[FOM] RE: Real Numbers

[Sean C Stidd]

John Baldwin wrote:

>  Flaws: Status of structuralist foundations unclear; creates ugly, 
> counterintuitive cross-theoretic identifications.
> 
> As a mathematician (model theorist) who just naively finds 
> `structuralism' natural, can you elaborate on the flaws you note above.
 
> I have no notion what the second half of the sentence means.

Certainly!

The first quibble is just the usual one: what is a structure and how do
you define it? If you were to define it as, say, a class of isomorphic
models, then if you construe those models as sets, structures appear to
just be sets of sets of a certain kind. But if you don't construe those
models as sets, what are they? Different ways of answering this question
may be found in the literature - Stewart Shapiro and Michael Resnik, two
leading structuralist philosophers of mathematics, wrote books defending
structuralism in 1997 giving somewhat different answers, and it could be
argued that category theory itself is an attempt to provide a
structuralist foundation for mathematics that predates philosophers' own
interest in the doctrine and which takes a different approach.
Furthermore, even after you've got the notion of structure clear and
given a plausible argument that all or some central portion of all or at
least current mathematics is concerned with something you want to call
structure, there's then the difficult task of showing that it provides a
foundation for mathematics in the technical sense. (Resnik, Shapiro, and
other structuralist writers have of course addressed some of these
problems in their books.)

The second problem depends a little on what version of structuralism you
adopt, but the easiest way to illustrate it is by example. Consider ZFC,
NGB, and AFA. At least some set theorists would naturally regard these as
theories of distinct kinds of mathematical object - the first of sets,
the second of sets and classes, the third perhaps of hypersets. They
might go on to say that the differences in axiomatization and the
differences in what sentences you can prove in each show that they are
concerned with different things. (Some would go on further to say that
since you can prove anything in NGB or AFA in ZFC there's no reason to
believe in classes or hypersets as fundamentally distinct sorts of
mathematical object, but that's a separate issue.)

But thoroughgoing model-theoretic structuralism changes the picture here
considerably. Given a model for any of these three theories, one also has
available a model for both of the others. Therefore, from at least one
kind of structuralist viewpoint - one which Shapiro endorses in his 1997
book (pp. 241-2), crediting the idea to a 1981 Philosophical Studies
article by Mark Wilson - these theories, by virtue of their very
interdefinability, are about the same objects, or have the same ontology.
(Just because having a model for one gets you models for the others as
well.)

ZFC, NGB, and AFA would then on this picture be construed as about the
'set structure', say, except that by the usual definitions that turns out
to be the same thing as the set-and-class structure and the
hyperset-structure. The 'face value' reading of the axioms of these
different theories and the statements you prove in them, which suggests
that they are about different objects that have different sorts of
properties, cannot be maintained if you also maintain this version of
structuralism. For they then seem to just be treatments of the same
fundamental mathematical 'stuff', though given to us in different
'syntactic guises'.

Similar arguably unpleasant identifications can be created throughout
mathematics, in a similar manner. The real numbers and the geometric
line, for instance, would appear to be literally the same thing to the
structuralist, not just distinct kinds of mathematical entity with the
same structure, and the real numbers between 0 and 1 and any given line
segment and probabilities the same things as well. I don't really want to
go much farther than this because one would have to make the form of
structuralism in question precise in order to continue the argument, and
then one would be in the realm of papers or books rather than FOM
postings, but this is the kind of worry I had in mind.