..... Physics is validated by experiment. Math is validated
by rigor. ....
That is what philosophers say. They want laymen think that in
mathematics proofs come before discoveries and in physics
discoveries are made in laboratories, according to "experimental
method". In reality, as pointed out by others on this list, neither
physicists nor mathematicians feel obliged to follow the paths
prescribed by philosophers. The proof is in the pudding, not in
the way in which it was prepared. Each of us can give examples
of accidental discoveries in science, or discoveries based on the
basis of who knows what.
What philosophers say is often true but not always. Yes, a logical
derivation is an agreed upon method of validation of mathematical
theorems but "applied math" people are not waiting for formal
proofs. They verify postulated relations numerically and use them
in applications. A proof is more often than not the last step which
has nothing to do with the way in which a relation was actually
discovered.
About a month ago I read "What is Mathematics, Really?" by
R. Hersh (Oxford University Press, 1997, ISBN 0-19-511368-1).
The author is a mathematician but his main point is to show that
the way in which mathematics is practiced is very different from
the ways in which it is presented to students. Real math, he
claims, is not so drastically different from real science. He also
reflects on the role of computers in mathematics. A good book
to have in your school library ($17 in paperback).
Ludwik Kowalski