From: Martin Hyland <M.Hyland@dpmms.cam.ac.uk>
To: Wesley Phoa <email@example.com>
Cc: Pedro Resende <firstname.lastname@example.org>,
categories list <email@example.com>
Subject: categories: Re: Terminology for point-free topology?
Date: Wed, 1 Feb 2023 09:41:37 +0000 [thread overview]
Message-ID: <E1pNJtt-0006VD-EE@rr.mta.ca> (raw)
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The lay audience is very sensible. Further in the direction of not expecting
any best formulation I add some remarks.
1. From maybe mid last century we came to see many structures where there
is a notion of point but where it is important that there are not enough points.
(There is more than one sense of that … .) It is a question in the history of
thought whether the idea of a space as made up of points predates set
theory. Bill Lawvere liked to stress that in Greek geometry there were
other figures - lines, triangles whatever.
2. Thinking of Bill suggests taking as a *modern* starting point the idea
that a space is an object in a category of spaces. That is parallel to
the idea that a vector is an element in a vector space. But of course that
idea has limitations as e.g. in the theory of forces on a rigid body.
Similarly a category of spaces may only get one so far. Wesley
mention points with symmetries as e.g. in the space of triangles.
We have yet to develop a background theory there?
None of that helps re nomenclature which we can influence though
hardly control. But I do not know what any of us can do beyond
stressing the value of abstract mathematics. Not easy in a
scornful world … .
> On 30 Jan 2023, at 21:59, Wesley Phoa <firstname.lastname@example.org> wrote:
> 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
> 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.
> *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 <
> email@example.com> 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… :)
>>> On Jan 21, 2023, at 7:42 PM, firstname.lastname@example.org 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/ ]
next prev parent reply other threads:[~2023-02-01 20:41 UTC|newest]
Thread overview: 19+ messages / expand[flat|nested] mbox.gz Atom feed top
2023-01-16 11:50 Steven Vickers
[not found] ` <email@example.com>
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 ` categories: " Wesley Phoa
2023-02-01 9:41 ` Martin Hyland [this message]
[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
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