Mathematicians From Hungary

"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 He gave me most cordial encouragement, let me have his unpublished manuscripts to study, and later took the initiative to recommend me to Marshall Stone for a B. P. Instructorship at Harvard.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."

"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."

"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."

"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."

"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"."

"Fantastic mind."

"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."

"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?"

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

"Banesh Hoffmann: Yes, he was incredible."

"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."

"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."

"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."

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

"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."

"Certainly the greatest mathematician of that time."

"Richard Rhodes: Was he as extraordinary a mind as he has been described?"

"George Kistiakowsky: Yes, an extraordinary, fast mind. Extraordinarily fast mind."

"We were all drawn by von Neumann."

"It must have been a shattering experience to have grown up with von Neumann however bright one is."

"He was the quickest mathematician i have ever known."

"He was the most remarkable man. I’m always utterly surprised that his name is not common, household. It is a name that should be known to every American—in fact, every person in the world, just as the name of Einstein is. I am always utterly surprised how come he’s almost totally unknown. In fact, did you know – you did know, all right, you are an unusually well informed person. All people who had met him and interacted with him realized that his brain was more powerful than anyone’s they have ever encountered. I remember Hans Bethe even said, only half in jest, that von Neumann’s brain was a new development of the human brain. Only a slight exaggeration."

"People today have a hard time to imagine how brilliant von Neumann was. If you talked to him, after three words, he took over. He understood in an instant what the problem was and had ideas. Everybody wanted to talk to him."

"Mrs. Szegő often recalled that Szegő came home with tears in his eyes from his first encounter with the young prodigy."

"To gain a measure of von Neumann’s achievements, consider that had he lived a normal span of years, he would certainly have been a recipient of a Nobel Prize in economics. And if there were Nobel Prizes in computer science and mathematics, he would have been honored by these, too. So the writer of these letters should be thought of as a triple Nobel laureate or, possibly, a ​3 1⁄2-fold winner, for his work in physics, in particular, quantum mechanics."

"Von Neumann was addicted to thinking, and in particular to thinking about mathematics."

"Von Neumann combined, in a unique fashion, extreme quickness, very broad interests, and a fearsome technical prowess."

"Most mathematicians prove what they can, von Neumann proves what he wants." Once in a discussion about the rapid growth of mathematics in modern times, von Neumann was heard to remark that whereas thirty years ago a mathematician could grasp all of mathematics, that is impossible today. Someone asked him: "What percentage of all mathematics might a person aspire to understand today?" Von Neumann went into one of his five-second thinking trances, and said: "About 28 percent."

"He was admired by the brightest stars at Los Alamos: Oppenheimer, Bethe, Feynman, Peierls, Teller and many others; they acknowledged him as their superior for sheer brain power."

"Most scintillating intellect of this century."

"The most powerful brain."

"Historians have noted how Baron Eötvös’s educational efforts led to an explosion of genius — such luminaries as the physicists Edward Teller, Eugene Wigner, Leo Szilard, and the mathematician John von Neumann all came out of Budapest during the Eötvös era. The production of Hungarian scientists and mathematicians in the early twentieth century was so prolific that many otherwise calm observers believe Budapest was settled by Martians in a plan to infiltrate and take over the planet."

"As a matter of fact, he is very good."

"Stone told me that the two mathematicians in all the world who could be most helpful to my development were John von Neumann and Frederick Riesz. (His Hungarian name, Frigyes, became Frederick when anglicized). Von Neumann’s name was well known to me, of course."

"He was brilliant, spoke very fast, his English was quite fluent, he made remarkably few errors. A characteristic one was to talk about “infinite serious” for infinite series. No one ventured to correct his few lapses. I had met him recently at a party. The high point of the evening was a recitation race between him and Norbert Wiener. Somehow, someone recited a line from Lewis Carroll’s “The Hunting of the Snark.” Norbert, with his usual ebullience and sonorous voice, began reciting from line 1. Johnny started off in pursuit. Norbert accelerated, but Johnny came up even. We held our breaths as the lines poured out, on and on until they reached the end in a dead heat."

"The most brilliant mathematician of his generation."

"John von Neumann was the acknowledged genius of modern mathematics."

"The greeting, to a man or to a lady, was raising the hat completely off the head, simultaneously making a pronounced bow, all the while continuing to walk briskly forward. This courteous greeting was a Hungarian custom ingrained firmly in youth, and not easily forgotten in later years. I remember receiving such a greeting in 1950 in Cambridge, Massachusetts, across a 70 meter avenue, from that most courteous genius, John von Neumann."

"Von Neumann was certainly a true giant of the twentieth century, a figure more unique than rare in his astonishing capacity to join a theoretical intelligence of extraordinary depth to a very concrete view of science."

"In 1990 a thirty-five-year-old professor told me that “von Neumann took the fear out of learning math for all the professors who taught me.”"

"Eleven-year-old Johnny taught him [Wigner] set-theory math during Sunday afternoon walks."

"Wigner and others recall that Ratz’s recognition of Johnny’s mathematical talents was instant."

"Ratz turned his student over to the mathematicians at Budapest University, themselves men of no small renown. Professor Joseph Kurschak soon wrote to a university tutor, Gabriel Szego, saying that the Lutheran School had a young boy of quite extraordinary talent. Would Szego, as was the Hungarian tradition with infant prodigies, give some university teaching to the lad?"

"After Szego had done the initial coaching in 1915-16, tuition of schoolboy Johnny was taken over by other prominent mathematicians at Budapest University. He had contact with Kurschak, some with the brilliant Alfred Haar, and a little with the internationally known Frigyes Riesz. He was taught more directly by Michael Fekete (whose surname in Hungarian means “black”) and Leopold Fejer (feher is Hungarian for “white”)."

"There was allegedly an exception when one German professor praised the habit of asking Ph.D. students “unsolvable question” at their oral exams. If the student instantly said, “That's unsolvable”, he was deemed to have the right sharp set of mind. The professor put his favorite unsolvable equations on the blackboard as an illustration. Johnny muttered at the ceiling for a few minutes, and then solved some of them. A more typical occasion was when one professor propounded a new discovery that was actually quite wrong. This wrongdoer handled all the questions at the seminar devastatingly well, and there was discussion of his discovery at a private dinner that night. Johnny demolished the whole discovery by saying that he should have been asked a, b, and c . “Why didn't you ask that?” said the seminar organizer desperately. Johnny intimated that he did not like to be publicly rude."

"Von Neumann got very excited when J. M. put production functions on the board and jumped up, wagging his finger at the blackboard, saying (approx): “But surely you want inequalities, not equations there?” Jascha said that it became difficult to carry the seminar to conclusion because von Neumann was on his feet, wandering around the table, etc., while making rapid and audible progress on the linear programming theory of production. “The rapidity with which he made the connection and developed it,” said Arrow, “is in line with many anecdotes of von Neumann’s mental speed.”"

"On what was probably August 7, 1944, Goldstine took Johnny to see the ENIAC at Philadelphia. Before this visit Eckert told Goldstine he would be able to “tell whether von Neumann was really a genius by his first question. If this was about the logical structure of the machine, he would believe in von Neumann, otherwise not. Of course this was von Neumann’s first query.”"

"While all the other computer makers were generally heading in the same direction, von Neumann’s genius clarified and described the paths better than anyone else."

"All three of these men—Strauss, Quarles, and Gardner—thought that America’s technology for war could best be advanced by the man whom they regarded as America’s quickest-thinking scientific genius."

"He was building his computer. He was not just a person who told other guys to build a computer. He was always about details, "How are you going to do this? Which kind of gadget are you going to use for memory?" He was extraordinarily precise in these matters. At the same time he had written a book on the foundation of quantum mechanics, which I read with terror but great interest. He had written this book about games theory, which looked then extremely promising but of course was just the beginning. He had done this work about logic. I mean, in a way Von Neumann was the person who had performed the miracle. He was for me the model above all models."