Why Mathematics?

Terry Tao points to a beautiful article written by Michael Harris for the Princeton Companion to Mathematics, titled Why Mathematics, You Might Ask.

The titular question is the point of departure for a fascinating discussion on the foundations of mathematics, on the philosophy of mathematics, on post-modernism, on the "anthropology" approach to social science studies of mathematics, and on what mathematicians think they are doing, and why.

In general, I find articles on philosophical issues in mathematics to be more readable and enlightening when written by mathematicians. Perhaps it's just that they lack the sophistication of the working philosopher, a sophistication which I mistake for unreadability. But I also find that mathematicians tend to bring up issues that matter more to me.

For example, the metaphysical discussions on the "reality" of mathematical objects and the "truth" of theorems are all well and good, but the really interesting questions seem to be different ones.

The formalist view of mathematics, for example, according to which mathematics is the derivation of theorems from axioms via formal proofs, or as Hilbert apparently put it, "a game played according to certain simple rules with meaningless marks on paper," does not begin to capture what mathematics, just as "writing one sentence after another" does not capture what poetry is. (The analogy is due to Giancarlo Rota.) Indeed one of the main fallacies that follow by taking the extreme formalist position as anything more than a self-deprecating joke is to consider mathematical work as tautological. That is, to see a mathematical theorem as implicit in the axioms and so its proof as not a discovery. (Some of the comments in this thread relate to this point.) Plus, the view does not account for the difference between "recreational" mathematics and "real" mathematics, a difference that I don't find it easy to explain in a few words, probably because I don't have a coherent view of what mathematics really is.

It's not quite related, but I am reminded of a conversation I had a long time ago with Professor X about faculty candidate Y.

[Not an actual transcript, but close enough]

X: so what do you think of theory candidate Y?
Me: he is not a theory candidate.
X: but his results have no conceivable application.
Me: there is more to doing theory than proving useless theorems.
X: that's interesting! Tell me more

I enjoyed Harris's suggestion that "ideas" are the basic units of mathematical work, and his semi-serious discussion of whether ideas "exist" and on their importance.

There are indeed a number of philosophical questions about mathematics that I think are extremely interesting and do not seem to figure prominently in the social studies of mathematics.

For example, and totally randomly:

1. When are two proofs essentially the same, and when are they genuinely different?
   
2. What makes a problem interesting? What is the role of connections in this determination?
   
3. What makes a theorem deep?
   
4. What does it mean when mathematicians say that a certain proof explains something, or when they say that it does not?