Date: Sun, 31 Jan 1999 20:55:47 -0500 (EST)
From: Dev Sinha <dps@math.brown.edu>
Subject: what is math? (to toplist)


I know I'm late on this, but I was away when the flood of e-mail hit.  And
I'd like to give my two cents as I've had some success in answering this
question when asked by non-mathematicians.  In particular, I try to give
my students a sense of this question (though more a sense of what its like
to really do mathematics) whenever I teach.

Instead of an elegant, concise answer such as Gottlieb's, I try to give
examples first and then move to general discussion.  One of my favorite
examples to give is the sums-of-squares theorem, as it is both simple and
striking.  By giving someone data first (3 isnt, 5 is, 7 isnt...) and then
giving the theorem, they get a sense of mathematics as akin to
experimental science.  Of course, this facet isn't the only facet of
mathematics.  So one can point to the development of algebra to highlight
how mathematics is a remarkably flexible tool.  And one can point to
examples of focus on the internal structure of mathematics, such as the
fact that there are infinitely many primes or the question of
"categorizing ways in which space can curve in on itself" which is how I
explain one of the fundamental problems of topology to non-mathematicians.

There was a second question which almost came up in this list's
discussion, which I am sometimes asked, namely, "Is mathematics part of
the `real world'?"  Of course, this begs the question "what is the real
world?"  The point I bring up is that the real world we experience is
experienced through constructs/context/abstraction.  For example, if I
slow down on the highway because of a police car behind me, its not that
I'm slowing down because of the police car but becuase of the _idea of_ a
police car - I would also slow down if I mistook a car with a roof-rack as
a police car.  So for me the idea of a finite group is much like the idea
of a police car, one of the main differences being that the idea of a
finite group doesn't have a physical instance in the same way that the
idea of a police car does.  Of course there are many more differences, and
here is where analysis such as "Gottlieb's razor" (of well-definedness)
fits in nicely.

I don't claim that these are complete analyses, but they have helped me
get off the ground in talking to non-mathematicians about what I do.

-Dev