Mathematics By Mathematicians Quotes

If things are nice there is probably a good reason why they are nice: and if you do not know at least one reason for this good fortune, then you still have work to do. ~ Richard Askey

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. ~ George Polya

Where there is matter, there is geometry. ~ Johannes Kepler

Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them. ~ Joseph Fourier

It would be better for the true physics if there were no mathematicians on earth. ~ Daniel Bernoulli

Mathematics is no more computation than typing is literature. ~ John Allen Paulos

Mathematics is concerned with "all possible worlds." ~ David Malet Armstrong

[Mathematics] is an independent world created out of pure intelligence. ~ William Wordsworth

If only I had the Theorems! Then I should find the proofs easily enough. ~ Bernhard Riemann

We used to think that if we knew one, we knew two, because one and one are two. We are finding that we must learn a great deal more about 'and. ~ Arthur Stanley Eddington

God exists since mathematics is consistent, and the Devil exists since we cannot prove it. ~ Andre Weil

I have hardly ever known a mathematician who was capable of reasoning. ~ Plato

It is easier to square a circle than to get round a mathematician. ~ Augustus De Morgan

Il n'est pas certain que tout soit incertain.
(Translation: It is not certain that everything is uncertain.) ~ Blaise Pascal

The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal. ~ William James

There is no study in the world which brings into more harmonious action all the faculties of the mind than [mathematics], ... or, like this, seems to raise them, by successive steps of initiation, to higher and higher states of conscious intellectual being ... ~ James Joseph Sylvester

Mathematics is entirely free in its development, and its concepts are only linked by the necessity of being consistent, and are co-ordinated with concepts introduced previously by means of precise definitions. ~ Georg Cantor

Now being in such grace and favor by reason I learned him some points of geometry and understanding of the art of mathematics with other things, I pleased him so that what I said he would not contrary. ~ William Adams

There was virtually no aspect of twentieth-century defense technology that had not been touched by the hands and minds of female mathematicians. ~ Margot Lee Shetterly

I cannot recall a period when I did not draw; and at school, the studies that were distasteful to me, mathematics and grammar, were retarded by the indulgence of teachers who were proud of my drawing faculties, and passed over my neglect of uncongenial subjects. ~ Jacob Epstein

In the last two months I have been very busy with my own mathematical speculations, which have cost me much time, without my having reached my original goal. Again and again I was enticed by the frequently interesting prospects from one direction to the other, sometimes even by will-o'-the-wisps, as is not rare in mathematic speculations. ~ Carl Friedrich Gauss

Unfortunately, I do not find Tegmark's line of reasoning to be extremely compelling. The leap from the existence of an external reality (independent of humans) to the conclusion that, in Tegmark's words, "You must believe in what I call the mathematical universe hypothesis: that our physical reality is a mathematical structure," involves, in my opinion, a sleight of hand. When Tegmark attempts to characterize what mathematics really is, he says: "To a modern logician, a mathematical structure is precisely this: a set of abstract entities with relations between them." But this modern logician is human! In other words, Tegmark never really proves that our mathematics is not invented by humans; he simply assumes it. Furthermore, as the French neurobiologist Jean-Pierre Changeaux has pointed out in response to a similar assertion: "To claim physical reality for mathematical objects, on a level of the natural phenomena we study in biology, poses a worrisome epistemological problem it seems to me. How can a physical state, internal to our brain, represent another physical state external to it? ~ Mario Livio

At the federal level, this problem could be greatly alleviated by abolishing the Electoral College system. It's the winner-take-all mathematics from state to state that delivers so much power to a relative handful of voters. It's as if in politics, as in economics, we have a privileged 1 percent. And the money from the financial 1 percent underwrites the microtargeting to secure the votes of the political 1 percent. Without the Electoral College, by contrast, every vote would be worth exactly the same. That would be a step toward democracy. ~ Cathy O'Neil

A good example of the archetypal ideas which the archetypes produce are natural numbers or integers. With the aid of the integers the shaping and ordering of our experiences becomes exact. Another example is mathematical group theory. ...important applications of group theory are symmetries which can be found in most different connections both in nature and among the 'artifacts' produced by human beings. Group theory also has important applications in mathematics and mathematical physics. For example, the theory of elementary particles and their interactions can in essential respects be reduced to abstract symmetries.
[The Message of the Atoms: Essays on Wolfgang Pauli and the Unspeakable] ~ Kalervo V. Laurikainen

At the risk of being forgotten completely by the media, I went to college and pursued a passion that had nothing to do with acting: mathematics. ~ Danica McKellar

Hypatia's case then was this. She lived in a time when her intellectual heritage, a seven-hundred-year-old tradition, was crumbling. The supports that had once seemed so secure - the Museum and the libraries - had all been swept away by the swell of ignorant dogmatism. Almost alone, virtually the last academic, she stood for the intellectual values, for rigorous mathematics, ascetic Neoplatonism, the crucial role of the mind, and the voice of temperance and moderation in civic life. ~ Michael A.B. Deakin

Music can be appreciated from several points of view: the listener, the performer, the composer. In mathematics there is nothing analogous to the listener; and even if there were, it would be the composer, rather than the performer, that would interest him. It is the creation of new mathematics, rather than its mundane practice, that is interesting. Mathematics is not about symbols and calculations. These are just tools of the tradequavers and crotchets and five-finger exercises. Mathematics is about ideas. In particular it is about the way that different ideas relate to each other. If certain information is known, what else must necessarily follow? The aim of mathematics is to understand such questions by stripping away the inessentials and penetrating to the core of the problem. It is not just a question of getting the right answer; more a matter of understanding why an answer is possible at all, and why it takes the form that it does. Good mathematics has an air of economy and an element of surprise. But, above all, it has significance. ~ Ian Stewart

Debts are subject to the laws of mathematics rather than physics. Unlike wealth, which is subject to the laws of thermodynamics, debts do not rot with old age and are not consumed in the process of living. On the contrary, they grow at so much per cent per annum, by the well-known mathematical laws of simple and compound interest ... It is this underlying confusion between wealth and debt which has made such a tragedy of the scientific era. ~ Frederick Soddy

I know that two and two make four - and should be glad to prove it too if I could - though I must say if by any sort of process I could convert 2 and 2 into five it would give me much greater pleasure. ~ George Gordon Byron

When a branch of mathematics ceases to interest any but the specialists, it is very near its death, or at any rate dangerously close to a paralysis, from which it can be rescued only by being plunged back into the vivifying source of the science. ~ Andre Weil

Unfortunately, the critics of economics have had a tendency to discuss the whole structure as a tissue of misconceptions. It is a critique that fails. The strength of economics is its considerable, if far from complete, understanding of the flows and comparative advantages that underlie trade, jobs, capital and incomes, and the logic of optimising behaviour, all backed by glittering accomplishment in mathematics. That makes it a powerful analytical instrument, so that just a few misconceptions – such as a failure to understand the informal economy or resource depletion – have leverage: like a baby monkey at the controls of a Ferrari, they can turn it into an instrument with extraordinarily destructive potential. If it were a tissue of errors, it would not be dangerous: it is its 90 percent brilliance which makes it so. ~ David Fleming

Mechanics is the paradise of the mathematical sciences because by means of it one comes to the fruits of mathematics. ~ Leonardo Da Vinci

Turing attended Wittgenstein's lectures on the philosophy of mathematics in Cambridge in 1939 and disagreed strongly with a line of argument that Wittgenstein was pursuing which wanted to allow contradictions to exist in mathematical systems. Wittgenstein argues that he can see why people don't like contradictions outside of mathematics but cannot see what harm they do inside mathematics. Turing is exasperated and points out that such contradictions inside mathematics will lead to disasters outside mathematics: bridges will fall down. Only if there are no applications will the consequences of contradictions be innocuous. Turing eventually gave up attending these lectures. His despair is understandable. The inclusion of just one contradiction (like 0 = 1) in an axiomatic system allows any statement about the objects in the system to be proved true (and also proved false). When Bertrand Russel pointed this out in a lecture he was once challenged by a heckler demanding that he show how the questioner could be proved to be the Pope if 2 + 2 = 5. Russel replied immediately that 'if twice 2 is 5, then 4 is 5, subtract 3; then 1 = 2. But you and the Pope are 2; therefore you and the Pope are 1'! A contradictory statement is the ultimate Trojan horse. ~ John D. Barrow

Over the centuries, monumental upheavals in science have emerged time and again from following the leads set out by mathematics. ~ Brian Greene

Highly complex numbers like the Comma of Pythagoras, Pi and Phi (sometimes called the Golden Proportion), are known as irrational numbers. They lie deep in the structure of the physical universe, and were seen by the Egyptians as the principles controlling creation, the principles by which matter is precipitated from the cosmic mind.

Today scientists recognize the Comma of Pythagoras, Pi and the Golden Proportion as well as the closely related Fibonacci sequence are universal constants that describe complex patterns in astronomy, music and physics. ...

To the Egyptians these numbers were also the secret harmonies of the cosmos and they incorporated them as rhythms and proportions in the construction of their pyramids and temples. ~ Jonathan Black

Silicon Valley, "the largest legal creation of wealth in history," was built largely by unprofessional amateurs using math, sand, and the institutions of freedom. The Soviet Union had the greatest mathematicians on earth, and plenty of sand, but without the institutions of freedom their brilliant mathematicians were not empowered to create those devices that are changing the world. ~ Michael Strong

Several Watford supporters disgracefully started leaving the ground, and the Arsenal surprised us by adding another chant to their repertoire – making a total of two chants if my mathematics serves me correctly. 'You might as well go home.' What they don't realise, of course, is that we are home. Watford's not a pretty place, but its home. I live a half-hour walk from Vicarage Road. The chant went up from our end, 'We support our local team,' which always shuts up Premiership supporters from Borehamwood, Radlett and Surrey, no matter which of the top four teams they follow. ~ Karl Wiggins

If you're looking for fine art or literature, you might want to read some stuff written by the Greeks. Because to create true fine art, slaves are a necessity. That's how the ancient Greeks felt, with slaves working the fields, cooking their meals, rowing their ships, all the while their citizens, under the Mediterranean Sun, indulged in poetry writing and grappled with mathematics. That was their idea of fine art. ~ Haruki Murakami

Therefore I would not have it unknown to Your Holiness, the the only thing which induced me to look for another way of reckoning the movements of the heavenly bodies was that I knew that mathematicians by no means agree in their investigation thereof. ~ Nicolaus Copernicus

One thing the American defense establishment has traditionally understood very well is that countries don't win wars just by being braver than the other side, or freer, or slightly preferred by God. The winners are usually the guys who get 5% fewer of their planes shot down, or use 5% less fuel, or get 5% more nutrition into their infantry at 95% of the cost. ~ Jordan Ellenberg

It now becomes clear that consistency is not a property of a formal system per se, but depends on the interpretation which is proposed for it. By the same token, inconsistency is not an intrinsic property of any formal system. ~ Douglas R. Hofstadter

I entered Princeton University as a graduate student in 1959, when the Department of Mathematics was housed in the old Fine Hall. This legendary facility was marvellous in stimulating interaction among the graduate students and between the graduate students and the faculty. The faculty offered few formal courses, and essentially none of them were at the beginning graduate level. Instead the students were expected to learn the necessary background material by reading books and papers and by organising seminars among themselves. It was a stimulating environment but not an easy one for a student like me, who had come with only a spotty background. Fortunately I had an excellent group of classmates, and in retrospect I think the "Princeton method" of that period was quite effective. ~ Phillip A. Griffiths

The standard "foundation" for mathematics starts with sets and their elements. It is possible to start differently, by axiomatising not elements of sets but functions between sets. This can be done by using the language of categories and universal constructions. ~ Saunders Mac Lane

However great a man's fear of life, suicide remains the courageous act, the clear-headed act of a mathematician. The suicide has judged by the laws of chance - so many odds against one that to live will be more miserable than to die. His sense of mathematics is greater than his sense of survival. But think how a sense of survival must clamor to be heard at the last moment, what excuses it must present of a totally unscientific nature. ~ Graham Greene

Most remarks made by children consist of correct ideas very badly expressed. A good teacher will be very wary of saying 'No, that's wrong.' Rather, he will try to discover the correct idea behind the inadequate expression. This is one of the most important principles in the whole of the art of teaching. ~ W.W. Sawyer

Instruction tables will have to be made up by mathematicians with computing experience and perhaps a certain puzzle-solving ability. There need be no real danger of it ever becoming a drudge, for any processes that are quite mechanical may be turned over to the machine itself. ~ Alan Turing

When the twins asked what cuff-links were for - "To link cuffs together," Ammu told them - they were thrilled by this morsel of logic in what had so far seemed an illogical language. Cuff+link = cuff-link. This, to them, rivaled the precision of logic and mathematics. Cuff-links gave them an inordinate (if exaggerated) satisfaction, and a real affection for the English language. ~ Arundhati Roy

At schools, the children who are too stupid or lazy to learn languages, mathematics and elementary science can be set to doing the things that children used to do in their spare time. Let them, for example, make mud pies and call it modelling. But all the time there must be no faintest hint that they are inferior to the children who are at work. Whatever nonsense they are engaged in must have - I believe the English already use the phrase - "parity of esteem." An even more drastic scheme is not impossible. Children who are fit to proceed to a higher class may be artificially kept back, because the others would get a trauma - Beelzebub, what a useful word! - by being left behind. The bright pupil thus remains democratically fettered to his own age group throughout his school career, and a boy who would be capable of tackling Aeschylus or Dante sits listening to his coeval's attempts to spell out 'A Cat Sat On A Mat'. ~ C.S. Lewis

The word "mathematics" is a Greek word and, by origin, it means "something that has been learned or understood," or perhaps "acquired knowledge," or perhaps even, somewhat against grammar, "acquirable knowledge," that is, "learnable knowledge," that is, "knowledge acquirable by learning." ~ Salomon Bochner

What is the origin of the urge, the fascincation that drives physicists, mathematicians, and presumably other scientists as well? Psychoanalysis suggests that it is sexual curiosity. You start by asking where little babies come from, one thing leads to another, and you find yourself preparing nitroglycerine or solving differential equations. This explanation is somewhat irritating, and therefore probably basically correct. ~ David Ruelle

They were very upset when I said that the thing of greatest importance to mathematics in Europe was the discovery by Tartaglia that you can solve a cubic equation-which, altho it is very little used, must have been psychologically wonderful because it showed a modern man could do something no ancient Greek could do, and therefore helped in the renaissance which was the freeing of man from the intimidation of the ancients-what they are learning in school is to be intimidated into thinking they have fallen so far below their super ancestors. ~ Richard Feynman

I doubt not, but from self-evident Propositions, by necessary Consequences, as incontestable as those in Mathematics, the measures of right and wrong might be made out. ~ John Locke

I neither believe nor disbelieve in anything. That which can be imagined is as much an approximation to truth as that which can be proved by mathematics. ~ Charlie Chaplin

But many, many stories were told; from what could be gathered, all fifty of the mine's inhabitants had reacted on each other, two by two, as in combinatorial analysis, that is to say, everyone with all the others, and especially every man with all the women, old maids or married, and every woman with all the men. All I had to do was to select two names at random, better if different sex, and ask a third person, "What happened with those two?" and lo and behold, a splendid story was unfolded for me, since everyone knew the story of everyone else. ~ Primo Levi

I was perplexed by the failure of teachers at school to address what seemed the most urgent matter of all: the bewildering, stomach-churning insecurity of being alive. The standard subjects of history, geography, mathematics, and English seemed perversely designed to ignore the questions that really mattered. As soon as I had some inkling of what 'philosophy' meant, I was puzzled as to why we were not taught it. And my skepticism about religion only grew as I failed to see what the vicars and priests I encountered gained from their faith. They struck me either as insincere, pious, and aloof or just bumblingly good-natured. (p. 10) ~ Stephen Batchelor

The word "art" means harmony for me. I never speak of mathematics and never bother with the Spirit. My only science is the choice of impressions that the light in the universe furnishes to my consciousness as an artisan which I try, by imposing an Order, and Art, an appropriate representative life, to organize ... ~ Robert Delaunay

The axiom of equality states that x always equals x: it assumes that if you have a conceptual thing named x, that it must always be equivalent to itself, that it has a uniqueness about it, that it is in possession of something so irreducible that we must assume it is absolutely, unchangeably equivalent to itself for all time, that its very elementalness can never be altered. But it is impossible to prove. Always, absolutes, nevers: these are the words, as much as numbers, that make up the world of mathematics. Not everyone liked the axiom of equality––Dr. Li had once called it coy and twee, a fan dance of an axiom––but he had always appreciated how elusive it was, how the beauty of the equation itself would always be frustrated by the attempts to prove it. I was the kind of axiom that could drive you mad, that could consume you, that could easily become an entire life.

But now he knows for certain how true the axiom is, because he himself––his very life––has proven it. The person I was will always be the person I am, he realizes. The context may have changed: he may be in this apartment, and he may have a job that he enjoys and that pays him well, and he may have parents and friends he loves. He may be respected; in court, he may even be feared. But fundamentally, he is the same person, a person who inspires disgust, a person meant to be hated. ~ Hanya Yanagihara

In music, musicians must be able to read musical notes and have developed the skill to follow the music from their studies. This skill allows them to read new musical notes and be capable of hearing most or all of the sounds (melodies, harmonies, etc.) in their head without having to play the music piece. By analogy, in Mathematics, we believe a scientist, engineer or mathematician must be able to read and understand mathematical codes (e.g., Maple, Mathematica) in their head without having to execute the problem. ~ Inna K. Shingareva

I see, in place of that empty figment of one linear history which can be kept up only by shutting one's eyes to the overwhelming multitude of facts, the drama of a number of mighty Cultures, each springing with primitive strength from the soil of a mother-region to which it remains firmly bound throughout it's whole life-cycle; each stamping its material, its mankind, in its own image; each having its own idea, its own passions, its own life, will and feelings, its own death. Here indeed are colours, lights, movements, that no intellectual eye has yet discovered.

Here the Cultures, peoples, languages, truths, gods, landscapes bloom and age as the oaks and the pines, the blossoms, twigs and leaves - but there is no ageing "Mankind." Each Culture has its own new possibilities of self-expression which arise, ripen, decay and never return. There is not one sculpture, one painting, one mathematics, one physics, but many, each in the deepest essence different from the others, each limited in duration and self-contained, just as each species of plant has its peculiar blossom or fruit, its special type of growth and decline. ~ Oswald Spengler