FOM: Connections between mathematics, physics and FOM

[Mark Steiner]

Just a footnote to Jeffrey's very useful, comprehensive catalogue.  (I
hope he is able to enlighten us on many of these problems.)

The idea that mathematics applies to the Universe because the universe
exemplifies mathematical structures is really a remark about the logical
form and the logical possibility of the description of the world by
mathematical structure; that is, set theoretical predicates.  It's an
explanation in the sphere of semantical application (similar to the
explanation by Frege of the applicability of arithmetic I described in
the last posting) or logic.  (Frege did not like to talk about
mathematical structures, because it reminded him of formalism, which he
hated.)

However, many people who read Gardiner and others, believe that the
statement "the Universe exemplifies mathematical structures" sheds some
light on the epistemological problems of mathematical applicability like
Wigner's or mine.  To this I say that the statement that the Universe
exemplifies mathematical structures (set theoretical predicates) is
simply a tautology. (I don't suggest Jeff says anything different, but I
believe Gardiner and others missed this point.)  It becomes an empirical
claim (or can become one) only when the set theoretical structures that
are mathematical are specified.

And, permit me once more to stress that in my treatment, it is not just
the fact that particular concepts of mathematics are used to describe
phenomena, but rather the concept "mathematics" itself (i.e. the
CLASSIFICATION of mathematical structures as "like" and "unlike" one
another BY mathematicians) plays a dominant role in this century in
DISCOVERING these descriptions.

Discovery is very important; a recent paper by a mathematical physicist
argues that for physics we don't need differentail equations over the
reals at all, we could use difference equations over the rationals,
given that we can represent all possible measurements by rational
numbers.  But, the physicist goes on, we could never make discoveries
without the mathematical form of the differential equations themselves.

Mark Steiner