(1) If the topic being considered were written up in a book and sent to
the library, would the librarians classify it as mathematics? (e.g.,
under QA in U.S. Library of Congress, or under 510 in Dewey Decimal).
How do librarians decide what should be classified as mathematics?
(I'm going to ask my school's math librarian about that.)
(2) If the topic being considered were written up in an article and
published in a journal, could it get reviewed in Math Reviews? How do
the editors of MR decide what is mathematics? (I'm going to ask a
colleague who I believe is a former editor of MR.)
(3) Perhaps someone else in this list can suggest some other operational
criteria.
Without such criteria, I see the question of "what is mathematics" as
rather amorphous. I don't see mathematics as having very sharp
boundaries; it blends into adjacent subjects. Different mathematicians
will have different views about what is mathematics. A couple of
centuries ago, physics and mathematics were essentially one subject;
it is only today that we see them as separate subjects. Even today,
mathematicians are not in agreement on certain issues of classification
-- e.g., should the computer-aided proofs of the Four Color Theorem be
considered legitimate mathematics?
Eric Schechter
http://www.math.vanderbilt.edu/~schectex/