We were exposed to chemistry, to physics, to biology; there were no electives when you were in secondary school. Secondary schools in Europe, in Poland, in France were in a certain sense harder than the university because you had to learn a prescribed curriculum. There was no nonsense. If you were in a certain type of school. you had to take six years of Latin and four years of Greek and no nonsense about taking soul courses or folk music, or all that. I have nothing against taking such courses, except that it has become a substitute. You had to take physics, you had to learn a certain amount of chemistry, of biology, and if you didn't like it, so it was. But if there was some kind of resonating note in you, then you were introduced to it early.
At the university you really specialized, although not entirely: every mathematician had, for example, to pass an exam in physics and even, God help me, go through a physics lab. That was one of my most expensive experiences because, being rather clumsy, I broke more Kundt's tubes than I could afford.
Stan [Ulam] made an extremely important point to which I can bring a little extra light. I heard probably one of the last speeches by von Neumann. It was in May 1955. (In October of that year, while I was in Geneva on leave, it was discovered that he had incurable cancer, and he died then sometime later in 1957.) He was the principal banquet speaker at the meeting, I believe, of the American Physical Society in Washington. I was there, and I went to the meeting, and after the speech we had a drink together. His speech was, "Why Am I Not a Physicist?" or something of the sort. He explained that he had contributed technical things to physics; for example, everybody knows what a density matrix is, and it was von Neumann who invented density matrices, as well as a hundred other things that are now, so to speak, textbook stuff for theoretical physicists.
But he, nevertheless, gave a charming and also moving talk about why he was not really a physicist. and one thing he mentioned was that he thought in terms of symbols rather than of objects; I am reminded that his friend Eugene Wigner hit on it correctly by saying that he would gladly give a Ph.D. in physics to anyone who could really teach freshman physics. I know what he meant. I would attempt, I wouldn't be very good at it, but I would attempt to teach a first semester course in quantum mechanics, and I would probably teach it reasonably well.
But I would not know how to teach a freshman course in physics, because mathematics is, in fact, a crutch. When you feel unsafe with something, with concepts, you say, "Well now. let's derive it." Correct? Here is the equation, and if you manipulate with it, you finally get it interpreted, and you're there. But if you have to tell it to people who don't know the symbols, you have to think in terms of concepts. That is in fact where the major breach between the two - how to say - the two lines of thought come in. You are either like von Neumann, and I am in that sense closer to him, or you are like Ulam, who when you say pressure, feels it. It is not the partial derivative of the free energy with respect to volume; it is really something you feel with your fingers, so to speak.
... the really good [mathematician has a very strong conceptual understanding of the things he is working on]. But then, you see, there is a gamut, a continuum. In fact, let me put this in because I would like to record it for posterity. I think there are two acts in mathematics. There is the ability to prove and the ability to understand. Now the actions of understanding and of proving are not identical. In fact, it is quite often that you understand something without being able to prove it. Now, of course, the height of happiness is that you understand it and you can prove it. The next stage is that you don't understand it, but you can prove it. That happens over and over again, and mathematics journals are full of such stuff. Then there is the opposite, that is, where you understand it, but you can't prove it. Fortunately, it then may get into a physics journal. Finally comes the ultimate of dismalness, which is in fact the usual situation, when you neither understand it nor can you prove it.
The way mathematics is taught now and the way it is practiced emphasize the logical and the formal rather than the intuitive, which goes with understanding. Now I think you would agree with me because, especially with things like geometry, of which Stan's a past master, seeing things - not always leading neatly to a proof, but certainly leading to the understanding - ultimately results in the correct conjecture. And then, of course, the ultimate has to be done also - because of union regulations, you also have to prove it.
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