categories: Re: Terminology for point-free topology?

[Martin Hyland <M.Hyland@dpmms.cam.ac.uk>]

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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 … .

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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