All Quotes

"I like mathematics because it is not human and has nothing particular to do with this planet or with the whole accidental universe – because, like Spinoza's God, it won't love us in return."

"Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say."

"Pi's face was masked, and it was understood that none could behold it and live. But piercing eyes looked out from the mask, inexorable, cold and enigmatic."

"In universities, mathematics is taught mainly to men who are going to teach mathematics to men who are going to teach mathematics to.... Sometimes, it is true, there is an escape from this treadmill. Archimedes used mathematics to kill Romans, Galileo to improve the Grand Duke of Tuscany's artillery, modern physicists (grown more ambitious) to exterminate the human race. It is usually on this account that the study of mathematics is commended to the general public as worthy of State support."

"Is Mathematics merely the fire that allows mankind to see the shadows of a more remote ultimate reality?"

"You cannot be cheated, you cannot be lied to. A thing is true or not true, and there is this notion of clarity on which you can base yourself."

"It’s really beautiful to observe, as you progress in your mathematical maturity, how everything is somehow connected."

"10^{50} is a long way from infinity."

"Mathematics is universal. But very little else is."

"There is a wide distinction between the degree of mathematical acquirement necessary for making discoveries, and that which is requisite for understanding what other have done."

"No part of mathematics came to birth in the form that it now appears in a modern textbook: mathematical creativity can be slow, sometimes messy, often frustrating."

"I discovered the works of Euler and my perception of the nature of mathematics underwent a dramatic transformation. I was de-Bourbakized, stopped believing in sets, and was expelled from the Cantorian paradise. I still believe in abstraction, but now I know that one ends with abstraction, not starts with it. I learned that one has to adapt abstractions to reality and not the other way around. Mathematics stopped being a science of theories but reappeared to me as a science of numbers and shapes."

"It doesn't matter what mathematical things are: it's what they do that counts. Thus mathematics hovers uneasily between the real and the not-real; its meaning does not reside in formal abstractions, but neither is it tangible. ...it is the great strength of mathematics—what I have elsewhere called its "unreal reality." Mathematics links the abstract world of mental concepts to the real world of physical things without being located in completely in either."

"Math ain't about numbers! If you think math is about numbers, you probably think that Shakespeare is all about words. You probably think that dancing is all about shoes. You probably think that music is all about notes. Math ain't about numbers! Math is about logic, it's about beauty, it's about connections, it's about how you get from one place to another."

"So, nat'ralists observe, a flea Hath smaller fleas that on him prey, And these have smaller still to bite 'em And so proceed ad infinitum."

"Researchers of the brain believe that mathematical truths make little sense to our mind, particularly..the examination of random outcomes."

"Mathematics is principally a tool to meditate, rather than to compute."

"If a mathematician wishes to disparage the work of one of his colleagues, say, A, the most effective method he finds for doing this is to ask where the results can be applied. The hard pressed man, with his back against the wall, finally unearths the researches of another mathematician B as the locus of the application of his own results. If next B is plagued with a similar question, he will refer to another mathematician C. After a few steps of this kind we find ourselves referred back to the researches of A, and in this way the chain closes."

"The subject matter of mathematics has increased so rapidly and extensively that there is some element of truth in maintaining that mathematics is not so much a subject as a way of studying any subject, not so much a science as a way of life."

"Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality."

"One occasionally hears the question, is mathematics invented or discovered?—or an answer. As David Wells points out... both answers... are appropriate. Once a game is invented, the consequences are discovered... as it would require a divine intelligence to know just from the rules how a complex game could best be played. When in practice rules are changed, one makes adjustments that will not alter the consequences too dramatically. Analogously, axioms are usually only adjusted and the altered consequences discovered."

"I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science, whatever the matter may be."

"Mathematics is the only true metaphysics."

"Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding."

"The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example any real reason that even such famous results as Fermat's Last Theorem, or the Poincaré conjecture, really matter? Their real importance is not in their specific statements, but their role in challenging our understanding, presenting challenges that led to mathematical developments that increased our understanding."

"Mathematics is a process of staring hard enough with enough perseverence at at the fog of muddle and confusion to eventually break through to improved clarity."

"Comparatively few engineers are good mathematicians; and... it is fortunate that such is the case; for nature rarely combines high mathematical talent, with that practical tact, and observation of outward things, so essential to a successful engineer. There have been... brilliant exceptions; but they are very rare. But few even of those who have been tolerable mathematicians when young, can, as they advance in years, and become engaged in business, spare the time necessary for retaining such accomplishments."

"For many individuals, as they approach the limit of their abilities, mathematics loses its fun aspect. When a topic is undeveloped, it is recreational to many. As the theory is developed and becomes more abstract, fewer persons find it recreational."

"Mathematical reasoning may be regarded rather schematically as the exercise of a combination of... intuition and ingenuity. The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasoning... The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings. ...We are leaving out of account that most important faculty which distinguishes topics of interest from others; in fact, we are regarding the function of the mathematician as simply to determine the truth or falsity of propositions."

"The mathematicians know a great deal about very little and the physicists very little about a great deal."

"Mathematics, and perhaps other sciences like physics, have the mission to prepare or improve the human brain, be it the brain of an individual or the collective brain of mankind, for developments yet to come. Just as animals play... in preparation for situations arising later in their lives, it may be that mathematics... is a collection of games. ...may be the only way to change the individual or collective human mind to prepare it for a future that no one can yet imagine. ...[L]ife appears inter alia as a sequence of chemical games... between individuals, or between groups... essentially of a mathematical nature... not... the von Neumann-Morgenstern theory of games, but more general games in the widest sense. This has perhaps... a direct biological role... a book... by... , ...Das Spiel (The Game) ...describes a number of mathematical games or puzzles and discusses the games molecules... play with each other. ... ...we have seen ...starting with a simple pattern and simple recursive rules can lead to unbelievably complicated configurations... that... defy analysis a priori."

"In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug. Chess sometimes plays a similar role. In their unhappiness over the events of this world, some immerse themselves in a kind of self-sufficiency in mathematics. (Some have engaged in it for this reason alone.)"

"What exactly is mathematics? Many have tried but nobody has really succeeded in defining mathematics; it is always something else. ...[P]eople know that it deals with numbers and figures, with patterns, relations, operations, and that its formal procedures involving axioms, proofs, lemmas, theorems have not changed since the time of Archimedes. ... that it purports to form the foundations of all rational thought. ...The aesthetic side of mathematics has been of overwhelming importance throughout its growth. It is not so much whether a theorem is useful that matters, but how elegant it is. ...One can ...look conversely at ...the homely side of mathematics ...having to be punctilious ...having to make sure of every step. ...[O]ne cannot stop at drawing with a big, wide brush; all the details have to be filled in ...Mathematicians ...fool themselves ...when they think their main business is to prove theorems without at least indicating why they may be important. If left entirely to the aesthetic criteria, doesn't it compound the mystery? ...[I]n the decades to come there will be more understanding ...of the degree of beauty, though ...the criteria may have shifted ...[to] a super beauty in unanalyzable higher levels. ...It has to appeal to connections with other theories of the external world or to the history of the development of the human brain, or else it is purely aesthetic and very subjective in the sense that music is. ...[E]ven the quality of music will be analyzable ...by mathematizing the idea of analogy."

"Mathematics may be a way of developing physically, that is anatomically, new connections in the brain."

"Like the crests on the heads of peacocks, like the gems on the hoods of the cobras, mathematics is at the top of the Vedanga sastras."

"It is but right... to apprise you that, diffident of my own decision, the favorable opinion I favor of your performance is founded rather on the explicit and ample testimonies of Gentlemen confessedly possessed of great mathematical knowledge, than on the partial and incompetent attention I have been able to pay to it myself.—But I must be permitted to remark that the subject, in my estimation, holds a higher rank in the literary scale than you are disposed to allow.—The science of figures, to a certain degree, is not only indispensably requisite in every walk of civilised life; but investigation of mathematical truths accustoms the mind to method and correctness in reasoning, and is an employment peculiarly worthy of rational beings. In a clouded state of existence, where so many things appear precarious to the bewildered research, it is here that the rational faculties find a firm foundation to rest upon. From the high ground of mathematical and philosophical demonstration, we are insensibly led to far nobler speculations and sublimer meditations."

"I enjoy being surprised by mathematics and its intrinsic difficulty. The moment I enjoy best is when the pieces of the puzzle fall into one coherent whole."

"[A]ll science as it grows towards perfection becomes mathematical in its ideas."

"By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases... mental power... Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that, under the influence of compulsory education, the whole population of Western Europe, from the highest to the lowest, could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility."

"Nothing is more impressive than the fact that as mathematics withdrew increasingly into the upper regions of ever greater extremes of abstract thought, it returned back to earth with a corresponding growth of importance for the analysis of concrete fact. ...The paradox is now fully established that the utmost abstractions are the true weapons with which to control our thought of concrete fact."

"The only reality mathematical concepts have is as cultural elements or artifacts."

"A mathematician is bound to be horrified by my mathematical comments, since he has always been trained to avoid indulging in thoughts and doubts of the kind I develop. He has learned to regard them as something contemptible and […] he has acquired a revulsion from them as infantile. That is to say, I trot out all the problems that a child learning arithmetic, etc., finds difficult, the problems that education represses without solving. I say to those repressed doubts: you are quite correct, go on asking, demand clarification!"

"There is no religious denomination in which so much sin has been committed through the misuse of metaphorical expressions as in mathematics."

"I like to find out as much about a mathematical object as possible, just as you might want to understand a person as well as possible."

"Math is perfect (in principle), but mathematicians are not (because they are humans), hence the mathematics that (human) mathematicians do is influenced by the weltanschauung of the people around them."

"All of mathematics is arbitrary. The very language of existence, mathematics, has no reason to be a certain way. But it is."

"Blindly and lawlessly they did all things, Until I taught them how the stars do rise And set in mystery, and devised for them Number, the inducer of philosophies, The synthesis of Letters, and, beside, The artificer of all things, Memory, That sweet Muse-mother."

"The land of easy mathematics where he who works adds up and he who retires subtracts."

"Susan: What is algebra exactly; is it those three-cornered things? Phoebe: It is x minus y equals z plus y and things like that. And all the time you are saying they are equal, you feel in your heart, why should they be."

"I can’t help it, gas escapes from my fundament on the least pretext, it’s hard not to mention it now and then, however great my distaste. One day I counted them. Three hundred and fifteen farts in nineteen hours, or an average of over sixteen farts an hour. After all it’s not excessive. Four farts every fifteen minutes. It’s nothing. Not even one fart every four minutes. It’s unbelievable. Damn it, I hardly fart at all, I should never have mentioned it. Extraordinary how mathematics help you to know yourself."