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 … . Martin
On 30 Jan 2023, at 21:59, Wesley Phoa <doctorwes@gmail.com> 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 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
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