Theorem Quotes

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.

Rachel bit her lip. "I hope you're right. I'm a little worried. What if someone asks what's on the next math test and I start spouting a prophecy in the middle of geometry class? The Pythagorean theorem shall be problem two...Gods, that would be embarrassing.

A mathematician is a device for turning coffee into theorems

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.

You forget all of it anyway. First, you forget everything you learned-the dates of the Hay-Herran Treaty and Pythagorean Theorem. You especially forget everything you didn't really learn, but just memorized the night before. You forget the names of all but one or two of your teachers, and eventually you'll forget those, too. You forget your junior class schedule and where you used to sit and your best friend's home phone number and the lyrics to that song you must have played a million times. For me, it was something by Simon & Garfunkel. Who knows what it will be for you? And eventually, but slowly, oh so slowly, you forget your humiliations-even the ones that seemed indelible just fade away. You forget who was cool and who was not, who was pretty, smart, athletic, and not. Who went to a good college. Who threw the best parties Who could get you pot. You forget all of them. Even the ones you said you loved, and even the ones you actually did. They're the last to go. And then once you've forgotten enough, you love someone else.

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.

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.

A human being without the proper empathy or feeling is the same as an android built so as to lack it, either by design or mistake. We mean, basically, someone who does not care about the fate which his fellow living creatures fall victim to; he stands detached, a spectator, acting out by his indifference John Donne's theorem that 'No man is an island,' but giving that theorem a twist: that which is a mental and a moral island is not a man.

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.

[...] provability is a weaker notion than truth

The Three Theorems of Psychohistorical Quantitivity: The population under scrutiny is oblivious to the existence of the science of Psychohistory. The time periods dealt with are in the region of 3 generations. The population must be in the billions (±75 billions) for a statistical probability to have a psychohistorical validity.

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.

Carnal embrace is sexual congress, which is the insertion of the male genital organ into the female genital organ for purposes of procreation and pleasure. Fermat’s last theorem, by contrast, asserts that when x, y and z are whole numbers each raised to power of n, the sum of the first two can never equal the third when n is greater than 2.

A Man who is doing his True Will has the inertia of the Universe to assist him.

... 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.

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.

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

However, one cannot really argue with a mathematical theorem.

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?

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.

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.

A Fairy must make her own way in the world, for the world will never make way for her. That, incidentally, is the First Theorem of Questing Physicks, which you’ll learn all about when you’re older and don’t care anymore.

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.

The future is the result of actions, and actions are the result of behavior, and behavior is the result of prediction.

It was, Elvi thought, like finding a sea turtle who thoroughly understood Godel’s incompleteness theorem, but didn’t have any sea-turtley application for it.

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.

Inaccessibility Theorem: THE INFORMATION YOU HAVE IS NOT THE INFORMATION YOU WANT. THE INFORMATION YOU WANT IS NOT THE INFORMATION YOU NEED. THE INFORMATION YOU NEED IS NOT THE INFORMATION YOU CAN OBTAIN. Rule

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.

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.

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.

Under Bayes' theorem, no theory is perfect. Rather, it is a work in progress, always subject to further refinement and testing.

A mathematician is a machine for turning coffee into theorems.

A learner that uses Bayes’ theorem and assumes the effects are independent given the cause is called a Naïve Bayes classifier. That’s because, well, that’s such a naïve assumption.

The seriousness of a theorem, of course, does not lie in its consequences, which are merely the evidence for its seriousness.

He walked straight out of college into the waiting arms of the Navy. They gave him an intelligence test. The first question on the math part had to do with boats on a river: Port Smith is 100 miles upstream of Port Jones. The river flows at 5 miles per hour. The boat goes through water at 10 miles per hour. How long does it take to go from Port Smith to Port Jones? How long to come back? Lawrence immediately saw that it was a trick question. You would have to be some kind of idiot to make the facile assumption that the current would add or subtract 5 miles per hour to or from the speed of the boat. Clearly, 5 miles per hour was nothing more than the average speed. The current would be faster in the middle of the river and slower at the banks. More complicated variations could be expected at bends in the river. Basically it was a question of hydrodynamics, which could be tackled using certain well-known systems of differential equations. Lawrence dove into the problem, rapidly (or so he thought) covering both sides of ten sheets of paper with calculations. Along the way, he realized that one of his assumptions, in combination with the simplified Navier Stokes equations, had led him into an exploration of a particularly interesting family of partial differential equations. Before he knew it, he had proved a new theorem. If that didn't prove his intelligence, what would? Then the time bell rang and the papers were collected. Lawrence managed to hang onto his scratch paper. He took it back to his dorm, typed it up, and mailed it to one of the more approachable math professors at Princeton, who promptly arranged for it to be published in a Parisian mathematics journal. Lawrence received two free, freshly printed copies of the journal a few months later, in San Diego, California, during mail call on board a large ship called the U.S.S. Nevada. The ship had a band, and the Navy had given Lawrence the job of playing the glockenspiel in it, because their testing procedures had proven that he was not intelligent enough to do anything else.

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

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.

Simons shared a few life lessons with the school’s audience: “Work with the smartest people you can, hopefully smarter than you . . . be persistent, don’t give up easily. Be guided by beauty . . . it can be the way a company runs, or the way an experiment comes out, or the way a theorem comes out, but there’s a sense of beauty when something is working well, almost an aesthetic to it.

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

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.

Whatever— the soup is getting cold. [Last sentence of a mathematical theorem in Leonardo da Vinci’s notebook, 1518]

Occasionally, I get a letter from someone who is in “contact” with extraterrestrials. I am invited to “ask them anything.” And so over the years I’ve prepared a little list of questions. The extraterrestrials are very advanced, remember. So I ask things like, “Please provide a short proof of Fermat’s Last Theorem.” Or the Goldbach Conjecture. And then I have to explain what these are, because extraterrestrials will not call it Fermat’s Last Theorem. So I write out the simple equation with the exponents. I never get an answer. On the other hand, if I ask something like “Should we be good?” I almost always get an answer.

Maths is one of the purest forms of thought, and to outsiders mathematicians may seem almost other-worldly.

Colin, I hate to fulfill the Theorem, but I don't think we should be involved romantically. The problem is that I secretly in love with Hassan. I can't help myself. I hold your bony shoulder blades in my hands and think of his fleshy back. I kiss your stomach and I think of his awe-inspiring gut. I like you, Colin, I really do. But-I'm sorry. It's just not going to work. I hope we can still be friends. Sincerely, Lindsey Lee Wells P.S. Just kidding.

In mathematics, in place of characters, you have variables or unknowns. If I'm trying to plot a theorem, I try to imagine these variables interacting with each other. The boundary of their interaction is the theorem.

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.

It may be appropriate to quote a statement of Poincare, who said (partly in jest no doubt) that there must be something mysterious about the normal law since mathematicians think it is a law of nature whereas physicists are convinced that it is a mathematical theorem.

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

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

She wanted to tell him so much, on the tarmac, the day he left. The world is run by brutal men and the surest proof is their armies. If they ask you to stand still, you should dance. If they ask you to burn the flag, wave it. If they ask you to murder, re-create. Theorem, anti-theorem, corollary, anti-corollary. Underline it twice. It’s all there in the numbers. Listen to your mother. Listen to me, Joshua. Look me in the eyes. I have something to tell you.

REJECTION MINIMIZATION THEOREM (RMT): Think about it: boys basically, want to kiss girls. Guys want to make out. Always. Hassan aside, there's a rarely a time when a boy is thinking, "Eh, I think I'd rather not kiss a girl today.

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.

Some mathematics problems look simple, and you try them for a year or so, and then you try them for a hundred years, and it turns out that they're extremely hard to solve. There's no reason why these problems shouldn't be easy, and yet they turn out to be extremely intricate. [Fermat's] Last Theorem is the most beautiful example of this.

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.

You can't live with the idea that someone might leave. So instead of being happy for me, like any normal person, you're pissed off because ooh, oh no, Hassan doesn't like me anymore. You're such a sitzpinkler. You're so goddamned scared of the idea that someone might dump you that your whole fugging life is built around not gettting left behind. Well, it doesn't work, kafir. I just - it's not just dumb, it's ineffective. Because then you're not being a good friend or a good boyfriend or whatever, because you're only thinking they-might-not-like-me-they-might-not-like-me, and guess what? When you act like that, no one likes you. There's your goddamned Theorem.

Once again I had to put myself in a vulnerable position in order to become stronger.

I realized that if one reversed the direction of time in Penrose’s theorem so that the collapse became an expansion, the conditions of his theorem would still hold, provided the universe were roughly like a Friedmann model on large scales at the present time.

Mathematics is not a deductive science - that's a cliche. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.

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.

The reading quieted his brain a little. Without Katherine and without the Theorem and without his hopes of mattering, he had very little. But he always had books. Books are the ultimate Dumpees: put them down and they'll wait for you forever; pay attention to them and they always love you back.

you can make a Theorem that explains why you won or lost past poker hands, but you can never make one to predict future poker hands. The past, like Lindsey had told him, is a logical story. It's the sense of what happened. But since it is not yet remembered, the future need not make any fugging sense a all.

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.

Atheist, homosexual, eccentric, marathon-running English mathematician, A. M. Turing was in large part responsible not only for the concept of computers, incisive theorems about their powers, and a clear vision of the possibility of computer minds, but also for the cracking of German ciphers during the Second World War.

But you can't prove God exists. And isn't that what all science is ultimately about? Proving theories about the universe?" "Provability is not truth, Caro. Godel's incompleteness theorem tells us that, if we didn't already know it intuitively, which we do.

philosopher Kurt Gödel reached a similar conclusion in his 1931 “incompleteness theorem.” We are thus left with the perplexing situation of being able to define a problem, to prove that a unique answer exists, and yet know that the answer can never be found.

I do not deny the power and the beauty of reductionist science, as exemplified in the axioms and theorems of abstract algebra....But I assert the equal power and beauty of constructive science as exemplified in Godel's construction of an undecidable proposition....

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.

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

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.

Putting these three things together we arrive at our own ‘central theorem’ of the extended phenotype: An animal’s behaviour tends to maximize the survival of the genes ‘for’ that behaviour, whether or not those genes happen to be in the body of the particular animal performing it.

If you can't tolerate the truth, never ask me a question.

Whenever something is engineered as complex, it is designed to keep you simple.

Why does Alexander the Great never tell us about the exact location of his tomb, Fermat about his Last Theorem, John Wilkes Booth about the Lincoln assassination conspiracy, Hermann Göring about the Reichstag fire? Why don’t Sophocles, Democritus, and Aristarchus dictate their lost books?

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. 47.

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.)

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'.

These dictates of Reason, men use to call by the name of Lawes; but improperly: for they are but Conclusions, or Theoremes concerning what conduceth to the conservation and defence of themselves; whereas Law, properly is the word of him, that by right hath command over others. But yet if we consider the same Theoremes, as delivered in the word of God, that by right commandeth all things; then are they properly called Lawes.

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.

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. And when persons came to him and desired to be recommended by him to philosophers, he took and recommended them, so well did he bear being overlooked. So that if ever any talk should happen among the unlearned concerning philosophic theorems, be you, for the most part, silent. For there is great danger in immediately throwing out what you have not digested. And, if anyone tells you that you know nothing, and you are not nettled at it, then you may 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.

Although this may seem a paradox, all exact science is dominated by the idea of approximation.

To be an engineer, and build a marvelous machine, and to see the beauty of its operation is as valid an experience of beauty as a mathematician's absorption in a wondrous theorem. One is not "more" beautiful than the other. To see a space shuttle standing on the launch pad, the vented gases escaping, and witness the thunderous blast-off as it climbs heavenward on a pillar of flame - this is beauty. Yet it is a prime example of applied mathematics.

The first [method] I might speak about is simplification. Suppose that you are given a problem to solve, I don't care what kind of problem-a machine to design, or a physical theory to develop, or a mathematical theorem to prove or something of that kind-probably a very powerful approach to this is to attempt to eliminate everything from the problem except the essentials; that is, cut is down to size. Almost every problem that you come across is befuddled with all kinds of extraneous data of one sort or another; and if you can bring this problem down into the main issues, you can see more clearly what you are trying to do an perhaps find a solution. Now in so doing you may have stripped away the problem you're after. You may have simplified it to the point that it doesn't even resemble the problem that you started with; but very often if you can solve this simple problem, you can add refinements to the solution of this until you get back to the solution of the one you started with.

I began to study trigonometry. There was solace in its strange formulas and equations. I was drawn to the Pythagorean theorem and its promise of a universal—the ability to predict the nature of any three points containing a right angle, anywhere, always. What I knew of physics I had learned in the junkyard, where the physical world often seemed unstable, capricious. But here was a principle through which the dimensions of life could be defined, captured. Perhaps reality was not wholly volatile. Perhaps it could be explained, predicted. Perhaps it could be made to make sense.

A tenth-century copy of Archimedes’s Method of Mechanical Theorems. In it, Archimedes had ingeniously applied mechanical laws, such as the law of the lever, to find the volume and area of geometric shapes. Two thousand years before Newton, he had come tantalizingly close to deriving calculus. However, in the thirteenth century this work was scraped off and overwritten with a prayer book.

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.

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.

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.

Pascal was even convinced that he could use his theories to justify a belief in God. He stated that ‘the excitement that a gambler feels when making a bet is equal to the amount he might win multiplied by the probability of winning it’. He then argued that the possible prize of eternal happiness has an infinite value and that the probability of entering heaven by leading a virtuous life, no matter how small, is certainly finite. Therefore, according to Pascal’s definition, religion was a game of infinite excitement and one worth playing, because multiplying an infinite prize by a finite probability results in infinity.

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.

And Colin thought: Because like say I tell someone about my feral hog hunt. Even if it's a dumb story, telling it changes other people just the slightest little bit, just as living the story changes me. An infinitesimal change ripples outward-ever smaller but everlasting. I will get forgotten, but the stories will last. And so we all matter-maybe less than a lot, but always more than none." "Having the correct graph from the start proved not that the Theorem was accurate, but that there's a place in the brain for knowing what cannot be remembered.

Though resident much of his life in the city of Cnidus on the coast of Asia Minor, Eudoxus was a student at Plato’s Academy, and returned later to teach there. No writings of Eudoxus survive, but he is credited with solving a great number of difficult mathematical problems, such as showing that the volume of a cone is one-third the volume of the cylinder with the same base and height. (I have no idea how Eudoxus could have done this without calculus.) But his greatest contribution to mathematics was the introduction of a rigorous style, in which theorems are deduced from clearly stated axioms.

What's the point of talking about philosophical questions? Because we're going to be doing a fair bit of it here – I mean, of philosophical bullshitting. Well, there's a standard answer, and it's that philosophy is an intellectual clean-up job – the janitors who come in after the scientists have made a mess, to try and pick up the pieces. So in this view, philosophers sit in their armchairs waiting for something surprising to happen in science – like quantum mechanics, like the Bell inequality, like Gödel's Theorem – and then (to switch metaphors) swoop in like vultures and say, ah, this is what it really meant. Well, on its face, that seems sort of boring. But as you get more accustomed to this sort of work, I think what you'll find is...it's still boring!

Proof is what lies at the heart of maths, and is what marks it out from other sciences. Other sciences have hypotheses that are tested against experimental evidence until they fail, and are overtaken by new hypotheses. In maths, absolute proof is the goal, and once something is proved, it is proved forever, with no room for change.

One conceivable way to discriminate between a scientific intellectual and a literary intellectual is by considering that a scientific intellectual can usually recognize the writing of another but that the literary intellectual would not be able to tell the difference between lines jotted down by a scientist and those by a glib nonscientist. This is even more apparent when the literary intellectual starts using scientific buzzwords, like “uncertainty principle,” “Godel’s theorem,” “parallel universe,” or “relativity,” either out of context or, as often, in exact opposition to the scientific meaning. I suggest reading the hilarious Fashionable Nonsense by Alan Sokal for an illustration of such practice (I was laughing so loudly and so frequently while reading it on a plane that other passengers kept whispering things about me).

1 You said ‘The world is going back to Paganism’. Oh bright Vision! I saw our dynasty in the bar of the House Spill from their tumblers a libation to the Erinyes, And Leavis with Lord Russell wreathed in flowers, heralded with flutes, Leading white bulls to the cathedral of the solemn Muses To pay where due the glory of their latest theorem. Hestia’s fire in every flat, rekindled, burned before The Lardergods. Unmarried daughters with obedient hands Tended it. By the hearth the white-armd venerable mother Domum servabat, lanam faciebat. At the hour Of sacrifice their brothers came, silent, corrected, grave Before their elders; on their downy cheeks easily the blush Arose (it is the mark of freemen’s children) as they trooped, Gleaming with oil, demurely home from the palaestra or the dance. Walk carefully, do not wake the envy of the happy gods, Shun Hubris. The middle of the road, the middle sort of men, Are best. Aidos surpasses gold. Reverence for the aged Is wholesome as seasonable rain, and for a man to die Defending the city in battle is a harmonious thing. Thus with magistral hand the Puritan Sophrosune Cooled and schooled and tempered our uneasy motions; Heathendom came again, the circumspection and the holy fears … You said it. Did you mean it? Oh inordinate liar, stop. 2 Or did you mean another kind of heathenry? Think, then, that under heaven-roof the little disc of the earth, Fortified Midgard, lies encircled by the ravening Worm. Over its icy bastions faces of giant and troll Look in, ready to invade it. The Wolf, admittedly, is bound; But the bond wil1 break, the Beast run free. The weary gods, Scarred with old wounds the one-eyed Odin, Tyr who has lost a hand, Will limp to their stations for the Last defence. Make it your hope To be counted worthy on that day to stand beside them; For the end of man is to partake of their defeat and die His second, final death in good company. The stupid, strong Unteachable monsters are certain to be victorious at last, And every man of decent blood is on the losing side. Take as your model the tall women with yellow hair in plaits Who walked back into burning houses to die with men, Or him who as the death spear entered into his vitals Made critical comments on its workmanship and aim. Are these the Pagans you spoke of? Know your betters and crouch, dogs; You that have Vichy water in your veins and worship the event Your goddess History (whom your fathers called the strumpet Fortune).

The poem or the discovery exists in two moments of vision: the moment of appreciation as much as that of creation; for the appreciator must see the movement, wake to the echo which was started in the creation of the work. In the moment of appreciation we live again the moment when the creator saw and held the hidden likeness. When a simile takes us aback and persuades us together, when we find a juxtaposition in a picture both odd and intriguing, when a theory is at once fresh and convincing, we do not merely nod over someone else's work. We re-enact the creative act, and we ourselves make the discovery again... ...Reality is not an exhibit for man's inspection, labeled: "Do not touch." There are no appearances to be photographed, no experiences to be copied, in which we do not take part. We re-make nature by the act of discovery, in the poem or in the theorem. And the great poem and the deep theorem are new to every reader, and yet are his own experiences, because he himself re-creates them. They are the marks of unity in variety; and in the instant when the mind seizes this for itself, in art or in science, the heart misses a beat.

I woke up with a start, a little after 5:30 in the morning, to that wonderful feeling that lasts only a fraction of a second, when you don’t know where you are—not even what continent you’re on! I jumped up from the futon and went over to my computer to make a note of the few fragments of the dream I could still hold on to before they completely melted away in the mind’s morning fog. The complexity and the confusion of the adventure put me in a good mood: I take such dreams as a sign that my brain is in good working order.

Insight, then. Wisdom. The quest for knowledge, the derivation of theorems, science and technology and all those exclusively human pursuits that must surely rest on a conscious foundation. Maybe that's what sentience would be for— if scientific breakthroughs didn't spring fully-formed from the subconscious mind, manifest themselves in dreams, as full-blown insights after a deep night's sleep. It's the most basic rule of the stymied researcher: stop thinking about the problem. Do something else. It will come to you if you just stop being conscious of it... Don't even try to talk about the learning curve. Don't bother citing the months of deliberate practice that precede the unconscious performance, or the years of study and experiment leading up to the gift-wrapped Eureka moment. So what if your lessons are all learned consciously? Do you think that proves there's no other way? Heuristic software's been learning from experience for over a hundred years. Machines master chess, cars learn to drive themselves, statistical programs face problems and design the experiments to solve them and you think that the only path to learning leads through sentience? You're Stone-age nomads, eking out some marginal existence on the veldt—denying even the possibility of agriculture, because hunting and gathering was good enough for your parents. Do you want to know what consciousness is for? Do you want to know the only real purpose it serves? Training wheels. You can't see both aspects of the Necker Cube at once, so it lets you focus on one and dismiss the other. That's a pretty half-assed way to parse reality. You're always better off looking at more than one side of anything. Go on, try. Defocus. It's the next logical step.

Chemistry, for me, had stopped being such a source. It led to the heart of Matter, and Matter was our ally precisely because the Spirit, dear to Fascism, was our enemy; but, having reached the fourth year of Pure Chemistry, I could no longer ignore the fact that chemistry itself, or at least that which we were being administered, did not answer my questions. To prepare phenyl bromide according to Gatterman was amusing, even exhilarating, but not very different from following Artusi's recipes. Why in that particular way and not in another? After having been force fed in liceo the truths revealed by Fascist Doctrine, all revealed, unproven truths either bored me stiff or aroused my suspicion. 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 or magicians; instead, at the origin of physics lay the strenuous clarity of the West – Archimedes and Euclid.

All this attempt to control... We are talking about Western attitudes that are five hundred years old... The basic idea of science - that there was a new way to look at reality, that it was objective, that it did not depend on your beliefs or your nationality, that it was rational - that idea was fresh and exciting back then. It offered promise and hope for the future, and it swept away the old medieval system, which was hundreds of years old. The medieval world of feudal politics and religious dogma and hateful superstitions fell before science. But, in truth, this was because the medieval world didn't really work any more. It didn't work economically, it didn't work intellectually, and it didn't fit the new world that was emerging... But now... science is the belief system that is hundreds of years old. And, like the medieval system before it, science is starting to not fit the world any more. Science has attained so much power that its practical limits begin to be apparent. Largely through science, billions of us live in one small world, densely packed and intercommunicating. But science cannot help us decide what to do with that world, or how to live. Science can make a nuclear reactor, but it can not tell us not to build it. Science can make pesticide, but cannot tell us not to use it. And our world starts to seem polluted in fundamental ways - air, and water, and land - because of ungovernable science... At the same time, the great intellectual justification of science has vanished. Ever since Newton and Descartes, science has explicitly offered us the vision of total control. Science has claimed the power to eventually control everything, through its understanding of natural laws. But in the twentieth century, that claim has been shattered beyond repair. First, Heisenberg's uncertainty principle set limits on what we could know about the subatomic world. Oh well, we say. None of us lives in a subatomic world. It doesn't make any practical difference as we go through our lives. Then Godel's theorem set similar limits to mathematics, the formal language of science. Mathematicians used to think that their language had some inherent trueness that derived from the laws of logic. Now we know what we call 'reason' is just an arbitrary game. It's not special, in the way we thought it was. And now chaos theory proves that unpredictability is built into our daily lives. It is as mundane as the rain storms we cannot predict. And so the grand vision of science, hundreds of years old - the dream of total control - has died, in our century. And with it much of the justification, the rationale for science to do what it does. And for us to listen to it. Science has always said that it may not know everything now but it will know, eventually. But now we see that isn't true. It is an idle boast. As foolish, and misguided, as the child who jumps off a building because he believes he can fly... We are witnessing the end of the scientific era. Science, like other outmoded systems, is destroying itself. As it gains in power, it proves itself incapable of handling the power. Because things are going very fast now... it will be in everyone's hands. It will be in kits for backyard gardeners. Experiments for schoolchildren. Cheap labs for terrorists and dictators. And that will force everyone to ask the same question - What should I do with my power? - which is the very question science says it cannot answer.