[FOM] real numbers

[Neil Tennant]

On Thu, 19 Jun 2003, Andreas Blass wrote:

> ... I can finally explain "can".  When I
> say that something, like "the real numbers are Dedekind cuts", can be
> true, I mean that it becomes true under such an interpretation.

I enjoyed your explication of these modalized statements, and I believe
something of the sort must be the right account. But it seems to me that
what you are explicating here is what philosophical logicians would call
the *de dicto* possibility. That is, you have explained how to understand
a statement of the form

	It is possible that:
	the real numbers are Dedekind cuts

where the modal operator "It is possible that" has wide scope. This means
that, as one visits your suggested analogue of a possible world, one will
be reading "...is a real number" according to the criteria within *that*
world (or, in your case, theory).  Thus it will not be the same *things*
(i.e. real numbers) which exist both in the actual world and in the
various possible worlds. So you will have no theoretical handle on the
Kripkean notion of rigid designation. The Kripkean likes to think of
getting a grip on some things or kind of things in the actual world, and
finding *them* again in other worlds, in which one can inquire after the
(possibly different) properties that they enjoy there compared with those
that they enjoy in the actual world.

So how would you try to accommodate the person who claims that real
numbers can be Dedekind cuts, but who wishes this to be interpreted *de
re*? This person is saying *of* the things in the actual world, which
*are* the real numbers, that they *could be* Dedekind cuts. Is this
de re possibility metaphysical or epistemic?

I am inclined to think that the metaphysical interpretation must be
untenable, since the real numbers, surely, have their mathematical and
constitutive properties necessarily. By this I mean that if the de re
possibility statement is true then the corresponding de re necessitate is
true also. That is, if

	The real numbers are possibly Dedekind cuts

is true, then

	The real numbers are necessarily Dedekind cuts

is true. Similarly, however, if

	The real numbers are possibly Cauchy sequences

is true, then

	The real numbers are necessarily Cauchy sequences

is true.

Together these will entail that, in the actual world, the real numbers are
both Dedekind cuts and Cauchy sequences, which surely cannot be the case.
Hence the metaphysical interpretation of the de re possibility statement
is ruled out.

That leaves the epistemic interpretation. On this interpretation, we have

	Concerning the real numbers: it is possible, for all we know,
	that they are Dedekind cuts

and

	Concerning the real numbers: it is possible, for all we know,
	that they are Cauchy sequences.

I believe we can accept both of these on the basis of our established
mathematical practices. There are developments of real number theory in
which reals are treated as Dedekind cuts, and the axioms of real number
theory accordingly shown to be true; and there are yet other developments
of real number theory in which reals are treated as Cauchy sequences, and
the axioms of real number theory are likewise shown to be true.

The structuralist response to this is to regard the reals as no more
than positions within the kind of structure that makes the axioms of
the theory true. On this view, to think of the reals as having some
further, as yet undisclosed, metaphysical essence (i.e., a true
constitution that will reveal whether they *really are* Dedekind
cuts, or Cauchy sequences, or whatever) is to think of them the way an 
empirical realist might think of Dinge an sich. Such would-be essences lie
beyond the limits of what is knowable.

I think that Andreas's analysis of these modal locutions can be extended
to the de re cases on an epistemic interpretation, and will generally
concord with the view of the structuralist. The structuralist, but not
Andreas (yet, anyway) seems to be willing to take the extra step of
"modding out" over the various structures that make the axioms of real
number theory true, so as to abstract to the "positions within
structures", and then suggest that the real numbers are simply such
positions, to be understood in some (particular structure)-invariant way.

On Andreas's view, it would seem, there would be no fact of the matter (or
at least, no discoverable fact of the matter) as to what the real numbers
really, necessarily, are. The structuralist, however, seems to start with
that intuition, with a catholicity about structures, and then take the
abstractive step to try to get at what real numbers really are. That
leaves hanging the question: Well, what about the structure formed by
*those* things, then? The things in it can't just be positions common to
all structures---or can they?

Neil Tennant