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=1.5 required=5.0 tests=DATE_IN_PAST_24_48, DKIM_ADSP_CUSTOM_MED,FORGED_GMAIL_RCVD,FREEMAIL_FROM,RCVD_IN_MSPIKE_H2 autolearn=no autolearn_force=no version=3.4.4 Received: (qmail 7664 invoked from network); 1 Feb 2023 01:10:57 -0000 Received: from smtp2.mta.ca (198.164.44.75) by inbox.vuxu.org with ESMTPUTF8; 1 Feb 2023 01:10:57 -0000 Received: from rr.mta.ca ([198.164.44.159]:40586) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1pN1eE-0004uO-HY; Tue, 31 Jan 2023 21:10:50 -0400 Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1pN1dG-0002uo-Hc for categories-list@rr.mta.ca; Tue, 31 Jan 2023 21:09:50 -0400 MIME-Version: 1.0 References: In-Reply-To: From: Wesley Phoa Date: Mon, 30 Jan 2023 13:59:48 -0800 Subject: categories: Re: Terminology for point-free topology? To: Pedro Resende Cc: categories list Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable Precedence: bulk Reply-To: Wesley Phoa Message-Id: 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=E2=80=99 can be taken t= o 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 :) > > 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/ ]