Chapter 14 The second part accidentally broke into Princeton-6

In the Princeton Research Institute, the physics and mathematics departments share a common room. Every afternoon at 4 o'clock, we have tea there.On the one hand, it imitates the style of British schools, and on the other hand, it is also a good way to relax.Everyone will sit down and play chess, or discuss some theory.Topology was a hot topic in those days. I still remember one guy sitting on the couch trying to figure it out, and another guy standing in front of him and saying, "So, this and this is true." "Why?" asked the man sitting on the sofa. "It's so simple! It's so simple!" said the standing man, and went on to spout a series of logical inferences, "First you assume this and this, then we use Kerkoff's theory of this and that; then there's Waffenstower's theorem, let's substitute this and compose that. Now you put the vectors here, and so on..." The guy sitting on the sofa struggled to digest all these things, while the standing guy He spoke quickly and urgently for 15 minutes in one breath!After he finished, the guy on the couch said, "Yeah, yeah! It's really easy."

Those of us who study physics all laughed crookedly, and couldn't understand the logic of these two people.In the end we agreed that "simple" equals "proven". So we joked with these mathematicians, "We found a new theorem -- mathematicians only know how to prove the theorems that are very simple, because every theorem that has been proved is very simple." Those mathematicians don't like our theorems very much, so I'll play a joke on them again.I say that there will never be surprises in the world - precisely because mathematicians only prove simple things.

For mathematicians, topology is not such a simple subject. There are a lot of strange possibilities in it, which are completely "counter-intuitive".So I came up with another idea.I challenged them: "I bet you, if you come up with a theorem—if you tell me in a way I can understand, what it assumes, what the theorem is, etc.—I can tell you right away that it is Right or wrong!" Then the following happens: They tell me, "Suppose you have an orange in your hand. Well, if you cut it into N pieces, N is not an infinite number. Now you put the pieces back together and it turns out to be as big as the sun.Is this statement true or false? "

"Not a single hole?" "Not even half a hole." "Impossible! There is no such thing!" "Ha! We've got him! Come and see! It's so-and-so's 'immeasurable' theorem!" Just when they thought they'd stumped me, I reminded them, "You were talking about oranges! And you can't cut an orange peel thinner than an atom and break it into pieces!" "But we can use the continuity condition: we can cut forever!" "No, no, you were talking about an orange, so I assume you're talking about a real orange."

So I always win.Best if I guess right.If I'm wrong, I always have a way of finding holes in their narratives. In fact, I am not just guessing randomly.I have a method that I still use even today when someone is explaining something to me and I am trying to understand it: go on and on with examples. For example, those who study mathematics come up with a theorem that sounds amazing, and everyone is very excited.When they told me the conditions of the theorem, I imagined the situations that would satisfy them.When they say "set" in mathematics When I think of a ball, two incompatible sets are two balls.Then depending on the situation, the ball might take on a different color, grow hair, or do other weird things.Finally, when they came up with that baby theorem, I declared, "No!"

They would be delighted if I said their theorem was true. But I only made them happy for a while, and then I presented my counterexample. "Oh, we forgot to tell you that this is Hausdorff's homomorphism theorem of the second kind." So I said, "Well, that's so easy, so simple!" At that point, even though I had no idea what Hausdorff homomorphisms were, I knew I was right.Although mathematicians think their topology theorems are counter-intuitive, the reason I get it right most of the time is that they are not as difficult to understand as they appear.Gradually, you get used to the odd nature of those fine divisions, and your guesses become more and more accurate.

However, although I often made trouble for this group of mathematicians, they have always been very kind to me.They were a happy bunch of guys who made it their job to come up with theories and had fun doing it.They often discuss those "simple, trivial" theories; and when you ask a simple question, they always try to explain it to you. The mathematician I share the bathroom with is a mathematician named Paul Olum.We became good friends and he always wanted to teach me math.I finally gave up when I got to the level of "homotopy groups"; but everything below that level I understand pretty well.

What I have never learned is "contour integration". Mr. Bede, a high school physics teacher, gave me a book. All the integral methods I know are learned from this book. The thing is this: one day after class, he asked me to stay. "Feynman," he said, "you talk too much in class, and your voice is too loud. I know you find these classes boring, and now I'm giving you this book. Then you sit in the back corner and read this book." Book, I will not allow you to speak until you understand everything." So every time I was in physics class, no matter what the teacher taught was Pascal's law or something else, I ignored it.I sat in the corner of the classroom and read this "Advanced Calculus" by Woods.Bader knew I'd read a little of Practical Calculus, so he gave me this really big tome—a textbook for second and third year college students.There are Fourier series, Bessel functions, determinants, elliptic functions—all kinds of wonderful things that I didn't know before.

That book also teaches you how to differentiate with respect to parameters within integral notation.I later found out that this technique isn't taught much in college courses, but I got the hang of it and used it again and again.Therefore, relying on self-study of that book, the way I do points is often different. What often happens is that my friends at MIT or Princeton get stumped with some points because the standard methods they learned in school don't work. If it had been a contour integral or a series expansion, they would have figured out how to figure it out; now they hit a wall.At this time, I resorted to the method of "differentiation within the integral sign"-this is because I have a different toolbox.When other people have run out of their tools and can't find an answer, leave it to me!