Mathematicians From The United States

"There were several times in my life that I’ve, one way or another, got that feeling, my gosh, here is a tremendous mathematician; for instance, Weil, von Neumann, Serre, Milnor, Atiyah. Well, those are obvious names."

"Unquestionably the nearest thing to a genius I have ever encountered."

"I was privileged to have known von Neumann personally and, like most mathematicians of my generation, I have been strongly influenced by his work and by his person."

"Getting to know von Neumann better was one of the delights of my stay in Princeton. Apart from being one of the greatest mathematicians of our century, he was a wonderful companion."

"As a mathematician, Steinhaus’s main strengths were his intelligence and an unerring instinct and taste in the choice of problems. In this respect he reminded me of John von Neumann, a mathematician whom he greatly liked and admired."

"Banesh Hoffmann: Yes, he was incredible."

"Albert William Tucker: No, but it's typical."

"Banesh Hoffmann: He thought very fast, yes, and he was extraordinarily subtle. He was most impressive. You've heard the story of Robertson driving van Neumann to somewhere. Von Neumann asked him what he was working on, and Robertson said such and such an equation. By the time they got to the end of the ride von Neumann had solved the equation in his head. Had you heard that?"

"His extraordinariness lay in his mental abilities. These were so dazzling that some of his admiring colleagues were at a loss to describe them in ordinary human terms."

"Fantastic mind."

"And about Johann von Neumann, the mathematics lecturers seem to have even told stories to their students during lectures, as Alexander Dinghas (1908–1974) vividly described in his memories: Thus, Issai Schur reported to students in a lecture that the student von Neumann, in a seminar where a proof of the "Minkowski theorem on the estimation of linear forms" was being treated, had stood up and "added great simplifications to the presented proof"."

"Fraenkel later reported impressively that he only managed with great effort to work through von Neumann's work, which "differed from everything that had appeared up to then on the axiomatization of set theory" and introduced completely new concepts, and that he was immediately convinced of von Neumann's quite extraordinary talent."

"Von Neumann did not seem particularly interested in studying chemistry. On the other hand, it is documented that he attended lectures by Haber, and the latter allegedly expressed to friends the wish that von Neumann should pursue an academic career in chemistry."

"Von Neumann was a true genius, the only one I’ve ever known. I’ve met Einstein and Oppenheimer and Teller and—who’s the mad genius from MIT? I don’t mean McCulloch, but a mathematician. Any-way, a whole bunch of those other guys. Von Neumann was the only genius I ever met. The others were supersmart .... And great prima donnas. But von Neumann’s mind was all-encompassing. He could solve problems in any domain. . . . And his mind was always working, always restless. He walked into my living room one night and a half dozen people were already having cocktails, and he disappeared into a corner and stood with his back to us, hands behind him, and after about two minutes turned to me and said, “About two thirds of a liter a week, Leon.” And I had to think about it for three or four minutes, and finally I said, “Yeah, Johnny, that’s just about right.” He’d walked up to the nine-gallon tropical fish aquarium that stood on a table in the corner, had noted the temperature of the water, had made an estimate of the surface area, had seen the gap that existed between the overhead light and the glass to keep the fish from jumping out, made an estimate of the particular escape velocity of the water molecules, integrated and found out how much added water was needed each week for that aquarium. And he was right within a few percent. That’s the kind of thing he did all the time. Another thing that he isn’t known well for was his sense of humor. He really enjoyed dirty limericks. And though we never said anything to each other deliberately, it sort of evolved that whenever we came together, whether it was an hour or a month later, the name of the game was to see who could rush up the fastest and unload the largest number of new limericks. It turned out to be a delightful game. He had oodles of them; I was hard put to keep up with him. His memory was just beyond conception, a photograph for everything he ever learned or saw. Lightning calculator and head screwed on to boot—he put all of those together with a huge creative talent."

"Professors at the university direct doctoral theses but those at the Institute do not. Unaware of this, in 1934 I asked von Neumann if he would direct my doctoral thesis. He replied Yes, and suggested the problem of identifying the Hilbert space closure and adjoint of nth-order linear differential operators. Marshall Stone, in his huge volume Linear transformations in Hilbert Space, had solved the case for first order and his methods generalized to higher orders. My not particularly outstanding thesis was accepted and I moved into an ardent study of continuous geometry. In 1936, as a postdoctoral Fellow at Yale, I found a partly new proof, with weaker axioms, for von Neumann's transitivity of perspectivity. Von Neumann invited me to visit Princeton and talk with him. He gave me most cordial encouragement, let me have his unpublished manuscripts to study, and later took the initiative to recommend me to Marshall Stone for a B. P. Instructorship at Harvard. This warm, generous concern made a deep impression on me."

"In this galaxy of stars von Neumann, a professor at the Institute, simply radiated excitement. His lectures on Hilbert Space, measure theory, rings of operators (called now von Neumann algebras), and continuous geometry, fascinated a large audience. At the daily afternoon tea he engaged some group in a most lively and stimulating discussion. With obvious delight he explained, clarified, and analyzed problems on the spot and gave help to one and all. But sometimes he would stand apart, deep in thought, his brown eyes staring into space, his lips moving silently and rapidly, and at such times no one ventured to disturb him."

"Throughout the world mathematicians and others had marvelled at the lightning speed with which von Neumann analyzed and solved complex problems."

"I was fascinated by whatever von Neumann did."

"I was absolutely fascinated with von Neumann; I still am."

"Keeping up with him was... impossible. The feeling was you were on a tricycle chasing a racing car."

"I became Johnny’s assistant. How was it? Scary. The most spectacular thing about Johnny was not his power as a mathematician, which was great, or his insight and his clarity, but his rapidity; he was very, very fast."

"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 \phi(x) he dealt also with something denoted by \phi((x)). The hair that was split to get there had to be split again a little later, and there was \phi(((x))), and, ultimately, \phi((((x)))). Equations such as"

"One famous story concerns a complicated expression that a young scientist at the Aberdeen Proving Ground needed to evaluate. He spent ten minutes on the first special case; the second computation took an hour of paper and pencil work; for the third he had to resort to a desk calculator, and even so took half a day. When Johnny came to town, the young man showed him the formula and asked him what to do. Johnny was glad to tackle it. "Let's see what happens for the first few cases. If we put n = 1, we get..." -- and he looked into space and mumbled for a minute. Knowing the answer, the young questioner put in "2.31?" Johnny gave him a funny look and said "Now if n = 2, ...", and once again voiced some of his thoughts as he worked. The young man, prepared, could of course follow what Johnny was doing, and, a few seconds before Johnny finished, he interrupted again, in a hesitant tone of voice: "7.49?" This time Johnny frowned, and hurried on: "If n = 3, then...". The same thing happened as before - Johnny muttered for several minutes, the young man eavesdropped, and, just before Johnny finished, the young man exclaimed: "11.06!" That was too much for Johnny. It couldn't be! No unknown beginner could outdo him! He was upset and he sulked till the practical joker confessed."

"When his electronic computer was ready for its first preliminary test, someone suggested a relatively simple problem involving powers of 2. (It was something of this kind: what is the smallest power of 2 with the property that its decimal digit fourth from the right is 7? This is a completely trivial problem for a present-day computer: it takes only a fraction of a second of machine time.) The machine and Johnny started at the same time, and Johnny finished first."

"The speed with which von Neumann could think was awe-inspiring."

"Most of the legends, from childhood on, tell about his phenomenal speed in absorbing ideas and solving problems. At the age of 6 he could divide two eight-digit numbers in his head; by 8 he had mastered the calculus; by 12 he had read and understood Borel’s Théorie des Fonctions."

"Now, just consider the smartest person who has ever lived. On almost everyone's shortlist here is John von Neumann... I mean, the impression that von Neumann made on the people around him, and this included the greatest mathematicians and physicists of his time, is fairly well-documented. If only half the stories about him are half true, there's no question he's one of the smartest people who has ever lived."

"The greatest polymath and fastest thinker of the 20th century."

"In a 1948 Princeton talk, replying to a frequent affirmation that it's impossible to build a machine that can replace the human mind, von Neumann said: You insist that there is something that a machine can't do. If you will tell me precisely what it is that a machine cannot do, then I can always make a machine which will do just that."

"Of previous publications those of von Neumann have most strongly influenced the work presented here."

"It is the hallmark of a great mathematician that his output is prodigious and von Neumann was indeed a great mathematician."

"So he had a remarkable mind, a really remarkable mind."

"I always remember one time, Bochner, von Neumann, and I were in a room, I guess Johnny's room in the Institute. Bochner was presenting material to us, and he got stuck. He hemmed and hawed for a while, and he said "If you'll wait a minute, I know where the book is that has the proof of this. I'll run upstairs and get it." Johnny said, "Don't do that, I don't know what book it's in, but I 'II prove it for you." And he did."

"At just about the time you could run your eye down the page, he would be turning it."

"He had another quality which I always thought was unbelievable. He and I worked at trying to prove something about bounds on eigen values one time without .any success. One day I saw in Math Reviews a statement that Kolmogorov or somebody had proved a theorem, and I said, "This is what so and so proved." He said, "Sure, this is how it goes." And he went to the blackboard and he proved it. Somehow, just knowing that it was true, and not just a conjecture of ours, made it possible for him to see the proof. I don't know how or why or what."

"Whenever you'd go into his office, having spent the last week working on something, and say, "Johnny, I've got an idea," and start to write, you'd get maybe the first half-a-line down before he'd say, "Yes, let me have the chalk." Then he'd get up there, and for the rest of the hour he would be putting it down in the way it ought to be done."

"He [Veblen] delighted in Johnny von Neumann."

"I guess one of [Veblen's] greatest mathematical accomplishments was finding Johnny von Neumann and bringing him to Princeton University. At least I suppose that was his greatest achievement among many achievements."

"Fantastic speed."

"One of his remarkable abilities was his power of absolute recall. As far as I could tell, von Neumann was able on once reading a book or article to quote it back verbatim; moreover, he could do it years later without hesitation. He could also translate it at no diminution in speed from its original language into English. On one occasion I tested his ability by asking him to tell me how A Tale of Two Cities started. Whereupon, without any pause, he immediately began to recite the first chapter and continued until asked to stop after about ten or fifteen minutes. Another time, I watched him lecture on some material written in German about twenty years earlier. In this performance von Neumann even used exactly the same letters and symbols he had in the original."

"While still very young, von Neumann showed tremendous intellectual and linguistic ability, and he once told the author that at six he and his father often joked with each other in classical Greek."

"One of the great mathematical universalists."

"He is regarded as one of the giants of modern mathematics."

"That von Neumann was brilliant, perhaps a good deal more than brilliant, had been clear even in childhood."

"The fact, however, remains that a lot of wonderful people never received the prize. Just take a few examples from among Hungarian physicists. Von Neumann never received the prize and neither did Szilard."

"Well, I was so flattered to be mentioned in a footnote by John von Neumann that it didn't occur to me that he hadn't actually credited us with what we were doing."

"[On Rayleigh–Taylor instability] So, Fermi said, "Let me make a model; I'll have a broad tongue which moves into the dense material; I'll have a narrow tongue that moves away from it, and I'll just solve this numerically." So, he did some of that, but he wasn't quite satisfied with the solution. One afternoon around 4:50 p.m., John von Neumann came by and saw what Fermi had on the blackboard and asked what he was doing. So, [[Enrico Fermi|Enrico] told him, and John von Neumann said, "That's very interesting." He came back about 15 minutes later and gave him the answer. Fermi leaned against his doorpost and told me, "You know, that man makes me feel I know no mathematics at all.""

"A more interesting activity during that time was my periodic contact with Albert Einstein, who, along with other prominent experts such as John von Neumann, served as a consultant for the High Explosive Division."

"IQ tests for geniuses have not yet been constructed, because one cannot expect the IQ-specialists to be geniuses, but one must suspect that the scale continues upwards to giddy heights of ability. Most of those who have known the mathematician John von Neumann have felt as slow and stupid in his presence as the dunce with the top of the form."

"I never ceased to be fascinated by electronic computers, and I feel that I have been privileged in having been initiated so marvellously by the Master himself. His mathematical achievements are far too subtle and technical for me to understand or to describe, but I can attest to the strength of his brain because I once saw him, for a bet, drink sixteen martinis in a row and then be still on his feet and quite lucid, though somewhat pessimistic in his utterances."