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
suchandsuch 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
thusandso, then we have Kerchoff's thisandthat; then there's
Waffenstoffer's Theorem, and we substitute this and construct that. Now
you put the vector which goes around here and then thusandso..."
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 Soandso'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 ultrafine 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 selftaught 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.