The world of mathematics and theoretical physics is hierarchical. That was my first exposure to it. There's a limit beyond which one cannot progress. The differences between the limiting abilities of those on successively higher steps of the pyramid are enormous.See also Out on the tail. People who work in "soft" fields (even in science) don't seem to understand this stark reality. I believe it is because their fields do not have ready access to right and wrong answers to deep questions. When those are available, huge differences in cognitive power are undeniable, as is the utility of this power.
Nick Metropolis on von Neumann. See also his classic essay The Age of Computing.
Metropolis interview: ... Fermi and von Neumann overlapped. They collaborated on problems of Taylor instabilities and they wrote a report. When Fermi went back to Chicago after that work he called in his very close collaborator, namely Herbert Anderson, a young Ph.D. student at Columbia, a collaboration that began from Fermi's very first days at Columbia and lasted up until the very last moment. Herb was an experimental physicist. (If you want to know about Fermi in great detail, you would do well to interview Herbert Anderson.) But, at any rate, when Fermi got back he called in Herb Anderson to his office and he said, "You know, Herb, how much faster I am in thinking than you are. That is how much faster von Neumann is compared to me."Herman Goldstine on von Neumann:
Goldstine interview: ... If you ever look at Gauss' collected works, you'll see Gauss also loved to calculate. You just find in Gauss' works huge calculations that he undertook. It was a form of recreation. Von Neumann loved to do these things. It was a kind of being in touch with reality in a peculiar way. He would live to play mathematical games, such as the question of whether the numbers on a box car were prime or composite. He did calculations in his head that nobody else could do. He loved to do things like that. It was just part of his make-up. So calculation was not something abhorrent. Again, if you look in Gauss' collected works, you'll find all kinds of tabular things that he did. In fact, it was probably relaxing for each one of them to turn to calculation just for the fun of it.Both interviews are worth reading in their entirety. See also Only he was fully awake.
... I think they were very alike. I think different people's minds are differently constituted. I never particularly noticed any geometrical interests on von Neumann's part. He once told me he knew nothing about topology. Of course these have got to be taken as relative things. When he said he didn't know anything about topology, that probably meant he knew more than most people. But I think he loved to calculate. If you look at his book on quantum mechanics you'll find a number of things that you might conceivably do by other methods, he did do them by not numerical calculation, but by algebraical calculation. He was masterful at it. He could take the most elaborate formulas and manipulate them down until they were a couple of terms. This he loved. This was part of his virtuosity. You know, there was just nothing he liked so much as to do that.