Mathematicians from Hungary

"I think that, as , a very clever young Israeli mathematician, once said, "I am an opportunist. I do what I can do." If there is anything in number theory that I can do, I certainly do it. But you see some of the problems in number theory are enormously difficult and many of these classic problems are very, very hard to make any progress in."

"I'm not competent to judge. But no doubt he was a great man."

"Another roof, another proof."

"Suppose aliens invade the earth and threaten to obliterate it in a year's time unless human beings can find the Ramsey number for red five and blue five. We could marshal the world's best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded the Ramsey number for red six and blue six, however, we would have no choice but to launch a preemptive attack."

"Television is something the Russians invented to destroy American education."

"The SF created us to enjoy our suffering. … The sooner we die, the sooner we defy His plans."

"Some French socialist said that private property was theft … I say that private property is a nuisance."

"My brain is open!"

"If numbers aren't beautiful, I don't know what is."

"It is not enough to be in the right place at the right time. You should also have an open mind at the right time."

"Végre nem butulok tovább"

"We'll continue tomorrow — if I live."

"This one's from the Book!"

"SF means Supreme Fascist — this would show that God is bad. I don't claim that this is correct, or that God exists, but it is just sort of half a joke. … As a joke I said, "What is the purpose of Life?" "Proof and conjecture, and keep the SF's score low." Now, the game with the SF is defined as follows: If you do something bad the SF gets at least two points. If you don't do something good which you could have done, the SF gets at least one point. And if nothing — if you are okay, then no one gets any point. And the aim is to keep the SF's score low."

"God may not play dice with the universe, but something strange is going on with the prime numbers."

"A mathematician is a machine for turning coffee into theorems."

"The first sign of senility is that a man forgets his theorems, the second sign is that he forgets to zip up, the third sign is that he forgets to zip down."

"Let n be an integer."

"Erdős knows about more problems than anybody else, and he not only knows about various problems and conjectures, but he also knows the tastes of various mathematicians. So if I get a letter from him giving me three of his conjectures and two of his problems, then it's sure that these are exactly the kind of conjectures and problems I'm interested in, and these are exactly the kind of questions I may be able to answer. Of course, this applies not only to me, but to everybody else. So Erdős has an amazing ability to match problems with people. Which is why so many mathematicians benefit from his presence. Every letter is likely to inspire you to do some work, or every phone call will give you some problems you are interested in."

"Paul Erdős is the consummate problem solver: his hallmark is the succinct and clever argument, often leading to a solution from "the book". He loves areas of mathematics which do not require an excessive amount of technical knowledge but give scope for ingenuity and surprise. The mathematics of Paul Erdős is the mathematics of beauty and insight."

"One of my greatest regrets is that I didn't know him when he was a million times faster than most people. When I knew him he was only hundreds of times faster."

"Hungarian mathematician Paul Erdős, although an atheist, spoke of an imaginary book, in which God has written down all the most beautiful mathematical proofs. When Erdős wanted to express particular appreciation of a proof, he would exclaim "This one's from the Book!". This viewpoint expresses the idea that mathematics, as the intrinsically true foundation on which the laws of our universe are built, is a natural candidate for what has been personified as God by different religious mystics."

"He was an absolutely wonderful man. He was interested in everything. You felt right away that you are not dealing with one of your colleagues or an average guy. He was a genius, his thoughts were all over the place. I've met very smart people. I have never met a genius before. … He basically disregarded any disciplined approach to anything."

"Probably the greatest mathematician of the twentieth century, Paul Erdős … was so eccentric that he made Einstein look normal. He was 11 before he ever tied his shoes, 21 before he ever buttered toast, and died without ever boiling an egg. Erdős lived on the road, traveling from conference to conference, owning nothing but math notebooks and a suitcase or two. His life consisted of math, nothing else."

"This book is dedicated to Paul Erdos, who not only possessed the art of asking the right question, but of asking it of the right person."

"The SF is the Supreme Fascist, the Number-One Guy Up There, God, who was always tormenting Erdős by hiding his glasses, stealing his Hungarian passport, or, worse yet, keeping to Himself the elegant solutions to all sorts of intriguing mathematical problems."

"He wrote or co-authored 1,475 academic papers, many of them monumental, and all of them substantial. It wasn't just the quantity of work that was impressive but the quality: "There is an old saying," said Erdős. "Non numerantur, sed ponderantur (They are not counted but weighed)."

"In a never-ending search for good mathematical problems and fresh mathematical talent, Erdős crisscrossed four continents at a frenzied pace, moving from one university or research center to the next. His modus operandi was to show up on the doorstep of a fellow mathematician, declare, "My brain is open," work with his host for a day or two, until he was bored or his host was run down, and then move on to another home. Erdős's motto was not "Other cities, other maidens" but "Another roof, another proof." He did mathematics in more than 25 different countries, completing important proofs in remote places and sometimes publishing them in equally obscure journals."

"His language had a special vocabulary — not just "the SF" [God] and "epsilon" [child] but also "bosses" (women), "slaves" (men), "captured" (married), "liberated" (divorced), "recaptured" (remarried), "noise" (music), "poison" (alcohol), "preaching" (giving a mathematics lecture), "Sam" (the United States), and "Joe" (the Soviet Union). When he said someone had "died," Erdős meant that the person had stopped doing mathematics. When he said someone had "left," the person had died."

"In the late 1980s Erdős heard of a promising high school student named Glen Whitney who wanted to study mathematics at Harvard but was a little short the tuition. Erdős arranged to see him and, convinced of the young man's talent, lent him $1,000. He asked Whitney to pay him back only when it would not cause financial strain. A decade later Graham heard from Whitney, who at last had the money to repay Erdős. "Did Erdős expect me to pay interest?" Whitney wondered. "What should I do?" he asked Graham. Graham talked to Erdős. "Tell him," Erdős said, "to do with the $1,000 what I did.""

"As a mathematician Erdös is what in other fields is called a "natural". If a problem can be stated in terms he can understand, though it may belong to a field with which he is not familiar, he is as likely as, or even more likely than, the experts to find a solution."

"Paul Erdős (1913–1996) was once told that a friend of his had shot and killed his wife. Without blinking an eye, Erdős said, "Well, she was probably interrupting him when he was trying to prove a theorem.""

"In the early 1960s, when I was a student at University College London … Erdős came to visit us for a year. After collecting his first month's salary he was accosted by a beggar on Euston station, asking for the price of a cup of tea. Erdős removed a small amount from the pay packet to cover his own frugal needs and gave the remainder to the beggar."

"A conjecture both deep and profound Is whether a circle is round. In a paper of Erdős Written in Kurdish A counterexample is found."

"He was the Bob Hope of mathematics, a kind of vaudeville performer who told the same jokes and the same stories a thousand times. … When he was scheduled to give yet another talk, no matter how tired he was, as soon as he was introduced to an audience, the adrenaline (or maybe amphetamine) would release into his system and he would bound onto the stage, full of energy, and do his routine for the 1001st time."

"Want to meet Erdos?" mathematicians would ask. "Just stay here and wait. He'll show up."

"He loved to play silly tricks to amuse children and to make sly jokes and thumb his nose at authority. But most of all, Erdős loved those who loved numbers, mathematicians."

"Twenty hours of work a day was not unusual. Upon arriving at a meeting, he would announce, in his thick Hungarian accent, "my brain is open." At parties, he would often stand alone oblivious to all else, deep in thought pondering some difficult argument."

"Nothing bothered Erdős more than political strictures which did not allow for complete freedom of expression and the ability to travel freely. … Always traveling with a single shabby suitcase which doubled as a briefcase, he had little need or interest in the material world."

"I think that it is a relatively good approximation to truth — which is much too complicated to allow anything but approximations — that mathematical ideas originate in empirics. But, once they are conceived, the subject begins to live a peculiar life of its own and is … governed by almost entirely aesthetical motivations. In other words, at a great distance from its empirical source, or after much "abstract" inbreeding, a mathematical subject is in danger of degeneration. At the inception the style is usually classical; when it shows signs of becoming baroque, then the danger signal is up. It would be easy to give examples, to trace specific evolutions into the baroque and the very high baroque... Whenever this stage is reached the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas."

"For progress there is no cure.... The only safety possible is relative, and it lies in an intelligent exercise of day-to-day judgement."

"Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin. For, as has been pointed out several times, there is no such thing as a random number — there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method."

"The total subject of mathematics is clearly too broad for any one of us. I do not think that any mathematician since Gauss has covered it fully and uniformly, even Hilbert did not, and all of us are of considerably lesser width (quite apart from the question of depth) than Hilbert. It would therefore, be quite unrealistic not to admit, that any address I could possibly give would not be biased towards some areas in mathematics in which I have had some experience, to the detriment of others which may be equally or more important. To be specific, I could not avoid a bias towards those parts of analysis, logics, and certain border areas of the applications of mathematics to other sciences in which I have worked. If your Committee feels that an address which is affected by such imperfections still fits into the program of the Congress, and if the very generous confidence in my ability to deliver continues, I shall be glad to undertake it."

"A large part of mathematics which becomes useful developed with absolutely no desire to be useful, and in a situation where nobody could possibly know in what area it would become useful; and there were no general indications that it ever would be so. By and large it is uniformly true in mathematics that there is a time lapse between a mathematical discovery and the moment when it is useful; and that this lapse of time can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful."

"The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work."

"It is exceptional that one should be able to acquire the understanding of a process without having previously acquired a deep familiarity with running it, with using it, before one has assimilated it in an instinctive and empirical way… Thus any discussion of the nature of intellectual effort in any field is difficult, unless it presupposes an easy, routine familiarity with that field. In mathematics this limitation becomes very severe."

"When we talk mathematics, we may be discussing a secondary language built on the primary language of the nervous system."

"It is just as foolish to complain that people are selfish and treacherous as it is to complain that the magnetic field does not increase unless the electric field has a curl. Both are laws of nature."

"You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, no one really knows what entropy really is, so in a debate you will always have the advantage."

"Young man, in mathematics you don't understand things. You just get used to them."

"You don't have to be responsible for the world that you're in."

"The goys have proven the following theorem…"

"The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking."

"With four parameters I can fit an elephant, and with five I can make him wiggle his trunk."

"You wake me up early in the morning to tell me that I'm right? Please wait until I'm wrong."

"If one has really technically penetrated a subject, things that previously seemed in complete contrast, might be purely mathematical transformations of each other."

"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."

"There probably is a God. Many things are easier to explain if there is than if there isn't."

"If you say why not bomb them tomorrow, I say why not today? If you say today at five o' clock, I say why not one o' clock?"

"Some people confess guilt to claim credit for the sin."

"It will not be sufficient to know that the enemy has only fifty possible tricks and that we can counter every one of them, but we must be able to counter them almost at the very instant they occur."

"One of the world's great mathematicians."

"John von Neumann was an enormous personality."

"Princeton was the place which had all these names—Einstein, Weyl, von Neumann—who were great figures at the time."

"I met him, but in a sense, he didn’t meet me. We were introduced at a game theory conference in 1955, two years before he died. I said, “Hello, Professor von Neumann,” and he was very cordial, but I don’t think he remembered me afterwards unless he was even more extraordinary than everybody says. I was a young person and he was a great star."

"I think I had some feeling that their minds [von Neumann and Weyl] were so far ahead of mine that it was difficult to follow their thoughts."

"If one applies an appropriately broad view of physics one must say that von Neumann had a quite outstanding insight into the problems of physics. Because he has done first-rate work, and he was the man who succeeded in giving a correct mathematical formulation of quantum mechanics, and this was the major theory in physics in the first half of the century."

"Nobody doubts that von Neumann was brilliant; everybody admits that."

"Nevertheless, it was generally agreed that von Neumann was the leading mathematical mind in the world at that time."

"By any standard, von Neumann, was one of the most creative and versatile scientists of the twentieth century."

"Von Neumann had a phenomenal capacity for doing mental computations of all kinds. His thought processes were extremely fast, and often he would see through to the end of someone’s argument almost before the speaker had got out the first few sentences. Recently, one of von Neumann’s colleagues said in affectionate explanation of von Neumann’s power, “You see, Johnny wasn’t human. But after living with humans for so long he learned how to do a remarkable imitation of one.”"

"One of the most brilliant mathematicians who ever lived."

"Von Neumann was considered to be the most brilliant of the young mathematicians."

"Von Neumann had an absolute paranoia about the Russians and favored a first nuclear strike. Einstein referred to him as a Denktier, a think animal."

"I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man."

"I always thought Johnny’s brain indicated that he belonged to a new species, an evolution beyond man."

"It’s impossible to truly understand the speed at which von Neumann’s brain worked and how he thought, even for the cleverest observers. I drop hints about how fast and clever he was, but I don’t pretend to fully understand or get to grips with his human side. I’m not sure if I can, as some of his friends even said that von Neumann was an alien, a superintelligent being that had studied humans and learned how to copy us perfectly."

"Probably the smartest man on Earth."

"He had the kind of mind that if you go in to see him with an idea, inside of five minutes he's five blocks ahead of you and sees exactly where it's going. His mind was just so fast and so accurate that there was no keeping up with him. There was nobody on earth, as far as I'm concerned, who was in his category."

"His mind was faster than anybody's."

"He stimulated people everywhere. Von Neumann was generous intellectually, because his resources were so enormous, that he never gave away anything that he couldn't do without. He was a fountainhead of information, and he didn't hold back because there was always much more in depth than he ever exposed at one time. I'm not exaggerating. This was indeed how he was. In the few decades that followed, I have had the experience of understanding what this was like, by working as a consultant for some groups who were at a technical level so far below what I knew myself, that what they felt was worth bickering about was in fact possible [?] that one could ignore it, because one knows one has much more in depth than that and that's not an important point. Let them bicker about it, or let them think that they did something great. He was precisely that way. He was in depth, more knowledgeable than any man, and I say this having worked with Wiener for four years, and Wiener was no slouch himself. Von Neumann was a giant. He was ahead of anybody."

"What Von Neumann contributed as far as the engineering was concerned, was simply the enormous confidence everybody had that a machine so simple, and with no more doodads on it could knock dead, so to speak, an enormous amount of the computation that needed to be done in this world for the next few decades. He never came over and said to make a circuit of this, but he did know so much more of the deeper aspects of mathematics and the practical aspects of computation than any of the rest of us. What he did essentially, was to serve as this unshakable confidence that said: "Go ahead, nothing else matters, get it running at this speed and this capability, and the rest of it is just a lot of nonsense.""

"John von Neumann's brilliant mind blazed over lattice theory like a meteor."

"I went in and started telling him about my thesis. He listened for about ten minutes and asked me a couple of questions, and then he started telling me about my thesis. What you have really done is this, and probably this is true, and you could have done it in a somewhat simpler way, and so on. He was a really remarkable man. He listened to me talk about this rather obscure subject and in ten minutes he knew more about it than I did. He was extremely quick. I think he may have wasted a certain amount of time, by the way, because he was so willing to listen to second- or third-rate people and think about their problems. I saw him do that on many occasions."

"At the age of 6 he was able to divide two eight-digit numbers in his head. By the age of 8 he had mastered college calculus and as a trick could memorize on sight a column in a telephone book and repeat back the names, addresses and numbers. History was only a “hobby,” but by the outbreak of World War I, when he was 10, his photographic mind had absorbed most of the contents of the 46-volume works edited by the German historian Oncken with a sophistication that startled his elders."

"Several years ago his wife gave him a 21-volume Cambridge History set, and she is sure he memorized every name and fact in the books. “He is a major expert on all the royal family trees in Europe,” a friend said once. “He can tell you who fell in love with whom, and why, what obscure cousin this or that czar married, how many illegitimate children he had and so on.” One night during the Princeton days a world-famous expert on Byzantine history came to the Von Neumann house for a party. “Johnny and the professor got into a corner and began discussing some obscure facet,” recalls a friend who was there. “Then an argument arose over a date. Johnny insisted it was this, the professor that. So Johnny said, ‘Let’s get the book.’ They looked it up and Johnny was right. A few weeks later the professor was invited to the Von Neumann house again. He called Mrs. von Neumann and said jokingly, ‘I’ll come if Johnny promises not to discuss Byzantine history. Everybody thinks I am the world’s greatest expert in it and I want them to keep on thinking that.'”"

"One day he was urgently summoned to the offices of the Rand Corporation, a government-sponsored scientific research organization in Santa Monica, Calif. Rand scientists had come up with a problem so complex that the electronic computers then in existence seemingly could not handle it. The scientists wanted Von Neumann to invent a new kind of computer. After listening to the scientists expound, Von Neumann broke in: “Well, gentlemen, suppose you tell me exactly what the problem is?” For the next two hours the men at Rand lectured, scribbled on blackboards, and brought charts and tables back and forth. Von Neumann sat with his head buried in his hands. When the presentation was completed, he scribbled on a pad, stared so blankly that a Rand scientist later said he looked as if “his mind had slipped his face out of gear,” then said, “Gentlemen, you do not need the computer. I have the answer.” While the scientists sat in stunned silence, Von Neumann reeled off the various steps which would provide the solution to the problem. Having risen to this routine challenge, Von Neumann followed up with a routine suggestion: “Let’s go to lunch.”"

"One day, during an ICBM meeting on the West Coast, a physicist employed by an aircraft company approached Von Neumann with a detailed plan for one phase of the project. It consisted of a tome several hundred pages long on which the physicist had worked for eight months. Von Neumann took the book and flipped through the first several pages. Then he turned it over and began reading from back to front. He jotted down a figure on a pad, then a second and a third. He looked out the window for several seconds, returned the book to the physicist and said, “It won’t work.” The physicist returned to his company. After two months of re-evaluation, he came to the same conclusion."

"After the last visitor had departed Von Neumann would retire to his second-floor study to work on the paper which he knew would be his last contribution to science. It was an attempt to formulate a concept shedding new light on the workings of the human brain. He believed that if such a concept could be stated with certainty, it would also be applicable to electronic computers and would permit man to make a major step forward in using these 'automata'. In principle, he reasoned, there was no reason why some day a machine might not be built which not only could perform most of the functions of the human brain but could actually reproduce itself, i.e., create more supermachines like it. He proposed to present this paper at Yale, where he had been invited to give the 1956 Silliman Lectures."

"Probably the greatest mathematician of the century."

""Johnny," as all his friends called him, was the only scientist of the era to whom the word "genius" was almost universally applied. He had an uncanny ability to handle complex mathematical calculations in seconds. When he was six years old he could divide one eight-digit number into another, entirely in his head."

"Then, of course, there was Neumann, who always knew everything anyhow."

"Weyl had a very tremendous respect for him, and I could see in this advanced, seminar when Weyl didn't know the answer he would say, "Neumann, how does that go?" We all realized this was a great mathematician."

"His effectiveness was largely due to his ever-present mental manipulatory quickness. He could literally "think on his feet," and much of his best work may have received its initial impulse in just this way. He had a prodigious memory, and legend has it that he knew all the facts and dates from many volumes of standard histories by heart."

"He was also a great reader of books on history throughout his life, and in both science and history his retentive memory was most remarkable."

"It was also well remembered about 25 years later by one of his colleagues here at the time, M. Plancherel, who mentioned it to me then as an example of the extraordinary ability H. Weyl had, shared only by J. von Neumann among the mathematicians he had known, to get into a new subject and bring an important contribution to it within a few months."

"Von Neumann was considered the leading mathematician in the United States."

"Great mathematician."

"He would seize on the fuzzy notions of others and, by dint of his prodigious mental powers, leap five blocks ahead of the pack. “You would tell him something garbled, and he’d say, ‘Oh, you mean the following,’ and it would come back beautifully stated,” said his onetime protégé, the Harvard mathematician Raoul Bott."

"Von Neumann is a great scientific hero to me because it seemed… he seemed to have something. And of course it may be envy rather than admiration, but it's good to envy someone like von Neumann."

"Mathematics is not a pompous activity, least of all in the hands of extraordinarily fast and penetrating minds like Johnny von Neumann."

"There was something endearing and personal about Johnny von Neumann. He was the cleverest man I ever knew, without exception. And he was a genius, in the sense that a genius is a man who has two great ideas. When he died in 1957 it was a great tragedy to us all."

"In a Silliman lecture ... John von Neumann, who was dying at the time, wrote some of the most splendid sentences he wrote in all his life ... He pointed out that there were good grounds merely in terms of electrical analysis to show that the mind, the brain itself, could not be working on a digital system. It did not have enough accuracy; or ... it did not have enough memory. ... And he wrote some classical sentences saying there is a statistical language in the brain ... different from any other statistical language that we use... this is what we have to discover. ...I think we shall make some progress along the lines of looking for what kind of statistical language would work."

"The greatest polymath of the 20th century."

"The crucial point: in Dr. von Neumann the Institute has perhaps the cleverest man in the world, and the really deciding factor in the end should, I am sure, be what he wants to do."

"Von Neumann was a very great mathematician. He made many important contributions in a wide range of fields."

"The manuscripts for both parts of the present volume were unfinished; indeed, they were both, in a sense, first drafts. There is one compensation in this: one can see von Neumann's powerful mind at work."

"He was about as likable a chap as you could imagine. There is just one short thing about him. I was riding in a Pullman one day in the lounge car after the war and I hadn't looked about me before I sat down; I was reading something and it had my full attention. From across came von Neumann. He sat down aside me and introduced himself. Well here was the man who, in my opinion, was the most able mathematician in the country in many ways and he felt that he needed to introduce himself to me. That's a type of modesty one can't help liking."

"He was a superb lecturer. Superb."

"He was incredible - the enormous perception that he had. For me, ever since, a standard of comparison has always been von Neumann. And if I say, "He reminds me of von Neumann," that's about the best compliment I can give anyone."

"He was incredibly perceptive."

"[He] thought so fast that he very often anticipated what one was going to say. . . . a pleasant agreeable person . . . the amazing logic of his thought processes."

"Von Neumann was capable of all sorts of remarkable things."

"The smartest man in the world."

"Genius of the highest order."

"Now the story doesn't end here. Before going on with it, however, I'd like to introduce you to Johnnie von Neumann, an incredible genius whose mind worked about as rapidly as the super high-speed computers he helped design."

"Bennie decided to approach Johnnie on the matter and arranged to travel to Princeton’s Institute for Advanced Study, headed up at the time by Oppenheimer, where Johnnie (and lesser geniuses such as Albert Einstein) was stationed."

"He did a tremendous amount of different things in mathematics, many of them revolutionary."

"Mr. von Neumann, in spite of his youth, is a completely exceptional personality ... who has already done very productive work ... and whose future development is being watched with great expectation in many places."

"Von Neumann I never could quite figure out. He was just too fast for me."

"Strange, contradictory, and controversial person; childish and good-humored, sophisticated and savage, brilliantly clever yet with very limited, almost primitive lack of ability to handle his emotions—an enigma of nature that will have to remain unsolved."

"[One early 1945 night,] he woke up and started talking at a speed which, even for him, was extraordinarily fast. “What we are creating now is a monster whose influence is going to change history, provided there is any history left, yet it would be impossible not to see it through, not only for the military reasons, but it would also be unethical from the point of view of the scientists not to do what they know is feasible, no matter what terrible consequences it may have. And this is only the beginning!” The concerns von Neumann voiced that night were less about nuclear weapons, and more about the growing powers of machines. “From here on, Johnny’s fascination and preoccupation with the shape of things to come never ceased,” concludes Klári’s account. For the next seven years he neglected mathematics and devoted himself to the advance of technology in all forms. “It was almost as if he knew that there was not very much time left.”"

"He had always done his writing at home during the night or at dawn. His capacity for work was practically unlimited."

"People would come to him because of his great insight."

"In von Neumann’s generation his ability to absorb and digest an enormous amount of extremely diverse material in a short time was exceptional; and in a profession where quick minds are somewhat commonplace, his amazing rapidity was proverbial."

"May have been the last representative of a once-flourishing and numerous group, the great mathematicians who were equally at home in pure and applied mathematics and who throughout their careers maintained a steady production in both directions."

"Perhaps an even greater genius than Einstein, of almost extraterrestrial brilliance."

"However, as noted earlier, one of his central objectives—as a mathematician—was to publish the generalized proof of the fixed point theorem. Was the economics merely a convenient vehicle for an essentially mathematical exercise for von Neumann? Genius that he was, perhaps that is all that he wanted to do at that time. Later, after meeting Oscar Morgenstern, he returns to economics, but only through their joint interest in the theory of games."

"Von Neumann was a great mathematician and had the reputation at that time of being the cleverest man in the world. He was supposed to be the intellectual force driving the whole development of computers. He was a great thinker and a great entrepreneur."

"I remember a talk that Von Neumann gave at Princeton around 1950, describing the glorious future which he then saw for his computers. Most of the people that he hired for his computer project in the early days were meteorologists. Meteorology was the big thing on his horizon. He said, as soon as we have good computers, we shall be able to divide the phenomena of meteorology cleanly into two categories, the stable and the unstable. The unstable phenomena are those which are upset by small disturbances, the stable phenomena are those which are resilient to small disturbances. He said, as soon as we have some large computers working, the problems of meteorology will be solved. All processes that are stable we shall predict. All processes that are unstable we shall control. He imagined that we needed only to identify the points in space and time at which unstable processes originated, and then a few airplanes carrying smoke generators could fly to those points and introduce the appropriate small disturbances to make the unstable processes flip into the desired directions. A central committee of computer experts and meteorologists would tell the airplanes where to go in order to make sure that no rain would fall on the Fourth of July picnic. This was John von Neumann's dream. This, and the hydrogen bomb, were the main practical benefits which he saw arising from the development of computers."

"Von Neumann compensated for these superhuman abilities with an earthy sense of humor and tireless social life, and tried, with mixed success, to blend in on a normal human scale."

"I got to know von Neumann and I thought he was very quick mentally in mathematics and things."

"I think he was a damn fast guy for figuring out what the other guy was doing and explaining it better."

"Johnny has a very good mind."

"The Alexanders gave humdinger, wonderful parties. I don't know whether they would be regarded as outlandish today, but they were certainly regarded as far out in those days. The phenomenal feature of von Neumann was that he could go to these parties and party and drink and whoop it up to the early hours of the morning, and then come in the next morning at 8:30, hold class, and give an absolutely lucid lecture. What happened is that some of the graduate students thought that the way to be like von Neumannn was to live like him, and they couldn't do it."

"Von Neumann was very impressive to talk with. He was very quick."

"In speed and understanding Von Neumann was certainly phenomenal. He could understand a proof even far from his own subject very fast. I remember once in Cambridge I told him a proof of interpolation that was not quite correct. By the time we met again I had a correct proof. Von Neumann told me, “Something seems to be wrong in that proof.” And it was really not his subject. He wasn’t that interested in it, but he was quite right."

"Our country’s greatest Jancsi."

"You know, Herb, Johnny can do calculations in his head ten times as fast as I can! And I can do them ten times as fast as you can, Herb, so you can see how impressive Johnny is!"

"He is really a professional, isn’t he!"

"Dr. von Neumann is one of the very few men about whom I have not heard a single critical remark. It is astonishing that so much equanimity and so much intelligence could be concentrated in a man of not extraordinary appearance."

"Johnny von Neumann was the greatest mathematician around."

"Finally there came in the mail an invitation from the Institute for Advanced Study: Einstein. . . von Neumann. . .Weyl. . . all these great minds!"

"At about that time Einstein had agreed to serve as a consultant to our group but did not want to travel to Washington. So there had to be a liaison person and I was given that opportunity. Since Einstein did not know me, there had to be someone to introduce us. It then happened that I was introduced to Einstein by John von Neumann, one of the most important mathematicians of all time, and who had also become a consultant to our group. It was a very great experience for a new Ph.D. to be introduced to Einstein by Von Neumann!"

"Von Neumann was one of the greatest geniuses that ever lived."

"[Addressing Albert Tucker] The story goes that von Neumann's parents had all been lawyers and they sort of hoped that Johnny would be a good, lawyer. When he was sixteen or so they sort of tolerated his fiddling around with chemistry and mathematics. Finally they found out he wanted to be a mathematician, or chemist, or some mixture. They were very upset. Well, their attitude was that it wasn't too bad if he was going to pe a good one. So they inquired around who the best mathematician in his part of the world was, and it turned out to be Siegel. They had lots of money, and they arranged for Siegel to talk to Johnny. Afterwards they asked him, "Well, do you think he has any potential?" He said, "He knows more mathematics than I do now.""

"Von Neumann was also a delight to be with. His brainpower stuck out in every direction."

"Intellectual brilliance."

"Von Neumann would engage in any subject you wanted to discuss and within five minutes be right at the heart of the issue, even when he started off by saying, “I can discuss that not prejudiced by any facts.”"

"The most outstanding and at the same time versatile mathematician in the world in the second quarter of the 20th century."

"Around 1922–23, being then professor at Marburg University, I received from Professor Erhard Schmidt, Berlin (on behalf of the Redaktion of the Mathematische Zeitschrift) a long manuscript of an author unknown to me, Johann von Neumann, with the title Die Axiomatisierung der Mengenlehre, this being his eventual doctor dissertation which appeared in the Zeitschrift only in 1928, (Vol. 27). I was asked to express my view since it seemed incomprehensible. I don’t maintain that I understood everything, but enough to see that this was an outstanding work and to recognize ex ungue leonem. While answering in this sense, I invited the young scholar to visit me (in Marburg) and discussed things with him, strongly advising him to prepare the ground for the understanding of so technical an essay by a more informal essay which should stress the new access to the problem and its fundamental consequences. He wrote such an essay under the title, Eine Axiomatisierung der Mengenlehre, and I published it in 1925 in the Journal für Mathematik (vol. 154) of which I was then Associate Editor."

"I will not attempt to describe John von Neumann's unique abilities, but only say a little about the impression he made on us. The most striking was the enormous speed with which his brain worked. One could believe he already knew beforehand everything one asked him about. A little story illustrates this. Stefan Bergman […] went around and posed a problem to people. If one approached it directly, it required some calculation and the summation of a geometric series. But if one gave it a suitable twist, the solution became immediately apparent. When von Neumann promptly gave the correct answer, Bergman said: "You are the first of those I have asked who did not sum the geometric series." "No!" answered von Neumann, "I summed it.""

"John von Neumann is a kind of legendary mind ... Many people say he's like one of the smartest humans ever."

"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."

"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."

"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."

"[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.""

"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."

"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."

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

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

"One of the great mathematical universalists."

"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 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."

"Fantastic speed."

"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."

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

"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 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."

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

"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."

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

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

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

"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."

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

"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."

"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."

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

"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."

"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."

"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"

"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."

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

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

"I was fascinated by whatever von Neumann did."

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

"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."

"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."

"Much later, I was to find that my view of Von Neumann as a great man was completely confirmed."

"He was becoming more concerned with defense than with science. But it seemed that he was living proof that one could do science without really belonging to a “guild.” In fact, he was under extreme pressure at Princeton. From there, he left for Washington and was not planning to return. Luckily, von Neumann had realized that, by having failed to claim admission to any guild, I was leading a very dangerous life. A foundation executive told me much later that von Neumann had specifically asked him to watch after me, and to help in case of trouble."

"Phenomenon."

"Johnny von Neumann was the genius."

"I think we were right, though, in thinking he was several leaps ahead of the rest of us."

"Johnny von Neumann was very, very good and very quick and very sharp. He just was a universalist. He was not a mathematician."

"Johnny von Neumann who was very, very quick—I mean, you have no idea how quickly he would infer things and extrapolate them. Well, he was fantastic."

"He knew so much about physics and philosophy and even things like history. He was very, very sharp. He worked all the time."

"His talents were so obvious and his cooperative spirit so stimulating that he garnered the interest of many of us."

"Later, Tucker told me that he had gone to von Neumann and said, ‘This seems like very interesting work, but I can’t evaluate it. I don’t know whether it should really be called mathematics.’ Von Neumann replied, ‘Well, if it isn’t now, it will be someday—let’s encourage it.’ So I got my Ph.D.”"

"It is worth emphasizing that as great a mathematician as J. von Neumann never tired of repeating the fact that the errors of observation are what really matter. This is entirely in the spirit of Gauss. It is also noteworthy that von Neumann was firmly convinced that the intimate contact between mathematics and reality would produce, from time to time, decisive progress in mathematics."

"He worked with tremendous energy and fantastic speed."

"The fastest mind I ever met."

"I tried at that time to cast the unifying Dirac-Jordan transformation theory into a simpler and more easily understandable form and to convey its essence to Hilbert. When von Neumann saw this he cast it in a few days into an elegant axiomatic form much to the liking of Hilbert. (This is the origin of the paper “Ober die Grundlagen der Quantenmechanik” by Hilbert, von Neumann and myself . . .). The method used was that of integral operators. . . . This work set von Neumann on his way to his definite studies on the foundations of quantum mechanics."

"We are in what can only be described as a desperate need of your help. We have a good many theoretical people working here, but I think that if your usual shrewdness is a guide to you about the probable nature of our problems you will see why even this staff is in some respects critically inadequate...I would like you to come as a permanent, and let me assure you, honored member of our staff. A visit will give you a better idea of this somewhat Buck Rogers project than any amount of correspondence."

"I remember that there was a feeling of excitement and interest both in Hilbert’s lecture and in the lecture of von Neumann on the foundations of set theory — a feeling that one now finally was coming to grips with both the axiomatic foundation of mathematics and with the reasons for the applications of mathematics in the natural sciences."

"Morgenstern was once asked how a scholar outside the mainstream of economic thinking could make a contribution as original, innovative, and decisive as Johnny's. He replied that Johnny had an extraordinary capacity for picking the brains of a person whom he engaged in casual conversations. Once he saw from these that there was a problem of sufficient mathematical interest to warrant his spending time on it, he homed on to that subject like a guided missile."

"In my life I have met men even greater than Johnny, but none as brilliant. He shone not only in mathematics but was also fluently multilingual and particularly well-versed in history. One of his most remarkable abilities I soon came to note was his power of absolute recall."

"Von Neumann's reputation and fame have grown steadily since his death. His fantastic brain, and the breadth of his interests and undertakings, have become almost legendary."

"If doing physics meant proving theorems, you’d be a great physicist."

""Johnny" von Neumann, as he was always known among scientists, achieved fame first of all as a pure mathematician. I am not qualified to describe his contributions to pure mathematics, which usually related to the most recent, and most abstruse, branches of the subject at the time, but they certainly placed him among the leaders of modern mathematics. In the 1920s, he was interested in the development of quantum mechanics, then in rapid growth, which caused difficulty to many because of the bold use of new mathematical techniques. Von Neumann contributed greatly to making this new subject "respectable"; he pointed out the precise mathematical significance of the new developments and, at the same time, helped greatly to clarify the physical content of the new ideas. He was, in fact, quicker than many physicists in grasping the changes that were then taking place in physics Later he was a frequent visitor to the Atomic Weapons Project at Los Alamos. Here his particular quality of combining powerful mathematical insight with a very practical interest in the problems became familiar to all those associated with the project. He was never satisfied with showing that a problem could be solved on paper, but he took a personal interest in its quantitative application and in its practical realization. His many contributions, particularly to the hydrodynamics of shock waves and detonation waves, which are important both in the design of atomic weapons and in an understanding of their effects, were vital to the success of the project. For a man to whom complicated mathematics presented no difficulty, he could explain his conclusions to the uninitiated with amazing lucidity. After a talk with him one always came away with a feeling that the problem was really simple and transparent. About the same time, he became interested in the application of computing techniques to mathematical problems, and this led him to design the computer now in operation at Princeton and to planning out its applications both to practical problems and to abstract problems in nonlinear equations. He was the antithesis of the conventional image of the "long-haired" mathematics don. Always well-groomed, he had as lively views on international politics and practical affairs as on mathematical problems. His book on the Theory of Games, "including the theory of bluffing at poker," which has proved fruitful for many applications going beyond the field of games of chance and skill, is another example of the happy combination of his command of mathematics with an interest in practical matters. For the last few years, he was a member of the Atomic Energy Commission, and it is worth recording that in a field beset with much controversy, he retained the universal respect and confidence of those who did not agree with his views on policy as well as those who did."

"Another frequent visitor was John von Neumann, a brilliant mathematician, whom I knew from Germany. Although he was Hungarian, he did not have the extreme superficial politeness of many Hungarians. He liked good living and a good story. His mathematics was of the purest and most abstract kind, but he also understood physics and had written a book about quantum mechanics. He was extremely fast in solving practical problems, and contributed many useful ideas to the work of Los Alamos."

"Remarkable mathematician."

"The only student of mine I was ever intimidated by. He was so quick. There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn't say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann."

"Von Neumann was a calculating prodigy as well. He could divide two eight-digit numbers in his head with little effort. Cuthbert Hurd of IBM told me of von Neumann’s uncanny ability to create and revise computer programs (as long as fifty lines of assembly-language code!) in his head."

"In 1956 Good Housekeeping magazine ran an article on Klara von Neumann and her husband with the improbable title, “Married to a Man Who Believes the Mind Can Move the World.” One of the stranger examples of 1950s women’s magazine journalism, it is a dogged attempt to humanize a not entirely promising subject. “What’s it like to suspect your husband of being the smartest man on earth?” the article asks. “When Klara von Neumann, a slender brunette of Washington, D.C., glances at her husband, a plump, cheerful man who was born in Hungary fifty-two years ago, the thought sometimes occurs to her that she may be married to the best brain in the world.”"

"It seems fair to say that if the influence of a scientist is interpreted broadly enough to include impact on fields beyond science proper, then John von Neumann was probably the most influential mathematician who ever lived."

"Under the force of Courant's plea, Trowbridge reconsidered Hilbert's request; and in the fall of 1926 von Neumann came to Göttingen as a Rockefeller Fellow. The young mathematicians there recognized that he was obviously a prodigy, but some were suspicious of what they saw as a certain “glibness” about him. They also found his mathematics “too abstract” for their taste. “We were wrong about that,” confessed Friedrichs, part of whose later work was to be strongly influenced by the work of von Neumann."

"All the mathematicians I have talked to have said that von Neumann had the quickest mind they ever knew."

"If someone gave a problem and von Neumann did not give an immediate solution, then it was an unsolvable problem."

"He was a multifaceted genius."

"There is also no objection to a mathematician’s doing physics, provided he is qualified. The prime example was von Neumann—when he did physics, he talked, thought, and calculated like a physicist (but faster). He understood all branches of physics (including elementary particles as they were known then), and chemistry and astronomy, and he had a talent for introducing those and only those mathematical ideas that were relevant to the physics at hand."

"Apart from my thesis, though, I cannot overlook the great influence on all of us of the sparkling lectures in real analysis given by Professor John von Neumann, a young man who had also come from Germany during this period. How well I remember his hurried arrivals in the classroom, a mere second late but wasting no time. With spectacular fluency he instantly made the hour come alive. No notes were ever needed, for his complete control and mastery of his subject and his lightning-fast blackboard-equations quickly reflected to us some of the greatness of his precocious mind. His audience will remember his beautifully complexioned cheeks that often radiated a cherubic smile, and his bright piercing brown eyes that seemed to glow with great vitality."

"No other mathematician in this century has had as deep and lasting an influence on the course of civilization."

"At this half-century birthday party I have two purposes. The first is to free the dynamic input/output paradigin from gratuitous misinterpretations. The second is to say something about the genius of John von Neumann, contrasting the fertility of his contributions to economics with that of past great mathematicians and non-economist celebrities. While memories are still green, we should preserve for the historical record some of the legends about this great genius."

"Evidence enough has been given for von Neumann’s genius and eminence in pure and applied mathematics."

"We economists are grateful for von Neumann’s genius. It is not for us to calculate whether he was a Gauss, or a Poincaré, or a Hilbert. He was the incomparable Johnny von Neumann. He darted briefly into our domain and it has never been the same since."

"A man so smart that he saw through himself."

"The author, who through his previous mathematical achievements has already placed himself in the forefront of German mathematicians, is only 23 years old and completed his studies in chemistry at the Eidgenössisches Polytechnikum in Zurich with the diploma examination. He combines penetrating abstract acumen with an astonishing speed in the productive assimilation of large bodies of scientific knowledge. This is undoubtedly an altogether extraordinary talent, which justifies unlimited hopes."

"Comparing this work with the habilitation thesis, one recognizes in how outstanding and promising a manner the highly gifted young researcher combines the ability for far-reaching abstraction with a powerful sense for constructive work and also for the advancement of concrete problems."

"Bethe, Fermi, and von Neumann could often be found sitting together in a quiet room inside the throbbing heart of the Theoretical Division, challenging each other to solve complex integral equations related to pressure waves. Sometimes Oppenheimer would join them. Von Neumann usually left these other three brilliant physicists in the dust."

"But when you were in real thinking trouble, you would go to von Neumann and nobody else."

"I remember having listened to Fermi’s discussions on hydrodynamics with von Neumann. (These took the strange form of competitions before Fermi’s office blackboard as each tried to solve the problem under study first; von Neumann, with his unmatched lightning-fast analytical skill, usually won)."

"The smartest person I've ever met."

"I was a graduate student—he was one of the great mathematicians of the world."

"Neumann was undoubtedly a genius. This meant among other things that be was able to learn a new subject in an incredibly short time. Before designing the computer, he took two weeks off to learn electronics, thus became able to supervise the construction of the hardware."

"One of the century’s most esteemed scientists."

"If any one person in the previous century personified the word polymath, it was von Neumann."

"His contributions to physics, mathematics, computer science, and economics rank him as one of the all-time intellectual giants of each field."

"A great mathematician in his or any era."

"A memory which seemed to operate with even more speed than his machines enabled him to bring up, from his vast and well-indexed mental filing system, stories appropriate to whatever occasion."

"John von Neumann was one of the greatest mathematicians of the 20th century."

"The first thing that people recall about John von Neumann is his phenomenal speed of thought. He didn’t have to remember things; he computed them. If he was asked a question and didn’t know the answer, he would think for three seconds and produce a response. Yet, fast thinking was not his most outstanding characteristic. He was also very deep. It is the breadth of his scientific heritage that amazes me the most."

"I think that in terms of mathematical intelligence, he was virtually unparalleled."

"Brilliant mathematician."

"Von Neumann was exceptionally widely known among mathematicians, and there are plenty of anecdotes related to him. I think that as a student, I heard from my professor A. Rényi the saying: ‘‘Other mathematicians prove what they can, Neumann what he wants.’’"

"He was one of the most attractive people I’ve ever known, attractive in the sense that he knew so much and could reason in front of people and show them what was going on so well, it was really quite wonderful. He also had a good sense of humor. ... He was wonderful, and I was really crushed when I found out that he had cancer."

"He had a real knack for calculatin."

"Two were in their early twenties: Eugene Wigner, who became a great theoretical physicist, and Johnny von Neumann, whose brilliance as a mathematician is internationally acknowledged."

"Johnny von Neumann was so valuable, not only as a mathematician but in virtually every field, that he was welcome to work with us even for very short periods. He was allowed to come and go freely."

"Johnny was the most versatile and brilliant scientist I have ever known. His mind operated at speeds that suggested neural superconductivity."

"I believe that if a mentally superhuman race ever develops, its members will resemble Johnny von Neumann."

"That deep, practically monomaniacal devotion to the thinking process is what set Johnny von Neumann apart from everyone else I have ever known."

"I have come to suspect that to most people thinking is painful. Some of us are addicted to thinking. Some of us find it a necessity. Johnny enjoyed it. I even have the suspicion that he enjoyed practically nothing else."

"Many people have wondered how Johnny von Neumann could think so fast and so effectively. How he could find so many original solutions, in areas where most people did not even notice the problems. I think I know a part of the answer, perhaps an important part, Johnny von Neumann enjoyed thinking. I have come to suspect that to most people, thinking is painful. Some of us are addicted to thinking. Some of us find it a necessity. Johnny enjoyed it. I even have a suspicion that he enjoyed practically nothing else. This explains a lot, because what you like, you do well. And he liked thinking, not just in mathematics. He liked thinking in the clear and complete manner of mathematicians, in every field; in mathematics, in physics, in the business world - his father was a banker - and in many other fields. He could and did talk to my 3-year-old son on his own terms, and I sometimes wondered whether his relation to the rest of us were a little bit similar. This also explains his effectiveness in connection with computing machines, because computing machines apply logical processes to fields: not only mathematics, but to others as yet untouched by the logical process. And it is very significant that this revolution, the revolution of the electronic brains, was practically initiated by Johnny von Neumann. I cannot think of Johnny now without remembering a very tragic circumstance when he was dying of cancer. His brain was affected. I visited him frequently and he was trying to do what he always tried to do. And he was trying to argue with me as he used to and it wasn't functioning anymore. And I think that he suffered from this loss more than I have seen any human to suffer in any other circumstance."

"I never could keep up with him."

"I'm sure that von Neumann threw off lots of ideas, as he went about, that led to Ph.D. theses."

"I feel that Weyl and von Neumann were the greatest mathematicians that I have known."

"Von Neumann was so terribly quick in lecturing that people had to slow him up by asking questions. It was understood in his classes. That people would ask questions to slow him up. I think he was quite aware of that and was grateful for this help from the audience. Von Neumann had a way of taking an idea that he had and explaining it very quickly and very clearly."

"I think the choice of von Neumann is clear by the criteria of getting the best mathematical talent in the world."

"No one that he had to compete with. But he nevertheless was a terrifically competitive person."

"[Addressing Banesh Hoffmann] He also thought very fast."

"Then in 1927, Zawirski told me a congress of mathematicians was to take place in Lwów and foreign scholars had been invited. He added that a youthful and extremely brilliant mathematician named John von Neumann was to give a lecture."

"Kuratowski also described von Neumann’s results and his personality. He told me how in a Berlin taxicab von Neumann had explained in a few sentences much more than he, Kuratowski, would have gotten by correspondence or conversation with other mathematicians about questions of set theory, measure theory, and real variables."

"He always demonstrated his fantastic and to some extent prophetic range of interests in mathematics and its applications and at the same time an objectivity which I admired enormously."

"As a mathematician, von Neumann was quick, brilliant, efficient, and enormously broad in scientific interests beyond mathematics itself. He knew his technical abilities; his virtuosity in following complicated reasoning and his insights were supreme; yet he lacked absolute self-confidence."

"For Wigner, von Neumann and thinking were synonymous."

"Quite aware that the criteria of value in mathematical work are, to some extent, purely aesthetic, he once expressed an apprehension that the values put on abstract scientific achievement in our present civilization might diminish: "The interests of humanity might change, the present curiosities in science may cease, and entirely different things may occupy the human mind in the future." One conversation centered on the ever accelerating progress of technology and changes in the mode of human life, which gives the appearance of approaching some essential singularity in the history of the race beyond which human affairs, as we know them, could not continue."

"Johnny was probably the most brilliant star in this constellation of scientists."

"I remember that in 1927, when he came to Lwów (in Poland) to attend a congress of mathematicians, his work in foundations of mathematics and set theory was already famous. This was already mentioned to us, a group of students, as an example of the work of a youthful genius."

"Von Neumann was a giant in the breadth of his knowledge."

"His quickness was quite remarkable."

"Quantum mechanics was very fortunate indeed to attract, in the very first years after its discovery in 1925, the interest of a mathematical genius of von Neumann's stature."

"It is indeed supremely difficult to effectively refute the claim that John von Neumann is likely the most intelligent person who has ever lived."

"Universal mind."

"John von Neumann became one of the world’s greatest mathematicians and went on to father the digital computer, a device that is revolutionizing all walks of life."

"Especially as it brought me back in association with John von Neumann, whose great skill in mathematics I had first observed in Europe when he was a boy of seventeen."

"The extensive work Mathematische Begründung der Quantentheorie also testifies to the extraordinary talent of the author in the appropriation and assimilation of a large area of material."

"Incredible rapidity."

"His memory and unlimited scope of universal interests was amazing. At that time we probably did not attach any further significance to this, nor did we evaluate or even could have evaluated the incredible multiplicity and diversification of the innumerable subjects so discussed. But later, perhaps decades later, many of these subjects reappeared in his scientific work (directly, or in the background), and he had no difficulty in recovering these or related ideas from his memory as they became relevant in specific instances."

"Considered the smartest man alive."

"Throughout much of his career, he led a double life: as an intellectual leader in the ivory tower of pure mathematics and as a man of action, in constant demand as an advisor, consultant and decision-maker to what is sometimes called the military-industrial complex of the United States. My own belief is that these two aspects of his double life, his wide-ranging activities as well as his strictly intellectual pursuits, were motivated by two profound convictions. The first was the overriding responsibility that each of us has to make full use of whatever intellectual capabilities we were endowed with. He had the scientist's passion for learning and discovery for its own sake and the genius's ego-driven concern for the significance and durability of his own contributions. The second was the critical importance of an environment of political freedom for the pursuit of the first, and for the welfare of mankind in general. I'm convinced, in fact, that all his involvements with the halls of power were driven by his sense of the fragility of that freedom. By the beginning of the 1930s, if not even earlier, he became convinced that the lights of civilization would be snuffed out all over Europe by the spread of totalitarianism from the right: Nazism and Fascism. So he made an unequivocal commitment to his home in the new world and to fight to preserve and reestablish freedom from that new beachhead. In the 1940s and 1950s, he was equally convinced that the threat to civilization now came from totalitarianism on the left, that is, Soviet Communism, and his commitment was just as unequivocal to fighting it with whatever weapons lay at hand, scientific and economic as well as military. It was a matter of utter indifference to him, I believe, whether the threat came from the right or from the left. What motivated both his intense involvement in the issues of the day and his uncompromisingly hardline attitude was his belief in the overriding importance of political freedom, his strong sense of its continuing fragility, and his conviction that it was in the United States, and the passionate defense of the United States, that its best hope lay."

"Many mathematicians have suffered in fact by comparing themselves with von Neumann."

"I spent the rest of 1936 preparing for my trip to the United States, where von Neumann, with whom I had enjoyed friendly relations at least since 1930, had arranged for me to spend the second semester (from January through May, 1937) at the Institute for Advanced Study in Princeton."

"The relationship with Weyl was so close that the 22-year-old student von Neumann finished his lecture on axiomatics in February 1925 when Weyl had to take a health-related leave. In doing so, Weyl reported to the Swiss School Council that von Neumann, alongside Hilbert, was "the most expert among present mathematicians" in this field and that he himself in his lecture had "presented the subject in that form which had emerged from the unpublished investigations of Mr. Neumann.""

"John von Neumann was one whose talents reached so widely, I could talk to him about the puzzles of the geometry around what we today call a black hole."

"But John von Neumann had a marvelous interest in history. He had read the Cambridge Medieval History, [the] Cambridge Ancient History, and he had a phenomenal memory, so he could recite whole paragraphs from the Cambridge Ancient History and tell me about the Council of Nicea, for instance. But to become a member of the Atomic Energy Commission, I'm sure he was very useful, but it was so far removed from making use of this marvelous scientific imagination of his that I keep wondering if we made the best use of him."

"Von Neumann and Fermi, in particular, were enormously helpful. Both had the most stunning ability to listen to a recital of current problems for only an hour or two and then provide comments or calculations that would show the way to overcoming the problems. They also enriched the life of the lab by giving colloquium talks on almost every visit."

"The two mathematicians now or recently active in America who have adopted a similar point of view are—and I believe not by coincidence—two of the greatest forces in modern mathematics, namely, Hermann Weyl and John von Neumann."

"Neumann is one of the two or three top mathematicians in the world, is totally without national or race prejudice, and has an enormously great gift for inspiring younger men and getting them to do research."

"Young men like Heisenberg himself, Dirac, Wolfgang Paul and John von Neumann were making new discoveries almost every day. This feverish atmosphere is not one in which I function well."

"I have known a great many intelligent people in my life. I knew Max Planck, Max von Laue, and Werner Heisenberg. Paul Dirac was my brother-in-law; Leo Szilard and Edward Teller have been among my closest friends; and Albert Einstein was a good friend, too. And I have known many of the brightest younger scientists. But none of them had a mind as quick and acute as Jancsi von Neumann. I have often remarked this in the presence of those men, and no one ever disputed me."

"He understood mathematical problems not only in their initial aspect, but in their full complexity."

"Johnny was a most unusual person, a marvellously quick thinker, and was recognized as such in high school."

"Nobody knows all science, not even von Neumann did. But as for mathematics, he contributed to every part of it except number theory and topology. That is, I think, something unique."

"A deep sense of humor and an unusual ability for telling stories and jokes endeared Johnny even to casual acquaintances. He could be blunt when necessary, but was never pompous. A mind of von Neumann's inexorable logic had to understand and accept much that most of us do not want to accept and do not even wish to understand. This fact colored many of von Neumann's moral judgments. … Only scientific intellectual dishonesty and misappropriation of scientific results could rouse his indignation and ire — but these did — and did almost equally whether he himself, or someone else, was wronged."

"The accuracy of his logic was, perhaps, the most decisive character of his mind. One had the impression of a perfect instrument whose gears were machined to mesh accurately to a thousandth of an inch. "If one listens to von Neumann, one understands how the human mind should work," was the verdict of one of our perceptive colleagues... "If he analyzed a problem, it was not necessary to discuss it any further. It was clear what had to be done," said the present chairman of the U. S. Atomic Energy Commission."

"Perhaps one could find a body of phenomena which would make our concept-building ability less of a single stark fact by studying the concept-forming ability of animals. Perhaps the consciousness of animals is more shadowy than ours and perhaps their perceptions are always dreamlike. On the opposite side, whenever I talked with the sharpest intellect whom I have known -- with von Neumann -- I always had the impression that only he was fully awake, that I was halfway in a dream."

"“You have a good memory?” asked Kuhn. “Not like von Neumann’s,” replied Wigner."

"When Ratz saw how intelligent Jancsi was, he began giving him private lessons. . . . [Ratz] felt so privileged to tutor a phenomenon like Jancsi that he refused any money for it. His compensation was more subtle: the brush with a special kind of mind; the privilege of training that mind in a discipline that both of them loved."

"Our teachers were just enormously good, but the mathematics teacher was fantastic. He gave private classes to Johnny von Neumann. He gave him private classes because he realized that this would be a great mathematician."

"From talking to many people who knew him, I think I’ve gradually built up a decent picture of John von Neumann as a man. He would have been fun to meet. He knew a lot, was very quick, always impressed people, and was lively, social and funny."

"Von Neumann was extremely intelligent, and curious about everything. He looked like a cherub and sometimes acted like one; my three and five-year-old daughters delighted in climbing on him when he came to call at the house. He was very powerful and productive in pure science and mathematics and at the same time had a remarkably strong streak of practicality. He was one of the earliest pioneers in the design and construction of large electronic computers, he developed a strong interest in the technology of nuclear and other weapons, and he made a number of elegant inventions in each of these fields. This combination of scientific ability and practicality gave him a credibility with military officers, engineers, industrialists, and scientists that no one else could match. He was the clearly dominant advisory figure in nuclear missilery at the time, and everyone took his statements about what could and should be done very seriously."

"Other smart people commonly said that John von Neumann was the smartest person they had ever known. Although I worked with him for only five years—1952-57, and even then on just an occasional basis—I came to know him well enough to feel the same way. His accomplishments generally confirmed this view."

"A deep sense of humor and an unusual ability for telling stories and jokes endeared Johnny even to casual acquaintances. He could be blunt when necessary, but was never pompous. A mind of von Neumann's inexorable logic had to understand and accept much that most of us do not want to accept and do not even wish to understand. This fact colored many of von Neumann's moral judgments. "It is just as foolish to complain that people are selfish and treacherous as it is to complain that the magnetic field does not increase unless the electric field has a curl. Both are laws of nature." Only scientific intellectual dishonesty and misappropriation of scientific results could rouse his indignation and ire — but these did — and did almost equally whether he himself, or someone else, was wronged."

"A possible explanation of the physicist's use of mathematics to formulate his laws of nature is that he is a somewhat irresponsible person. As a result, when he finds a connection between two quantities which resembles a connection well-known from mathematics, he will jump at the conclusion that the connection is that discussed in mathematics simply because he does not know of any other similar connection."

"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve."

"In science, it is not speed that is the most important. It is the dedication, the commitment, the interest and the will to know something and to understand it — these are the things that come first."

"Where in the Schrödinger equation do you put the joy of being alive?"

"[M]athematicians and physicists think alike; they are led, and sometimes misled, by the same patterns of ."

""Groping" and "muddling through" is usually described as a solution by trial and error. ...a series of trials, each of which attempts to correct the error committed by the preceding and, on the whole, the errors diminished as we proceed and the successive trials come closer and closer to the desired final result. ...we may wish a better characterization ..."successive trials" or "successive corrections" or "successive approximations." ...You use successive approximations when ...looking for a word in the dictionary ...A mathematician may apply the term ...to a highly sophisticated procedure ...to treat some very advanced problem ...that he cannot treat otherwise. The term even applies to science as a whole; the scientific theories which succeed each other, each claiming a better explanation ...may appear as successive approximations to the truth. Therefore, the teacher should not discourage his students from using trial and error—on the contrary, he should encourage the intelligent use of the fundamental method of successive approximations. Yet he should convincingly show that for ...many ... situations, straightforward algebra is more efficient than successive approximations."

"Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper."

"There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn't say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann."

"Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements."

"Euclid's manner of exposition, progressing relentlessly from the data to the unknown and from the hypothesis to the conclusion, is perfect for checking the argument in detail but far from being perfect for making understandable the main line of the argument."

"The best of ideas is hurt by uncritical acceptance and thrives on critical examination."

"We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building."

"Pedantry and mastery are opposite attitudes toward rules. To apply a rule to the letter, rigidly, unquestioningly, in cases where it fits and in cases where it does not fit, is pedantry. … To apply a rule with natural ease, with judgment, noticing the cases where it fits, and without ever letting the words of the rule obscure the purpose of the action or the opportunities of the situation, is mastery."

"To write and speak correctly is certainly necessary; but it is not sufficient. A derivation correctly presented in the book or on the blackboard may be inaccessible and uninstructive, if the purpose of the successive steps is incomprehensible, if the reader or listener cannot understand how it was humanly possible to find such an argument...."

"The cookbook gives a detailed description of ingredients and procedures but no proofs for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. … Mathematics cannot be tested in exactly the same manner as a pudding; if all sorts of reasoning are debarred, a course of calculus may easily become an incoherent inventory of indigestible information."

"Demonstrative reasoning penetrates the sciences just as far as mathematics does, but it is in itself (as mathematics is in itself) incapable of yielding essentially new knowledge about the world around us. Anything new that we learn about the world involves plausible reasoning, which is the only kind of reasoning for which we care in everyday affairs."

"Everyone knows that mathematics offers an excellent opportunity to learn demonstrative reasoning, but I contend also that there is no other subject in the usual curricula of the schools that affords a comparable opportunity to learn plausible reasoning. ...let us learn proving, but also let us learn guessing."

"The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference."

"In plausible reasoning the principal thing is to distinguish... a more reasonable guess from a less reasonable guess."

"The general or amateur student should also get a taste of demonstrative reasoning... he should acquire a standard with which he can compare alleged evidence of all sorts aimed at him in modern life."

"The efficient use of plausible reasoning is a practical skill and it is learned... by imitation and practice. ...what I can offer are only examples for imitation and opportunity for practice."

"I shall often discuss mathematical discoveries... I shall try to make up a likely story how the discovery could have happened. I shall try to emphasize the motives underlying the discovery, the plausible inferences that led to it... everything that deserves imitation."

"I... present also examples of historic interest, examples of real mathematical beauty, and examples illustrating the parallelism of the procedures in other sciences, or in everyday life."

"For many of the stories told the final form resulted from a sort of informal psychological experiment. I discussed the subject with several different classes... Several passages... have been suggested by answers of my students, or... modified... by the reaction of my audience."

"In my presentation I... follow the genetic method. The essential idea... is that the order in which knowledge has been acquired by the human race will be a good teacher for its acquisition by the individual. The sciences came in a certain order; an order determined by human interest and inherent difficulty. Mathematics and astronomy were the first sciences really worth the name; later came mechanics, optics, and so on. At each stage of its development the human race has had a certain climate of opinion, a way of looking, conceptually, at the world. The next glimmer of fresh understanding had to grow out of what was already understood. The next move forward, halting shuffle, faltering step, or stride with some confidence, was developed upon how well the [human] race could then walk. As for the human race, so for the human child. But this is not to say that to teach science we must repeat the thousand and one errors of the past, each ill-directed shuffle. It is to say that the sequence in which the major strides forward were made is a good sequence in which to teach them. The genetic method is a guide to, not a substitute for, judgement."

"Why should the typical student be interested in those wretched triangles? ...He is to be brought to see that without the knowledge of triangles there is not trigonometry; that without trigonometry we put back the clock millennia to Standard Darkness Time and antedate the Greeks."

"Good approximations often lead to better ones."

"The volume of the cone was discovered by Democritus... He did not prove it, he guessed it... not a blind guess, rather it was reasoned conjecture. As Archimedes has remarked, great credit is due to Democritus for his conjecture since this made proof much easier. Eudoxes... a pupil of Plato, subsequently gave a rigorous proof. Surely the labor or writing limited his manuscript to a few copies; none has survived. In those days editions did not run to thousands or hundreds of thousands of copies as modern books—especially, bad books—do. However, the substance of what he wrote is nevertheless available to us. ...Euclid's great achievement was the systematization of the works of his predecessors. The Elements preserve several of Eudoxes' proofs."

"Mathematics succeeds in dealing with tangible reality by being conceptual. We cannot cope with the full physical complexity; we must idealize."

"We wish to see... the typical attitude of the scientist who uses mathematics to understand the world around us. ...In the solution of a problem ...there are typically three phases. The first phase is entirely or almost entirely a matter of physics; the third, a matter of mathematics; and the intermediate phase, a transition from physics to mathematics. The first phase is the formulation of the physical hypothesis or conjecture; the second, its translation into equations; the third, the solution of the equations. Each phase calls for a different kind of work and demands a different attitude."

"Facing any part of the observable reality, we are never in possession of complete knowledge, nor in a state of complete ignorance, although usually much closer to the latter state."

"If we deal with our problem not knowing, or pretending not to know the general theory encompassing the concrete case before us, if we tackle the problem "with bare hands", we have a better chance to understand the scientist's attitude in general, and especially the task of the applied mathematician."

"If you cannot solve the proposed problem, try to solve first a simpler related problem."

"\frac {dy}{dx} = \frac {\omega^2x}{g}...The first derivative, the result of the differentiation of y with respect to x, was written by Leibniz in the form \frac {dy}{dx}...Leibniz's notation ...is both extremely useful and dangerous. Today, as the concepts of limit and derivative are sufficiently clarified, the use of the notation... need not be dangerous. Yet, the situation was different in the 150 years between the discovery of calculus by Newton and Leibniz and the time of Cauchy. The derivative \frac {dy}{dx} was considered as the ratio of two "infinitely small quanitites", of the infinitesimals dy and dx. ...it greatly facilitated the systematization of the rules of the calculus and gave intuitive meaning to its formulas. Yet this consideration was also obscure... it brought mathematics into disrepute... some of the best minds... such as... Berkeley, complained that calculus is incomprehensible. ...\frac {dy}{dx} is the limit of a ratio of dy to dx... Once we have realized this sufficiently clearly, we may, under certain circumstances, treat \frac {dy}{dx} so as if it were a ratio... and multiply by dx to achieve the separation of variables. We get {dy} = \frac {\omega^2x}{g}xdx"

"Simplicity is worth buying if we do not have to pay too great a loss of precision for it."

"Even if without the Scott's proverbial thrift, the difficulty of solving differential equations is an incentive to using them parsimoniously. Happily here is a commodity of which a little may be made to go a long way. ...the equation of small oscillations of a pendulum also holds for other vibrational phenomena. In investigating swinging pendulums we were, albeit unwittingly, also investigating vibrating tuning forks."

"The differential equation of the first order \frac {dy}{dx} = f(x,y) ...prescribes the slope \frac {dy}{dx} at each point of the plane (or at each point of a certain region of the plane we call the field"). ...a differential equation of the first order... can be conceived intuitively as a problem about the steady flow of a river: Being given the direction of the flow at each point, find the streamlines. ...It leaves open the choice between the two possible directions in the line of a given slope. Thus... we should say specifically "direction of an unoriented straight line" and not merely "direction.""

"Life is full of surprises: our approximate condition for the fall of a body through a resisting medium is precisely analogous to the exact condition for the flow of an electric current through a resisting wire [of an induction coil]. ... m\frac {dv}{dt} = mg - Kv This is the form most convenient for making an analogy with the "fall", i.e., flow, of an electric current. ...in order from left to right, mass m, rate of change of velocity \frac {dv}{dt}, gravitational force mg, and velocity v. What are the electrical counterparts? ...To press the switch, to allow current to start flowing is the analogue of opening the fingers, to allow the body to start falling. The fall of the body is caused by the force mg due to gravity; the flow of the current is caused by the electromotive force or tension E due to the battery. The falling body has to overcome the frictional resistance of the air; the flowing current has to overcome the electrical resistance of the wire. Air resistance is proportional to the body's velocity v; electrical resistance is proportional to the current i. And consequently rate of change of velocity \frac {dv}{dt} corresponds to rate of change of current \frac {di}{dt}. ...The electromagnetic induction L opposes the change of current... And doesn't the inertia or mass m..? Isn't L, so to speak, an electromagnetic inertia? L\frac {di}{dt} = E - Ki"

"People tell you that wishful thinking is bad. Do not believe it, this is just one of those generally accepted errors."

"If we could be any mathematician in the history of the world (besides ourselves), who would we rather be? ...we narrowed the choice down to Euler and Pólya, and finally settled on George Pólya because of the sheer enjoyment of mathematics that he has conveyed by so many examples."

"For mathematics education and the world of problem solving it marked a line of demarcation between two eras, problem solving before and after Polya."

"A parallelákat azon az útan ne próbáld: tudom én azt az utat is mind végig — megmértem azt a feneketlen éjszakát én, és az életemnek minden világossága, minden öröme kialudt benne..."

"No monument should stand over my grave, only an apple-tree, in memory of the three apples; the two of Eve and Paris, which made hell out of earth, and that of I. Newton, which elevated the earth again into the circle of heavenly bodies."

"In 1823, non-Euclidean geometry was discovered simultaneously, in one of those inexplicable coincidences, by a Hungarian mathematician, Janos (or Johann) Bolyai, aged twenty-one, and a Russian mathematician, Nikolai Lobachevsky, aged thirty. And, ironically, in that same year, the great French mathematician Adrien-Marie Legendre came up with what he was sure was a proof of Euclid's fifth postulate, very much along the lines of Saccheri. Incidentally, Bolyai's father, Farkas (or Wolfgang) Bolyai, a close friend of the great Gauss, invested much effort in trying to prove Euclid's fifth postulate."