From mboxrd@z Thu Jan 1 00:00:00 1970 X-Spam-Checker-Version: SpamAssassin 3.4.4 (2020-01-24) on inbox.vuxu.org X-Spam-Level: * X-Spam-Status: No, score=1.1 required=5.0 tests=DATE_IN_PAST_06_12, RCVD_IN_MSPIKE_H2 autolearn=no autolearn_force=no version=3.4.4 Received: (qmail 24778 invoked from network); 1 Feb 2023 20:41:32 -0000 Received: from smtp2.mta.ca (198.164.44.75) by inbox.vuxu.org with ESMTPUTF8; 1 Feb 2023 20:41:32 -0000 Received: from rr.mta.ca ([198.164.44.159]:40826) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1pNJv3-0001Zv-9r; Wed, 01 Feb 2023 16:41:25 -0400 Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1pNJtt-0006VD-EE for categories-list@rr.mta.ca; Wed, 01 Feb 2023 16:40:13 -0400 MIME-Version: 1.0 (Mac OS X Mail 16.0 \(3696.120.41.1.1\)) Subject: categories: Re: Terminology for point-free topology? From: Martin Hyland In-Reply-To: Date: Wed, 1 Feb 2023 09:41:37 +0000 Cc: Pedro Resende , categories list Content-Transfer-Encoding: quoted-printable References: To: Wesley Phoa Precedence: bulk Reply-To: Martin Hyland Message-Id: The lay audience is very sensible. Further in the direction of not = expecting any best formulation I add some remarks. 1. =46rom 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 =E2=80=A6 .) 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=20 stressing the value of abstract mathematics. Not easy in a=20 scornful world =E2=80=A6 . Martin > On 30 Jan 2023, at 21:59, Wesley Phoa wrote: >=20 > I've been out of mathematics for three decades, so I feel qualified to > represent the lay audience in this discussion. >=20 > Mathematicians use the word "space" to refer to three concepts which, = to a > lay person, seem completely unrelated: >=20 > 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 >=20 > These concepts have quite different (lay) intuitions associated with = them:* >=20 > 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 >=20 > 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! >=20 > 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. >=20 > Wesley >=20 > *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*. >=20 > On Mon, Jan 23, 2023 at 2:18 PM Pedro Resende < > pedro.m.a.resende@tecnico.ulisboa.pt> wrote: >=20 >> In addition to all the deeper reasons, `pointless=E2=80=99 can be = taken to be >> derogatory, so preferably it should be used only when in = tongue-in-cheek >> mode. At least that=E2=80=99s what I tell my students =E2=80=94 just = as I ask them not to >> say `abstract nonsense=E2=80=99 too enthusiastically=E2=80=A6 :) >>=20 >> Pedro >>=20 >>> On Jan 21, 2023, at 7:42 PM, ptj@maths.cam.ac.uk wrote: >>>=20 >>> 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'. >>>=20 >>> 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. >>>=20 >>> Peter Johnstone >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]