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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
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