paint on an old chair, and the painter began to mix the paint. He put a little more red, he put a little more white–it still looked pink to me–and he mixed some more. Then he mumbled something like, “I used to have a little tube of yellow here to sharpen it up a bit–then this’ll be yellow.”
“Oh!” I said. “Of course! You add yellow, and you can get yellow, but you couldn’t do it without the yellow.”
The painter went back upstairs to paint.
The restaurant owner said, “That guy has his nerve, arguing with a guy who’s studied light all his life!”
But that shows you how much I trusted these “real guys.” The painter had told me so much stuff that was reasonable that I was ready to give a certain chance that there was an odd phenomenon I didn’t know. I was expecting pink, but my set of thoughts were, “The only way to get yellow will be something new and interesting, and I’ve got to see this.”
I’ve very often made mistakes in my physics by thinking the theory isn’t as good as it really is, thinking that there are lots of complications that are going to spoil it–an attitude that anything can happen, in spite of what you’re pretty sure should happen.
————————
A Different Box of Tools
————————
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
“Oh!” I said. “Of course! You add yellow, and you can get yellow, but you couldn’t do it without the yellow.”
The painter went back upstairs to paint.
The restaurant owner said, “That guy has his nerve, arguing with a guy who’s studied light all his life!”
But that shows you how much I trusted these “real guys.” The painter had told me so much stuff that was reasonable that I was ready to give a certain chance that there was an odd phenomenon I didn’t know. I was expecting pink, but my set of thoughts were, “The only way to get yellow will be something new and interesting, and I’ve got to see this.”
I’ve very often made mistakes in my physics by thinking the theory isn’t as good as it really is, thinking that there are lots of complications that are going to spoil it–an attitude that anything can happen, in spite of what you’re pretty sure should happen.
————————
A Different Box of Tools
————————
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
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