Mathematicians From Hungary

"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."