Here is some of my personal experience with mathematics. It reflects some of my personal opinions. I am sure it is not a good example for everybody.

some years ago, during the full power days of string theory (or should I call it geometrical engineering), it seems that smartness is measured by the amount of mathematics one knows. You always hear things like this fiber, that cycle, this category, that functor, during PHYSICS talks. Finally, I decided that I had enough of it. I was going to take a serious math course (maybe more),taught by a real mathematician, and learn all those stuff once and for all.

I signed up for a graduate course in algebraic topology in math department. Let me not go into details about how I struggled through it. I came away with the conviction that I won't do that again, for the following reasons

1) For a physicist like me, I am far more interested in conclusions than the intermediate steps. I actually don't care there are at least 5 systematical ways of calculating the homotopy group. I just want to know what they are. Mathematicains are exactly the opposite. They seems care much less about the answer than building up complicated machineries to compute. One has to prove 20 theorems before one can say the fundamental group of a circle is Z.

Anything I said from my intuition is considered to be wrong. One has to invoke this extension theorem, that separation theorem in order to do anything. I did pretty well on the final exam since I finally learned their way of saying things.

2) Because this overwhelming emphasis on the formal proof, I did not learn that much things that is actually useful. We spent a lot of time on completely useless spaces like Hawaiian ear ring (it seems to be the counter example of every wrong theoerems). By the end of the semester, I could compute homology groups using 3 methods and so on. However, I still did not learn nearly as much of the useful result as later when I just quickly flipped through the book by Nakahara.

Even after that, I still held the belief that really good mathematicians must could somehow branch through both physics and mathematics (as I was told so so many times). I even had William Fulton on my thesis committee. My belief was shaken a little bit when I met Michael Atiyah. He is promised to be one of the foremost person who did something very important both in mathematics and physics. He gave us 3 lectures, ,not about his famous index theorem, but about K-theory. He promised to teach us K-theory from a very physical point of view. Instead, he spent 3 lectures give 3 different proofs of Bott periodicity theorem. the only physics is the first 15 minutes of the first lecture on Coulomb's law. Unfortunately, I didn't learn K-theory from him (I did get some ideas of it later on from d-brane annhilations).

Anyway, I decided that I am just too stupid to become a grand master of both mathematics and physics. And actually, I am not too depressed either. I have since noticed that almost all the good ideas in physics are indeed very physical and not mathematical at all. Mathematics comes in handy in formalizing them. It is of course true that knowing a lot of math will help and one cannot do anything purely without math. However, these days, I often excuse myself of being stupid on mathematics for the reason that I think one can still be a good physicist without thinking like a mathematician.

二零零五年六月十五日