04 17:55 ` Steve Awodey 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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