A long time ago, Kristy posted something questioning the value of the million dollar Clay Institute prizes for solving the major unsolved mathematical problems. I meant to post something responding to that, but I forgot, and then it seemed really late, but whatever. Here's the post now. Bit of a disclaimer-- I can't find the actual post, so I'm relying on Will's response (sounds like a problem in historical research-- the original is lost and only some 10th century monastical interpolation survives. But I digress). If I'm misrepresenting Kristy's post, I'm sorry.

Kristy questioned the necessity of giving the prize at all. Couldn't the money be better spent? Well, yes and no. I mean, I don't believe that the prize increases the desire of people to solve these problems. If you're a research mathematician, you're not a research mathematician because of the chance of getting a million dollar prize for solving some unsolved problem. As Hardy said in The Mathematician's Apology "The case for my life, then or for that of any one else who has been a mathematician in the same sense in which I have been one, is this: that I have added something to knowledge, and helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any other artists, great or small, who have left some kind of memorial behind them." If you do research on the level of Jaffee or Lafforgue or Hardy, you doing that research because you can't imagine doing anything else. So the million dollar prize isn't a motivator. On the other hand, it does make things a bit easier. You've got this money with no strings attached. You don't have to justify a trip to Russia to talk with a possible future collaborator or spending six months just thinking about a problem. If it doesn't pan out, it doesn't pan out, but it might.

As for the usefulness of the research, the short answer is that we don't know whether it will be useful or not. It's possible that there will be no practical applications of the solution of the Poincare conjecture (and since I believe most mathematicians believed it was true and acted accordingly, I kind of think this will be the case). However, pure mathematics definitely has impact on the real world. Just a few examples I know: one of my friends last summer worked on applications of algebraic topology to quantum computing (I didn't understand it really, so don't ask me how it worked, though if you want to read his paper, I have a copy) and ideas from number theory will generate the next advances in codebreaking and codemaking.

I think ultimately what math can do, beyond its immediate practical applications, is help us to understand the universe. Whatever applications of GR that we've used, I think that the fundamental understanding of spacetime is more important for those, both because of the longterm applications and because I believe in the search for knowledge for its own sake.

Ultimately, though, I see these prizes as a reward. The people who manage to solve these problems have done something great in the advancement of pure knowledge. They've done something most people can't do. How much do we pay pro atheletes? A really talented mathematician is much rarer than a really talented quarterback. Shouldn't we pay them accordingly?

I think mathematicians kind of get screwed. Most people can see the beauty in a Bach fugue or in a perfect pitch. They can't (or believe they can't) see this beauty in a proof. So they don't see mathematicians as artists; nor do they really understand what mathematicians do or why they do it. Hell, I think my relatives think I sit around computing derivatives all day, and people I meet at parties seem to think I should be good at calculating gratuities. So most people kind of brush off mathematicians as "those weird people who sit around doing calculus all day." Weird? probably justified. The rest maybe not so much. Mathematicians create proofs to explain things that are true. When I was in high school, I was told "if you want to understand how something works, be an engineer. If you want to understand why it works, be a physicist." To that I'd add "If you want to understand why we can even think about it in this way, be a mathematician."