23 July 2006

Math is Fun

...

B: I wonder why this mathematics is so exciting now, when it used to be so dull in school. Do you remember old Professor Landau's lectures? I used to really hate that class: theorem, proof, lemma, remark, theorem, proof, what a total drag.

A: Yes, I remember having a tough time staying awake. But look, wouldn't our beautiful discoveries be just about the same?

B: True. I've got this mad urge to get up before a class and present our results: theorem, proof, lemma, remark. I'd make it so slick, nobody would be able to guess how we did it, and everyone would be so impressed.

A: Or bored.

B: Yes, there's that. I guess the excitement and the beauty comes in the discovering, not the hearing.

A: But it is beautiful. And I enjoyed hearing your discoveries almost as much as making my own. So what's the real difference?

B: I guess you're right, at that. I was able to appreciate what you did, because I had already been struggling with the same problem myself.

A: It was dull before, because we weren't involved at all; we were just being told to absorb what somebody else did, and for all we knew there was nothing special about it.

B: From now on whenever I read a math book, I'm going to try to figure out by myself how everything was done, before looking at the solution. Even if I don't figure it out, I think I'll be able to see the beauty of a proof then.

A: And I think we should also try to guess what theorems are comming up; or at least, to figure out how and why anybody would try to prove such theorems in the first place. We should imagine ourselves in the discoverer's place. The creative part is really more interesting than the deductive part. Instead on concentrating on finding good answers to questions, it's more important to learn how to find good questions!

B: You've got something there. I wish our teachers would give us problems like, "Find something interesting about x," instead of "Prove x."

A: Exactly. But teachers are so conservative, they'd be afraid of scaring off the "grind" type of students who obediently and mechanically do all the homework. Besides, they wouldn't like the extra work of grading the answers to nondirected questions. The traditional way is to put off all creative aspects until the last part of graduate school. For seventeen or more years students are taught examsmanship; then suddenly after passing enough exams in graduate school they're told to do something original.

B: Right. I doubt if many of the original students have stuck around that long.

A: Oh, I don't know, maybe they're original enough to find a way to enjoy the system. Like putting themselves into the subject, as we were saying. That would make the traditional college courses tolerable, maybe even fun.

B: You always were an optimist. I'm afraid you're painting too rosy a picture. But look, the rain has stopped. Let's lug this rock back to the camp and see what it says.

Taken from D. Knuth, Surreal Numbers.