[FOM] Real Numbers

[Sean C Stidd]

> By any definition of numbers I know of, the natural number 3 is not
> identical to the rational number 3, and neither is identical to the 
> real number 3.  

It depends which constructions you identify with the number-types in
question. Mathematics does not seem to me to settle this question
unambiguously.

If we identify the naturals (or, say, the counting numbers, if you prefer
to reserve the word 'naturals' for the constructions we get in our logic
and set theory courses) with a certain subset of the reals (or, say, the
measuring numbers, ditto), then the natural 2 is the real 2. If you
identify the natural 2 with a certain set of sets and the real 2 as an
equivalence class of certain constructions out of equivalence classes of
ordered pairs of natural numbers, then obviously they aren't the same
thing.

How would we decide which of these identifications was the right one? The
classical order of exposition of the logical construction of the
number-systems one finds in Russell's classic texts, or in Feferman's The
Number Systems, suggests the latter. On the other hand, the only argument
for this identification that I know of is that at those stages in the
logical construction of numbers you can already do the mathematics
related to that kind of number. This is a necessary but not a sufficient
condition for those logical constructions to be the kinds of number in
question. 

I actually favor the former, since I believe that the natural 2 and the
real 2 are the same number. (So I think for example that the natural 2
does have a square root, but that it isn't among the natural numbers.) I
understand that I don't have to do all the stages in the construction to
get to the reals in order to have objects that do the work of the natural
numbers, but that doesn't settle the question in my mind.

Some people like to say that this sort of question is 'meaningless', but
I don't see why a straightforward question about whether one object is
identical to another or not should be judged meaningless. I understand
that you can do math either way; all that tells me is that the bare fact
of whether you can do math with a certain definition of an object doesn't
settle the identity-question; also that mathematicians don't need the
identity-question settled for their work. Caring about this sort of thing
explains how a vaguely promising mathematical physics B.S. winds up as a
philosopher. That said, a substantial number of my colleagues in
philosophy (I make no claim about even rough percentages) do in fact
regard this kind of question as meaningless or uninteresting as well.