Style. As a writer of mathematics von Neumann was clear, but not clean; he was powerful but not elegant. He seemed to love fussy detail, needless repetition, and notation so explicit as to be confusing. To maintain a logically valid but perfectly transparent and unimportant distinction, in one paper he introduced an extension of the usual functional notation: along with the standard φ(x) he dealt also with something denoted by φ((x)). The hair that was split to get there had to be split again a little later, and there was φ(((x))), and, ultimately, φ((((x)))). Equations such as
(φ((((a))))^2 = φ(((a))))
have to be peeled before they can be digested; some irreverent students referred to this paper as von Neumann’s onion.
Perhaps one reason for von Neumann’s attention to detail was that he found it quicker to hack through the underbrush himself than to trace references and see what others had done. The result was that sometimes he appeared ignorant of the standard literature. If he needed facts, well-known facts, from Lebesgue integration theory, he waded in, defined the basic notions, and developed the theory to the point where he could use it. If, in a later paper, he needed integration theory again, he would go back to the beginning and do the same thing again. He saw nothing wrong with long strings of suffixes, and subscripts on subscripts; his papers abound in avoidable algebraic computations. The reason, probably, is that he saw the large picture; the trees did not conceal the forest from him. He saw and he relished all parts of the mathematics he was thinking about. He never wrote “down” to an audience; he told it as he saw it. The practice caused no harm; the main result was that, quite a few times, it gave lesser men an opportunity to publish “improvements” of von Neumann.
(tangent: I’m a bit peeved by Halmos’ “lesser men” throwaway remark, mainly because I think interpretive research labor and distillation is very valuable, very hard to do well, somewhat orthogonal to vN-style competence, and very underappreciated and undersupplied.)
von Neumann was also courageous, Halmos wrote, in the following way:
Another notable and enviable trait of von Neumann’s was his mathematical courage. If, in the middle of a search for a counterexample, an infinite series came up, with a lot of exponentials that had quadratic exponents, many mathematicians would start with a clean sheet of paper and look for another counterexample. Not Johnny! When that happened to him, he cheerfully said: “Oh, yes, a theta function...’’, and plowed ahead with the mountainous computations. He wasn’t afraid of anything.
More specifically, one thing I learned from Terry that I was not taught in school is the importance of bad proofs. I would say “I think this is true”, work on it, see that there was no nice proof, and give up. Terry would say “Here’s a criterion that eliminates most of the problem. Then in what’s left, here’s a worse one that handles most of the detritus. One or two more epicycles. At that point it comes down to fourteen cases, and I checked them.” Yuck. But we would know it was true, and we would move on. (Usually these would get cleaned up a fair bit before publication.) …
Sometimes we’d really be on the same page, at the same letter of the same word even; one extreme case was when I needed to read his computer code and found it as easy to do as if I’d written it myself. But more often we’d bring different strengths. Since we were working in my field of expertise rather than his, I knew better what the interesting questions were, and could translate them into combinatorics, then sic Terry on them. He would beat them to a bloody death as described above, and then it would be my job to dress the carcass for public viewing back in the original field.
von Neumann also had endless capacity for work. Halmos:
Work habits. Von Neumann was not satisfied with seeing things quickly and clearly; he also worked very hard. His wife said “’he had always done his writing at home during the night or at dawn. His capacity for work was practically unlimited.” In addition to his work at home, he worked hard at his office. He arrived early, he stayed late, and he never wasted any time. He was systematic in both large things and small; he was, for instance, a meticulous proofreader. He would correct a manuscript, record on the first page the page numbers where he found errors, and, by appropriate tallies, record the number of errors that he had marked on each of those pages. Another example: when requested to prepare an abstract of not more than 200 words, he would not be satisfied with a statistical check — there are roughly 20 lines with about 10 words each — but he would count every word.
I thought this was striking: why waste time on such seeming trivialities? But I guess if you’re John von Neumann you just have such a glut of brain cycles that you can spend it in ridiculously poorly-optimised ways like this instead of needing to 80⁄20 and still get your many, many jobs done.
I have this experience with @ryan_greenblatt—he’s got an incredible ability to keep really large and complicated argument trees in his head, so he feels much less need to come up with slightly-lossy abstractions and categorizations than e.g. I do. This is part of why his work often feels like huge, mostly unstructured lists. (The lists are more unstructured before his pre-release commenters beg him to structure them more.) (His code often also looks confusing to me, for similar reasons.)
Interesting anecdote on “von Neumann’s onion” and his general style, from P. R. Halmos’ The Legend of John von Neumann:
(tangent: I’m a bit peeved by Halmos’ “lesser men” throwaway remark, mainly because I think interpretive research labor and distillation is very valuable, very hard to do well, somewhat orthogonal to vN-style competence, and very underappreciated and undersupplied.)
von Neumann was also courageous, Halmos wrote, in the following way:
Terry Tao is similar, according to Allen Knutson:
von Neumann also had endless capacity for work. Halmos:
I thought this was striking: why waste time on such seeming trivialities? But I guess if you’re John von Neumann you just have such a glut of brain cycles that you can spend it in ridiculously poorly-optimised ways like this instead of needing to 80⁄20 and still get your many, many jobs done.
I have this experience with @ryan_greenblatt—he’s got an incredible ability to keep really large and complicated argument trees in his head, so he feels much less need to come up with slightly-lossy abstractions and categorizations than e.g. I do. This is part of why his work often feels like huge, mostly unstructured lists. (The lists are more unstructured before his pre-release commenters beg him to structure them more.) (His code often also looks confusing to me, for similar reasons.)