Theorem Quotes

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

And now of a sudden my illusion vanished. What was my body to me? A kind of flunkey in my service. Let my anger wax hot, my love grow exalted, my hatred collect in me, and the boasted solidarity between me and my body was gone.

Your son is in a burning house. Nobody can hold you back. You may burn up, but what do you think of that? You are ready to bequeath the rags of your body to any man who will take them. You discover that what you set so much store by is trash. You would sell your hand, if need be, to give a hand to a friend. It is in your act that you exist, not in your body. Your act is yourself, and there is no other you. Your body belongs to you: it is not you. Are you about to strike an enemy? No threat of bodily harm can hold you back. You? It is the death of your enemy that is you. You? It is the rescue of your child that is you. In that moment you exchange yourself against something else; and you have no feeling tat you lost by the exchange. Your members? Tools. A tool snaps in your hand: how important is that tool? You exchange yourself against the death of your enemy, the rescue of your child, the recovery of your patient, the perfection of your theorem...Your true significance becomes dazzlingly evident. Your true name is duty, hatred, love, child, theorem. There is no other you than this. ~ Antoine De Saint Exupery

The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules ... One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems. It will be shown below that this is not the case, that on the contrary there are in the two systems mentioned relatively simple problems in the theory of integers that cannot be decided on the basis of the axioms. ~ Kurt Godel

A felicitous but unproved conjecture may be of much more consequence for mathematics than the proof of many a respectable theorem. ~ Atle Selberg

The Limbaugh Theorem was not about me giving me credit for something. It was simply sharing with you when the light went off. ~ Rush Limbaugh

Any good theorem should have several proofs, the more the better ~ Michael Atiyah

You're right: if there's sentient life behind the border, it probably won't share my goals. Unlike the people in this room, who all want exactly the same things in life as I do, and have precisely the same tastes in food, art, music, and sex. Unlike the people of Schur, and Cartan, and Zapata - who I came here in the hope of protecting, after losing my own home - who doubtless celebrate all the same festivals, delight in the same songs and stories, and gather every fortieth night to watch actors perform the same plays, in the same language, from the same undisputed canon, as the people I left behind.

"If there's sentient life behind the border, of course we couldn't empathize with it. These creatures are unlikely to possess cute mammalian neonate faces, or anything else we might mistake for human features. None of us could have the imagination to get over such insurmountable barriers, or the wit to apply such difficult abstractions as the General Intelligence theorem - though since every twelve-year-old on my home world was required to master that result, it must be universally known on this side of the border.

"You're right: we should give up responsibility for making any difficult moral judgments, and surrender to the dictates of natural selection. Evolution cares so much about our happiness that no one who's obeyed an inherited urge has ever suffered a moment's regret for it. History is full of joyful case studies of people who followed their na ~ Greg Egan

I loved doing problems in school. I'd take them home and make up new ones of my own. But the best problem I ever found, I found in my local public library. I was just browsing through the section of math books and I found this one book, which was all about one particular problem - Fermat's Last Theorem. ~ Andrew Wiles

Therefore Godel's theorem had an electrifying effect upon logicians, mathematicians, and philosophers interested in the foundations of mathematics, for it showed that no fixed system, no matter how complicated could represent the complexity of the whole numbers: 0,1,2,3,..... ~ Douglas R. Hofstadter

If you hold there is a 100 percent probability that God exists, or a 0 percent probability, then under Bayes's theorem, no amount of evidence could persuade you otherwise. ~ Nate Silver

In that moment, the future - uncontainable by any Theorem mathematical or otherwise - stretched out before Colin: infinite and unknowable and beautiful. ~ John Green

I carried this problem around in my head basically the whole time. I would wake up with it first thing in the morning, I would be thinking about it all day, and I would be thinking about it when I went to sleep. Without distraction I would have the same thing going round and round in my mind.
(Recalling the degree of focus and determination that eventually yielded the proof of Fermat's Last Theorem.) ~ Andrew John Wiles

Words are the children of reason and, therefore, can't explain it. They really can't translate feeling because they're not part of it. That's why it bugs me when people try to analyze jazz as an intellectual theorem. It's not. It's feeling. ~ Bill Evans

Science in its everyday practice is much closer to art than to philosophy. When I look at Godel's proof of his undecidability theorem, I do not see a philosophical argument. The proof is a soaring piece of architecture, as unique and as lovely as Chartres Cathedral. Godel took Hilbert's formalized axioms of mathematics as his building blocks and built out of them a lofty structure of ideas into which he could finally insert his undecidable arithmetical statement as the keystone of the arch. The proof is a great work of art. It is a construction, not a reduction. It destroyed Hilbert's dream of reducing all mathematics to a few equations, and replaced it with a greater dream of mathematics as an endlessly growing realm of ideas. Godel proved that in mathematics the whole is always greater than the sum of the parts. Every formalization of mathematics raises questions that reach beyond the limits of the formalism into unexplored territory. ~ Freeman Dyson

Bells theorem dealt a shattering blow to Einsteins position by showing that the conception of reality as consisting of separate parts, joined by local connections, is incompatible with quantum theory ... Bells theorem demonstrates that the universe is fundamentally interconnected, interdependent, and inseparable. ~ Fritjof Capra

Combinatorial analysis, in the trivial sense of manipulating binomial and multinomial coefficients, and formally expanding powers of infinite series by applications ad libitum and ad nauseamque of the multinomial theorem, represented the best that academic mathematics could do in the Germany of the late 18th century. ~ Richard Askey

Max Boot theorem: Don't invade without planning for a long occupation. ~ Max Boot

Mathematical knowledge is unlike any other knowledge. While our perception of the physical world can always be distorted, our perception of mathematical truths can't be. They are objective, persistent, necessary truths. A mathematical formula or theorem means the same thing to anyone anywhere – no matter what gender, religion, or skin color; it will mean the same thing to anyone a thousand years from now. And what's also amazing is that we own all of them. No one can patent a mathematical formula, it's ours to share. There is nothing in this world that is so deep and exquisite and yet so readily available to all. That such a reservoir of knowledge really exists is nearly unbelievable. It's too precious to be given away to the "initiated few." It belongs to all of us. ~ Edward Frenkel

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

Carnot's theorem: The most efficient heat engine is one that operates reversibly. ~ Don S. Lemons

Science is the one culture that's truly global - protons, proteins and Pythagoras's Theorem are the same from China to Peru. It should transcend all barriers of nationality. It should straddle all faiths, too. ~ Martin Rees

The essence of Hilbert's program was to find a decision process that would operate on symbols in a purely mechanical fashion, without requiring any understanding of their meaning. Since mathematics was reduced to a collection of marks on paper, the decision process should concern itself only with the marks and not with the fallible human intuitions out of which the marks were reduced. In spite of the prolonged efforts of Hilbert and his disciples, the Entscheidungsproblem was never solved. Success was achieved only in highly restricted domains of mathematics, excluding all the deeper and more interesting concepts. Hilbert never gave up hope, but as the years went by his program became an exercise in formal logic having little connection with real mathematics. Finally, when Hilbert was seventy years old, Kurt Godel proved by a brilliant analysis that the Entscheindungsproblem as Hilbert formulated it cannot be solved.

Godel proved that in any formulation of mathematics, including the rules of ordinary arithmetic, a formal process for separating statements into true and false cannot exist. He proved the stronger result which is now known as Godel's theorem, that in any formalization of mathematics including the rules of ordinary arithmetic there are meaningful arithmetical statements that cannot be proved true or false. Godel's theorem shows conclusively that in pure mathematics reductionism does not work. To decide whether a mathematical statement is true, it is not ~ Freeman Dyson

Our offense is like the pythagorean theorem: There is no answer! ~ Shaquille O'Neal

If you have to prove a theorem, do not rush. First of all, understand fully what the theorem says, try to see clearly what it means. Then check the theorem; it could be false. Examine the consequences, verify as many particular instances as are needed to convince yourself of the truth. When you have satisfied yourself that the theorem is true, you can start proving it. ~ George Polya

In 1975, ... [speaking with Shiing Shen Chern], I told him I had finally learned ... the beauty of fiber-bundle theory and the profound Chern-Weil theorem. I said I found it amazing that gauge fields are exactly connections on fiber bundles, which the mathematicians developed without reference to the physical world. I added, "this is both thrilling and puzzling, since you mathematicians dreamed up these concepts out of nowhere." He immediately protested: "No, no. These concepts were not dreamed up. They were natural and real." ~ Chen-Ning Yang

We live in a society where we're not taught how to deal with our weaknesses and frailties as human beings. We're not taught how to speak to our difficulties and challenges. We're taught the Pythagorean theorem and chemistry and biology and history. We're not taught anger management. We're not taught dissolution of fear and how to process shame and guilt. I've never in my life ever used the Pythagorean theorem! ~ Iyanla Vanzant

No matter how correct a mathematical theorem may appear to be, one ought never to be satisfied that there was not something imperfect about it until it also gives the impression of being beautiful. ~ George Boole

The paraphrase of Gödel's Theorem says that for any record player, there are records which it cannot play because they will cause its indirect self-destruction. ~ Douglas R. Hofstadter

One cannot really argue with a mathematical theorem. ~ Stephen Hawking

The missing piece in his stomach hurt so much-and eventually he stopped thinking about the Theorem and wondered only how something that isn't there can hurt you. ~ John Green

That story is proof of the theorem that then as today in Chicago, the mysterious equation of whiskey plus music equals what can only be called happiness. ~ Sarah Vowell

The theorem can be likened to a pearl, and the method of proof to an oyster. The pearl is prized for its luster and simplicity; the oyster is a complex living beast whose innards give rise to this mysteriously simple gem. ~ Douglas R. Hofstadter

I realized that anything to do with Fermat's Last Theorem generates too much interest. ~ Andrew Wiles

There is no answer to the Pythagorean theorem. Well, there is an answer, but by the time you figure it out, I got 40 points, 10 rebounds and then we're planning for the parade. ~ Shaquille O'Neal

The primes are the raw material out of which we have to build arithmetic, and Euclid's theorem assures us that we have plenty of material for the task. ~ G.H. Hardy

Fain would I turn back the clock and devote to French or some other language the hours I spent upon algebra, geometry, and trigonometry, of which not one principle remains with me. Stay! There is one theorem painfully drummed into my head which seems to have inhabited some corner of my brain since that early time: "The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides!" There it sticks, but what of it, ye gods, what of it? ~ Jessie Belle Rittenhouse

Context enables us to determine which of several meanings is in play in a particular text. The verb in "I see" means something quite different if uttered by a formerly blind man healed with spittle and dust, by a student who has just received an extended explanation of a difficult mathematical theorem, or by a skeptical wife whose husband offers a lame explanation for the lipstick on his collar. In the first context, see refers to physical sight, while in the latter two it refers to understanding, and in the last it could hardly be said without a heap of sarcasm. ~ Peter J. Leithart

Any effect, constant, theorem or equation named after Professor X was first discovered by Professor Y , for some value of Y not equal to X. ~ John C. Baez

if a theorem is geometrically obvious why prove it? This was exactly the attitude taken in the eighteenth century. The result, in the nineteenth century, was chaos and confusion: for intuition, unsupported by logic, habitually assumes that everything is much nicer behaved than it really is. Good ~ Ian Stewart

To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well. ~ Albert Camus

By now, he was also a 'Protestant Atheist', which he remained all his life. ~ John Ellis

In 1931, Kurt Godel proved in his famous second incompleteness theorem that there could be no finitary proof of the consistency of arithmetic. He had killed Hilbert's program with a single stroke.

So should you be worried that all of mathematics might collapse tomorrow afternoon? For what it's worth, I'm not. I do believe in infinite sets, and I find the proofs of consistency that use infinite sets to be convincing enough to let me sleep at night. ~ Jordan Ellenberg

I tend to regard the Coase theorem as a stepping stone on the way to an analysis of an economy with positive transaction costs. ~ Ronald Coase

Those who have more power are liable to sin more; no theorem in geometry is more certain than this. ~ Lord Acton

Identifying a problem is the real theorem and fixing it is only a corollary. ~ Balan Gothandaraman

The overall structure of the calculus is simple. The subject is defined by a fantastic leading idea, one basic axiom, a calm and profound intellectual invention, a deep property, two crucial definitions, one ancillary definition, one major theorem, and the fundamental theorem of the calculus.
The fantastic leading idea: the real world may be understood in terms of the real numbers.
The basic axiom: brings the real numbers into existence.
The calm and profound invention: the mathematical function.
The deep property: continuity.
The crucial definitions: instantaneous speed and the area underneath a curve.
The ancillary definition: a limit
The major theorem: the mean value theorem.
The fundamental theorem of the calculus is the fundamental theorem of the calculus.
These are the massive load-bearing walls and buttresses of the subject. ~ David Berlinski

Can the difficulty of an exam be measured by how many bits of information a student would need to pass it? This may not be so absurd in the encyclopedic subjects but in mathematics it doesn't make any sense since things follow from each other and, in principle, whoever knows the bases knows everything. All of the results of a mathematical theorem are in the axioms of mathematics in embryonic form, aren't they? ~ Alfred Renyi

Mean-Value Theorem for Integrals, 123 but for Hal's synoptic purposes here it's enough to say that megatonnage is distributed among Combatants according to an integrally regressed ratio of (a) ~ David Foster Wallace

Kurt Gödel's achievement in modern logic is singular and monumental – indeed it is more than a monument, it is a landmark which will remain visible far in space and time. ... The subject of logic has certainly completely changed its nature and possibilities with Gödel's achievement." - John von Neumann ~ John Von Neumann

It is a common misconception, in the spirit of the sentiments expressed in Q16, that Godel's theorem shows that there are many different kinds of arithmetic, each of which is equally valid. The particular arithmetic that we may happen to choose to work with would, accordingly, be defined merely by some arbitrarily chosen formal system. Godel's theorem shows that none of these formal systems, if consistent, can be complete; so-it is argued-we can keep adjoining new axioms, according to our whim, and obtain all kinds of alternative consistent systems within which we may choose to work. The comparison is sometimes made with the situation that occurred with Euclidean geometry. For some 21 centuries it was believed that Euclidean geometry was the only geometry possible. But when, in the eighteenth century, mathematicians such as Gauss, Lobachevsky, and Bolyai showed that indeed there are alternatives that are equally possible, the matter of geometry was seemingly removed from the absolute to the arbitrary. Likewise, it is often argued, Godel showed that arithmetic, also, is a matter of arbitrary choice, any one set of consistent axioms being as good as any other. ~ Roger Penrose

Dullards would have you believe that once you have eliminated the impossible, whatever remains, however improbable, must be the truth ... but to a mathematical mind, the impossible is simply a theorem yet to be solved. We must not eliminate the impossible, we must conquer it, suborn it to our purpose. ~ Kim Newman

But he could no longer disbelieve in the reality of love, since God Himself had loved his individual soul with divine love from all eternity. Gradually, as his soul was enriched with spiritual knowledge, he saw the whole world forming one vast symmetrical expression of God's power and love. Life became a divine gift for every moment and sensation of which, were it even the sight of a single leaf hanging on the twig of a tree, his soul should praise and thank the Giver. The world for all its solid substance and complexity no longer existed for his soul save as a theorem of divine power and love and universality. ~ James Joyce

The analysis of variance is not a mathematical theorem, but rather a convenient method of arranging the arithmetic. ~ Ronald Fisher

The ancient Greek mathematician Ptolemy was born some time at the end of the first century. Ptolemy based his version of trigonometry on the relationships between the chords of circles and the corresponding central angles of those chords. Ptolemy came up with a theorem involving four-sided figures that you can construct with the chords. In the meantime, mathematicians in India decided to use the measure of half a chord and half the angle to try to figure out these relationships. Drawing a radius from the center of a circle through the middle of a chord (halving it) forms a right angle, which is important in the definitions of the trig functions. These half-measures were the beginning of the sine function in trigonometry. In fact, the word sine actually comes from the Hindu name jiva. ~ Mary Jane Sterling

The elegance of a mathematical theorem is directly proportional to the number of independent ideas one can see in the theorem and inversely proportional to the effort it takes to see them. ~ George Polya

Here are the basic principles of Constructivism as practiced by Kronecker and codified by J.H. Poincare and L.E.J. Brouwer and other major figures in Intuitionism: (1) Any mathematical statement or theorem that is more complicated or abstract than plain old integer-style arithmetic must be explicitly derived (i.e. 'constructed') from integer arithmetic via a finite number of purely deductive steps. (2) The only valid proofs in math are constructive ones, with the adjective here meaning that the proof provides a method for finding (i.e., 'constructing') whatever mathematical entities it's concerned with. ~ David Foster Wallace

The axiom of conditioned repetition, like the binomial theorem, is nothing but a piece of insolence. ~ Edward Abbey

That is why one day I said my game will be like the Pythagorean Theorem - hard to figure out. A lot of people really don't know the Pythagorean Theory. They don't make them like me anymore. They don't want to make them like that anymore. ~ Shaquille O'Neal

Explanation is always incomplete: we can always raise another Why-questions. And the new why-questions may lead to a new theory which not only "explains" the old theory but corrects it.
This is why the evolution of Physics is likely to be an endless process of correction and better approximation. And even if one day we should reach a stage where our theories were no longer open to correction, because they are simply true, they would still not be complete - and we should know it. For Godel's famous incompleteness theorem would come into play: in view of the Mathematical background of Physics, at best an infinite sequence of such true theories would be needed in order to answer the problems which any given (formalized) theory would be undecidable.
Such considerations do not prove that the objective physical world is incomplete, or undetermined: they only show the essential incompleteness of our efforts. But they also show that it's barely possible (if possible at all) for science to reach a stage in which it can provide genuine support for the view that the physical world is deterministic. Why, the, should we not accept the verdict of common sense- at least until these arguments have been refuted? ~ Karl Popper

The proof of Fermat's Last Theorem underscores how stable mathematics is through the centuries - how mathematics is one of humanity's long continuous conversations with itself. ~ Barry Mazur

And I believe that the Binomial Theorem and a Bach Fugue are, in the long run, more important than all the battles of history. ~ James Hilton

About Thomas Hobbes: He was 40 years old before he looked on geometry; which happened accidentally. Being in a gentleman's library, Euclid's Elements lay open, and "twas the 47 El. libri I" [Pythagoras' Theorem]. He read the proposition "By God", sayd he, "this is impossible:" So he reads the demonstration of it, which referred him back to such a proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps, that at last he was demonstratively convinced of that truth. This made him in love with geometry. ~ John Aubrey

I am the number one Ninja and I have killed all the Shoguns in front of me. ~ Shaquille O'Neal

A mathematician experiments, amasses information, makes a conjecture, finds out that it does not work, gets confused and then tries to recover. A good mathematician eventually does so - and proves a theorem. ~ Steven G. Krantz

Fourier's theorem is not only one of the most beautiful results of modern analysis, but it may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics. ~ Lord Kelvin

The beauty of a mathematical theorem depends a great deal on its seriousness, as even in poetry the beauty of a line may depend to some extent on the significance of the ideas which it contains. ~ G.H. Hardy

A reply to Olbers' attempt in 1816 to entice him to work on Fermat's Theorem. I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of. [] ~ Carl Friedrich Gauss

There's no answer for my offense, just like the polythagorean theorem. ~ Shaquille O'Neal

Note for a Textbook

The question is never answered, never resolved,
The circle of love and anger never squared,
The stubborn instinct not translated yet
In decimals, accurate and predictable
In union dues or payroll cuts or blood...

Always a symbol lost in the lovely theorem,
A fraction that will not fit in the sum of the system,
A jutting thrust in the graph of the commissar's forecast,
A troublesome blank in gauleiter's careful accounting,
An awkward hitch in the plans of the second vice-president.

The answers worked on the slate are never the same,
And the answers proved in the back of the book are wrong. ~ Charles Bruce

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

There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn't say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann. ~ George Polya

Immortality is often ridiculous or cruel: few of us would have chosen to be Og or Ananias or Gallio. Even in mathematics, history sometimes plays strange tricks; Rolle figures in the textbooks of elementary calculus as if he had been a mathematician like Newton; Farey is immortal because he failed to understand a theorem which Haros had proved perfectly fourteen years before; the names of five worthy Norwegians still stand in Abel's Life, just for one act of conscientious imbecility, dutifully performed at the expense of their country's greatest man. But on the whole the history of science is fair, and this is particularly true in mathematics. No other subject has such clear-cut or unanimously accepted standards, and the men who are remembered are almost always the men who merit it. Mathematical fame, if you have the cash to pay for it, is one of the soundest and steadiest of investments. ~ G.H. Hardy

Oh, Anyone can make a quilt,' she said modestly. 'It's just scraps, from the clothes you've sewn.'
'Yes, but the talent is in joining the pieces, the way you have.'
'Look,' Om pointed, 'look at that - the poplin from our very first job.'
'You remember,' said Dina, pleased. 'And how fast you finished those first dresses. I thought I had two geniuses.'
'Hungry stomachs were driving our fingers,' chuckled Ishvar.
'Then came that yellow calico with orange strips. And what a hard time this young fellow gave me. Fighting and arguing about everything.'
'Me?Argue?Never.'
...
He steeped back, pleased with himself, as though he had elucidated an intricate theorem. 'So that's the rule to remember, the whole quilt is much more important than the square'. ~ Rohinton Mistry

I think mathematics is a vast territory. The outskirts of mathematics are the outskirts of mathematical civilization. There are certain subjects that people learn about and gather together. Then there is a sort of inevitable development in those fields. You get to the point where a certain theorem is bound to be proved, independent of any particular individual, because it is just in the path of development. ~ William Thurston

Schelling's God is the totality of Nature struggling towards consciousness, and Man is as far as the struggle has got, with the animals not too far behind, vegetables somewhat lagging, and rocks nowhere as yet. Do we believe this? Does it matter? Think of it as a poem or a painting. Art doesn't have to be true like a theorem. It can be true in other ways. This truth says there is a meaning to it all, and Man is where the meaning begins to show. ~ Tom Stoppard

There are lots of things I don't understand - say, the latest debates over whether neutrinos have mass or the way that Fermat's last theorem was (apparently) proven recently. But from 50 years in this game, I have learned two things: (1) I can ask friends who work in these areas to explain it to me at a level that I can understand, and they can do so, without particular difficulty; (2) if I'm interested, I can proceed to learn more so that I will come to understand it. Now Derrida, Lacan, Lyotard, Kristeva, etc. -- even Foucault, whom I knew and liked, and who was somewhat different from the rest -- write things that I also don't understand, but (1) and (2) don't hold: no one who says they do understand can explain it to me and I haven't a clue as to how to proceed to overcome my failures. That leaves one of two possibilities: (a) some new advance in intellectual life has been made, perhaps some sudden genetic mutation, which has created a form of "theory" that is beyond quantum theory, topology, etc., in depth and profundity; or (b) ... I won't spell it out. ~ Noam Chomsky

[...] provability is a weaker notion than truth ~ Douglas R. Hofstadter

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

(As a rule of thumb, the sample size must be at least 30 for the central limit theorem to hold true.) This ~ Charles Wheelan

What we do may be small, but it has a certain character of permanence; and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something utterly beyond the powers of the vast majority of men. ~ G.H. Hardy

But my shift to the serious study of economics gradually weakened my belief in Major Douglas's A+B theorem, which was replaced in my thought by the expression MV = PT. ~ James Meade

The most famous Diophantine equation in history is the one known as Fermat's last Theorem, the celebrated statement by Pierre de Fermat (1601-55) that there are no whole number solutions to the equation x^n + y^n = z^n, where n is any number greater than 2. When n = 2, there are many solutions (in fact an infinite number). For instance, 3^2 + 4^2 = 5^2 (9 + 16 = 25); or 12^2 +5^2 = 13^2 (144 + 25 = 169). Miraculously, when we go from n = 2 to n = 3, there are no whole numbers x,y,z that satisfy x^3 + y^3 = z^3, and the same is true for any other value of n that is greater than 2. Appropriately, it was in the margin of the second book of Diophantus's Arithmetica, which Fermat was eagerly reading, that he wrote his extraordinary claim-one that took no fewer than 356 years to prove. ~ Mario Livio

Before beginning [to try to prove Fermat's Last Theorem] I should have to put in three years of intensive study, and I haven't that much time to squander on a probable failure. ~ David Hilbert

His sense of humor was permanently replaced by the Pythagorean Theorem. ~ Dima Zales

Contemporary physicists come in two varieties. Type 1 physicists are bothered by EPR and Bell's Theorem. Type 2 (the majority) are not, but one has to distinguish two subvarieties. Type 2a physicists explain why they are not bothered. Their explanations tend either to miss the point entirely (like Born's to Einstein) or to contain physical assertions that can be shown to be false. Type 2b are not bothered and refuse to explain why. ~ David Mermin

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

You wouldn't think there was a need for a Coase Theorem, really. ~ Ronald Coase

Do people believe in human rights because such rights actually exist, like mathematical truths, sitting on a cosmic shelf next to the Pythagorean theorem just waiting to be discovered by Platonic reasoners? Or do people feel revulsion and sympathy when they read accounts of torture, and then invent a story about universal rights to help justify their feelings? ~ Jonathan Haidt

Scramble Books were written prior to the personal computer. For the most part they were used to supplement text books as a teaching and testing tool. I wrote a scramble book to help students understand the "Pythagorean theorem or Law of Pythagoras." What made these books different from text books was that the answers to questions would lead you to different pages, which in turn would confirm that either your answer was right or it would direct you to another page explaining how to arrive at the correct answer. ~ Hank Bracker

Poetry is a theorem of a yellow-silk handkerchief knotted with riddles, sealed in a balloon tied to the tail of a kite flying in a white wind against a blue sky in spring. ~ Carl Sandburg

MacPherson told me that my theorem can be viewed as blah blah blah Grothendieck blah blah blah, which makes it much more respectable. ~ Jim Propp

Gradually, at various points in our childhoods, we discover different forms of conviction. There's the rock-hard certainty of personal experience ("I put my finger in the fire and it hurt,"), which is probably the earliest kind we learn. Then there's the logically convincing, which we probably come to first through maths, in the context of Pythagoras's theorem or something similar, and which, if we first encounter it at exactly the right moment, bursts on our minds like sunrise with the whole universe playing a great chord of C Major. ~ Philip Pullman

I took care to replace the Compendium in its correct pamphlet, and in doing so dislodged a slim pamphlet by Grastrom, one of the most eccentric authors in Solarist literature. I had read the pamphlet, which was dictated by the urge to understand what lies beyond the individual, man, and the human species. It was the abstract, acidulous work of an autodidact who had previously made a series of unusual contributions to various marginal and rarefied branches of quantum physics. In this fifteen-page booklet (his magnum opus!), Grastrom set out to demonstrate that the most abstract achievements of science, the most advanced theories and victories of mathematics represented nothing more than a stumbling, one or two-step progression from our rude, prehistoric, anthropomorphic understanding of the universe around us. He pointed out correspondences with the human body-the projections of our sense, the structure of our physical organization, and the physiological limitations of man-in the equations of the theory of relativity, the theorem of magnetic fields and the various unified field theories. Grastrom's conclusion was that there neither was, nor could be any question of 'contact' between mankind and any nonhuman civilization. This broadside against humanity made no specific mention of the living ocean, but its constant presence and scornful, victorious silence could be felt between every line, at any rate such had been my own impression. It was Gibarian who drew it to my attention, ~ Stanisław Lem

The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas. Thus a serious mathematical theorem, a theorem which connects significant ideas, is likely to lead to important advances in mathematics itself and even in other sciences. ~ G.H. Hardy

I swear the older I get, the more I value bad examples over good ones. It's a good thing too, because most people are egotistical, neurotic, self-absorbed peons, insistent on wearing near-sighted glasses in a far-sighted world. And it's this exact sort of myopic ignorance that has led to my groundbreaking new theory. I call it Mim's Theorem of Monkey See Monkey Don't, and what it boils down to is this: it is my belief that there are some people whose sole purpose of existence is to show the rest of how not to act. ~ David Arnold

Stephen Hawking has proven a general theorem stating that all solutions of Einstein's equations that allow faster-than-light travel must involve negative matter or energy. ~ Michio Kaku

We may consequently state the fundamental theorem of Natural Selection in the form: The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time. ~ Ronald Fisher

I know this may sound like an excuse," he said. "But tensor functions in higher differential topology, as exemplified by application of the Gauss-Bonnett Theorem to Todd Polynomials, indicate that cohometric axial rotation in nonadiabatic thermal upwelling can, by random inference derived from translational equilibrium aggregates, array in obverse transitional order the thermodynamic characteristics of a transactional plasma undergoing negative entropy conversions."
"Why don't you just shut up," said Hardesty. ~ Mark Helprin

In any event, Socrates' proof of prenatal immortality is that one of Meno's uneducated slave boys actually comes up with the Pythagorean theorem without ever having studied geometry! Therefore, he must be remembering it. You recall that theorem: in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Huh? We can barely remember that from tenth grade, let alone from before we were born. ~ Thomas Cathcart

Geometry has two great treasures; one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel. ~ Johannes Kepler