Mathematicians From The United States

"Infinite product spaces are the natural habitat of probability theory."

"To every event defined for the original random walk there corresponds an event of equal probability in the dual random walk, and in this way almost every probability relation has its dual."

"The theory of independent experiments is the analytically simplest and most advanced part of probability theory."

"When a coin is tossed, it does not necessarily fall heads or tails; it can roll away or stand on its edge."

"This means that if in a city seven accidents occur each week, then (assuming that all possible distributions are equally likely) practically all weeks will contain days with two or more accidents, and on the average only one week out of 165 will show a uniform distribution of one accident per day."

"Simple methods will soon lead us to results of far reaching theoretical and practical importance. We shall encounter theoretical conclusions which not only are unexpected but actually come as a shock to intuition and common sense. They will reveal that commonly accepted notions concerning chance fluctuations are without foundation and that the implications of the law of large numbers are widely misconstrued."

"The manner in which mathematical theories are applied does not depend on preconceived ideas; it is a purposeful technique depending on, and changing with, experience."

"The philosophy of the foundations of probability must be divorced from mathematics and statistics, exactly as the discussion of our intuitive space concept is now divorced from geometry."

"The bewildered novice in chess moves cautiously, recalling individual rules, whereas the experienced player absorbs a complicated situation at a glance and is unable to account rationally for his intuition. In like manner mathematical intuition grows with experience, and it is possible to develop a natural feeling for concepts such as four dimensional space."

"Historically, the original purpose of the theory of probability was to describe the exceedingly narrow domain of experience connected with games of chance, and the main effort was directed to the calculation of certain probabilities."

"When this book was first conceived (more than 25 years ago) few mathematicians outside the Soviet Union recognized probability as a legitimate branch of mathematics."

"Only yesterday the practical things of today were decried as impractical, and the theories which will be practical tomorrow will always be branded as valueless games by the practical man of today."

"The results concerning fluctuations in coin tossing show the widely held beliefs about the law of large numbers are fallacious. They were so amazing and so at variance with common intuition that even sophisticated colleagues doubted that coins actually misbehave as theory predicts."

"three repairman per twenty machines are much more economical than one repairman per six machines."

"Anyone writing a probability text today owes a great debt to William Feller, who taught us all how to make probability come alive as a subject matter. If you find an example, an application, or an exercise that you really like, it probably had its origin in Feller's classic text, An Introduction to Probability Theory And (Its) Applications."

"The fact that the mean recurrence time is infinite implies that the chance fluctuations in an individual prolonged coin-tossing game are far removed from the familiar pattern governed by the normal distribution."

"Physical irreversibility manifests itself in the fact that, whenever the system is in a state far removed from equilibrium, it is much more likely to move toward equilibrium, than in the opposite direction."

"It is not necessary to think of gambling places; the statistician who applies statistical tests is engaged in a dignified sort of gambling, and in his case the distribution of the random variables changes from occasion to occasion."

"It is seen that continued shuffling may reasonably be expected to produce perfect "randomness" and to eliminate all traces of the original order. It should be noted, however, that the number of operations required for this purpose is extremely large."

"The classical theory of probability was devoted mainly to a study of the gamble's gain, which is again a random variable; in fact, every random variable can be interpreted as the gain of a real or imaginary gambler in a suitable game."

"Much harm was done by the misleading suggestive power of this name. It must be understood that a fair game may be distinctly unfavorable to the player."

"In stochastic processes the future is not uniquely determined, but we have at least probability relations enabling us to make predictions."

"This faulty intuition as well as many modern applications of probability theory are under the strong influence of traditional misconceptions concerning the meaning of the law of large numbers and of a popular mystique concerning a so-called law of averages."

"Warning. It is usual to read into the law of large numbers things which it definitely does not imply. If Peter and Paul toss a perfect coin 10,000 times, it is customary to expect that Peter will be in the lead roughly half the time. This is not true. In a large number of different coin-tossing games it is reasonable to expect that any fixed moment heads will be in the lead in roughly half of all cases. But it is quiet likely that the player who ends at the winning side has been in the lead for practically the whole duration of the game. Thus, contrary to widespread belief, the time average for any individual game has nothing to do with the ensemble average at any given moment."

"The painful experience of many gamblers has taught us the lesson that no system of betting is successful in improving the gambler's chances. If the theory of probability is true to life, this experience must correspond to a provable statement."

"The man in the street, and also the philosopher K. Marbe, believe that after a run of seventeen heads tail becomes more probable. This argument has nothing to do with imperfections of physical coins; it endows nature with memory, or, in our terminology, it denies the stochastic independence of successive trials. Marbe's theory cannot be refuted by logic but is rejected because of empirical support."

"Note the situation is different when the player is permitted to vary his stakes. In this case there exist advantageous strategies, and the game depends on the strategy."

"It has been suggested that an army of monkeys might be trained to pound typewriters at random in the hope that ultimately great works of literature would be produced. Using a coin for the same purpose may save feeding and training expenses and free the monkeys for other monkey business."

"It should be noted that this duration is considerably longer than we naively expect. If two players with 500 dollars each toss a coin until one is ruined, the average duration of the game is 250,000 trials. If a gambler has only one dollar and his adversary has 1000, the average duration is 1000 trials."

"It is a common fallacy to believe that the law of large numbers acts as a force endowed with memory seeking to return to the original state, and many wrong conclusions have been drawn from this assumption."

"There are two kinds of geniuses: the ‘ordinary’ and the ‘magicians.’ an ordinary genius is a fellow whom you and I would be just as good as, if we were only many times better. There is no mystery as to how his mind works. Once we understand what they’ve done, we feel certain that we, too, could have done it. It is different with the magicians... Feynman is a magician of the highest caliber."

"Unrestricted abstraction tends to divert attention from whole areas of application whose very discovery depends of the features that the abstract point of view rules out as being accidental."

"Creative people live in two worlds. One is the ordinary world which they share with others and in which they are not in any special way set apart from their fellow men. The other is private and it is in this world that the creative acts take place."

"I didn't even try to penetrate the comics, though many years later I came, somewhat grudgingly, to admire Pogo."

"I prefer concrete things and I don't like to learn more about abstract stuff than I absolutely have to."

"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."

"Actually, my solution generated considerable further work and the "dog-flea" model keeps cropping up from time to time in unexpected contexts."

"There was hardly a page in Markov's book which did not feature the normal law and it cast a spell over me from which I have never fully recovered. Adding to the fascination was the impression that somehow the normal law was the key to mysterious and elusive world of chance phenomena."

"As an introduction to America, my ten months in Baltimore were superb. I find it difficult to find words to convey the feeling of decompression, of freedom, of being caught in a sweep of unimagined and unimaginable grandeur. It was a life on a different scale with more of everything - more air to breathe, more things to see, more people to know."

"I then reached for a time honored tactic used by mathematicians: if you can't solve the real problem, change it into one you can solve."

"Where there is independence there must be the normal law."

"I had a phenomenal memory and could recite long poems by Russian poets, mainly Pushkin. Except for an unusual memory, I was not precocious in any respect and somewhat later, to the chagrin of my father, I was inordinately slow learning the multiplication tables."

"When one is young, and seventeen is very young, one lives in the present. The future, even the near future, is cloaked in unreality."

"In the summer of 1930 my academic future, however, was not uppermost in my mind. I had been stricken by an acute attack of a disease which at regular intervals afflicts all mathematicians and, for that matter, all scientists: I became obsessed by a problem."

"There are, roughly speaking, two kinds of mathematical creativity. One, akin to conquering a mountain peak, consists of solving a problem which has remained unsolved for a long time and has commanded the attention of many mathematicians. The other is exploring new territory."

"Independence is the central concept of probability theory and few would believe today that understanding what it meant was ever a problem."

"Mathematics is an ancient discipline. For as long as we can reliably reach into the past, we find its development intimately connected with the development of the whole of our civilization. For as long as we have a record of man's curiosity and his quest for understanding, we find mathematics cultivated and cherished, practiced and taught. Throughout the ages it has stood as an ultimate in rational thought and as a monument to man's desire to probe the workings of his own mind."

"In science, as well as in other fields of human endeavor, there are two kinds of geniuses: the “ordinary” and the “magicians.’’ An ordinary genius is a fellow that you and I would be just as good as, if we were only many times better. There is no mystery as to how his mind works. Once we understand what he has done, we feel certain that we, too, could have done it. It is different with the magicians. They are, to use mathematical jargon, in the orthogonal complement of where we are and the working of their minds is for all intents and purposes incomprehensible. Even after we understand what they have done, the process by which they have done it is completely dark. They seldom, if ever, have students because they cannot be emulated and it must be terribly frustrating for a brilliant young mind to cope with the mysterious ways in which the magician’s mind works. Richard Feynman is a magician of the highest caliber. Hans Bethe, whom Dyson considers to be his teacher, is an “‘ordinary genius’’; so much so that one may gain the erroneous impression that he is not a genius at all. But it was Feynman, only slightly older than Dyson, who captured the young man’s imagination. To be a physicist must have meant to him to be like Feynman and this, alas, was impossible. And so Dyson fell back on the source of strength he always had in reserve: the mastery of mathematical technique."

"Historians often do not mention the truly important features of daily life, like games, concentrating instead on tedious political maneuvering."

"The creative scientist lives in a 'wildness of logic,' where reason is the handmaiden and not the master."