Mathematics

"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."

"The Koch Curve - Triangles outside triangles outside triangles ad infinitum the Koch curve goes, it's infinitely infinitesimal, this self-similarity shows. A length too long to measure, an area too small to see, what else can this contradiction be, behold fractal geometry."

"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."

"They who study mathematiks only to fix their minds, and render them the steadyer to apply to all other things, as there are many who profess to do, are as wise as those who think by rowing boats, to learn to swim."

"Mathematical development in England was at a low ebb in the early decades of the nineteenth century, with Cambridge stagnating in the shadow of Newton, who had produced his mathematics nearly a century and a half earlier. This dead hand of tradition, which stifled much initiative and originality, was in sharp contrast to the situation in France."

"Advanced mathematics opens doors to many fields of study. More importantly, it expands the way you think."

"Yeah, Silver and his math are jokes, because math has a liberal bias. After all, math is the reason Mitt Romney's tax plan doesn't add up."

"I was an atheist, finding no reason to postulate the existence of any truths outside of mathematics, physics and chemistry. But then I went to medical school, and encountered life and death issues at the bedsides of my patients. Challenged by one of those patients, who asked "What do you believe, doctor?", I began searching for answers."

"Magick, mysticism, and the mathematics are triplets."

"In the pure mathematics, we contemplate absolute truths, which existed in the Divine Mind before the morning stars sang together, and which will continue to exist there, when the last of their radiant host shall have fallen from heaven."

"A man with all the algebra in the world is often only an ass when he knows nothing else. Perhaps in ten years society may derive advantage from the curves which these visionary algebraists will have laboriously squared. I congratulate posterity beforehand. But to tell you the truth I see nothing but a scientific extravagance in all these calculations. That which is neither useful nor agreeable is worthless. And as for useful things, they have all been discovered; and to those which are agreeable, I hope that good taste will not admit algebra among them."

"As to your Newton, I confess I do not understand his void and his gravity; I admit he has demonstrated the movement of the heavenly bodies with more exactitude than his forerunners; but you will admit it is an absurdity to maintain the existence of Nothing."

"Euler calculated the force of the wheels necessary to raise the water in a reservoir … My mill was carried out geometrically and could not raise a drop of water fifty yards from the reservoir. Vanity of vanities! Vanity of geometry!"

"Measuring is one of the more practical uses of mathematics, but the ability and desire to measure aren't always wrapped up with the need to know useful answers."

"Letting numbers take us where we can't go in person — whether that's to the top of a windmill or to the origin and borders of the universe — has been and still is one of humankind's favorite intellectual adventures."

"The Devil: Okay, boys...tonight's homework. Algebra. Xn + Yn = Zn. You're never gonna use that, are you? Imperialism and the First World War. What was done is done. No point thinking about it now. German, French, Spanish. Ja, ja, oui, oui, s, s. It's nonsense. Everyone speaks English anyway. And if they don't, they ought to. So, no homework tonight. But I want you to watch a lot of TV, don't neglect your video games...and I'll see you in the morning. Shall we say 10:00, 10:30? No point in getting up too early."

"Mathematics is really a liberal art if you look at it from a slightly different point of view."

"The mathematical genius can only carry on from the point which mathematical knowledge within his culture has already reached. Thus if Einstein had been born into a primitive tribe which was unable to count beyond three, life-long application to mathematics probably would not have carried him beyond the development of a decimal system based on fingers and toes."

"So far, all of reality seems to be described by exquisite, elegant mathematical equations. We can’t stop now – it’s got to be beautiful all the way down!"

"Poetry is a sort of inspired mathematics, which gives us equations, not for abstract figures, triangles, squares, and the like, but for the human emotions. If one has a mind which inclines to magic rather than science, one will prefer to speak of these equations as spells or incantations; it sounds more arcane, mysterious, recondite."

"Halakhic man, well furnished with rules, judgments, and fundamental principles, draws near the world with an a priori relation. His approach begins with an ideal creation and concludes with a real one. To whom may he be compared? To a mathematician who fashions an ideal world and then uses it for the purpose of establishing a relationship between it and the real world. ... The essence of the Halakhah, which was received from God, consists in creating an ideal world and cognizing the relationship between that ideal world and our concrete environment."

"Mathematics is a versatile art; it can be applied to widely different purposes. Math has no morality; it does not care what it counts or what it proves."

"I had been to school most all the time and could spell and read and write just a little, and could say the multiplication table up to six times seven is thirty-five, and I don't reckon I could ever get any further than that if I was to live forever. I don't take no stock in mathematics anyway."

"School children and students who love God should never say: “For my part I like mathematics”; “I like French”; “I like Greek.” They should learn to like all these subjects, because all of them develop that faculty of attention which, directed toward God, is the very substance of prayer."

"The number 2 thought of by one man cannot be added to the number 2 thought of by another man so us to make up the number 4."

"The good Christian should beware of mathematicians, and all those who make empty prophesies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell."