Friday, April 17, 2009

Math Is About Isomorphism

I previously argued that math is about definitions. I am now ready to revise my position to the one in the title: math is about isomorphism. I feel that in some ways this idea is more advanced, and in other ways more primitive.

To say that things are "isomorphic" is to say that they have the same structure (lit: shape) in some important and useful ways. On the primitive side, of course math is about this! Consider these situations:

Mary has three chandeliers. John would like to borrow them, and promises Mary that if she lets him borrow them for a few months, he'll give them back along with a fourth chandelier.

I put \$3,000 in a CD. Several years later, the CD will pay out \$4,000.
These situations are isomorphic (in my sense, at least) and even the most elementary math is about recognizing isomorphisms so that we can use mathematical procedures across a variety of situations that are superficially dissimilar. If I have 3 apples and someone gives me 4 more apples, then how many apples will I have? 3 + 4 = 7.

Ed and I were talking about the coordinates being used in my geometry class, and how some aspects of projective spaces are isomorphic to linear algebra, and he made the obvious remark, "Linear algebra is isomorphic to a huge number of things."

And then he made the less obvious remark, which is coming to seem more and more true to me over time (although still not so very true), "It's amazing how few different things there are in math."

So, there you have it. Math is about isomorphisms.