FOM: Philosophy and platonism

[Mark Steiner]

        There have been a large number of postings on platonism in
mathematics
on this list.  Most of these postings seem to be unaware that there is
an extremely rich literature on this subject in recent philosophy of
mathematics(most of it apparently published by Oxford UP).  The authors
of this literature are very competent philosophers most of whom have a
good background in the foundations of mathematics as well and may even
have contributed also to the latter.  I believe that the lack of
reference to this advanced literature might help to explain the silence
of the philosophers on this topic (I'm assuming that there are
philosophers "out there" on the list).
        In particular, anti-platonists who think of themselves as tough
minded
scientists, ought to read the literature inspired by Quine's earlier
defense (with Putnam) of a form of platonism.  (The Quine-Putnam
arguments seem also be unknown to some of the posters here.) This
defense goes: since the mathematical sciences are indispensable for
science, they must be regarded as any other set of scientific claims.
Numbers are no more or less "fictional" than quarks or even atoms.
Science itself is the measure of fiction and reality (argues Quine) and
mathematics is part of science.  There is no stance outside of science
which could be adopted to claim that bona fide entities of a bona fide
indispensable scientific field like mathematics are "fictions."
        This argument can be resisted in various ways, for example, by
countering that mathematics is not indispensable to science, and, as I
say, there is a very big literature even if you only count recent books
published by Oxford UP.
        I really do not want to embroil myself in an extensive
discussion on
the Quine-Putnam thesis and related issues.  My appeal here (for those
interested in this subject and who are unaware of the literature) is to
take the integrity of philosophy seriously and do a little reading on
this subject--just as f.o.m. people have a justified complaint when
scholars outside their field write on subjects that have been treated in
depth by foundationalists.
        I believe a good introduction (which gives a lot of information
in a
fairly nonpartisan manner) is *A Subject with No Object* by Burgess and
Rosen (published by guess who).  It is an f.o.m.-sophisticated treatment
of the Quine-Putnam arguments and strategies to defeat it.  Their long
introduction is a very useful way to bring yourself up to the frontier
of philosophical research in this field.  I'm not disparaging the worth
of other books on this question like those of Field, Maddy, Shapiro,
Azzouni, Balaguer, Resnik, Hale, Katz, Chihara, and others, which take
particular
stands on the question.  Also, I'm not claiming that this is the only
aspect of Platonism that needs discussing, or that Platonism is the only
important subject in the philosophy of mathematics.  But it happens to
be an issue that has come up for discussion here, and I think that the
discussion would be enhanced if participants knew what philosophers--on
both sides of the question--have come up with.

Mark Steiner