Re: Questioning the defintions of set and element.

On Thu, 1 May 2008 17:13:09 +0000 (UTC), magidin@xxxxxxxxxxxxxxxxx
(Arturo Magidin) wrote:

In article <pjqj14ps39u6jacq45m2jbn9r8mrlo9bf9@xxxxxxx>,
Angus Rodgers <twirlip@xxxxxxxxxxx> wrote:

Isn't there a dilemma here? When a mathematician thinks about e.g.
the set of zeros of the Riemann zeta function, must we suppose that
(unlike Riemann himself) he is "really" thinking about some element
of some model of some fixed, formal axiomatic theory of sets, which
must therefore be taken as foundational for all of mathematics?

There is of course no telling how one "thinks" about something;
and the source of my (or your) intuition is irrelevant provided that
you can present an acceptable correct proof following the usual
canons, regardless of whether you came up with it by thinking about
some element in some specific model of some formal theory, or you came
up with it by thinking of functions as sheep jumping a
fence. Whatever representation goes on in your mind is irrelevant so
long as you can "map" that representation into something formal enough
to constitute a proof (that is why one tries to avoid "obvious" as
part of an argument).

But I think (!) that the thinking here is shared, i.e. it is not
just a question of what idiosyncratic neural representations may
exist in the wobbly grey matter inside one mathematician's skull.

I have the firm impression that I have much the same "idea" of
what sets are as any other competent or semi-competent student
of mathematics does - and that we all have great difficulty in
saying what this idea is - this difficulty also being shared,
and not particularly idiosyncratic (although each of us almost
certainly puts his or her own individual spin on the matter).

Most mathematicians are not actually doing mathematics within a
formalized axiomatic set theory (just think how many hundreds of pages
it took Russell and Whitehead to prove that 1+1=2). Instead, they work
within a more "naive" set theory (or even other foundational model,
such as categories, or arithmetic, or even the theory of real
numbers). For some people, specific models are a good way to get a
handle of the objects they are dealing with; for others, formalism is
the best way of thinking about it. I really don't see a "dilemma".

I certainly agree with the first part of that. But it seems to
imply that one cannot reply to a question about the definition
of sets - which surely belongs to the (shared) "naive" theory -
by referring to the idea of an axiomatic theory of sets (any
more than one could reply to a question about points of space
or spacetime in physics by referring to an axiomatic theory of
point-set topology). That is, the question specifically refers
to the "naive" model, not to any uninterpreted formal theory.

But the honest answer has already been given (by you, as well as by
several others): viz. that requests for definitions must end, after
a finite number of iterations, perhaps with some sort of "ostensive
definition", or other alternative to verbal or symbolic definition.

(No doubt analytic philosophers from Frege and Wittgenstein onwards
have written a lot on this topic, but I don't know much about it.)

Or
is he (as I tend to presuppose) thinking, rather, of a collection of
elements of a set R[i], where R is a model of some theory of the real
number system? In the former case, isn't the picture of mathematics
as a whole somewhat unrealistic?

Whose picture? Which of their pictures?

I mean, the picture (which you have already rejected) of mathematics
as depending, in practice, on some formal axiomatic theory (of sets).

And in the latter case, isn't there
a real question of explaining what a "collection" of elements of a
"set" constructed from a "model" is?

If you are thinking exclusively about the Riemann zeta function in the
context of real numbers and complex numbers, then do you really need
an explanation of what the generic notion of "collection", "set" and
"model" are? No. You just need an explanation of what "set of complex
numbers" is, of what "function" is within that context, etc. There is
no need to be informed of all the technical details of axiomatic set
theory, nor to have a model for axiomatic set theory as a whole in
order to do calculus, either; an informal understanding is sufficient,
provided you avoid the error of using the informal understanding as
part of your argument to establish something.

I agree, really. And it may be that the OP is not worried about the
definition of sets on this level at all; he just got confused because
he thought the process of definition would somehow reduce the concept
of a "set" to something else that isn't obviously just the same thing
in different words. So I probably shouldn't worry too much, either!
(Not until much later in my studies.)

After all, isn't a "model" (of
/any/ theory - whether of undefined entities called "sets", undefined
entities called "real numbers", or another kind of undefined entities)
itself a kind of "collection" or "set"?

Depends on how you found your theory. You can found your theory on
categories instead of sets. [...]

I've never understood how that gets around the apparent dependence
on some form of naive set theory. But I fear that I exasperated
even the very patient Colin McLarty by going on and on about this in
sci.math, a long time ago, and as I haven't made any progress since
then, I'd better not try to go over it all again now!

Also, isn't it true that even
though axiomatic theories of sets play a vital role in mathematics,

I would call it a pervasive rather than a vital role. It is possible
to do mathematics without any axiomatic set theory on hand, though
much of what you will read assumes at least some background in the
basic notions of such a theory.

I agree; I was just conceding as much as I could, while denying a
fundamental role to such theories in practice.

Still, I can't
deny that there seems to me, at least, to be a real problem here.

There may be a philosophical issue; which is why you still have
vigorous philosophical debates about the nature of mathematics. But
then, there is a philosophical issue about what "mind" is, and yet
that does not seem to create a problem in terms of not allowing us to
think!

Indeed, and I always bear in mind the fable of the centipede who
forgot how to walk when he tried to work out how he did it.

--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
.

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