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```From: Wesley Phoa <doctorwes@gmail.com>
To: Pedro Resende <pedro.m.a.resende@tecnico.ulisboa.pt>
Cc: categories list <categories@mta.ca>
Subject: categories: Re: Terminology for point-free topology?
Date: Mon, 30 Jan 2023 13:59:48 -0800	[thread overview]
Message-ID: <E1pN1dG-0002uo-Hc@rr.mta.ca> (raw)

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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```

```next prev parent reply	other threads:[~2023-02-01  1:10 UTC|newest]

Thread overview: 19+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2023-01-16 11:50 Steven Vickers
2023-01-18 12:12   ` Steven Vickers
2023-01-20  3:06     ` David Yetter
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             ` Wesley Phoa [this message]
2023-02-01  9:41               ` categories: " Martin Hyland
2023-01-23 13:47       ` Steven Vickers
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
2023-01-22 21:32   ` Terminology for point-free topology? Vaughan Pratt
2023-01-23 13:25   ` Steven Vickers
2023-01-23 23:17   ` categories: " Vaughan Pratt
2023-01-24 11:45   ` Steven Vickers
```

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