Richard Feynman - about the differences between mathematics and physics

Now I would like to make a number of remarks on the relation of mathematics and physics, which is a little more general. The first is that the mathematicians only are dealing with the structure of the reasoning, and they do not really care about what they're talking. They don't even need to know what they're talking about, or as they themselves say, or whether what they say is true. Now I explain that. If you state the axioms, you say such and such and so and such and such and so and such and such and so, what then? Then the logic can be carried out without knowing what the such and such words mean. That is, if the statements about the axioms are true, I mean, are carefully formulated and complete enough, it is not necessary for the man who's doing the reasoning to have any knowledge of the meaning of these words, and he'll be able to deduce in the same language new conclusions. If I use the word triangle in one of the axioms, there'd be some statement about triangles in the conclusion, whereas the man who's doing the reasoning might not know what the triangle is. But then I can read his thing back and say, oh, a triangle, that's just a three-sided what-have-you that's a so-and-so, and so I know this new fact. In other words, mathematicians prepare abstract reasoning that's ready to be used if you will only have a set of axioms about the real world. But the physicist has meaning to all the phrases, and there's a very important thing that a lot of people who study physics and come from mathematics don't appreciate. The physics is not mathematics, and mathematics is not physics. One helps the other. But you have to have some understanding of the connection of the words with the real world. It's necessary at the end to translate what you figured out into English, into the world, into the blocks of copper and glass that you're going to do the experiments with to find out whether the consequences are true. And this is a problem which is not a problem of mathematics at all. I've already mentioned on the other relationship that, of course, it's obvious how the mathematical reasonings which have been developed are of great power and use for physicists. That the, on the other hand, sometimes a physicist's reasoning is useful for mathematicians. Mathematicians also like to make their reasoning as general as possible. If you say, I have a three-dimensional space, an ordinary space. I want to talk about ordinary space. You know, you're in it, and you measure distances, and there are three numbers you need to tell where something is. You go breadth, width, and height. Three-dimensional space, and you begin to ask them about theorems. Then they say, now look, if you had a space of n dimensions, then here are the theorems. Well, I say, yeah, but I only want the case three. Well, substitute n equals three. And then it turns out, and then it turns out that very many of the complicated theorems they have are much simpler because it happens to be a special case. Now, the physicist is always interested in a special case. He's never interested in the general case. He does, he's talking about something. He's not talking abstractly about anything. He knows what he's talking about. He wants to discuss the gravity law. He doesn't want the arbitrary force case. He wants the gravity law. And so there's a certain amount of reducing because the mathematicians have prepared these things for a wide range of problems, which is very useful. And later on, it always turns out that the poor physicist has to come back and says, excuse me, when you wanted to tell me about the four dimensions. Now, another item that's interesting in this relationship is the question of how to do new physics. Is it important to have a feeling, a kind of intuition? Oh, I must mention one other item. When you know what it is you're talking about, that these things are forces and these are masses and this is inertia and this is so on, then you can use an awful lot of common sense, seat-of-the-pants feeling about the world. You've seen various things. You know more or less how the phenomenon is going to behave. Well, the poor mathematician, he translates it into equations and the symbols don't mean anything to him, and he has no guide but precise mathematical rigor and care in the argument. Whereas a physicist who knows more or less how the answer is going to come out can sort of guess partway and go right along rather rapidly. The mathematical rigor of great precision is not very useful in the physics, nor is the modern attitude in mathematics to look at axioms. Now, mathematicians can do what they want to do. One should not criticize them because they are not slaves to physics. It is not necessary that just because this would be useful to you, they have to do it that way. They can do what they will. It's their own job. And if you want something else, then you work it out yourself. The next point is the question of whether we should guess when we try to get a new law, whether we should use the seat-of-the-pants feeling and philosophical principles. I don't like a minimum principle, or I do like a minimum principle, or I don't like action at a distance, or I do like action. The question is, to what extent models help? And it's a very interesting thing. Very often models help, and most physics teachers try to teach how to use these models and get a good physical feel for how things are going to work out. But the greatest discoveries, it always turns out, abstract away from the model, it never did any good. Maxwell's discovery of electrodynamics was first made with a lot of imaginary wheels and idlers and everything else in space. If you got rid of all the idlers and everything else in space, the thing was okay. Dirac discovered the correct laws of quantum mechanics for relativity, quantum mechanics, simply by guessing the equation. And the method of guessing the equation seems to be a pretty effective way of guessing new laws. This shows, again, that mathematics is a deep way of expressing nature, and attempts to express nature in philosophical principles or in seat-of-the-pants mechanical feelings is not an efficient way. I must say that there is possible, and I've often made the hypothesis, that physics ultimately will not require a mathematical statement, that the machinery ultimately will be revealed, just a prejudice like one of these other prejudices. It always bothers me that in spite of all this local business, what goes on in a tiny, no matter how tiny a region of space, and no matter how tiny a region of time, according to the laws as we understand them today, takes a computing machine an infinite number of logical operations to figure out. Now how can all that be going on in that tiny space? That why should it take an infinite amount of logic to figure out what one stinky, tiny bit of space-time is going to do? And so I made a hypothesis often that the laws are going to turn out to be, in the end, simple like the checkerboard, and that all the complexity is from size. But that is of the same nature as the other speculations that other people make. It says, I like it, you don't like it. It's not good to be too prejudiced about this thing. To summarize, I would use the words of Jeans, who said that the great architect seems to be a mathematician. And for you who don't know mathematics, it's really quite difficult to get a real feeling across up to the beauty of the deepest beauty of nature. C.P. Snow talked about two cultures. I really think that those two cultures are people who do, and people who have had and people who have not had this experience of understanding mathematics well enough to appreciate nature once. It's too bad that it has to be mathematics, and that mathematics for some people is hard. When one of the, it's reputed, I don't know if it's true, that when one of the kings was trying to learn geometry from Euclid, he complained that it was difficult. And Euclid said that there's no royal road to geometry. And there's no royal road. It's not the, we cannot, as people who've looked at this thing as a physicist, cannot convert this thing to any other language. You have, if you want to discuss nature, to learn about nature, to appreciate nature, it's necessary to find out the language that she speaks in. She offers her information only in one form. We are not so unhumble as to say, the man that she changed before we pay any attention. It seems to me that it's like all the intellectual arguments that you can make would not in any way, or very, very little, will communicate to deaf ears what music, the experience of music really is. And all the intellectual arguments in the world will not convince those of the other culture. The philosophers who try to teach you by telling you qualitatively about this thing. Me, who's trying to describe it to you, but it's not getting across because it's impossible. I'm talking, we're talking to deaf ears. And it's when they, it's perhaps that the horizons are limited, which permits such people to imagine that the center of the universe of interest is man. Thank you.