Taken from “Surely You're Joking, Mr.
Feynman!” Adventures of a Curious Character by Richard Phillips Feynman as told to Ralph Leighton edited by
Edward Hutchings
At
the Princeton graduate school, the physics department and the math department
shared a common lounge, 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! It’s 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 that theorem, and I teased them about it. I said
there are never any surprises—that the mathematicians only prove things that
are obvious.
Topology
was not at all obvious to the mathematicians. There were all kinds of weird
possibilities that were “counterintuitive.” 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 a thing.”
“Ha!
We got him! Everybody gather around! It’s So-and-so’s theorem of immeasurable
measure!”
Just
when they think they’ve got 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 an orange, so I assumed
that you meant a real
orange.”
So
I always won. If I guessed it 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 a 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 halls). 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 that 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 that 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 very 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 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 tools was different from
everybody else’s, and they had tried all their tools on it before giving the
problem to me.