Quotes & Sayings About Theorems

Paul Erdos has a theory that God has a book containing all the theorems of mathematics with their absolutely most beautiful proofs, and when he wants to express particular appreciation of a proof he exclaims, "This is from the book!" — Ross Honsberger

God has the Big Book, the beautiful proofs of mathematical theorems are listed here. — Paul Erdos

A dozen more questions occurred to me. Not to mention twenty-two possible solutions to each one, sixteen resulting hypotheses and counter-theorems, eight abstract speculations, a quadrilateral equation, two axioms, and a limerick. That's raw intelligence for you. — Jonathan Stroud

Mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It's the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation. — Ivars Peterson

And, if anyone tells you that you know nothing, and you are not nettled at it, then you be sure that you have begun your business. For sheep don't throw up the grass to show the shepherds how much they have eaten; but, inwardly digesting their food, they outwardly produce wool and milk. Thus, therefore, do you likewise not show theorems to the unlearned, but the actions produced by them after they have been digested. — Epictetus

Did chemistry theorems exist? No: therefore you had to go further, not be satisfied with the quia, go back to the origins, to mathematics and physics. The origins of chemistry were ignoble, or at least equivocal: the dens of the alchemists, their abominable hodgepodge of ideas and language, their confessed interest in gold, their Levantine swindles typical of charlatans and magicians; instead, at the origin of physics lay the strenuous clarity of the West-Archimedes and Euclid. — Primo Levi

I compare arithmetic with a tree that unfolds upwards in a multitude of techniques and theorems while the root drives into the depths. — Gottlob Frege

Which is to say that culture is not a reflex of political economy, but
that society is now a reflex of key shifts in music theory and practice ...
[Sampladelia is] the sound made by those early-twentieth-century discoveries
in particle physics and relativiity theory, the projection of the minds of
Einstein, Heisenbery, and Bohr, their fateful explorations of liquid time,
curving space, uncertainty fields and relativity theorems, into densely
configured and fully ambivalent android music tracks — Arthur Kroker

Free will is an idealization of human beings that makes the ethics game playable. Euclidean geometry requires idealizations like infinite straight lines and perfect circles, and its deductions are sound and useful even though the world does not really have infinite straight lines or perfect circles. The world is close enough to the idealization that the theorems can usefully be applied. Similarly, ethical theory requires idealizations like free, sentient, rational, equivalent agents whose behavior is uncaused, and its conclusions can be sound and useful even though the world, as seen by science, does not really have uncaused events. As long as there is no outright coercion or gross malfunction of reasoning, the world is close enough to the idealization of free will that moral theory can meaningfully be applied to it. — Steven Pinker

The job of economic theorists is to prove theorems. The job of policy economists is to figure out which theorems to apply. — N. Gregory Mankiw

Phyllis explained to him, trying to give of her deeper self, 'Don't you find it so beautiful, math? Like an endless sheet of gold chains, each link locked into the one before it, the theorems and functions, one thing making the next inevitable. It's music, hanging there in the middle of space, meaning nothing but itself, and so moving ... ' — John Updike

All theorems have three names: a French name, a German name, and a Russian name, each nationality having claimed to discover it first. Once in a while there's an English name, too, but it's always Newton. — Arthur Mattuck

Math does come easily to me, but I was always much more interested in what theorems imply about the world than in proving them. — Antony Garrett Lisi

Tengo's lectures took on uncommon warmth, and the students found themselves swept up in his eloquence. He taught them how to practically and effectively solve mathematical problems while simultaneously presenting a spectacular display of the romance concealed in the questions it posed. Tengo saw admiration in the eyes of several of his female students, and he realized that he was seducing these seventeen- or eighteen-year-olds through mathematics. His eloquence was a kind of intellectual foreplay. Mathematical functions stroked their backs; theorems sent warm breath into their ears. — Haruki Murakami

In the Principia Mathematica, Bertrand Russell and Alfred Whitehead attempted to give a rigorous foundation to mathematics using formal logic as their basis. They began with what they considered to be axioms, and used those to derive theorems of increasing complexity. By page 362, they had established enough to prove 1 + 1 = 2. — Ted Chiang

The most probable of all my theorems, is that life is ordered by the principles of some religion so peculiar and obscure it has no followers, and none may fathom it, nor know the rituals by which to court its favour. — Alan Moore

A mathematician is a device for turning coffee into theorems — Paul Erdos

My dad once told me that Winstone Churchill said that Russia was riddle, wrapped in a mystery, inside an enigma. According to my dad, Churchill had been talking about my mother. This was before the divorce, and he said it half-bitterly, half-respectfully. Because even when he hated her, he admired her.
I think he would have stayed with her forever, trying to figure out the mystery. He was a puzzle solver, the kind of person who likes theorems, theories. X always had to equal something. It couldn't just be X.
To me, my mother wasn't that mysterious. She was my mother. Always reasonable, always sure of herself. To me, she was about as mysterious as a glass fo water. She knew what she wanted; she knew what she didn't want. And that was to be married to my father. I wasn't sure if it was that she fell our of love or if it was that she just never was. in love, I mean. — Jenny Han

We decided that 'trivial' means 'proved'. So we joked with the mathematicians: We have a new theorem- that mathematicians can prove only trivial theorems, because every theorem that's proved is trivial. — Richard P. Feynman

Without computers we will be stuck only proving theorems that have short proofs. — Kenneth Appel

It is not so much whether a theorem is useful that matters, but how elegant it is. — Stanislaw Ulam

to be grave without affectation: to observe carefully the several dispositions of my friends, not to be offended with idiots, nor unseasonably to set upon those that are carried with the vulgar opinions, with the theorems, and tenets of philosophers: his conversation being an example how a man might accommodate himself to all men and companies; so that though his company were sweeter and more pleasing than any flatterer's cogging and fawning; yet was it at the same time most respected and reverenced: — Marcus Aurelius

In science, every question answered leads to 10 more. I love that science can never, ever be finished. From a young age, people think, 'Science is hard and boring.' We don't tell children, 'Yes, you have to learn these formulae and theorems, but then you go on to learn about nuclear reactions and stars.' — Elise Andrew

For what is important when we give children a theorem to use is not that they should memorize it. What matters most is that by growing up with a few very powerful theorems one comes to appreciate how certain ideas can be used as tools to think with over a lifetime. One learns to enjoy and to respect the power of powerful ideas. One learns that the most powerful idea of all is the idea of powerful ideas. — Seymour Papert

Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems - general and specific statements - can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences. — Christopher Zeeman

Theorems are fun especially when you are the prover, but then the pleasure fades. What keeps us going are the unsolved problems. — Carl Pomerance

If market pricing is the only legitimate test of quality, why are we still bothering with proven theorems? Why don't we just have a vote on whether a theorem is true? To make it better we'll have everyone vote on it, especially the hundreds of millions of people who don't understand the math. Would that satisfy you? — Jaron Lanier

There are three signs of senility. The first sign is that a man forgets his theorems. The second sign is that he forgets to zip up. The third sign is that he forgets to zip down. — Paul Erdos

He was thinking of Marco, Daisy, Sono
Oguki, Madeleine, the Pontritters, and now and then of the difference between ancient and modern
tragedy according to Hegel, the inner experience of the heart and the deepening of individual
character in the modern age. His own individual character cut off at times both from facts and from
values. But modern character is inconstant, divided, vacillating, lacking the stone-like certitude of
archaic man, also deprived of the firm ideas of the seventeenth century, clear, hard theorems. — Saul Bellow

It is a matter for considerable regret that Fermat, who cultivated the theory of numbers with so much success, did not leave us with the proofs of the theorems he discovered. In truth, Messrs Euler and Lagrange, who have not disdained this kind of research, have proved most of these theorems, and have even substituted extensive theories for the isolated propositions of Fermat. But there are several proofs which have resisted their efforts. — Adrien-Marie Legendre

Just solving certain theorems makes waves in the Platonic over-space. Pump lots of power through a grid tuned carefully in accordance with the right parameters - which fall naturally out of the geometry curve I mentioned, which in turn falls easily out of the Turing theorem - and you can actually amplify these waves, until they rip honking great holes in spacetime and let congruent segments of otherwise-separate universes merge. You really don't want to be standing at ground zero when that happens. — Charles Stross

Papers should include more side remarks, open questions, and such. Very often, these are more interesting than the theorems actually proved. Alas, most people are afraid to admit that they don't know the answer to some question, and as a consequence they refrain from mentioning the question, even if it is a very natural one. What a pity! As for myself, I enjoy saying 'I do not know'. — Jean-Pierre Serre

There are infinitely many variations of the initial situation and therefore no doubt indefinitely many theorems of moral geometry. — John Rawls

An axiomatic system comprises axioms and theorems and requires a certain amount of hand-eye coordination before it works. A formal system comprises an explicit list of symbols, an explicit set of rules governing their cohabitation, an explicit list of axioms, and, above all, an explicit list of rules explicitly governing the steps that the mathematician may take in going from assumptions to conclusions. No appeal to meaning nor to intuition. Symbols lose their referential powers; inferences become mechanical. — David Berlinski

I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our "creations," are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it. — G.H. Hardy

The pursuit of pretty formulas and neat theorems can no doubt quickly degenerate into a silly vice, but so can the quest for austere generalities which are so very general indeed that they are incapable of application to any particular. — E. T. Bell

A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. — Stefan Banach

Fact: She was alone with Quint in his bedchamber.
Fact: He was talking of theorems, secret messages, and mathematical probability.
Fact: She was incredibly, distractingly aroused. — Joanna Shupe

I was a capable boy, intelligent, well-liked, but powerfully afraid. And I felt, vaguely, wordlessly, that for a child to be marked off for such a life, to be forced to live in fear was a great injustice. And what was the source of this fear? What was hiding behind the smoke screen of streets and schools? And what did it mean that number 2 pencils, conjugations without context, Pythagorean theorems, handshakes, and head nods were the difference between life and death, were the curtains drawing down between the world and me? — Ta-Nehisi Coates

The theory of numbers, more than any other branch of mathematics, began by being an experimental science. Its most famous theorems have all been conjectured, sometimes a hundred years or more before they were proved; and they have been suggested by the evidence of a mass of computations. — G.H. Hardy

One can be enlightened about proofs as well as theorems. Without enlightenment, one is merely reduced to memorizing proofs. With enlightenment about a proof, its flow becomes clear and it can become an item of astonishing beauty. In addition, the need to memorize disappears because the proof has become part of your soul. — Herbert S. Gaskill

History cannot teach us any general rule, principle, or law. There is no means to abstract from a historical experience a posteriori any theories or theorems concerning human conduct and policies. The — Ludwig Von Mises

If we didn't have bodies, we couldn't feel the sun on our faces or smell the earthy, mushroom-y rich smell of the ground right after the rain. If we didn't have bodies, we couldn't wrap our arms around the people we love or taste a perfect tomato right at the height of summer. I'm so thankful to live in this physical, messy, blood-and-guts world. I don't want to live in a world that's all dry ideas and theorems. Food is one of the ways we acknowledge our humanity, our appetites, our need for nourishment. And so it may seem trivial or peripheral to some people, but to me, when I'm telling a story, the part about what we ate really does matter. — Shauna Niequist

There are very few theorems in advanced analysis which have been demonstrated in a logically tenable manner. Everywhere one finds this miserable way of concluding from the special to the general and it is extremely peculiar that such a procedure has led to so few of the so-called paradoxes. — Niels Henrik Abel

One may characterize physics as the doctrine of the repeatable, be it a succession in time or the co-existence in space. The validity of physical theorems is founded on this repeatability. — Friedrich Hund

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

Less depends upon the choice of words than upon this, that their introduction shall be justified by pregnant theorems. — Carl Friedrich Gauss

He knew by heart every last minute crack on its surface. He had made maps of the ceiling and gone exploring on them; rivers, islands, and continents. He had made guessing games of it and discovered hidden objects; faces, birds, and fishes. He made mathematical calculations of it and rediscovered his childhood; theorems, angles, and triangles. There was practically nothing else he could do but look at it. He hated the sight of it. — Josephine Tey

Imagine a life-form whose brainpower is to ours as ours is to a chimpanzee's. To such a species, our highest mental achievements would be trivial. Their toddlers, instead of learning their ABCs on Sesame Street, would learn multivariable calculus on Boolean Boulevard. Our most complex theorems, our deepest philosophies, the cherished works of our most creative artists, would be projects their schoolkids bring home for Mom and Dad to display on the refrigerator door. — Neil DeGrasse Tyson

Theorems are not to mathematics what successful courses are to a meal. — Gian-Carlo Rota

There is a theorem that colloquially translates, You cannot comb the hair on a bowling ball ... Clearly, none of these mathematicians had Afros, because to comb an Afro is to pick it straight away from the scalp. If bowling balls had Afros, then yes, they could be combed without violation of mathematical theorems. — Neil DeGrasse Tyson

How many theorems in geometry which have seemed at first impracticable are in time successfully worked out! — Archimedes

An axiomatic system establishes a reverberating relationship between what a mathematician assumes (the axioms) and what he or she can derive (the theorems). In the best of circumstances, the relationship is clear enough so that the mathematician can submit his or her reasoning to an informal checklist, passing from step to step with the easy confidence the steps are small enough so that he cannot be embarrassed nor she tripped up. — David Berlinski

Though determinants and matrices received a great deal of attention in the nineteenth century and thousands of papers were written on these subjects, they do not constitute great innovations in mathematics ... Neither determinants nor matrices have influenced deeply the course of mathematics despite their utility as compact expressions and despite the suggestiveness of matrices as concrete groups for the discernment of general theorems of group theory ... — Morris Kline

Now we can see what makes mathematics unique. Only in mathematics is there no significant correction - only extension. Once the Greeks had developed the deductive method, they were correct in what they did, correct for all time. Euclid was incomplete and his work has been extended enormously, but it has not had to be corrected. His theorems are, every one of them, valid to this day. — Carl B. Boyer

The fundamental laws of the universe which correspond to the two fundamental theorems of the mechanical theory of heat.
1. The energy of the universe is constant.
2. The entropy of the universe tends to a maximum. — Rudolf Clausius

Mathematicians have, according to Wright, been "unreasonably successful" in finding applications to apparently useless theorems, and often years after the theorems were first discovered. — Alex Bellos

I have found a very great number of exceedingly beautiful theorems. — Pierre De Fermat

Never call yourself a philosopher, nor talk a great deal among the unlearned about theorems, but act conformably to them. Thus, at an entertainment, don't talk how persons ought to eat, but
eat as you ought. For remember that in this manner Socrates also universally avoided all ostentation. — Epictetus

Economists often like startling theorems, results which seem to run counter to conventional wisdom. — Joseph Stiglitz

It is both mysterious and miraculous that roughly the same intelligence necessary to flake a barbed spearpoint is sufficient to discover the theorems of mathematics. In a different universe, it might have been otherwise. And so human beings would have been spared the tragedy of existing half as ape and half as god. — David Zindell

Mathematics does not grow through a monotonous increase of the number of indubitably established theorems but through the incessant improvement of guesses by speculation and criticism, by the logic of proofs and refutations. — Imre Lakatos

It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds. — G.H. Hardy

A peculiarity of the higher arithmetic is the great difficulty which has often been experienced in proving simple general theorems which had been suggested quite naturally by numerical evidence. — Harold Davenport

We often hear that mathematics consists mainly of 'proving theorems.' Is a writer's job mainly that of 'writing sentences? — Gian-Carlo Rota

Men propound mathematical theorems in besieged cities, conduct metaphysical arguments in condemned cells, make jokes on the scaffold, discuss a new poem while advancing to the walls of Quebec, and comb their hair at Thermopylae. This is not panache; it is our nature. — C.S. Lewis

Unlike mathematical theorems, scientific results can't be proved. They can only be tested again and again, until only a fool would not believe them. I cannot prove that electrons exist ... if you don't believe in them I have a high voltage cattle prod I'm willing to apply as an argument on their behalf. Electrons speak for themselves. — Seth Lloyd

Obviously, the final goal of scientists and mathematicians is not simply the accumulation of facts and lists of formulas, but rather they seek to understand the patterns, organizing principles, and relationships between these facts to form theorems and entirely new branches of human thought. For me, mathematics cultivates a perpetual state of wonder about the nature of mind, the limits of thoughts, and our place in this vast cosmos. — Clifford A. Pickover

Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means ... — Henri Poincare

Response, when asked how he came upon his theorems. — Carl Friedrich Gauss

Mathematicians can and do fill in gaps, correct errors, and supply more detail and more careful scholarship when they are called on or motivated to do so. Our system is quite good at producing reliable theorems that can be solidly backed up. It's just that the reliability does not primarily come from mathematicians formally checking formal arguments; it comes from mathematicians thinking carefully and critically about mathematical ideas. — William Thurston

For even sheep do not vomit up their grass and show to the shepherds how much they have eaten; but when they have internally digested the pasture, they produce externally wool and milk. Do you also show not your theorems to the uninstructed, but show the acts which come from their digestion. — Epictetus

To leave the atom constituted as it was but to interfere with the probability of its undetermined behaviour, does not seem quite so drastic an interference with natural law as other modes of mental interference that have been suggested. (Perhaps that is only because we do not understand enough about these probabilities to realize the heinousness of our suggestion.) Unless it belies its name, probability can be modified in ways which ordinary physical entities would not admit of. There can be no unique probability attached to any event or behaviour; we can only speak of 'probability in the light of certain given information,' and the probability alters according to the extent of the information. It is, I think, one of the most unsatisfactory features of the new quantum theory in its present stage that it scarcely seems to recognize this fact, and leaves us to guess at the basis of information to which its probability theorems are supposed to refer. — Arthur Stanley Eddington

I am not qualified to say whether or not God exists. I kind of doubt He does. Nevertheless I'm always saying that the SF( The SF is the supreme Fascist, the Number-One guy up there) has this transfinite book-transfinite being a concept in mathematics that is larger than infinite-that contains the best proofs of all mathematical theorems, proofs that are elegant and perfect. — Paul Erdos

A mathematician," he liked to say, "is a machine for turning coffee into theorems. — Mason Currey

One of the most important lessons, perhaps, is the fact that SOFTWARE IS HARD. From now on I shall have significantly greater respect for every successful software tool that I encounter. During the past decade I was surprised to learn that the writing of programs for TeX and Metafont proved to be much more difficult than all the other things I had done (like proving theorems or writing books). The creation of good software demand a significiantly higher standard of accuracy than those other things do, and it requires a longer attention span than other intellectual tasks. — Donald Knuth

If all sentient beings in the universe disappeared, there would remain a sense in which mathematical objects and theorems would continue to exist even though there would be no one around to write or talk about them. Huge prime numbers would continue to be prime, even if no one had proved them prime. — Martin Gardner

You can learn to find unknowns in equations, draw equidistant lines and demonstrate theorems, but in real life there's nothing to position, calculate, or guess. — Delphine De Vigan

The product of mathematics is clarity and understanding. Not theorems, by themselves ... In short, mathematics only exists in a living community of mathematicians that spreads understanding and breathes life into ideas both old and new. — William Thurston

That's the problem with false proofs of true theorems; it's not easy to produce a counterexample. — Jeffrey Shallit

Finance is concerned with the relations between the values of securities and their risk, and with the behavior of those values. It aspires to be a practical, like physics or chemistry or electrical engineering. As John Maynard Keynes once remarked about economics, "If economists could manage to get themselves thought of as humble, competent people on a level with dentists, that would be splendid." Dentists rely on science, engineering, empirical knowledge, and heuristics, and there are no theorems in dentistry. Similarly, one would hope that nance would be concerned with laws rather than theorems, with behavior rather than assumptions. One doesn't seriously describe the behavior of a market with theorems. — Emanuel Derman

Humans like to think of themselves as unusual. We've got big brains that make it possible for us to think, and we think that we have free will and that our behavior can't be described by some mechanistic set of theorems or ideas. But even in terms of much of our behavior, we really aren't very different from other animals. — Mark Pagel

they propound mathematical theorems in beleaguered cities, conduct metaphysical arguments in condemned cells, make jokes on scaffolds. — C.S. Lewis

Credit expansion results in the recurrence of economic crisis and periods of depression. Inflation makes the prices of all commodities and services soar. The attempts to enforce wage rates higher than those the unhampered market would have determined produce mass unemployment prolonged year after year. Price ceilings result in a drop in the supply of commodities affected. The economists have proved these theorems in an irrefutable way. No — Ludwig Von Mises

For hundreds of pages the closely-reasoned arguments unroll, axioms and theorems interlock. And what remains with us in the end? A general sense that the world can be expressed in closely-reasoned arguments, in interlocking axioms and theorems. — Michael Frayn

I am persuaded that this method [for calculating the volume of a sphere] will be of no little service to mathematics. For I foresee that once it is understood and established, it will be used to discover other theorems which have not yet occurred to me, by other mathematicians, now living or yet unborn. — Archimedes

The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. — G.H. Hardy

Come back down here, heat supply," I commanded. "I'm going to close my eyes and you are
going to tell me about math so I can fall asleep. Tell me some theorems. Is that what you called them?
Tell me how Einstein knew e equals mc squared. And start with once upon a time ... okay?"
"You're a little bossy, you know that?"
"I know. I have to be. It's to make up for not being born with a calculator. Now share your wisdom,
Infinity."
"Once upon a time - "
I giggled and Finn immediately shushed me, continuing on with his "story. — Amy Harmon

I think that mathematics can benefit by acknowledging that the creation of good models is just as important as proving deep theorems. — David Mumford

To many, mathematics is a collection of theorems. For me, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples ... — John B. Conway

Theorems often tell us complex truths about the simple things, but only rarely tell us simple truths about the complex ones. To believe otherwise is wishful thinking or mathematics envy. — Marvin Minsky

It is my dearest hope that one day I shall be the one to discover the GUT-the Grand Unified Tale, the one which will bind together all our Theorems and Laws, leaving out not one Orphan Girl or Youngest Son or Cup of Life and Death. Not one Descent or Ascent, not one Riddle or Puzzle or Trick. One perfect golden map that can guide any soul to its desire and back again. I will be the one to do it, I know it. I hope I know it. I know I hope it. — Catherynne M Valente

The Mean Value Theorem is the midwife of calculus - not very important or glamorous by itself, but often helping to deliver other theorems that are of major significance. — Edward Mills Purcell

Murphy's Law, that brash proletarian restatement of Godel's Theorem ... — Thomas Pynchon

Isolated, so-called "pretty theorems" have even less value in the eyes of a modern mathematician than the discovery of a new "pretty flower" has to the scientific botanist, though the layman finds in these the chief charm of the respective sciences. — Hermann Hankel

Young men should prove theorems, old men should write books. — G.H. Hardy