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A Different

Box of

Tools

AT THE Princeton graduate school, the physics department and the math department shared a common longue, and every day at four o'clock we would have tea. It was a way of relaxing in the afternoon, in addition to imitating an English college. People would sit around playing Go, or discussing theorems. In those days Topology was the big thing.

I still remember a guy sitting on the couch, thinking very hard, and another guy standing in front of him, saying, "And therefore such-and-such is true."

"Why is that?" the guy on the couch asks.

"It's trivial! Itís trivial!" the standing guy says and he rapidly reels off a series of logical steps: " First you assume thus-and-so, then we have Kerchoff's this-and-that; then there's Waffenstoffer's Theorem, and we substitute this and construct that. Now you put the vector which goes around here and then thus-and-so..." the guy on the couch is struggling to understand all this stuff, which goes on at high speed for about fifteen minutes.

Finally the standing guy comes out the other end and the guy on the couch says, "Yeah, yeah. It's trivial."

We physicists were laughing, trying to figure them out. We decided that "trivial" means "proved". So we joked with the mathematicians: "We have a new theorem- that mathematicians can prove only trivial theorems, because every theorem that's proved is trivial."

 

The mathematicians didn't like the theorem, and I teased them about it. I said there are never any surprises - that the mathematicians prove only things that are obvious.

Topology was not at all obvious to the mathematicians. There were all kinds of weird possibilities that were "counterintutive". Then I got an idea. I challenged them: " I bet there isn't a single theorem that you can tell me- what the assumptions are and what the theorem is in terms I can understand - where I can't tell you right away whether it's true or false."

It often went like this: They would explain to me, "You've got an orange, OK? Now you cut the orange into a finite number of pieces, put it back together, and it's as big as the sun. True or False?"

"No holes?"

"No holes."

"Impossible! There ain't no such thing."

"Ha! We got him! Everybody gather around! It's So-and-so's theorem of immeasurable measure!"

Just when they think they've hot me, I remind them. "But you said an orange! You can't cut the orange peel any thinner than the atoms."

"But we have the condition of continuity: We can keep on cutting!"

"No, you said and orange, so I assumed that you meant a real orange."

So I always won. If I guessed right, great. If I guessed it wrong, there was always something I could find in their simplification that they left out.

Actually, there was a certain amount of genuine quality to my guesses. I had a scheme, which I still use today when somebody is explaining something that I'm trying to understand: I keep making up examples. For instance, the mathematicians would come in with terrific theorem, and they're all excited. As they're telling me the conditions of the theorem, I construct something which fits all the conditions. You know, you have a set (one ball) -disjoint (two balls). Then the balls turn colors, grow hairs, or whatever, in my head as they put more conditions on. Finally they state the theorem, which is some dumb thing about the ball which isn't true for my hairy green ball thing, so I say, "False!"

If it's true, they get all excited, and I let them go on for a while. Then I point out my counterexample.

"Oh. We forgot to tell you that it's Class 2 Hausdorff homomorphic."

"Well, then," I say, "It's trivial! It's trivial!" By that time I know which way it goes, even though I don't know what Hausdorff homomorphic means.

I guessed right most of the time because although the mathematicians thought their topology theorems were counterintuitive, they weren't really as difficult as they looked. You can get used to the funny properties of this ultra-fine cutting business and do a pretty good job of guessing how it will come out.

Although I gave the mathematicians a lot of trouble, they were always very kind to me. They were a happy bunch of boys who were developing things, and they were terrifically excited about it. They would discuss their "trivial" theorems, and always try to explain something to you if you asked a simple question.

Paul Olum and I shared a bathroom. We got to be good friends, and he tried to teach me mathematics. He got me up to homotopy groups, and at that point I gave up. But the things below I understood fairly well.

One thing I never did learn was Contour integration. I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr.Bader had given me.

One day he told me to stay after class."Feynman," he said, "you talk too much and you make too much noise. I know why. You're bored. So I'm going to give you a book. You go up there in the back, in the corner, and study this book, and when you know everything that's in this book, you can talk again."

So every physics class, I paid no attention to what was going on with Pascal's Law or whatever they were doing. I was up in the back with this book: Advanced Calculus, by Woods. Bader knew I had studied Calculus for the Practical Man a little bit, so he gave me the real works - it was for a junior or senior course in college. It had Fourier series, Bessel functions, determinants, elliptic functions - all kinds of wonderful stuff I didn't know anything about.

That book also showed how to differentiate parameters under the integral sign - it's a certain operation. It turns out that's not taught much in the universities; they don't emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals.

The result was, when the guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn't do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tool was different from everybody else's, and they had tries all their tools on it before giving the problem to me.