[FOM] the stuff that [numbers] are made of

[Sean C Stidd]

I wouldn't mind leaving this discussion behind either, FWIW. But

(a) It's not clear that they 'don't care in the least' in an absolute
sense, only that they don't need an answer to go on with their proofs. Do
you really mean to suggest that if mathematicians got a clear, coherent
answer to the question 'what is a number?' that handled all or most
practical cases they'd not adopt it? Such a position seems dangerously
close to irrationalism. Now, it may be that this is a topic better
handled by logic and/or philosophy than real, 'core' mathematics, but do
you really mean to suggest that there's just no question here?

I suppose what you probably mean is that you think that by indicating the
general pattern through its instantiations you already have such a clear,
coherent understanding. It's interesting that some mathematicians and
philosophers find such an understanding perfectly sufficient, while
others have semantic and metaphysical worries. I admit to falling into
the latter camp.

(b) The hexagonal tiles are not adequately analogous to mathematical
entities, unless you think that your tile salesman is committed not only
to all the hexagonal tile patterns in production, but all possible
hexagonal tile patterns, including those that will never be produced. The
tile salesman can get by with an Aristotelian conception of form; the
mathematician cannot, at least not without substantial work of the type
undertaken by Field Hellman et. al..

(c)

> But once one understands clearly that this construct is just one of 
> infinitely possible legitimate representations to be selected only on
the basis 
> of convenience, what philosophical issues remain?

Representations of what? Do we have an unambiguous definition? Indicating
patterns from instantiations of them would not seem to provide that. We
all agree that we can indicate various 'instantiations' or perhaps
'representations' of this 'pattern', but talking about patterns directly
turns out to be hard. It's been causing trouble since Plato. Mathematics,
as a central part of our knowledge of the physical world, is the place
where this philosophical problem comes up most today, because it's the
one place where we appear forced to talk seriously about abstract
entities even within the scientific world-picture.

Again, I'm not really indicating a desire to take all this up on the
list: just standing up for the philosophical interest of philosophical
questions, if not necessarily the mathematical. Serious discussion of
these issues is hard. If a mathematician or logician tells me that they
don't have interest because they have tons of problems they can work on
without solutions, more power to them. But I don't believe it to be the
case that practitioner common sense solves all philosophical problems:
even on Wittgenstein's view you have to understand the forms of language
that make the problems appear to be serious in order to deflate them.
That perhaps takes us out of mathematics into semantics or metaphysics,
but such problems are nonetheless not resolved by simple analogies and
repetitions of textbook definitions with which we are for the most part
familiar.