categories - Category Theory list
 help / color / mirror / Atom feed
From: Wesley Phoa <doctorwes@gmail.com>
To: Pedro Resende <pedro.m.a.resende@tecnico.ulisboa.pt>
Cc: categories list <categories@mta.ca>
Subject: categories: Re: Terminology for point-free topology?
Date: Mon, 30 Jan 2023 13:59:48 -0800	[thread overview]
Message-ID: <E1pN1dG-0002uo-Hc@rr.mta.ca> (raw)
In-Reply-To: <E1pK57j-00046C-UW@rr.mta.ca>

I've been out of mathematics for three decades, so I feel qualified to
represent the lay audience in this discussion.

Mathematicians use the word "space" to refer to three concepts which, to a
lay person, seem completely unrelated:

1. a space of parameters: e.g. a space of moduli, a configuration space,
parameter space for a neural network
2. a thing with a shape: e.g. a doughnut, a coffee cup, a Klein bottle, a
tesseract
3. empty space: e.g. Euclidean space, curved spacetimes, the higher
dimensional spaces in string theory

These concepts have quite different (lay) intuitions associated with them:*

1. this kind of space obviously has points, but it's tricky to grasp what
cohesion means
2. this kind of space is obviously cohesive, but it's a leap to think of it
as made up of points
3. it doesn't obviously/naively make sense to talk about either points or
cohesion when there's nothing there

The fact that there are formalisms in which #1 and #2 are "the same thing"
is surprising, amazing and powerful. And the fact that there are several
formalisms, even more so!

So you wouldn't expect there to be a single language that feels natural to
everyone, in all three settings.* Any more than you would expect to find a
single "best" formalism.

Wesley

*Further confusion ensues as some of these concepts ramify further, e.g.
"cohesion" into continuity, smoothness etc., "point" as a bare point, a
point with symmetries, a point with an extent... We've gotten used to
regarding these as living inside different subject areas within
mathematics, but that wasn't obvious* ex ante*.

On Mon, Jan 23, 2023 at 2:18 PM Pedro Resende <
pedro.m.a.resende@tecnico.ulisboa.pt> wrote:

> In addition to all the deeper reasons, `pointless’ can be taken to be
> derogatory, so preferably it should be used only when in tongue-in-cheek
> mode. At least that’s what I tell my students — just as I  ask them not to
> say `abstract nonsense’ too enthusiastically… :)
>
> Pedro
>
>> On Jan 21, 2023, at 7:42 PM, ptj@maths.cam.ac.uk wrote:
>>
>> I was wondering how long it would be before someone in this thread
>> referred to my `point of pointless topology' paper! Perhaps not so many
>> people know that the title was a conscious echo of an earlier paper
>> by Mike Barr called `The point of the empty set', which began with the
>> words (I quote from memory) `The point is, there isn't any point there;
>> that's exactly the point'.
>>
>> As Steve says, to fit that title I had to use the word `pointless', but
>> on the whole I prefer `pointfree'; it carries the implication that you
>> are free to work without points or to use them (in a generalized sense),
>> as you prefer.
>>
>> Peter Johnstone

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

  reply	other threads:[~2023-02-01  1:10 UTC|newest]

Thread overview: 19+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2023-01-16 11:50 Steven Vickers
     [not found] ` <a23cbfc0-7433-eec1-115e-af2041d4cfd1@uu.nl>
2023-01-18 12:12   ` Steven Vickers
2023-01-20  3:06     ` David Yetter
     [not found]     ` <SN6PR05MB5213EBE225CB83D101EA0F57A2C59@SN6PR05MB5213.namprd05.prod.outlook.com>
2023-01-20 11:50       ` Steven Vickers
2023-01-21 19:42         ` ptj
2023-01-23 11:44           ` Pedro Resende
2023-01-30 21:59             ` Wesley Phoa [this message]
2023-02-01  9:41               ` categories: " Martin Hyland
     [not found]     ` <18E1AA5F-0054-4CA3-B231-BD9B799B03A2@tecnico.ulisboa.pt>
2023-01-23 13:47       ` Steven Vickers
     [not found]     ` <YQXPR01MB26464DF33EAE7481847A4F82E5C99@YQXPR01MB2646.CANPRD01.PROD.OUTLOOK.COM>
2023-01-24 12:20       ` categories: " Robert Pare
2023-01-27 17:55     ` Pedro Resende
2023-01-28  5:43       ` Patrik Eklund
2023-01-29 23:16         ` dawson
2023-01-28 10:48       ` categories: complete Galois groups Clemens Berger
2023-01-30 17:34         ` categories: " Eduardo J. Dubuc
     [not found] ` <LNXP265MB1049E00AEC9CE5BE1233CCEF95C69@LNXP265MB1049.GBRP265.PROD.OUTLOOK.COM>
2023-01-22 21:32   ` Terminology for point-free topology? Vaughan Pratt
     [not found] ` <CAL7kZqCz081nyFQuo_QFfaGzGe+UBORJHOZWsO9VgNzpDJ9_Gw@mail.gmail.com>
2023-01-23 13:25   ` Steven Vickers
     [not found] ` <LNXP265MB104912A7940157738582CE2595C89@LNXP265MB1049.GBRP265.PROD.OUTLOOK.COM>
2023-01-23 23:17   ` categories: " Vaughan Pratt
     [not found] ` <CAL7kZqAPzmzf=wt=qKNBjjeb_dGtG4eDu7tv5Eku-AVZD7wWtw@mail.gmail.com>
2023-01-24 11:45   ` Steven Vickers

Reply instructions:

You may reply publicly to this message via plain-text email
using any one of the following methods:

* Save the following mbox file, import it into your mail client,
  and reply-to-all from there: mbox

  Avoid top-posting and favor interleaved quoting:
  https://en.wikipedia.org/wiki/Posting_style#Interleaved_style

* Reply using the --to, --cc, and --in-reply-to
  switches of git-send-email(1):

  git send-email \
    --in-reply-to=E1pN1dG-0002uo-Hc@rr.mta.ca \
    --to=doctorwes@gmail.com \
    --cc=categories@mta.ca \
    --cc=pedro.m.a.resende@tecnico.ulisboa.pt \
    /path/to/YOUR_REPLY

  https://kernel.org/pub/software/scm/git/docs/git-send-email.html

* If your mail client supports setting the In-Reply-To header
  via mailto: links, try the mailto: link
Be sure your reply has a Subject: header at the top and a blank line before the message body.
This is a public inbox, see mirroring instructions
for how to clone and mirror all data and code used for this inbox;
as well as URLs for NNTP newsgroup(s).