Re: ordered pair

From: Jim Burns (burns.87_at_osu.edu)
Date: 09/01/04 

Date: Wed, 01 Sep 2004 00:29:51 -0400

My thanks to you and everyone else who has responded to
my cry for "More light!" I've been given much to think about.

James Dolan wrote:
>
> in article <4133d119.cc7dc52@osu.edu>, jim burns <burns.87@osu.edu>
> wrote:
>
[...]
> |structure, but absurd according to how they are _intended_ to be
> |understood, like "(x,x) = {{x}}" or "1 = {0}".
> |
> |How are these absurdities kept from influencing more difficult
> |results?
>
[...]
> similarly, "structured mathematics" is the solution to the problem
> that you mention. roughly speaking, this means that the
> foundations of mathematics are more category-theoretic than
> set-theoretic, but there's no great need to rush out and learn
> more category theory than however much of it you feel like
> learning for non-foundational purposes, because category-theoretic
> foundations are what mathematicians pretty much naturally use
> anyway when left to their own devices. (you've already done a
> reasonable job of pointing out how much of a fiction it is to
> pretend that mathematicians use set-theoretic foundations.)

Right this moment, I favor category-theoretic foundations, with a
dash of set-theoretic foundations for flavoring.

My current take on all this is that set-theoretic existence theorems
aren't required in areas other than set theory for any logical
reason. Let's say a particular existence theorem links a
category-theoretic object with a set-theoretic structure, for
example, complete ordered fields with a partition of Cauchy sequences
of rationals. Such a theorem may be useful only for informally
persuasive reasons (while the theorem itself is completely rigorous),
but it does not, it should not expand the range of theorems
describing the category-theoretic object. What an existence theorem
persuades us of is that its topic, the category-theoretic object,
is at least minimally interesting.

Suppose I wish to study square circles, and, after discovering
several nice theorems about them, I realize, Alas!, that all the
square circles fit in the empty set, with room left over. My lovely
theorems are not made invalid; they are made uninteresting.

It seems to me that this minimal interest doesn't have to be provided
by a set-theoretic construct. If we had a well-developed theory of
blobs, say, which were neither able to model sets nor able to be
modeled by them, then I think mathematicians would be satisfied that
a particular mathematical structure were "founded" if we had either
a set-theoretic model or a blob-theoretic model for it.

Set theory, in fact, is uniquely valuable to us, but it is so by
being the big Erector set out of which we make these persuasive
models. It helps that we've already been using it for a long time
for that purpose, so that we've got a lot of pre-assembled pieces
lying around with which we can build new stuff.

[...]
>
> |Certainly Math does not care about to our _intentions_.
>
> notice that you suddenly sound a bit like jsh here, which is a clue
> that you're probably veering off in the direction of nonsense.

Ah, but such productive nonsense, if it got me eight (at last count)
thoughtful responses to my question. I wouldn't say JSH knows
nothing: he seems to be very good at eliciting thoughtful responses
from others.

And yet, however it sounded, I wasn't trying to produce nonsense.
I had only hoped it would be a bit amusing to personify Mathematics
(I call her Math for short) in order to point out Mathematics
is not a person. If that were all JSH did, I would have to agree
with him.

Jim Burns

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